Spatstat Manual

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Package ‘spatstat’
January 29, 2018
Version 1.55-0
Date 2018-01-29
Title Spatial Point Pattern Analysis, Model-Fitting, Simulation, Tests
Author Adrian Baddeley ,
Rolf Turner 
and Ege Rubak ,
with substantial contributions of code by
Kasper Klitgaard Berthelsen;
Ottmar Cronie;
Yongtao Guan;
Ute Hahn;
Abdollah Jalilian;
Marie-Colette van Lieshout;
Greg McSwiggan;
Tuomas Rajala;
Suman Rakshit;
Dominic Schuhmacher;
Rasmus Waagepetersen;
and Hangsheng Wang.
Additional contributions
by M. Adepeju;
C. Anderson;
Q.W. Ang;
M. Austenfeld;
S. Azaele;
M. Baddeley;
C. Beale;
M. Bell;
R. Bernhardt;
T. Bendtsen;
A. Bevan;
B. Biggerstaff;
A. Bilgrau;
L. Bischof;
C. Biscio;
R. Bivand;
J.M. Blanco Moreno;
F. Bonneu;
J. Burgos;
1

2
S. Byers;
Y.M. Chang;
J.B. Chen;
I. Chernayavsky;
Y.C. Chin;
B. Christensen;
J.-F. Coeurjolly;
K. Colyvas;
R. Constantine;
R. Corria Ainslie;
R. Cotton;
M. de la Cruz;
P. Dalgaard;
M. D'Antuono;
S. Das;
T. Davies;
P.J. Diggle;
P. Donnelly;
I. Dryden;
S. Eglen;
A. El-Gabbas;
B. Fandohan;
O. Flores;
E.D. Ford;
P. Forbes;
S. Frank;
J. Franklin;
N. Funwi-Gabga;
O. Garcia;
A. Gault;
J. Geldmann;
M. Genton;
S. Ghalandarayeshi;
J. Gilbey;
J. Goldstick;
P. Grabarnik;
C. Graf;
U. Hahn;
A. Hardegen;
M.B. Hansen;
M. Hazelton;
J. Heikkinen;
M. Hering;
M. Herrmann;
P. Hewson;
K. Hingee;
K. Hornik;
P. Hunziker;
J. Hywood;
R. Ihaka;
C. Icos;
A. Jammalamadaka;

3
R. John-Chandran;
D. Johnson;
M. Khanmohammadi;
R. Klaver;
P. Kovesi;
L. Kozmian-Ledward;
M. Kuhn;
J. Laake;
F. Lavancier;
T. Lawrence;
R.A. Lamb;
J. Lee;
G.P. Leser;
H.T. Li;
G. Limitsios;
A. Lister;
B. Madin;
M. Maechler;
J. Marcus;
K. Marchikanti;
R. Mark;
J. Mateu;
P. McCullagh;
U. Mehlig;
F. Mestre;
S. Meyer;
X.C. Mi;
L. De Middeleer;
R.K. Milne;
E. Miranda;
J. Moller;
M. Moradi;
V. Morera Pujol;
E. Mudrak;
G.M. Nair;
N. Najari;
N. Nava;
L.S. Nielsen;
F. Nunes;
J.R. Nyengaard;
J. Oehlschlaegel;
T. Onkelinx;
S. O'Riordan;
E. Parilov;
J. Picka;
N. Picard;
M. Porter;
S. Protsiv;
A. Raftery;
S. Rakshit;
B. Ramage;
P. Ramon;

4
X. Raynaud;
N. Read;
M. Reiter;
I. Renner;
T.O. Richardson;
B.D. Ripley;
E. Rosenbaum;
B. Rowlingson;
J. Rudokas;
J. Rudge;
C. Ryan;
F. Safavimanesh;
A. Sarkka;
C. Schank;
K. Schladitz;
S. Schutte;
B.T. Scott;
O. Semboli;
F. Semecurbe;
V. Shcherbakov;
G.C. Shen;
P. Shi;
H.-J. Ship;
T.L. Silva;
I.-M. Sintorn;
Y. Song;
M. Spiess;
M. Stevenson;
K. Stucki;
M. Sumner;
P. Surovy;
B. Taylor;
T. Thorarinsdottir;
L. Torres;
B. Turlach;
T. Tvedebrink;
K. Ummer;
M. Uppala;
A. van Burgel;
T. Verbeke;
M. Vihtakari;
A. Villers;
F. Vinatier;
S. Voss;
S. Wagner;
H. Wang;
H. Wendrock;
J. Wild;
C. Witthoft;
S. Wong;
M. Woringer;
M.E. Zamboni

5
and
A. Zeileis.
Maintainer Adrian Baddeley 
Depends R (>= 3.3.0), spatstat.data (>= 1.2-0), stats, graphics, grDevices, utils, methods, nlme, rpart
Imports spatstat.utils (>= 1.8-0), mgcv, Matrix, deldir (>= 0.0-21), abind, tensor, polyclip (>= 1.50), goftest
Suggests sm, maptools, gsl, locfit, spatial, rpanel, tkrplot, RandomFields (>= 3.1.24.1), RandomFieldsUtils(>= 0.3.3.1), fftwtools (>= 0.9-8)
Description Comprehensive open-source toolbox for analysing Spatial Point Patterns. Focused mainly on two-dimensional point patterns, including multitype/marked points, in any spatial region. Also supports three-dimensional point patterns, space-time point patterns in any number of dimensions, point patterns on a linear network, and patterns of other geometrical objects. Supports spatial covariate data such as pixel images.
Contains over 2000 functions for plotting spatial data, exploratory data analysis, modelfitting, simulation, spatial sampling, model diagnostics, and formal inference.
Data types include point patterns, line segment patterns, spatial windows, pixel images, tessellations, and linear networks.
Exploratory methods include quadrat counts, K-functions and their simulation envelopes, nearest neighbour distance and empty space statistics, Fry plots, pair correlation function, kernel smoothed intensity, relative risk estimation with cross-validated bandwidth selection, mark correlation functions, segregation indices, mark dependence diagnostics, and kernel estimates of covariate effects. Formal hypothesis tests of random pattern (chi-squared, Kolmogorov-Smirnov, Monte Carlo, Diggle-Cressie-Loosmore-Ford, Dao-Genton, twostage Monte Carlo) and tests for covariate effects (Cox-Berman-Waller-Lawson, KolmogorovSmirnov, ANOVA) are also supported.
Parametric models can be fitted to point pattern data using the functions ppm(), kppm(), slrm(), dppm() similar to glm(). Types of models include Poisson, Gibbs and Cox point processes, Neyman-Scott cluster processes, and determinantal point processes. Models may involve dependence on covariates, inter-point interaction, cluster formation and dependence on marks. Models are fitted by maximum likelihood, logistic regression, minimum contrast, and composite likelihood methods.
A model can be fitted to a list of point patterns (replicated point pattern data) using the function mppm(). The model can include random effects and fixed effects depending on the experimental design, in addition to all the features listed above.
Fitted point process models can be simulated, automatically. Formal hypothesis tests of a fitted model are supported (likelihood ratio test, analysis of deviance, Monte Carlo tests) along with basic tools for model selection (stepwise(), AIC()). Tools for validating the fitted model include simulation envelopes, residuals, residual plots and Q-Q plots, leverage and influence diagnostics, partial residuals, and added variable plots.
License GPL (>= 2)
URL http://www.spatstat.org
LazyData true
NeedsCompilation yes
ByteCompile true
BugReports https://github.com/spatstat/spatstat/issues

R topics documented:

6

R topics documented:
spatstat-package . . . . .
adaptive.density . . . . .
add.texture . . . . . . .
addvar . . . . . . . . . .
affine . . . . . . . . . .
affine.im . . . . . . . . .
affine.linnet . . . . . . .
affine.lpp . . . . . . . .
affine.owin . . . . . . .
affine.ppp . . . . . . . .
affine.psp . . . . . . . .
affine.tess . . . . . . . .
allstats . . . . . . . . . .
alltypes . . . . . . . . .
angles.psp . . . . . . . .
anova.lppm . . . . . . .
anova.mppm . . . . . . .
anova.ppm . . . . . . . .
anova.slrm . . . . . . . .
anylist . . . . . . . . . .
anyNA.im . . . . . . . .
append.psp . . . . . . .
applynbd . . . . . . . .
area.owin . . . . . . . .
areaGain . . . . . . . . .
AreaInter . . . . . . . .
areaLoss . . . . . . . . .
as.box3 . . . . . . . . .
as.boxx . . . . . . . . .
as.data.frame.envelope .
as.data.frame.hyperframe
as.data.frame.im . . . . .
as.data.frame.owin . . .
as.data.frame.ppp . . . .
as.data.frame.psp . . . .
as.data.frame.tess . . . .
as.function.fv . . . . . .
as.function.im . . . . . .
as.function.leverage.ppm
as.function.owin . . . . .
as.function.tess . . . . .
as.fv . . . . . . . . . . .
as.hyperframe . . . . . .
as.hyperframe.ppx . . . .
as.im . . . . . . . . . . .
as.interact . . . . . . . .
as.layered . . . . . . . .
as.linfun . . . . . . . . .
as.linim . . . . . . . . .
as.linnet.linim . . . . . .

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58
59
61
62
63
64
67
68
70
72
74
75
76
77
78
80
82
83
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
104
105
109
110
112
113
114

R topics documented:
as.linnet.psp . . .
as.lpp . . . . . .
as.mask . . . . .
as.mask.psp . . .
as.matrix.im . . .
as.matrix.owin . .
as.owin . . . . .
as.polygonal . . .
as.ppm . . . . . .
as.ppp . . . . . .
as.psp . . . . . .
as.rectangle . . .
as.solist . . . . .
as.tess . . . . . .
auc . . . . . . . .
BadGey . . . . .
bc.ppm . . . . .
bdist.pixels . . .
bdist.points . . .
bdist.tiles . . . .
beachcolours . .
beginner . . . . .
begins . . . . . .
berman.test . . .
bind.fv . . . . . .
bits.test . . . . .
blur . . . . . . .
border . . . . . .
bounding.box.xy
boundingbox . .
boundingcircle .
box3 . . . . . . .
boxx . . . . . . .
branchlabelfun .
bugfixes . . . . .
bw.diggle . . . .
bw.frac . . . . .
bw.pcf . . . . . .
bw.ppl . . . . . .
bw.relrisk . . . .
bw.scott . . . . .
bw.smoothppp . .
bw.stoyan . . . .
by.im . . . . . .
by.ppp . . . . . .
cauchy.estK . . .
cauchy.estpcf . .
cbind.hyperframe
CDF . . . . . . .
cdf.test . . . . . .
cdf.test.mppm . .
centroid.owin . .

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116
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187
189

R topics documented:

8
chop.tess . . . . .
circdensity . . . .
clarkevans . . . .
clarkevans.test . .
clickbox . . . . .
clickdist . . . . .
clickjoin . . . . .
clicklpp . . . . .
clickpoly . . . .
clickppp . . . . .
clip.infline . . . .
closepairs . . . .
closepairs.pp3 . .
closetriples . . .
closing . . . . . .
clusterfield . . . .
clusterfit . . . . .
clusterkernel . . .
clusterradius . . .
clusterset . . . .
coef.mppm . . .
coef.ppm . . . .
coef.slrm . . . .
collapse.fv . . . .
colourmap . . . .
colouroutputs . .
colourtools . . .
commonGrid . .
compareFit . . .
compatible . . . .
compatible.fasp .
compatible.fv . .
compatible.im . .
compileK . . . .
complement.owin
concatxy . . . . .
Concom . . . . .
connected . . . .
connected.linnet .
connected.lpp . .
connected.ppp . .
contour.im . . . .
contour.imlist . .
convexhull . . . .
convexhull.xy . .
convexify . . . .
convolve.im . . .
coords . . . . . .
corners . . . . . .
covering . . . . .
crossdist . . . . .
crossdist.default .

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R topics documented:
crossdist.lpp . . . . . .
crossdist.pp3 . . . . .
crossdist.ppp . . . . .
crossdist.ppx . . . . .
crossdist.psp . . . . . .
crossing.linnet . . . . .
crossing.psp . . . . . .
cut.im . . . . . . . . .
cut.lpp . . . . . . . . .
cut.ppp . . . . . . . .
data.ppm . . . . . . . .
dclf.progress . . . . . .
dclf.sigtrace . . . . . .
dclf.test . . . . . . . .
default.dummy . . . .
default.expand . . . . .
default.rmhcontrol . .
delaunay . . . . . . . .
delaunayDistance . . .
delaunayNetwork . . .
deletebranch . . . . . .
deltametric . . . . . .
density.lpp . . . . . . .
density.ppp . . . . . .
density.psp . . . . . .
density.splitppp . . . .
deriv.fv . . . . . . . .
detpointprocfamilyfun
dfbetas.ppm . . . . . .
dg.envelope . . . . . .
dg.progress . . . . . .
dg.sigtrace . . . . . . .
dg.test . . . . . . . . .
diagnose.ppm . . . . .
diameter . . . . . . . .
diameter.box3 . . . . .
diameter.boxx . . . . .
diameter.linnet . . . .
diameter.owin . . . . .
DiggleGatesStibbard .
DiggleGratton . . . . .
dilated.areas . . . . . .
dilation . . . . . . . .
dim.detpointprocfamily
dimhat . . . . . . . . .
dirichlet . . . . . . . .
dirichletAreas . . . . .
dirichletVertices . . . .
dirichletWeights . . . .
disc . . . . . . . . . .
discpartarea . . . . . .
discretise . . . . . . .

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256
257
258
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260
261
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263
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271
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279
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331

R topics documented:

10
discs . . . . . . .
distcdf . . . . . .
distfun . . . . . .
distfun.lpp . . . .
distmap . . . . .
distmap.owin . .
distmap.ppp . . .
distmap.psp . . .
divide.linnet . . .
dkernel . . . . .
dmixpois . . . .
domain . . . . .
dppapproxkernel
dppapproxpcf . .
dppBessel . . . .
dppCauchy . . .
dppeigen . . . . .
dppGauss . . . .
dppkernel . . . .
dppm . . . . . .
dppMatern . . . .
dppparbounds . .
dppPowerExp . .
dppspecden . . .
dppspecdenrange
dummify . . . . .
dummy.ppm . . .
duplicated.ppp . .
edge.Ripley . . .
edge.Trans . . . .
edges . . . . . .
edges2triangles .
edges2vees . . .
edit.hyperframe .
edit.ppp . . . . .
eem . . . . . . .
effectfun . . . . .
ellipse . . . . . .
Emark . . . . . .
emend . . . . . .
emend.ppm . . .
endpoints.psp . .
envelope . . . . .
envelope.envelope
envelope.lpp . . .
envelope.pp3 . .
envelopeArray . .
eroded.areas . . .
erosion . . . . . .
erosionAny . . .
eval.fasp . . . . .
eval.fv . . . . . .

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333
334
335
337
338
339
341
342
343
344
345
346
349
350
350
351
352
353
354
354
358
359
359
360
361
362
363
364
365
367
369
370
371
372
373
374
375
377
378
380
381
382
384
393
395
398
401
402
403
404
405
407

R topics documented:
eval.im . . . . . . . .
eval.linim . . . . . .
ewcdf . . . . . . . .
exactMPLEstrauss .
expand.owin . . . . .
Extract.anylist . . . .
Extract.fasp . . . . .
Extract.fv . . . . . .
Extract.hyperframe .
Extract.im . . . . . .
Extract.influence.ppm
Extract.layered . . .
Extract.leverage.ppm
Extract.linim . . . .
Extract.linnet . . . .
Extract.listof . . . . .
Extract.lpp . . . . . .
Extract.msr . . . . .
Extract.owin . . . . .
Extract.ppp . . . . .
Extract.ppx . . . . .
Extract.psp . . . . .
Extract.quad . . . . .
Extract.solist . . . .
Extract.splitppp . . .
Extract.tess . . . . .
F3est . . . . . . . . .
fardist . . . . . . . .
fasp.object . . . . . .
Fest . . . . . . . . .
Fiksel . . . . . . . .
Finhom . . . . . . .
fitin.ppm . . . . . . .
fitted.lppm . . . . . .
fitted.mppm . . . . .
fitted.ppm . . . . . .
fitted.slrm . . . . . .
fixef.mppm . . . . .
flipxy . . . . . . . .
FmultiInhom . . . .
foo . . . . . . . . . .
formula.fv . . . . . .
formula.ppm . . . . .
fourierbasis . . . . .
Frame . . . . . . . .
fryplot . . . . . . . .
funxy . . . . . . . .
fv . . . . . . . . . .
fv.object . . . . . . .
fvnames . . . . . . .
G3est . . . . . . . .
gauss.hermite . . . .

11
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409
410
412
413
414
415
416
417
419
420
423
424
426
427
428
429
430
431
432
433
436
437
439
440
441
442
443
445
446
447
451
453
455
456
457
459
461
462
463
464
465
466
467
468
469
470
472
473
476
477
478
480

R topics documented:

12
Gcom . . . . . .
Gcross . . . . . .
Gdot . . . . . . .
Gest . . . . . . .
Geyer . . . . . .
Gfox . . . . . . .
Ginhom . . . . .
Gmulti . . . . . .
GmultiInhom . .
Gres . . . . . . .
gridcentres . . . .
gridweights . . .
grow.boxx . . . .
grow.rectangle . .
Hardcore . . . .
harmonic . . . .
harmonise . . . .
harmonise.fv . .
harmonise.im . .
harmonise.msr . .
harmonise.owin .
has.close . . . . .
headtail . . . . .
Hest . . . . . . .
hextess . . . . . .
HierHard . . . .
hierpair.family . .
HierStrauss . . .
HierStraussHard .
hist.funxy . . . .
hist.im . . . . . .
hopskel . . . . .
Hybrid . . . . . .
hybrid.family . .
hyperframe . . .
identify.ppp . . .
identify.psp . . .
idw . . . . . . .
Iest . . . . . . . .
im . . . . . . . .
im.apply . . . . .
im.object . . . .
imcov . . . . . .
improve.kppm . .
incircle . . . . .
increment.fv . . .
infline . . . . . .
influence.ppm . .
inforder.family .
insertVertices . .
inside.boxx . . .
inside.owin . . .

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481
484
487
490
493
494
496
498
500
502
503
504
505
506
507
509
510
511
512
513
514
515
517
518
520
522
523
524
525
527
528
529
531
532
533
534
535
536
538
540
542
543
544
545
547
548
549
550
552
552
554
555

R topics documented:
integral.im . . . . . .
integral.linim . . . .
integral.msr . . . . .
intensity . . . . . . .
intensity.dppm . . . .
intensity.lpp . . . . .
intensity.ppm . . . .
intensity.ppp . . . . .
intensity.ppx . . . . .
intensity.psp . . . . .
intensity.quadratcount
interp.colourmap . .
interp.im . . . . . . .
intersect.owin . . . .
intersect.tess . . . . .
invoke.symbolmap .
iplot . . . . . . . . .
ippm . . . . . . . . .
is.connected . . . . .
is.connected.ppp . .
is.convex . . . . . .
is.dppm . . . . . . .
is.empty . . . . . . .
is.hybrid . . . . . . .
is.im . . . . . . . . .
is.lpp . . . . . . . . .
is.marked . . . . . .
is.marked.ppm . . . .
is.marked.ppp . . . .
is.multitype . . . . .
is.multitype.ppm . .
is.multitype.ppp . . .
is.owin . . . . . . . .
is.ppm . . . . . . . .
is.ppp . . . . . . . .
is.rectangle . . . . .
is.stationary . . . . .
is.subset.owin . . . .
istat . . . . . . . . .
Jcross . . . . . . . .
Jdot . . . . . . . . .
Jest . . . . . . . . .
Jinhom . . . . . . . .
Jmulti . . . . . . . .
K3est . . . . . . . .
kaplan.meier . . . . .
Kcom . . . . . . . .
Kcross . . . . . . . .
Kcross.inhom . . . .
Kdot . . . . . . . . .
Kdot.inhom . . . . .
kernel.factor . . . . .

13
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R topics documented:

14
kernel.moment . . .
kernel.squint . . . . .
Kest . . . . . . . . .
Kest.fft . . . . . . .
Kinhom . . . . . . .
km.rs . . . . . . . .
Kmark . . . . . . . .
Kmeasure . . . . . .
Kmodel . . . . . . .
Kmodel.dppm . . . .
Kmodel.kppm . . . .
Kmodel.ppm . . . .
Kmulti . . . . . . . .
Kmulti.inhom . . . .
kppm . . . . . . . .
Kres . . . . . . . . .
Kscaled . . . . . . .
Ksector . . . . . . .
LambertW . . . . . .
laslett . . . . . . . .
latest.news . . . . . .
layered . . . . . . . .
layerplotargs . . . .
layout.boxes . . . . .
Lcross . . . . . . . .
Lcross.inhom . . . .
Ldot . . . . . . . . .
Ldot.inhom . . . . .
lengths.psp . . . . .
LennardJones . . . .
Lest . . . . . . . . .
levelset . . . . . . .
leverage.ppm . . . .
lgcp.estK . . . . . .
lgcp.estpcf . . . . . .
lineardirichlet . . . .
lineardisc . . . . . .
linearK . . . . . . .
linearKcross . . . . .
linearKcross.inhom .
linearKdot . . . . . .
linearKdot.inhom . .
linearKinhom . . . .
linearmarkconnect . .
linearmarkequal . . .
linearpcf . . . . . . .
linearpcfcross . . . .
linearpcfcross.inhom
linearpcfdot . . . . .
linearpcfdot.inhom .
linearpcfinhom . . .
linequad . . . . . . .

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629
630
631
635
636
641
642
644
647
648
649
650
651
654
657
662
664
667
668
669
671
672
673
674
675
677
678
680
681
682
684
685
686
688
691
694
695
696
697
699
700
702
703
705
707
708
709
711
712
714
715
717

R topics documented:
linfun . . . . . .
Linhom . . . . .
linim . . . . . . .
linnet . . . . . .
lintess . . . . . .
lixellate . . . . .
localK . . . . . .
localKinhom . .
localpcf . . . . .
logLik.dppm . . .
logLik.kppm . . .
logLik.mppm . .
logLik.ppm . . .
logLik.slrm . . .
lohboot . . . . .
lpp . . . . . . . .
lppm . . . . . . .
lurking . . . . . .
lurking.mppm . .
lut . . . . . . . .
markconnect . . .
markcorr . . . . .
markcrosscorr . .
marks . . . . . .
marks.psp . . . .
marks.tess . . . .
markstat . . . . .
marktable . . . .
markvario . . . .
matchingdist . . .
matclust.estK . .
matclust.estpcf .
Math.im . . . . .
Math.imlist . . .
Math.linim . . .
matrixpower . . .
maxnndist . . . .
mean.im . . . . .
mean.linim . . .
measureVariation
mergeLevels . . .
methods.box3 . .
methods.boxx . .
methods.dppm . .
methods.fii . . .
methods.funxy .
methods.kppm . .
methods.layered .
methods.linfun .
methods.linim . .
methods.linnet . .
methods.lpp . . .

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718
719
720
722
723
724
726
728
730
732
733
735
737
739
740
742
744
745
749
751
752
754
758
760
762
763
764
766
767
769
770
772
775
776
778
780
781
782
783
784
785
786
788
789
790
791
792
793
795
796
798
800

R topics documented:

16
methods.lppm . . . . .
methods.objsurf . . . .
methods.pp3 . . . . . .
methods.ppx . . . . . .
methods.rho2hat . . . .
methods.rhohat . . . .
methods.slrm . . . . .
methods.ssf . . . . . .
methods.unitname . . .
methods.zclustermodel
midpoints.psp . . . . .
mincontrast . . . . . .
MinkowskiSum . . . .
miplot . . . . . . . . .
model.depends . . . .
model.frame.ppm . . .
model.images . . . . .
model.matrix.mppm . .
model.matrix.ppm . . .
model.matrix.slrm . . .
mppm . . . . . . . . .
msr . . . . . . . . . .
MultiHard . . . . . . .
multiplicity.ppp . . . .
MultiStrauss . . . . . .
MultiStraussHard . . .
nearest.raster.point . .
nearestsegment . . . .
nestsplit . . . . . . . .
nnclean . . . . . . . .
nncorr . . . . . . . . .
nncross . . . . . . . .
nncross.lpp . . . . . .
nncross.pp3 . . . . . .
nndensity.ppp . . . . .
nndist . . . . . . . . .
nndist.lpp . . . . . . .
nndist.pp3 . . . . . . .
nndist.ppx . . . . . . .
nndist.psp . . . . . . .
nnfromvertex . . . . .
nnfun . . . . . . . . .
nnfun.lpp . . . . . . .
nnmap . . . . . . . . .
nnmark . . . . . . . .
nnorient . . . . . . . .
nnwhich . . . . . . . .
nnwhich.lpp . . . . . .
nnwhich.pp3 . . . . . .
nnwhich.ppx . . . . . .
nobjects . . . . . . . .
npfun . . . . . . . . .

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801
803
804
805
806
807
809
810
812
813
814
815
817
818
820
821
823
825
826
828
829
832
834
835
837
838
840
841
842
843
845
848
850
852
855
856
858
860
861
863
864
865
866
867
869
871
872
875
876
877
878
879

R topics documented:
npoints . . . . .
nsegments . . .
nvertices . . . .
objsurf . . . . .
opening . . . .
Ops.msr . . . .
Ord . . . . . .
ord.family . . .
OrdThresh . . .
overlap.owin . .
owin . . . . . .
owin.object . .
padimage . . .
pairdist . . . .
pairdist.default
pairdist.lpp . .
pairdist.pp3 . .
pairdist.ppp . .
pairdist.ppx . .
pairdist.psp . .
pairorient . . .
PairPiece . . .
pairs.im . . . .
pairs.linim . . .
pairsat.family .
Pairwise . . . .
pairwise.family
panel.contour .
parameters . . .
parres . . . . .
pcf . . . . . . .
pcf.fasp . . . .
pcf.fv . . . . .
pcf.ppp . . . .
pcf3est . . . . .
pcfcross . . . .
pcfcross.inhom
pcfdot . . . . .
pcfdot.inhom .
pcfinhom . . .
pcfmulti . . . .
Penttinen . . .
perimeter . . .
periodify . . . .
persp.im . . . .
perspPoints . .
pixelcentres . .
pixellate . . . .
pixellate.owin .
pixellate.ppp . .
pixellate.psp . .
pixelquad . . .

17
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880
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951

R topics documented:

18
plot.anylist . . . .
plot.bermantest . .
plot.cdftest . . . .
plot.colourmap . .
plot.dppm . . . . .
plot.envelope . . .
plot.fasp . . . . . .
plot.fv . . . . . . .
plot.hyperframe . .
plot.im . . . . . . .
plot.imlist . . . . .
plot.influence.ppm
plot.kppm . . . . .
plot.laslett . . . . .
plot.layered . . . .
plot.leverage.ppm .
plot.linim . . . . .
plot.linnet . . . . .
plot.lintess . . . . .
plot.listof . . . . .
plot.lpp . . . . . .
plot.lppm . . . . .
plot.mppm . . . . .
plot.msr . . . . . .
plot.onearrow . . .
plot.owin . . . . .
plot.plotppm . . . .
plot.pp3 . . . . . .
plot.ppm . . . . . .
plot.ppp . . . . . .
plot.psp . . . . . .
plot.quad . . . . .
plot.quadratcount .
plot.quadrattest . .
plot.rppm . . . . .
plot.scan.test . . .
plot.slrm . . . . . .
plot.solist . . . . .
plot.splitppp . . . .
plot.ssf . . . . . . .
plot.symbolmap . .
plot.tess . . . . . .
plot.textstring . . .
plot.texturemap . .
plot.yardstick . . .
points.lpp . . . . .
pointsOnLines . . .
Poisson . . . . . .
polynom . . . . . .
pool . . . . . . . .
pool.anylist . . . .
pool.envelope . . .

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R topics documented:
pool.fasp . . . . . .
pool.fv . . . . . . .
pool.quadrattest . .
pool.rat . . . . . .
pp3 . . . . . . . .
ppm . . . . . . . .
ppm.object . . . .
ppm.ppp . . . . . .
ppmInfluence . . .
ppp . . . . . . . .
ppp.object . . . . .
pppdist . . . . . . .
pppmatching . . .
pppmatching.object
PPversion . . . . .
ppx . . . . . . . .
predict.dppm . . .
predict.kppm . . .
predict.lppm . . . .
predict.mppm . . .
predict.ppm . . . .
predict.rppm . . . .
predict.slrm . . . .
print.im . . . . . .
print.owin . . . . .
print.ppm . . . . .
print.ppp . . . . . .
print.psp . . . . . .
print.quad . . . . .
profilepl . . . . . .
progressreport . . .
project2segment . .
project2set . . . . .
prune.rppm . . . .
pseudoR2 . . . . .
psib . . . . . . . .
psp . . . . . . . . .
psp.object . . . . .
psst . . . . . . . .
psstA . . . . . . .
psstG . . . . . . .
qqplot.ppm . . . .
quad.object . . . .
quad.ppm . . . . .
quadrat.test . . . .
quadrat.test.mppm .
quadrat.test.splitppp
quadratcount . . .
quadratresample . .
quadrats . . . . . .
quadscheme . . . .
quadscheme.logi .

19
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. 1035
. 1036
. 1037
. 1038
. 1039
. 1040
. 1046
. 1048
. 1058
. 1060
. 1063
. 1065
. 1068
. 1069
. 1071
. 1072
. 1073
. 1074
. 1075
. 1076
. 1078
. 1083
. 1084
. 1085
. 1086
. 1087
. 1088
. 1089
. 1090
. 1091
. 1093
. 1095
. 1096
. 1097
. 1098
. 1099
. 1100
. 1101
. 1102
. 1104
. 1107
. 1109
. 1113
. 1114
. 1116
. 1119
. 1121
. 1122
. 1125
. 1126
. 1127
. 1129

R topics documented:

20
quantess . . . . .
quantile.density .
quantile.ewcdf . .
quantile.im . . .
quasirandom . . .
rags . . . . . . .
ragsAreaInter . .
ragsMultiHard . .
ranef.mppm . . .
range.fv . . . . .
raster.x . . . . . .
rat . . . . . . . .
rCauchy . . . . .
rcell . . . . . . .
rcellnumber . . .
rDGS . . . . . .
rDiggleGratton .
rdpp . . . . . . .
reach . . . . . . .
reach.dppm . . .
reduced.sample .
reflect . . . . . .
regularpolygon .
relevel.im . . . .
reload.or.compute
relrisk . . . . . .
relrisk.ppm . . .
relrisk.ppp . . . .
Replace.im . . .
Replace.linim . .
requireversion . .
rescale . . . . . .
rescale.im . . . .
rescale.owin . . .
rescale.ppp . . .
rescale.psp . . . .
rescue.rectangle .
residuals.dppm .
residuals.kppm .
residuals.mppm .
residuals.ppm . .
rex . . . . . . . .
rGaussPoisson . .
rgbim . . . . . .
rHardcore . . . .
rho2hat . . . . .
rhohat . . . . . .
ripras . . . . . .
rjitter . . . . . .
rknn . . . . . . .
rlabel . . . . . .
rLGCP . . . . . .

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. 1131
. 1133
. 1134
. 1135
. 1136
. 1137
. 1138
. 1140
. 1141
. 1142
. 1143
. 1144
. 1145
. 1147
. 1149
. 1150
. 1151
. 1153
. 1154
. 1156
. 1157
. 1158
. 1159
. 1160
. 1161
. 1162
. 1163
. 1165
. 1168
. 1170
. 1171
. 1172
. 1173
. 1174
. 1176
. 1177
. 1178
. 1179
. 1180
. 1181
. 1182
. 1185
. 1186
. 1187
. 1188
. 1190
. 1191
. 1195
. 1197
. 1198
. 1199
. 1200

R topics documented:
rlinegrid . . . . .
rlpp . . . . . . .
rMatClust . . . .
rMaternI . . . . .
rMaternII . . . .
rmh . . . . . . .
rmh.default . . .
rmh.ppm . . . . .
rmhcontrol . . . .
rmhexpand . . .
rmhmodel . . . .
rmhmodel.default
rmhmodel.list . .
rmhmodel.ppm .
rmhstart . . . . .
rMosaicField . .
rMosaicSet . . .
rmpoint . . . . .
rmpoispp . . . .
rNeymanScott . .
rnoise . . . . . .
roc . . . . . . . .
rose . . . . . . .
rotate . . . . . .
rotate.im . . . . .
rotate.infline . . .
rotate.owin . . .
rotate.ppp . . . .
rotate.psp . . . .
rotmean . . . . .
round.ppp . . . .
rounding . . . . .
rPenttinen . . . .
rpoint . . . . . .
rpoisline . . . . .
rpoislinetess . . .
rpoislpp . . . . .
rpoispp . . . . .
rpoispp3 . . . . .
rpoisppOnLines .
rpoisppx . . . . .
rPoissonCluster .
rppm . . . . . . .
rQuasi . . . . . .
rshift . . . . . . .
rshift.ppp . . . .
rshift.psp . . . .
rshift.splitppp . .
rSSI . . . . . . .
rstrat . . . . . . .
rStrauss . . . . .
rStraussHard . .

21
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. 1202
. 1203
. 1204
. 1207
. 1208
. 1209
. 1210
. 1220
. 1223
. 1227
. 1229
. 1230
. 1237
. 1239
. 1241
. 1242
. 1243
. 1244
. 1248
. 1251
. 1254
. 1255
. 1256
. 1258
. 1259
. 1260
. 1261
. 1262
. 1263
. 1264
. 1266
. 1267
. 1268
. 1270
. 1271
. 1272
. 1273
. 1274
. 1276
. 1277
. 1279
. 1280
. 1282
. 1283
. 1284
. 1285
. 1287
. 1289
. 1290
. 1292
. 1293
. 1295

R topics documented:

22
rsyst . . . . . . . .
rtemper . . . . . .
rthin . . . . . . . .
rThomas . . . . . .
run.simplepanel . .
runifdisc . . . . . .
runiflpp . . . . . .
runifpoint . . . . .
runifpoint3 . . . .
runifpointOnLines .
runifpointx . . . .
rVarGamma . . . .
SatPiece . . . . . .
Saturated . . . . .
scalardilate . . . .
scaletointerval . . .
scan.test . . . . . .
scanLRTS . . . . .
scanpp . . . . . . .
sdr . . . . . . . . .
sdrPredict . . . . .
segregation.test . .
selfcrossing.psp . .
selfcut.psp . . . . .
sessionLibs . . . .
setcov . . . . . . .
sharpen . . . . . .
shift . . . . . . . .
shift.im . . . . . .
shift.owin . . . . .
shift.ppp . . . . . .
shift.psp . . . . . .
sidelengths.owin .
simplepanel . . . .
simplify.owin . . .
simulate.dppm . . .
simulate.kppm . . .
simulate.lppm . . .
simulate.mppm . .
simulate.ppm . . .
simulate.slrm . . .
slrm . . . . . . . .
Smooth . . . . . .
Smooth.fv . . . . .
Smooth.msr . . . .
Smooth.ppp . . . .
Smooth.ssf . . . .
Smoothfun.ppp . .
Softcore . . . . . .
solapply . . . . . .
solist . . . . . . . .
solutionset . . . . .

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. 1296
. 1297
. 1299
. 1300
. 1302
. 1305
. 1306
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. 1326
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. 1329
. 1330
. 1331
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. 1334
. 1335
. 1336
. 1337
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. 1339
. 1342
. 1343
. 1345
. 1347
. 1348
. 1349
. 1351
. 1352
. 1354
. 1355
. 1356
. 1358
. 1360
. 1361
. 1362
. 1364
. 1365
. 1366

R topics documented:
spatdim . . . . .
spatialcdf . . . .
spatstat.options .
split.hyperframe .
split.im . . . . .
split.msr . . . . .
split.ppp . . . . .
split.ppx . . . . .
spokes . . . . . .
square . . . . . .
ssf . . . . . . . .
stieltjes . . . . .
stienen . . . . . .
stratrand . . . . .
Strauss . . . . . .
StraussHard . . .
studpermu.test . .
subfits . . . . . .
subset.hyperframe
subset.ppp . . . .
subspaceDistance
suffstat . . . . . .
summary.anylist .
summary.im . . .
summary.kppm .
summary.listof .
summary.owin . .
summary.ppm . .
summary.ppp . .
summary.psp . .
summary.quad . .
summary.solist .
summary.splitppp
sumouter . . . .
superimpose . . .
superimpose.lpp .
symbolmap . . .
tess . . . . . . .
test.crossing.psp .
text.ppp . . . . .
texturemap . . .
textureplot . . . .
thinNetwork . . .
thomas.estK . . .
thomas.estpcf . .
tile.areas . . . . .
tile.lengths . . . .
tileindex . . . . .
tilenames . . . .
tiles . . . . . . .
tiles.empty . . . .
timed . . . . . .

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. 1368
. 1369
. 1370
. 1374
. 1375
. 1376
. 1378
. 1380
. 1382
. 1383
. 1384
. 1385
. 1386
. 1387
. 1389
. 1390
. 1392
. 1394
. 1395
. 1396
. 1398
. 1399
. 1401
. 1402
. 1403
. 1404
. 1405
. 1406
. 1407
. 1408
. 1409
. 1410
. 1411
. 1412
. 1413
. 1416
. 1417
. 1419
. 1421
. 1422
. 1423
. 1424
. 1425
. 1427
. 1429
. 1431
. 1432
. 1433
. 1434
. 1435
. 1436
. 1437

R topics documented:

24
timeTaken . . . . . . . . .
transect.im . . . . . . . . .
transmat . . . . . . . . . .
treebranchlabels . . . . . .
treeprune . . . . . . . . .
triangulate.owin . . . . . .
trim.rectangle . . . . . . .
triplet.family . . . . . . .
Triplets . . . . . . . . . .
Tstat . . . . . . . . . . . .
tweak.colourmap . . . . .
union.quad . . . . . . . .
unique.ppp . . . . . . . .
unitname . . . . . . . . .
unmark . . . . . . . . . .
unnormdensity . . . . . .
unstack.msr . . . . . . . .
unstack.ppp . . . . . . . .
update.detpointprocfamily
update.interact . . . . . . .
update.kppm . . . . . . . .
update.ppm . . . . . . . .
update.rmhcontrol . . . . .
update.symbolmap . . . .
valid . . . . . . . . . . . .
valid.detpointprocfamily .
valid.ppm . . . . . . . . .
varblock . . . . . . . . . .
varcount . . . . . . . . . .
vargamma.estK . . . . . .
vargamma.estpcf . . . . .
vcov.kppm . . . . . . . . .
vcov.mppm . . . . . . . .
vcov.ppm . . . . . . . . .
vcov.slrm . . . . . . . . .
vertices . . . . . . . . . .
volume . . . . . . . . . .
weighted.median . . . . .
where.max . . . . . . . . .
whichhalfplane . . . . . .
whist . . . . . . . . . . . .
will.expand . . . . . . . .
Window . . . . . . . . . .
WindowOnly . . . . . . .
with.fv . . . . . . . . . . .
with.hyperframe . . . . . .
with.msr . . . . . . . . . .
with.ssf . . . . . . . . . .
yardstick . . . . . . . . . .
zapsmall.im . . . . . . . .
zclustermodel . . . . . . .
[.ssf . . . . . . . . . . . .

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spatstat-package

25

Index

1503

spatstat-package

The Spatstat Package

Description
This is a summary of the features of spatstat, a package in R for the statistical analysis of spatial
point patterns.
Details
spatstat is a package for the statistical analysis of spatial data. Its main focus is the analysis of
spatial patterns of points in two-dimensional space. The points may carry auxiliary data (‘marks’),
and the spatial region in which the points were recorded may have arbitrary shape.
The package is designed to support a complete statistical analysis of spatial data. It supports
• creation, manipulation and plotting of point patterns;
• exploratory data analysis;
• spatial random sampling;
• simulation of point process models;
• parametric model-fitting;
• non-parametric smoothing and regression;
• formal inference (hypothesis tests, confidence intervals);
• model diagnostics.
Apart from two-dimensional point patterns and point processes, spatstat also supports point patterns in three dimensions, point patterns in multidimensional space-time, point patterns on a linear
network, patterns of line segments in two dimensions, and spatial tessellations and random sets in
two dimensions.
The package can fit several types of point process models to a point pattern dataset:
• Poisson point process models (by Berman-Turner approximate maximum likelihood or by
spatial logistic regression)
• Gibbs/Markov point process models (by Baddeley-Turner approximate maximum pseudolikelihood, Coeurjolly-Rubak logistic likelihood, or Huang-Ogata approximate maximum likelihood)
• Cox/cluster point process models (by Waagepetersen’s two-step fitting procedure and minimum contrast, composite likelihood, or Palm likelihood)
• determinantal point process models (by Waagepetersen’s two-step fitting procedure and minimum contrast, composite likelihood, or Palm likelihood)
The models may include spatial trend, dependence on covariates, and complicated interpoint interactions. Models are specified by a formula in the R language, and are fitted using a function
analogous to lm and glm. Fitted models can be printed, plotted, predicted, simulated and so on.

26

spatstat-package

Getting Started
For a quick introduction to spatstat, read the package vignette Getting started with spatstat installed
with spatstat. To read that document, you can either
• visit cran.r-project.org/web/packages/spatstat and click on Getting Started with Spatstat
• start R, type library(spatstat) and vignette('getstart')
• start R, type help.start() to open the help browser, and navigate to Packages > spatstat > Vignettes.
Once you have installed spatstat, start R and type library(spatstat). Then type beginner for
a beginner’s introduction, or demo(spatstat) for a demonstration of the package’s capabilities.
For a complete course on spatstat, and on statistical analysis of spatial point patterns, read the
book by Baddeley, Rubak and Turner (2015). Other recommended books on spatial point process
methods are Diggle (2014), Gelfand et al (2010) and Illian et al (2008).
The spatstat package includes over 50 datasets, which can be useful when learning the package.
Type demo(data) to see plots of all datasets available in the package. Type vignette('datasets')
for detailed background information on these datasets, and plots of each dataset.
For information on converting your data into spatstat format, read Chapter 3 of Baddeley, Rubak
and Turner (2015). This chapter is available free online, as one of the sample chapters at the book
companion website, spatstat.github.io/book.
For information about handling data in shapefiles, see Chapter 3, or the Vignette Handling shapefiles in the spatstat package, installed with spatstat, accessible as vignette('shapefiles').
Updates
New versions of spatstat are released every 8 weeks. Users are advised to update their installation
of spatstat regularly.
Type latest.news to read the news documentation about changes to the current installed version
of spatstat.
See the Vignette Summary of recent updates, installed with spatstat, which describes the main
changes to spatstat since the book (Baddeley, Rubak and Turner, 2015) was published. It is accessible as vignette('updates').
Type news(package="spatstat") to read news documentation about all previous versions of the
package.
FUNCTIONS AND DATASETS
Following is a summary of the main functions and datasets in the spatstat package. Alternatively
an alphabetical list of all functions and datasets is available by typing library(help=spatstat).
For further information on any of these, type help(name) or ?name where name is the name of the
function or dataset.
CONTENTS:
I.
II.
III.
IV.
V.
VI.
VII.

Creating and manipulating data
Exploratory Data Analysis
Model fitting (Cox and cluster models)
Model fitting (Poisson and Gibbs models)
Model fitting (determinantal point processes)
Model fitting (spatial logistic regression)
Simulation

spatstat-package

27
VIII.
IX.

Tests and diagnostics
Documentation

I. CREATING AND MANIPULATING DATA
Types of spatial data:
The main types of spatial data supported by spatstat are:
ppp
owin
im
psp
tess
pp3
ppx
lpp

point pattern
window (spatial region)
pixel image
line segment pattern
tessellation
three-dimensional point pattern
point pattern in any number of dimensions
point pattern on a linear network

To create a point pattern:
ppp

as.ppp
clickppp
marks<-, %mark%

create a point pattern from (x, y) and window information
ppp(x, y, xlim, ylim) for rectangular window
ppp(x, y, poly) for polygonal window
ppp(x, y, mask) for binary image window
convert other types of data to a ppp object
interactively add points to a plot
attach/reassign marks to a point pattern

To simulate a random point pattern:
runifpoint
rpoint
rmpoint
rpoispp
rmpoispp
runifdisc
rstrat
rsyst
rjitter
rMaternI
rMaternII
rSSI
rStrauss
rHardcore
rStraussHard
rDiggleGratton
rDGS
rPenttinen
rNeymanScott
rPoissonCluster
rMatClust

generate n independent uniform random points
generate n independent random points
generate n independent multitype random points
simulate the (in)homogeneous Poisson point process
simulate the (in)homogeneous multitype Poisson point process
generate n independent uniform random points in disc
stratified random sample of points
systematic random sample of points
apply random displacements to points in a pattern
simulate the Matérn Model I inhibition process
simulate the Matérn Model II inhibition process
simulate Simple Sequential Inhibition process
simulate Strauss process (perfect simulation)
simulate Hard Core process (perfect simulation)
simulate Strauss-hard core process (perfect simulation)
simulate Diggle-Gratton process (perfect simulation)
simulate Diggle-Gates-Stibbard process (perfect simulation)
simulate Penttinen process (perfect simulation)
simulate a general Neyman-Scott process
simulate a general Poisson cluster process
simulate the Matérn Cluster process

28

spatstat-package
rThomas
rGaussPoisson
rCauchy
rVarGamma
rthin
rcell
rmh
simulate.ppm
runifpointOnLines
rpoisppOnLines

simulate the Thomas process
simulate the Gauss-Poisson cluster process
simulate Neyman-Scott Cauchy cluster process
simulate Neyman-Scott Variance Gamma cluster process
random thinning
simulate the Baddeley-Silverman cell process
simulate Gibbs point process using Metropolis-Hastings
simulate Gibbs point process using Metropolis-Hastings
generate n random points along specified line segments
generate Poisson random points along specified line segments

To randomly change an existing point pattern:
rshift
rjitter
rthin
rlabel
quadratresample

random shifting of points
apply random displacements to points in a pattern
random thinning
random (re)labelling of a multitype point pattern
block resampling

Standard point pattern datasets:
Datasets in spatstat are lazy-loaded, so you can simply type the name of the dataset to use it; there
is no need to type data(amacrine) etc.
Type demo(data) to see a display of all the datasets installed with the package.
Type vignette('datasets') for a document giving an overview of all datasets, including background information, and plots.
amacrine
anemones
ants
bdspots
bei
betacells
bramblecanes
bronzefilter
cells
chicago
chorley
clmfires
copper
dendrite
demohyper
demopat
finpines
flu
gordon
gorillas
hamster
humberside
hyytiala
japanesepines
lansing

Austin Hughes’ rabbit amacrine cells
Upton-Fingleton sea anemones data
Harkness-Isham ant nests data
Breakdown spots in microelectrodes
Tropical rainforest trees
Waessle et al. cat retinal ganglia data
Bramble Canes data
Bronze Filter Section data
Crick-Ripley biological cells data
Chicago crimes
Chorley-Ribble cancer data
Castilla-La Mancha forest fires
Berman-Huntington copper deposits data
Dendritic spines
Synthetic point patterns
Synthetic point pattern
Finnish Pines data
Influenza virus proteins
People in Gordon Square, London
Gorilla nest sites
Aherne’s hamster tumour data
North Humberside childhood leukaemia data
Mixed forest in Hyytiälä, Finland
Japanese Pines data
Lansing Woods data

spatstat-package
longleaf
mucosa
murchison
nbfires
nztrees
osteo
paracou
ponderosa
pyramidal
redwood
redwoodfull
residualspaper
shapley
simdat
spiders
sporophores
spruces
swedishpines
urkiola
waka
waterstriders

29
Longleaf Pines data
Cells in gastric mucosa
Murchison gold deposits
New Brunswick fires data
Mark-Esler-Ripley trees data
Osteocyte lacunae (3D, replicated)
Kimboto trees in Paracou, French Guiana
Getis-Franklin ponderosa pine trees data
Pyramidal neurons from 31 brains
Strauss-Ripley redwood saplings data
Strauss redwood saplings data (full set)
Data from Baddeley et al (2005)
Galaxies in an astronomical survey
Simulated point pattern (inhomogeneous, with interaction)
Spider webs on mortar lines of brick wall
Mycorrhizal fungi around a tree
Spruce trees in Saxonia
Strand-Ripley Swedish pines data
Urkiola Woods data
Trees in Waka national park
Insects on water surface

To manipulate a point pattern:
plot.ppp
iplot
edit.ppp
[.ppp
subset.ppp
superimpose
by.ppp
cut.ppp
split.ppp
unmark
npoints
coords
marks
rotate
shift
flipxy
reflect
periodify
affine
scalardilate
density.ppp
Smooth.ppp
nnmark
sharpen.ppp
identify.ppp
unique.ppp
duplicated.ppp
connected.ppp

plot a point pattern (e.g. plot(X))
plot a point pattern interactively
interactive text editor
extract or replace a subset of a point pattern
pp[subset] or pp[subwindow]
extract subset of point pattern satisfying a condition
combine several point patterns
apply a function to sub-patterns of a point pattern
classify the points in a point pattern
divide pattern into sub-patterns
remove marks
count the number of points
extract coordinates, change coordinates
extract marks, change marks or attach marks
rotate pattern
translate pattern
swap x and y coordinates
reflect in the origin
make several translated copies
apply affine transformation
apply scalar dilation
kernel estimation of point pattern intensity
kernel smoothing of marks of point pattern
mark value of nearest data point
data sharpening
interactively identify points
remove duplicate points
determine which points are duplicates
find clumps of points

30

spatstat-package
dirichlet
delaunay
delaunayDistance
convexhull
discretise
pixellate.ppp
as.im.ppp

compute Dirichlet-Voronoi tessellation
compute Delaunay triangulation
graph distance in Delaunay triangulation
compute convex hull
discretise coordinates
approximate point pattern by pixel image
approximate point pattern by pixel image

See spatstat.options to control plotting behaviour.
To create a window:
An object of class "owin" describes a spatial region (a window of observation).
owin

Window
Frame
as.owin
square
disc
ellipse
ripras
convexhull
letterR
clickpoly
clickbox

Create a window object
owin(xlim, ylim) for rectangular window
owin(poly) for polygonal window
owin(mask) for binary image window
Extract window of another object
Extract the containing rectangle (’frame’) of another object
Convert other data to a window object
make a square window
make a circular window
make an elliptical window
Ripley-Rasson estimator of window, given only the points
compute convex hull of something
polygonal window in the shape of the R logo
interactively draw a polygonal window
interactively draw a rectangle

To manipulate a window:
plot.owin
boundingbox
erosion
dilation
closing
opening
border
complement.owin
simplify.owin
rotate
flipxy
shift
periodify
affine
as.data.frame.owin

plot a window.
plot(W)
Find a tight bounding box for the window
erode window by a distance r
dilate window by a distance r
close window by a distance r
open window by a distance r
difference between window and its erosion/dilation
invert (swap inside and outside)
approximate a window by a simple polygon
rotate window
swap x and y coordinates
translate window
make several translated copies
apply affine transformation
convert window to data frame

Digital approximations:
as.mask
as.im.owin

Make a discrete pixel approximation of a given window
convert window to pixel image

spatstat-package
pixellate.owin
commonGrid
nearest.raster.point
raster.x
raster.y
raster.xy
as.polygonal

31
convert window to pixel image
find common pixel grid for windows
map continuous coordinates to raster locations
raster x coordinates
raster y coordinates
raster x and y coordinates
convert pixel mask to polygonal window

See spatstat.options to control the approximation
Geometrical computations with windows:
edges
intersect.owin
union.owin
setminus.owin
inside.owin
area.owin
perimeter
diameter.owin
incircle
inradius
connected.owin
eroded.areas
dilated.areas
bdist.points
bdist.pixels
bdist.tiles
distmap.owin
distfun.owin
centroid.owin
is.subset.owin
is.convex
convexhull
triangulate.owin
as.mask
as.polygonal
is.rectangle
is.polygonal
is.mask
setcov
pixelcentres
clickdist

extract boundary edges
intersection of two windows
union of two windows
set subtraction of two windows
determine whether a point is inside a window
compute area
compute perimeter length
compute diameter
find largest circle inside a window
radius of incircle
find connected components of window
compute areas of eroded windows
compute areas of dilated windows
compute distances from data points to window boundary
compute distances from all pixels to window boundary
boundary distance for each tile in tessellation
distance transform image
distance transform
compute centroid (centre of mass) of window
determine whether one window contains another
determine whether a window is convex
compute convex hull
decompose into triangles
pixel approximation of window
polygonal approximation of window
test whether window is a rectangle
test whether window is polygonal
test whether window is a mask
spatial covariance function of window
extract centres of pixels in mask
measure distance between two points clicked by user

Pixel images: An object of class "im" represents a pixel image. Such objects are returned by some
of the functions in spatstat including Kmeasure, setcov and density.ppp.
im
as.im
pixellate
as.matrix.im
as.data.frame.im
as.function.im

create a pixel image
convert other data to a pixel image
convert other data to a pixel image
convert pixel image to matrix
convert pixel image to data frame
convert pixel image to function

32

spatstat-package
plot.im
contour.im
persp.im
rgbim
hsvim
[.im
[<-.im
rotate.im
shift.im
affine.im
X
summary(X)
hist.im
mean.im
integral.im
quantile.im
cut.im
is.im
interp.im
blur
Smooth.im
connected.im
compatible.im
harmonise.im
commonGrid
eval.im
scaletointerval
zapsmall.im
levelset
solutionset
imcov
convolve.im
transect.im
pixelcentres
transmat
rnoise

plot a pixel image on screen as a digital image
draw contours of a pixel image
draw perspective plot of a pixel image
create colour-valued pixel image
create colour-valued pixel image
extract a subset of a pixel image
replace a subset of a pixel image
rotate pixel image
apply vector shift to pixel image
apply affine transformation to image
print very basic information about image X
summary of image X
histogram of image
mean pixel value of image
integral of pixel values
quantiles of image
convert numeric image to factor image
test whether an object is a pixel image
interpolate a pixel image
apply Gaussian blur to image
apply Gaussian blur to image
find connected components
test whether two images have compatible dimensions
make images compatible
find a common pixel grid for images
evaluate any expression involving images
rescale pixel values
set very small pixel values to zero
level set of an image
region where an expression is true
spatial covariance function of image
spatial convolution of images
line transect of image
extract centres of pixels
convert matrix of pixel values
to a different indexing convention
random pixel noise

Line segment patterns
An object of class "psp" represents a pattern of straight line segments.
psp
as.psp
edges
is.psp
plot.psp
print.psp
summary.psp
[.psp
as.data.frame.psp
marks.psp
marks<-.psp

create a line segment pattern
convert other data into a line segment pattern
extract edges of a window
determine whether a dataset has class "psp"
plot a line segment pattern
print basic information
print summary information
extract a subset of a line segment pattern
convert line segment pattern to data frame
extract marks of line segments
assign new marks to line segments

spatstat-package

33

unmark.psp
midpoints.psp
endpoints.psp
lengths.psp
angles.psp
superimpose
flipxy
rotate.psp
shift.psp
periodify
affine.psp
pixellate.psp
as.mask.psp
distmap.psp
distfun.psp
density.psp
selfcrossing.psp
selfcut.psp
crossing.psp
nncross
nearestsegment
project2segment
pointsOnLines
rpoisline
rlinegrid

delete marks from line segments
compute the midpoints of line segments
extract the endpoints of line segments
compute the lengths of line segments
compute the orientation angles of line segments
combine several line segment patterns
swap x and y coordinates
rotate a line segment pattern
shift a line segment pattern
make several shifted copies
apply an affine transformation
approximate line segment pattern by pixel image
approximate line segment pattern by binary mask
compute the distance map of a line segment pattern
compute the distance map of a line segment pattern
kernel smoothing of line segments
find crossing points between line segments
cut segments where they cross
find crossing points between two line segment patterns
find distance to nearest line segment from a given point
find line segment closest to a given point
find location along a line segment closest to a given point
generate points evenly spaced along line segment
generate a realisation of the Poisson line process inside a window
generate a random array of parallel lines through a window

Tessellations
An object of class "tess" represents a tessellation.
tess
quadrats
hextess
quantess
as.tess
plot.tess
tiles
[.tess
[<-.tess
intersect.tess
chop.tess
dirichlet
delaunay
rpoislinetess
tile.areas
bdist.tiles

create a tessellation
create a tessellation of rectangles
create a tessellation of hexagons
quantile tessellation
convert other data to a tessellation
plot a tessellation
extract all the tiles of a tessellation
extract some tiles of a tessellation
change some tiles of a tessellation
intersect two tessellations
or restrict a tessellation to a window
subdivide a tessellation by a line
compute Dirichlet-Voronoi tessellation of points
compute Delaunay triangulation of points
generate tessellation using Poisson line process
area of each tile in tessellation
boundary distance for each tile in tessellation

Three-dimensional point patterns
An object of class "pp3" represents a three-dimensional point pattern in a rectangular box. The box
is represented by an object of class "box3".
pp3

create a 3-D point pattern

34

spatstat-package
plot.pp3
coords
as.hyperframe
subset.pp3
unitname.pp3
npoints
runifpoint3
rpoispp3
envelope.pp3
box3
as.box3
unitname.box3
diameter.box3
volume.box3
shortside.box3
eroded.volumes

plot a 3-D point pattern
extract coordinates
extract coordinates
extract subset of 3-D point pattern
name of unit of length
count the number of points
generate uniform random points in 3-D
generate Poisson random points in 3-D
generate simulation envelopes for 3-D pattern
create a 3-D rectangular box
convert data to 3-D rectangular box
name of unit of length
diameter of box
volume of box
shortest side of box
volumes of erosions of box

Multi-dimensional space-time point patterns
An object of class "ppx" represents a point pattern in multi-dimensional space and/or time.
ppx
coords
as.hyperframe
subset.ppx
unitname.ppx
npoints
runifpointx
rpoisppx
boxx
diameter.boxx
volume.boxx
shortside.boxx
eroded.volumes.boxx

create a multidimensional space-time point pattern
extract coordinates
extract coordinates
extract subset
name of unit of length
count the number of points
generate uniform random points
generate Poisson random points
define multidimensional box
diameter of box
volume of box
shortest side of box
volumes of erosions of box

Point patterns on a linear network
An object of class "linnet" represents a linear network (for example, a road network).
linnet
clickjoin
iplot.linnet
simplenet
lineardisc
delaunayNetwork
dirichletNetwork
methods.linnet
vertices.linnet
pixellate.linnet

create a linear network
interactively join vertices in network
interactively plot network
simple example of network
disc in a linear network
network of Delaunay triangulation
network of Dirichlet edges
methods for linnet objects
nodes of network
approximate by pixel image

An object of class "lpp" represents a point pattern on a linear network (for example, road accidents
on a road network).

spatstat-package

35
lpp
methods.lpp
subset.lpp
rpoislpp
runiflpp
chicago
dendrite
spiders

create a point pattern on a linear network
methods for lpp objects
method for subset
simulate Poisson points on linear network
simulate random points on a linear network
Chicago crime data
Dendritic spines data
Spider webs on mortar lines of brick wall

Hyperframes
A hyperframe is like a data frame, except that the entries may be objects of any kind.
hyperframe
as.hyperframe
plot.hyperframe
with.hyperframe
cbind.hyperframe
rbind.hyperframe
as.data.frame.hyperframe
subset.hyperframe
head.hyperframe
tail.hyperframe

create a hyperframe
convert data to hyperframe
plot hyperframe
evaluate expression using each row of hyperframe
combine hyperframes by columns
combine hyperframes by rows
convert hyperframe to data frame
method for subset
first few rows of hyperframe
last few rows of hyperframe

Layered objects
A layered object represents data that should be plotted in successive layers, for example, a background and a foreground.
layered
plot.layered
[.layered

create layered object
plot layered object
extract subset of layered object

Colour maps
A colour map is a mechanism for associating colours with data. It can be regarded as a function,
mapping data to colours. Using a colourmap object in a plot command ensures that the mapping
from numbers to colours is the same in different plots.
colourmap
plot.colourmap
tweak.colourmap
interp.colourmap
beachcolourmap

create a colour map
plot the colour map only
alter individual colour values
make a smooth transition between colours
one special colour map

II. EXPLORATORY DATA ANALYSIS
Inspection of data:
summary(X)
X
any(duplicated(X))
istat(X)
View(X)

print useful summary of point pattern X
print basic description of point pattern X
check for duplicated points in pattern X
Interactive exploratory analysis
spreadsheet-style viewer

36

spatstat-package
Classical exploratory tools:
clarkevans
fryplot
miplot

Clark and Evans aggregation index
Fry plot
Morisita Index plot

Smoothing:
density.ppp
relrisk
Smooth.ppp
bw.diggle
bw.ppl
bw.scott
bw.relrisk
bw.smoothppp
bw.frac
bw.stoyan

kernel smoothed density/intensity
kernel estimate of relative risk
spatial interpolation of marks
cross-validated bandwidth selection for density.ppp
likelihood cross-validated bandwidth selection for density.ppp
Scott’s rule of thumb for density estimation
cross-validated bandwidth selection for relrisk
cross-validated bandwidth selection for Smooth.ppp
bandwidth selection using window geometry
Stoyan’s rule of thumb for bandwidth for pcf

Modern exploratory tools:
clusterset
nnclean
sharpen.ppp
rhohat
rho2hat
spatialcdf
roc

Allard-Fraley feature detection
Byers-Raftery feature detection
Choi-Hall data sharpening
Kernel estimate of covariate effect
Kernel estimate of effect of two covariates
Spatial cumulative distribution function
Receiver operating characteristic curve

Summary statistics for a point pattern: Type demo(sumfun) for a demonstration of many of the
summary statistics.
intensity
quadratcount
intensity.quadratcount
Fest
Gest
Jest
Kest
Lest
Tstat
allstats
pcf
Kinhom
Linhom
pcfinhom
Finhom
Ginhom
Jinhom
localL
localK
localpcf

Mean intensity
Quadrat counts
Mean intensity in quadrats
empty space function F
nearest neighbour distribution function G
J-function J = (1 − G)/(1 − F )
Ripley’s K-function
Besag L-function
Third order T -function
all four functions F , G, J, K
pair correlation function
K for inhomogeneous point patterns
L for inhomogeneous point patterns
pair correlation for inhomogeneous patterns
F for inhomogeneous point patterns
G for inhomogeneous point patterns
J for inhomogeneous point patterns
Getis-Franklin neighbourhood density function
neighbourhood K-function
local pair correlation function

spatstat-package
localKinhom
localLinhom
localpcfinhom
Ksector
Kscaled
Kest.fft
Kmeasure
envelope
varblock
lohboot

37
local K for inhomogeneous point patterns
local L for inhomogeneous point patterns
local pair correlation for inhomogeneous patterns
Directional K-function
locally scaled K-function
fast K-function using FFT for large datasets
reduced second moment measure
simulation envelopes for a summary function
variances and confidence intervals
for a summary function
bootstrap for a summary function

Related facilities:
plot.fv
eval.fv
harmonise.fv
eval.fasp
with.fv
Smooth.fv
deriv.fv
pool.fv
nndist
nnwhich
pairdist
crossdist
nncross
exactdt
distmap
distfun
nnmap
nnfun
density.ppp
Smooth.ppp
relrisk
sharpen.ppp
rknn

plot a summary function
evaluate any expression involving summary functions
make functions compatible
evaluate any expression involving an array of functions
evaluate an expression for a summary function
apply smoothing to a summary function
calculate derivative of a summary function
pool several estimates of a summary function
nearest neighbour distances
find nearest neighbours
distances between all pairs of points
distances between points in two patterns
nearest neighbours between two point patterns
distance from any location to nearest data point
distance map image
distance map function
nearest point image
nearest point function
kernel smoothed density
spatial interpolation of marks
kernel estimate of relative risk
data sharpening
theoretical distribution of nearest neighbour distance

Summary statistics for a multitype point pattern: A multitype point pattern is represented by an
object X of class "ppp" such that marks(X) is a factor.
relrisk
scan.test
Gcross,Gdot,Gmulti
Kcross,Kdot, Kmulti
Lcross,Ldot
Jcross,Jdot,Jmulti
pcfcross
pcfdot
pcfmulti
markconnect
alltypes

kernel estimation of relative risk
spatial scan test of elevated risk
multitype nearest neighbour distributions Gij , Gi•
multitype K-functions Kij , Ki•
multitype L-functions Lij , Li•
multitype J-functions Jij , Ji•
multitype pair correlation function gij
multitype pair correlation function gi•
general pair correlation function
marked connection function pij
estimates of the above for all i, j pairs

38

spatstat-package
Iest
Kcross.inhom,Kdot.inhom
Lcross.inhom,Ldot.inhom
pcfcross.inhom,pcfdot.inhom

multitype I-function
inhomogeneous counterparts of Kcross, Kdot
inhomogeneous counterparts of Lcross, Ldot
inhomogeneous counterparts of pcfcross, pcfdot

Summary statistics for a marked point pattern: A marked point pattern is represented by an
object X of class "ppp" with a component X$marks. The entries in the vector X$marks may be
numeric, complex, string or any other atomic type. For numeric marks, there are the following
functions:
markmean
markvar
markcorr
markcrosscorr
markvario
Kmark
Emark
Vmark
nnmean
nnvario

smoothed local average of marks
smoothed local variance of marks
mark correlation function
mark cross-correlation function
mark variogram
mark-weighted K function
mark independence diagnostic E(r)
mark independence diagnostic V (r)
nearest neighbour mean index
nearest neighbour mark variance index

For marks of any type, there are the following:
Gmulti
Kmulti
Jmulti

multitype nearest neighbour distribution
multitype K-function
multitype J-function

Alternatively use cut.ppp to convert a marked point pattern to a multitype point pattern.
Programming tools:
applynbd
markstat
marktable
pppdist

apply function to every neighbourhood in a point pattern
apply function to the marks of neighbours in a point pattern
tabulate the marks of neighbours in a point pattern
find the optimal match between two point patterns

Summary statistics for a point pattern on a linear network:
These are for point patterns on a linear network (class lpp). For unmarked patterns:
linearK
linearKinhom
linearpcf
linearpcfinhom

K function on linear network
inhomogeneous K function on linear network
pair correlation function on linear network
inhomogeneous pair correlation on linear network

For multitype patterns:
linearKcross
linearKdot
linearKcross.inhom
linearKdot.inhom
linearmarkconnect

K function between two types of points
K function from one type to any type
Inhomogeneous version of linearKcross
Inhomogeneous version of linearKdot
Mark connection function on linear network

spatstat-package

39

linearmarkequal
linearpcfcross
linearpcfdot
linearpcfcross.inhom
linearpcfdot.inhom

Mark equality function on linear network
Pair correlation between two types of points
Pair correlation from one type to any type
Inhomogeneous version of linearpcfcross
Inhomogeneous version of linearpcfdot

Related facilities:
pairdist.lpp
crossdist.lpp
nndist.lpp
nncross.lpp
nnwhich.lpp
nnfun.lpp
density.lpp
distfun.lpp
envelope.lpp
rpoislpp
runiflpp

distances between pairs
distances between pairs
nearest neighbour distances
nearest neighbour distances
find nearest neighbours
find nearest data point
kernel smoothing estimator of intensity
distance transform
simulation envelopes
simulate Poisson points on linear network
simulate random points on a linear network

It is also possible to fit point process models to lpp objects. See Section IV.
Summary statistics for a three-dimensional point pattern:
These are for 3-dimensional point pattern objects (class pp3).
F3est
G3est
K3est
pcf3est

empty space function F
nearest neighbour function G
K-function
pair correlation function

Related facilities:
envelope.pp3
pairdist.pp3
crossdist.pp3
nndist.pp3
nnwhich.pp3
nncross.pp3

simulation envelopes
distances between all pairs of points
distances between points in two patterns
nearest neighbour distances
find nearest neighbours
find nearest neighbours in another pattern

Computations for multi-dimensional point pattern:
These are for multi-dimensional space-time point pattern objects (class ppx).
pairdist.ppx
crossdist.ppx
nndist.ppx
nnwhich.ppx

distances between all pairs of points
distances between points in two patterns
nearest neighbour distances
find nearest neighbours

Summary statistics for random sets:
These work for point patterns (class ppp), line segment patterns (class psp) or windows (class owin).
Hest

spherical contact distribution H

40

spatstat-package
Gfox
Jfox

Foxall G-function
Foxall J-function

III. MODEL FITTING (COX AND CLUSTER MODELS)
Cluster process models (with homogeneous or inhomogeneous intensity) and Cox processes can be
fitted by the function kppm. Its result is an object of class "kppm". The fitted model can be printed,
plotted, predicted, simulated and updated.
kppm
plot.kppm
summary.kppm
fitted.kppm
predict.kppm
update.kppm
improve.kppm
simulate.kppm
vcov.kppm
coef.kppm
formula.kppm
parameters
clusterfield
clusterradius
Kmodel.kppm
pcfmodel.kppm

Fit model
Plot the fitted model
Summarise the fitted model
Compute fitted intensity
Compute fitted intensity
Update the model
Refine the estimate of trend
Generate simulated realisations
Variance-covariance matrix of coefficients
Extract trend coefficients
Extract trend formula
Extract all model parameters
Compute offspring density
Radius of support of offspring density
K function of fitted model
Pair correlation of fitted model

For model selection, you can also use the generic functions step, drop1 and AIC on fitted point
process models.
The theoretical models can also be simulated, for any choice of parameter values, using rThomas,
rMatClust, rCauchy, rVarGamma, and rLGCP.
Lower-level fitting functions include:
lgcp.estK
lgcp.estpcf
thomas.estK
thomas.estpcf
matclust.estK
matclust.estpcf
cauchy.estK
cauchy.estpcf
vargamma.estK
vargamma.estpcf
mincontrast

fit a log-Gaussian Cox process model
fit a log-Gaussian Cox process model
fit the Thomas process model
fit the Thomas process model
fit the Matern Cluster process model
fit the Matern Cluster process model
fit a Neyman-Scott Cauchy cluster process
fit a Neyman-Scott Cauchy cluster process
fit a Neyman-Scott Variance Gamma process
fit a Neyman-Scott Variance Gamma process
low-level algorithm for fitting models
by the method of minimum contrast

IV. MODEL FITTING (POISSON AND GIBBS MODELS)
Types of models
Poisson point processes are the simplest models for point patterns. A Poisson model assumes that
the points are stochastically independent. It may allow the points to have a non-uniform spatial density. The special case of a Poisson process with a uniform spatial density is often called Complete
Spatial Randomness.

spatstat-package

41

Poisson point processes are included in the more general class of Gibbs point process models. In a
Gibbs model, there is interaction or dependence between points. Many different types of interaction
can be specified.
For a detailed explanation of how to fit Poisson or Gibbs point process models to point pattern data
using spatstat, see Baddeley and Turner (2005b) or Baddeley (2008).
To fit a Poisson or Gibbs point process model:
Model fitting in spatstat is performed mainly by the function ppm. Its result is an object of class
"ppm".
Here are some examples, where X is a point pattern (class "ppp"):
command
ppm(X)
ppm(X ~ 1)
ppm(X ~ x)
ppm(X ~ 1, Strauss(0.1))
ppm(X ~ x, Strauss(0.1))

model
Complete Spatial Randomness
Complete Spatial Randomness
Poisson process with
intensity loglinear in x coordinate
Stationary Strauss process
Strauss process with
conditional intensity loglinear in x

It is also possible to fit models that depend on other covariates.
Manipulating the fitted model:
plot.ppm
predict.ppm
coef.ppm
parameters
formula.ppm
intensity.ppm
Kmodel.ppm
pcfmodel.ppm
fitted.ppm
residuals.ppm
update.ppm
vcov.ppm
rmh.ppm
simulate.ppm
print.ppm
summary.ppm
effectfun
logLik.ppm
anova.ppm
model.frame.ppm
model.images
model.depends
as.interact
fitin
is.hybrid
valid.ppm
project.ppm

Plot the fitted model
Compute the spatial trend and conditional intensity
of the fitted point process model
Extract the fitted model coefficients
Extract all model parameters
Extract the trend formula
Compute fitted intensity
K function of fitted model
pair correlation of fitted model
Compute fitted conditional intensity at quadrature points
Compute point process residuals at quadrature points
Update the fit
Variance-covariance matrix of estimates
Simulate from fitted model
Simulate from fitted model
Print basic information about a fitted model
Summarise a fitted model
Compute the fitted effect of one covariate
log-likelihood or log-pseudolikelihood
Analysis of deviance
Extract data frame used to fit model
Extract spatial data used to fit model
Identify variables in the model
Interpoint interaction component of model
Extract fitted interpoint interaction
Determine whether the model is a hybrid
Check the model is a valid point process
Ensure the model is a valid point process

42

spatstat-package
For model selection, you can also use the generic functions step, drop1 and AIC on fitted point
process models.
See spatstat.options to control plotting of fitted model.
To specify a point process model:
The first order “trend” of the model is determined by an R language formula. The formula specifies
the form of the logarithm of the trend.
X ~ 1
X ~ x
X ~ polynom(x,y,3)
X ~ harmonic(x,y,2)
X ~ Z

No trend (stationary)
Loglinear trend λ(x, y) = exp(α + βx)
where x, y are Cartesian coordinates
Log-cubic polynomial trend
Log-harmonic polynomial trend
Loglinear function of covariate Z
λ(x, y) = exp(α + βZ(x, y))

The higher order (“interaction”) components are described by an object of class "interact". Such
objects are created by:
Poisson()
AreaInter()
BadGey()
Concom()
DiggleGratton()
DiggleGatesStibbard()
Fiksel()
Geyer()
Hardcore()
HierHard()
HierStrauss()
HierStraussHard()
Hybrid()
LennardJones()
MultiHard()
MultiStrauss()
MultiStraussHard()
OrdThresh()
Ord()
PairPiece()
Pairwise()
Penttinen()
SatPiece()
Saturated()
Softcore()
Strauss()
StraussHard()
Triplets()

the Poisson point process
Area-interaction process
multiscale Geyer process
connected component interaction
Diggle-Gratton potential
Diggle-Gates-Stibbard potential
Fiksel pairwise interaction process
Geyer’s saturation process
Hard core process
Hierarchical multiype hard core process
Hierarchical multiype Strauss process
Hierarchical multiype Strauss-hard core process
Hybrid of several interactions
Lennard-Jones potential
multitype hard core process
multitype Strauss process
multitype Strauss/hard core process
Ord process, threshold potential
Ord model, user-supplied potential
pairwise interaction, piecewise constant
pairwise interaction, user-supplied potential
Penttinen pairwise interaction
Saturated pair model, piecewise constant potential
Saturated pair model, user-supplied potential
pairwise interaction, soft core potential
Strauss process
Strauss/hard core point process
Geyer triplets process

Note that it is also possible to combine several such interactions using Hybrid.
Finer control over model fitting:
A quadrature scheme is represented by an object of class "quad". To create a quadrature scheme,
typically use quadscheme.

spatstat-package

43
quadscheme

default quadrature scheme
using rectangular cells or Dirichlet cells
quadrature scheme based on image pixels
create an object of class "quad"

pixelquad
quad
To inspect a quadrature scheme:
plot(Q)
print(Q)
summary(Q)

plot quadrature scheme Q
print basic information about quadrature scheme Q
summary of quadrature scheme Q

A quadrature scheme consists of data points, dummy points, and weights. To generate dummy
points:
default.dummy
gridcentres
rstrat
spokes
corners

default pattern of dummy points
dummy points in a rectangular grid
stratified random dummy pattern
radial pattern of dummy points
dummy points at corners of the window

To compute weights:
gridweights
dirichletWeights

quadrature weights by the grid-counting rule
quadrature weights are Dirichlet tile areas

Simulation and goodness-of-fit for fitted models:
rmh.ppm
simulate.ppm
envelope

simulate realisations of a fitted model
simulate realisations of a fitted model
compute simulation envelopes for a fitted model

Point process models on a linear network:
An object of class "lpp" represents a pattern of points on a linear network. Point process models
can also be fitted to these objects. Currently only Poisson models can be fitted.
lppm
anova.lppm
envelope.lppm
fitted.lppm
predict.lppm
linim
plot.linim
eval.linim
linfun
methods.linfun

point process model on linear network
analysis of deviance for
point process model on linear network
simulation envelopes for
point process model on linear network
fitted intensity values
model prediction on linear network
pixel image on linear network
plot a pixel image on linear network
evaluate expression involving images
function defined on linear network
conversion facilities

V. MODEL FITTING (DETERMINANTAL POINT PROCESS MODELS)
Code for fitting determinantal point process models has recently been added to spatstat.

44

spatstat-package
For information, see the help file for dppm.

VI. MODEL FITTING (SPATIAL LOGISTIC REGRESSION)
Logistic regression
Pixel-based spatial logistic regression is an alternative technique for analysing spatial point patterns
that is widely used in Geographical Information Systems. It is approximately equivalent to fitting a
Poisson point process model.
In pixel-based logistic regression, the spatial domain is divided into small pixels, the presence
or absence of a data point in each pixel is recorded, and logistic regression is used to model the
presence/absence indicators as a function of any covariates.
Facilities for performing spatial logistic regression are provided in spatstat for comparison purposes.
Fitting a spatial logistic regression
Spatial logistic regression is performed by the function slrm. Its result is an object of class "slrm".
There are many methods for this class, including methods for print, fitted, predict, simulate,
anova, coef, logLik, terms, update, formula and vcov.
For example, if X is a point pattern (class "ppp"):
command
slrm(X ~ 1)
slrm(X ~ x)
slrm(X ~ Z)

model
Complete Spatial Randomness
Poisson process with
intensity loglinear in x coordinate
Poisson process with
intensity loglinear in covariate Z

Manipulating a fitted spatial logistic regression
anova.slrm
coef.slrm
vcov.slrm
fitted.slrm
logLik.slrm
plot.slrm
predict.slrm
simulate.slrm

Analysis of deviance
Extract fitted coefficients
Variance-covariance matrix of fitted coefficients
Compute fitted probabilities or intensity
Evaluate loglikelihood of fitted model
Plot fitted probabilities or intensity
Compute predicted probabilities or intensity with new data
Simulate model

There are many other undocumented methods for this class, including methods for print, update,
formula and terms. Stepwise model selection is possible using step or stepAIC.
VII. SIMULATION
There are many ways to generate a random point pattern, line segment pattern, pixel image or
tessellation in spatstat.
Random point patterns:
runifpoint
rpoint
rmpoint
rpoispp

generate n independent uniform random points
generate n independent random points
generate n independent multitype random points
simulate the (in)homogeneous Poisson point process

spatstat-package

45

rmpoispp
runifdisc
rstrat
rsyst
rMaternI
rMaternII
rSSI
rHardcore
rStrauss
rStraussHard
rDiggleGratton
rDGS
rPenttinen
rNeymanScott
rMatClust
rThomas
rLGCP
rGaussPoisson
rCauchy
rVarGamma
rcell
runifpointOnLines
rpoisppOnLines

simulate the (in)homogeneous multitype Poisson point process
generate n independent uniform random points in disc
stratified random sample of points
systematic random sample (grid) of points
simulate the Matérn Model I inhibition process
simulate the Matérn Model II inhibition process
simulate Simple Sequential Inhibition process
simulate hard core process (perfect simulation)
simulate Strauss process (perfect simulation)
simulate Strauss-hard core process (perfect simulation)
simulate Diggle-Gratton process (perfect simulation)
simulate Diggle-Gates-Stibbard process (perfect simulation)
simulate Penttinen process (perfect simulation)
simulate a general Neyman-Scott process
simulate the Matérn Cluster process
simulate the Thomas process
simulate the log-Gaussian Cox process
simulate the Gauss-Poisson cluster process
simulate Neyman-Scott process with Cauchy clusters
simulate Neyman-Scott process with Variance Gamma clusters
simulate the Baddeley-Silverman cell process
generate n random points along specified line segments
generate Poisson random points along specified line segments

Resampling a point pattern:
quadratresample
rjitter
rshift
rthin

block resampling
apply random displacements to points in a pattern
random shifting of (subsets of) points
random thinning

See also varblock for estimating the variance of a summary statistic by block resampling, and
lohboot for another bootstrap technique.
Fitted point process models:
If you have fitted a point process model to a point pattern dataset, the fitted model can be simulated.
Cluster process models are fitted by the function kppm yielding an object of class "kppm". To
generate one or more simulated realisations of this fitted model, use simulate.kppm.
Gibbs point process models are fitted by the function ppm yielding an object of class "ppm". To
generate a simulated realisation of this fitted model, use rmh. To generate one or more simulated
realisations of the fitted model, use simulate.ppm.
Other random patterns:
rlinegrid
rpoisline
rpoislinetess
rMosaicSet
rMosaicField

generate a random array of parallel lines through a window
simulate the Poisson line process within a window
generate random tessellation using Poisson line process
generate random set by selecting some tiles of a tessellation
generate random pixel image by assigning random values in each tile of a tessellation

Simulation-based inference
envelope

critical envelope for Monte Carlo test of goodness-of-fit

46

spatstat-package
qqplot.ppm
scan.test
studpermu.test
segregation.test

diagnostic plot for interpoint interaction
spatial scan statistic/test
studentised permutation test
test of segregation of types

VIII. TESTS AND DIAGNOSTICS
Hypothesis tests:
quadrat.test
clarkevans.test
cdf.test
berman.test
envelope
scan.test
dclf.test
mad.test
anova.ppm

χ2 goodness-of-fit test on quadrat counts
Clark and Evans test
Spatial distribution goodness-of-fit test
Berman’s goodness-of-fit tests
critical envelope for Monte Carlo test of goodness-of-fit
spatial scan statistic/test
Diggle-Cressie-Loosmore-Ford test
Mean Absolute Deviation test
Analysis of Deviance for point process models

More recently-developed tests:
dg.test
bits.test
dclf.progress
mad.progress

Dao-Genton test
Balanced independent two-stage test
Progress plot for DCLF test
Progress plot for MAD test

Sensitivity diagnostics:
Classical measures of model sensitivity such as leverage and influence have been adapted to point
process models.
leverage.ppm
influence.ppm
dfbetas.ppm

Leverage for point process model
Influence for point process model
Parameter influence

Diagnostics for covariate effect:
Classical diagnostics for covariate effects have been adapted to point process models.
parres
addvar
rhohat
rho2hat

Partial residual plot
Added variable plot
Kernel estimate of covariate effect
Kernel estimate of covariate effect (bivariate)

Residual diagnostics:
Residuals for a fitted point process model, and diagnostic plots based on the residuals, were introduced in Baddeley et al (2005) and Baddeley, Rubak and Møller (2011).
Type demo(diagnose) for a demonstration of the diagnostics features.
diagnose.ppm
qqplot.ppm

diagnostic plots for spatial trend
diagnostic Q-Q plot for interpoint interaction

spatstat-package

47
residualspaper
Kcom
Gcom
Kres
Gres
psst
psstA
psstG
compareFit

examples from Baddeley et al (2005)
model compensator of K function
model compensator of G function
score residual of K function
score residual of G function
pseudoscore residual of summary function
pseudoscore residual of empty space function
pseudoscore residual of G function
compare compensators of several fitted models

Resampling and randomisation procedures
You can build your own tests based on randomisation and resampling using the following capabilities:
quadratresample
rjitter
rshift
rthin

block resampling
apply random displacements to points in a pattern
random shifting of (subsets of) points
random thinning

IX. DOCUMENTATION
The online manual entries are quite detailed and should be consulted first for information about a
particular function.
The book Baddeley, Rubak and Turner (2015) is a complete course on analysing spatial point patterns, with full details about spatstat.
Older material (which is now out-of-date but is freely available) includes Baddeley and Turner
(2005a), a brief overview of the package in its early development; Baddeley and Turner (2005b),
a more detailed explanation of how to fit point process models to data; and Baddeley (2010), a
complete set of notes from a 2-day workshop on the use of spatstat.
Type citation("spatstat") to get a list of these references.
Licence
This library and its documentation are usable under the terms of the "GNU General Public License",
a copy of which is distributed with the package.
Acknowledgements
Kasper Klitgaard Berthelsen, Ottmar Cronie, Yongtao Guan, Ute Hahn, Abdollah Jalilian, MarieColette van Lieshout, Greg McSwiggan, Tuomas Rajala, Suman Rakshit, Dominic Schuhmacher,
Rasmus Waagepetersen and Hangsheng Wang made substantial contributions of code.
Additional contributions and suggestions from Monsuru Adepeju, Corey Anderson, Ang Qi Wei,
Marcel Austenfeld, Sandro Azaele, Malissa Baddeley, Guy Bayegnak, Colin Beale, Melanie Bell,
Thomas Bendtsen, Ricardo Bernhardt, Andrew Bevan, Brad Biggerstaff, Anders Bilgrau, Leanne
Bischof, Christophe Biscio, Roger Bivand, Jose M. Blanco Moreno, Florent Bonneu, Julian Burgos,
Simon Byers, Ya-Mei Chang, Jianbao Chen, Igor Chernayavsky, Y.C. Chin, Bjarke Christensen,
Jean-Francois Coeurjolly, Kim Colyvas, Rochelle Constantine, Robin Corria Ainslie, Richard Cotton, Marcelino de la Cruz, Peter Dalgaard, Mario D’Antuono, Sourav Das, Tilman Davies, Peter
Diggle, Patrick Donnelly, Ian Dryden, Stephen Eglen, Ahmed El-Gabbas, Belarmain Fandohan,
Olivier Flores, David Ford, Peter Forbes, Shane Frank, Janet Franklin, Funwi-Gabga Neba, Oscar Garcia, Agnes Gault, Jonas Geldmann, Marc Genton, Shaaban Ghalandarayeshi, Julian Gilbey,

48

spatstat-package
Jason Goldstick, Pavel Grabarnik, C. Graf, Ute Hahn, Andrew Hardegen, Martin Bøgsted Hansen,
Martin Hazelton, Juha Heikkinen, Mandy Hering, Markus Herrmann, Paul Hewson, Kassel Hingee,
Kurt Hornik, Philipp Hunziker, Jack Hywood, Ross Ihaka, C̆enk Içös, Aruna Jammalamadaka,
Robert John-Chandran, Devin Johnson, Mahdieh Khanmohammadi, Bob Klaver, Lily KozmianLedward, Peter Kovesi, Mike Kuhn, Jeff Laake, Frederic Lavancier, Tom Lawrence, Robert Lamb,
Jonathan Lee, George Leser, Li Haitao, George Limitsios, Andrew Lister, Ben Madin, Martin
Maechler, Kiran Marchikanti, Jeff Marcus, Robert Mark, Peter McCullagh, Monia Mahling, Jorge
Mateu Mahiques, Ulf Mehlig, Frederico Mestre, Sebastian Wastl Meyer, Mi Xiangcheng, Lore
De Middeleer, Robin Milne, Enrique Miranda, Jesper Møller, Mehdi Moradi, Virginia Morera Pujol, Erika Mudrak, Gopalan Nair, Nader Najari, Nicoletta Nava, Linda Stougaard Nielsen, Felipe
Nunes, Jens Randel Nyengaard, Jens Oehlschlägel, Thierry Onkelinx, Sean O’Riordan, Evgeni Parilov, Jeff Picka, Nicolas Picard, Mike Porter, Sergiy Protsiv, Adrian Raftery, Suman Rakshit, Ben
Ramage, Pablo Ramon, Xavier Raynaud, Nicholas Read, Matt Reiter, Ian Renner, Tom Richardson, Brian Ripley, Ted Rosenbaum, Barry Rowlingson, Jason Rudokas, John Rudge, Christopher
Ryan, Farzaneh Safavimanesh, Aila Särkkä, Cody Schank, Katja Schladitz, Sebastian Schutte,
Bryan Scott, Olivia Semboli, François Sémécurbe, Vadim Shcherbakov, Shen Guochun, Shi Peijian, Harold-Jeffrey Ship, Tammy L Silva, Ida-Maria Sintorn, Yong Song, Malte Spiess, Mark
Stevenson, Kaspar Stucki, Michael Sumner, P. Surovy, Ben Taylor, Thordis Linda Thorarinsdottir, Leigh Torres, Berwin Turlach, Torben Tvedebrink, Kevin Ummer, Medha Uppala, Andrew van
Burgel, Tobias Verbeke, Mikko Vihtakari, Alexendre Villers, Fabrice Vinatier, Sasha Voss, Sven
Wagner, Hao Wang, H. Wendrock, Jan Wild, Carl G. Witthoft, Selene Wong, Maxime Woringer,
Mike Zamboni and Achim Zeileis.

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
References
Baddeley, A. (2010) Analysing spatial point patterns in R. Workshop notes, Version 4.1. Online
technical publication, CSIRO. https://research.csiro.au/software/wp-content/uploads/
sites/6/2015/02/Rspatialcourse_CMIS_PDF-Standard.pdf
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
Baddeley, A. and Turner, R. (2005a) Spatstat: an R package for analyzing spatial point patterns.
Journal of Statistical Software 12:6, 1–42. URL: www.jstatsoft.org, ISSN: 1548-7660.
Baddeley, A. and Turner, R. (2005b) Modelling spatial point patterns in R. In: A. Baddeley, P. Gregori, J. Mateu, R. Stoica, and D. Stoyan, editors, Case Studies in Spatial Point Pattern Modelling,
Lecture Notes in Statistics number 185. Pages 23–74. Springer-Verlag, New York, 2006. ISBN:
0-387-28311-0.
Baddeley, A., Turner, R., Møller, J. and Hazelton, M. (2005) Residual analysis for spatial point
processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
Baddeley, A., Rubak, E. and Møller, J. (2011) Score, pseudo-score and residual diagnostics for
spatial point process models. Statistical Science 26, 613–646.
Baddeley, A., Turner, R., Mateu, J. and Bevan, A. (2013) Hybrids of Gibbs point process models
and their implementation. Journal of Statistical Software 55:11, 1–43. http://www.jstatsoft.
org/v55/i11/
Diggle, P.J. (2003) Statistical analysis of spatial point patterns, Second edition. Arnold.
Diggle, P.J. (2014) Statistical Analysis of Spatial and Spatio-Temporal Point Patterns, Third edition.
Chapman and Hall/CRC.

adaptive.density

49

Gelfand, A.E., Diggle, P.J., Fuentes, M. and Guttorp, P., editors (2010) Handbook of Spatial Statistics. CRC Press.
Huang, F. and Ogata, Y. (1999) Improvements of the maximum pseudo-likelihood estimators in
various spatial statistical models. Journal of Computational and Graphical Statistics 8, 510–530.
Illian, J., Penttinen, A., Stoyan, H. and Stoyan, D. (2008) Statistical Analysis and Modelling of
Spatial Point Patterns. Wiley.
Waagepetersen, R. An estimating function approach to inference for inhomogeneous Neyman-Scott
processes. Biometrics 63 (2007) 252–258.

adaptive.density

Intensity Estimate of Point Pattern Using Tessellation

Description
Computes an adaptive estimate of the intensity function of a point pattern.
Usage
adaptive.density(X, f = 0.1, ..., nrep = 1, verbose=TRUE)
Arguments
X

Point pattern dataset (object of class "ppp").

f

Fraction (between 0 and 1 inclusive) of the data points that will be removed from
the data and used to determine a tessellation for the intensity estimate.

...

Arguments passed to as.im determining the pixel resolution of the result.

nrep

Number of independent repetitions of the randomised procedure.

verbose

Logical value indicating whether to print progress reports.

Details
This function is an alternative to density.ppp. It computes an estimate of the intensity function of
a point pattern dataset. The result is a pixel image giving the estimated intensity,
If f=1, the Voronoi estimate (Barr and Schoenberg, 2010) is computed: the point pattern X is used
to construct a Voronoi/Dirichlet tessellation (see dirichlet); the areas of the Dirichlet tiles are
computed; the estimated intensity in each tile is the reciprocal of the tile area.
If f=0, the intensity estimate at every location is equal to the average intensity (number of points
divided by window area).
If f is strictly between 0 and 1, the dataset X is randomly split into two patterns A and B containing
a fraction f and 1-f, respectively, of the original data. The subpattern A is used to construct a
Dirichlet tessellation, while the subpattern B is retained for counting. For each tile of the Dirichlet
tessellation, we count the number of points of B falling in the tile, and divide by the area of the same
tile, to obtain an estimate of the intensity of the pattern B in the tile. This estimate is divided by
1-f to obtain an estimate of the intensity of X in the tile. The result is a pixel image of intensity
estimates which are constant on each tile of the tessellation.
If nrep is greater than 1, this randomised procedure is repeated nrep times, and the results are
averaged.
This technique has been used by Ogata et al. (2003), Ogata (2004) and Baddeley (2007).

50

add.texture

Value
A pixel image (object of class "im") whose values are estimates of the intensity of X.
Author(s)
Adrian Baddeley  and Rolf Turner 
References
Baddeley, A. (2007) Validation of statistical models for spatial point patterns. In J.G. Babu and
E.D. Feigelson (eds.) SCMA IV: Statistical Challenges in Modern Astronomy IV, volume 317 of
Astronomical Society of the Pacific Conference Series, San Francisco, California USA, 2007. Pages
22–38.
Barr, C., and Schoenberg, F.P. (2010). On the Voronoi estimator for the intensity of an inhomogeneous planar Poisson process. Biometrika 97 (4), 977–984.
Ogata, Y. (2004) Space-time model for regional seismicity and detection of crustal stress changes.
Journal of Geophysical Research, 109, 2004.
Ogata, Y., Katsura, K. and Tanemura, M. (2003). Modelling heterogeneous space-time occurrences
of earthquakes and its residual analysis. Applied Statistics 52 499–509.
See Also
density.ppp, dirichlet, im.object.
Examples
plot(adaptive.density(nztrees, 1), main="Voronoi estimate")
nr <- if(interactive()) 100 else 5
plot(adaptive.density(nztrees, nrep=nr), main="Adaptive estimate")

add.texture

Fill Plot With Texture

Description
Draws a simple texture inside a region on the plot.
Usage
add.texture(W, texture = 4, spacing = NULL, ...)
Arguments
W

Window (object of class "owin") inside which the texture should be drawn.

texture

Integer from 1 to 8 identifying the type of texture. See Details.

spacing

Spacing between elements of the texture, in units of the current plot.

...

Further arguments controlling the plot colour, line width etc.

addvar

51

Details
The chosen texture, confined to the window W, will be added to the current plot. The available
textures are:
texture=1: Small crosses arranged in a square grid.
texture=2: Parallel vertical lines.
texture=3: Parallel horizontal lines.
texture=4: Parallel diagonal lines at 45 degrees from the horizontal.
texture=5: Parallel diagonal lines at 135 degrees from the horizontal.
texture=6: Grid of horizontal and vertical lines.
texture=7: Grid of diagonal lines at 45 and 135 degrees from the horizontal.
texture=8: Grid of hexagons.
Author(s)
Adrian Baddeley  and Rolf Turner 
See Also
owin, plot.owin, textureplot, texturemap.
Examples
W <- Window(chorley)
plot(W, main="")
add.texture(W, 7)

addvar

Added Variable Plot for Point Process Model

Description
Computes the coordinates for an Added Variable Plot for a fitted point process model.
Usage
addvar(model, covariate, ...,
subregion=NULL,
bw="nrd0", adjust=1,
from=NULL, to=NULL, n=512,
bw.input = c("points", "quad"),
bw.restrict = FALSE,
covname, crosscheck=FALSE)

52

addvar

Arguments
model

Fitted point process model (object of class "ppm").

covariate

The covariate to be added to the model. Either a pixel image, a function(x,y),
or a character string giving the name of a covariate that was supplied when the
model was fitted.

subregion

Optional. A window (object of class "owin") specifying a subset of the spatial
domain of the data. The calculation will be confined to the data in this subregion.

bw

Smoothing bandwidth or bandwidth rule (passed to density.default).

adjust

Smoothing bandwidth adjustment factor (passed to density.default).

n, from, to

Arguments passed to density.default to control the number and range of values at which the function will be estimated.

...

Additional arguments passed to density.default.

bw.input

Character string specifying the input data used for automatic bandwidth selection.

bw.restrict

Logical value, specifying whether bandwidth selection is performed using data
from the entire spatial domain or from the subregion.

covname

Optional. Character string to use as the name of the covariate.

crosscheck

For developers only. Logical value indicating whether to perform cross-checks
on the validity of the calculation.

Details
This command generates the plot coordinates for an Added Variable Plot for a spatial point process
model.
Added Variable Plots (Cox, 1958, sec 4.5; Wang, 1985) are commonly used in linear models and
generalized linear models, to decide whether a model with response y and predictors x would be
improved by including another predictor z.
In a (generalised) linear model with response y and predictors x, the Added Variable Plot for a new
covariate z is a plot of the smoothed Pearson residuals from the original model against the scaled
residuals from a weighted linear regression of z on x. If this plot has nonzero slope, then the new
covariate z is needed. For general advice see Cook and Weisberg(1999); Harrell (2001).
Essentially the same technique can be used for a spatial point process model (Baddeley et al, 2012).
The argument model should be a fitted spatial point process model (object of class "ppm").
The argument covariate identifies the covariate that is to be considered for addition to the model.
It should be either a pixel image (object of class "im") or a function(x,y) giving the values of
the covariate at any spatial location. Alternatively covariate may be a character string, giving the
name of a covariate that was supplied (in the covariates argument to ppm) when the model was
fitted, but was not used in the model.
The result of addvar(model, covariate) is an object belonging to the classes "addvar" and
"fv". Plot this object to generate the added variable plot.
Note that the plot method shows the pointwise significance bands for a test of the null model, i.e.
the null hypothesis that the new covariate has no effect.
The smoothing bandwidth is controlled by the arguments bw, adjust, bw.input and bw.restrict.
If bw is a numeric value, then the bandwidth is taken to be adjust * bw. If bw is a string representing
a bandwidth selection rule (recognised by density.default) then the bandwidth is selected by this
rule.

addvar

53

The data used for automatic bandwidth selection are specified by bw.input and bw.restrict. If
bw.input="points" (the default) then bandwidth selection is based on the covariate values at the
points of the original point pattern dataset to which the model was fitted. If bw.input="quad" then
bandwidth selection is based on the covariate values at every quadrature point used to fit the model.
If bw.restrict=TRUE then the bandwidth selection is performed using only data from inside the
subregion.
Value
An object of class "addvar" containing the coordinates for the added variable plot. There is a plot
method.
Slow computation
In a large dataset, computation can be very slow if the default settings are used, because the smoothing bandwidth is selected automatically. To avoid this, specify a numerical value for the bandwidth
bw. One strategy is to use a coarser subset of the data to select bw automatically. The selected
bandwidth can be read off the print output for addvar.
Internal data
The return value has an attribute "spatial" which contains the internal data: the computed values
of the residuals, and of all relevant covariates, at each quadrature point of the model. It is an object
of class "ppp" with a data frame of marks.
Author(s)
Adrian Baddeley , Rolf Turner ,
Ya-Mei Chang and Yong Song.
References
Baddeley, A., Chang, Y.-M., Song, Y. and Turner, R. (2013) Residual diagnostics for covariate
effects in spatial point process models. Journal of Computational and Graphical Statistics, 22,
886–905.
Cook, R.D. and Weisberg, S. (1999) Applied regression, including computing and graphics. New
York: Wiley.
Cox, D.R. (1958) Planning of Experiments. New York: Wiley.
Harrell, F. (2001) Regression Modeling Strategies. New York: Springer.
Wang, P. (1985) Adding a variable in generalized linear models. Technometrics 27, 273–276.
See Also
parres, rhohat, rho2hat.
Examples
X <- rpoispp(function(x,y){exp(3+3*x)})
model <- ppm(X, ~y)
adv <- addvar(model, "x")
plot(adv)
adv <- addvar(model, "x", subregion=square(0.5))

54

affine.im

affine

Apply Affine Transformation

Description
Applies any affine transformation of the plane (linear transformation plus vector shift) to a plane
geometrical object, such as a point pattern or a window.
Usage
affine(X, ...)
Arguments
X

Any suitable dataset representing a two-dimensional object, such as a point pattern (object of class "ppp"), a line segment pattern (object of class "psp"), a
window (object of class "owin") or a pixel image (object of class "im").

...

Arguments determining the affine transformation.

Details
This is generic. Methods are provided for point patterns (affine.ppp) and windows (affine.owin).
Value
Another object of the same type, representing the result of applying the affine transformation.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
affine.ppp, affine.psp, affine.owin, affine.im, flipxy, reflect, rotate, shift

affine.im

Apply Affine Transformation To Pixel Image

Description
Applies any affine transformation of the plane (linear transformation plus vector shift) to a pixel
image.
Usage
## S3 method for class 'im'
affine(X, mat=diag(c(1,1)), vec=c(0,0), ...)

affine.linnet

55

Arguments
X

Pixel image (object of class "im").

mat

Matrix representing a linear transformation.

vec

Vector of length 2 representing a translation.

...

Optional arguments passed to as.mask controlling the pixel resolution of the
transformed image.

Details
The image is subjected first to the linear transformation represented by mat (multiplying on the left
by mat), and then the result is translated by the vector vec.
The argument mat must be a nonsingular 2 × 2 matrix.
This is a method for the generic function affine.
Value
Another pixel image (of class "im") representing the result of applying the affine transformation.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
affine, affine.ppp, affine.psp, affine.owin, rotate, shift
Examples
X <- setcov(owin())
stretch <- diag(c(2,3))
Y <- affine(X, mat=stretch)
shear <- matrix(c(1,0,0.6,1),ncol=2, nrow=2)
Z <- affine(X, mat=shear)

affine.linnet

Apply Geometrical Transformations to a Linear Network

Description
Apply geometrical transformations to a linear network.

56

affine.linnet

Usage
## S3 method for class 'linnet'
affine(X, mat=diag(c(1,1)), vec=c(0,0), ...)
## S3 method for class 'linnet'
shift(X, vec=c(0,0), ..., origin=NULL)
## S3 method for class 'linnet'
rotate(X, angle=pi/2, ..., centre=NULL)
## S3 method for class 'linnet'
scalardilate(X, f, ...)
## S3 method for class 'linnet'
rescale(X, s, unitname)
Arguments
X

Linear network (object of class "linnet").

mat

Matrix representing a linear transformation.

vec

Vector of length 2 representing a translation.

angle

Rotation angle in radians.

f

Scalar dilation factor.

s

Unit conversion factor: the new units are s times the old units.

...

Arguments passed to other methods.

origin

Character string determining a location that will be shifted to the origin. Options
are "centroid", "midpoint" and "bottomleft". Partially matched.

centre

Centre of rotation. Either a vector of length 2, or a character string (partially
matched to "centroid", "midpoint" or "bottomleft"). The default is the
coordinate origin c(0,0).

unitname

Optional. New name for the unit of length. A value acceptable to the function
unitname<-

Details
These functions are methods for the generic functions affine, shift, rotate, rescale and scalardilate
applicable to objects of class "linnet".
All of these functions perform geometrical transformations on the object X, except for rescale,
which simply rescales the units of length.
Value
Another linear network (of class "linnet") representing the result of applying the geometrical
transformation.
Author(s)
Adrian Baddeley 
and Rolf Turner 

affine.lpp

57

See Also
linnet and as.linnet.
Generic functions affine, shift, rotate, scalardilate, rescale.
Examples
U <- rotate(simplenet, pi)
stretch <- diag(c(2,3))
Y <- affine(simplenet, mat=stretch)
shear <- matrix(c(1,0,0.6,1),ncol=2, nrow=2)
Z <- affine(simplenet, mat=shear, vec=c(0, 1))

affine.lpp

Apply Geometrical Transformations to Point Pattern on a Linear Network

Description
Apply geometrical transformations to a point pattern on a linear network.
Usage
## S3 method for class 'lpp'
affine(X, mat=diag(c(1,1)), vec=c(0,0), ...)
## S3 method for class 'lpp'
shift(X, vec=c(0,0), ..., origin=NULL)
## S3 method for class 'lpp'
rotate(X, angle=pi/2, ..., centre=NULL)
## S3 method for class 'lpp'
scalardilate(X, f, ...)
## S3 method for class 'lpp'
rescale(X, s, unitname)
Arguments
X

Point pattern on a linear network (object of class "lpp").

mat

Matrix representing a linear transformation.

vec

Vector of length 2 representing a translation.

angle

Rotation angle in radians.

f

Scalar dilation factor.

s

Unit conversion factor: the new units are s times the old units.

...

Arguments passed to other methods.

origin

Character string determining a location that will be shifted to the origin. Options
are "centroid", "midpoint" and "bottomleft". Partially matched.

58

affine.owin
centre

Centre of rotation. Either a vector of length 2, or a character string (partially
matched to "centroid", "midpoint" or "bottomleft"). The default is the
coordinate origin c(0,0).

unitname

Optional. New name for the unit of length. A value acceptable to the function
unitname<-

Details
These functions are methods for the generic functions affine, shift, rotate, rescale and scalardilate
applicable to objects of class "lpp".
All of these functions perform geometrical transformations on the object X, except for rescale,
which simply rescales the units of length.
Value
Another point pattern on a linear network (object of class "lpp") representing the result of applying
the geometrical transformation.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
lpp.
Generic functions affine, shift, rotate, scalardilate, rescale.
Examples
X <- rpoislpp(2, simplenet)
U <- rotate(X, pi)
stretch <- diag(c(2,3))
Y <- affine(X, mat=stretch)
shear <- matrix(c(1,0,0.6,1),ncol=2, nrow=2)
Z <- affine(X, mat=shear, vec=c(0, 1))

affine.owin

Apply Affine Transformation To Window

Description
Applies any affine transformation of the plane (linear transformation plus vector shift) to a window.
Usage
## S3 method for class 'owin'
affine(X, mat=diag(c(1,1)), vec=c(0,0), ..., rescue=TRUE)

affine.ppp

59

Arguments
X

Window (object of class "owin").

mat

Matrix representing a linear transformation.

vec

Vector of length 2 representing a translation.

rescue

Logical. If TRUE, the transformed window will be processed by rescue.rectangle.

...

Optional arguments passed to as.mask controlling the pixel resolution of the
transformed window, if X is a binary pixel mask.

Details
The window is subjected first to the linear transformation represented by mat (multiplying on the
left by mat), and then the result is translated by the vector vec.
The argument mat must be a nonsingular 2 × 2 matrix.
This is a method for the generic function affine.
Value
Another window (of class "owin") representing the result of applying the affine transformation.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
affine, affine.ppp, affine.psp, affine.im, rotate, shift
Examples
# shear transformation
shear <- matrix(c(1,0,0.6,1),ncol=2)
X <- affine(owin(), shear)
## Not run:
plot(X)
## End(Not run)
data(letterR)
affine(letterR, shear, c(0, 0.5))
affine(as.mask(letterR), shear, c(0, 0.5))

affine.ppp

Apply Affine Transformation To Point Pattern

Description
Applies any affine transformation of the plane (linear transformation plus vector shift) to a point
pattern.

60

affine.ppp

Usage
## S3 method for class 'ppp'
affine(X, mat=diag(c(1,1)), vec=c(0,0), ...)
Arguments
X

Point pattern (object of class "ppp").

mat

Matrix representing a linear transformation.

vec

Vector of length 2 representing a translation.

...

Arguments passed to affine.owin affecting the handling of the observation
window, if it is a binary pixel mask.

Details
The point pattern, and its window, are subjected first to the linear transformation represented by mat
(multiplying on the left by mat), and are then translated by the vector vec.
The argument mat must be a nonsingular 2 × 2 matrix.
This is a method for the generic function affine.
Value
Another point pattern (of class "ppp") representing the result of applying the affine transformation.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
affine, affine.owin, affine.psp, affine.im, flipxy, rotate, shift
Examples
data(cells)
# shear transformation
X <- affine(cells, matrix(c(1,0,0.6,1),ncol=2))
## Not run:
plot(X)
# rescale y coordinates by factor 1.3
plot(affine(cells, diag(c(1,1.3))))
## End(Not run)

affine.psp

affine.psp

61

Apply Affine Transformation To Line Segment Pattern

Description
Applies any affine transformation of the plane (linear transformation plus vector shift) to a line
segment pattern.
Usage
## S3 method for class 'psp'
affine(X, mat=diag(c(1,1)), vec=c(0,0), ...)
Arguments
X

Line Segment pattern (object of class "psp").

mat

Matrix representing a linear transformation.

vec

Vector of length 2 representing a translation.

...

Arguments passed to affine.owin affecting the handling of the observation
window, if it is a binary pixel mask.

Details
The line segment pattern, and its window, are subjected first to the linear transformation represented
by mat (multiplying on the left by mat), and are then translated by the vector vec.
The argument mat must be a nonsingular 2 × 2 matrix.
This is a method for the generic function affine.
Value
Another line segment pattern (of class "psp") representing the result of applying the affine transformation.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
affine, affine.owin, affine.ppp, affine.im, flipxy, rotate, shift
Examples
oldpar <- par(mfrow=c(2,1))
X <- psp(runif(10), runif(10), runif(10), runif(10), window=owin())
plot(X, main="original")
# shear transformation
Y <- affine(X, matrix(c(1,0,0.6,1),ncol=2))
plot(Y, main="transformed")
par(oldpar)

62

affine.tess
#
# rescale y coordinates by factor 0.2
affine(X, diag(c(1,0.2)))

affine.tess

Apply Geometrical Transformation To Tessellation

Description
Apply various geometrical transformations of the plane to each tile in a tessellation.
Usage
## S3 method for class 'tess'
reflect(X)
## S3 method for class 'tess'
shift(X, ...)
## S3 method for class 'tess'
rotate(X, angle=pi/2, ..., centre=NULL)
## S3 method for class 'tess'
scalardilate(X, f, ...)
## S3 method for class 'tess'
affine(X, mat=diag(c(1,1)), vec=c(0,0), ...)
Arguments
X

Tessellation (object of class "tess").

angle

Rotation angle in radians (positive values represent anticlockwise rotations).

mat

Matrix representing a linear transformation.

vec

Vector of length 2 representing a translation.

f

Positive number giving scale factor.

...

Arguments passed to other methods.

centre

Centre of rotation. Either a vector of length 2, or a character string (partially
matched to "centroid", "midpoint" or "bottomleft"). The default is the
coordinate origin c(0,0).

Details
These are method for the generic functions reflect, shift, rotate, scalardilate, affine for
tessellations (objects of class "tess").
The individual tiles of the tessellation, and the window containing the tessellation, are all subjected
to the same geometrical transformation.
The transformations are performed by the corresponding method for windows (class "owin") or
images (class "im") depending on the type of tessellation.
If the argument origin is used in shift.tess it is interpreted as applying to the window containing
the tessellation. Then all tiles are shifted by the same vector.

allstats

63

Value
Another tessellation (of class "tess") representing the result of applying the geometrical transformation.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
Generic functions reflect, shift, rotate, scalardilate, affine.
Methods for windows: reflect.default, shift.owin, rotate.owin, scalardilate.owin, affine.owin.
Methods for images: reflect.im, shift.im, rotate.im, scalardilate.im, affine.im.
Examples
live <- interactive()
if(live) {
H <- hextess(letterR, 0.2)
plot(H)
plot(reflect(H))
plot(rotate(H, pi/3))
} else H <- hextess(letterR, 0.6)
# shear transformation
shear <- matrix(c(1,0,0.6,1),2,2)
sH <- affine(H, shear)
if(live) plot(sH)

allstats

Calculate four standard summary functions of a point pattern.

Description
Calculates the F , G, J, and K summary functions for an unmarked point pattern. Returns them as
a function array (of class "fasp", see fasp.object).
Usage
allstats(pp, ..., dataname=NULL, verb=FALSE)
Arguments
pp

The observed point pattern, for which summary function estimates are required.
An object of class "ppp". It must not be marked.

...

Optional arguments passed to the summary functions Fest, Gest, Jest and
Kest.

dataname

A character string giving an optional (alternative) name for the point pattern.

verb

A logical value meaning “verbose”. If TRUE, progress reports are printed during
calculation.

64

alltypes

Details
This computes four standard summary statistics for a point pattern: the empty space function F (r),
nearest neighbour distance distribution function G(r), van Lieshout-Baddeley function J(r) and
Ripley’s function K(r). The real work is done by Fest, Gest, Jest and Kest respectively. Consult
the help files for these functions for further information about the statistical interpretation of F , G,
J and K.
If verb is TRUE, then “progress reports” (just indications of completion) are printed out when the
calculations are finished for each of the four function types.
The overall title of the array of four functions (for plotting by plot.fasp) will be formed from the
argument dataname. If this is not given, it defaults to the expression for pp given in the call to
allstats.
Value
A list of length 4 containing the F , G, J and K functions respectively.
The list can be plotted directly using plot (which dispatches to plot.solist).
Each list entry retains the format of the output of the relevant estimating routine Fest, Gest, Jest
or Kest. Thus each entry in the list is a function value table (object of class "fv", see fv.object).
The default formulae for plotting these functions are cbind(km,theo) ~ r for F, G, and J, and
cbind(trans,theo) ~ r for K.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
plot.solist, plot.fv, fv.object, Fest, Gest, Jest, Kest
Examples
data(swedishpines)
a <- allstats(swedishpines,dataname="Swedish Pines")
## Not run:
plot(a)
plot(a, subset=list("r<=15","r<=15","r<=15","r<=50"))
## End(Not run)

alltypes

Calculate Summary Statistic for All Types in a Multitype Point Pattern

Description
Given a marked point pattern, this computes the estimates of a selected summary function (F ,G,
J, K etc) of the pattern, for all possible combinations of marks, and returns these functions in an
array.

alltypes

65

Usage
alltypes(X, fun="K", ...,
dataname=NULL,verb=FALSE,envelope=FALSE,reuse=TRUE)
Arguments
X

The observed point pattern, for which summary function estimates are required.
An object of class "ppp" or "lpp".

fun

The summary function. Either an R function, or a character string indicating
the summary function required. Options for strings are "F", "G", "J", "K", "L",
"pcf", "Gcross", "Jcross", "Kcross", "Lcross", "Gdot", "Jdot", "Kdot",
"Ldot".

...

Arguments passed to the summary function (and to the function envelope if
appropriate)

dataname

Character string giving an optional (alternative) name to the point pattern, different from what is given in the call. This name, if supplied, may be used by
plot.fasp() in forming the title of the plot. If not supplied it defaults to the
parsing of the argument supplied as X in the call.

verb

Logical value. If verb is true then terse “progress reports” (just the values of
the mark indices) are printed out when the calculations for that combination of
marks are completed.

envelope

Logical value. If envelope is true, then simulation envelopes of the summary
function will also be computed. See Details.

reuse

Logical value indicating whether the envelopes in each panel should be based
on the same set of simulated patterns (reuse=TRUE) or on different, independent
sets of simulated patterns (reuse=FALSE).

Details
This routine is a convenient way to analyse the dependence between types in a multitype point
pattern. It computes the estimates of a selected summary function of the pattern, for all possible
combinations of marks. It returns these functions in an array (an object of class "fasp") amenable
to plotting by plot.fasp().
The argument fun specifies the summary function that will be evaluated for each type of point, or
for each pair of types. It may be either an R function or a character string.
Suppose that the points have possible types 1, 2, . . . , m and let Xi denote the pattern of points of
type i only.
If fun="F" then this routine calculates, for each possible type i, an estimate of the Empty Space
Function Fi (r) of Xi . See Fest for explanation of the empty space function. The estimate is
computed by applying Fest to Xi with the optional arguments ....
If fun is "Gcross", "Jcross", "Kcross" or "Lcross", the routine calculates, for each pair of types
(i, j), an estimate of the “i-toj” cross-type function Gij (r), Jij (r), Kij (r) or Lij (r) respectively
describing the dependence between Xi and Xj . See Gcross, Jcross, Kcross or Lcross respectively for explanation of these functions. The estimate is computed by applying the relevant function
(Gcross etc) to X using each possible value of the arguments i,j, together with the optional arguments ....
If fun is "pcf" the routine calculates the cross-type pair correlation function pcfcross between
each pair of types.

66

alltypes
If fun is "Gdot", "Jdot", "Kdot" or "Ldot", the routine calculates, for each type i, an estimate
of the “i-to-any” dot-type function Gi• (r), Ji• (r) or Ki• (r) or Li• (r) respectively describing the
dependence between Xi and X. See Gdot, Jdot, Kdot or Ldot respectively for explanation of these
functions. The estimate is computed by applying the relevant function (Gdot etc) to X using each
possible value of the argument i, together with the optional arguments ....
The letters "G", "J", "K" and "L" are interpreted as abbreviations for Gcross, Jcross, Kcross and
Lcross respectively, assuming the point pattern is marked. If the point pattern is unmarked, the
appropriate function Fest, Jest, Kest or Lest is invoked instead.
If envelope=TRUE, then as well as computing the value of the summary function for each combination of types, the algorithm also computes simulation envelopes of the summary function for each
combination of types. The arguments ... are passed to the function envelope to control the number of simulations, the random process generating the simulations, the construction of envelopes,
and so on.

Value
A function array (an object of class "fasp", see fasp.object). This can be plotted using plot.fasp.
If the pattern is not marked, the resulting “array” has dimensions 1 × 1. Otherwise the following is
true:
If fun="F", the function array has dimensions m × 1 where m is the number of different marks
in the point pattern. The entry at position [i,1] in this array is the result of applying Fest to the
points of type i only.
If fun is "Gdot", "Jdot", "Kdot" or "Ldot", the function array again has dimensions m × 1. The
entry at position [i,1] in this array is the result of Gdot(X, i), Jdot(X, i) Kdot(X, i) or
Ldot(X, i) respectively.
If fun is "Gcross", "Jcross", "Kcross" or "Lcross" (or their abbreviations "G", "J", "K" or "L"),
the function array has dimensions m × m. The [i,j] entry of the function array (for i 6= j) is the
result of applying the function Gcross, Jcross, Kcross orLcross to the pair of types (i,j). The
diagonal [i,i] entry of the function array is the result of applying the univariate function Gest,
Jest, Kest or Lest to the points of type i only.
If envelope=FALSE, then each function entry fns[[i]] retains the format of the output of the
relevant estimating routine Fest, Gest, Jest, Kest, Lest, Gcross, Jcross ,Kcross, Lcross, Gdot,
Jdot, Kdot or Ldot The default formulae for plotting these functions are cbind(km,theo) ~ r for
F, G, and J functions, and cbind(trans,theo) ~ r for K and L functions.
If envelope=TRUE, then each function entry fns[[i]] has the same format as the output of the
envelope command.
Note
Sizeable amounts of memory may be needed during the calculation.
Author(s)
Adrian Baddeley  and Rolf Turner .
See Also
plot.fasp, fasp.object, Fest, Gest, Jest, Kest, Lest, Gcross, Jcross, Kcross, Lcross, Gdot,
Jdot, Kdot, envelope.

angles.psp

67

Examples
# bramblecanes (3 marks).
bram <- bramblecanes
bF <- alltypes(bram,"F",verb=TRUE)
plot(bF)
if(interactive()) {
plot(alltypes(bram,"G"))
plot(alltypes(bram,"Gdot"))
}
# Swedishpines (unmarked).
swed <- swedishpines
plot(alltypes(swed,"K"))
plot(alltypes(amacrine, "pcf"), ylim=c(0,1.3))
# A setting where you might REALLY want to use dataname:
## Not run:
xxx <- alltypes(ppp(Melvin$x,Melvin$y,
window=as.owin(c(5,20,15,50)),marks=clyde),
fun="F",verb=TRUE,dataname="Melvin")
## End(Not run)
# envelopes
bKE <- alltypes(bram,"K",envelope=TRUE,nsim=19)
## Not run:
bFE <- alltypes(bram,"F",envelope=TRUE,nsim=19,global=TRUE)
## End(Not run)
# extract one entry
as.fv(bKE[1,1])

angles.psp

Orientation Angles of Line Segments

Description
Computes the orientation angle of each line segment in a line segment pattern.
Usage
angles.psp(x, directed=FALSE)
Arguments
x

A line segment pattern (object of class "psp").

directed

Logical flag. See details.

68

anova.lppm

Details
For each line segment, the angle of inclination to the x-axis (in radians) is computed, and the angles
are returned as a numeric vector.
If directed=TRUE, the directed angle of orientation is computed. The angle respects the sense of
direction from (x0,y0) to (x1,y1). The values returned are angles in the full range from −π to π.
The angle is computed as atan2(y1-y0,x1-x0). See atan2.
If directed=FALSE, the undirected angle of orientation is computed. Angles differing by π are
regarded as equivalent. The values returned are angles in the range from 0 to π. These angles are
computed by first computing the directed angle, then adding π to any negative angles.
Value
Numeric vector.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
summary.psp, midpoints.psp, lengths.psp
Examples
a <- psp(runif(10), runif(10), runif(10), runif(10), window=owin())
b <- angles.psp(a)

anova.lppm

ANOVA for Fitted Point Process Models on Linear Network

Description
Performs analysis of deviance for two or more fitted point process models on a linear network.
Usage
## S3 method for class 'lppm'
anova(object, ..., test=NULL)
Arguments
object

A fitted point process model on a linear network (object of class "lppm").

...

One or more fitted point process models on the same linear network.

test

Character string, partially matching one of "Chisq", "F" or "Cp".

anova.lppm

69

Details
This is a method for anova for fitted point process models on a linear network (objects of class
"lppm", usually generated by the model-fitting function lppm).
If the fitted models are all Poisson point processes, then this function performs an Analysis of
Deviance of the fitted models. The output shows the deviance differences (i.e. 2 times log likelihood
ratio), the difference in degrees of freedom, and (if test="Chi") the two-sided p-values for the chisquared tests. Their interpretation is very similar to that in anova.glm.
If some of the fitted models are not Poisson point processes, then the deviance difference is replaced
by the adjusted composite likelihood ratio (Pace et al, 2011; Baddeley et al, 2014).
Value
An object of class "anova", or NULL.
Errors and warnings
models not nested: There may be an error message that the models are not “nested”. For an Analysis of Deviance the models must be nested, i.e. one model must be a special case of the
other. For example the point process model with formula ~x is a special case of the model
with formula ~x+y, so these models are nested. However the two point process models with
formulae ~x and ~y are not nested.
If you get this error message and you believe that the models should be nested, the problem
may be the inability of R to recognise that the two formulae are nested. Try modifying the
formulae to make their relationship more obvious.
different sizes of dataset: There may be an error message from anova.glmlist that “models were
not all fitted to the same size of dataset”. This generally occurs when the point process models
are fitted on different linear networks.
Author(s)
Adrian Baddeley 
References
Ang, Q.W. (2010) Statistical methodology for events on a network. Master’s thesis, School of
Mathematics and Statistics, University of Western Australia.
Ang, Q.W., Baddeley, A. and Nair, G. (2012) Geometrically corrected second-order analysis of
events on a linear network, with applications to ecology and criminology. Scandinavian Journal of
Statistics 39, 591–617.

Baddeley, A., Turner, R. and Rubak, E. (2015) Adjusted composite likelihood ratio test for Gibbs
point processes. Journal of Statistical Computation and Simulation 86 (5) 922–941. DOI: 10.1080/00949655.2015.10445
McSwiggan, G., Nair, M.G. and Baddeley, A. (2012) Fitting Poisson point process models to events
on a linear network. Manuscript in preparation.
Pace, L., Salvan, A. and Sartori, N. (2011) Adjusting composite likelihood ratio statistics. Statistica
Sinica 21, 129–148.
See Also
lppm

70

anova.mppm

Examples
X <- runiflpp(10, simplenet)
mod0 <- lppm(X ~1)
modx <- lppm(X ~x)
anova(mod0, modx, test="Chi")

anova.mppm

ANOVA for Fitted Point Process Models for Replicated Patterns

Description
Performs analysis of deviance for one or more point process models fitted to replicated point pattern
data.
Usage
## S3 method for class 'mppm'
anova(object, ...,
test=NULL, adjust=TRUE,
fine=FALSE, warn=TRUE)
Arguments
object

Object of class "mppm" representing a point process model that was fitted to
replicated point patterns.

...

Optional. Additional objects of class "mppm".

test

Type of hypothesis test to perform. A character string, partially matching one of
"Chisq", "LRT", "Rao", "score", "F" or "Cp", or NULL indicating that no test
should be performed.

adjust

Logical value indicating whether to correct the pseudolikelihood ratio when
some of the models are not Poisson processes.

fine

Logical value passed to vcov.ppm indicating whether to use a quick estimate
(fine=FALSE, the default) or a slower, more accurate estimate (fine=TRUE) of
the variance of the fitted coefficients of each model. Relevant only when some
of the models are not Poisson and adjust=TRUE.

warn

Logical value indicating whether to issue warnings if problems arise.

Details
This is a method for anova for comparing several fitted point process models of class "mppm",
usually generated by the model-fitting function mppm).
If the fitted models are all Poisson point processes, then this function performs an Analysis of
Deviance of the fitted models. The output shows the deviance differences (i.e. 2 times log likelihood
ratio), the difference in degrees of freedom, and (if test="Chi") the two-sided p-values for the chisquared tests. Their interpretation is very similar to that in anova.glm.
If some of the fitted models are not Poisson point processes, the ‘deviance’ differences in this
table are ’pseudo-deviances’ equal to 2 times the differences in the maximised values of the log
pseudolikelihood (see ppm). It is not valid to compare these values to the chi-squared distribution.
In this case, if adjust=TRUE (the default), the pseudo-deviances will be adjusted using the method

anova.mppm

71

of Pace et al (2011) and Baddeley, Turner and Rubak (2015) so that the chi-squared test is valid. It
is strongly advisable to perform this adjustment.
The argument test determines which hypothesis test, if any, will be performed to compare the
models. The argument test should be a character string, partially matching one of "Chisq", "F"
or "Cp", or NULL. The first option "Chisq" gives the likelihood ratio test based on the asymptotic
chi-squared distribution of the deviance difference. The meaning of the other options is explained
in anova.glm. For random effects models, only "Chisq" is available, and again gives the likelihood
ratio test.
Value
An object of class "anova", or NULL.
Error messages
An error message that reports system is computationally singular indicates that the determinant of
the Fisher information matrix of one of the models was either too large or too small for reliable
numerical calculation. See vcov.ppm for suggestions on how to handle this.
Author(s)

Adrian Baddeley, Ida-Maria Sintorn and Leanne Bischoff. Implemented by Adrian Baddeley  and Ege Rubak .
References
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. London: Chapman and Hall/CRC Press.

Baddeley, A., Turner, R. and Rubak, E. (2015) Adjusted composite likelihood ratio test for Gibbs
point processes. Journal of Statistical Computation and Simulation 86 (5) 922–941. DOI: 10.1080/00949655.2015.10445
Pace, L., Salvan, A. and Sartori, N. (2011) Adjusting composite likelihood ratio statistics. Statistica
Sinica 21, 129–148.
See Also
mppm
Examples
H <- hyperframe(X=waterstriders)
mod0 <- mppm(X~1, data=H, Poisson())
modx <- mppm(X~x, data=H, Poisson())
anova(mod0, modx, test="Chi")
mod0S <- mppm(X~1, data=H, Strauss(2))
modxS <- mppm(X~x, data=H, Strauss(2))
anova(mod0S, modxS, test="Chi")

72

anova.ppm

anova.ppm

ANOVA for Fitted Point Process Models

Description
Performs analysis of deviance for one or more fitted point process models.
Usage
## S3 method for class 'ppm'
anova(object, ..., test=NULL,
adjust=TRUE, warn=TRUE, fine=FALSE)
Arguments
object

A fitted point process model (object of class "ppm").

...

Optional. Additional objects of class "ppm".

test

Character string, partially matching one of "Chisq", "LRT", "Rao", "score",
"F" or "Cp", or NULL indicating that no test should be performed.

adjust

Logical value indicating whether to correct the pseudolikelihood ratio when
some of the models are not Poisson processes.

warn

Logical value indicating whether to issue warnings if problems arise.

fine

Logical value, passed to vcov.ppm, indicating whether to use a quick estimate
(fine=FALSE, the default) or a slower, more accurate estimate (fine=TRUE) of
variance terms. Relevant only when some of the models are not Poisson and
adjust=TRUE.

Details
This is a method for anova for fitted point process models (objects of class "ppm", usually generated
by the model-fitting function ppm).
If the fitted models are all Poisson point processes, then by default, this function performs an Analysis of Deviance of the fitted models. The output shows the deviance differences (i.e. 2 times log
likelihood ratio), the difference in degrees of freedom, and (if test="Chi" or test="LRT") the twosided p-values for the chi-squared tests. Their interpretation is very similar to that in anova.glm. If
test="Rao" or test="score", the score test (Rao, 1948) is performed instead.
If some of the fitted models are not Poisson point processes, the ‘deviance’ differences in this
table are ’pseudo-deviances’ equal to 2 times the differences in the maximised values of the log
pseudolikelihood (see ppm). It is not valid to compare these values to the chi-squared distribution.
In this case, if adjust=TRUE (the default), the pseudo-deviances will be adjusted using the method
of Pace et al (2011) and Baddeley et al (2015) so that the chi-squared test is valid. It is strongly
advisable to perform this adjustment.
Value
An object of class "anova", or NULL.

anova.ppm

73

Errors and warnings
models not nested: There may be an error message that the models are not “nested”. For an Analysis of Deviance the models must be nested, i.e. one model must be a special case of the
other. For example the point process model with formula ~x is a special case of the model
with formula ~x+y, so these models are nested. However the two point process models with
formulae ~x and ~y are not nested.
If you get this error message and you believe that the models should be nested, the problem
may be the inability of R to recognise that the two formulae are nested. Try modifying the
formulae to make their relationship more obvious.
different sizes of dataset: There may be an error message from anova.glmlist that “models were
not all fitted to the same size of dataset”. This implies that the models were fitted using different quadrature schemes (see quadscheme) and/or with different edge corrections or different
values of the border edge correction distance rbord.
To ensure that models are comparable, check the following:
• the models must all have been fitted to the same point pattern dataset, in the same window.
• all models must have been fitted by the same fitting method as specified by the argument
method in ppm.
• If some of the models depend on covariates, then they should all have been fitted using
the same list of covariates, and using allcovar=TRUE to ensure that the same quadrature
scheme is used.
• all models must have been fitted using the same edge correction as specified by the arguments correction and rbord. If you did not specify the value of rbord, then it may
have taken a different value for different models. The default value of rbord is equal to
zero for a Poisson model, and otherwise equals the reach (interaction distance) of the interaction term (see reach). To ensure that the models are comparable, set rbord to equal
the maximum reach of the interactions that you are fitting.
Error messages
An error message that reports system is computationally singular indicates that the determinant of
the Fisher information matrix of one of the models was either too large or too small for reliable
numerical calculation. See vcov.ppm for suggestions on how to handle this.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
References

Baddeley, A., Turner, R. and Rubak, E. (2015) Adjusted composite likelihood ratio test for Gibbs
point processes. Journal of Statistical Computation and Simulation 86 (5) 922–941. DOI: 10.1080/00949655.2015.10445
Pace, L., Salvan, A. and Sartori, N. (2011) Adjusting composite likelihood ratio statistics. Statistica
Sinica 21, 129–148.
Rao, C.R. (1948) Large sample tests of statistical hypotheses concerning several parameters with
applications to problems of estimation. Proceedings of the Cambridge Philosophical Society 44,
50–57.
See Also
ppm, vcov.ppm

74

anova.slrm

Examples
mod0 <- ppm(swedishpines ~1)
modx <- ppm(swedishpines ~x)
# Likelihood ratio test
anova(mod0, modx, test="Chi")
# Score test
anova(mod0, modx, test="Rao")
# Single argument
modxy <- ppm(swedishpines ~x + y)
anova(modxy, test="Chi")
# Adjusted composite likelihood ratio test
modP <- ppm(swedishpines ~1, rbord=9)
modS <- ppm(swedishpines ~1, Strauss(9))
anova(modP, modS, test="Chi")

anova.slrm

Analysis of Deviance for Spatial Logistic Regression Models

Description
Performs Analysis of Deviance for two or more fitted Spatial Logistic Regression models.
Usage
## S3 method for class 'slrm'
anova(object, ..., test = NULL)
Arguments
object

a fitted spatial logistic regression model. An object of class "slrm".

...

additional objects of the same type (optional).

test

a character string, (partially) matching one of "Chisq", "F" or "Cp", indicating
the reference distribution that should be used to compute p-values.

Details
This is a method for anova for fitted spatial logistic regression models (objects of class "slrm",
usually obtained from the function slrm).
The output shows the deviance differences (i.e. 2 times log likelihood ratio), the difference in
degrees of freedom, and (if test="Chi") the two-sided p-values for the chi-squared tests. Their
interpretation is very similar to that in anova.glm.
Value
An object of class "anova", inheriting from class "data.frame", representing the analysis of deviance table.

anylist

75

Author(s)
Adrian Baddeley  
and Rolf Turner 
See Also
slrm
Examples
X <- rpoispp(42)
fit0 <- slrm(X ~ 1)
fit1 <- slrm(X ~ x+y)
anova(fit0, fit1, test="Chi")

anylist

List of Objects

Description
Make a list of objects of any type.
Usage
anylist(...)
as.anylist(x)
Arguments
...

Any number of arguments of any type.

x

A list.

Details
An object of class "anylist" is a list of objects that the user intends to treat in a similar fashion.
For example it may be desired to plot each of the objects side-by-side: this can be done using the
function plot.anylist.
The objects can belong to any class; they may or may not all belong to the same class.
In the spatstat package, various functions produce an object of class "anylist".
Value
A list, belonging to the class "anylist", containing the original objects.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 

76

anyNA.im

See Also
solist, as.solist, anylapply.
Examples
anylist(cells, intensity(cells), Kest(cells))

anyNA.im

Check Whether Image Contains NA Values

Description
Checks whether any pixel values in a pixel image are NA (meaning that the pixel lies outside the
domain of definition of the image).
Usage
## S3 method for class 'im'
anyNA(x, recursive = FALSE)
Arguments
x

A pixel image (object of class "im").

recursive

Ignored.

Details
The function anyNA is generic: anyNA(x) is a faster alternative to any(is.na(x)).
This function anyNA.im is a method for the generic anyNA defined for pixel images. It returns the
value TRUE if any of the pixel values in x are NA, and and otherwise returns FALSE.
Value
A single logical value.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
im.object
Examples
anyNA(as.im(letterR))

append.psp

77

append.psp

Combine Two Line Segment Patterns

Description
Combine two line segment patterns into a single pattern.
Usage
append.psp(A, B)
Arguments
A,B

Line segment patterns (objects of class "psp").

Details
This function is used to superimpose two line segment patterns A and B.
The two patterns must have identical windows. If one pattern has marks, then the other must also
have marks of the same type. It the marks are data frames then the number of columns of these data
frames, and the names of the columns must be identical.
(To combine two point patterns, see superimpose).
Value
Another line segment pattern (object of class "psp").
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
psp, as.psp, superimpose,
Examples
X <- psp(runif(20), runif(20), runif(20), runif(20), window=owin())
Y <- psp(runif(5), runif(5), runif(5), runif(5), window=owin())
append.psp(X,Y)

78

applynbd

applynbd

Apply Function to Every Neighbourhood in a Point Pattern

Description
Visit each point in a point pattern, find the neighbouring points, and apply a given function to them.
Usage
applynbd(X, FUN, N=NULL, R=NULL, criterion=NULL, exclude=FALSE, ...)
Arguments
X

Point pattern. An object of class "ppp", or data which can be converted into this
format by as.ppp.

FUN

Function to be applied to each neighbourhood. The arguments of FUN are described under Details.

N

Integer. If this argument is present, the neighbourhood of a point of X is defined
to consist of the N points of X which are closest to it.

R

Nonnegative numeric value. If this argument is present, the neighbourhood of a
point of X is defined to consist of all points of X which lie within a distance R of
it.

criterion

Function. If this argument is present, the neighbourhood of a point of X is determined by evaluating this function. See under Details.

exclude

Logical. If TRUE then the point currently being visited is excluded from its own
neighbourhood.

...

extra arguments passed to the function FUN. They must be given in the form
name=value.

Details
This is an analogue of apply for point patterns. It visits each point in the point pattern X, determines which points of X are “neighbours” of the current point, applies the function FUN to this
neighbourhood, and collects the values returned by FUN.
The definition of “neighbours” depends on the arguments N, R and criterion. Also the argument
exclude determines whether the current point is excluded from its own neighbourhood.
• If N is given, then the neighbours of the current point are the N points of X which are closest to
the current point (including the current point itself unless exclude=TRUE).
• If R is given, then the neighbourhood of the current point consists of all points of X which lie
closer than a distance R from the current point.
• If criterion is given, then it must be a function with two arguments dist and drank which
will be vectors of equal length. The interpretation is that dist[i] will be the distance of a
point from the current point, and drank[i] will be the rank of that distance (the three points
closest to the current point will have rank 1, 2 and 3). This function must return a logical
vector of the same length as dist and drank whose i-th entry is TRUE if the corresponding
point should be included in the neighbourhood. See the examples below.

applynbd

79

• If more than one of the arguments N, R and criterion is given, the neighbourhood is defined
as the intersection of the neighbourhoods specified by these arguments. For example if N=3
and R=5 then the neighbourhood is formed by finding the 3 nearest neighbours of current point,
and retaining only those neighbours which lie closer than 5 units from the current point.
When applynbd is executed, each point of X is visited, and the following happens for each point:
• the neighbourhood of the current point is determined according to the chosen rule, and stored
as a point pattern Y;
• the function FUN is called as:
FUN(Y=Y, current=current, dists=dists, dranks=dranks, ...)
where current is the location of the current point (in a format explained below), dists is a
vector of distances from the current point to each of the points in Y, dranks is a vector of the
ranks of these distances with respect to the full point pattern X, and ... are the arguments
passed from the call to applynbd;
• The result of the call to FUN is stored.
The results of each call to FUN are collected and returned according to the usual rules for apply and
its relatives. See the Value section of this help file.
The format of the argument current is as follows. If X is an unmarked point pattern, then current
is a vector of length 2 containing the coordinates of the current point. If X is marked, then current is
a point pattern containing exactly one point, so that current$x is its x-coordinate and current$marks
is its mark value. In either case, the coordinates of the current point can be referred to as current$x
and current$y.
Note that FUN will be called exactly as described above, with each argument named explicitly. Care
is required when writing the function FUN to ensure that the arguments will match up. See the
Examples.
See markstat for a common use of this function.
To simply tabulate the marks in every R-neighbourhood, use marktable.
Value
Similar to the result of apply. If each call to FUN returns a single numeric value, the result is a
vector of dimension npoints(X), the number of points in X. If each call to FUN returns a vector of
the same length m, then the result is a matrix of dimensions c(m,n); note the transposition of the
indices, as usual for the family of apply functions. If the calls to FUN return vectors of different
lengths, the result is a list of length npoints(X).
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
ppp.object, apply, markstat, marktable
Examples
redwood
# count the number of points within radius 0.2 of each point of X
nneighbours <- applynbd(redwood, R=0.2, function(Y, ...){npoints(Y)-1})

80

area.owin
# equivalent to:
nneighbours <- applynbd(redwood, R=0.2, function(Y, ...){npoints(Y)}, exclude=TRUE)
# compute the distance to the second nearest neighbour of each point
secondnndist <- applynbd(redwood, N = 2,
function(dists, ...){max(dists)},
exclude=TRUE)
# marked point pattern
trees <- longleaf
# compute the median of the marks of all neighbours of a point
# (see also 'markstat')
dbh.med <- applynbd(trees, R=90, exclude=TRUE,
function(Y, ...) { median(marks(Y))})
# ANIMATION explaining the definition of the K function
# (arguments `fullpicture' and 'rad' are passed to FUN)
if(interactive()) {
showoffK <- function(Y, current, dists, dranks, fullpicture,rad) {
plot(fullpicture, main="")
points(Y, cex=2)
ux <- current[["x"]]
uy <- current[["y"]]
points(ux, uy, pch="+",cex=3)
theta <- seq(0,2*pi,length=100)
polygon(ux + rad * cos(theta), uy+rad*sin(theta))
text(ux + rad/3, uy + rad/2,npoints(Y),cex=3)
if(interactive()) Sys.sleep(if(runif(1) < 0.1) 1.5 else 0.3)
return(npoints(Y))
}
applynbd(redwood, R=0.2, showoffK, fullpicture=redwood, rad=0.2, exclude=TRUE)
# animation explaining the definition of the G function
showoffG <- function(Y, current, dists, dranks, fullpicture) {
plot(fullpicture, main="")
points(Y, cex=2)
u <- current
points(u[1],u[2],pch="+",cex=3)
v <- c(Y$x[1],Y$y[1])
segments(u[1],u[2],v[1],v[2],lwd=2)
w <- (u + v)/2
nnd <- dists[1]
text(w[1],w[2],round(nnd,3),cex=2)
if(interactive()) Sys.sleep(if(runif(1) < 0.1) 1.5 else 0.3)
return(nnd)
}
applynbd(cells, N=1, showoffG, exclude=TRUE, fullpicture=cells)
}

area.owin

Area of a Window

area.owin

81

Description
Computes the area of a window
Usage
area(w)
## S3 method for class 'owin'
area(w)
## Default S3 method:
area(w)
## S3 method for class 'owin'
volume(x)
Arguments
w

A window, whose area will be computed. This should be an object of class owin,
or can be given in any format acceptable to as.owin().

x

Object of class owin

Details
If the window w is of type "rectangle" or "polygonal", the area of this rectangular window is
computed by analytic geometry. If w is of type "mask" the area of the discrete raster approximation
of the window is computed by summing the binary image values and adjusting for pixel size.
The function volume.owin is identical to area.owin except for the argument name. It is a method
for the generic function volume.
Value
A numerical value giving the area of the window.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
perimeter, diameter.owin, owin.object, as.owin
Examples
w <- unit.square()
area(w)
# returns 1.00000
k <- 6
theta <- 2 * pi * (0:(k-1))/k
co <- cos(theta)
si <- sin(theta)

82

areaGain
mas <- owin(c(-1,1), c(-1,1), poly=list(x=co, y=si))
area(mas)
# returns approx area of k-gon
mas <- as.mask(square(2), eps=0.01)
X <- raster.x(mas)
Y <- raster.y(mas)
mas$m <- ((X - 1)^2 + (Y - 1)^2 <= 1)
area(mas)
# returns 3.14 approx

areaGain

Difference of Disc Areas

Description
Computes the area of that part of a disc that is not covered by other discs.
Usage
areaGain(u, X, r, ..., W=as.owin(X), exact=FALSE,
ngrid=spatstat.options("ngrid.disc"))
Arguments
u

Coordinates of the centre of the disc of interest. A vector of length 2. Alternatively, a point pattern (object of class "ppp").

X

Locations of the centres of other discs. A point pattern (object of class "ppp").

r

Disc radius, or vector of disc radii.

...

Arguments passed to distmap to determine the pixel resolution, when exact=FALSE.

W

Window (object of class "owin") in which the area should be computed.

exact

Choice of algorithm. If exact=TRUE, areas are computed exactly using analytic
geometry. If exact=FALSE then a faster algorithm is used to compute a discrete
approximation to the areas.

ngrid

Integer. Number of points in the square grid used to compute the discrete approximation, when exact=FALSE.

Details
This function computes the area of that part of the disc of radius r centred at the location u that
is not covered by any of the discs of radius r centred at the points of the pattern X. This area is
important in some calculations related to the area-interaction model AreaInter.
If u is a point pattern and r is a vector, the result is a matrix, with one row for each point in u and
one column for each entry of r. The [i,j] entry in the matrix is the area of that part of the disc
of radius r[j] centred at the location u[i] that is not covered by any of the discs of radius r[j]
centred at the points of the pattern X.
If W is not NULL, then the areas are computed only inside the window W.

AreaInter

83

Value
A matrix with one row for each point in u and one column for each value in r.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
AreaInter, areaLoss
Examples
data(cells)
u <- c(0.5,0.5)
areaGain(u, cells, 0.1)

AreaInter

The Area Interaction Point Process Model

Description
Creates an instance of the Area Interaction point process model (Widom-Rowlinson penetrable
spheres model) which can then be fitted to point pattern data.
Usage
AreaInter(r)
Arguments
r

The radius of the discs in the area interaction process

Details
This function defines the interpoint interaction structure of a point process called the WidomRowlinson penetrable sphere model or area-interaction process. It can be used to fit this model
to point pattern data.
The function ppm(), which fits point process models to point pattern data, requires an argument
of class "interact" describing the interpoint interaction structure of the model to be fitted. The
appropriate description of the area interaction structure is yielded by the function AreaInter().
See the examples below.
In standard form, the area-interaction process (Widom and Rowlinson, 1970; Baddeley and Van
Lieshout, 1995) with disc radius r, intensity parameter κ and interaction parameter γ is a point
process with probability density
f (x1 , . . . , xn ) = ακn(x) γ −A(x)
for a point pattern x, where x1 , . . . , xn represent the points of the pattern, n(x) is the number of
points in the pattern, and A(x) is the area of the region formed by the union of discs of radius r
centred at the points x1 , . . . , xn . Here α is a normalising constant.

84

AreaInter
The interaction parameter γ can be any positive number. If γ = 1 then the model reduces to a
Poisson process with intensity κ. If γ < 1 then the process is regular, while if γ > 1 the process is
clustered. Thus, an area interaction process can be used to model either clustered or regular point
patterns. Two points interact if the distance between them is less than 2r.
The standard form of the model, shown above, is a little complicated to interpret in practical ap2
plications. For example, each isolated point of the pattern x contributes a factor κγ −πr to the
probability density.
In spatstat, the model is parametrised in a different form, which is easier to interpret. In canonical
scale-free form, the probability density is rewritten as
f (x1 , . . . , xn ) = αβ n(x) η −C(x)
where β is the new intensity parameter, η is the new interaction parameter, and C(x) = B(x)−n(x)
is the interaction potential. Here
A(x)
B(x) =
πr2
is the normalised area (so that the discs have unit area). In this formulation, each isolated point
of the pattern contributes a factor β to the probability density (so the first order trend is β). The
quantity C(x) is a true interaction potential, in the sense that C(x) = 0 if the point pattern x does
not contain any points that lie close together (closer than 2r units apart).
When a new point u is added to an existing point pattern x, the rescaled potential −C(x) increases
by a value between 0 and 1. The increase is zero if u is not close to any point of x. The increase is
1 if the disc of radius r centred at u is completely contained in the union of discs of radius r centred
at the data points xi . Thus, the increase in potential is a measure of how close the new point u is
to the existing pattern x. Addition of the point u contributes a factor βη δ to the probability density,
where δ is the increase in potential.
The old parameters κ, γ of the standard form are related to the new parameters β, η of the canonical
scale-free form, by
2

β = κγ −πr = κ/η
and
η = γ πr

2

provided γ and κ are positive and finite.
In the canonical scale-free form, the parameter η can take any nonnegative value. The value η = 1
again corresponds to a Poisson process, with intensity β. If η < 1 then the process is regular, while
if η > 1 the process is clustered. The value η = 0 corresponds to a hard core process with hard core
radius r (interaction distance 2r).
The nonstationary area interaction process is similar except that the contribution of each individual
point xi is a function β(xi ) of location, rather than a constant beta.
Note the only argument of AreaInter() is the disc radius r. When r is fixed, the model becomes
an exponential family. The canonical parameters log(β) and log(η) are estimated by ppm(), not
fixed in AreaInter().
Value
An object of class "interact" describing the interpoint interaction structure of the area-interaction
process with disc radius r.

AreaInter

85

Warnings
The interaction distance of this process is equal to 2 * r. Two discs of radius r overlap if their
centres are closer than 2 * r units apart.
The estimate of the interaction parameter η is unreliable if the interaction radius r is too small or
too large. In these situations the model is approximately Poisson so that η is unidentifiable. As a
rule of thumb, one can inspect the empty space function of the data, computed by Fest. The value
F (r) of the empty space function at the interaction radius r should be between 0.2 and 0.8.
Author(s)
Adrian Baddeley  and Rolf Turner 
References
Baddeley, A.J. and Van Lieshout, M.N.M. (1995). Area-interaction point processes. Annals of the
Institute of Statistical Mathematics 47 (1995) 601–619.
Widom, B. and Rowlinson, J.S. (1970). New model for the study of liquid-vapor phase transitions.
The Journal of Chemical Physics 52 (1970) 1670–1684.
See Also
ppm, pairwise.family, ppm.object
ragsAreaInter and rmh for simulation of area-interaction models.
Examples

# prints a sensible description of itself
AreaInter(r=0.1)
# Note the reach is twice the radius
reach(AreaInter(r=1))
# Fit the stationary area interaction process to Swedish Pines data
data(swedishpines)
ppm(swedishpines, ~1, AreaInter(r=7))
# Fit the stationary area interaction process to `cells'
data(cells)
ppm(cells, ~1, AreaInter(r=0.06))
# eta=0 indicates hard core process.
# Fit a nonstationary area interaction with log-cubic polynomial trend
## Not run:
ppm(swedishpines, ~polynom(x/10,y/10,3), AreaInter(r=7))
## End(Not run)

86

areaLoss

areaLoss

Difference of Disc Areas

Description
Computes the area of that part of a disc that is not covered by other discs.
Usage
areaLoss(X, r, ..., W=as.owin(X), subset=NULL,
exact=FALSE,
ngrid=spatstat.options("ngrid.disc"))
Arguments
X

Locations of the centres of discs. A point pattern (object of class "ppp").

r

Disc radius, or vector of disc radii.

...

Ignored.

W

Optional. Window (object of class "owin") inside which the area should be
calculated.

subset

Optional. Index identifying a subset of the points of X for which the area difference should be computed.

exact

Choice of algorithm. If exact=TRUE, areas are computed exactly using analytic
geometry. If exact=FALSE then a faster algorithm is used to compute a discrete
approximation to the areas.

ngrid

Integer. Number of points in the square grid used to compute the discrete approximation, when exact=FALSE.

Details
This function computes, for each point X[i] in X and for each radius r, the area of that part of the
disc of radius r centred at the location X[i] that is not covered by any of the other discs of radius r
centred at the points X[j] for j not equal to i. This area is important in some calculations related
to the area-interaction model AreaInter.
The result is a matrix, with one row for each point in X and one column for each entry of r.
Value
A matrix with one row for each point in X (or X[subset]) and one column for each value in r.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
AreaInter, areaGain, dilated.areas

as.box3

87

Examples
data(cells)
areaLoss(cells, 0.1)

as.box3

Convert Data to Three-Dimensional Box

Description
Interprets data as the dimensions of a three-dimensional box.
Usage
as.box3(...)
Arguments
...

Data that can be interpreted as giving the dimensions of a three-dimensional
box. See Details.

Details
This function converts data in various formats to an object of class "box3" representing a threedimensional box (see box3). The arguments ... may be
• an object of class "box3"
• arguments acceptable to box3

• a numeric vector of length 6, interpreted as c(xrange[1],xrange[2],yrange[1],yrange[2],zrange[1],zrange
• an object of class "pp3" representing a three-dimensional point pattern contained in a box.
Value
Object of class "box3".
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
box3, pp3
Examples
X <- c(0,10,0,10,0,5)
as.box3(X)
X <- pp3(runif(42),runif(42),runif(42), box3(c(0,1)))
as.box3(X)

88

as.boxx

as.boxx

Convert Data to Multi-Dimensional Box

Description
Interprets data as the dimensions of a multi-dimensional box.
Usage
as.boxx(..., warn.owin = TRUE)
Arguments
...

Data that can be interpreted as giving the dimensions of a multi-dimensional
box. See Details.

warn.owin

Logical value indicating whether to print a warning if a non-rectangular window
(object of class "owin") is supplied.

Details
Either a single argument should be provided which is one of the following:
• an object of class "boxx"
• an object of class "box3"
• an object of class "owin"
• a numeric vector of even length, specifying the corners of the box. See Examples
or a list of arguments acceptable to boxx.
Value
A "boxx" object.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
Examples
# Convert unit square to two dimensional box.
W <- owin()
as.boxx(W)
# Make three dimensional box [0,1]x[0,1]x[0,1] from numeric vector
as.boxx(c(0,1,0,1,0,1))

as.data.frame.envelope

89

as.data.frame.envelope
Coerce Envelope to Data Frame

Description
Converts an envelope object to a data frame.
Usage
## S3 method for class 'envelope'
as.data.frame(x, ..., simfuns=FALSE)
Arguments
x

Envelope object (class "envelope").

...

Ignored.

simfuns

Logical value indicating whether the result should include the values of the simulated functions that were used to build the envelope.

Details
This is a method for the generic function as.data.frame for the class of envelopes (see envelope.
The result is a data frame with columns containing the values of the function argument (usually
named r), the function estimate for the original point pattern data (obs), the upper and lower envelope limits (hi and lo), and possibly additional columns.
If simfuns=TRUE, the result also includes columns of values of the simulated functions that were
used to compute the envelope. This is possible only when the envelope was computed with the
argument savefuns=TRUE in the call to envelope.
Value
A data frame.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
Examples
E <- envelope(cells, nsim=5, savefuns=TRUE)
tail(as.data.frame(E))
tail(as.data.frame(E, simfuns=TRUE))

90

as.data.frame.hyperframe

as.data.frame.hyperframe
Coerce Hyperframe to Data Frame

Description
Converts a hyperframe to a data frame.
Usage
## S3 method for class 'hyperframe'
as.data.frame(x, row.names = NULL,
optional = FALSE, ...,
discard=TRUE, warn=TRUE)
Arguments
x

Hyperframe (object of class "hyperframe").

row.names

Optional character vector of row names.

optional

Argument passed to as.data.frame controlling what happens to row names.

...

Ignored.

discard

Logical. Whether to discard columns of the hyperframe that do not contain
atomic data. See Details.

warn

Logical. Whether to issue a warning when columns are discarded.

Details
This is a method for the generic function as.data.frame for the class of hyperframes (see hyperframe.
If discard=TRUE, any columns of the hyperframe that do not contain atomic data will be removed
(and a warning will be issued if warn=TRUE). If discard=FALSE, then such columns are converted
to strings indicating what class of data they originally contained.
Value
A data frame.
Author(s)
Adrian Baddeley 
and Rolf Turner 
Examples
h <- hyperframe(X=1:3, Y=letters[1:3], f=list(sin, cos, tan))
as.data.frame(h, discard=TRUE, warn=FALSE)
as.data.frame(h, discard=FALSE)

as.data.frame.im

as.data.frame.im

91

Convert Pixel Image to Data Frame

Description
Convert a pixel image to a data frame

Usage
## S3 method for class 'im'
as.data.frame(x, ...)
Arguments
x

A pixel image (object of class "im").

...

Further arguments passed to as.data.frame.default to determine the row
names and other features.

Details
This function takes the pixel image x and returns a data frame with three columns containing the
pixel coordinates and the pixel values.
The data frame entries are automatically sorted in increasing order of the x coordinate (and in
increasing order of y within x).
Value
A data frame.

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
Examples
# artificial image
Z <- setcov(square(1))
Y <- as.data.frame(Z)
head(Y)

92

as.data.frame.owin

as.data.frame.owin

Convert Window to Data Frame

Description
Converts a window object to a data frame.
Usage
## S3 method for class 'owin'
as.data.frame(x, ..., drop=TRUE)
Arguments
x

Window (object of class "owin").

...

Further arguments passed to as.data.frame.default to determine the row
names and other features.

drop

Logical value indicating whether to discard pixels that are outside the window,
when x is a binary mask.

Details
This function returns a data frame specifying the coordinates of the window.
If x is a binary mask window, the result is a data frame with columns x and y containing the spatial
coordinates of each pixel. If drop=TRUE (the default), only pixels inside the window are retained. If
drop=FALSE, all pixels are retained, and the data frame has an extra column inside containing the
logical value of each pixel (TRUE for pixels inside the window, FALSE for outside).
If x is a rectangle or a polygonal window, the result is a data frame with columns x and y containing
the spatial coordinates of the vertices of the window. If the boundary consists of several polygons,
the data frame has additional columns id, identifying which polygon is being traced, and sign,
indicating whether the polygon is an outer or inner boundary (sign=1 and sign=-1 respectively).
Value
A data frame with columns named x and y, and possibly other columns.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
as.data.frame.im

as.data.frame.ppp

93

Examples
as.data.frame(square(1))
holey <- owin(poly=list(
list(x=c(0,10,0), y=c(0,0,10)),
list(x=c(2,2,4,4), y=c(2,4,4,2))))
as.data.frame(holey)

as.data.frame.ppp

Coerce Point Pattern to a Data Frame

Description
Extracts the coordinates of the points in a point pattern, and their marks if any, and returns them in
a data frame.
Usage
## S3 method for class 'ppp'
as.data.frame(x, row.names = NULL, ...)
Arguments
x

Point pattern (object of class "ppp").

row.names

Optional character vector of row names.

...

Ignored.

Details
This is a method for the generic function as.data.frame for the class "ppp" of point patterns.
It extracts the coordinates of the points in the point pattern, and returns them as columns named x
and y in a data frame. If the points were marked, the marks are returned as a column named marks
with the same type as in the point pattern dataset.
Value
A data frame.
Author(s)
Adrian Baddeley 
and Rolf Turner 
Examples
data(amacrine)
df <- as.data.frame(amacrine)
df[1:5,]

94

as.data.frame.psp

as.data.frame.psp

Coerce Line Segment Pattern to a Data Frame

Description
Extracts the coordinates of the endpoints in a line segment pattern, and their marks if any, and
returns them in a data frame.
Usage
## S3 method for class 'psp'
as.data.frame(x, row.names = NULL, ...)
Arguments
x

Line segment pattern (object of class "psp").

row.names

Optional character vector of row names.

...

Ignored.

Details
This is a method for the generic function as.data.frame for the class "psp" of line segment
patterns.
It extracts the coordinates of the endpoints of the line segments, and returns them as columns named
x0, y0, x1 and y1 in a data frame. If the line segments were marked, the marks are appended as an
extra column or columns to the data frame which is returned. If the marks are a vector then a single
column named marks is appended. in the data frame, with the same type as in the line segment
pattern dataset. If the marks are a data frame, then the columns of this data frame are appended
(retaining their names).
Value
A data frame with 4 or 5 columns.
Author(s)
Adrian Baddeley 
and Rolf Turner 
Examples
data(copper)
df <- as.data.frame(copper$Lines)

as.data.frame.tess

95

as.data.frame.tess

Convert Tessellation to Data Frame

Description
Converts a spatial tessellation object to a data frame.
Usage
## S3 method for class 'tess'
as.data.frame(x, ...)
Arguments
x

Tessellation (object of class "tess").

...

Further arguments passed to as.data.frame.owin or as.data.frame.im and
ultimately to as.data.frame.default to determine the row names and other
features.

Details
This function converts the tessellation x to a data frame.
If x is a pixel image tessellation (a pixel image with factor values specifying the tile membership
of each pixel) then this pixel image is converted to a data frame by as.data.frame.im. The result
is a data frame with columns x and y giving the pixel coordinates, and Tile identifying the tile
containing the pixel.
If x is a tessellation consisting of a rectangular grid of tiles or a list of polygonal tiles, then each tile
is converted to a data frame by as.data.frame.owin, and these data frames are joined together,
yielding a single large data frame containing columns x, y giving the coordinates of vertices of the
polygons, and Tile identifying the tile.
Value
A data frame with columns named x, y, Tile, and possibly other columns.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
as.data.frame.owin, as.data.frame.im
Examples
Z <- as.data.frame(dirichlet(cells))
head(Z, 10)

96

as.function.fv

as.function.fv

Convert Function Value Table to Function

Description
Converts an object of class "fv" to an R language function.
Usage
## S3 method for class 'fv'
as.function(x, ..., value=".y", extrapolate=FALSE)
## S3 method for class 'rhohat'
as.function(x, ..., value=".y", extrapolate=TRUE)
Arguments
x

Object of class "fv" or "rhohat".

...

Ignored.

value

Optional. Character string or character vector selecting one or more of the
columns of x for use as the function value. See Details.

extrapolate

Logical, indicating whether to extrapolate the function outside the domain of x.
See Details.

Details
A function value table (object of class "fv") is a convenient way of storing and plotting several
different estimates of the same function. Objects of this class are returned by many commands in
spatstat, such as Kest which returns an estimate of Ripley’s K-function for a point pattern dataset.
Sometimes it is useful to convert the function value table to a function in the R language. This is
done by as.function.fv. It converts an object x of class "fv" to an R function f.
If f <- as.function(x) then f is an R function that accepts a numeric argument and returns
a corresponding value for the summary function by linear interpolation between the values in the
table x.
Argument values lying outside the range of the table yield an NA value (if extrapolate=FALSE) or
the function value at the nearest endpoint of the range (if extrapolate = TRUE). To apply different
rules to the left and right extremes, use extrapolate=c(TRUE,FALSE) and so on.
Typically the table x contains several columns of function values corresponding to different edge
corrections. Auxiliary information for the table identifies one of these columns as the recommended
value. By default, the values of the function f <- as.function(x) are taken from this column
of recommended values. This default can be changed using the argument value, which can be a
character string or character vector of names of columns of x. Alternatively value can be one of
the abbreviations used by fvnames.
If value specifies a single column of the table, then the result is a function f(r) with a single
numeric argument r (with the same name as the orginal argument of the function table).
If value specifies several columns of the table, then the result is a function f(r,what) where r is
the numeric argument and what is a character string identifying the column of values to be used.

as.function.im

97

The formal arguments of the resulting function are f(r, what=value), which means that in a call
to this function f, the permissible values of what are the entries of the original vector value; the
default value of what is the first entry of value.
The command as.function.fv is a method for the generic command as.function.
Value
A function(r) or function(r,what) where r is the name of the original argument of the function
table.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
fv, fv.object, fvnames, plot.fv, Kest
Examples
K <- Kest(cells)
f <- as.function(K)
f
f(0.1)
g <- as.function(K, value=c("iso", "trans"))
g
g(0.1, "trans")

as.function.im

Convert Pixel Image to Function of Coordinates

Description
Converts a pixel image to a function of the x and y coordinates.
Usage
## S3 method for class 'im'
as.function(x, ...)
Arguments
x

Pixel image (object of class "im").

...

Ignored.

Details
This command converts a pixel image (object of class "im") to a function(x,y) where the arguments x and y are (vectors of) spatial coordinates. This function returns the pixel values at the
specified locations.

98

as.function.leverage.ppm

Value
A function in the R language, also belonging to the class "funxy".
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
[.im
Examples
d <- density(cells)
f <- as.function(d)
f(0.1, 0.3)

as.function.leverage.ppm
Convert Leverage Object to Function of Coordinates

Description
Converts an object of class "leverage.ppm" to a function of the x and y coordinates.
Usage
## S3 method for class 'leverage.ppm'
as.function(x, ...)
Arguments
x

Object of class "leverage.ppm" produced by leverage.ppm.

...

Ignored.

Details
An object of class "leverage.ppm" represents the leverage function of a fitted point process model.
This command converts the object to a function(x,y) where the arguments x and y are (vectors of)
spatial coordinates. This function returns the leverage values at the specified locations (calculated
by referring to the nearest location where the leverage has been computed).
Value
A function in the R language, also belonging to the class "funxy".
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .

as.function.owin

99

See Also
as.im.leverage.ppm
Examples
X <- rpoispp(function(x,y) { exp(3+3*x) })
fit <- ppm(X ~x+y)
lev <- leverage(fit)
f <- as.function(lev)
f(0.2, 0.3)
y <- f(X)

# evaluate at (x,y) coordinates
# evaluate at a point pattern

as.function.owin

Convert Window to Indicator Function

Description
Converts a spatial window to a function of the x and y coordinates returning the value 1 inside the
window and 0 outside.
Usage
## S3 method for class 'owin'
as.function(x, ...)
Arguments
x

Pixel image (object of class "owin").

...

Ignored.

Details
This command converts a spatial window (object of class "owin") to a function(x,y) where the
arguments x and y are (vectors of) spatial coordinates. This is the indicator function of the window:
it returns the value 1 for locations inside the window, and returns 0 for values outside the window.
Value
A function in the R language.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
as.im.owin

100

as.function.tess

Examples
W <- Window(humberside)
f <- as.function(W)
f(5000, 4500)
f(123456, 78910)
X <- runifpoint(5, Frame(humberside))
f(X)

as.function.tess

Convert a Tessellation to a Function

Description
Convert a tessellation into a function of the x and y coordinates. The default function values are
factor levels specifying which tile of the tessellation contains the point (x, y).
Usage
## S3 method for class 'tess'
as.function(x,...,values=NULL)
Arguments
x

A tessellation (object of class "tess").

values

Optional. A vector giving the values of the function for each tile of x.

...

Ignored.

Details
This command converts a tessellation (object of class "tess") to a function(x,y) where the arguments x and y are (vectors of) spatial coordinates. The corresponding function values are factor
levels identifying which tile of the tessellation contains each point. Values are NA if the corresponding point lies outside the tessellation.
If the argument values is given, then it determines the value of the function in each tile of x.
Value
A function in the R language, also belonging to the class "funxy".
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
tileindex for the low-level calculation of tile index.
cut.ppp and split.ppp to divide up the points of a point pattern according to a tessellation.

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101

Examples
X <- runifpoint(7)
V <- dirichlet(X)
f <- as.function(V)
f(0.1, 0.4)
plot(f)

as.fv

Convert Data To Class fv

Description
Converts data into a function table (an object of class "fv").
Usage
as.fv(x)
## S3 method for class 'fv'
as.fv(x)
## S3 method for class 'data.frame'
as.fv(x)
## S3 method for class 'matrix'
as.fv(x)
## S3 method for class 'fasp'
as.fv(x)
## S3 method for class 'minconfit'
as.fv(x)
## S3 method for class 'dppm'
as.fv(x)
## S3 method for class 'kppm'
as.fv(x)
## S3 method for class 'bw.optim'
as.fv(x)
Arguments
x

Data which will be converted into a function table

Details
This command converts data x, that could be interpreted as the values of a function, into a function
value table (object of the class "fv" as described in fv.object). This object can then be plotted
easily using plot.fv.
The dataset x may be any of the following:

102

as.hyperframe
• an object of class "fv";
• a matrix or data frame with at least two columns;
• an object of class "fasp", representing an array of "fv" objects.
• an object of class "minconfit", giving the results of a minimum contrast fit by the command
mincontrast. The
• an object of class "kppm", representing a fitted Cox or cluster point process model, obtained
from the model-fitting command kppm;
• an object of class "dppm", representing a fitted determinantal point process model, obtained
from the model-fitting command dppm;
• an object of class "bw.optim", representing an optimal choice of smoothing bandwidth by a
cross-validation method, obtained from commands like bw.diggle.
The function as.fv is generic, with methods for each of the classes listed above. The behaviour is
as follows:
• If x is an object of class "fv", it is returned unchanged.
• If x is a matrix or data frame, the first column is interpreted as the function argument, and
subsequent columns are interpreted as values of the function computed by different methods.
• If x is an object of class "fasp" representing an array of "fv" objects, these are combined
into a single "fv" object.
• If x is an object of class "minconfit", or an object of class "kppm" or "dppm", the result is a
function table containing the observed summary function and the best fit summary function.
• If x is an object of class "bw.optim", the result is a function table of the optimisation criterion
as a function of the smoothing bandwidth.

Value
An object of class "fv" (see fv.object).
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
Examples
r <- seq(0, 1, length=101)
x <- data.frame(r=r, y=r^2)
as.fv(x)

as.hyperframe

Convert Data to Hyperframe

Description
Converts data from any suitable format into a hyperframe.

as.hyperframe

103

Usage
as.hyperframe(x, ...)
## Default S3 method:
as.hyperframe(x, ...)
## S3 method for class 'data.frame'
as.hyperframe(x, ..., stringsAsFactors=FALSE)
## S3 method for class 'hyperframe'
as.hyperframe(x, ...)
## S3 method for class 'listof'
as.hyperframe(x, ...)
## S3 method for class 'anylist'
as.hyperframe(x, ...)
Arguments
x

Data in some other format.

...

Optional arguments passed to hyperframe.

stringsAsFactors
Logical. If TRUE, any column of the data frame x that contains character strings
will be converted to a factor. If FALSE, no such conversion will occur.
Details
A hyperframe is like a data frame, except that its entries can be objects of any kind.
The generic function as.hyperframe converts any suitable kind of data into a hyperframe.
There are methods for the classes data.frame, listof, anylist and a default method, all of which
convert data that is like a hyperframe into a hyperframe object. (The method for the class listof
and anylist converts a list of objects, of arbitrary type, into a hyperframe with one column.) These
methods do not discard any information.
There are also methods for other classes (see as.hyperframe.ppx) which extract the coordinates
from a spatial dataset. These methods do discard some information.
Value
An object of class "hyperframe" created by hyperframe.
Conversion of Strings to Factors
Note that as.hyperframe.default will convert a character vector to a factor. It behaves like
as.data.frame.
However as.hyperframe.data.frame does not convert strings to factors; it respects the structure
of the data frame x.
The behaviour can be changed using the argument stringsAsFactors.

104

as.hyperframe.ppx

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
hyperframe, as.hyperframe.ppx
Examples
df <- data.frame(x=runif(4),y=letters[1:4])
as.hyperframe(df)
sims <- list()
for(i in 1:3) sims[[i]] <- rpoispp(42)
as.hyperframe(as.listof(sims))
as.hyperframe(as.solist(sims))

as.hyperframe.ppx

Extract coordinates and marks of multidimensional point pattern

Description
Given any kind of spatial or space-time point pattern, extract the coordinates and marks of the
points.
Usage
## S3 method for class 'ppx'
as.hyperframe(x, ...)
## S3 method for class 'ppx'
as.data.frame(x, ...)
## S3 method for class 'ppx'
as.matrix(x, ...)
Arguments
x

A general multidimensional space-time point pattern (object of class "ppx").

...

Ignored.

Details
An object of class "ppx" (see ppx) represents a marked point pattern in multidimensional space
and/or time. There may be any number of spatial coordinates, any number of temporal coordinates,
and any number of mark variables. The individual marks may be atomic (numeric values, factor
values, etc) or objects of any kind.
The function as.hyperframe.ppx extracts the coordinates and the marks as a "hyperframe" (see
hyperframe) with one row of data for each point in the pattern. This is a method for the generic
function as.hyperframe.

as.im

105

The function as.data.frame.ppx discards those mark variables which are not atomic values, and
extracts the coordinates and the remaining marks as a data.frame with one row of data for each
point in the pattern. This is a method for the generic function as.data.frame.
Finally as.matrix(x) is equivalent to as.matrix(as.data.frame(x)) for an object of class
"ppx". Be warned that, if there are any columns of non-numeric data (i.e. if there are mark variables
that are factors), the result will be a matrix of character values.
Value
A hyperframe, data.frame or matrix as appropriate.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
ppx, hyperframe, as.hyperframe.
Examples
df <- data.frame(x=runif(4),y=runif(4),t=runif(4))
X <- ppx(data=df, coord.type=c("s","s","t"))
as.data.frame(X)
val <- runif(4)
E <- lapply(val, function(s) { rpoispp(s) })
hf <- hyperframe(t=val, e=as.listof(E))
Z <- ppx(data=hf, domain=c(0,1))
as.hyperframe(Z)
as.data.frame(Z)

as.im

Convert to Pixel Image

Description
Converts various kinds of data to a pixel image
Usage
as.im(X, ...)
## S3 method for class 'im'
as.im(X, W=NULL, ...,
eps=NULL, dimyx=NULL, xy=NULL,
na.replace=NULL)
## S3 method for class 'owin'
as.im(X, W=NULL, ...,
eps=NULL, dimyx=NULL, xy=NULL,
na.replace=NULL, value=1)

106

as.im

## S3 method for class 'matrix'
as.im(X, W=NULL, ...)
## S3 method for class 'tess'
as.im(X, W=NULL, ...,
eps=NULL, dimyx=NULL, xy=NULL,
na.replace=NULL)
## S3 method for class 'function'
as.im(X, W=NULL, ...,
eps=NULL, dimyx=NULL, xy=NULL,
na.replace=NULL, strict=FALSE)
## S3 method for class 'funxy'
as.im(X, W=Window(X), ...)
## S3 method for class 'distfun'
as.im(X, W=NULL, ...,
eps=NULL, dimyx=NULL, xy=NULL,
na.replace=NULL, approx=TRUE)
## S3 method for class 'nnfun'
as.im(X, W=NULL, ...,
eps=NULL, dimyx=NULL, xy=NULL,
na.replace=NULL)
## S3 method for class 'Smoothfun'
as.im(X, W=NULL, ...)
## S3 method for class 'leverage.ppm'
as.im(X, ..., what=c("smooth", "nearest"))
## S3 method for class 'data.frame'
as.im(X, ..., step, fatal=TRUE, drop=TRUE)
## Default S3 method:
as.im(X, W=NULL, ...,
eps=NULL, dimyx=NULL, xy=NULL,
na.replace=NULL)
Arguments
X

Data to be converted to a pixel image.

W

Window object which determines the spatial domain and pixel array geometry.

...

Additional arguments passed to X when X is a function.

eps,dimyx,xy

Optional parameters passed to as.mask which determine the pixel array geometry. See as.mask.

na.replace

Optional value to replace NA entries in the output image.

value

Optional. The value to be assigned to pixels inside the window, if X is a window.

as.im

107

strict

Logical value indicating whether to match formal arguments of X when X is a
function. If strict=FALSE (the default), all the ... arguments are passed to X.
If strict=TRUE, only named arguments are passed, and only if they match the
names of formal arguments of X.

step

Optional. A single number, or numeric vector of length 2, giving the grid step
lengths in the x and y directions.

fatal

Logical value indicating what to do if the resulting image would be too large for
available memory. If fatal=TRUE (the default), an error occurs. If fatal=FALSE,
a warning is issued and NULL is returned.

drop

Logical value indicating what to do when X is a data frame with 3 columns. If
drop=TRUE (the default), the result is a pixel image. If drop=FALSE, the result is
a list containing one image.

approx

Logical value indicating whether to compute an approximate result at faster
speed, by using distmap, when X is a distance function.

what

Character string (partially matched) specifying which image data should be extracted. See plot.leverage.ppm for explanation.

Details
This function converts the data X into a pixel image object of class "im" (see im.object). The
function as.im is generic, with methods for the classes listed above.
Currently X may be any of the following:
• a pixel image object, of class "im".
• a window object, of class "owin" (see owin.object). The result is an image with all pixel
entries equal to value inside the window X, and NA outside.
• a matrix.
• a tessellation (object of class "tess"). The result is a factor-valued image, with one factor
level corresponding to each tile of the tessellation. Pixels are classified according to the tile of
the tessellation into which they fall.
• a single number (or a single logical, complex, factor or character value). The result is an image
with all pixel entries equal to this constant value inside the window W (and NA outside, unless
the argument na.replace is given). Argument W is required.
• a function of the form function(x, y, ...) which is to be evaluated to yield the image pixel
values. In this case, the additional argument W must be present. This window will be converted
to a binary image mask. Then the function X will be evaluated in the form X(x, y, ...)
where x and y are vectors containing the x and y coordinates of all the pixels in the image
mask, and ... are any extra arguments given. This function must return a vector or factor of
the same length as the input vectors, giving the pixel values.
• an object of class "funxy" representing a function(x,y,...)
• an object of class "distfun" representing a distance function (created by the command distfun).
• an object of class "nnfun" representing a nearest neighbour function (created by the command
nnfun).
• a list with entries x, y, z in the format expected by the standard R functions image.default
and contour.default. That is, z is a matrix of pixel values, x and y are vectors of x and y
coordinates respectively, and z[i,j] is the pixel value for the location (x[i],y[j]).
• a point pattern (object of class "ppp"). See the separate documentation for as.im.ppp.

108

as.im
• A data frame with at least three columns. Columns named x, y and z, if present, will be
assumed to contain the spatial coordinates and the pixel values, respectively. Otherwise the x
and y coordinates will be taken from the first two columns of the data frame, and any remaining
columns will be interpreted as pixel values.
The spatial domain (enclosing rectangle) of the pixel image is determined by the argument W. If W is
absent, the spatial domain is determined by X. When X is a function, a matrix, or a single numerical
value, W is required.
The pixel array dimensions of the final resulting image are determined by (in priority order)
• the argument eps, dimyx or xy if present;
• the pixel dimensions of the window W, if it is present and if it is a binary mask;
• the pixel dimensions of X if it is an image, a binary mask, or a list(x,y,z);
• the default pixel dimensions, controlled by spatstat.options.
Note that if eps, dimyx or xy is given, this will override the pixel dimensions of X if it has them.
Thus, as.im can be used to change an image’s pixel dimensions.
If the argument na.replace is given, then all NA entries in the image will be replaced by this value.
The resulting image is then defined everwhere on the full rectangular domain, instead of a smaller
window. Here na.replace should be a single value, of the same type as the other entries in the
image.
If X is a pixel image that was created by an older version of spatstat, the command X <- as.im(X)
will repair the internal format of X so that it conforms to the current version of spatstat.
If X is a data frame with m columns, then m-2 columns of data are interpreted as pixel values, yielding
m-2 pixel images. The result of as.im.data.frame is a list of pixel images, belonging to the class
"imlist". If m = 3 and drop=TRUE (the default), then the result is a pixel image rather than a list
containing this image.

Value
A pixel image (object of class "im"), or a list of pixel images, or NULL if the conversion failed.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
Separate documentation for as.im.ppp
Examples
data(demopat)
# window object
W <- Window(demopat)
plot(W)
Z <- as.im(W)
image(Z)
# function
Z <- as.im(function(x,y) {x^2 + y^2}, unit.square())
image(Z)
# function with extra arguments

as.interact

109

f <- function(x, y, x0, y0) {
sqrt((x - x0)^2 + (y-y0)^2)
}
Z <- as.im(f, unit.square(), x0=0.5, y0=0.5)
image(Z)
# Revisit the Sixties
data(letterR)
Z <- as.im(f, letterR, x0=2.5, y0=2)
image(Z)
# usual convention in S
stuff <- list(x=1:10, y=1:10, z=matrix(1:100, nrow=10))
Z <- as.im(stuff)
# convert to finer grid
Z <- as.im(Z, dimyx=256)
# pixellate the Dirichlet tessellation
Di <- dirichlet(runifpoint(10))
plot(as.im(Di))
plot(Di, add=TRUE)
# as.im.data.frame is the reverse of as.data.frame.im
grad <- bei.extra$grad
slopedata <- as.data.frame(grad)
slope <- as.im(slopedata)
unitname(slope) <- c("metre","metres")
all.equal(slope, grad) # TRUE

as.interact

Extract Interaction Structure

Description
Extracts the interpoint interaction structure from a point pattern model.
Usage
as.interact(object)
## S3 method for class 'fii'
as.interact(object)
## S3 method for class 'interact'
as.interact(object)
## S3 method for class 'ppm'
as.interact(object)
Arguments
object

A fitted point process model (object of class "ppm") or an interpoint interaction
structure (object of class "interact").

Details
The function as.interact extracts the interpoint interaction structure from a suitable object.
An object of class "interact" describes an interpoint interaction structure, before it has been fitted
to point pattern data. The irregular parameters of the interaction (such as the interaction range) are

110

as.layered
fixed, but the regular parameters (such as interaction strength) are undetermined. Objects of this
class are created by the functions Poisson, Strauss and so on. The main use of such objects is in
a call to ppm.
The function as.interact is generic, with methods for the classes "ppm", "fii" and "interact".
The result is an object of class "interact" which can be printed.

Value
An object of class "interact" representing the interpoint interaction. This object can be printed
and plotted.
Note on parameters
This function does not extract the fitted coefficients of the interaction. To extract the fitted interaction including the fitted coefficients, use fitin.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
fitin, ppm.
Examples
data(cells)
model <- ppm(cells, ~1, Strauss(0.07))
f <- as.interact(model)
f

as.layered

Convert Data To Layered Object

Description
Converts spatial data into a layered object.
Usage
as.layered(X)
## Default S3 method:
as.layered(X)
## S3 method for class 'ppp'
as.layered(X)
## S3 method for class 'splitppp'
as.layered(X)

as.layered

111

## S3 method for class 'solist'
as.layered(X)
## S3 method for class 'listof'
as.layered(X)
## S3 method for class 'msr'
as.layered(X)
Arguments
X

Some kind of spatial data.

Details
This function converts the object X into an object of class "layered".
The argument X should contain some kind of spatial data such as a point pattern, window, or pixel
image.
If X is a simple object then it will be converted into a layered object containing only one layer
which is equivalent to X.
If X can be interpreted as consisting of multiple layers of data, then the result will be a layered
object consisting of these separate layers of data.
• if X is a list of class "listof" or "solist", then as.layered(X) consists of several layers,
one for each entry in the list X;
• if X is a multitype point pattern, then as.layered(X) consists of several layers, each containing the sub-pattern consisting of points of one type;
• if X is a vector-valued measure, then as.layered(X) consists of several layers, each containing a scalar-valued measure.
Value
An object of class "layered" (see layered).
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
layered, split.ppp
Examples
as.layered(cells)
as.layered(amacrine)
P <- rpoispp(100)
fit <- ppm(P ~ x+y)
rs <- residuals(fit, type="score")
as.layered(rs)

112

as.linfun

as.linfun

Convert Data to a Function on a Linear Network

Description
Convert some kind of data to an object of class "linfun" representing a function on a linear network.
Usage
as.linfun(X, ...)
## S3 method for class 'linim'
as.linfun(X, ...)
## S3 method for class 'lintess'
as.linfun(X, ..., values, navalue=NA)
Arguments
X

Some kind of data to be converted.

...

Other arguments passed to methods.

values

Optional. Vector of function values, one entry associated with each tile of the
tessellation.

navalue

Optional. Function value associated with locations that do not belong to a tile
of the tessellation.

Details
An object of class "linfun" represents a function defined on a linear network.
The function as.linfun is generic. The method as.linfun.linim converts objects of class "linim"
(pixel images on a linear network) to functions on the network.
The method as.linfun.lintess converts a tessellation on a linear network into a function with a
different value on each tile of the tessellation. If the argument values is missing or null, then the
function returns factor values identifying which tile contains each given point. If values is given,
it should be a vector with one entry for each tile of the tessellation: any point lying in tile number i
will return the value v[i].
Value
Object of class "linfun".
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
linfun

as.linim

113

Examples
X <- runiflpp(2, simplenet)
Y <- runiflpp(5, simplenet)
# image on network
D <- density(Y, 0.1, verbose=FALSE)
f <- as.linfun(D)
f
f(X)
# tessellation on network
Z <- lineardirichlet(Y)
g <- as.linfun(Z)
g(X)
h <- as.linfun(Z, values = runif(5))
h(X)

as.linim

Convert to Pixel Image on Linear Network

Description
Converts various kinds of data to a pixel image on a linear network.
Usage
as.linim(X, ...)
## S3 method for class 'linim'
as.linim(X, ...)
## Default S3 method:
as.linim(X, L, ...,

eps = NULL, dimyx = NULL, xy = NULL,
delta=NULL)

## S3 method for class 'linfun'
as.linim(X, L=domain(X), ...,
eps = NULL, dimyx = NULL, xy = NULL,
delta=NULL)
Arguments
X

Data to be converted to a pixel image on a linear network.

L

Linear network (object of class "linnet").

...

Additional arguments passed to X when X is a function.

eps,dimyx,xy

Optional arguments passed to as.mask to control the pixel resolution.

delta

Optional. Numeric value giving the approximate distance (in coordinate units)
between successive sample points along each segment of the network.

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as.linnet.linim

Details
This function converts the data X into a pixel image on a linear network, an object of class "linim"
(see linim).
The argument X may be any of the following:
• a function on a linear network, an object of class "linfun".
• a pixel image on a linear network, an object of class "linim".
• a pixel image, an object of class "im".
• any type of data acceptable to as.im, such as a function, numeric value, or window.
First X is converted to a pixel image object Y (object of class "im"). The conversion is performed
by as.im. The arguments eps, dimyx and xy determine the pixel resolution.
Next Y is converted to a pixel image on a linear network using linim. The argument L determines
the linear network. If L is missing or NULL, then X should be an object of class "linim", and L
defaults to the linear network on which X is defined.
In addition to converting the function to a pixel image, the algorithm also generates a fine grid
of sample points evenly spaced along each segment of the network (with spacing at most delta
coordinate units). The function values at these sample points are stored in the resulting object
as a data frame (the argument df of linim). This mechanism allows greater accuracy for some
calculations (such as integral.linim).
Value
An image object on a linear network; an object of class "linim".
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
as.im
Examples
f <- function(x,y){ x + y }
plot(as.linim(f, simplenet))

as.linnet.linim

Extract Linear Network from Data on a Linear Network

Description
Given some kind of data on a linear network, the command as.linnet extracts the linear network
itself.

as.linnet.linim

115

Usage
## S3 method for class 'linim'
as.linnet(X, ...)
## S3 method for class 'linfun'
as.linnet(X, ...)
## S3 method for class 'lintess'
as.linnet(X, ...)
## S3 method for class 'lpp'
as.linnet(X, ..., fatal=TRUE, sparse)
Arguments
X

Data on a linear network. A point pattern (class "lpp"), pixel image (class
"linim"), function (class "linfun") or tessellation (class "lintess") on a linear network.

...

Ignored.

fatal

Logical value indicating whether data in the wrong format should lead to an
error (fatal=TRUE) or a warning (fatal=FALSE).

sparse

Logical value indicating whether to use a sparse matrix representation, as explained in linnet. Default is to keep the same representation as in X.

Details
These are methods for the generic as.linnet for various classes.
The network on which the data are defined is extracted.
Value
A linear network (object of class "linnet").
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
linnet, methods.linnet.
Examples
# make some data
xcoord <- linfun(function(x,y,seg,tp) { x }, simplenet)
as.linnet(xcoord)
X <- as.linim(xcoord)
as.linnet(X)

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as.linnet.psp

as.linnet.psp

Convert Line Segment Pattern to Linear Network

Description
Converts a line segment pattern to a linear network.
Usage
## S3 method for class 'psp'
as.linnet(X, ..., eps, sparse=FALSE)
Arguments
X

Line segment pattern (object of class "psp").

...

Ignored.

eps

Optional. Distance threshold. If two segment endpoints are closer than eps units
apart, they will be treated as the same point, and will become a single vertex in
the linear network.

sparse

Logical value indicating whether to use a sparse matrix representation, as explained in linnet.

Details
This command converts any collection of line segments into a linear network by guessing the connectivity of the network, using the distance threshold eps.
If any segments in X cross over each other, they are first cut into pieces using selfcut.psp.
Then any pair of segment endpoints lying closer than eps units apart, is treated as a single vertex.
The linear network is then constructed using linnet.
It would be wise to check the result by plotting the degree of each vertex, as shown in the Examples.
If X has marks, then these are stored in the resulting linear network Y <- as.linnet(X), and can
be extracted as marks(as.psp(Y)) or marks(Y$lines).
Value
A linear network (object of class "linnet").
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
linnet, selfcut.psp, methods.linnet.

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117

Examples
# make some data
A <- psp(0.09, 0.55, 0.79, 0.80, window=owin())
B <- superimpose(A, as.psp(simplenet))
# convert to a linear network
D <- as.linnet(B)
# check validity
D
plot(D)
text(vertices(D), labels=vertexdegree(D))

as.lpp

Convert Data to a Point Pattern on a Linear Network

Description
Convert various kinds of data to a point pattern on a linear network.
Usage
as.lpp(x=NULL, y=NULL, seg=NULL, tp=NULL, ...,
marks=NULL, L=NULL, check=FALSE, sparse)
Arguments
x,y

Vectors of cartesian coordinates, or any data acceptable to xy.coords. Alternatively x can be a point pattern on a linear network (object of class "lpp") or a
planar point pattern (object of class "ppp").

seg,tp

Optional local coordinates. Vectors of the same length as x,y. See Details.

...

Ignored.

marks

Optional marks for the point pattern. A vector or factor with one entry for each
point, or a data frame or hyperframe with one row for each point.

L

Linear network (object of class "linnet") on which the points lie.

check

Logical. Whether to check the validity of the spatial coordinates.

sparse

Optional logical value indicating whether to store the linear network data in a
sparse matrix representation or not. See linnet.

Details
This function converts data in various formats into a point pattern on a linear network (object of
class "lpp").
The possible formats are:
• x is already a point pattern on a linear network (object of class "lpp"). Then x is returned
unchanged.
• x is a planar point pattern (object of class "ppp"). Then x is converted to a point pattern on
the linear network L using lpp.

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as.mask
• x,y,seg,tp are vectors of equal length. These specify that the ith point has Cartesian coordinates (x[i],y[i]), and lies on segment number seg[i] of the network L, at a fractional
position tp[i] along that segment (with tp=0 representing one endpoint and tp=1 the other
endpoint of the segment).
• x,y are missing and seg,tp are vectors of equal length as described above.
• seg,tp are NULL, and x,y are data in a format acceptable to xy.coords specifying the Cartesian coordinates.

Value
A point pattern on a linear network (object of class "lpp").
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
lpp.
Examples
A <- as.psp(simplenet)
X <- runifpointOnLines(10, A)
is.ppp(X)
Y <- as.lpp(X, L=simplenet)

as.mask

Pixel Image Approximation of a Window

Description
Obtain a discrete (pixel image) approximation of a given window
Usage
as.mask(w, eps=NULL, dimyx=NULL, xy=NULL)
Arguments
w

A window (object of class "owin") or data acceptable to as.owin.

eps

(optional) width and height of pixels.

dimyx

(optional) pixel array dimensions

xy

(optional) data containing pixel coordinates

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119

Details
This function generates a rectangular grid of locations in the plane, tests whether each of these
locations lies inside the window w, and stores the results as a binary pixel image or ‘mask’ (an
object of class "owin", see owin.object).
The most common use of this function is to approximate the shape of another window w by a binary
pixel image. In this case, we will usually want to have a very fine grid of pixels.
This function can also be used to generate a coarsely-spaced grid of locations inside a window, for
purposes such as subsampling and prediction.
The grid spacing and location are controlled by the arguments eps, dimyx and xy, which are mutually incompatible.
If eps is given, then it determines the grid spacing. If eps is a single number, then the grid spacing
will be approximately eps in both the x and y directions. If eps is a vector of length 2, then the grid
spacing will be approximately eps[1] in the x direction and eps[2] in the y direction.
If dimyx is given, then the pixel grid will be an m × n rectangular grid where m, n are given by
dimyx[2], dimyx[1] respectively. Warning: dimyx[1] is the number of pixels in the y direction,
and dimyx[2] is the number in the x direction.
If xy is given, then this should be some kind of data specifing the coordinates of a pixel grid. It may
be
• a list or structure containing elements x and y which are numeric vectors of equal length.
These will be taken as x and y coordinates of the margins of the grid. The pixel coordinates
will be generated from these two vectors.
• a pixel image (object of class "im").
• a window (object of class "owin") which is of type "mask" so that it contains pixel coordinates.
If xy is given, w may be omitted.
If neither eps nor dimyx nor xy is given, the pixel raster dimensions are obtained from spatstat.options("npixel").
There is no inverse of this function. However, the function as.polygonal will compute a polygonal
approximation of a binary mask.
Value
A window (object of class "owin") of type "mask" representing a binary pixel image.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
owin.object, as.rectangle, as.polygonal, spatstat.options
Examples
w <- owin(c(0,10),c(0,10), poly=list(x=c(1,2,3,2,1), y=c(2,3,4,6,7)))
## Not run: plot(w)
m <- as.mask(w)
## Not run: plot(m)

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as.mask.psp
x <- 1:9
y <- seq(0.25, 9.75, by=0.5)
m <- as.mask(w, xy=list(x=x, y=y))

as.mask.psp

Convert Line Segment Pattern to Binary Pixel Mask

Description
Converts a line segment pattern to a binary pixel mask by determining which pixels intersect the
lines.
Usage
as.mask.psp(x, W=NULL, ...)
Arguments
x

Line segment pattern (object of class "psp").

W

Optional window (object of class "owin") determining the pixel raster.

...

Optional extra arguments passed to as.mask to determine the pixel resolution.

Details
This function converts a line segment pattern to a binary pixel mask by determining which pixels
intersect the lines.
The pixel raster is determined by W and the optional arguments .... If W is missing or NULL, it
defaults to the window containing x. Then W is converted to a binary pixel mask using as.mask.
The arguments ... are passed to as.mask to control the pixel resolution.
Value
A window (object of class "owin") which is a binary pixel mask (type "mask").
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
pixellate.psp, as.mask.
Use pixellate.psp if you want to measure the length of line in each pixel.
Examples
X <- psp(runif(10), runif(10), runif(10), runif(10), window=owin())
plot(as.mask.psp(X))
plot(X, add=TRUE, col="red")

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121

as.matrix.im

Convert Pixel Image to Matrix or Array

Description
Converts a pixel image to a matrix or an array.
Usage
## S3 method for class 'im'
as.matrix(x, ...)
## S3 method for class 'im'
as.array(x, ...)
Arguments
x

A pixel image (object of class "im").

...

See below.

Details
The function as.matrix.im converts the pixel image x into a matrix containing the pixel values. It
is handy when you want to extract a summary of the pixel values. See the Examples.
The function as.array.im converts the pixel image to an array. By default this is a three-dimensional
array of dimension n by m by 1. If the extra arguments ... are given, they will be passed to array,
and they may change the dimensions of the array.
Value
A matrix or array.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
as.matrix.owin
Examples
# artificial image
Z <- setcov(square(1))
M <- as.matrix(Z)
median(M)
## Not run:
# plot the cumulative distribution function of pixel values
plot(ecdf(as.matrix(Z)))

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as.matrix.owin

## End(Not run)

as.matrix.owin

Convert Pixel Image to Matrix

Description
Converts a pixel image to a matrix.
Usage
## S3 method for class 'owin'
as.matrix(x, ...)
Arguments
x

A window (object of class "owin").

...

Arguments passed to as.mask to control the pixel resolution.

Details
The function as.matrix.owin converts a window to a logical matrux.
It first converts the window x into a binary pixel mask using as.mask. It then extracts the pixel
entries as a logical matrix.
The resulting matrix has entries that are TRUE if the corresponding pixel is inside the window, and
FALSE if it is outside.
The function as.matrix is generic. The function as.matrix.owin is the method for windows
(objects of class "owin").
Use as.im to convert a window to a pixel image.
Value
A logical matrix.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
as.matrix.im, as.im
Examples
m <- as.matrix(letterR)

as.owin

as.owin

123

Convert Data To Class owin

Description
Converts data specifying an observation window in any of several formats, into an object of class
"owin".
Usage
as.owin(W, ..., fatal=TRUE)
## S3 method for class 'owin'
as.owin(W, ..., fatal=TRUE)
## S3 method for class 'ppp'
as.owin(W, ..., fatal=TRUE)
## S3 method for class 'ppm'
as.owin(W, ..., from=c("points", "covariates"), fatal=TRUE)
## S3 method for class 'kppm'
as.owin(W, ..., from=c("points", "covariates"), fatal=TRUE)
## S3 method for class 'dppm'
as.owin(W, ..., from=c("points", "covariates"), fatal=TRUE)
## S3 method for class 'lpp'
as.owin(W, ..., fatal=TRUE)
## S3 method for class 'lppm'
as.owin(W, ..., fatal=TRUE)
## S3 method for class 'msr'
as.owin(W, ..., fatal=TRUE)
## S3 method for class 'psp'
as.owin(W, ..., fatal=TRUE)
## S3 method for class 'quad'
as.owin(W, ..., fatal=TRUE)
## S3 method for class 'quadratcount'
as.owin(W, ..., fatal=TRUE)
## S3 method for class 'quadrattest'
as.owin(W, ..., fatal=TRUE)
## S3 method for class 'tess'
as.owin(W, ..., fatal=TRUE)

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as.owin
## S3 method for class 'im'
as.owin(W, ..., fatal=TRUE)
## S3 method for class 'layered'
as.owin(W, ..., fatal=TRUE)
## S3 method for class 'data.frame'
as.owin(W, ..., step, fatal=TRUE)
## S3 method for class 'distfun'
as.owin(W, ..., fatal=TRUE)
## S3 method for class 'nnfun'
as.owin(W, ..., fatal=TRUE)
## S3 method for class 'funxy'
as.owin(W, ..., fatal=TRUE)
## S3 method for class 'boxx'
as.owin(W, ..., fatal=TRUE)
## S3 method for class 'rmhmodel'
as.owin(W, ..., fatal=FALSE)
## S3 method for class 'leverage.ppm'
as.owin(W, ..., fatal=TRUE)
## S3 method for class 'influence.ppm'
as.owin(W, ..., fatal=TRUE)
## Default S3 method:
as.owin(W, ..., fatal=TRUE)

Arguments
W
fatal
...
from
step

Data specifying an observation window, in any of several formats described under Details below.
Logical flag determining what to do if the data cannot be converted to an observation window. See Details.
Ignored.
Character string. See Details.
Optional. A single number, or numeric vector of length 2, giving the grid step
lengths in the x and y directions.

Details
The class "owin" is a way of specifying the observation window for a point pattern. See owin.object
for an overview.
This function converts data in any of several formats into an object of class "owin" for use by the
spatstat package. The function as.owin is generic, with methods for different classes of objects,
and a default method.
The argument W may be

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125
• an object of class "owin"
• a structure with entries xrange, yrange specifying the x and y dimensions of a rectangle
• a four-element vector (interpreted as (xmin, xmax, ymin, ymax)) specifying the x and y
dimensions of a rectangle
• a structure with entries xl, xu, yl, yu specifying the x and y dimensions of a rectangle as
(xmin, xmax) = (xl, xu) and (ymin, ymax) = (yl, yu). This will accept objects of
class spp used in the Venables and Ripley spatial library.
• an object of class "ppp" representing a point pattern. In this case, the object’s window structure
will be extracted.
• an object of class "psp" representing a line segment pattern. In this case, the object’s window
structure will be extracted.
• an object of class "tess" representing a tessellation. In this case, the object’s window structure
will be extracted.
• an object of class "quad" representing a quadrature scheme. In this case, the window of the
data component will be extracted.
• an object of class "im" representing a pixel image. In this case, a window of type "mask" will
be returned, with the same pixel raster coordinates as the image. An image pixel value of NA,
signifying that the pixel lies outside the window, is transformed into the logical value FALSE,
which is the corresponding convention for window masks.
• an object of class "ppm", "kppm" or "dppm" representing a fitted point process model. In
this case, if from="data" (the default), as.owin extracts the original point pattern data to
which the model was fitted, and returns the observation window of this point pattern. If
from="covariates" then as.owin extracts the covariate images to which the model was
fitted, and returns a binary mask window that specifies the pixel locations.
• an object of class "lpp" representing a point pattern on a linear network. In this case, as.owin
extracts the linear network and returns a window containing this network.
• an object of class "lppm" representing a fitted point process model on a linear network. In this
case, as.owin extracts the linear network and returns a window containing this network.
• A data.frame with exactly three columns. Each row of the data frame corresponds to one
pixel. Each row contains the x and y coordinates of a pixel, and a logical value indicating
whether the pixel lies inside the window.
• A data.frame with exactly two columns. Each row of the data frame contains the x and y
coordinates of a pixel that lies inside the window.
• an object of class "distfun", "nnfun" or "funxy" representing a function of spatial location,
defined on a spatial domain. The spatial domain of the function will be extracted.
• an object of class "rmhmodel" representing a point process model that can be simulated using
rmh. The window (spatial domain) of the model will be extracted. The window may be NULL
in some circumstances (indicating that the simulation window has not yet been determined).
This is not treated as an error, because the argument fatal defaults to FALSE for this method.
• an object of class "layered" representing a list of spatial objects. See layered. In this case,
as.owin will be applied to each of the objects in the list, and the union of these windows will
be returned.

If the argument W is not in one of these formats and cannot be converted to a window, then an error
will be generated (if fatal=TRUE) or a value of NULL will be returned (if fatal=FALSE).
When W is a data frame, the argument step can be used to specify the pixel grid spacing; otherwise,
the spacing will be guessed from the data.

126

as.polygonal

Value
An object of class "owin" (see owin.object) specifying an observation window.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
owin.object, owin
Examples
w <- as.owin(c(0,1,0,1))
w <- as.owin(list(xrange=c(0,5),yrange=c(0,10)))
# point pattern
data(demopat)
w <- as.owin(demopat)
# image
Z <- as.im(function(x,y) { x + 3}, unit.square())
w <- as.owin(Z)
# Venables & Ripley 'spatial' package
require(spatial)
towns <- ppinit("towns.dat")
w <- as.owin(towns)
detach(package:spatial)

as.polygonal

Convert a Window to a Polygonal Window

Description
Given a window W of any geometric type (rectangular, polygonal or binary mask), this function
returns a polygonal window that represents the same spatial domain.
Usage
as.polygonal(W, repair=FALSE)
Arguments
W

A window (object of class "owin").

repair

Logical value indicating whether to check the validity of the polygon data and
repair it, if W is already a polygonal window.

as.ppm

127

Details
Given a window W of any geometric type (rectangular, polygonal or binary mask), this function
returns a polygonal window that represents the same spatial domain.
If W is a rectangle, it is converted to a polygon with 4 vertices.
If W is already polygonal, it is returned unchanged, by default. However if repair=TRUE then the
validity of the polygonal coordinates will be checked (for example to check the boundary is not
self-intersecting) and repaired if necessary, so that the result could be different from W.
If W is a binary mask, then each pixel in the mask is replaced by a small square or rectangle, and the
union of these squares or rectangles is computed. The result is a polygonal window that has only
horizontal and vertical edges. (Use simplify.owin to remove the staircase appearance, if desired).
Value
A polygonal window (object of class "owin" and of type "polygonal").
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
owin, as.owin, as.mask, simplify.owin
Examples
data(letterR)
m <- as.mask(letterR, dimyx=32)
p <- as.polygonal(m)
if(interactive()) {
plot(m)
plot(p, add=TRUE, lwd=2)
}

as.ppm

Extract Fitted Point Process Model

Description
Extracts the fitted point process model from some kind of fitted model.
Usage
as.ppm(object)
## S3 method for class 'ppm'
as.ppm(object)
## S3 method for class 'profilepl'
as.ppm(object)

128

as.ppm
## S3 method for class 'kppm'
as.ppm(object)
## S3 method for class 'dppm'
as.ppm(object)

Arguments
object

An object that includes a fitted Poisson or Gibbs point process model. An object
of class "ppm", "profilepl", "kppm" or "dppm" or possibly other classes.

Details
The function as.ppm extracts the fitted point process model (of class "ppm") from a suitable object.
The function as.ppm is generic, with methods for the classes "ppm", "profilepl", "kppm" and
"dppm", and possibly for other classes.
For the class "profilepl" of models fitted by maximum profile pseudolikelihood, the method
as.ppm.profilepl extracts the fitted point process model (with the optimal values of the irregular
parameters).
For the class "kppm" of models fitted by minimum contrast (or Palm or composite likelihood) using
Waagepetersen’s two-step estimation procedure (see kppm), the method as.ppm.kppm extracts the
Poisson point process model that is fitted in the first stage of the procedure.
The behaviour for the class "dppm" is analogous to the "kppm" case above.
Value
An object of class "ppm".
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
ppm, profilepl.
Examples
# fit a model by profile maximum pseudolikelihood
rvals <- data.frame(r=(1:10)/100)
pfit <- profilepl(rvals, Strauss, cells, ~1)
# extract the fitted model
fit <- as.ppm(pfit)

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129

as.ppp

Convert Data To Class ppp

Description
Tries to coerce any reasonable kind of data to a spatial point pattern (an object of class "ppp") for
use by the spatstat package).
Usage
as.ppp(X, ..., fatal=TRUE)
## S3 method for class 'ppp'
as.ppp(X, ..., fatal=TRUE)
## S3 method for class 'psp'
as.ppp(X, ..., fatal=TRUE)
## S3 method for class 'quad'
as.ppp(X, ..., fatal=TRUE)
## S3 method for class 'matrix'
as.ppp(X, W=NULL, ..., fatal=TRUE)
## S3 method for class 'data.frame'
as.ppp(X, W=NULL, ..., fatal=TRUE)
## S3 method for class 'influence.ppm'
as.ppp(X, ...)
## Default S3 method:
as.ppp(X, W=NULL, ..., fatal=TRUE)
Arguments
X

Data which will be converted into a point pattern

W

Data which define a window for the pattern, when X does not contain a window.
(Ignored if X contains window information.)

...

Ignored.

fatal

Logical value specifying what to do if the data cannot be converted. See Details.

Details
Converts the dataset X to a point pattern (an object of class "ppp"; see ppp.object for an overview).
This function is normally used to convert an existing point pattern dataset, stored in another format,
to the "ppp" format. To create a new point pattern from raw data such as x, y coordinates, it is
normally easier to use the creator function ppp.
The function as.ppp is generic, with methods for the classes "ppp", "psp", "quad", "matrix",
"data.frame" and a default method.
The dataset X may be:

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as.ppp
• an object of class "ppp"
• an object of class "psp"
• a point pattern object created by the spatial library
• an object of class "quad" representing a quadrature scheme (see quad.object)
• a matrix or data frame with at least two columns
• a structure with entries x, y which are numeric vectors of equal length
• a numeric vector of length 2, interpreted as the coordinates of a single point.
In the last three cases, we need the second argument W which is converted to a window object by the
function as.owin. In the first four cases, W will be ignored.
If X is a line segment pattern (an object of class psp) the point pattern returned consists of the
endpoints of the segments. If X is marked then the point pattern returned will also be marked, the
mark associated with a point being the mark of the segment of which that point was an endpoint.
If X is a matrix or data frame, the first and second columns will be interpreted as the x and y
coordinates respectively. Any additional columns will be interpreted as marks.
The argument fatal indicates what to do when W is missing and X contains no information about the
window. If fatal=TRUE, a fatal error will be generated; if fatal=FALSE, the value NULL is returned.
In the spatial library, a point pattern is represented in either of the following formats:
• (in spatial versions 1 to 6) a structure with entries x, y xl, xu, yl, yu
• (in spatial version 7) a structure with entries x, y and area, where area is a structure with
entries xl, xu, yl, yu
where x and y are vectors of equal length giving the point coordinates, and xl, xu, yl, yu are
numbers giving the dimensions of a rectangular window.
Point pattern datasets can also be created by the function ppp.

Value
An object of class "ppp" (see ppp.object) describing the point pattern and its window of observation. The value NULL may also be returned; see Details.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
ppp, ppp.object, as.owin, owin.object
Examples
xy <- matrix(runif(40), ncol=2)
pp <- as.ppp(xy, c(0,1,0,1))
# Venables-Ripley format
# check for 'spatial' package
spatialpath <- system.file(package="spatial")
if(nchar(spatialpath) > 0) {
require(spatial)

as.psp

}

131
towns <- ppinit("towns.dat")
pp <- as.ppp(towns) # converted to our format
detach(package:spatial)

xyzt <- matrix(runif(40), ncol=4)
Z <- as.ppp(xyzt, square(1))

as.psp

Convert Data To Class psp

Description
Tries to coerce any reasonable kind of data object to a line segment pattern (an object of class
"psp") for use by the spatstat package.
Usage
as.psp(x, ..., from=NULL, to=NULL)
## S3 method for class 'psp'
as.psp(x, ..., check=FALSE, fatal=TRUE)
## S3 method for class 'data.frame'
as.psp(x, ..., window=NULL, marks=NULL,
check=spatstat.options("checksegments"), fatal=TRUE)
## S3 method for class 'matrix'
as.psp(x, ..., window=NULL, marks=NULL,
check=spatstat.options("checksegments"), fatal=TRUE)
## Default S3 method:
as.psp(x, ..., window=NULL, marks=NULL,
check=spatstat.options("checksegments"), fatal=TRUE)
Arguments
x

Data which will be converted into a line segment pattern

window

Data which define a window for the pattern.

...

Ignored.

marks

(Optional) vector or data frame of marks for the pattern

check

Logical value indicating whether to check the validity of the data, e.g. to check
that the line segments lie inside the window.

fatal

Logical value. See Details.

from,to

Point patterns (object of class "ppp") containing the first and second endpoints
(respectively) of each segment. Incompatible with x.

132

as.psp

Details
Converts the dataset x to a line segment pattern (an object of class "psp"; see psp.object for an
overview).
This function is normally used to convert an existing line segment pattern dataset, stored in another
format, to the "psp" format. To create a new point pattern from raw data such as x, y coordinates,
it is normally easier to use the creator function psp.
The dataset x may be:
• an object of class "psp"
• a data frame with at least 4 columns
• a structure (list) with elements named x0, y0,
and possibly a fifth element named marks

x1, y1 or elements named xmid, ymid, length, angle

If x is a data frame the interpretation of its columns is as follows:
• If there are columns named x0, y0, x1, y1 then these will be interpreted as the coordinates
of the endpoints of the segments and used to form the ends component of the psp object to be
returned.
• If there are columns named xmid, ymid, length, angle then these will be interpreted as
the coordinates of the segment midpoints, the lengths of the segments, and the orientations of
the segments in radians and used to form the ends component of the psp object to be returned.
• If there is a column named marks then this will be interpreted as the marks of the pattern
provided that the argument marks of this function is NULL. If argument marks is not NULL then
the value of this argument is taken to be the marks of the pattern and the column named marks
is ignored (with a warning). In either case the column named marks is deleted and omitted
from further consideration.
• If there is no column named marks and if the marks argument of this function is NULL, and if
after interpreting 4 columns of x as determining the ends component of the psp object to be
returned, there remain other columns of x, then these remaining columns will be taken to form
a data frame of marks for the psp object to be returned.
If x is a structure (list) with elements named x0, y0, x1, y1, marks or xmid, ymid, length, angle, marks,
then the element named marks will be interpreted as the marks of the pattern provide that the argument marks of this function is NULL. If this argument is non-NULL then it is interpreted as the marks
of the pattern and the element marks of x is ignored — with a warning.
Alternatively, you may specify two point patterns from and to containing the first and second
endpoints of the line segments.
The argument window is converted to a window object by the function as.owin.
The argument fatal indicates what to do when the data cannot be converted to a line segment
pattern. If fatal=TRUE, a fatal error will be generated; if fatal=FALSE, the value NULL is returned.
The function as.psp is generic, with methods for the classes "psp", "data.frame", "matrix" and
a default method.
Point pattern datasets can also be created by the function psp.
Value
An object of class "psp" (see psp.object) describing the line segment pattern and its window of
observation. The value NULL may also be returned; see Details.

as.rectangle

133

Warnings
If only a proper subset of the names x0,y0,x1,y1 or xmid,ymid,length,angle appear amongst
the names of the columns of x where x is a data frame, then these special names are ignored.
For example if the names of the columns were xmid,ymid,length,degrees, then these columns
would be interpreted as if the represented x0,y0,x1,y1 in that order.
Whether it gets used or not, column named marks is always removed from x before any attempt to
form the ends component of the psp object that is returned.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
psp, psp.object, as.owin, owin.object.
See edges for extracting the edges of a polygonal window as a "psp" object.
Examples
mat <- matrix(runif(40), ncol=4)
mx <- data.frame(v1=sample(1:4,10,TRUE),
v2=factor(sample(letters[1:4],10,TRUE),levels=letters[1:4]))
a <- as.psp(mat, window=owin(),marks=mx)
mat <- cbind(as.data.frame(mat),mx)
b <- as.psp(mat, window=owin()) # a and b are identical.
stuff <- list(xmid=runif(10),
ymid=runif(10),
length=rep(0.1, 10),
angle=runif(10, 0, 2 * pi))
a <- as.psp(stuff, window=owin())
b <- as.psp(from=runifpoint(10), to=runifpoint(10))

as.rectangle

Window Frame

Description
Extract the window frame of a window or other spatial dataset
Usage
as.rectangle(w, ...)
Arguments
w

A window, or a dataset that has a window. Either a window (object of class
"owin"), a pixel image (object of class "im") or other data determining such a
window.

...

Optional. Auxiliary data to help determine the window. If w does not belong to
a recognised class, the arguments w and ... are passed to as.owin to determine
the window.

134

as.solist

Details
This function is the quickest way to determine a bounding rectangle for a spatial dataset.
If w is a window, the function just extracts the outer bounding rectangle of w as given by its elements
xrange,yrange.
The function can also be applied to any spatial dataset that has a window: for example, a point
pattern (object of class "ppp") or a line segment pattern (object of class "psp"). The bounding
rectangle of the window of the dataset is extracted.
Use the function boundingbox to compute the smallest bounding rectangle of a dataset.
Value
A window (object of class "owin") of type "rectangle" representing a rectangle.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
owin, as.owin, boundingbox
Examples
w <- owin(c(0,10),c(0,10), poly=list(x=c(1,2,3,2,1), y=c(2,3,4,6,7)))
r <- as.rectangle(w)
# returns a 10 x 10 rectangle
data(lansing)
as.rectangle(lansing)
data(copper)
as.rectangle(copper$SouthLines)

as.solist

Convert List of Two-Dimensional Spatial Objects

Description
Given a list of two-dimensional spatial objects, convert it to the class "solist".
Usage
as.solist(x, ...)
Arguments
x

A list of objects, each representing a two-dimensional spatial dataset.

...

Additional arguments passed to solist.

as.tess

135

Details
This command makes the list x into an object of class "solist" (spatial object list). See solist
for details.
The entries in the list x should be two-dimensional spatial datasets (not necessarily of the same
class).
Value
A list, usually of class "solist".
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
solist, as.anylist, solapply.
Examples
x <- list(cells, density(cells))
y <- as.solist(x)

as.tess

Convert Data To Tessellation

Description
Converts data specifying a tessellation, in any of several formats, into an object of class "tess".
Usage
as.tess(X)
## S3 method
as.tess(X)
## S3 method
as.tess(X)
## S3 method
as.tess(X)
## S3 method
as.tess(X)
## S3 method
as.tess(X)
## S3 method
as.tess(X)

for class 'tess'
for class 'im'
for class 'owin'
for class 'quadratcount'
for class 'quadrattest'
for class 'list'

Arguments
X

Data to be converted to a tessellation.

136

as.tess

Details
A tessellation is a collection of disjoint spatial regions (called tiles) that fit together to form a larger
spatial region. This command creates an object of class "tess" that represents a tessellation.
This function converts data in any of several formats into an object of class "tess" for use by the
spatstat package. The argument X may be
• an object of class "tess". The object will be stripped of any extraneous attributes and returned.
• a pixel image (object of class "im") with pixel values that are logical or factor values. Each
level of the factor will determine a tile of the tessellation.
• a window (object of class "owin"). The result will be a tessellation consisting of a single tile.
• a set of quadrat counts (object of class "quadratcount") returned by the command quadratcount.
The quadrats used to generate the counts will be extracted and returned as a tessellation.
• a quadrat test (object of class "quadrattest") returned by the command quadrat.test. The
quadrats used to perform the test will be extracted and returned as a tessellation.
• a list of windows (objects of class "owin") giving the tiles of the tessellation.
The function as.tess is generic, with methods for various classes, as listed above.
Value
An object of class "tess" specifying a tessellation.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
tess
Examples
# pixel image
v <- as.im(function(x,y){factor(round(5 * (x^2 + y^2)))}, W=owin())
levels(v) <- letters[seq(length(levels(v)))]
as.tess(v)
# quadrat counts
data(nztrees)
qNZ <- quadratcount(nztrees, nx=4, ny=3)
as.tess(qNZ)

auc

137

auc

Area Under ROC Curve

Description
Compute the AUC (area under the Receiver Operating Characteristic curve) for a fitted point process
model.
Usage
auc(X, ...)
## S3 method for class 'ppp'
auc(X, covariate, ..., high = TRUE)
## S3 method for class 'ppm'
auc(X, ...)
## S3 method for class 'kppm'
auc(X, ...)
## S3 method for class 'lpp'
auc(X, covariate, ..., high = TRUE)
## S3 method for class 'lppm'
auc(X, ...)
Arguments
X

Point pattern (object of class "ppp" or "lpp") or fitted point process model
(object of class "ppm" or "kppm" or "lppm").

covariate

Spatial covariate. Either a function(x,y), a pixel image (object of class "im"),
or one of the strings "x" or "y" indicating the Cartesian coordinates.

...

Arguments passed to as.mask controlling the pixel resolution for calculations.

high

Logical value indicating whether the threshold operation should favour high or
low values of the covariate.

Details
This command computes the AUC, the area under the Receiver Operating Characteristic curve. The
ROC itself is computed by roc.
For a point pattern X and a covariate Z, the AUC is a numerical index that measures the ability of
the covariate to separate the spatial domain into areas of high and low density of points. Let xi be
a randomly-chosen data point from X and U a randomly-selected location in the study region. The
AUC is the probability that Z(xi ) > Z(U ) assuming high=TRUE. That is, AUC is the probability
that a randomly-selected data point has a higher value of the covariate Z than does a randomlyselected spatial location. The AUC is a number between 0 and 1. A value of 0.5 indicates a
complete lack of discriminatory power.
For a fitted point process model X, the AUC measures the ability of the fitted model intensity to
separate the spatial domain into areas of high and low density of points. Suppose λ(u) is the

138

BadGey
intensity function of the model. The AUC is the probability that λ(xi ) > λ(U ). That is, AUC is the
probability that a randomly-selected data point has higher predicted intensity than does a randomlyselected spatial location. The AUC is not a measure of the goodness-of-fit of the model (Lobo et
al, 2007).

Value
Numeric. For auc.ppp and auc.lpp, the result is a single number giving the AUC value. For
auc.ppm, auc.kppm and auc.lppm, the result is a numeric vector of length 2 giving the AUC value
and the theoretically expected AUC value for this model.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
References
Lobo, J.M., Jiménez-Valverde, A. and Real, R. (2007) AUC: a misleading measure of the performance of predictive distribution models. Global Ecology and Biogeography 17(2) 145–151.
Nam, B.-H. and D’Agostino, R. (2002) Discrimination index, the area under the ROC curve. Pages
267–279 in Huber-Carol, C., Balakrishnan, N., Nikulin, M.S. and Mesbah, M., Goodness-of-fit tests
and model validity, Birkhäuser, Basel.
See Also
roc
Examples
fit <- ppm(swedishpines ~ x+y)
auc(fit)
auc(swedishpines, "x")

BadGey

Hybrid Geyer Point Process Model

Description
Creates an instance of the Baddeley-Geyer point process model, defined as a hybrid of several Geyer
interactions. The model can then be fitted to point pattern data.
Usage
BadGey(r, sat)
Arguments
r

vector of interaction radii

sat

vector of saturation parameters, or a single common value of saturation parameter

BadGey

139

Details
This is Baddeley’s generalisation of the Geyer saturation point process model, described in Geyer,
to a process with multiple interaction distances.
The BadGey point process with interaction radii r1 , . . . , rk , saturation thresholds s1 , . . . , sk , intensity parameter β and interaction parameters γ1 , . . . , gammak , is the point process in which each
point xi in the pattern X contributes a factor
v (xi ,X)

βγ1 1

v (xi ,X)

. . . gammakk

to the probability density of the point pattern, where
vj (xi , X) = min(sj , tj (xi , X))
where tj (xi , X) denotes the number of points in the pattern X which lie within a distance rj from
the point xi .
BadGey is used to fit this model to data. The function ppm(), which fits point process models to
point pattern data, requires an argument of class "interact" describing the interpoint interaction
structure of the model to be fitted. The appropriate description of the piecewise constant Saturated
pairwise interaction is yielded by the function BadGey(). See the examples below.
The argument r specifies the vector of interaction distances. The entries of r must be strictly
increasing, positive numbers.
The argument sat specifies the vector of saturation parameters that are applied to the point counts
tj (xi , X). It should be a vector of the same length as r, and its entries should be nonnegative
numbers. Thus sat[1] is applied to the count of points within a distance r[1], and sat[2] to the
count of points within a distance r[2], etc. Alternatively sat may be a single number, and this
saturation value will be applied to every count.
Infinite values of the saturation parameters are also permitted; in this case vj (xi , X) = tj (xi , X)
and there is effectively no ‘saturation’ for the distance range in question. If all the saturation parameters are set to Inf then the model is effectively a pairwise interaction process, equivalent to
PairPiece (however the interaction parameters γ obtained from BadGey have a complicated relationship to the interaction parameters γ obtained from PairPiece).
If r is a single number, this model is virtually equivalent to the Geyer process, see Geyer.
Value
An object of class "interact" describing the interpoint interaction structure of a point process.
Author(s)
Adrian Baddeley  and Rolf Turner 
in collaboration with Hao Wang and Jeff Picka
See Also
ppm, pairsat.family, Geyer, PairPiece, SatPiece
Examples
BadGey(c(0.1,0.2), c(1,1))
# prints a sensible description of itself
BadGey(c(0.1,0.2), 1)
data(cells)

140

bc.ppm

# fit a stationary Baddeley-Geyer model
ppm(cells, ~1, BadGey(c(0.07, 0.1, 0.13), 2))
# nonstationary process with log-cubic polynomial trend
## Not run:
ppm(cells, ~polynom(x,y,3), BadGey(c(0.07, 0.1, 0.13), 2))
## End(Not run)

bc.ppm

Bias Correction for Fitted Model

Description
Applies a first-order bias correction to a fitted model.
Usage
bc(fit, ...)
## S3 method for class 'ppm'
bc(fit, ..., nfine = 256)
Arguments
fit

A fitted point process model (object of class "ppm") or a model of some other
class.

...

Additional arguments are currently ignored.

nfine

Grid dimensions for fine grid of locations. An integer, or a pair of integers. See
Details.

Details
This command applies the first order Newton-Raphson bias correction method of Baddeley and
Turner (2014, sec 4.2) to a fitted model. The function bc is generic, with a method for fitted point
process models of class "ppm".
A fine grid of locations, of dimensions nfine * nfine or nfine[2] * nfine[1], is created
over the original window of the data, and the intensity or conditional intensity of the fitted model is
calculated on this grid. The result is used to update the fitted model parameters once by a NewtonRaphson update.
This is only useful if the quadrature points used to fit the original model fit are coarser than the
grid of points specified by nfine.
Value
A numeric vector, of the same length as coef(fit), giving updated values for the fitted model
coefficients.
Author(s)
Adrian Baddeley  and Rolf Turner .

bdist.pixels

141

References
Baddeley, A. and Turner, R. (2014) Bias correction for parameter estimates of spatial point process
models. Journal of Statistical Computation and Simulation 84, 1621–1643. DOI: 10.1080/00949655.2012.755976
See Also
rex
Examples
fit <- ppm(cells ~ x, Strauss(0.07))
coef(fit)
if(!interactive()) {
bc(fit, nfine=64)
} else {
bc(fit)
}

bdist.pixels

Distance to Boundary of Window

Description
Computes the distances from each pixel in a window to the boundary of the window.
Usage
bdist.pixels(w, ..., style="image", method=c("C", "interpreted"))
Arguments
w

A window (object of class "owin").

...

Arguments passed to as.mask to determine the pixel resolution.

style

Character string determining the format of the output: either "matrix", "coords"
or "image".

method

Choice of algorithm to use when w is polygonal.

Details
This function computes, for each pixel u in the window w, the shortest distance d(u, W c ) from u to
the boundary of W .
If the window is a binary mask then the distance from each pixel to the boundary is computed using
the distance transform algorithm distmap.owin. The result is equivalent to distmap(W, invert=TRUE).
If the window is a rectangle or a polygonal region, the grid of pixels is determined by the arguments
"..." passed to as.mask. The distance from each pixel to the boundary is calculated exactly, using
analytic geometry. This is slower but more accurate than in the case of a binary mask.
For software testing purposes, there are two implementations available when w is a polygon: the
default is method="C" which is much faster than method="interpreted".

142

bdist.points

Value
If style="image", a pixel image (object of class "im") containing the distances from each pixel in
the image raster to the boundary of the window.
If style="matrix", a matrix giving the distances from each pixel in the image raster to the boundary of the window. Rows of this matrix correspond to the y coordinate and columns to the x
coordinate.
If style="coords", a list with three components x,y,z, where x,y are vectors of length m, n
giving the x and y coordinates respectively, and z is an m × n matrix such that z[i,j] is the
distance from (x[i],y[j]) to the boundary of the window. Rows of this matrix correspond to the
x coordinate and columns to the y coordinate. This result can be plotted with persp, image or
contour.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
owin.object, erosion, bdist.points, bdist.tiles, distmap.owin.
Examples
u <- owin(c(0,1),c(0,1))
d <- bdist.pixels(u, eps=0.01)
image(d)
d <- bdist.pixels(u, eps=0.01, style="matrix")
mean(d >= 0.1)
# value is approx (1 - 2 * 0.1)^2 = 0.64

bdist.points

Distance to Boundary of Window

Description
Computes the distances from each point of a point pattern to the boundary of the window.
Usage
bdist.points(X)
Arguments
X

A point pattern (object of class "ppp").

Details
This function computes, for each point xi in the point pattern X, the shortest distance d(xi , W c )
from xi to the boundary of the window W of observation.
If the window Window(X) is of type "rectangle" or "polygonal", then these distances are computed by analytic geometry and are exact, up to rounding errors. If the window is of type "mask"
then the distances are computed using the real-valued distance transform, which is an approximation
with maximum error equal to the width of one pixel in the mask.

bdist.tiles

143

Value
A numeric vector, giving the distances from each point of the pattern to the boundary of the window.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
bdist.pixels, bdist.tiles, ppp.object, erosion
Examples
data(cells)
d <- bdist.points(cells)

bdist.tiles

Distance to Boundary of Window

Description
Computes the shortest distances from each tile in a tessellation to the boundary of the window.
Usage
bdist.tiles(X)
Arguments
X

A tessellation (object of class "tess").

Details
This function computes, for each tile si in the tessellation X, the shortest distance from si to the
boundary of the window W containing the tessellation.
Value
A numeric vector, giving the shortest distance from each tile in the tessellation to the boundary of
the window. Entries of the vector correspond to the entries of tiles(X).
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
tess, bdist.points, bdist.pixels

144

beachcolours

Examples
P <- runifpoint(15)
X <- dirichlet(P)
plot(X, col="red")
B <- bdist.tiles(X)
# identify tiles that do not touch the boundary
plot(X[B > 0], add=TRUE, col="green", lwd=3)

beachcolours

Create Colour Scheme for a Range of Numbers

Description
Given a range of numerical values, this command creates a colour scheme that would be appropriate
if the numbers were altitudes (elevation above or below sea level).
Usage
beachcolours(range, sealevel = 0, monochrome = FALSE,
ncolours = if (monochrome) 16 else 64,
nbeach = 1)
beachcolourmap(range, ...)
Arguments
range

Range of numerical values to be mapped. A numeric vector of length 2.

sealevel

Value that should be treated as zero. A single number, lying between range[1]
and range[2].

monochrome

Logical. If TRUE then a greyscale colour map is constructed.

ncolours

Number of distinct colours to use.

nbeach

Number of colours that will be yellow.

...

Arguments passed to beachcolours.

Details
Given a range of numerical values, these commands create a colour scheme that would be appropriate if the numbers were altitudes (elevation above or below sea level).
Numerical values close to zero are portrayed in green (representing the waterline). Negative values
are blue (representing water) and positive values are yellow to red (representing land). At least,
these are the colours of land and sea in Western Australia. This colour scheme was proposed by
Baddeley et al (2005).
The function beachcolours returns these colours as a character vector, while beachcolourmap
returns a colourmap object.
The argument range should be a numeric vector of length 2 giving a range of numerical values.
The argument sealevel specifies the height value that will be treated as zero, and mapped to the
colour green. A vector of ncolours colours will be created, of which nbeach colours will be green.
The argument monochrome is included for convenience when preparing publications. If monochrome=TRUE
the colour map will be a simple grey scale containing ncolours shades from black to white.

beginner

145

Value
For beachcolours, a character vector of length ncolours specifying colour values. For beachcolourmap,
a colour map (object of class "colourmap").
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Baddeley, A., Turner, R., Møller, J. and Hazelton, M. (2005) Residual analysis for spatial point
processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
See Also
colourmap, colourtools.
Examples
plot(beachcolourmap(c(-2,2)))

beginner

Print Introduction For Beginners

Description
Prints an introduction for beginners to the spatstat package, or another specified package.
Usage
beginner(package = "spatstat")
Arguments
package

Name of package.

Details
This function prints an introduction for beginners to the spatstat package.
The function can be executed simply by typing beginner without parentheses.
If the argument package is given, then the function prints the beginner’s help file BEGINNER.txt
from the specified package (if it has one).
Value
Null.
Author(s)
Adrian Baddeley 
and Rolf Turner 

146

begins

See Also
latest.news
Examples
beginner

begins

Check Start of Character String

Description
Checks whether a character string begins with a particular prefix.
Usage
begins(x, firstbit)
Arguments
x

Character string, or vector of character strings, to be tested.

firstbit

A single character string.

Details
This simple wrapper function checks whether (each entry in) x begins with the string firstbit,
and returns a logical value or logical vector with one entry for each entry of x. This function is
useful mainly for reducing complexity in model formulae.
Value
Logical vector of the same length as x.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
Examples
begins(c("Hello", "Goodbye"), "Hell")
begins("anything", "")

berman.test

147

berman.test

Berman’s Tests for Point Process Model

Description
Tests the goodness-of-fit of a Poisson point process model using methods of Berman (1986).
Usage
berman.test(...)
## S3 method for class 'ppp'
berman.test(X, covariate,
which = c("Z1", "Z2"),
alternative = c("two.sided", "less", "greater"), ...)
## S3 method for class 'ppm'
berman.test(model, covariate,
which = c("Z1", "Z2"),
alternative = c("two.sided", "less", "greater"), ...)
## S3 method for class 'lpp'
berman.test(X, covariate,
which = c("Z1", "Z2"),
alternative = c("two.sided", "less", "greater"), ...)
## S3 method for class 'lppm'
berman.test(model, covariate,
which = c("Z1", "Z2"),
alternative = c("two.sided", "less", "greater"), ...)
Arguments
X

A point pattern (object of class "ppp" or "lpp").

model

A fitted point process model (object of class "ppm" or "lppm").

covariate

The spatial covariate on which the test will be based. An image (object of class
"im") or a function.

which

Character string specifying the choice of test.

alternative

Character string specifying the alternative hypothesis.

...

Additional arguments controlling the pixel resolution (arguments dimyx and eps
passed to as.mask) or other undocumented features.

Details
These functions perform a goodness-of-fit test of a Poisson point process model fitted to point
pattern data. The observed distribution of the values of a spatial covariate at the data points, and
the predicted distribution of the same values under the model, are compared using either of two test
statistics Z1 and Z2 proposed by Berman (1986). The Z1 test is also known as the Lawson-Waller
test.

148

berman.test
The function berman.test is generic, with methods for point patterns ("ppp" or "lpp") and point
process models ("ppm" or "lppm").
• If X is a point pattern dataset (object of class "ppp" or "lpp"), then berman.test(X, ...)
performs a goodness-of-fit test of the uniform Poisson point process (Complete Spatial Randomness, CSR) for this dataset.
• If model is a fitted point process model (object of class "ppm" or "lppm") then berman.test(model, ...)
performs a test of goodness-of-fit for this fitted model. In this case, model should be a Poisson
point process.
The test is performed by comparing the observed distribution of the values of a spatial covariate
at the data points, and the predicted distribution of the same covariate under the model. Thus, you
must nominate a spatial covariate for this test.
The argument covariate should be either a function(x,y) or a pixel image (object of class "im"
containing the values of a spatial function. If covariate is an image, it should have numeric values,
and its domain should cover the observation window of the model. If covariate is a function,
it should expect two arguments x and y which are vectors of coordinates, and it should return a
numeric vector of the same length as x and y.
First the original data point pattern is extracted from model. The values of the covariate at these
data points are collected.
Next the values of the covariate at all locations in the observation window are evaluated. The
point process intensity of the fitted model is also evaluated at all locations in the window.
• If which="Z1", the test statistic Z1 is computed as follows. The sum S of the covariate values
at all data points is evaluated. The predicted mean µ and variance σ 2 of S are computed from
the values of the covariate at all locations in the window. Then we compute Z1 = (S − µ)/σ.
Closely-related tests were proposed independently by Waller et al (1993) and Lawson (1993)
so this test is often termed the Lawson-Waller test in epidemiological literature.
• If which="Z2", the test statistic Z2 is computed as follows. The values of the covariate at
all locations in the observation window, weighted by the point process intensity, are compiled
into a cumulative distribution function F . The probability integral transformation is then applied: the values of the covariate at the original data points are transformed by the predicted
cumulative distribution function F into numbers between 0 and 1. If the model is correct,
these numbers are i.i.d. uniform random numbers. The standardised sample mean of these
numbers is the statistic Z2 .
In both cases the null distribution of the test statistic is the standard normal distribution, approximately.
The return value is an object of class "htest" containing the results of the hypothesis test. The
print method for this class gives an informative summary of the test outcome.

Value
An object of class "htest" (hypothesis test) and also of class "bermantest", containing the results
of the test. The return value can be plotted (by plot.bermantest) or printed to give an informative
summary of the test.
Warning
The meaning of a one-sided test must be carefully scrutinised: see the printed output.

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149

Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak .
References
Berman, M. (1986) Testing for spatial association between a point process and another stochastic
process. Applied Statistics 35, 54–62.
Lawson, A.B. (1993) On the analysis of mortality events around a prespecified fixed point. Journal
of the Royal Statistical Society, Series A 156 (3) 363–377.
Waller, L., Turnbull, B., Clark, L.C. and Nasca, P. (1992) Chronic Disease Surveillance and testing
of clustering of disease and exposure: Application to leukaemia incidence and TCE-contaminated
dumpsites in upstate New York. Environmetrics 3, 281–300.
See Also
cdf.test, quadrat.test, ppm
Examples
# Berman's data
data(copper)
X <- copper$SouthPoints
L <- copper$SouthLines
D <- distmap(L, eps=1)
# test of CSR
berman.test(X, D)
berman.test(X, D, "Z2")

bind.fv

Combine Function Value Tables

Description
Advanced Use Only. Combine objects of class "fv", or glue extra columns of data onto an existing
"fv" object.
Usage
## S3 method for class 'fv'
cbind(...)
bind.fv(x, y, labl = NULL, desc = NULL, preferred = NULL, clip=FALSE)
Arguments
...

Any number of arguments, which are objects of class "fv".

x

An object of class "fv".

y

Either a data frame or an object of class "fv".

labl

Plot labels (see fv) for columns of y. A character vector.

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bind.fv
desc

Descriptions (see fv) for columns of y. A character vector.

preferred

Character string specifying the column which is to be the new recommended
value of the function.

clip

Logical value indicating whether each object must have exactly the same domain, that is, the same sequence of values of the function argument (clip=FALSE,
the default) or whether objects with different domains are permissible and will
be restricted to a common domain (clip=TRUE).

Details
This documentation is provided for experienced programmers who want to modify the internal
behaviour of spatstat.
The function cbind.fv is a method for the generic R function cbind. It combines any number of
objects of class "fv" into a single object of class "fv". The objects must be compatible, in the
sense that they have identical values of the function argument.
The function bind.fv is a lower level utility which glues additional columns onto an existing object
x of class "fv". It has two modes of use:
• If the additional dataset y is an object of class "fv", then x and y must be compatible as
described above. Then the columns of y that contain function values will be appended to the
object x.
• Alternatively if y is a data frame, then y must have the same number of rows as x. All columns
of y will be appended to x.
The arguments labl and desc provide plot labels and description strings (as described in fv) for
the new columns. If y is an object of class "fv" then labl and desc are optional, and default to the
relevant entries in the object y. If y is a data frame then labl and desc must be provided.
Value
An object of class "fv".
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
fv, with.fv.
Undocumented functions for modifying an "fv" object include fvnames, fvnames<-, tweak.fv.entry
and rebadge.fv.
Examples
data(cells)
K1 <- Kest(cells, correction="border")
K2 <- Kest(cells, correction="iso")
# remove column 'theo' to avoid duplication
K2 <- K2[, names(K2) != "theo"]
cbind(K1, K2)

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151
bind.fv(K1, K2, preferred="iso")
# constrain border estimate to be monotonically increasing
bm <- cumsum(c(0, pmax(0, diff(K1$border))))
bind.fv(K1, data.frame(bmono=bm),
"%s[bmo](r)",
"monotone border-corrected estimate of %s",
"bmono")

bits.test

Balanced Independent Two-Stage Monte Carlo Test

Description
Performs a Balanced Independent Two-Stage Monte Carlo test of goodness-of-fit for spatial pattern.
Usage
bits.test(X, ...,
exponent = 2, nsim=19,
alternative=c("two.sided", "less", "greater"),
leaveout=1, interpolate = FALSE,
savefuns=FALSE, savepatterns=FALSE,
verbose = TRUE)
Arguments
X

Either a point pattern dataset (object of class "ppp", "lpp" or "pp3") or a fitted
point process model (object of class "ppm", "kppm", "lppm" or "slrm").

...

Arguments passed to dclf.test or mad.test or envelope to control the conduct of the test. Useful arguments include fun to determine the summary function, rinterval to determine the range of r values used in the test, and use.theory
described under Details.

exponent

Exponent used in the test statistic. Use exponent=2 for the Diggle-CressieLoosmore-Ford test, and exponent=Inf for the Maximum Absolute Deviation
test.

nsim

Number of replicates in each stage of the test. A total of nsim * (nsim + 1)
simulated point patterns will be generated, and the p-value will be a multiple of
1/(nsim+1).

alternative

Character string specifying the alternative hypothesis. The default (alternative="two.sided")
is that the true value of the summary function is not equal to the theoretical
value postulated under the null hypothesis. If alternative="less" the alternative hypothesis is that the true value of the summary function is lower than the
theoretical value.

leaveout

Optional integer 0, 1 or 2 indicating how to calculate the deviation between the
observed summary function and the nominal reference value, when the reference
value must be estimated by simulation. See Details.

interpolate

Logical value indicating whether to interpolate the distribution of the test statistic by kernel smoothing, as described in Dao and Genton (2014, Section 5).

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bits.test
savefuns

Logical flag indicating whether to save the simulated function values (from the
first stage).

savepatterns

Logical flag indicating whether to save the simulated point patterns (from the
first stage).

verbose

Logical value indicating whether to print progress reports.

Details
Performs the Balanced Independent Two-Stage Monte Carlo test proposed by Baddeley et al (2017),
an improvement of the Dao-Genton (2014) test.
If X is a point pattern, the null hypothesis is CSR.
If X is a fitted model, the null hypothesis is that model.
The argument use.theory passed to envelope determines whether to compare the summary function for the data to its theoretical value for CSR (use.theory=TRUE) or to the sample mean of
simulations from CSR (use.theory=FALSE).
The argument leaveout specifies how to calculate the discrepancy between the summary function
for the data and the nominal reference value, when the reference value must be estimated by simulation. The values leaveout=0 and leaveout=1 are both algebraically equivalent (Baddeley et al,
2014, Appendix) to computing the difference observed - reference where the reference is the
mean of simulated values. The value leaveout=2 gives the leave-two-out discrepancy proposed by
Dao and Genton (2014).
Value
A hypothesis test (object of class "htest" which can be printed to show the outcome of the test.
Author(s)

Adrian Baddeley, Andrew Hardegen, Tom Lawrence, Robin Milne, Gopalan Nair and Suman Rakshit. Implemented by Adrian Baddeley , Rolf Turner .
References
Dao, N.A. and Genton, M. (2014) A Monte Carlo adjusted goodness-of-fit test for parametric models describing spatial point patterns. Journal of Graphical and Computational Statistics 23, 497–
517.
Baddeley, A., Diggle, P.J., Hardegen, A., Lawrence, T., Milne, R.K. and Nair, G. (2014) On tests of
spatial pattern based on simulation envelopes. Ecological Monographs 84 (3) 477–489.
Baddeley, A., Hardegen, A., Lawrence, L., Milne, R.K., Nair, G.M. and Rakshit, S. (2017) On twostage Monte Carlo tests of composite hypotheses. Computational Statistics and Data Analysis, in
press.
See Also
dg.test, dclf.test, mad.test

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153

Examples
ns <- if(interactive()) 19 else 4
bits.test(cells, nsim=ns)
bits.test(cells, alternative="less", nsim=ns)
bits.test(cells, nsim=ns, interpolate=TRUE)

blur

Apply Gaussian Blur to a Pixel Image

Description
Applies a Gaussian blur to a pixel image.
Usage
blur(x, sigma = NULL, ..., normalise=FALSE, bleed = TRUE, varcov=NULL)
## S3 method for class 'im'
Smooth(X, sigma = NULL, ...,
normalise=FALSE, bleed = TRUE, varcov=NULL)
Arguments
x,X

The pixel image. An object of class "im".

sigma

Standard deviation of isotropic Gaussian smoothing kernel.

...

Ignored.

normalise

Logical flag indicating whether the output values should be divided by the corresponding blurred image of the window itself. See Details.

bleed

Logical flag indicating whether to allow blur to extend outside the original domain of the image. See Details.

varcov

Variance-covariance matrix of anisotropic Gaussian kernel. Incompatible with
sigma.

Details
This command applies a Gaussian blur to the pixel image x.
Smooth.im is a method for the generic Smooth for pixel images. It is currently identical to blur,
apart from the name of the first argument.
The blurring kernel is the isotropic Gaussian kernel with standard deviation sigma, or the anisotropic
Gaussian kernel with variance-covariance matrix varcov. The arguments sigma and varcov are
incompatible. Also sigma may be a vector of length 2 giving the standard deviations of two independent Gaussian coordinates, thus equivalent to varcov = diag(sigma^2).
If the pixel values of x include some NA values (meaning that the image domain does not completely
fill the rectangular frame) then these NA values are first reset to zero.
The algorithm then computes the convolution x ∗ G of the (zero-padded) pixel image x with the
specified Gaussian kernel G.
If normalise=FALSE, then this convolution x ∗ G is returned. If normalise=TRUE, then the convolution x ∗ G is normalised by dividing it by the convolution w ∗ G of the image domain w with

154

border
the same Gaussian kernel. Normalisation ensures that the result can be interpreted as a weighted
average of input pixel values, without edge effects due to the shape of the domain.
If bleed=FALSE, then pixel values outside the original image domain are set to NA. Thus the output
is a pixel image with the same domain as the input. If bleed=TRUE, then no such alteration is
performed, and the result is a pixel image defined everywhere in the rectangular frame containing
the input image.
Computation is performed using the Fast Fourier Transform.

Value
A pixel image with the same pixel array as the input image x.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
interp.im for interpolating a pixel image to a finer resolution, density.ppp for blurring a point
pattern, Smooth.ppp for interpolating marks attached to points.
Examples
data(letterR)
Z <- as.im(function(x,y) { 4 * x^2 + 3 * y }, letterR)
par(mfrow=c(1,3))
plot(Z)
plot(letterR, add=TRUE)
plot(blur(Z, 0.3, bleed=TRUE))
plot(letterR, add=TRUE)
plot(blur(Z, 0.3, bleed=FALSE))
plot(letterR, add=TRUE)
par(mfrow=c(1,1))

border

Border Region of a Window

Description
Computes the border region of a window, that is, the region lying within a specified distance of the
boundary of a window.
Usage
border(w, r, outside=FALSE, ...)

border

155

Arguments
w

A window (object of class "owin") or something acceptable to as.owin.

r

Numerical value.

outside

Logical value determining whether to compute the border outside or inside w.

...

Optional arguments passed to erosion (if outside=FALSE) or to dilation (if
outside=TRUE).

Details
By default (if outside=FALSE), the border region is the subset of w lying within a distance r of the
boundary of w. It is computed by eroding w by the distance r (using erosion) and subtracting this
eroded window from the original window w.
If outside=TRUE, the border region is the set of locations outside w lying within a distance r of w. It
is computed by dilating w by the distance r (using dilation) and subtracting the original window
w from the dilated window.

Value
A window (object of class "owin").

Author(s)
Adrian Baddeley 
and Rolf Turner 

See Also
erosion, dilation

Examples
# rectangle
u <- unit.square()
border(u, 0.1)
border(u, 0.1, outside=TRUE)
# polygon
data(letterR)
plot(letterR)
plot(border(letterR, 0.1), add=TRUE)
plot(border(letterR, 0.1, outside=TRUE), add=TRUE)

156

bounding.box.xy

bounding.box.xy

Convex Hull of Points

Description
Computes the smallest rectangle containing a set of points.
Usage
bounding.box.xy(x, y=NULL)
Arguments
x

vector of x coordinates of observed points, or a 2-column matrix giving x,y
coordinates, or a list with components x,y giving coordinates (such as a point
pattern object of class "ppp".)

y

(optional) vector of y coordinates of observed points, if x is a vector.

Details
Given an observed pattern of points with coordinates given by x and y, this function finds the
smallest rectangle, with sides parallel to the coordinate axes, that contains all the points, and returns
it as a window.
Value
A window (an object of class "owin").
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
owin, as.owin, convexhull.xy, ripras
Examples
x <- runif(30)
y <- runif(30)
w <- bounding.box.xy(x,y)
plot(owin(), main="bounding.box.xy(x,y)")
plot(w, add=TRUE)
points(x,y)
X <- rpoispp(30)
plot(X, main="bounding.box.xy(X)")
plot(bounding.box.xy(X), add=TRUE)

boundingbox

157

boundingbox

Bounding Box of a Window, Image, or Point Pattern

Description
Find the smallest rectangle containing a given window(s), image(s) or point pattern(s).
Usage
boundingbox(...)
## Default S3 method:
boundingbox(...)
## S3 method for class 'im'
boundingbox(...)
## S3 method for class 'owin'
boundingbox(...)
## S3 method for class 'ppp'
boundingbox(...)
## S3 method for class 'psp'
boundingbox(...)
## S3 method for class 'lpp'
boundingbox(...)
## S3 method for class 'linnet'
boundingbox(...)
## S3 method for class 'solist'
boundingbox(...)
Arguments
...

One or more windows (objects of class "owin"), pixel images (objects of class
"im") or point patterns (objects of class "ppp" or "lpp") or line segment patterns
(objects of class "psp") or linear networks (objects of class "linnet") or any
combination of such objects. Alternatively, the argument may be a list of such
objects, of class "solist".

Details
This function finds the smallest rectangle (with sides parallel to the coordinate axes) that contains
all the given objects.
For a window (object of class "owin"), the bounding box is the smallest rectangle that contains all
the vertices of the window (this is generally smaller than the enclosing frame, which is returned by
as.rectangle).

158

boundingcircle
For a point pattern (object of class "ppp" or "lpp"), the bounding box is the smallest rectangle that
contains all the points of the pattern. This is usually smaller than the bounding box of the window
of the point pattern.
For a line segment pattern (object of class "psp") or a linear network (object of class "linnet"),
the bounding box is the smallest rectangle that contains all endpoints of line segments.
For a pixel image (object of class "im"), the image will be converted to a window using as.owin,
and the bounding box of this window is obtained.
If the argument is a list of several objects, then this function finds the smallest rectangle that contains
all the bounding boxes of the objects.

Value
owin, as.owin, as.rectangle
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
Examples
w <- owin(c(0,10),c(0,10), poly=list(x=c(1,2,3,2,1), y=c(2,3,4,6,7)))
r <- boundingbox(w)
# returns rectangle [1,3] x [2,7]
w2 <- unit.square()
r <- boundingbox(w, w2)
# returns rectangle [0,3] x [0,7]

boundingcircle

Smallest Enclosing Circle

Description
Find the smallest circle enclosing a spatial window or other object. Return its radius, or the location
of its centre, or the circle itself.
Usage
boundingradius(x, ...)
boundingcentre(x, ...)
boundingcircle(x, ...)
## S3 method for class 'owin'
boundingradius(x, ...)
## S3 method for class 'owin'
boundingcentre(x, ...)

boundingcircle

159

## S3 method for class 'owin'
boundingcircle(x, ...)
## S3 method for class 'ppp'
boundingradius(x, ...)
## S3 method for class 'ppp'
boundingcentre(x, ...)
## S3 method for class 'ppp'
boundingcircle(x, ...)
Arguments
x

A window (object of class "owin"), or another spatial object.

...

Arguments passed to as.mask to determine the pixel resolution for the calculation.

Details
The boundingcircle of a spatial region W is the smallest circle that contains W . The boundingradius
is the radius of this circle, and the boundingcentre is the centre of the circle.
The functions boundingcircle, boundingcentre and boundingradius are generic. There are
methods for objects of class "owin", "ppp" and "linnet".
Value
The result of boundingradius is a single numeric value.
The result of boundingcentre is a point pattern containing a single point.
The result of boundingcircle is a window representing the boundingcircle.
Author(s)
Adrian Baddeley 
See Also
boundingradius.linnet
Examples
boundingradius(letterR)
plot(grow.rectangle(Frame(letterR), 0.2), main="", type="n")
plot(letterR, add=TRUE, col="grey")
plot(boundingcircle(letterR), add=TRUE, border="green", lwd=2)
plot(boundingcentre(letterR), pch="+", cex=2, col="blue", add=TRUE)
X <- runifpoint(5)
plot(X)
plot(boundingcircle(X), add=TRUE)
plot(boundingcentre(X), pch="+", cex=2, col="blue", add=TRUE)

160

box3

box3

Three-Dimensional Box

Description
Creates an object representing a three-dimensional box.
Usage
box3(xrange = c(0, 1), yrange = xrange, zrange = yrange, unitname = NULL)
Arguments
xrange, yrange, zrange
Dimensions of the box in the x, y, z directions. Each of these arguments should
be a numeric vector of length 2.
unitname

Optional. Name of the unit of length. See Details.

Details
This function creates an object representing a three-dimensional rectangular parallelepiped (box)
with sides parallel to the coordinate axes.
The object can be used to specify the domain of a three-dimensional point pattern (see pp3) and in
various geometrical calculations (see volume.box3, diameter.box3, eroded.volumes).
The optional argument unitname specifies the name of the unit of length. See unitname for valid
formats.
The function as.box3 can be used to convert other kinds of data to this format.
Value
An object of class "box3". There is a print method for this class.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
as.box3, pp3, volume.box3, diameter.box3, eroded.volumes.
Examples
box3()
box3(c(0,10),c(0,10),c(0,5), unitname=c("metre","metres"))
box3(c(-1,1))

boxx

161

boxx

Multi-Dimensional Box

Description
Creates an object representing a multi-dimensional box.
Usage
boxx(..., unitname = NULL)
Arguments
...

Dimensions of the box. Vectors of length 2.

unitname

Optional. Name of the unit of length. See Details.

Details
This function creates an object representing a multi-dimensional rectangular parallelepiped (box)
with sides parallel to the coordinate axes.
The object can be used to specify the domain of a multi-dimensional point pattern (see ppx) and in
various geometrical calculations (see volume.boxx, diameter.boxx, eroded.volumes).
The optional argument unitname specifies the name of the unit of length. See unitname for valid
formats.
Value
An object of class "boxx". There is a print method for this class.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
ppx, volume.boxx, diameter.boxx, eroded.volumes.boxx.
Examples
boxx(c(0,10),c(0,10),c(0,5),c(0,1), unitname=c("metre","metres"))

162

branchlabelfun

branchlabelfun

Tree Branch Membership Labelling Function

Description
Creates a function which returns the tree branch membership label for any location on a linear
network.
Usage
branchlabelfun(L, root = 1)
Arguments
L

Linear network (object of class "linnet"). The network must have no loops.

root

Root of the tree. An integer index identifying which point in vertices(L) is
the root of the tree.

Details
The linear network L must be an acyclic graph (i.e. must not contain any loops) so that it can be
interpreted as a tree.
The result of f <- branchlabelfun(L, root) is a function f which gives, for each location on
the linear network L, the tree branch label at that location.
Tree branch labels are explained in treebranchlabels.
The result f also belongs to the class "linfun". It can be called using several different kinds of
data, as explained in the help for linfun. The values of the function are character strings.
Value
A function (of class "linfun").
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
treebranchlabels, linfun
Examples
# make a simple tree
m <- simplenet$m
m[8,10] <- m[10,8] <- FALSE
L <- linnet(vertices(simplenet), m)
# make function
f <- branchlabelfun(L, 1)
plot(f)

bugfixes

163

X <- runiflpp(5, L)
f(X)

bugfixes

List Recent Bug Fixes

Description
List all bug fixes in a package, starting from a certain date or version of the package. Fixes are
sorted alphabetically by the name of the affected function. The default is to list bug fixes in the
latest version of the spatstat package.
Usage
bugfixes(sinceversion = NULL, sincedate = NULL,
package = "spatstat", show = TRUE)
Arguments
sinceversion

Earliest version of package for which bugs should be listed. The default is the
current installed version.

sincedate

Earliest release date of package for which bugs should be listed. A character
string or a date-time object.

package

Character string. The name of the package for which bugs are to be listed.

show

Logical value indicating whether to display the bug table on the terminal.

Details
Bug reports are extracted from the NEWS file of the specified package. Only those after a specified
date, or after a specified version of the package, are retained. The bug reports are then sorted
alphabetically, so that all bugs affecting a particular function are listed consecutively. Finally the
table of bug reports is displayed (if show=TRUE) and returned invisibly.
The argument sinceversion should be a character string like "1.2-3". The default is the current installed version of the package. The argument sincedata should be a character string like
"2015-05-27", or a date-time object.
Typing bugfixes without parentheses will display a table of all bug fixes in the current installed
version of spatstat.
Value
A data frame, belonging to the class "bugtable", which has its own print method.
Author(s)
Adrian Baddeley .
See Also
latest.news, news.

164

bw.diggle

Examples
# show all bugs reported after publication of the spatstat book
if(interactive()) bugfixes("1.42-0")

bw.diggle

Cross Validated Bandwidth Selection for Kernel Density

Description
Uses cross-validation to select a smoothing bandwidth for the kernel estimation of point process
intensity.
Usage
bw.diggle(X, ..., correction="good", hmax=NULL, nr=512)
Arguments
X

A point pattern (object of class "ppp").

...

Ignored.

correction

Character string passed to Kest determining the edge correction to be used to
calculate the K function.

hmax

Numeric. Maximum value of bandwidth that should be considered.

nr

Integer. Number of steps in the distance value r to use in computing numerical
integrals.

Details
This function selects an appropriate bandwidth sigma for the kernel estimator of point process
intensity computed by density.ppp.
The bandwidth σ is chosen to minimise the mean-square error criterion defined by Diggle (1985).
The algorithm uses the method of Berman and Diggle (1989) to compute the quantity
M (σ) =

MSE(σ)
− g(0)
λ2

as a function of bandwidth σ, where MSE(σ) is the mean squared error at bandwidth σ, while λ is
the mean intensity, and g is the pair correlation function. See Diggle (2003, pages 115-118) for a
summary of this method.
The result is a numerical value giving the selected bandwidth. The result also belongs to the class
"bw.optim" which can be plotted to show the (rescaled) mean-square error as a function of sigma.
Value
A numerical value giving the selected bandwidth. The result also belongs to the class "bw.optim"
which can be plotted.

bw.frac

165

Definition of bandwidth
The smoothing parameter sigma returned by bw.diggle (and displayed on the horizontal axis of
the plot) corresponds to h/2, where h is the smoothing parameter described in Diggle (2003, pages
116-118) and Berman and Diggle (1989). In those references, the smoothing kernel is the uniform
density on the disc of radius h. In density.ppp, the smoothing kernel is the isotropic Gaussian
density with standard deviation sigma. When replacing one kernel by another, the usual practice is
to adjust the bandwidths so that the kernels have equal variance (cf. Diggle 2003, page 118). This
implies that sigma = h/2.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Berman, M. and Diggle, P. (1989) Estimating weighted integrals of the second-order intensity of a
spatial point process. Journal of the Royal Statistical Society, series B 51, 81–92.
Diggle, P.J. (1985) A kernel method for smoothing point process data. Applied Statistics (Journal
of the Royal Statistical Society, Series C) 34 (1985) 138–147.
Diggle, P.J. (2003) Statistical analysis of spatial point patterns, Second edition. Arnold.
See Also
density.ppp, bw.ppl, bw.scott
Examples
data(lansing)
attach(split(lansing))
b <- bw.diggle(hickory)
plot(b, ylim=c(-2, 0), main="Cross validation for hickories")
plot(density(hickory, b))

bw.frac

Bandwidth Selection Based on Window Geometry

Description
Select a smoothing bandwidth for smoothing a point pattern, based only on the geometry of the
spatial window. The bandwidth is a specified quantile of the distance between two independent
random points in the window.
Usage
bw.frac(X, ..., f=1/4)

166

bw.pcf

Arguments
X

A window (object of class "owin") or point pattern (object of class "ppp") or
other data which can be converted to a window using as.owin.

...

Arguments passed to distcdf.

f

Probability value (between 0 and 1) determining the quantile of the distribution.

Details
This function selects an appropriate bandwidth sigma for the kernel estimator of point process
intensity computed by density.ppp.
The bandwidth σ is computed as a quantile of the distance between two independent random points
in the window. The default is the lower quartile of this distribution.
If F (r) is the cumulative distribution function of the distance between two independent random
points uniformly distributed in the window, then the value returned is the quantile with probability
f . That is, the bandwidth is the value r such that F (r) = f .
The cumulative distribution function F (r) is computed using distcdf. We then we compute the
smallest number r such that F (r) ≥ f .
Value
A numerical value giving the selected bandwidth. The result also belongs to the class "bw.frac"
which can be plotted to show the cumulative distribution function and the selected quantile.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
density.ppp, bw.diggle, bw.ppl, bw.relrisk, bw.scott, bw.smoothppp, bw.stoyan
Examples
h <- bw.frac(letterR)
h
plot(h, main="bw.frac(letterR)")

bw.pcf

Cross Validated Bandwidth Selection for Pair Correlation Function

Description
Uses composite likelihood or generalized least squares cross-validation to select a smoothing bandwidth for the kernel estimation of pair correlation function.

bw.pcf

167

Usage
bw.pcf(X, rmax=NULL, lambda=NULL, divisor="r",
kernel="epanechnikov", nr=10000, bias.correct=TRUE,
cv.method=c("compLik", "leastSQ"), simple=TRUE, srange=NULL,
..., verbose=FALSE)
Arguments
X

A point pattern (object of class "ppp").

rmax

Numeric. Maximum value of the spatial lag distance r for which g(r) should be
evaluated.

lambda

Optional. Values of the estimated intensity function. A vector giving the intensity values at the points of the pattern X.

divisor

Choice of divisor in the estimation formula: either "r" (the default) or "d". See
pcf.ppp.

kernel

Choice of smoothing kernel, passed to density; see pcf and pcfinhom.

nr

Integer. Number of subintervals for discretization of [0, rmax] to use in computing numerical integrals.

bias.correct

Logical. Whether to use bias corrected version of the kernel estimate. See
Details.

cv.method

Choice of cross validation method: either "compLik" or "leastSQ" (partially
matched).

simple

Logical. Whether to use simple removal of spatial lag distances. See Details.

srange

Optional. Numeric vector of length 2 giving the range of bandwidth values that
should be searched to find the optimum bandwidth.

...

Other arguments, passed to pcf or pcfinhom.

verbose

Logical value indicating whether to print progress reports during the optimization procedure.

Details
This function selects an appropriate bandwidth bw for the kernel estimator of the pair correlation
function of a point process intensity computed by pcf.ppp (homogeneous case) or pcfinhom (inhomogeneous case).
With cv.method="leastSQ", the bandwidth h is chosen to minimise an unbiased estimate of the
integrated mean-square error criterion M (h) defined in equation (4) in Guan (2007a).
With cv.method="compLik", the bandwidth h is chosen to maximise a likelihood cross-validation
criterion CV (h) defined in equation (6) of Guan (2007b).
M (b) =

MSE(σ)
− g(0)
λ2

The result is a numerical value giving the selected bandwidth.
Value
A numerical value giving the selected bandwidth. The result also belongs to the class "bw.optim"
which can be plotted.

168

bw.ppl

Definition of bandwidth
The bandwidth bw returned by bw.pcf corresponds to the standard deviation of the smoothoing
kernel. As mentioned in the documentation of density.default and pcf.ppp, this differs from
the scale parameter h of the smoothing kernel which is often considered in the literature as the
bandwidth of the kernel function. For example for the Epanechnikov kernel, bw=h/sqrt(h).
Author(s)

Rasmus Waagepetersen and Abdollah Jalilian. Adapted for spatstat by Adrian Baddeley  and Ege Rubak .
References
Guan, Y. (2007a). A composite likelihood cross-validation approach in selecting bandwidth for the
estimation of the pair correlation function. Scandinavian Journal of Statistics, 34(2), 336–346.
Guan, Y. (2007b). A least-squares cross-validation bandwidth selection approach in pair correlation
function estimations. Statistics & Probability Letters, 77(18), 1722–1729.
See Also
pcf.ppp, pcfinhom
Examples
b <- bw.pcf(redwood)
plot(pcf(redwood, bw=b))

bw.ppl

Likelihood Cross Validation Bandwidth Selection for Kernel Density

Description
Uses likelihood cross-validation to select a smoothing bandwidth for the kernel estimation of point
process intensity.
Usage
bw.ppl(X, ..., srange=NULL, ns=16, sigma=NULL, weights=NULL)
Arguments
X

A point pattern (object of class "ppp").

...

Ignored.

srange

Optional numeric vector of length 2 giving the range of values of bandwidth to
be searched.

ns

Optional integer giving the number of values of bandwidth to search.

sigma

Optional. Vector of values of the bandwidth to be searched. Overrides the values
of ns and srange.

weights

Optional. Numeric vector of weights for the points of X. Argument passed to
density.ppp.

bw.ppl

169

Details
This function selects an appropriate bandwidth sigma for the kernel estimator of point process
intensity computed by density.ppp.
The bandwidth σ is chosen to maximise the point process likelihood cross-validation criterion
LCV(σ) =

X

Z
log λ̂−i (xi ) −

i

λ̂(u) du
W

where the sum is taken over all the data points xi , where λ̂−i (xi ) is the leave-one-out kernelsmoothing estimate of the intensity at xi with smoothing bandwidth σ, and λ̂(u) is the kernelsmoothing estimate of the intensity at a spatial location u with smoothing bandwidth σ. See
Loader(1999, Section 5.3).
The value of LCV(σ) is computed directly, using density.ppp, for ns different values of σ between
srange[1] and srange[2].
The result is a numerical value giving the selected bandwidth. The result also belongs to the class
"bw.optim" which can be plotted to show the (rescaled) mean-square error as a function of sigma.
Value
A numerical value giving the selected bandwidth. The result also belongs to the class "bw.optim"
which can be plotted.

Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Loader, C. (1999) Local Regression and Likelihood. Springer, New York.

See Also
density.ppp, bw.diggle, bw.scott
Examples
b <- bw.ppl(redwood)
plot(b, main="Likelihood cross validation for redwoods")
plot(density(redwood, b))

170

bw.relrisk

bw.relrisk

Cross Validated Bandwidth Selection for Relative Risk Estimation

Description
Uses cross-validation to select a smoothing bandwidth for the estimation of relative risk.
Usage
bw.relrisk(X, method = "likelihood", nh = spatstat.options("n.bandwidth"),
hmin=NULL, hmax=NULL, warn=TRUE)
Arguments
X

A multitype point pattern (object of class "ppp" which has factor valued marks).

method

Character string determining the cross-validation method. Current options are
"likelihood", "leastsquares" or "weightedleastsquares".

nh

Number of trial values of smoothing bandwith sigma to consider. The default is
32.

hmin, hmax

Optional. Numeric values. Range of trial values of smoothing bandwith sigma
to consider. There is a sensible default.

warn

Logical. If TRUE, issue a warning if the minimum of the cross-validation criterion occurs at one of the ends of the search interval.

Details
This function selects an appropriate bandwidth for the nonparametric estimation of relative risk
using relrisk.
Consider the indicators yij which equal 1 when data point xi belongs to type j, and equal 0 otherwise. For a particular value of smoothing bandwidth, let p̂j (u) be the estimated probabilities that
a point at location u will belong to type j. Then the bandwidth is chosen to minimise either the
likelihood, the squared error, or the approximately standardised squared error, of the indicators yij
relative to the fitted values p̂j (xi ). See Diggle (2003).
The result is a numerical value giving the selected bandwidth sigma. The result also belongs to
the class "bw.optim" allowing it to be printed and plotted. The plot shows the cross-validation
criterion as a function of bandwidth.
The range of values for the smoothing bandwidth sigma is set by the arguments hmin, hmax. There
is a sensible default, based on multiples of Stoyan’s rule of thumb bw.stoyan.
If the optimal bandwidth is achieved at an endpoint of the interval [hmin, hmax], the algorithm
will issue a warning (unless warn=FALSE). If this occurs, then it is probably advisable to expand the
interval by changing the arguments hmin, hmax.
Computation time depends on the number nh of trial values considered, and also on the range
[hmin, hmax] of values considered, because larger values of sigma require calculations involving
more pairs of data points.
Value
A numerical value giving the selected bandwidth. The result also belongs to the class "bw.optim"
which can be plotted.

bw.scott

171

Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Diggle, P.J. (2003) Statistical analysis of spatial point patterns, Second edition. Arnold.
Kelsall, J.E. and Diggle, P.J. (1995) Kernel estimation of relative risk. Bernoulli 1, 3–16.
See Also
relrisk, bw.stoyan
Examples
data(urkiola)
b <- bw.relrisk(urkiola)
b
plot(b)
b <- bw.relrisk(urkiola, hmax=20)
plot(b)

bw.scott

Scott’s Rule for Bandwidth Selection for Kernel Density

Description
Use Scott’s rule of thumb to determine the smoothing bandwidth for the kernel estimation of point
process intensity.
Usage
bw.scott(X)
Arguments
X

A point pattern (object of class "ppp").

Details
This function selects a bandwidth sigma for the kernel estimator of point process intensity computed
by density.ppp.
The bandwidth σ is computed by the rule of thumb of Scott (1992, page 152). It is very fast to
compute.
This rule is designed for density estimation, and typically produces a larger bandwidth than bw.diggle.
It is useful for estimating gradual trend.
Value
A numerical vector of two elements giving the selected bandwidths in the x and y directions.

172

bw.smoothppp

Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Scott, D.W. (1992) Multivariate Density Estimation. Theory, Practice and Visualization. New York:
Wiley.
See Also
density.ppp, bw.diggle, bw.ppl, bw.frac.
Examples
data(lansing)
attach(split(lansing))
b <- bw.scott(hickory)
b
plot(density(hickory, b))

bw.smoothppp

Cross Validated Bandwidth Selection for Spatial Smoothing

Description
Uses least-squares cross-validation to select a smoothing bandwidth for spatial smoothing of marks.
Usage
bw.smoothppp(X, nh = spatstat.options("n.bandwidth"),
hmin=NULL, hmax=NULL, warn=TRUE)
Arguments
X

A marked point pattern with numeric marks.

nh

Number of trial values of smoothing bandwith sigma to consider. The default is
32.

hmin, hmax

Optional. Numeric values. Range of trial values of smoothing bandwith sigma
to consider. There is a sensible default.

warn

Logical. If TRUE, issue a warning if the minimum of the cross-validation criterion occurs at one of the ends of the search interval.

bw.smoothppp

173

Details
This function selects an appropriate bandwidth for the nonparametric smoothing of mark values
using Smooth.ppp.
The argument X must be a marked point pattern with a vector or data frame of marks. All mark
values must be numeric.
The bandwidth is selected by least-squares cross-validation. Let yi be the mark value at the ith
data point. For a particular choice of smoothing bandwidth, let ŷi be the smoothed value at the ith
data
the bandwidth is chosen to minimise the squared error of the smoothed values
P point. Then
2
i (yi − ŷi ) .
The result of bw.smoothppp is a numerical value giving the selected bandwidth sigma. The result
also belongs to the class "bw.optim" allowing it to be printed and plotted. The plot shows the
cross-validation criterion as a function of bandwidth.
The range of values for the smoothing bandwidth sigma is set by the arguments hmin, hmax. There
is a sensible default, based on the nearest neighbour distances.
If the optimal bandwidth is achieved at an endpoint of the interval [hmin, hmax], the algorithm
will issue a warning (unless warn=FALSE). If this occurs, then it is probably advisable to expand the
interval by changing the arguments hmin, hmax.
Computation time depends on the number nh of trial values considered, and also on the range
[hmin, hmax] of values considered, because larger values of sigma require calculations involving
more pairs of data points.

Value
A numerical value giving the selected bandwidth. The result also belongs to the class "bw.optim"
which can be plotted.

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
Smooth.ppp
Examples
data(longleaf)
b <- bw.smoothppp(longleaf)
b
plot(b)

174

bw.stoyan

bw.stoyan

Stoyan’s Rule of Thumb for Bandwidth Selection

Description
Computes a rough estimate of the appropriate bandwidth for kernel smoothing estimators of the
pair correlation function and other quantities.
Usage
bw.stoyan(X, co=0.15)
Arguments
X

A point pattern (object of class "ppp").

co

Coefficient appearing in the rule of thumb. See Details.

Details
Estimation of the pair correlation function and other quantities by smoothing methods requires a
choice of the smoothing bandwidth. Stoyan and Stoyan (1995, equation (15.16), page 285) proposed
a rule of thumb for choosing the smoothing bandwidth.
√
For the Epanechnikov kernel, the rule of thumb is to set the kernel’s half-width h to 0.15/ λ where
λ is the estimated intensity of the point pattern, typically computed as the number of points of X
divided by the area of the window containing X.
For a √
general kernel, the corresponding rule is to set the standard deviation of the kernel to σ =
0.15/ 5λ.
The coefficient 0.15 can be tweaked using the argument co.
Value
A numerical value giving the selected bandwidth (the standard deviation of the smoothing kernel).
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Stoyan, D. and Stoyan, H. (1995) Fractals, random shapes and point fields: methods of geometrical
statistics. John Wiley and Sons.
See Also
pcf, bw.relrisk
Examples
data(shapley)
bw.stoyan(shapley)

by.im

175

by.im

Apply Function to Image Broken Down by Factor

Description
Splits a pixel image into sub-images and applies a function to each sub-image.
Usage
## S3 method for class 'im'
by(data, INDICES, FUN, ...)
Arguments
data

A pixel image (object of class "im").

INDICES

Grouping variable. Either a tessellation (object of class "tess") or a factorvalued pixel image.

FUN

Function to be applied to each sub-image of data.

...

Extra arguments passed to FUN.

Details
This is a method for the generic function by for pixel images (class "im").
The pixel image data is first divided into sub-images according to INDICES. Then the function FUN
is applied to each subset. The results of each computation are returned in a list.
The grouping variable INDICES may be either
• a tessellation (object of class "tess"). Each tile of the tessellation delineates a subset of the
spatial domain.
• a pixel image (object of class "im") with factor values. The levels of the factor determine
subsets of the spatial domain.
Value
A list containing the results of each evaluation of FUN.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
split.im, tess, im

176

by.ppp

Examples
W <- square(1)
X <- as.im(function(x,y){sqrt(x^2+y^2)}, W)
Y <- dirichlet(runifpoint(12, W))
# mean pixel value in each subset
unlist(by(X, Y, mean))
# trimmed mean
unlist(by(X, Y, mean, trim=0.05))

by.ppp

Apply a Function to a Point Pattern Broken Down by Factor

Description
Splits a point pattern into sub-patterns, and applies the function to each sub-pattern.
Usage
## S3 method for class 'ppp'
by(data, INDICES=marks(data), FUN, ...)
Arguments
data

Point pattern (object of class "ppp").

INDICES

Grouping variable. Either a factor, a pixel image with factor values, or a tessellation.

FUN

Function to be applied to subsets of data.

...

Additional arguments to FUN.

Details
This is a method for the generic function by for point patterns (class "ppp").
The point pattern data is first divided into subsets according to INDICES. Then the function FUN is
applied to each subset. The results of each computation are returned in a list.
The argument INDICES may be
• a factor, of length equal to the number of points in data. The levels of INDICES determine
the destination of each point in data. The ith point of data will be placed in the sub-pattern
split.ppp(data)$l where l = f[i].
• a pixel image (object of class "im") with factor values. The pixel value of INDICES at each
point of data will be used as the classifying variable.
• a tessellation (object of class "tess"). Each point of data will be classified according to the
tile of the tessellation into which it falls.
If INDICES is missing, then data must be a multitype point pattern (a marked point pattern whose
marks vector is a factor). Then the effect is that the points of each type are separated into different
point patterns.

cauchy.estK

177

Value
A list (also of class "anylist" or "solist" as appropriate) containing the results returned from
FUN for each of the subpatterns.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
ppp, split.ppp, cut.ppp, tess, im.
Examples
# multitype point pattern, broken down by type
data(amacrine)
by(amacrine, FUN=density)
by(amacrine, FUN=function(x) { min(nndist(x)) } )
# how to pass additional arguments to FUN
by(amacrine, FUN=clarkevans, correction=c("Donnelly","cdf"))
# point pattern broken down by tessellation
data(swedishpines)
tes <- quadrats(swedishpines, 5, 5)
B <- by(swedishpines, tes, clarkevans, correction="Donnelly")
unlist(lapply(B, as.numeric))

cauchy.estK

Fit the Neyman-Scott cluster process with Cauchy kernel

Description
Fits the Neyman-Scott Cluster point process with Cauchy kernel to a point pattern dataset by the
Method of Minimum Contrast.
Usage
cauchy.estK(X, startpar=c(kappa=1,scale=1), lambda=NULL,
q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...)
Arguments
X
startpar
lambda
q,p
rmin, rmax
...

Data to which the model will be fitted. Either a point pattern or a summary
statistic. See Details.
Vector of starting values for the parameters of the model.
Optional. An estimate of the intensity of the point process.
Optional. Exponents for the contrast criterion.
Optional. The interval of r values for the contrast criterion.
Optional arguments passed to optim to control the optimisation algorithm. See
Details.

178

cauchy.estK

Details
This algorithm fits the Neyman-Scott cluster point process model with Cauchy kernel to a point
pattern dataset by the Method of Minimum Contrast, using the K function.
The argument X can be either
a point pattern: An object of class "ppp" representing a point pattern dataset. The K function of
the point pattern will be computed using Kest, and the method of minimum contrast will be
applied to this.
a summary statistic: An object of class "fv" containing the values of a summary statistic, computed for a point pattern dataset. The summary statistic should be the K function, and this
object should have been obtained by a call to Kest or one of its relatives.
The algorithm fits the Neyman-Scott cluster point process with Cauchy kernel to X, by finding the
parameters of the Matern Cluster model which give the closest match between the theoretical K
function of the Matern Cluster process and the observed K function. For a more detailed explanation of the Method of Minimum Contrast, see mincontrast.
The model is described in Jalilian et al (2013). It is a cluster process formed by taking a pattern
of parent points, generated according to a Poisson process with intensity κ, and around each parent
point, generating a random number of offspring points, such that the number of offspring of each
parent is a Poisson random variable with mean µ, and the locations of the offspring points of one
parent follow a common distribution described in Jalilian et al (2013).
If the argument lambda is provided, then this is used as the value of the point process intensity λ.
Otherwise, if X is a point pattern, then λ will be estimated from X. If X is a summary statistic and
lambda is missing, then the intensity λ cannot be estimated, and the parameter µ will be returned
as NA.
The remaining arguments rmin,rmax,q,p control the method of minimum contrast; see mincontrast.
The corresponding model can be simulated using rCauchy.
For computational reasons, the optimisation procedure uses the parameter eta2, which is equivalent
to 4 * scale^2 where scale is the scale parameter for the model as used in rCauchy.
Homogeneous or inhomogeneous Neyman-Scott/Cauchy models can also be fitted using the function kppm and the fitted models can be simulated using simulate.kppm.
The optimisation algorithm can be controlled through the additional arguments "..." which are
passed to the optimisation function optim. For example, to constrain the parameter values to a
certain range, use the argument method="L-BFGS-B" to select an optimisation algorithm that respects box constraints, and use the arguments lower and upper to specify (vectors of) minimum
and maximum values for each parameter.
Value
An object of class "minconfit". There are methods for printing and plotting this object. It contains
the following main components:
par

Vector of fitted parameter values.

fit

Function value table (object of class "fv") containing the observed values of the
summary statistic (observed) and the theoretical values of the summary statistic
computed from the fitted model parameters.

Author(s)

Abdollah Jalilian and Rasmus Waagepetersen. Adapted for spatstat by Adrian Baddeley 
and Rolf Turner 
See Also
hyperframe, as.hyperframe
Examples
lambda <- runif(5, min=10, max=30)
X <- lapply(as.list(lambda), function(x) { rpoispp(x) })
h <- hyperframe(lambda=lambda, X=X)
g <- hyperframe(id=letters[1:5], Y=rev(X))
gh <- cbind(h, g)
hh <- rbind(h, h)

CDF

Cumulative Distribution Function From Kernel Density Estimate

Description
Given a kernel estimate of a probability density, compute the corresponding cumulative distribution
function.
Usage
CDF(f, ...)
## S3 method for class 'density'
CDF(f, ..., warn = TRUE)
Arguments
f

Density estimate (object of class "density").

...

Ignored.

warn

Logical value indicating whether to issue a warning if the density estimate f had
to be renormalised because it was computed in a restricted interval.

Details
CDF is generic, with a method for class "density".
This calculates the cumulative distribution function whose probability density has been estimated
and stored in the object f. The object f must belong to the class "density", and would typically
have been obtained from a call to the function density.

cdf.test

183

Value
A function, which can be applied to any numeric value or vector of values.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
density, quantile.density
Examples
b <- density(runif(10))
f <- CDF(b)
f(0.5)
plot(f)

cdf.test

Spatial Distribution Test for Point Pattern or Point Process Model

Description
Performs a test of goodness-of-fit of a point process model. The observed and predicted distributions of the values of a spatial covariate are compared using either the Kolmogorov-Smirnov test,
Cramér-von Mises test or Anderson-Darling test. For non-Poisson models, a Monte Carlo test is
used.
Usage
cdf.test(...)
## S3 method for class 'ppp'
cdf.test(X, covariate, test=c("ks", "cvm", "ad"), ...,
interpolate=TRUE, jitter=TRUE)
## S3 method for class 'ppm'
cdf.test(model, covariate, test=c("ks", "cvm", "ad"), ...,
interpolate=TRUE, jitter=TRUE, nsim=99, verbose=TRUE)
## S3 method for class 'lpp'
cdf.test(X, covariate, test=c("ks", "cvm", "ad"), ...,
interpolate=TRUE, jitter=TRUE)
## S3 method for class 'lppm'
cdf.test(model, covariate, test=c("ks", "cvm", "ad"),
...,
interpolate=TRUE, jitter=TRUE, nsim=99, verbose=TRUE)
## S3 method for class 'slrm'
cdf.test(model, covariate, test=c("ks", "cvm", "ad"), ..., modelname=NULL, covname=NULL)

184

cdf.test

Arguments
X

A point pattern (object of class "ppp" or "lpp").

model

A fitted point process model (object of class "ppm" or "lppm") or fitted spatial
logistic regression (object of class "slrm").

covariate

The spatial covariate on which the test will be based. A function, a pixel image
(object of class "im"), a list of pixel images, or one of the characters "x" or "y"
indicating the Cartesian coordinates.

test

Character string identifying the test to be performed: "ks" for KolmogorovSmirnov test, "cvm" for Cramér-von Mises test or "ad" for Anderson-Darling
test.

...

Arguments passed to ks.test (from the stats package) or cvm.test or ad.test
(from the goftest package) to control the test.

interpolate

Logical flag indicating whether to interpolate pixel images. If interpolate=TRUE,
the value of the covariate at each point of X will be approximated by interpolating the nearby pixel values. If interpolate=FALSE, the nearest pixel value will
be used.

jitter

Logical flag. If jitter=TRUE, values of the covariate will be slightly perturbed
at random, to avoid tied values in the test.
modelname,covname
Character strings giving alternative names for model and covariate to be used
in labelling plot axes.
nsim

Number of simulated realisations from the model to be used for the Monte Carlo
test, when model is not a Poisson process.

verbose

Logical value indicating whether to print progress reports when performing a
Monte Carlo test.

Details
These functions perform a goodness-of-fit test of a Poisson or Gibbs point process model fitted
to point pattern data. The observed distribution of the values of a spatial covariate at the data
points, and the predicted distribution of the same values under the model, are compared using the
Kolmogorov-Smirnov test, the Cramér-von Mises test or the Anderson-Darling test. For Gibbs
models, a Monte Carlo test is performed using these test statistics.
The function cdf.test is generic, with methods for point patterns ("ppp" or "lpp"), point process
models ("ppm" or "lppm") and spatial logistic regression models ("slrm").
• If X is a point pattern dataset (object of class "ppp"), then cdf.test(X, ...) performs
a goodness-of-fit test of the uniform Poisson point process (Complete Spatial Randomness,
CSR) for this dataset. For a multitype point pattern, the uniform intensity is assumed to depend on the type of point (sometimes called Complete Spatial Randomness and Independence,
CSRI).
• If model is a fitted point process model (object of class "ppm" or "lppm") then cdf.test(model, ...)
performs a test of goodness-of-fit for this fitted model.
• If model is a fitted spatial logistic regression (object of class "slrm") then cdf.test(model, ...)
performs a test of goodness-of-fit for this fitted model.
The test is performed by comparing the observed distribution of the values of a spatial covariate
at the data points, and the predicted distribution of the same covariate under the model, using a
classical goodness-of-fit test. Thus, you must nominate a spatial covariate for this test.

cdf.test

185

If X is a point pattern that does not have marks, the argument covariate should be either a
function(x,y) or a pixel image (object of class "im" containing the values of a spatial function, or one of the characters "x" or "y" indicating the Cartesian coordinates. If covariate is an
image, it should have numeric values, and its domain should cover the observation window of the
model. If covariate is a function, it should expect two arguments x and y which are vectors of
coordinates, and it should return a numeric vector of the same length as x and y.
If X is a multitype point pattern, the argument covariate can be either a function(x,y,marks),
or a pixel image, or a list of pixel images corresponding to each possible mark value, or one of the
characters "x" or "y" indicating the Cartesian coordinates.
First the original data point pattern is extracted from model. The values of the covariate at these
data points are collected.
The predicted distribution of the values of the covariate under the fitted model is computed as
follows. The values of the covariate at all locations in the observation window are evaluated,
weighted according to the point process intensity of the fitted model, and compiled into a cumulative
distribution function F using ewcdf.
The probability integral transformation is then applied: the values of the covariate at the original
data points are transformed by the predicted cumulative distribution function F into numbers between 0 and 1. If the model is correct, these numbers are i.i.d. uniform random numbers. The A
goodness-of-fit test of the uniform distribution is applied to these numbers using stats::ks.test,
goftest::cvm.test or goftest::ad.test.
This test was apparently first described (in the context of spatial data, and using KolmogorovSmirnov) by Berman (1986). See also Baddeley et al (2005).
If model is not a Poisson process, then a Monte Carlo test is performed, by generating nsim point
patterns which are simulated realisations of the model, re-fitting the model to each simulated point
pattern, and calculating the test statistic for each fitted model. The Monte Carlo p value is determined by comparing the simulated values of the test statistic with the value for the original data.
The return value is an object of class "htest" containing the results of the hypothesis test. The
print method for this class gives an informative summary of the test outcome.
The return value also belongs to the class "cdftest" for which there is a plot method plot.cdftest.
The plot method displays the empirical cumulative distribution function of the covariate at the data
points, and the predicted cumulative distribution function of the covariate under the model, plotted
against the value of the covariate.
The argument jitter controls whether covariate values are randomly perturbed, in order to avoid
ties. If the original data contains any ties in the covariate (i.e. points with equal values of the
covariate), and if jitter=FALSE, then the Kolmogorov-Smirnov test implemented in ks.test will
issue a warning that it cannot calculate the exact p-value. To avoid this, if jitter=TRUE each value
of the covariate will be perturbed by adding a small random value. The perturbations are normally
distributed with standard deviation equal to one hundredth of the range of values of the covariate.
This prevents ties, and the p-value is still correct. There is a very slight loss of power.
Value
An object of class "htest" containing the results of the test. See ks.test for details. The return
value can be printed to give an informative summary of the test.
The value also belongs to the class "cdftest" for which there is a plot method.
Warning
The outcome of the test involves a small amount of random variability, because (by default) the
coordinates are randomly perturbed to avoid tied values. Hence, if cdf.test is executed twice, the

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cdf.test
p-values will not be exactly the same. To avoid this behaviour, set jitter=FALSE.

Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Baddeley, A., Turner, R., Møller, J. and Hazelton, M. (2005) Residual analysis for spatial point
processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
Berman, M. (1986) Testing for spatial association between a point process and another stochastic
process. Applied Statistics 35, 54–62.
See Also
plot.cdftest, quadrat.test, berman.test, ks.test, cvm.test, ad.test, ppm
Examples
op <- options(useFancyQuotes=FALSE)
# test of CSR using x coordinate
cdf.test(nztrees, "x")
cdf.test(nztrees, "x", "cvm")
cdf.test(nztrees, "x", "ad")
# test of CSR using a function of x and y
fun <- function(x,y){2* x + y}
cdf.test(nztrees, fun)
# test of CSR using an image covariate
funimage <- as.im(fun, W=Window(nztrees))
cdf.test(nztrees, funimage)
# fit inhomogeneous Poisson model and test
model <- ppm(nztrees ~x)
cdf.test(model, "x")
if(interactive()) {
# synthetic data: nonuniform Poisson process
X <- rpoispp(function(x,y) { 100 * exp(x) }, win=square(1))
# fit uniform
fit0 <- ppm(X
# fit correct
fit1 <- ppm(X

}

Poisson process
~1)
nonuniform Poisson process
~x)

# test wrong model
cdf.test(fit0, "x")
# test right model
cdf.test(fit1, "x")

# multitype point pattern
cdf.test(amacrine, "x")

cdf.test.mppm

187

yimage <- as.im(function(x,y){y}, W=Window(amacrine))
cdf.test(ppm(amacrine ~marks+y), yimage)
options(op)

cdf.test.mppm

Spatial Distribution Test for Multiple Point Process Model

Description
Performs a spatial distribution test of a point process model fitted to multiple spatial point patterns.
The test compares the observed and predicted distributions of the values of a spatial covariate, using
either the Kolmogorov-Smirnov, Cramér-von Mises or Anderson-Darling test of goodness-of-fit.

Usage
## S3 method for class 'mppm'
cdf.test(model, covariate, test=c("ks", "cvm", "ad"), ...,
nsim=19, verbose=TRUE, interpolate=FALSE, fast=TRUE, jitter=TRUE)
Arguments
model

An object of class "mppm" representing a point process model fitted to multiple
spatial point patterns.

covariate

The spatial covariate on which the test will be based. A function, a pixel image,
a list of functions, a list of pixel images, a hyperframe, a character string containing the name of one of the covariates in model, or one of the strings "x" or
"y".

test

Character string identifying the test to be performed: "ks" for KolmogorovSmirnov test, "cvm" for Cramér-von Mises test or "ad" for Anderson-Darling
test.

...

Arguments passed to cdf.test to control the test.

nsim

Number of simulated realisations which should be generated, if a Monte Carlo
test is required.

verbose

Logical flag indicating whether to print progress reports.

interpolate

Logical flag indicating whether to interpolate between pixel values when codecovariate is a pixel image. See Details.

fast

Logical flag. If TRUE, values of the covariate are only sampled at the original
quadrature points used to fit the model. If FALSE, values of the covariate are
sampled at all pixels, which can be slower by three orders of magnitude.

jitter

Logical flag. If TRUE, observed values of the covariate are perturbed by adding
small random values, to avoid tied observations.

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cdf.test.mppm

Details
This function is a method for the generic function cdf.test for the class mppm.
This function performs a goodness-of-fit test of a point process model that has been fitted to multiple
point patterns. The observed distribution of the values of a spatial covariate at the data points, and
the predicted distribution of the same values under the model, are compared using the KolmogorovSmirnov, Cramér-von Mises or Anderson-Darling test of goodness-of-fit. These are exact tests if the
model is Poisson; otherwise, for a Gibbs model, a Monte Carlo p-value is computed by generating
simulated realisations of the model and applying the selected goodness-of-fit test to each simulation.
The argument model should be a fitted point process model fitted to multiple point patterns (object
of class "mppm").
The argument covariate contains the values of a spatial function. It can be
• a function(x,y)
• a pixel image (object of class "im"
• a list of function(x,y), one for each point pattern
• a list of pixel images, one for each point pattern
• a hyperframe (see hyperframe) of which the first column will be taken as containing the
covariate
• a character string giving the name of one of the covariates in model
• one of the character strings "x" or "y", indicating the spatial coordinates.
If covariate is an image, it should have numeric values, and its domain should cover the observation window of the model. If covariate is a function, it should expect two arguments x and y
which are vectors of coordinates, and it should return a numeric vector of the same length as x and
y.
First the original data point pattern is extracted from model. The values of the covariate at these
data points are collected.
The predicted distribution of the values of the covariate under the fitted model is computed as
follows. The values of the covariate at all locations in the observation window are evaluated,
weighted according to the point process intensity of the fitted model, and compiled into a cumulative
distribution function F using ewcdf.
The probability integral transformation is then applied: the values of the covariate at the original data points are transformed by the predicted cumulative distribution function F into numbers between 0 and 1. If the model is correct, these numbers are i.i.d. uniform random numbers. A goodness-of-fit test of the uniform distribution is applied to these numbers using ks.test,
cvm.test or ad.test.
The argument interpolate determines how pixel values will be handled when codecovariate is a
pixel image. The value of the covariate at a data point is obtained by looking up the value of the
nearest pixel if interpolate=FALSE, or by linearly interpolating between the values of the four
nearest pixels if interpolate=TRUE. Linear interpolation is slower, but is sometimes necessary to
avoid tied values of the covariate arising when the pixel grid is coarse.
If model is a Poisson point process, then the Kolmogorov-Smirnov, Cramér-von Mises and AndersonDarling tests are theoretically exact. This test was apparently first described (in the context of spatial
data, and for Kolmogorov-Smirnov) by Berman (1986). See also Baddeley et al (2005).
If model is not a Poisson point process, then the Kolmogorov-Smirnov, Cramér-von Mises and
Anderson-Darling tests are biased. Instead they are used as the basis of a Monte Carlo test. First
nsim simulated realisations of the model will be generated. Each simulated realisation consists of
a list of simulated point patterns, one for each of the original data patterns. This can take a very

centroid.owin

189

long time. The model is then re-fitted to each simulation, and the refitted model is subjected to the
goodness-of-fit test described above. A Monte Carlo p-value is then computed by comparing the
p-value of the original test with the p-values obtained from the simulations.
Value
An object of class "cdftest" and "htest" containing the results of the test. See cdf.test for
details.
Author(s)
Adrian Baddeley , Ida-Maria Sintorn and Leanne Bischoff.
Implemented by Adrian Baddeley , Rolf Turner 
and Ege Rubak .
References
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. London: Chapman and Hall/CRC Press.
Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2005) Residual analysis for spatial point
processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
Berman, M. (1986) Testing for spatial association between a point process and another stochastic
process. Applied Statistics 35, 54–62.
See Also
cdf.test, quadrat.test, mppm
Examples
# three i.i.d. realisations of nonuniform Poisson process
lambda <- as.im(function(x,y) { 300 * exp(x) }, square(1))
dat <- hyperframe(X=list(rpoispp(lambda), rpoispp(lambda), rpoispp(lambda)))
# fit uniform Poisson process
fit0 <- mppm(X~1, dat)
# fit correct nonuniform Poisson process
fit1 <- mppm(X~x, dat)
# test wrong model
cdf.test(fit0, "x")
# test right model
cdf.test(fit1, "x")

centroid.owin

Centroid of a window

Description
Computes the centroid (centre of mass) of a window

190

centroid.owin

Usage
centroid.owin(w, as.ppp = FALSE)
Arguments
w

A window

as.ppp

Logical flag indicating whether to return the centroid as a point pattern (ppp
object)

Details
The centroid of the window w is computed. The centroid (“centre of mass”) is the point whose x
and y coordinates are the mean values of the x and y coordinates of all points in the window.
The argument w should be a window (an object of class "owin", see owin.object for details) or
can be given in any format acceptable to as.owin().
The calculation uses an exact analytic formula for the case of polygonal windows.
Note that the centroid of a window is not necessarily inside the window, unless the window is
convex. If as.ppp=TRUE and the centroid of w lies outside w, then the window of the returned point
pattern will be a rectangle containing the original window (using as.rectangle.
Value
Either a list with components x, y, or a point pattern (of class ppp) consisting of a single point,
giving the coordinates of the centroid of the window w.
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak 
See Also
owin, as.owin
Examples
w <- owin(c(0,1),c(0,1))
centroid.owin(w)
# returns 0.5, 0.5
data(demopat)
w <- Window(demopat)
# an irregular window
cent <- centroid.owin(w, as.ppp = TRUE)
## Not run:
plot(cent)
# plot the window and its centroid
## End(Not run)
wapprox <- as.mask(w)
# pixel approximation of window

chop.tess

191

## Not run:
points(centroid.owin(wapprox))
# should be indistinguishable
## End(Not run)

chop.tess

Subdivide a Window or Tessellation using a Set of Lines

Description
Divide a given window into tiles delineated by a set of infinite straight lines, obtaining a tessellation
of the window. Alternatively, given a tessellation, divide each tile of the tessellation into sub-tiles
delineated by the lines.
Usage
chop.tess(X, L)
Arguments
X

A window (object of class "owin") or tessellation (object of class "tess") to be
subdivided by lines.

L

A set of infinite straight lines (object of class "infline")

Details
The argument L should be a set of infinite straight lines in the plane (stored in an object L of class
"infline" created by the function infline).
If X is a window, then it is divided into tiles delineated by the lines in L.
If X is a tessellation, then each tile of X is subdivided into sub-tiles delineated by the lines in L.
The result is a tessellation.
Value
A tessellation (object of class "tess").
Warning
If X is a non-convex window, or a tessellation containing non-convex tiles, then chop.tess(X,L)
may contain a tile which consists of several unconnected pieces.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
infline, clip.infline

192

circdensity

Examples
L <- infline(p=1:3, theta=pi/4)
W <- square(4)
chop.tess(W, L)

circdensity

Density Estimation for Circular Data

Description
Computes a kernel smoothed estimate of the probability density for angular data.
Usage
circdensity(x, sigma = "nrd0", ...,
bw = NULL,
weights=NULL, unit = c("degree", "radian"))
Arguments
x
sigma
bw
...
weights
unit

Numeric vector, containing angular data.
Smoothing bandwidth, or bandwidth selection rule, passed to density.default.
Alternative to sigma for consistency with other functions.
Additional arguments passed to density.default, such as kernel and weights.
Optional numeric vector of weights for the data in x.
The unit of angle in which x is expressed.

Details
The angular values x are smoothed using (by default) the wrapped Gaussian kernel with standard
deviation sigma.
Value
An object of class "density" (produced by density.default) which can be plotted by plot or
by rose.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
density.default), rose.
Examples
ang <- runif(1000, max=360)
rose(circdensity(ang, 12))

clarkevans

clarkevans

193

Clark and Evans Aggregation Index

Description
Computes the Clark and Evans aggregation index R for a spatial point pattern.
Usage
clarkevans(X, correction=c("none", "Donnelly", "cdf"),
clipregion=NULL)
Arguments
X

A spatial point pattern (object of class "ppp").

correction

Character vector. The type of edge correction(s) to be applied.

clipregion

Clipping region for the guard area correction. A window (object of class "owin").
See Details.

Details
The Clark and Evans (1954) aggregation index R is a crude measure of clustering or ordering of a
point pattern. It is the ratio of the observed mean nearest neighbour distance in the pattern to that
expected for a Poisson point process of the same intensity. A value R > 1 suggests ordering, while
R < 1 suggests clustering.
Without correction for edge effects, the value of R will be positively biased. Edge effects arise
because, for a point of X close to the edge of the window, the true nearest neighbour may actually
lie outside the window. Hence observed nearest neighbour distances tend to be larger than the true
nearest neighbour distances.
The argument correction specifies an edge correction or several edge corrections to be applied.
It is a character vector containing one or more of the options "none", "Donnelly", "guard" and
"cdf" (which are recognised by partial matching). These edge corrections are:
"none": No edge correction is applied.
"Donnelly": Edge correction of Donnelly (1978), available for rectangular windows only. The
theoretical expected value of mean nearest neighbour distance under a Poisson process is
adjusted for edge effects by the edge correction of Donnelly (1978). The value of R is the
ratio of the observed mean nearest neighbour distance to this adjusted theoretical mean.
"guard": Guard region or buffer area method. The observed mean nearest neighbour distance for
the point pattern X is re-defined by averaging only over those points of X that fall inside the
sub-window clipregion.
"cdf": Cumulative Distribution Function method. The nearest neighbour distance distribution
function G(r) of the stationary point process is estimated by Gest using the Kaplan-Meier
type edge correction. Then the mean of the distribution is calculated from the cdf.
Alternatively correction="all" selects all options.
If the argument clipregion is given, then the selected edge corrections will be assumed to include
correction="guard".
To perform a test based on the Clark-Evans index, see clarkevans.test.

194

clarkevans

Value
A numeric value, or a numeric vector with named components
naive

R without edge correction

Donnelly

R using Donnelly edge correction

guard

R using guard region

cdf

R using cdf method

(as selected by correction). The value of the Donnelly component will be NA if the window of X
is not a rectangle.

Author(s)
John Rudge  with modifications by Adrian Baddeley 

References
Clark, P.J. and Evans, F.C. (1954) Distance to nearest neighbour as a measure of spatial relationships
in populations Ecology 35, 445–453.
Donnelly, K. (1978) Simulations to determine the variance and edge-effect of total nearest neighbour distance. In I. Hodder (ed.) Simulation studies in archaeology, Cambridge/New York: Cambridge University Press, pp 91–95.

See Also
clarkevans.test, hopskel, nndist, Gest

Examples
# Example of a clustered pattern
clarkevans(redwood)
# Example of an ordered pattern
clarkevans(cells)
# Random pattern
X <- rpoispp(100)
clarkevans(X)
# How to specify a clipping region
clip1 <- owin(c(0.1,0.9),c(0.1,0.9))
clip2 <- erosion(Window(cells), 0.1)
clarkevans(cells, clipregion=clip1)
clarkevans(cells, clipregion=clip2)

clarkevans.test

clarkevans.test

195

Clark and Evans Test

Description
Performs the Clark-Evans test of aggregation for a spatial point pattern.
Usage
clarkevans.test(X, ...,
correction="none",
clipregion=NULL,
alternative=c("two.sided", "less", "greater",
"clustered", "regular"),
nsim=999)
Arguments
X
...
correction
clipregion
alternative
nsim

A spatial point pattern (object of class "ppp").
Ignored.
Character string. The type of edge correction to be applied. See clarkevans
Clipping region for the guard area correction. A window (object of class "owin").
See clarkevans
String indicating the type of alternative for the hypothesis test. Partially matched.
Number of Monte Carlo simulations to perform, if a Monte Carlo p-value is
required.

Details
This command uses the Clark and Evans (1954) aggregation index R as the basis for a crude test of
clustering or ordering of a point pattern.
The Clark-Evans index is computed by the function clarkevans. See the help for clarkevans for
information about the Clark-Evans index R and about the arguments correction and clipregion.
This command performs a hypothesis test of clustering or ordering of the point pattern X. The
null hypothesis is Complete Spatial Randomness, i.e.\ a uniform Poisson process. The alternative
hypothesis is specified by the argument alternative:
• alternative="less" or alternative="clustered": the alternative hypothesis is that R <
1 corresponding to a clustered point pattern;
• alternative="greater" or alternative="regular": the alternative hypothesis is that R >
1 corresponding to a regular or ordered point pattern;
• alternative="two.sided": the alternative hypothesis is that R 6= 1 corresponding to a
clustered or regular pattern.
The Clark-Evans index R is computed for the data as described in clarkevans.
If correction="none" and nsim is missing, the p-value for the test is computed by standardising R as proposed by Clark and Evans (1954) and referring the statistic to the standard Normal
distribution.
Otherwise, the p-value for the test is computed by Monte Carlo simulation of nsim realisations of
Complete Spatial Randomness conditional on the observed number of points.

196

clickbox

Value
An object of class "htest" representing the result of the test.
Author(s)
Adrian Baddeley 
References
Clark, P.J. and Evans, F.C. (1954) Distance to nearest neighbour as a measure of spatial relationships
in populations. Ecology 35, 445–453.
Donnelly, K. (1978) Simulations to determine the variance and edge-effect of total nearest neighbour distance. In Simulation methods in archaeology, Cambridge University Press, pp 91–95.
See Also
clarkevans, hopskel.test
Examples
# Redwood data - clustered
clarkevans.test(redwood)
clarkevans.test(redwood, alternative="clustered")

clickbox

Interactively Define a Rectangle

Description
Allows the user to specify a rectangle by point-and-click in the display.
Usage
clickbox(add=TRUE, ...)
Arguments
add

Logical value indicating whether to create a new plot (add=FALSE) or draw over
the existing plot (add=TRUE).

...

Graphics arguments passed to polygon to plot the box.

Details
This function allows the user to create a rectangular window by interactively clicking on the screen
display.
The user is prompted to point the mouse at any desired locations for two corners of the rectangle,
and click the left mouse button to add each point.
The return value is a window (object of class "owin") representing the rectangle.
This function uses the R command locator to input the mouse clicks. It only works on screen
devices such as ‘X11’, ‘windows’ and ‘quartz’.

clickdist

197

Value
A window (object of class "owin") representing the selected rectangle.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
clickpoly, clickppp, clickdist, locator

clickdist

Interactively Measure Distance

Description
Measures the distance between two points which the user has clicked on.
Usage
clickdist()
Details
This function allows the user to measure the distance between two spatial locations, interactively,
by clicking on the screen display.
When clickdist() is called, the user is expected to click two points in the current graphics device.
The distance between these points will be returned.
This function uses the R command locator to input the mouse clicks. It only works on screen
devices such as ‘X11’, ‘windows’ and ‘quartz’.
Value
A single nonnegative number.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
locator, clickppp, clicklpp, clickpoly, clickbox

198

clickjoin

clickjoin

Interactively join vertices on a plot

Description
Given a point pattern representing a set of vertices, this command gives a point-and-click interface
allowing the user to join pairs of selected vertices by edges.
Usage
clickjoin(X, ..., add = TRUE, m = NULL, join = TRUE)
Arguments
X

Point pattern of vertices. An object of class "ppp".

...

Arguments passed to segments to control the plotting of the new edges.

add

Logical. Whether the point pattern X should be added to the existing plot (add=TRUE)
or a new plot should be created (add=FALSE).

m

Optional. Logical matrix specifying an initial set of edges. There is an edge
between vertices i and j if m[i,j] = TRUE.

join

Optional. If TRUE, then each user click will join a pair of vertices. If FALSE, then
each user click will delete an existing edge. This is only relevant if m is supplied.

Details
This function makes it easier for the user to create a linear network or a planar graph, given a set of
vertices.
The function first displays the point pattern X, then repeatedly prompts the user to click on a pair of
points in X. Each selected pair of points will be joined by an edge. The function returns a logical
matrix which has entries equal to TRUE for each pair of vertices joined by an edge.
The selection of points is performed using identify.ppp which typically expects the user to click
the left mouse button. This point-and-click interaction continues until the user terminates it, by
pressing the middle mouse button, or pressing the right mouse button and selecting stop.
The return value can be used in linnet to create a linear network.
Value
Logical matrix m with value m[i,j] = TRUE for every pair of vertices X[i] and X[j] that should
be joined by an edge.
Author(s)
Adrian Baddeley .
See Also
linnet, clickppp

clicklpp

clicklpp

199

Interactively Add Points on a Linear Network

Description
Allows the user to create a point pattern on a linear network by point-and-click in the display.
Usage
clicklpp(L, n=NULL, types=NULL, ...,
add=FALSE, main=NULL, hook=NULL)
Arguments
L

Linear network on which the points will be placed. An object of class "linnet".

n

Number of points to be added (if this is predetermined).

types

Vector of types, when creating a multitype point pattern.

...

Optional extra arguments to be passed to locator to control the display.

add

Logical value indicating whether to create a new plot (add=FALSE) or draw over
the existing plot (add=TRUE).

main

Main heading for plot.

hook

For internal use only. Do not use this argument.

Details
This function allows the user to create a point pattern on a linear network by interactively clicking
on the screen display.
First the linear network L is plotted on the current screen device. Then the user is prompted to point
the mouse at any desired locations and click the left mouse button to add each point. Interactive
input stops after n clicks (if n was given) or when the middle mouse button is pressed.
The return value is a point pattern on the network L, containing the locations of all the clicked
points, after they have been projected onto the network L. Any points that were clicked outside the
bounding window of the network will be ignored.
If the argument types is given, then a multitype point pattern will be created. The user is prompted
to input the locations of points of type type[i], for each successive index i. (If the argument n was
given, there will be n points of each type.) The return value is a multitype point pattern on a linear
network.
This function uses the R command locator to input the mouse clicks. It only works on screen devices such as ‘X11’, ‘windows’ and ‘quartz’. Arguments that can be passed to locator through ...
include pch (plotting character), cex (character expansion factor) and col (colour). See locator
and par.
Value
A point pattern (object of class "lpp").

200

clickpoly

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak , based on an idea by Dominic Schuhmacher.
See Also
clickppp, identify.lpp, locator, clickpoly, clickbox, clickdist

clickpoly

Interactively Define a Polygon

Description
Allows the user to create a polygon by point-and-click in the display.
Usage
clickpoly(add=FALSE, nv=NULL, np=1, ...)
Arguments
add

Logical value indicating whether to create a new plot (add=FALSE) or draw over
the existing plot (add=TRUE).

nv

Number of vertices of the polygon (if this is predetermined).

np

Number of polygons to create.

...

Arguments passed to locator to control the interactive plot, and to polygon to
plot the polygons.

Details
This function allows the user to create a polygonal window by interactively clicking on the screen
display.
The user is prompted to point the mouse at any desired locations for the polygon vertices, and click
the left mouse button to add each point. Interactive input stops after nv clicks (if nv was given) or
when the middle mouse button is pressed.
The return value is a window (object of class "owin") representing the polygon.
This function uses the R command locator to input the mouse clicks. It only works on screen devices such as ‘X11’, ‘windows’ and ‘quartz’. Arguments that can be passed to locator through ...
include pch (plotting character), cex (character expansion factor) and col (colour). See locator
and par.
Multiple polygons can also be drawn, by specifying np > 1. The polygons must be disjoint. The
result is a single window object consisting of all the polygons.
Value
A window (object of class "owin") representing the polygon.
Author(s)
Adrian Baddeley  and Rolf Turner .

clickppp

201

See Also
identify.ppp, clickbox, clickppp, clickdist, locator

clickppp

Interactively Add Points

Description
Allows the user to create a point pattern by point-and-click in the display.
Usage
clickppp(n=NULL, win=square(1), types=NULL, ..., add=FALSE,
main=NULL, hook=NULL)
Arguments
n

Number of points to be added (if this is predetermined).

win

Window in which to create the point pattern. An object of class "owin".

types

Vector of types, when creating a multitype point pattern.

...

Optional extra arguments to be passed to locator to control the display.

add

Logical value indicating whether to create a new plot (add=FALSE) or draw over
the existing plot (add=TRUE).

main

Main heading for plot.

hook

For internal use only. Do not use this argument.

Details
This function allows the user to create a point pattern by interactively clicking on the screen display.
First the window win is plotted on the current screen device. Then the user is prompted to point the
mouse at any desired locations and click the left mouse button to add each point. Interactive input
stops after n clicks (if n was given) or when the middle mouse button is pressed.
The return value is a point pattern containing the locations of all the clicked points inside the original
window win, provided that all of the clicked locations were inside this window. Otherwise, the
window is expanded to a box large enough to contain all the points (as well as containing the
original window).
If the argument types is given, then a multitype point pattern will be created. The user is prompted
to input the locations of points of type type[i], for each successive index i. (If the argument n was
given, there will be n points of each type.) The return value is a multitype point pattern.
This function uses the R command locator to input the mouse clicks. It only works on screen devices such as ‘X11’, ‘windows’ and ‘quartz’. Arguments that can be passed to locator through ...
include pch (plotting character), cex (character expansion factor) and col (colour). See locator
and par.
Value
A point pattern (object of class "ppp").

202

clip.infline

Author(s)
Original by Dominic Schuhmacher. Adapted by Adrian Baddeley 
and Rolf Turner .
See Also
identify.ppp, locator, clickpoly, clickbox, clickdist

clip.infline

Intersect Infinite Straight Lines with a Window

Description
Take the intersection between a set of infinite straight lines and a window, yielding a set of line
segments.
Usage
clip.infline(L, win)
Arguments
L

Object of class "infline" specifying a set of infinite straight lines in the plane.

win

Window (object of class "owin").

Details
This function computes the intersection between a set of infinite straight lines in the plane (stored
in an object L of class "infline" created by the function infline) and a window win. The result
is a pattern of line segments. Each line segment carries a mark indicating which line it belongs to.
Value
A line segment pattern (object of class "psp") with a single column of marks.
Author(s)
Adrian Baddeley  and Rolf Turner .
See Also
infline,psp.
To divide a window into pieces using infinite lines, use chop.tess.
Examples
L <- infline(p=1:3, theta=pi/4)
W <- square(4)
clip.infline(L, W)

closepairs

203

closepairs

Close Pairs of Points

Description
Low-level functions to find all close pairs of points.
Usage
closepaircounts(X, r)
crosspaircounts(X, Y, r)
closepairs(X, rmax, ...)
## S3 method for class 'ppp'
closepairs(X, rmax, twice=TRUE,
what=c("all","indices","ijd"),
distinct=TRUE, neat=TRUE, ...)
crosspairs(X, Y, rmax, ...)
## S3 method for class 'ppp'
crosspairs(X, Y, rmax, what=c("all", "indices", "ijd"), ...)
Arguments
X,Y

Point patterns (objects of class "ppp").

r,rmax

Maximum distance between pairs of points to be counted as close pairs.

twice

Logical value indicating whether all ordered pairs of close points should be returned. If twice=TRUE (the default), each pair will appear twice in the output,
as (i,j) and again as (j,i). If twice=FALSE, then each pair will appear only
once, as the pair (i,j) with i < j.

what

String specifying the data to be returned for each close pair of points. If what="all"
(the default) then the returned information includes the indices i,j of each pair,
their x,y coordinates, and the distance between them. If what="indices" then
only the indices i,j are returned. If what="ijd" then the indices i,j and the
distance d are returned.

distinct

Logical value indicating whether to return only the pairs of points with different
indices i and j (distinct=TRUE, the default) or to also include the pairs where
i=j (distinct=FALSE).

neat

Logical value indicating whether to ensure that i < j in each output pair, when
twice=FALSE.

...

Extra arguments, ignored by methods.

Details
These are the efficient low-level functions used by spatstat to find all close pairs of points in a point
pattern or all close pairs between two point patterns.

204

closepairs
closepaircounts(X,r) counts the number of neighbours for each point in the pattern X. That is,
for each point X[i], it counts the number of other points X[j] with j != i such that d(X[i],X[j]) <= r
where d denotes Euclidean distance. The result is an integer vector v such that v[i] is the number
of neighbours of X[i].
crosspaircounts(X,Y,r) counts, for each point in the pattern X, the number of neighbours in the
pattern Y. That is, for each point X[i], it counts the number of points Y[j] such that d(X[i],X[j]) <= r.
The result is an integer vector v such that v[i] is the number of neighbours of X[i] in the pattern
Y.
closepairs(X,rmax) identifies all pairs of distinct neighbours in the pattern X and returns them.
The result is a list with the following components:
i Integer vector of indices of the first point in each pair.
j Integer vector of indices of the second point in each pair.
xi,yi Coordinates of the first point in each pair.
xj,yj Coordinates of the second point in each pair.
dx Equal to xj-xi
dy Equal to yj-yi
d Euclidean distance between each pair of points.
If what="indices" then only the components i and j are returned. This is slightly faster and more
efficient with use of memory.
crosspairs(X,rmax) identifies all pairs of neighbours (X[i], Y[j]) between the patterns X and
Y, and returns them. The result is a list with the same format as for closepairs.

Value
For closepaircounts and crosspaircounts, an integer vector of length equal to the number of
points in X.
For closepairs and crosspairs, a list with components i and j, and possibly other components
as described under Details.
Warning about accuracy
The results of these functions may not agree exactly with the correct answer (as calculated by a
human) and may not be consistent between different computers and different installations of R. The
discrepancies arise in marginal cases where the interpoint distance is equal to, or very close to, the
threshold rmax.
Floating-point numbers in a computer are not mathematical Real Numbers: they are approximations
using finite-precision binary arithmetic. The approximation is accurate to a tolerance of about
.Machine$double.eps.
If the true interpoint distance d and the threshold rmax are equal, or if their difference is no more
than .Machine$double.eps, the result may be incorrect.
Author(s)
Adrian Baddeley 
and Rolf Turner 

closepairs.pp3

205

See Also
closepairs.pp3 for the corresponding functions for 3D point patterns.
Kest, Kcross, nndist, nncross, applynbd, markstat for functions which use these capabilities.
Examples
a <- closepaircounts(cells, 0.1)
sum(a)
Y <- split(amacrine)
b <- crosspaircounts(Y$on, Y$off, 0.1)
d <- closepairs(cells, 0.1)
e <- crosspairs(Y$on, Y$off, 0.1)

closepairs.pp3

Close Pairs of Points in 3 Dimensions

Description
Low-level functions to find all close pairs of points in three-dimensional point patterns.
Usage
## S3 method for class 'pp3'
closepairs(X, rmax, twice=TRUE,
what=c("all", "indices"),
distinct=TRUE, neat=TRUE, ...)
## S3 method for class 'pp3'
crosspairs(X, Y, rmax, what=c("all", "indices"), ...)
Arguments
X,Y

Point patterns in three dimensions (objects of class "pp3").

rmax

Maximum distance between pairs of points to be counted as close pairs.

twice

Logical value indicating whether all ordered pairs of close points should be returned. If twice=TRUE, each pair will appear twice in the output, as (i,j) and
again as (j,i). If twice=FALSE, then each pair will appear only once, as the
pair (i,j) such that i < j.

what

String specifying the data to be returned for each close pair of points. If what="all"
(the default) then the returned information includes the indices i,j of each pair,
their x,y,z coordinates, and the distance between them. If what="indices"
then only the indices i,j are returned.

distinct

Logical value indicating whether to return only the pairs of points with different
indices i and j (distinct=TRUE, the default) or to also include the pairs where
i=j (distinct=FALSE).

neat

Logical value indicating whether to ensure that i < j in each output pair, when
twice=FALSE.

...

Ignored.

206

closepairs.pp3

Details
These are the efficient low-level functions used by spatstat to find all close pairs of points in a
three-dimensional point pattern or all close pairs between two point patterns in three dimensions.
closepairs(X,rmax) identifies all pairs of neighbours in the pattern X and returns them. The result
is a list with the following components:
i Integer vector of indices of the first point in each pair.
j Integer vector of indices of the second point in each pair.
xi,yi,zi Coordinates of the first point in each pair.
xj,yj,zj Coordinates of the second point in each pair.
dx Equal to xj-xi
dy Equal to yj-yi
dz Equal to zj-zi
d Euclidean distance between each pair of points.
If what="indices" then only the components i and j are returned. This is slightly faster.
crosspairs(X,rmax) identifies all pairs of neighbours (X[i], Y[j]) between the patterns X and
Y, and returns them. The result is a list with the same format as for closepairs.
Value
A list with components i and j, and possibly other components as described under Details.
Warning about accuracy
The results of these functions may not agree exactly with the correct answer (as calculated by a
human) and may not be consistent between different computers and different installations of R. The
discrepancies arise in marginal cases where the interpoint distance is equal to, or very close to, the
threshold rmax.
Floating-point numbers in a computer are not mathematical Real Numbers: they are approximations
using finite-precision binary arithmetic. The approximation is accurate to a tolerance of about
.Machine$double.eps.
If the true interpoint distance d and the threshold rmax are equal, or if their difference is no more
than .Machine$double.eps, the result may be incorrect.
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak .
See Also
closepairs
Examples
X
Y
a
b

<<<<-

pp3(runif(10), runif(10), runif(10), box3(c(0,1)))
pp3(runif(10), runif(10), runif(10), box3(c(0,1)))
closepairs(X, 0.1)
crosspairs(X, Y, 0.1)

closetriples

207

closetriples

Close Triples of Points

Description
Low-level function to find all close triples of points.

Usage
closetriples(X, rmax)

Arguments
X

Point pattern (object of class "ppp" or "pp3").

rmax

Maximum distance between each pair of points in a triple.

Details
This low-level function finds all triples of points in a point pattern in which each pair lies closer
than rmax.

Value
A data frame with columns i,j,k giving the indices of the points in each triple, and a column diam
giving the diameter (maximum pairwise distance) in the triple.

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .

See Also
closepairs, Tstat.

Examples
closetriples(redwoodfull, 0.02)
closetriples(redwoodfull, 0.005)

208

closing

closing

Morphological Closing

Description
Perform morphological closing of a window, a line segment pattern or a point pattern.
Usage
closing(w, r, ...)
## S3 method for class 'owin'
closing(w, r, ..., polygonal=NULL)
## S3 method for class 'ppp'
closing(w, r, ..., polygonal=TRUE)
## S3 method for class 'psp'
closing(w, r, ..., polygonal=TRUE)
Arguments
w

A window (object of class "owin" or a line segment pattern (object of class
"psp") or a point pattern (object of class "ppp").

r

positive number: the radius of the closing.

...

extra arguments passed to as.mask controlling the pixel resolution, if a pixel
approximation is used

polygonal

Logical flag indicating whether to compute a polygonal approximation to the
erosion (polygonal=TRUE) or a pixel grid approximation (polygonal=FALSE).

Details
The morphological closing (Serra, 1982) of a set W by a distance r > 0 is the set of all points that
cannot be separated from W by any circle of radius r. That is, a point x belongs to the closing W ∗
if it is impossible to draw any circle of radius r that has x on the inside and W on the outside. The
closing W ∗ contains the original set W .
For a small radius r, the closing operation has the effect of smoothing out irregularities in the boundary of W . For larger radii, the closing operation smooths out concave features in the boundary. For
very large radii, the closed set W ∗ becomes more and more convex.
The algorithm applies dilation followed by erosion.
Value
If r > 0, an object of class "owin" representing the closed region. If r=0, the result is identical to
w.
Author(s)
Adrian Baddeley 
and Rolf Turner 

clusterfield

209

References
Serra, J. (1982) Image analysis and mathematical morphology. Academic Press.
See Also
opening for the opposite operation.
dilation, erosion for the basic operations.
owin, as.owin for information about windows.
Examples
v <- closing(letterR, 0.25)
plot(v, main="closing")
plot(letterR, add=TRUE)

clusterfield

Field of clusters

Description
Calculate the superposition of cluster kernels at the location of a point pattern.
Usage
clusterfield(model, locations = NULL, ...)
## S3 method for class 'character'
clusterfield(model, locations = NULL, ...)
## S3 method for class 'function'
clusterfield(model, locations = NULL, ..., mu = NULL)
## S3 method for class 'kppm'
clusterfield(model, locations = NULL, ...)
Arguments
model

Cluster model. Either a fitted cluster model (object of class "kppm"), a character
string specifying the type of cluster model, or a function defining the cluster
kernel. See Details.

locations

A point pattern giving the locations of the kernels. Defaults to the centroid of
the observation window for the "kppm" method and to the center of a unit square
otherwise.

...

Additional arguments passed to density.ppp or the cluster kernel. See Details.

mu

Mean number of offspring per cluster. A single number or a pixel image.

210

clusterfield

Details
The actual calculations are preformed by density.ppp and ... arguments are passed thereto for
control over the pixel resolution etc. (These arguments are then passed on to pixellate.ppp and
as.mask.)
For the function method the given kernel function should accept vectors of x and y coordinates as
its first two arguments. Any additional arguments may be passed through the ....
The function method also accepts the optional parameter mu (defaulting to 1) specifying the mean
number of points per cluster (as a numeric) or the inhomogeneous reference cluster intensity (as
an "im" object or a function(x,y)). The interpretation of mu is as explained in the simulation
functions referenced in the See Also section below.
For the character method model must be one of: model="Thomas" for the Thomas process, model="MatClust"
for the Matern cluster process, model="Cauchy" for the Neyman-Scott cluster process with Cauchy
kernel, or model="VarGamma" for the Neyman-Scott cluster process with Variance Gamma kernel.
For all these models the parameter scale is required and passed through ... as well as the parameter nu when model="VarGamma". This method calls clusterfield.function so the parameter mu
may also be passed through ... and will be interpreted as explained above.
The kppm method extracts the relevant information from the fitted model (including mu) and calls
clusterfield.function.
Value
A pixel image (object of class "im").
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak  .
See Also
density.ppp and kppm
Simulation algorithms for cluster models: rCauchy rMatClust rThomas rVarGamma
Examples
# method for fitted model
fit <- kppm(redwood~1, "Thomas")
clusterfield(fit, eps = 0.01)
# method for functions
kernel <- function(x,y,scal) {
r <- sqrt(x^2 + y^2)
ifelse(r > 0,
dgamma(r, shape=5, scale=scal)/(2 * pi * r),
0)
}
X <- runifpoint(10)
clusterfield(kernel, X, scal=0.05)

clusterfit

clusterfit

211

Fit Cluster or Cox Point Process Model via Minimum Contrast

Description
Fit a homogeneous or inhomogeneous cluster process or Cox point process model to a point pattern
by the Method of Minimum Contrast.
Usage
clusterfit(X, clusters, lambda = NULL, startpar = NULL,
q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...,
statistic = NULL, statargs = NULL, algorithm="Nelder-Mead")
Arguments
X

Data to which the cluster or Cox model will be fitted. Either a point pattern or a
summary statistic. See Details.

clusters

Character string determining the cluster or Cox model. Partially matched. Options are "Thomas", "MatClust", "Cauchy", "VarGamma" and "LGCP".

lambda

Optional. An estimate of the intensity of the point process. Either a single
numeric specifying a constant intensity, a pixel image (object of class "im")
giving the intensity values at all locations, a fitted point process model (object
of class "ppm" or "kppm") or a function(x,y) which can be evaluated to give
the intensity value at any location.

startpar

Vector of initial values of the parameters of the point process mode. If X is a
point pattern sensible defaults are used. Otherwise rather arbitrary values are
used.

q,p

Optional. Exponents for the contrast criterion.

rmin, rmax

Optional. The interval of r values for the contrast criterion.

...

Additional arguments passed to mincontrast.

statistic

Optional. Name of the summary statistic to be used for minimum contrast estimation: either "K" or "pcf".

statargs

Optional list of arguments to be used when calculating the statistic. See
Details.

algorithm

Character string determining the mathematical optimisation algorithm to be used
by optim. See the argument method of optim.

Details
This function fits the clustering parameters of a cluster or Cox point process model by the Method
of Minimum Contrast, that is, by matching the theoretical K-function of the model to the empirical
K-function of the data, as explained in mincontrast.
If statistic="pcf" (or X appears to be an estimated pair correlation function) then instead of using
the K-function, the algorithm will use the pair correlation function.
If X is a point pattern of class "ppp" an estimate of the summary statistic specfied by statistic
(defaults to "K") is first computed before minimum contrast estimation is carried out as described

212

clusterfit
above. In this case the argument statargs can be used for controlling the summary statistic estimation. The precise algorithm for computing the summary statistic depends on whether the intensity
specification (lambda) is:
homogeneous: If lambda is NUll or a single numeric the pattern is considered homogeneous and
either Kest or pcf is invoked. In this case lambda is not used for anything when estimating
the summary statistic.
inhomogeneous: If lambda is a pixel image (object of class "im"), a fitted point process model (object of class "ppm" or "kppm") or a function(x,y) the pattern is considered inhomogeneous.
In this case either Kinhom or pcfinhom is invoked with lambda as an argument.
After the clustering parameters of the model have been estimated by minimum contrast lambda (if
non-null) is used to compute the additional model parameter µ.

Value
An object of class "minconfit". There are methods for printing and plotting this object. See
mincontrast.
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak 
References
Diggle, P.J. and Gratton, R.J. (1984) Monte Carlo methods of inference for implicit statistical models. Journal of the Royal Statistical Society, series B 46, 193 – 212.
Møller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point
Processes. Chapman and Hall/CRC, Boca Raton.
Waagepetersen, R. (2007). An estimating function approach to inference for inhomogeneous NeymanScott processes. Biometrics 63 (2007) 252–258.
See Also
kppm
Examples
fit <- clusterfit(redwood, "Thomas")
fit
if(interactive()){
plot(fit)
}

clusterkernel

213

clusterkernel

Extract Cluster Offspring Kernel

Description
Given a cluster point process model, this command returns the probability density of the cluster
offspring.
Usage
clusterkernel(model, ...)
## S3 method for class 'kppm'
clusterkernel(model, ...)
## S3 method for class 'character'
clusterkernel(model, ...)
Arguments
model

Cluster model. Either a fitted cluster or Cox model (object of class "kppm"), or
a character string specifying the type of cluster model.

...

Parameter values for the model, when model is a character string.

Details
Given a specification of a cluster point process model, this command returns a function(x,y)
giving the two-dimensional probability density of the cluster offspring points assuming a cluster
parent located at the origin.
Value
A function in the R\ language with arguments x,y,....
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
clusterfield, kppm
Examples
fit <- kppm(redwood ~ x, "MatClust")
f <- clusterkernel(fit)
f(0.1, 0.2)

214

clusterradius

clusterradius

Compute or Extract Effective Range of Cluster Kernel

Description
Given a cluster point process model, this command returns a value beyond which the the probability
density of the cluster offspring is neglible.
Usage
clusterradius(model, ...)
## S3 method for class 'kppm'
clusterradius(model, ..., thresh = NULL, precision = FALSE)
## S3 method for class 'character'
clusterradius(model, ..., thresh = NULL, precision = FALSE)
Arguments
model
...
thresh

precision

Cluster model. Either a fitted cluster or Cox model (object of class "kppm"), or
a character string specifying the type of cluster model.
Parameter values for the model, when model is a character string.
Numerical threshold relative to the cluster kernel value at the origin (parent location) determining when the cluster kernel will be considered neglible. A sensible
default is provided.
Logical. If precision=TRUE the precision of the calculated range is returned as
an attribute to the range. See details.

Details
Given a cluster model this function by default returns the effective range of the model with the given
parameters as used in spatstat. For the Matern cluster model (see e.g. rMatClust) this is simply the
finite radius of the offsring density given by the paramter scale irrespective of other options given
to this function. The remaining models in spatstat have infinite theoretical range, and an effective
finite value is given as follows: For the Thomas model (see e.g. rThomas the default is 4*scale
where scale is the scale or standard deviation parameter of the model. If thresh is given the value
is instead found as described for the other models below.
For the Cauchy model (see e.g. rCauchy) and the Variance Gamma (Bessel) model (see e.g.
rVarGamma) the value of thresh defaults to 0.001, and
pthen this is used to compute the range
numerically as follows. If k(x, y) = k0 (r) with r = (x2 + y 2 ) denotes the isotropic cluster
kernel then f (r) = 2πrk0 (r) is the density function of the offspring distance from the parent. The
range is determined as the value of r where f (r) falls below thresh times k0 (r).
If precision=TRUE the precision related to the chosen range is returned as an attribute. Here the
precision is defined as the polar integral of the kernel from distance 0 to the calculated range. Ideally
this should be close to the value 1 which would be obtained for the true theretical infinite range.
Value
A positive numeric.
Additionally, the precision related to this range value is returned as an attribute "prec", if precision=TRUE.

clusterset

215

Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
clusterkernel, kppm, rMatClust, rThomas, rCauchy, rVarGamma, rNeymanScott.
Examples
fit <- kppm(redwood ~ x, "MatClust")
clusterradius(fit)
clusterradius("Thomas", scale = .1)
clusterradius("Thomas", scale = .1, thresh = 0.001)
clusterradius("VarGamma", scale = .1, nu = 2, precision = TRUE)

clusterset

Allard-Fraley Estimator of Cluster Feature

Description
Detect high-density features in a spatial point pattern using the (unrestricted) Allard-Fraley estimator.
Usage
clusterset(X, what=c("marks", "domain"),
..., verbose=TRUE,
fast=FALSE,
exact=!fast)
Arguments
X

A dimensional spatial point pattern (object of class "ppp").

what

Character string or character vector specifying the type of result. See Details.

verbose

Logical value indicating whether to print progress reports.

fast

Logical. If FALSE (the default), the Dirichlet tile areas will be computed exactly
using polygonal geometry, so that the optimal choice of tiles will be computed
exactly. If TRUE, the Dirichlet tile areas will be approximated using pixel counting, so the optimal choice will be approximate.

exact

Logical. If TRUE, the Allard-Fraley estimator of the domain will be computed
exactly using polygonal geometry. If FALSE, the Allard-Fraley estimator of the
domain will be approximated by a binary pixel mask. The default is initially set
to FALSE.

...

Optional arguments passed to as.mask to control the pixel resolution if exact=FALSE.

216

clusterset

Details
Allard and Fraley (1997) developed a technique for recognising features of high density in a spatial
point pattern in the presence of random clutter.
This algorithm computes the unrestricted Allard-Fraley estimator. The Dirichlet (Voronoi) tessellation of the point pattern X is computed. The smallest m Dirichlet cells are selected, where the
number m is determined by a maximum likelihood criterion.
• If fast=FALSE (the default), the areas of the tiles of the Dirichlet tessellation will be computed
exactly using polygonal geometry. This ensures that the optimal selection of tiles is computed
exactly.
• If fast=TRUE, the Dirichlet tile areas will be approximated by counting pixels. This is faster,
and is usually correct (depending on the pixel resolution, which is controlled by the arguments
...).
The type of result depends on the character vector what.
• If what="marks" the result is the point pattern X with a vector of marks labelling each point
with a value yes or no depending on whether the corresponding Dirichlet cell is selected by
the Allard-Fraley estimator. In other words each point of X is labelled as either a cluster point
or a non-cluster point.
• If what="domain", the result is the Allard-Fraley estimator of the cluster feature set, which is
the union of all the selected Dirichlet cells, represented as a window (object of class "owin").
• If what=c("marks", "domain") the result is a list containing both of the results described
above.
Computation of the Allard-Fraley set estimator depends on the argument exact.
• If exact=TRUE (the default), the Allard-Fraley set estimator will be computed exactly using
polygonal geometry. The result is a polygonal window.
• If exact=FALSE, the Allard-Fraley set estimator will be approximated by a binary pixel mask.
This is faster than the exact computation. The result is a binary mask.
Value
If what="marks", a multitype point pattern (object of class "ppp").
If what="domain", a window (object of class "owin").
If what=c("marks", "domain") (the default), a list consisting of a multitype point pattern and a
window.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Allard, D. and Fraley, C. (1997) Nonparametric maximum likelihood estimation of features in spatial point processes using Voronoi tessellation. Journal of the American Statistical Association 92,
1485–1493.
See Also
nnclean, sharpen

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217

Examples
opa <- par(mfrow=c(1,2))
W <- grow.rectangle(as.rectangle(letterR), 1)
X <- superimpose(runifpoint(300, letterR),
runifpoint(50, W), W=W)
plot(W, main="clusterset(X, 'm')")
plot(clusterset(X, "marks", fast=TRUE), add=TRUE, chars=c(1, 3), cols=1:2)
plot(letterR, add=TRUE)
plot(W, main="clusterset(X, 'd')")
plot(clusterset(X, "domain", exact=FALSE), add=TRUE)
plot(letterR, add=TRUE)
par(opa)

coef.mppm

Coefficients of Point Process Model Fitted to Multiple Point Patterns

Description
Given a point process model fitted to a list of point patterns, extract the coefficients of the fitted
model. A method for coef.
Usage
## S3 method for class 'mppm'
coef(object, ...)
Arguments
object
...

The fitted point process model (an object of class "mppm")
Ignored.

Details
This function is a method for the generic function coef.
The argument object must be a fitted point process model (object of class "mppm") produced by
the fitting algorithm mppm). This represents a point process model that has been fitted to a list of
several point pattern datasets. See mppm for information.
This function extracts the vector of coefficients of the fitted model. This is the estimate of the
parameter vector θ such that the conditional intensity of the model is of the form
λ(u, x) = exp(θS(u, x))
where S(u, x) is a (vector-valued) statistic.
For example, if the model object is the uniform Poisson process, then coef(object) will yield a
single value (named "(Intercept)") which is the logarithm of the fitted intensity of the Poisson
process.
If the fitted model includes random effects (i.e. if the argument random was specified in the call
to mppm), then the fitted coefficients are different for each point pattern in the original data, so
coef(object) is a data frame with one row for each point pattern, and one column for each parameter. Use fixef.mppm to extract the vector of fixed effect coefficients, and ranef.mppm to extract
the random effect coefficients at each level.
Use print.mppm to print a more useful description of the fitted model.

218

coef.ppm

Value
Either a vector containing the fitted coefficients, or a data frame containing the fitted coefficients
for each point pattern.
Author(s)
Adrian Baddeley, Ida-Maria Sintorn and Leanne Bischoff. Implemented in spatstat by Adrian
Baddeley , Rolf Turner  and
Ege Rubak .
References
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. London: Chapman and Hall/CRC Press.
See Also
fixef.mppm and ranef.mppm for the fixed and random effect coefficients in a model that includes
random effects.
print.mppm, mppm
Examples
H <- hyperframe(X=waterstriders)
fit.Poisson <- mppm(X ~ 1, H)
coef(fit.Poisson)
# The single entry "(Intercept)"
# is the log of the fitted intensity of the Poisson process
fit.Strauss <- mppm(X~1, H, Strauss(7))
coef(fit.Strauss)
# The two entries "(Intercept)" and "Interaction"
# are respectively log(beta) and log(gamma)
# in the usual notation for Strauss(beta, gamma, r)
# Tweak data to exaggerate differences
H$X[[1]] <- rthin(H$X[[1]], 0.3)
# Model with random effects
fitran <- mppm(X ~ 1, H, random=~1|id)
coef(fitran)

coef.ppm

Coefficients of Fitted Point Process Model

Description
Given a point process model fitted to a point pattern, extract the coefficients of the fitted model. A
method for coef.

coef.ppm

219

Usage
## S3 method for class 'ppm'
coef(object, ...)
Arguments
object

The fitted point process model (an object of class "ppm")

...

Ignored.

Details
This function is a method for the generic function coef.
The argument object must be a fitted point process model (object of class "ppm"). Such objects
are produced by the maximum pseudolikelihood fitting algorithm ppm).
This function extracts the vector of coefficients of the fitted model. This is the estimate of the
parameter vector θ such that the conditional intensity of the model is of the form
λ(u, x) = exp(θS(u, x))
where S(u, x) is a (vector-valued) statistic.
For example, if the model object is the uniform Poisson process, then coef(object) will yield a
single value (named "(Intercept)") which is the logarithm of the fitted intensity of the Poisson
process.
Use print.ppm to print a more useful description of the fitted model.
Value
A vector containing the fitted coefficients.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
print.ppm, ppm.object, ppm
Examples
data(cells)
poi <- ppm(cells, ~1, Poisson())
coef(poi)
# This is the log of the fitted intensity of the Poisson process
stra <- ppm(cells, ~1, Strauss(r=0.07))
coef(stra)
# The two entries "(Intercept)" and "Interaction"
# are respectively log(beta) and log(gamma)
# in the usual notation for Strauss(beta, gamma, r)

220

coef.slrm

coef.slrm

Coefficients of Fitted Spatial Logistic Regression Model

Description
Extracts the coefficients (parameters) from a fitted Spatial Logistic Regression model.
Usage
## S3 method for class 'slrm'
coef(object, ...)
Arguments
object

a fitted spatial logistic regression model. An object of class "slrm".

...

Ignored.

Details
This is a method for coef for fitted spatial logistic regression models (objects of class "slrm",
usually obtained from the function slrm).
It extracts the fitted canonical parameters, i.e.\ the coefficients in the linear predictor of the spatial
logistic regression.
Value
Numeric vector of coefficients.
Author(s)
Adrian Baddeley  
and Rolf Turner 
See Also
slrm
Examples
X <- rpoispp(42)
fit <- slrm(X ~ x+y)
coef(fit)

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221

collapse.fv

Collapse Several Function Tables into One

Description
Combines several function tables (objects of class "fv") into a single function table, merging
columns that are identical and relabelling columns that are different.
Usage
## S3 method for class 'fv'
collapse(object, ..., same = NULL, different = NULL)
## S3 method for class 'anylist'
collapse(object, ..., same = NULL, different = NULL)
Arguments
object

An object of class "fv", or a list of such objects.

...

Additional objects of class "fv".

same

Character string or character vector specifying a column or columns, present in
each "fv" object, that are identical in each object. This column or columns will
be included only once.

different

Character string or character vector specifying a column or columns, present in
each "fv" object, that contain different values in each object. Each of these
columns of data will be included, with labels that distinguish them from each
other.

Details
This is a method for the generic function collapse.
It combines the data in several function tables (objects of class "fv", see fv.object) to make a
single function table. It is essentially a smart wrapper for cbind.fv.
A typical application is to calculate the same summary statistic (such as the K function) for different
point patterns, and then to use collapse.fv to combine the results into a single object that can
easily be plotted. See the Examples.
The arguments object and ... should be function tables (objects of class "fv", see fv.object)
that are compatible in the sense that they have the same values of the function argument.
The argument same identifies any columns that are present in each function table, and which are
known to contain exactly the same values in each table. This column or columns will be included
only once in the result.
The argument different identifies any columns that are present in each function table, and which
contain different numerical values in each table. Each of these columns will be included, with labels
to distinguish them.
Columns that are not named in same or different will not be included.
Value
Object of class "fv".

222

colourmap

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
fv.object, cbind.fv
Examples
# generate simulated data
X <- replicate(3, rpoispp(100), simplify=FALSE)
names(X) <- paste("Simulation", 1:3)
# compute K function estimates
Klist <- anylapply(X, Kest)
# collapse
K <- collapse(Klist, same="theo", different="iso")
K

colourmap

Colour Lookup Tables

Description
Create a colour map (colour lookup table).
Usage
colourmap(col, ..., range=NULL, breaks=NULL, inputs=NULL, gamma=1)
Arguments
col

Vector of values specifying colours

...

Ignored.

range

Interval to be mapped. A numeric vector of length 2, specifying the endpoints
of the range of values to be mapped. Incompatible with breaks or inputs.

inputs

Values to which the colours are associated. A factor or vector of the same length
as col. Incompatible with breaks or range.

breaks

Breakpoints for the colour map. A numeric vector of length equal to length(col)+1.
Incompatible with range or inputs.

gamma

Exponent for the gamma correction, when range is given. A single positive
number. See Details.

colourmap

223

Details
A colour map is a mechanism for associating colours with data. It can be regarded as a function,
mapping data to colours.
The command colourmap creates an object representing a colour map, which can then be used
to control the plot commands in the spatstat package. It can also be used to compute the colour
assigned to any data value.
The argument col specifies the colours to which data values will be mapped. It should be a vector
whose entries can be interpreted as colours by the standard R graphics system. The entries can be
string names of colours like "red", or integers that refer to colours in the standard palette, or strings
containing six-letter hexadecimal codes like "#F0A0FF".
Exactly one of the arguments range, inputs or breaks must be specified by name.
If inputs is given, then it should be a vector or factor, of the same length as col. The entries of
inputs can be any atomic type (e.g. numeric, logical, character, complex) or factor values. The
resulting colour map associates the value inputs[i] with the colour col[i].
If range is given, then it determines the interval of the real number line that will be mapped. It
should be a numeric vector of length 2. The interval will be divided evenly into bands, each of
which is assigned one of the colours in col. (If gamma is given, then the bands are equally spaced
on a scale where the original values are raised to the power gamma.)
If breaks is given, then it determines the precise intervals of the real number line which are mapped
to each colour. It should be a numeric vector, of length at least 2, with entries that are in increasing
order. Infinite values are allowed. Any number in the range between breaks[i] and breaks[i+1]
will be mapped to the colour col[i].
The result is an object of class "colourmap". There are print and plot methods for this class.
Some plot commands in the spatstat package accept an object of this class as a specification of the
colour map.
The result is also a function f which can be used to compute the colour assigned to any data value.
That is, f(x) returns the character value of the colour assigned to x. This also works for vectors of
data values.
Value
A function, which is also an object of class "colourmap".
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
The plot method plot.colourmap.
See the R help file on colours for information about the colours that R recognises, and how to
manipulate them.
To make a smooth transition between colours, see interp.colourmap. To alter individual colour
values, see tweak.colourmap. To extract or replace all colour values, see colouroutputs.
See colourtools for more tools to manipulate colour values.
See lut for lookup tables.

224

colouroutputs

Examples
# colour map for real numbers, using breakpoints
cr <- colourmap(c("red", "blue", "green"), breaks=c(0,5,10,15))
cr
cr(3.2)
cr(c(3,5,7))
# a large colour map
co <- colourmap(rainbow(100), range=c(-1,1))
co(0.2)
# colour map for discrete set of values
ct <- colourmap(c("red", "green"), inputs=c(FALSE, TRUE))
ct(TRUE)

colouroutputs

Extract or Assign Colour Values in a Colour Map

Description
Extract the colour values in a colour map, or assign new colour values.
Usage
colouroutputs(x)
colouroutputs(x) <- value
Arguments
x

A colour map (object of class "colourmap").

value

A vector of values that can be interpreted as colours.

Details
An object of class "colourmap" is effectively a function that maps its inputs (numbers or factor
levels) to colour values.
The command colouroutputs(x) extracts the colour values in the colour map x.
The assignment colouroutputs(x) <- value replaces the colour values in the colour map x by the
entries in value. The replacement vector value should have the same length as colouroutputs(x),
and its entries should be interpretable as colours.
To change only some of the colour values in a colour map, it may be easier to use tweak.colourmap.
Value
The result of colouroutputs is a character vector of colour values. The result of the assignment
colouroutputs(x) <- value is another colour map (object of class "colourmap").
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .

colourtools

225

See Also
colourmap, interp.colourmap, tweak.colourmap, colourtools.
Examples
m <- colourmap(rainbow(5), range=c(0,1))
m
# reverse order of colours
colouroutputs(m) <- rev(colouroutputs(m))
m

colourtools

Convert and Compare Colours in Different Formats

Description
These functions convert between different formats for specifying a colour in R, determine whether
colours are equivalent, and convert colour to greyscale.
Usage
col2hex(x)
rgb2hex(v, maxColorValue=255)
rgb2hsva(red, green=NULL, blue=NULL, alpha=NULL, maxColorValue=255)
paletteindex(x)
samecolour(x,y)
complementarycolour(x)
interp.colours(x, length.out=512)
is.colour(x)
to.grey(x, weights=c(0.299, 0.587, 0.114), transparent=FALSE)
is.grey(x)
to.opaque(x)
to.transparent(x, fraction)
to.saturated(x, s=1)
Arguments
x,y

Any valid specification for a colour or sequence of colours accepted by col2rgb.

v

A numeric vector of length 3, giving the RGB values of a single colour, or a 3column matrix giving the RGB values of several colours. Alternatively a vector
of length 4 or a matrix with 4 columns, giving the RGB and alpha (transparency)
values.
red,green,blue,alpha
Arguments acceptable to rgb determining the red, green, blue channels and optionally the alpha (transparency) channel. Note that red can also be a matrix
with 3 rows giving the RGB values, or a matrix with 4 rows giving RGB and
alpha values.
maxColorValue

Number giving the maximum possible value for the entries in v or red,green,blue,alpha.

weights

Numeric vector of length 3 giving relative weights for the red, green, and blue
channels respectively.

226

colourtools
transparent

fraction
length.out
s

Logical value indicating whether transparent colours should be converted to
transparent grey values (transparent=TRUE) or converted to opaque grey values (transparent=FALSE, the default).
Transparency fraction. Numerical value or vector of values between 0 and 1,
giving the opaqueness of a colour. A fully opaque colour has fraction=1.
Integer. Length of desired sequence.
Saturation value (between 0 and 1).

Details
is.colour(x) can be applied to any kind of data x and returns TRUE if x can be interpreted as a
colour or colours. The remaining functions expect data that can be interpreted as colours.
col2hex converts colours specified in any format into their hexadecimal character codes.
rgb2hex converts RGB colour values into their hexadecimal character codes. It is a very minor
extension to rgb. Arguments to rgb2hex should be similar to arguments to rgb.
rgb2hsva converts RGB colour values into HSV colour values including the alpha (transparency)
channel. It is an extension of rgb2hsv. Arguments to rgb2hsva should be similar to arguments to
rgb2hsv.
paletteindex checks whether the colour or colours specified by x are available in the default
palette returned by palette(). If so, it returns the index or indices of the colours in the palette. If
not, it returns NA.
samecolour decides whether two colours x and y are equivalent.
is.grey determines whether each entry of x is a greyscale colour, and returns a logical vector.
to.grey converts the colour data in x to greyscale colours. Alternatively x can be an object of class
"colourmap" and to.grey(x) is the modified colour map.
to.opaque converts the colours in x to opaque (non-transparent) colours, and to.transparent
converts them to transparent colours with a specified transparency value. Note that to.transparent(x,1)
is equivalent to to.opaque(x).
For to.grey, to.opaque and to.transparent, if all the data in x specifies colours from the standard palette, and if the result would be equivalent to x, then the result is identical to x.
to.saturated converts each colour in x to its fully-saturated equivalent. For example, pink is
mapped to red. Shades of grey are converted to black; white is unchanged.
complementarycolour replaces each colour by its complementary colour in RGB space (the colour
obtained by replacing RGB values (r, g, b) by (255-r, 255-g, 255-b)). The transparency
value is not changed. Alternatively x can be an object of class "colourmap" and complementarycolour(x)
is the modified colour map.
interp.colours interpolates between each successive pair of colours in a sequence of colours,
to generate a more finely-spaced sequence. It uses linear interpolation in HSV space (with hue
represented as a two-dimensional unit vector).
Value
For col2hex and rgb2hex a character vector containing hexadecimal colour codes.
For to.grey, to.opaque and to.transparent, either a character vector containing hexadecimal
colour codes, or a value identical to the input x.
For rgb2hsva, a matrix with 3 or 4 rows containing HSV colour values.
For paletteindex, an integer vector, possibly containing NA values.
For samecolour and is.grey, a logical value or logical vector.

commonGrid

227

Warning
paletteindex("green") returns NA because the green colour in the default palette is called "green3".
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
col2rgb, rgb2hsv, palette.
See also the class of colour map objects in the spatstat package: colourmap, interp.colourmap,
tweak.colourmap.
Examples
samecolour("grey", "gray")
paletteindex("grey")
col2hex("orange")
to.grey("orange")
to.saturated("orange")
complementarycolour("orange")
is.grey("lightgrey")
is.grey(8)
to.transparent("orange", 0.5)
to.opaque("red")
interp.colours(c("orange", "red", "violet"), 5)

commonGrid

Determine A Common Spatial Domain And Pixel Resolution

Description
Determine a common spatial domain and pixel resolution for several spatial objects such as images,
masks, windows and point patterns.
Usage
commonGrid(...)
Arguments
...

Any number of pixel images (objects of class "im"), binary masks (objects of
class "owin" of type "mask") or data which can be converted to binary masks
by as.mask.

228

compareFit

Details
This function determines a common spatial resolution and spatial domain for several spatial objects.
The arguments ... may be pixel images, binary masks, or other spatial objects acceptable to
as.mask.
The common pixel grid is determined by inspecting all the pixel images and binary masks in the
argument list, finding the pixel grid with the highest spatial resolution, and extending this pixel grid
to cover the bounding box of all the spatial objects.
The return value is a binary mask M, representing the bounding box at the chosen pixel resolution.
Use as.im(X, W=M) to convert a pixel image X to this new pixel resolution. Use as.mask(W, xy=M)
to convert a window W to a binary mask at this new pixel resolution. See the Examples.
Value
A binary mask (object of class "owin" and type "mask").
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
harmonise.im, compatible.im, as.im
Examples
A , Adrian Baddeley 
and Jesper Møller.
See Also
ppm, Kcom, Kres, Gcom, Gres, psst, psstA, psstG, collapse.fv

230

compatible

Examples
nd <- 40
ilist <- list(Poisson(), Geyer(7, 2), Strauss(7))
iname <- c("Poisson", "Geyer", "Strauss")
K <- compareFit(swedishpines, Kcom, interaction=ilist, rbord=9,
correction="translate",
same="trans", different="tcom", modelnames=iname, nd=nd)
K

compatible

Test Whether Objects Are Compatible

Description
Tests whether two or more objects of the same class are compatible.
Usage
compatible(A, B, ...)
Arguments
A,B,...

Two or more objects of the same class

Details
This generic function is used to check whether the objects A and B (and any additional objects ...)
are compatible.
What is meant by ‘compatible’ depends on the class of object.
There are methods for the classes "fv", "fasp", "im" and "unitname".
Value
Logical value: TRUE if the objects are compatible, and FALSE if they are not.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
compatible.fv, compatible.fasp, compatible.im, compatible.unitname

compatible.fasp

compatible.fasp

231

Test Whether Function Arrays Are Compatible

Description
Tests whether two or more function arrays (class "fasp") are compatible.
Usage
## S3 method for class 'fasp'
compatible(A, B, ...)
Arguments
A,B,...

Two or more function arrays (object of class "fasp").

Details
An object of class "fasp" can be regarded as an array of functions. Such objects are returned by
the command alltypes.
This command tests whether such objects are compatible (so that, for example, they could be added
or subtracted). It is a method for the generic command compatible.
The function arrays are compatible if the arrays have the same dimensions, and the corresponding
elements in each cell of the array are compatible as defined by compatible.fv.
Value
Logical value: TRUE if the objects are compatible, and FALSE if they are not.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
eval.fasp

compatible.fv

Test Whether Function Objects Are Compatible

Description
Tests whether two or more function objects (class "fv") are compatible.
Usage
## S3 method for class 'fv'
compatible(A, B, ...)

232

compatible.im

Arguments
A,B,...

Two or more function value objects (class "fv").

Details
An object of class "fv" is essentially a data frame containing several different statistical estimates
of the same function. Such objects are returned by Kest and its relatives.
This command tests whether such objects are compatible (so that, for example, they could be added
or subtracted). It is a method for the generic command compatible.
The functions are compatible if they have been evaluated at the same sequence of values of the
argument r, and if the statistical estimates have the same names.
Value
Logical value: TRUE if the objects are compatible, and FALSE if they are not.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
eval.fv

compatible.im

Test Whether Pixel Images Are Compatible

Description
Tests whether two or more pixel image objects have compatible dimensions.
Usage
## S3 method for class 'im'
compatible(A, B, ..., tol=1e-6)
Arguments
A,B,...

Two or more pixel images (objects of class "im").

tol

Tolerance factor

Details
This function tests whether the pixel images A and B (and any additional images ...) have compatible pixel dimensions. They are compatible if they have the same number of rows and columns, the
same physical pixel dimensions, and occupy the same rectangle in the plane.
The argument tol specifies the maximum tolerated error in the pixel coordinates, expressed as a
fraction of the dimensions of a single pixel.

compileK

233

Value
Logical value: TRUE if the images are compatible, and FALSE if they are not.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
eval.im, harmonise.im, commonGrid

compileK

Generic Calculation of K Function and Pair Correlation Function

Description
Low-level functions which calculate the estimated K function and estimated pair correlation function (or any similar functions) from a matrix of pairwise distances and optional weights.
Usage
compileK(D, r, weights = NULL, denom = 1,
check = TRUE, ratio = FALSE, fname = "K")
compilepcf(D, r, weights = NULL, denom = 1,
check = TRUE, endcorrect = TRUE, ratio=FALSE,
..., fname = "g")
Arguments
D

A square matrix giving the distances between all pairs of points.

r

An equally spaced, finely spaced sequence of distance values.

weights

Optional numerical weights for the pairwise distances. A numeric matrix with
the same dimensions as D. If absent, the weights are taken to equal 1.

denom

Denominator for the estimator. A single number, or a numeric vector with the
same length as r. See Details.

check

Logical value specifying whether to check that D is a valid matrix of pairwise
distances.

ratio

Logical value indicating whether to store ratio information. See Details.

...

Optional arguments passed to density.default controlling the kernel smoothing.

endcorrect

Logical value indicating whether to apply End Correction of the pair correlation
estimate at r=0.

fname

Character string giving the name of the function being estimated.

234

compileK

Details
These low-level functions construct estimates of the K function or pair correlation function, or any
similar functions, given only the matrix of pairwise distances and optional weights associated with
these distances.
These functions are useful for code development and for teaching, because they perform a common
task, and do the housekeeping required to make an object of class "fv" that represents the estimated
function. However, they are not very efficient.
compileK calculates the weighted estimate of the K function,
XX
K̂(r) = (1/v(r))
1{dij ≤ r}wij
i

j

and compilepcf calculates the weighted estimate of the pair correlation function,
XX
ĝ(r) = (1/v(r))
κ(dij − r)wij
i

j

where dij is the distance between spatial points i and j, with corresponding weight wij , and v(r) is
a specified denominator. Here κ is a fixed-bandwidth smoothing kernel.
For a point pattern in two dimensions, the usual denominator v(r) is constant for the K function,
and proportional to r for the pair correlation function. See the Examples.
The result is an object of class "fv" representing the estimated function. This object has only one
column of function values. Additional columns (such as a column giving the theoretical value) must
be added by the user, with the aid of bind.fv.
If ratio=TRUE, the result also belongs to class "rat" and has attributes containing the numerator
and denominator of the function estimate. This allows function estimates from several datasets to
be pooled using pool.
Value
An object of class "fv" representing the estimated function.
Author(s)
Adrian Baddeley 
See Also
Kest, pcf for definitions of the K function and pair correlation function.
bind.fv to add more columns.
Examples
X <- japanesepines
D <- pairdist(X)
Wt <- edge.Ripley(X, D)
lambda <- intensity(X)
a <- (npoints(X)-1) * lambda
r <- seq(0, 0.25, by=0.01)
K <- compileK(D=D, r=r, weights=Wt, denom=a)
g <- compilepcf(D=D, r=r, weights=Wt, denom= a * 2 * pi * r)

complement.owin

235

complement.owin

Take Complement of a Window

Description
Take the set complement of a window, within its enclosing rectangle or in a larger rectangle.
Usage
complement.owin(w, frame=as.rectangle(w))
Arguments
w

an object of class "owin" describing a window of observation for a point pattern.

frame

Optional. The enclosing rectangle, with respect to which the set complement is
taken.

Details
This yields a window object (of class "owin", see owin.object) representing the set complement
of w with respect to the rectangle frame.
By default, frame is the enclosing box of w (originally specified by the arguments xrange and
yrange given to owin when w was created). If frame is specified, it must be a rectangle (an object
of class "owin" whose type is "rectangle") and it must be larger than the enclosing box of w. This
rectangle becomes the enclosing box for the resulting window.
If w is a rectangle, then frame must be specified. Otherwise an error will occur (since the complement of w in itself is empty).
For rectangular and polygonal windows, the complement is computed by reversing the sign of each
boundary polygon, while for binary masks it is computed by negating the pixel values.
Value
Another object of class "owin" representing the complement of the window, i.e. the inside of the
window becomes the outside.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
owin, owin.object

236

concatxy

Examples
# rectangular
a <- owin(c(0,1),c(0,1))
b <- owin(c(-1,2),c(-1,2))
bmina <- complement.owin(a, frame=b)
# polygonal
data(demopat)
w <- Window(demopat)
outside <- complement.owin(w)
# mask
w <- as.mask(Window(demopat))
outside <- complement.owin(w)

concatxy

Concatenate x,y Coordinate Vectors

Description
Concatenate any number of pairs of x and y coordinate vectors.
Usage
concatxy(...)
Arguments
...

Any number of arguments, each of which is a structure containing elements x
and y.

Details
This function can be used to superimpose two or more point patterns of unmarked points (but see
also superimpose which is recommended).
It assumes that each of the arguments in ... is a structure containing (at least) the elements x and y.
It concatenates all the x elements into a vector x, and similarly for y, and returns these concatenated
vectors.
Value
A list with two components x and y, which are the concatenations of all the corresponding x and y
vectors in the argument list.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
superimpose, quadscheme

Concom

237

Examples
dat <- runifrect(30)
xy <- list(x=runif(10),y=runif(10))
new <- concatxy(dat, xy)

Concom

The Connected Component Process Model

Description
Creates an instance of the Connected Component point process model which can then be fitted to
point pattern data.
Usage
Concom(r)
Arguments
r

Threshold distance

Details
This function defines the interpoint interaction structure of a point process called the connected
component process. It can be used to fit this model to point pattern data.
The function ppm(), which fits point process models to point pattern data, requires an argument of
class "interact" describing the interpoint interaction structure of the model to be fitted. The appropriate description of the connected component interaction is yielded by the function Concom().
See the examples below.
In standard form, the connected component process (Baddeley and Møller, 1989) with disc radius
r, intensity parameter κ and interaction parameter γ is a point process with probability density
f (x1 , . . . , xn ) = ακn(x) γ −C(x)
for a point pattern x, where x1 , . . . , xn represent the points of the pattern, n(x) is the number of
points in the pattern, and C(x) is defined below. Here α is a normalising constant.
To define the term C(x), suppose that we construct a planar graph by drawing an edge between each
pair of points xi , xj which are less than r units apart. Two points belong to the same connected
component of this graph if they are joined by a path in the graph. Then C(x) is the number of
connected components of the graph.
The interaction parameter γ can be any positive number. If γ = 1 then the model reduces to a
Poisson process with intensity κ. If γ < 1 then the process is regular, while if γ > 1 the process is
clustered. Thus, a connected-component interaction process can be used to model either clustered
or regular point patterns.
In spatstat, the model is parametrised in a different form, which is easier to interpret. In canonical
form, the probability density is rewritten as
f (x1 , . . . , xn ) = αβ n(x) γ −U (x)
where β is the new intensity parameter and U (x) = C(x) − n(x) is the interaction potential. In
this formulation, each isolated point of the pattern contributes a factor β to the probability density

238

Concom
(so the first order trend is β). The quantity U (x) is a true interaction potential, in the sense that
U (x) = 0 if the point pattern x does not contain any points that lie close together.
When a new point u is added to an existing point pattern x, the rescaled potential −U (x) increases
by zero or a positive integer. The increase is zero if u is not close to any point of x. The increase is
a positive integer k if there are k different connected components of x that lie close to u. Addition
of the point u contributes a factor βη δ to the probability density, where δ is the increase in potential.
If desired, the original parameter κ can be recovered from the canonical parameter by κ = βγ.
The nonstationary connected component process is similar except that the contribution of each
individual point xi is a function β(xi ) of location, rather than a constant beta.
Note the only argument of Concom() is the threshold distance r. When r is fixed, the model becomes an exponential family. The canonical parameters log(β) and log(γ) are estimated by ppm(),
not fixed in Concom().

Value
An object of class "interact" describing the interpoint interaction structure of the connected component process with disc radius r.
Edge correction
The interaction distance of this process is infinite. There are no well-established procedures for
edge correction for fitting such models, and accordingly the model-fitting function ppm will give an
error message saying that the user must specify an edge correction. A reasonable solution is to use
the border correction at the same distance r, as shown in the Examples.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
References
Baddeley, A.J. and Møller, J. (1989) Nearest-neighbour Markov point processes and random sets.
International Statistical Review 57, 89–121.
See Also
ppm, pairwise.family, ppm.object
Examples
# prints a sensible description of itself
Concom(r=0.1)
# Fit the stationary connected component process to redwood data
ppm(redwood, ~1, Concom(r=0.07), rbord=0.07)
# Fit the stationary connected component process to `cells' data
ppm(cells, ~1, Concom(r=0.06), rbord=0.06)
# eta=0 indicates hard core process.
# Fit a nonstationary connected component model
# with log-cubic polynomial trend
## Not run:

connected

239

ppm(swedishpines, ~polynom(x/10,y/10,3), Concom(r=7), rbord=7)
## End(Not run)

connected

Connected components

Description
Finds the topologically-connected components of a spatial object, such as the connected clumps of
pixels in a binary image.
Usage
connected(X, ...)
## S3 method for class 'owin'
connected(X, ..., method="C")
## S3 method for class 'im'
connected(X, ..., background = NA, method="C")
Arguments
X

A spatial object such as a pixel image (object of class "im") or a window (object
of class "owin").

background

Optional. Treat pixels with this value as being part of the background.

method

String indicating the algorithm to be used. Either "C" or "interpreted". See
Details.

...

Arguments passed to as.mask to determine the pixel resolution.

Details
The function connected is generic, with methods for pixel images (class "im") and windows (class
"owin") described here. There is also a method for point patterns described in connected.ppp.
The functions described here compute the connected component transform (Rosenfeld and Pfalz,
1966) of a binary image or binary mask. The argument X is first converted into a pixel image with
logical values. Then the algorithm identifies the connected components (topologically-connected
clumps of pixels) in the foreground.
Two pixels belong to the same connected component if they have the value TRUE and if they are
neighbours (in the 8-connected sense). This rule is applied repeatedly until it terminates. Then
each connected component contains all the pixels that can be reached by stepping from neighbour
to neighbour.
If method="C", the computation is performed by a compiled C language implementation of the
classical algorithm of Rosenfeld and Pfalz (1966). If method="interpreted", the computation is
performed by an R implementation of the algorithm of Park et al (2000).
The result is a factor-valued image, with levels that correspond to the connected components. The
Examples show how to extract each connected component as a separate window object.

240

connected

Value
A pixel image (object of class "im") with factor values. The levels of the factor correspond to the
connected components.

Warnings
It may be hard to distinguish different components in the default plot because the colours of nearby
components may be very similar. See the Examples for a randomised colour map.
The algorithm for method="interpreted" can be very slow for large images (or images where the
connected components include a large number of pixels).

Author(s)
Original R code by Julian Burgos, University of Washington. Adapted for spatstat by Adrian
Baddeley  and Rolf Turner .
References
Park, J.-M., Looney, C.G. and Chen, H.-C. (2000) Fast connected component labeling algorithm
using a divide and conquer technique. Pages 373-376 in S.Y. Shin (ed) Computers and Their Applications: Proceedings of the ISCA 15th International Conference on Computers and Their Applications, March 29-31, 2000, New Orleans, Louisiana USA. ISCA 2000, ISBN 1-880843-32-3.
Rosenfeld, A. and Pfalz, J.L. (1966) Sequential operations in digital processing. Journal of the
Association for Computing Machinery 13 471-494.

See Also
connected.ppp, im.object, tess
Examples
d <- distmap(cells, dimyx=256)
X <- levelset(d, 0.07)
plot(X)
Z <- connected(X)
plot(Z)
# or equivalently
Z <- connected(d <= 0.07)
# number of components
nc <- length(levels(Z))
# plot with randomised colour map
plot(Z, col=hsv(h=sample(seq(0,1,length=nc), nc)))
# how to extract the components as a list of windows
W <- tiles(tess(image=Z))

connected.linnet

connected.linnet

241

Connected Components of a Linear Network

Description
Find the topologically-connected components of a linear network.
Usage
## S3 method for class 'linnet'
connected(X, ..., what = c("labels", "components"))
Arguments
X
...
what

A linear network (object of class "linnet").
Ignored.
Character string specifying the kind of result.

Details
The function connected is generic. This is the method for linear networks (objects of class
"linnet").
Two vertices of the network are connected if they are joined by a path in the network. This function
divides the network into subsets, such that all points in a subset are connected to each other.
If what="labels" the return value is a factor with one entry for each vertex of X, identifying which
connected component the vertex belongs to.
If what="components" the return value is a list of linear networks, which are the connected components of X.
Value
If what="labels", a factor. If what="components", a list of linear networks.
Author(s)
Adrian Baddeley  and Suman Rakshit.
See Also
thinNetwork
Examples
# remove some edges from a network to make it disconnected
plot(simplenet, col="grey", main="", lty=2)
A <- thinNetwork(simplenet, retainedges=-c(3,5))
plot(A, add=TRUE, lwd=2)
# find the connected components
connected(A)
cA <- connected(A, what="components")
plot(cA[[1]], add=TRUE, col="green", lwd=2)
plot(cA[[2]], add=TRUE, col="blue", lwd=2)

242

connected.lpp

connected.lpp

Connected Components of a Point Pattern on a Linear Network

Description
Finds the topologically-connected components of a point pattern on a linear network, when all pairs
of points closer than a threshold distance are joined.
Usage
## S3 method for class 'lpp'
connected(X, R=Inf, ..., dismantle=TRUE)
Arguments
X

A linear network (object of class "lpp").

R

Threshold distance. Pairs of points will be joined together if they are closer than
R units apart, measured by the shortest path in the network. The default R=Inf
implies that points will be joined together if they are mutually connected by any
path in the network.

dismantle

Logical. If TRUE (the default), the network itself will be divided into its pathconnected components using connected.linnet.

...

Ignored.

Details
The function connected is generic. This is the method for point patterns on a linear network
(objects of class "lpp"). It divides the point pattern X into one or more groups of points.
If R=Inf (the default), then X is divided into groups such that any pair of points in the same group
can be joined by a path in the network.
If R is a finite number, then two points of X are declared to be R-close if they lie closer than R units
apart, measured by the length of the shortest path in the network. Two points are R-connected if
they can be reached by a series of steps between R-close pairs of points of X. Then X is divided into
groups such that any pair of points in the same group is R-connected.
If dismantle=TRUE (the default) the algorithm first checks whether the network is connected (i.e.
whether any pair of vertices can be joined by a path in the network), and if not, the network is
decomposed into its connected components.
Value
A point pattern (of class "lpp") with marks indicating the grouping, or a list of such point patterns.
Author(s)
Adrian Baddeley .
See Also
thinNetwork

connected.ppp

243

Examples
# remove some edges from a network to make it disconnected
plot(simplenet, col="grey", main="", lty=2)
A <- thinNetwork(simplenet, retainedges=-c(3,5))
plot(A, add=TRUE, lwd=2)
X <- runiflpp(10, A)
# find the connected components
cX <- connected(X)
plot(cX[[1]], add=TRUE, col="blue", lwd=2)

connected.ppp

Connected Components of a Point Pattern

Description
Finds the topologically-connected components of a point pattern, when all pairs of points closer
than a threshold distance are joined.
Usage
## S3 method for class 'ppp'
connected(X, R, ...)
## S3 method for class 'pp3'
connected(X, R, ...)
Arguments
X

A point pattern (object of class "ppp" or "pp3").

R

Threshold distance. Pairs of points closer than R units apart will be joined together.

...

Other arguments, not recognised by these methods.

Details
This function can be used to identify clumps of points in a point pattern.
The function connected is generic. This file documents the methods for point patterns in dimension
two or three (objects of class "ppp" or "pp3").
The point pattern X is first converted into an abstract graph by joining every pair of points that lie
closer than R units apart. Then the connected components of this graph are identified.
Two points in X belong to the same connected component if they can be reached by a series of steps
between points of X, each step being shorter than R units in length.
The result is a vector of labels for the points of X where all the points in a connected component
have the same label.
Value
A point pattern, equivalent to X except that the points have factor-valued marks, with levels corresponding to the connected components.

244

contour.im

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
connected.im, im.object, tess
Examples
Y <- connected(redwoodfull, 0.1)
if(interactive()) {
plot(Y, cols=1:length(levels(marks(Y))),
main="connected(redwoodfull, 0.1)")
}
X <- osteo$pts[[1]]
Z <- connected(X, 32)
if(interactive()) {
plot(Z, col=marks(Z), main="")
}

contour.im

Contour plot of pixel image

Description
Generates a contour plot of a pixel image.
Usage
## S3 method for class 'im'
contour(x, ..., main,
axes=FALSE, add=FALSE, col=par("fg"),
clipwin=NULL, show.all=!add, do.plot=TRUE)
Arguments
x

Pixel image to be plotted. An object of class "im".

main

Character string to be displayed as the main title.

axes

Logical. If TRUE, coordinate axes are plotted (with tick marks) around a region
slightly larger than the image window. If FALSE (the default), no axes are plotted,
and a box is drawn tightly around the image window. Ignored if add=TRUE.

add

Logical. If FALSE, a new plot is created. If TRUE, the contours are drawn over
the existing plot.

col

Colour in which to draw the contour lines. Either a single value that can be
interpreted as a colour value, or a colourmap object.

clipwin

Optional. A window (object of class "owin"). Only this subset of the data will
be displayed.

...

Other arguments passed to contour.default controlling the contour plot; see
Details.

contour.im

245

show.all

Logical value indicating whether to display all plot elements including the main
title, bounding box, and (if axis=TRUE) coordinate axis markings. Default is
TRUE for new plots and FALSE for added plots.

do.plot

Logical value indicating whether to actually perform the plot.

Details
This is a method for the generic contour function, for objects of the class "im".
An object of class "im" represents a pixel image; see im.object.
This function displays the values of the pixel image x as a contour plot on the current plot device,
using equal scales on the x and y axes.
The appearance of the plot can be modified using any of the arguments listed in the help for
contour.default. Useful ones include:
nlevels Number of contour levels to plot.
drawlabels Whether to label the contour lines with text.
col,lty,lwd Colour, type, and width of contour lines.
See contour.default for a full list of these arguments.
The defaults for any of the abovementioned arguments can be reset using spatstat.options("par.contour").
If col is a colour map (object of class "colourmap", see colourmap) then the contours will be
plotted in different colours as determined by the colour map. The contour at level z will be plotted
in the colour col(z) associated with this level in the colour map.
Value
none.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
im.object, plot.im, persp.im
Examples
# an image
Z <- setcov(owin())
contour(Z, axes=TRUE)
contour(Z)
co <- colourmap(rainbow(100), range=c(0,1))
contour(Z, col=co, lwd=2)

246

contour.imlist

contour.imlist

Array of Contour Plots

Description
Generates an array of contour plots.
Usage
## S3 method for class 'imlist'
contour(x, ...)
## S3 method for class 'listof'
contour(x, ...)
Arguments
x

An object of the class "imlist" representing a list of pixel images. Alternatively
x may belong to the outdated class "listof".

...

Arguments passed to plot.solist to control the spatial arrangement of panels,
and arguments passed to contour.im to control the display of each panel.

Details
This is a method for the generic command contour for the class "imlist". An object of class
"imlist" represents a list of pixel images.
(The outdated class "listof" is also handled.)
Each entry in the list x will be displayed as a contour plot, in an array of panels laid out on the same
graphics display, using plot.solist. Invididual panels are plotted by contour.im.
Value
Null.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
plot.solist, contour.im
Examples
# Multitype point pattern
contour(D <- density(split(amacrine)))

convexhull

convexhull

247

Convex Hull

Description
Computes the convex hull of a spatial object.

Usage
convexhull(x)

Arguments
x

a window (object of class "owin"), a point pattern (object of class "ppp"), a line
segment pattern (object of class "psp"), or an object that can be converted to a
window by as.owin.

Details
This function computes the convex hull of the spatial object x.

Value
A window (an object of class "owin").

Author(s)
Adrian Baddeley 
and Rolf Turner 

See Also
owin, convexhull.xy, is.convex

Examples
data(demopat)
W <- Window(demopat)
plot(convexhull(W), col="lightblue", border=NA)
plot(W, add=TRUE, lwd=2)

248

convexhull.xy

convexhull.xy

Convex Hull of Points

Description
Computes the convex hull of a set of points in two dimensions.
Usage
convexhull.xy(x, y=NULL)
Arguments
x

vector of x coordinates of observed points, or a 2-column matrix giving x,y
coordinates, or a list with components x,y giving coordinates (such as a point
pattern object of class "ppp".)

y

(optional) vector of y coordinates of observed points, if x is a vector.

Details
Given an observed pattern of points with coordinates given by x and y, this function computes the
convex hull of the points, and returns it as a window.
Value
A window (an object of class "owin").
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
owin, as.owin, convexhull, bounding.box.xy, ripras
Examples
x <- runif(30)
y <- runif(30)
w <- convexhull.xy(x,y)
plot(owin(), main="convexhull.xy(x,y)", lty=2)
plot(w, add=TRUE)
points(x,y)
X <- rpoispp(30)
plot(X, main="convexhull.xy(X)")
plot(convexhull.xy(X), add=TRUE)

convexify

249

convexify

Weil’s Convexifying Operation

Description
Converts the window W into a convex set by rearranging the edges, preserving spatial orientation of
each edge.
Usage
convexify(W, eps)
Arguments
W

A window (object of class "owin").

eps

Optional. Minimum edge length of polygonal approximation, if W is not a polygon.

Details
Weil (1995) defined a convexification operation for windows W that belong to the convex ring (that
is, for any W which is a finite union of convex sets). Note that this is not the same as the convex
hull.
The convexified set f (W ) has the same total boundary length as W and the same distribution of
orientations of the boundary. If W is a polygonal set, then the convexification f (W ) is obtained by
rearranging all the edges of W in order of their spatial orientation.
The argument W must be a window. If it is not already a polygonal window, it is first converted
to one, using simplify.owin. The edges are sorted in increasing order of angular orientation and
reassembled into a convex polygon.
Value
A window (object of class "owin").
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Weil, W. (1995) The estimation of mean particle shape and mean particle number in overlapping
particle systems in the plane. Advances in Applied Probability 27, 102–119.
See Also
convexhull for the convex hull of a window.

250

convolve.im

Examples
opa <- par(mfrow=c(1,2))
plot(letterR)
plot(convexify(letterR))
par(opa)

convolve.im

Convolution of Pixel Images

Description
Computes the convolution of two pixel images.
Usage
convolve.im(X, Y=X, ..., reflectX=FALSE, reflectY=FALSE)
Arguments
X

A pixel image (object of class "im".

Y

Optional. Another pixel image.

...
Ignored.
reflectX,reflectY
Logical values specifying whether the images X and Y (respectively) should be
reflected in the origin before computing the convolution.
Details
The convolution of two pixel images X and Y in the plane is the function C(v) defined for each
vector v as
Z
C(v) = X(u)Y (v − u) du
where the integral is over all spatial locations u, and where X(u) and Y (u) denote the pixel values
of X and Y respectively at location u.
This command computes a discretised approximation to the convolution, using the Fast Fourier
Transform. The return value is another pixel image (object of class "im") whose greyscale values
are values of the convolution.
If reflectX = TRUE then the pixel image X is reflected in the origin (see reflect) before the
convolution is computed, so that convolve.im(X,Y,reflectX=TRUE) is mathematically equivalent
to convolve.im(reflect(X), Y). (These two commands are not exactly equivalent, because the
reflection is performed in the Fourier domain in the first command, and reflection is performed in
the spatial domain in the second command).
Similarly if reflectY = TRUE then the pixel image Y is reflected in the origin before the convolution is computed, so that convolve.im(X,Y,reflectY=TRUE) is mathematically equivalent to
convolve.im(X, reflect(Y)).
Value
A pixel image (an object of class "im") representing the convolution of X and Y.

coords

251

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
imcov, reflect
Examples
X <- as.im(letterR)
Y <- as.im(square(1))
plot(convolve.im(X, Y))
plot(convolve.im(X, Y, reflectX=TRUE))
plot(convolve.im(X))

coords

Extract or Change Coordinates of a Spatial or Spatiotemporal Point
Pattern

Description
Given any kind of spatial or space-time point pattern, this function extracts the (space and/or time
and/or local) coordinates of the points and returns them as a data frame.
Usage
coords(x, ...)
## S3 method for class 'ppp'
coords(x, ...)
## S3 method for class 'ppx'
coords(x, ..., spatial = TRUE,
coords(x, ...) <- value
## S3 replacement method for
coords(x, ...) <- value
## S3 replacement method for
coords(x, ..., spatial = TRUE,

temporal = TRUE, local=TRUE)
class 'ppp'
class 'ppx'
temporal = TRUE, local=TRUE) <- value

Arguments
x

A point pattern: either a two-dimensional point pattern (object of class "ppp"),
a three-dimensional point pattern (object of class "pp3"), or a general multidimensional space-time point pattern (object of class "ppx").

...
Further arguments passed to methods.
spatial,temporal,local
Logical values indicating whether to extract spatial, temporal and local coordinates, respectively. The default is to return all such coordinates. (Only relevant
to ppx objects).
value

New values of the coordinates. A numeric vector with one entry for each point
in x, or a numeric matrix or data frame with one row for each point in x.

252

corners

Details
The function coords extracts the coordinates from a point pattern. The function coords<- replaces
the coordinates of the point pattern with new values.
Both functions coords and coords<- are generic, with methods for the classes "ppp") and "ppx".
An object of class "pp3" also inherits from "ppx" and is handled by the method for "ppx".
Value
coords returns a data.frame with one row for each point, containing the coordinates. coords
and Rolf Turner 
See Also
ppx, pp3, ppp, as.hyperframe.ppx, as.data.frame.ppx.
Examples
df <- data.frame(x=runif(4),y=runif(4),t=runif(4))
X <- ppx(data=df, coord.type=c("s","s","t"))
coords(X)
coords(X, temporal=FALSE)
coords(X) <- matrix(runif(12), ncol=3)

corners

Corners of a rectangle

Description
Returns the four corners of a rectangle
Usage
corners(window)
Arguments
window

A window. An object of class owin, or data in any format acceptable to as.owin().

Details
This trivial function is occasionally convenient. If window is of type "rectangle" this returns the
four corners of the window itself; otherwise, it returns the corners of the bounding rectangle of the
window.

covering

253

Value
A list with two components x and y, which are numeric vectors of length 4 giving the coordinates
of the four corner points of the (bounding rectangle of the) window.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
quad.object, quadscheme
Examples
w <- unit.square()
corners(w)
# returns list(x=c(0,1,0,1),y=c(0,0,1,1))

covering

Cover Region with Discs

Description
Given a spatial region, this function finds an efficient covering of the region using discs of a chosen
radius.
Usage
covering(W, r, ..., giveup=1000)
Arguments
W

A window (object of class "owin").

r

positive number: the radius of the covering discs.

...

extra arguments passed to as.mask controlling the pixel resolution for the calculations.

giveup

Maximum number of attempts to place additional discs.

Details
This function finds an efficient covering of the window W using discs of the given radius r. The
result is a point pattern giving the centres of the discs.
The algorithm tries to use as few discs as possible, but is not guaranteed to find the minimal number
of discs. It begins by placing a hexagonal grid of points inside W, then adds further points until every
location inside W lies no more than r units away from one of the points.
Value
A point pattern (object of class "ppp") giving the centres of the discs.

254

crossdist

Author(s)
Adrian Baddeley 
Examples
rr <- 0.5
X <- covering(letterR, rr)
plot(grow.rectangle(Frame(X), rr), type="n", main="")
plot(X, pch=16, add=TRUE, col="red")
plot(letterR, add=TRUE, lwd=3)
plot(X %mark% (2*rr), add=TRUE, markscale=1)

crossdist

Pairwise distances

Description
Computes the distances between pairs of ‘things’ taken from two different datasets.
Usage
crossdist(X, Y, ...)
Arguments
X,Y

Two objects of the same class.

...

Additional arguments depending on the method.

Details
Given two datasets X and Y (representing either two point patterns or two line segment patterns)
crossdist computes the Euclidean distance from each thing in the first dataset to each thing in the
second dataset, and returns a matrix containing these distances.
The function crossdist is generic, with methods for point patterns (objects of class "ppp"),
line segment patterns (objects of class "psp"), and a default method. See the documentation for
crossdist.ppp, crossdist.psp or crossdist.default for further details.
Value
A matrix whose [i,j] entry is the distance from the i-th thing in the first dataset to the j-th thing
in the second dataset.
Author(s)
Adrian Baddeley 
See Also
crossdist.ppp, crossdist.psp, crossdist.default, pairdist, nndist

crossdist.default

crossdist.default

255

Pairwise distances between two different sets of points

Description
Computes the distances between each pair of points taken from two different sets of points.
Usage
## Default S3 method:
crossdist(X, Y, x2, y2, ...,
period=NULL, method="C", squared=FALSE)
Arguments
X,Y

Numeric vectors of equal length specifying the coordinates of the first set of
points.

x2,y2

Numeric vectors of equal length specifying the coordinates of the second set of
points.

...

Ignored.

period

Optional. Dimensions for periodic edge correction.

method

String specifying which method of calculation to use. Values are "C" and "interpreted".

squared

Logical. If squared=TRUE, the squared distances are returned instead (this computation is faster).

Details
Given two sets of points, this function computes the Euclidean distance from each point in the first
set to each point in the second set, and returns a matrix containing these distances.
This is a method for the generic function crossdist.
This function expects X and Y to be numeric vectors of equal length specifying the coordinates of
the first set of points. The arguments x2,y2 specify the coordinates of the second set of points.
Alternatively if period is given, then the distances will be computed in the ‘periodic’ sense (also
known as ‘torus’ distance). The points will be treated as if they are in a rectangle of width
period[1] and height period[2]. Opposite edges of the rectangle are regarded as equivalent.
The argument method is not normally used. It is retained only for checking the validity of the
software. If method = "interpreted" then the distances are computed using interpreted R code
only. If method="C" (the default) then C code is used. The C code is faster by a factor of 4.
Value
A matrix whose [i,j] entry is the distance from the i-th point in the first set of points to the j-th
point in the second set of points.
Author(s)
Pavel Grabarnik  and Adrian Baddeley 

256

crossdist.lpp

See Also
crossdist, crossdist.ppp, crossdist.psp, pairdist, nndist, Gest
Examples
d <- crossdist(runif(7), runif(7), runif(12), runif(12))
d <- crossdist(runif(7), runif(7), runif(12), runif(12), period=c(1,1))

crossdist.lpp

Pairwise distances between two point patterns on a linear network

Description
Computes the distances between pairs of points taken from two different point patterns on the same
linear network.
Usage
## S3 method for class 'lpp'
crossdist(X, Y, ..., method="C")
Arguments
X,Y

Point patterns on a linear network (objects of class "lpp"). They must lie on the
same network.

...

Ignored.

method

String specifying which method of calculation to use. Values are "C" and "interpreted".

Details
Given two point patterns on a linear network, this function computes the Euclidean distance from
each point in the first pattern to each point in the second pattern, measuring distance by the shortest
path in the network.
This is a method for the generic function crossdist for point patterns on a linear network (objects
of class "lpp").
This function expects two point pattern objects X and Y on the same linear network, and returns the
matrix whose [i,j] entry is the shortest-path distance from X[i] to Y[j].
The argument method is not normally used. It is retained only for checking the validity of the
software. If method = "interpreted" then the distances are computed using interpreted R code
only. If method="C" (the default) then C code is used. The C code is much faster.
If two points cannot be joined by a path, the distance between them is infinite (Inf).
Value
A matrix whose [i,j] entry is the distance from the i-th point in X to the j-th point in Y. Matrix
entries are nonnegative numbers or infinity (Inf).
Author(s)
Adrian Baddeley .

crossdist.pp3

257

See Also
crossdist, crossdist.ppp, pairdist, nndist
Examples
v
X
Y
d

<<<<-

split(chicago)
v$cartheft
v$burglary
crossdist(X, Y)

crossdist.pp3

Pairwise distances between two different three-dimensional point patterns

Description
Computes the distances between pairs of points taken from two different three-dimensional point
patterns.
Usage
## S3 method for class 'pp3'
crossdist(X, Y, ..., periodic=FALSE, squared=FALSE)
Arguments
X,Y

Point patterns in three dimensions (objects of class "pp3").

...

Ignored.

periodic

Logical. Specifies whether to apply a periodic edge correction.

squared

Logical. If squared=TRUE, the squared distances are returned instead (this computation is faster).

Details
Given two point patterns in three-dimensional space, this function computes the Euclidean distance
from each point in the first pattern to each point in the second pattern, and returns a matrix containing these distances.
This is a method for the generic function crossdist for three-dimensional point patterns (objects
of class "pp3").
This function expects two point patterns X and Y, and returns the matrix whose [i,j] entry is the
distance from X[i] to Y[j].
Alternatively if periodic=TRUE, then provided the windows containing X and Y are identical and
are rectangular, then the distances will be computed in the ‘periodic’ sense (also known as ‘torus’
distance): opposite edges of the rectangle are regarded as equivalent. This is meaningless if the
window is not a rectangle.
Value
A matrix whose [i,j] entry is the distance from the i-th point in X to the j-th point in Y.

258

crossdist.ppp

Author(s)
Adrian Baddeley 
based on code for two dimensions by Pavel Grabarnik.
See Also
crossdist, pairdist, nndist, G3est
Examples
X
Y
d
d

<<<<-

runifpoint3(20)
runifpoint3(30)
crossdist(X, Y)
crossdist(X, Y, periodic=TRUE)

crossdist.ppp

Pairwise distances between two different point patterns

Description
Computes the distances between pairs of points taken from two different point patterns.
Usage
## S3 method for class 'ppp'
crossdist(X, Y, ..., periodic=FALSE, method="C", squared=FALSE)
Arguments
X,Y

Point patterns (objects of class "ppp").

...

Ignored.

periodic

Logical. Specifies whether to apply a periodic edge correction.

method

String specifying which method of calculation to use. Values are "C" and "interpreted".

squared

Logical. If squared=TRUE, the squared distances are returned instead (this computation is faster).

Details
Given two point patterns, this function computes the Euclidean distance from each point in the first
pattern to each point in the second pattern, and returns a matrix containing these distances.
This is a method for the generic function crossdist for point patterns (objects of class "ppp").
This function expects two point patterns X and Y, and returns the matrix whose [i,j] entry is the
distance from X[i] to Y[j].
Alternatively if periodic=TRUE, then provided the windows containing X and Y are identical and
are rectangular, then the distances will be computed in the ‘periodic’ sense (also known as ‘torus’
distance): opposite edges of the rectangle are regarded as equivalent. This is meaningless if the
window is not a rectangle.
The argument method is not normally used. It is retained only for checking the validity of the
software. If method = "interpreted" then the distances are computed using interpreted R code
only. If method="C" (the default) then C code is used. The C code is faster by a factor of 4.

crossdist.ppx

259

Value
A matrix whose [i,j] entry is the distance from the i-th point in X to the j-th point in Y.
Author(s)
Pavel Grabarnik  and Adrian Baddeley 
See Also
crossdist, crossdist.default, crossdist.psp, pairdist, nndist, Gest
Examples
data(cells)
d <- crossdist(cells, runifpoint(6))
d <- crossdist(cells, runifpoint(6), periodic=TRUE)

crossdist.ppx

Pairwise Distances Between Two Different Multi-Dimensional Point
Patterns

Description
Computes the distances between pairs of points taken from two different multi-dimensional point
patterns.
Usage
## S3 method for class 'ppx'
crossdist(X, Y, ...)
Arguments
X,Y

Multi-dimensional point patterns (objects of class "ppx").

...

Arguments passed to coords.ppx to determine which coordinates should be
used.

Details
Given two point patterns in multi-dimensional space, this function computes the Euclidean distance from each point in the first pattern to each point in the second pattern, and returns a matrix
containing these distances.
This is a method for the generic function crossdist for three-dimensional point patterns (objects
of class "ppx").
This function expects two multidimensional point patterns X and Y, and returns the matrix whose
[i,j] entry is the distance from X[i] to Y[j].
By default, both spatial and temporal coordinates are extracted. To obtain the spatial distance
between points in a space-time point pattern, set temporal=FALSE.

260

crossdist.psp

Value
A matrix whose [i,j] entry is the distance from the i-th point in X to the j-th point in Y.
Author(s)
Adrian Baddeley 
See Also
crossdist, pairdist, nndist
Examples
df <- data.frame(x=runif(3),y=runif(3),z=runif(3),w=runif(3))
X <- ppx(data=df)
df <- data.frame(x=runif(5),y=runif(5),z=runif(5),w=runif(5))
Y <- ppx(data=df)
d <- crossdist(X, Y)

crossdist.psp

Pairwise distances between two different line segment patterns

Description
Computes the distances between all pairs of line segments taken from two different line segment
patterns.
Usage
## S3 method for class 'psp'
crossdist(X, Y, ..., method="C", type="Hausdorff")
Arguments
X,Y

Line segment patterns (objects of class "psp").

...

Ignored.

method

String specifying which method of calculation to use. Values are "C" and "interpreted".
Usually not specified.

type

Type of distance to be computed. Options are "Hausdorff" and "separation".
Partial matching is used.

Details
This is a method for the generic function crossdist.
Given two line segment patterns, this function computes the distance from each line segment in
the first pattern to each line segment in the second pattern, and returns a matrix containing these
distances.
The distances between line segments are measured in one of two ways:

crossing.linnet

261

• if type="Hausdorff", distances are computed in the Hausdorff metric. The Hausdorff distance between two line segments is the maximum distance from any point on one of the segments to the nearest point on the other segment.
• if type="separation", distances are computed as the minimum distance from a point on one
line segment to a point on the other line segment. For example, line segments which cross
over each other have separation zero.
The argument method is not normally used. It is retained only for checking the validity of the
software. If method = "interpreted" then the distances are computed using interpreted R code
only. If method="C" (the default) then compiled C code is used. The C code is several times faster.
Value
A matrix whose [i,j] entry is the distance from the i-th line segment in X to the j-th line segment
in Y.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
pairdist, nndist, Gest
Examples
L1 <- psp(runif(5), runif(5), runif(5), runif(5), owin())
L2 <- psp(runif(10), runif(10), runif(10), runif(10), owin())
D <- crossdist(L1, L2)
#result is a 5 x 10 matrix
S <- crossdist(L1, L2, type="sep")

crossing.linnet

Crossing Points between Linear Network and Other Lines

Description
Find all the crossing-points between a linear network and another pattern of lines or line segments.
Usage
crossing.linnet(X, Y)
Arguments
X

Linear network (object of class "linnet").

Y

A linear network, or a spatial pattern of line segments (class "psp") or infinite
lines (class "infline").

Details
All crossing-points between X and Y are determined. The result is a point pattern on the network X.

262

crossing.psp

Value
Point pattern on a linear network (object of class "lpp").
Author(s)
Adrian Baddeley .
See Also
crossing.psp
Examples
plot(simplenet, main="")
L <- infline(p=runif(3), theta=runif(3, max=pi/2))
plot(L, col="red")
Y <- crossing.linnet(simplenet, L)
plot(Y, add=TRUE, cols="blue")

crossing.psp

Crossing Points of Two Line Segment Patterns

Description
Finds any crossing points between two line segment patterns.
Usage
crossing.psp(A,B,fatal=TRUE,details=FALSE)
Arguments
A,B

Line segment patterns (objects of class "psp").

details

Logical value indicating whether to return additional information. See below.

fatal

Logical value indicating what to do if the windows of A and B do not overlap.
See Details.

Details
This function finds any crossing points between the line segment patterns A and B.
A crossing point occurs whenever one of the line segments in A intersects one of the line segments
in B, at a nonzero angle of intersection.
The result is a point pattern consisting of all the intersection points.
If details=TRUE, additional information is computed, specifying where each intersection point
came from. The resulting point pattern has a data frame of marks, with columns named iA, jB, tA, tB.
The marks iA and jB are the indices of the line segments in A and B, respectively, which produced
each intersection point. The marks tA and tB are numbers between 0 and 1 specifying the position
of the intersection point along the original segments.
If the windows Window(A) and Window(B) do not overlap, then an error will be reported if fatal=TRUE,
while if fatal=FALSE an error will not occur and the result will be NULL.

cut.im

263

Value
Point pattern (object of class "ppp").
Author(s)
Adrian Baddeley  and Rolf Turner 
See Also
selfcrossing.psp, psp.object, ppp.object.
Examples
a <- psp(runif(10), runif(10), runif(10), runif(10), window=owin())
b <- psp(runif(10), runif(10), runif(10), runif(10), window=owin())
plot(a, col="green", main="crossing.psp")
plot(b, add=TRUE, col="blue")
P <- crossing.psp(a,b)
plot(P, add=TRUE, col="red")
as.data.frame(crossing.psp(a,b,details=TRUE))

cut.im

Convert Pixel Image from Numeric to Factor

Description
Transform the values of a pixel image from numeric values into a factor.
Usage
## S3 method for class 'im'
cut(x, ...)
Arguments
x

A pixel image. An object of class "im".

...

Arguments passed to cut.default. They determine the breakpoints for the
mapping from numerical values to factor values. See cut.default.

Details
This simple function applies the generic cut operation to the pixel values of the image x. The range
of pixel values is divided into several intervals, and each interval is associated with a level of a
factor. The result is another pixel image, with the same window and pixel grid as x, but with the
numeric value of each pixel discretised by replacing it by the factor level.
This function is a convenient way to inspect an image and to obtain summary statistics. See the
examples.
To select a subset of an image, use the subset operator [.im instead.
Value
A pixel image (object of class "im") with pixel values that are a factor. See im.object.

264

cut.lpp

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
cut, im.object
Examples
# artificial image data
Z <- setcov(square(1))
Y <- cut(Z, 3)
Y <- cut(Z, breaks=seq(0,1,length=5))
# cut at the quartiles
# (divides the image into 4 equal areas)
Y <- cut(Z, quantile(Z))

cut.lpp

Classify Points in a Point Pattern on a Network

Description
For a point pattern on a linear network, classify the points into distinct types according to the
numerical marks in the pattern, or according to another variable.
Usage
## S3 method for class 'lpp'
cut(x, z=marks(x), ...)
Arguments
x

A point pattern on a linear network (object of class "lpp").

z

Data determining the classification. A numeric vector, a factor, a pixel image on a linear network (class "linim"), a function on a linear network (class
"linfun"), a tessellation on a linear network (class "lintess"), a string giving
the name of a column of marks, or one of the coordinate names "x", "y", "seg"
or "tp".

...

Arguments passed to cut.default. They determine the breakpoints for the
mapping from numerical values in z to factor values in the output. See cut.default.

Details
This function has the effect of classifying each point in the point pattern x into one of several
possible types. The classification is based on the dataset z, which may be either
• a factor (of length equal to the number of points in z) determining the classification of each
point in x. Levels of the factor determine the classification.

cut.lpp

265
• a numeric vector (of length equal to the number of points in z). The range of values of z will
be divided into bands (the number of bands is determined by ...) and z will be converted to
a factor using cut.default.
• a pixel image on a network (object of class "linim"). The value of z at each point of x will
be used as the classifying variable.
• a function on a network (object of class "linfun", see linfun). The value of z at each point
of x will be used as the classifying variable.
• a tessellation on a network (object of class "lintess", see lintess). Each point of x will be
classified according to the tile of the tessellation into which it falls.
• a character string, giving the name of one of the columns of marks(x), if this is a data frame.
• a character string identifying one of the coordinates: the spatial coordinates "x", "y" or the
segment identifier "seg" or the fractional coordinate along the segment, "tp".

The default is to take z to be the vector of marks in x (or the first column in the data frame of marks
of x, if it is a data frame). If the marks are numeric, then the range of values of the numerical marks
is divided into several intervals, and each interval is associated with a level of a factor. The result
is a marked point pattern, on the same linear network, with the same point locations as x, but with
the numeric mark of each point discretised by replacing it by the factor level. This is a convenient
way to transform a marked point pattern which has numeric marks into a multitype point pattern,
for example to plot it or analyse it. See the examples.
To select some points from x, use the subset operators [.lpp or subset.lpp instead.

Value
A multitype point pattern on the same linear network, that is, a point pattern object (of class "lpp")
with a marks vector that is a factor.

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .

See Also
cut, lpp, lintess, linfun, linim

Examples
X <- runiflpp(20, simplenet)
f <- linfun(function(x,y,seg,tp) { x }, simplenet)
plot(cut(X, f, breaks=4))
plot(cut(X, "x", breaks=4))
plot(cut(X, "seg"))

266

cut.ppp

cut.ppp

Classify Points in a Point Pattern

Description
Classifies the points in a point pattern into distinct types according to the numerical marks in the
pattern, or according to another variable.
Usage
## S3 method for class 'ppp'
cut(x, z=marks(x), ...)
Arguments
x

A two-dimensional point pattern. An object of class "ppp".

z

Data determining the classification. A numeric vector, a factor, a pixel image, a
window, a tessellation, or a string giving the name of a column of marks or the
name of a spatial coordinate.

...

Arguments passed to cut.default. They determine the breakpoints for the
mapping from numerical values in z to factor values in the output. See cut.default.

Details
This function has the effect of classifying each point in the point pattern x into one of several
possible types. The classification is based on the dataset z, which may be either
• a factor (of length equal to the number of points in z) determining the classification of each
point in x. Levels of the factor determine the classification.
• a numeric vector (of length equal to the number of points in z). The range of values of z will
be divided into bands (the number of bands is determined by ...) and z will be converted to
a factor using cut.default.
• a pixel image (object of class "im"). The value of z at each point of x will be used as the
classifying variable.
• a tessellation (object of class "tess", see tess). Each point of x will be classified according
to the tile of the tessellation into which it falls.
• a window (object of class "owin"). Each point of x will be classified according to whether it
falls inside or outside this window.
• a character string, giving the name of one of the columns of marks(x), if this is a data frame.
• a character string "x" or "y" identifying one of the spatial coordinates.
The default is to take z to be the vector of marks in x (or the first column in the data frame of marks
of x, if it is a data frame). If the marks are numeric, then the range of values of the numerical marks
is divided into several intervals, and each interval is associated with a level of a factor. The result is
a marked point pattern, with the same window and point locations as x, but with the numeric mark
of each point discretised by replacing it by the factor level. This is a convenient way to transform a
marked point pattern which has numeric marks into a multitype point pattern, for example to plot it
or analyse it. See the examples.
To select some points from a point pattern, use the subset operators [.ppp or subset.ppp instead.

cut.ppp

267

Value
A multitype point pattern, that is, a point pattern object (of class "ppp") with a marks vector that is
a factor.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
cut, ppp.object, tess
Examples
# (1) cutting based on numeric marks of point pattern
trees <- longleaf
# Longleaf Pines data
# the marks are positive real numbers indicating tree diameters.
## Not run:
plot(trees)
## End(Not run)
# cut the range of tree diameters into three intervals
long3 <- cut(trees, breaks=3)
## Not run:
plot(long3)
## End(Not run)
# adult trees defined to have diameter at least 30 cm
long2 <- cut(trees, breaks=c(0,30,100), labels=c("Sapling", "Adult"))
plot(long2)
plot(long2, cols=c("green","blue"))
# (2) cutting based on another numeric vector
# Divide Swedish Pines data into 3 classes
# according to nearest neighbour distance
swedishpines
plot(cut(swedishpines, nndist(swedishpines), breaks=3))
# (3) cutting based on tessellation
# Divide Swedish Pines study region into a 4 x 4 grid of rectangles
# and classify points accordingly
tes <- tess(xgrid=seq(0,96,length=5),ygrid=seq(0,100,length=5))
plot(cut(swedishpines, tes))
plot(tes, lty=2, add=TRUE)
# (4) multivariate marks
finpines

268

data.ppm
cut(finpines, "height", breaks=4)

data.ppm

Extract Original Data from a Fitted Point Process Model

Description
Given a fitted point process model, this function extracts the original point pattern dataset to which
the model was fitted.
Usage
data.ppm(object)
Arguments
object

fitted point process model (an object of class "ppm").

Details
An object of class "ppm" represents a point process model that has been fitted to data. It is typically
produced by the model-fitting algorithm ppm. The object contains complete information about the
original data point pattern to which the model was fitted. This function extracts the original data
pattern.
See ppm.object for a list of all operations that can be performed on objects of class "ppm".
Value
A point pattern (object of class "ppp").
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
ppm.object, ppp.object
Examples
data(cells)
fit <- ppm(cells, ~1, Strauss(r=0.1))
X <- data.ppm(fit)
# 'X' is identical to 'cells'

dclf.progress

dclf.progress

269

Progress Plot of Test of Spatial Pattern

Description
Generates a progress plot (envelope representation) of the Diggle-Cressie-Loosmore-Ford test or
the Maximum Absolute Deviation test for a spatial point pattern.
Usage
dclf.progress(X, ...)
mad.progress(X, ...)
mctest.progress(X, fun = Lest, ...,
exponent = 1, nrank = 1,
interpolate = FALSE, alpha, rmin=0)
Arguments
X

Either a point pattern (object of class "ppp", "lpp" or other class), a fitted point
process model (object of class "ppm", "kppm" or other class) or an envelope
object (class "envelope").

...

Arguments passed to mctest.progress or to envelope. Useful arguments include fun to determine the summary function, nsim to specify the number of
Monte Carlo simulations, alternative to specify one-sided or two-sided envelopes, and verbose=FALSE to turn off the messages.

fun

Function that computes the desired summary statistic for a point pattern.

exponent

Positive number. The exponent of the Lp distance. See Details.

nrank

Integer. The rank of the critical value of the Monte Carlo test, amongst the nsim
simulated values. A rank of 1 means that the minimum and maximum simulated
values will become the critical values for the test.

interpolate

Logical value indicating how to compute the critical value. If interpolate=FALSE
(the default), a standard Monte Carlo test is performed, and the critical value
is the largest simulated value of the test statistic (if nrank=1) or the nrank-th
largest (if nrank is another number). If interpolate=TRUE, kernel density estimation is applied to the simulated values, and the critical value is the upper
alpha quantile of this estimated distribution.

alpha

Optional. The significance level of the test. Equivalent to nrank/(nsim+1)
where nsim is the number of simulations.

rmin

Optional. Left endpoint for the interval of r values on which the test statistic is
calculated.

Details
The Diggle-Cressie-Loosmore-Ford test and the Maximum Absolute Deviation test for a spatial
point pattern are described in dclf.test. These tests depend on the choice of an interval of distance
values (the argument rinterval). A progress plot or envelope representation of the test (Baddeley
et al, 2014) is a plot of the test statistic (and the corresponding critical value) against the length of
the interval rinterval.

270

dclf.progress
The command dclf.progress performs dclf.test on X using all possible intervals of the form
[0, R], and returns the resulting values of the test statistic, and the corresponding critical values of
the test, as a function of R.
Similarly mad.progress performs mad.test using all possible intervals and returns the test statistic
and critical value.
More generally, mctest.progress performs a test based on the Lp discrepancy between the curves.
The deviation between two curves is measured by the pth root of the integral of the pth power of the
absolute value of the difference between the two curves. The exponent p is given by the argument
exponent. The case exponent=2 is the Cressie-Loosmore-Ford test, while exponent=Inf is the
MAD test.
If the argument rmin is given, it specifies the left endpoint of the interval defining the test statistic:
the tests are performed using intervals [rmin , R] where R ≥ rmin .
The result of each command is an object of class "fv" that can be plotted to obtain the progress
plot. The display shows the test statistic (solid black line) and the Monte Carlo acceptance region
(grey shading).
The significance level for the Monte Carlo test is nrank/(nsim+1). Note that nsim defaults to 99,
so if the values of nrank and nsim are not given, the default is a test with significance level 0.01.
If X is an envelope object, then some of the data stored in X may be re-used:
• If X is an envelope object containing simulated functions, and fun=NULL, then the code will
re-use the simulated functions stored in X.
• If X is an envelope object containing simulated point patterns, then fun will be applied to the
stored point patterns to obtain the simulated functions. If fun is not specified, it defaults to
Lest.
• Otherwise, new simulations will be performed, and fun defaults to Lest.

Value
An object of class "fv" that can be plotted to obtain the progress plot.
Author(s)
Adrian Baddeley 
, Andrew Hardegen, Tom Lawrence, Gopal Nair and Robin Milne.
References
Baddeley, A., Diggle, P., Hardegen, A., Lawrence, T., Milne, R. and Nair, G. (2014) On tests of
spatial pattern based on simulation envelopes. Ecological Monographs 84 (3) 477–489.
See Also
dclf.test and mad.test for the tests.
See plot.fv for information on plotting objects of class "fv".
Examples
plot(dclf.progress(cells, nsim=19))

dclf.sigtrace

dclf.sigtrace

271

Significance Trace of Cressie-Loosmore-Ford or Maximum Absolute
Deviation Test

Description
Generates a Significance Trace of the Diggle(1986)/ Cressie (1991)/ Loosmore and Ford (2006) test
or the Maximum Absolute Deviation test for a spatial point pattern.
Usage
dclf.sigtrace(X, ...)
mad.sigtrace(X, ...)
mctest.sigtrace(X, fun=Lest, ...,
exponent=1, interpolate=FALSE, alpha=0.05,
confint=TRUE, rmin=0)
Arguments
X

Either a point pattern (object of class "ppp", "lpp" or other class), a fitted point
process model (object of class "ppm", "kppm" or other class) or an envelope
object (class "envelope").

...

Arguments passed to envelope or mctest.progress. Useful arguments include fun to determine the summary function, nsim to specify the number of
Monte Carlo simulations, alternative to specify a one-sided test, and verbose=FALSE
to turn off the messages.

fun

Function that computes the desired summary statistic for a point pattern.

exponent

Positive number. The exponent of the Lp distance. See Details.

interpolate

Logical value specifying whether to calculate the p-value by interpolation. If
interpolate=FALSE (the default), a standard Monte Carlo test is performed,
yielding a p-value of the form (k + 1)/(n + 1) where n is the number of simulations and k is the number of simulated values which are more extreme than
the observed value. If interpolate=TRUE, the p-value is calculated by applying kernel density estimation to the simulated values, and computing the tail
probability for this estimated distribution.

alpha

Significance level to be plotted (this has no effect on the calculation but is simply
plotted as a reference value).

confint

Logical value indicating whether to compute a confidence interval for the ‘true’
p-value.

rmin

Optional. Left endpoint for the interval of r values on which the test statistic is
calculated.

Details
The Diggle (1986)/ Cressie (1991)/Loosmore and Ford (2006) test and the Maximum Absolute
Deviation test for a spatial point pattern are described in dclf.test. These tests depend on the
choice of an interval of distance values (the argument rinterval). A significance trace (Bowman
and Azzalini, 1997; Baddeley et al, 2014, 2015) of the test is a plot of the p-value obtained from the
test against the length of the interval rinterval.

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The command dclf.sigtrace performs dclf.test on X using all possible intervals of the form
[0, R], and returns the resulting p-values as a function of R.
Similarly mad.sigtrace performs mad.test using all possible intervals and returns the p-values.
More generally, mctest.sigtrace performs a test based on the Lp discrepancy between the curves.
The deviation between two curves is measured by the pth root of the integral of the pth power of the
absolute value of the difference between the two curves. The exponent p is given by the argument
exponent. The case exponent=2 is the Cressie-Loosmore-Ford test, while exponent=Inf is the
MAD test.
If the argument rmin is given, it specifies the left endpoint of the interval defining the test statistic:
the tests are performed using intervals [rmin , R] where R ≥ rmin .
The result of each command is an object of class "fv" that can be plotted to obtain the significance
trace. The plot shows the Monte Carlo p-value (solid black line), the critical value 0.05 (dashed
red line), and a pointwise 95% confidence band (grey shading) for the ‘true’ (Neyman-Pearson)
p-value. The confidence band is based on the Agresti-Coull (1998) confidence interval for a binomial proportion (when interpolate=FALSE) or the delta method and normal approximation (when
interpolate=TRUE).
If X is an envelope object and fun=NULL then the code will re-use the simulated functions stored in
X.

Value
An object of class "fv" that can be plotted to obtain the significance trace.
Author(s)
Adrian Baddeley , Andrew Hardegen, Tom Lawrence, Robin
Milne, Gopalan Nair and Suman Rakshit. Implemented by Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Agresti, A. and Coull, B.A. (1998) Approximate is better than “Exact” for interval estimation of
binomial proportions. American Statistician 52, 119–126.
Baddeley, A., Diggle, P., Hardegen, A., Lawrence, T., Milne, R. and Nair, G. (2014) On tests of
spatial pattern based on simulation envelopes. Ecological Monographs 84(3) 477–489.
Baddeley, A., Hardegen, A., Lawrence, L., Milne, R.K., Nair, G.M. and Rakshit, S. (2015) Pushing
the envelope: extensions of graphical Monte Carlo tests. Submitted for publication.
Bowman, A.W. and Azzalini, A. (1997) Applied smoothing techniques for data analysis: the kernel
approach with S-Plus illustrations. Oxford University Press, Oxford.
See Also
dclf.test for the tests; dclf.progress for progress plots.
See plot.fv for information on plotting objects of class "fv".
See also dg.sigtrace.
Examples
plot(dclf.sigtrace(cells, Lest, nsim=19))

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273

dclf.test

Diggle-Cressie-Loosmore-Ford and Maximum Absolute Deviation
Tests

Description
Perform the Diggle (1986) / Cressie (1991) / Loosmore and Ford (2006) test or the Maximum
Absolute Deviation test for a spatial point pattern.
Usage
dclf.test(X, ..., alternative=c("two.sided", "less", "greater"),
rinterval = NULL, leaveout=1,
scale=NULL, clamp=FALSE, interpolate=FALSE)
mad.test(X, ...,

alternative=c("two.sided", "less", "greater"),
rinterval = NULL, leaveout=1,
scale=NULL, clamp=FALSE, interpolate=FALSE)

Arguments
X

Data for the test. Either a point pattern (object of class "ppp", "lpp" or other
class), a fitted point process model (object of class "ppm", "kppm" or other
class), a simulation envelope (object of class "envelope") or a previous result
of dclf.test or mad.test.

...

Arguments passed to envelope. Useful arguments include fun to determine
the summary function, nsim to specify the number of Monte Carlo simulations,
verbose=FALSE to turn off the messages, savefuns or savepatterns to save
the simulation results, and use.theory described under Details.

alternative

The alternative hypothesis. A character string. The default is a two-sided alternative. See Details.

rinterval

Interval of values of the summary function argument r over which the maximum
absolute deviation, or the integral, will be computed for the test. A numeric
vector of length 2.

leaveout

Optional integer 0, 1 or 2 indicating how to calculate the deviation between the
observed summary function and the nominal reference value, when the reference
value must be estimated by simulation. See Details.

scale

Optional. A function in the R language which determines the relative scale of
deviations, as a function of distance r. Summary function values for distance r
will be divided by scale(r) before the test statistic is computed.

clamp

Logical value indicating how to compute deviations in a one-sided test. Deviations of the observed summary function from the theoretical summary function
are initially evaluated as signed real numbers, with large positive values indicating consistency with the alternative hypothesis. If clamp=FALSE (the default),
these values are not changed. If clamp=TRUE, any negative values are replaced
by zero.

interpolate

Logical value specifying whether to calculate the p-value by interpolation. If
interpolate=FALSE (the default), a standard Monte Carlo test is performed,

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dclf.test
yielding a p-value of the form (k + 1)/(n + 1) where n is the number of simulations and k is the number of simulated values which are more extreme than
the observed value. If interpolate=TRUE, the p-value is calculated by applying kernel density estimation to the simulated values, and computing the tail
probability for this estimated distribution.

Details
These functions perform hypothesis tests for goodness-of-fit of a point pattern dataset to a point
process model, based on Monte Carlo simulation from the model.
dclf.test performs the test advocated by Loosmore and Ford (2006) which is also described
in Diggle (1986), Cressie (1991, page 667, equation (8.5.42)) and Diggle (2003, page 14). See
Baddeley et al (2014) for detailed discussion.
mad.test performs the ‘global’ or ‘Maximum Absolute Deviation’ test described by Ripley (1977,
1981). See Baddeley et al (2014).
The type of test depends on the type of argument X.
• If X is some kind of point pattern, then a test of Complete Spatial Randomness (CSR) will be
performed. That is, the null hypothesis is that the point pattern is completely random.
• If X is a fitted point process model, then a test of goodness-of-fit for the fitted model will be
performed. The model object contains the data point pattern to which it was originally fitted.
The null hypothesis is that the data point pattern is a realisation of the model.
• If X is an envelope object generated by envelope, then it should have been generated with
savefuns=TRUE or savepatterns=TRUE so that it contains simulation results. These simulations will be treated as realisations from the null hypothesis.
• Alternatively X could be a previously-performed test of the same kind (i.e. the result of calling
dclf.test or mad.test). The simulations used to perform the original test will be re-used
to perform the new test (provided these simulations were saved in the original test, by setting
savefuns=TRUE or savepatterns=TRUE).
The argument alternative specifies the alternative hypothesis, that is, the direction of deviation
that will be considered statistically significant. If alternative="two.sided" (the default), both
positive and negative deviations (between the observed summary function and the theoretical function) are significant. If alternative="less", then only negative deviations (where the observed
summary function is lower than the theoretical function) are considered. If alternative="greater",
then only positive deviations (where the observed summary function is higher than the theoretical
function) are considered.
In all cases, the algorithm will first call envelope to generate or extract the simulated summary
functions. The number of simulations that will be generated or extracted, is determined by the
argument nsim, and defaults to 99. The summary function that will be computed is determined by
the argument fun (or the first unnamed argument in the list ...) and defaults to Kest (except when
X is an envelope object generated with savefuns=TRUE, when these functions will be taken).
The choice of summary function fun affects the power of the test. It is normally recommended
to apply a variance-stabilising transformation (Ripley, 1981). If you are using the K function,
the normal practice is to replace this by the L function (Besag, 1977) computed by Lest. If you
are using the F or G √
functions, the recommended practice is to apply Fisher’s variance-stabilising
transformation sin−1 x using the argument transform. See the Examples.
The argument rinterval specifies the interval of distance values r which will contribute to the test
statistic (either maximising over this range of values for mad.test, or integrating over this range
of values for dclf.test). This affects the power of the test. General advice and experiments in
Baddeley et al (2014) suggest that the maximum r value should be slightly larger than the maximum

dclf.test

275

possible range of interaction between points. The dclf.test is quite sensitive to this choice, while
the mad.test is relatively insensitive.
It is also possible to specify a pointwise test (i.e. taking a single, fixed value of distance r) by
specifing rinterval = c(r,r).
The argument use.theory passed to envelope determines whether to compare the summary function for the data to its theoretical value for CSR (use.theory=TRUE) or to the sample mean of
simulations from CSR (use.theory=FALSE).
The argument leaveout specifies how to calculate the discrepancy between the summary function
for the data and the nominal reference value, when the reference value must be estimated by simulation. The values leaveout=0 and leaveout=1 are both algebraically equivalent (Baddeley et al,
2014, Appendix) to computing the difference observed - reference where the reference is the
mean of simulated values. The value leaveout=2 gives the leave-two-out discrepancy proposed by
Dao and Genton (2014).
Value
An object of class "htest". Printing this object gives a report on the result of the test. The p-value
is contained in the component p.value.
Handling Ties
If the observed value of the test statistic is equal to one or more of the simulated values (called a
tied value), then the tied values will be assigned a random ordering, and a message will be printed.
Author(s)
Adrian Baddeley 
, Andrew Hardegen and Suman Rakshit.
References
Baddeley, A., Diggle, P.J., Hardegen, A., Lawrence, T., Milne, R.K. and Nair, G. (2014) On tests of
spatial pattern based on simulation envelopes. Ecological Monographs 84(3) 477–489.
Baddeley, A., Hardegen, A., Lawrence, T., Milne, R.K. and Nair, G. (2015) Pushing the envelope.
In preparation.
Besag, J. (1977) Discussion of Dr Ripley’s paper. Journal of the Royal Statistical Society, Series B,
39, 193–195.
Cressie, N.A.C. (1991) Statistics for spatial data. John Wiley and Sons, 1991.
Dao, N.A. and Genton, M. (2014) A Monte Carlo adjusted goodness-of-fit test for parametric models describing spatial point patterns. Journal of Graphical and Computational Statistics 23, 497–
517.
Diggle, P. J. (1986). Displaced amacrine cells in the retina of a rabbit : analysis of a bivariate spatial
point pattern. J. Neuroscience Methods 18, 115–125.
Diggle, P.J. (2003) Statistical analysis of spatial point patterns, Second edition. Arnold.
Loosmore, N.B. and Ford, E.D. (2006) Statistical inference using the G or K point pattern spatial
statistics. Ecology 87, 1925–1931.
Ripley, B.D. (1977) Modelling spatial patterns (with discussion). Journal of the Royal Statistical
Society, Series B, 39, 172 – 212.
Ripley, B.D. (1981) Spatial statistics. John Wiley and Sons.

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default.dummy

See Also
envelope, dclf.progress
Examples
dclf.test(cells, Lest, nsim=39)
m <- mad.test(cells, Lest, verbose=FALSE, rinterval=c(0, 0.1), nsim=19)
m
# extract the p-value
m$p.value
# variance stabilised G function
dclf.test(cells, Gest, transform=expression(asin(sqrt(.))),
verbose=FALSE, nsim=19)
## one-sided test
ml <- mad.test(cells, Lest, verbose=FALSE, nsim=19, alternative="less")
## scaled
mad.test(cells, Kest, verbose=FALSE, nsim=19,
rinterval=c(0.05, 0.2),
scale=function(r) { r })

default.dummy

Generate a Default Pattern of Dummy Points

Description
Generates a default pattern of dummy points for use in a quadrature scheme.
Usage
default.dummy(X, nd, random=FALSE, ntile=NULL, npix=NULL,
quasi=FALSE, ..., eps=NULL, verbose=FALSE)
Arguments
X
nd
random
quasi
ntile
npix
...
eps

verbose

The observed data point pattern. An object of class "ppp" or in a format recognised by as.ppp()
Optional. Integer, or integer vector of length 2, specifying an nd * nd or
nd[1] * nd[2] rectangular array of dummy points.
Logical value. If TRUE, the dummy points are generated randomly.
Logical value. If TRUE, the dummy points are generated by a quasirandom sequence.
Optional. Integer or pair of integers specifying the number of rows and columns
of tiles used in the counting rule.
Optional. Integer or pair of integers specifying the number of rows and columns
of pixels used in computing approximate areas.
Ignored.
Optional. Grid spacing. A positive number, or a vector of two positive numbers,
giving the horizontal and vertical spacing, respectively, of the grid of dummy
points. Incompatible with nd.
If TRUE, information about the construction of the quadrature scheme is printed.

default.expand

277

Details
This function provides a sensible default for the dummy points in a quadrature scheme.
A quadrature scheme consists of the original data point pattern, an additional pattern of dummy
points, and a vector of quadrature weights for all these points. See quad.object for further information about quadrature schemes.
If random and quasi are both false (the default), then the function creates dummy points in a regular
nd[1] by nd[1] rectangular grid. If random is true and quasi is false, then the frame of the window
is divided into an nd[1] by nd[1] array of tiles, and one dummy point is generated at random inside
each tile. If quasi is true, a quasirandom pattern of nd[1] * nd[2] points is generated. In all cases,
the four corner points of the frame of the window are added. Then if the window is not rectangular,
any dummy points lying outside it are deleted.
If nd is missing, a default value (depending on the data pattern X) is computed by default.ngrid.
Alternative functions for creating dummy patterns include corners, gridcentres, stratrand and
spokes.
Value
A point pattern (an object of class "ppp", see ppp.object) containing the dummy points.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
quad.object, quadscheme, corners, gridcentres, stratrand, spokes
Examples
data(simdat)
P <- simdat
D <- default.dummy(P, 100)
## Not run: plot(D)
Q <- quadscheme(P, D, "grid")
## Not run: plot(union.quad(Q))

default.expand

Default Expansion Rule for Simulation of Model

Description
Defines the default expansion window or expansion rule for simulation of a fitted point process
model.
Usage
default.expand(object, m=2, epsilon=1e-6, w=Window(object))

278

default.expand

Arguments
object

A point process model (object of class "ppm" or "rmhmodel").

m

A single numeric value. The window will be expanded by a distance m * reach(object)
along each side.

epsilon

Threshold argument passed to reach to determine reach(object).

w

Optional. The un-expanded window in which the model is defined. The resulting simulated point patterns will lie in this window.

Details
This function computes a default value for the expansion rule (the argument expand in rmhcontrol)
given a fitted point process model object. This default is used by envelope, qqplot.ppm, simulate.ppm
and other functions.
Suppose we wish to generate simulated realisations of a fitted point process model inside a window
w. It is advisable to first simulate the pattern on a larger window, and then clip it to the original
window w. This avoids edge effects in the simulation. It is called expansion of the simulation
window.
Accordingly, for the Metropolis-Hastings simulation algorithm rmh, the algorithm control parameters specified by rmhcontrol include an argument expand that determines the expansion of the
simulation window.
The function default.expand determines the default expansion rule for a fitted point process
model object.
If the model is Poisson, then no expansion is necessary. No expansion is performed by default, and
default.expand returns a rule representing no expansion. The simulation window is the original
window w = Window(object).
If the model depends on external covariates (i.e.\ covariates other than the Cartesian covariates x
and y and the marks) then no expansion is feasible, in general, because the spatial domain of the
covariates is not guaranteed to be large enough. default.expand returns a rule representing no
expansion. The simulation window is the original window w = Window(object).
If the model depends on the Cartesian covariates x and y, it would be feasible to expand the simulation window, and this was the default for spatstat version 1.24-1 and earlier. However this
sometimes produces artefacts (such as an empty point pattern) or memory overflow, because the fitted trend, extrapolated outside the original window of the data, may become very large. In spatstat
version 1.24-2 and later, the default rule is not to expand if the model depends on x or y. Again
default.expand returns a rule representing no expansion.
Otherwise, expansion will occur. The original window w = Window(object) is expanded by a
distance m * rr, where rr is the interaction range of the model, computed by reach. If w is a
rectangle then each edge of w is displaced outward by distance m * rr. If w is not a rectangle then
w is dilated by distance m * rr using dilation.
Value
A window expansion rule (object of class "rmhexpand").
Author(s)
Adrian Baddeley 
and Rolf Turner 

default.rmhcontrol

279

See Also
rmhexpand, rmhcontrol, rmh, envelope, qqplot.ppm
Examples
data(cells)
fit <- ppm(cells, ~1, Strauss(0.07))
default.expand(fit)
mod <- rmhmodel(cif="strauss", par=list(beta=100, gamma=0.5, r=0.07))
default.expand(fit)

default.rmhcontrol

Set Default Control Parameters for Metropolis-Hastings Algorithm.

Description
Given a fitted point process model, this command sets appropriate default values of the parameters
controlling the iterative behaviour of the Metropolis-Hastings algorithm.
Usage
default.rmhcontrol(model, w=NULL)
Arguments
model

A fitted point process model (object of class "ppm")

w

Optional. Window for the resulting simulated patterns.

Details
This function sets the values of the parameters controlling the iterative behaviour of the MetropolisHastings simulation algorithm. It uses default values that would be appropriate for the fitted point
process model model.
The expansion parameter expand is set to default.expand(model, w).
All other parameters revert to their defaults given in rmhcontrol.default.
See rmhcontrol for the full list of control parameters. To override default parameters, use update.rmhcontrol.
Value
An object of class "rmhcontrol". See rmhcontrol.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
rmhcontrol, update.rmhcontrol, ppm, default.expand

280

delaunay

Examples
fit <- ppm(cells, ~1, Strauss(0.1))
default.rmhcontrol(fit)
default.rmhcontrol(fit, w=square(2))

delaunay

Delaunay Triangulation of Point Pattern

Description
Computes the Delaunay triangulation of a spatial point pattern.
Usage
delaunay(X)
Arguments
X

Spatial point pattern (object of class "ppp").

Details
The Delaunay triangulation of a spatial point pattern X is defined as follows. First the Dirichlet/Voronoi tessellation of X computed; see dirichlet. Then two points of X are defined to be
Delaunay neighbours if their Dirichlet/Voronoi tiles share a common boundary. Every pair of Delaunay neighbours is joined by a straight line. The result is a tessellation, consisting of disjoint
triangles. The union of these triangles is the convex hull of X.
Value
A tessellation (object of class "tess"). The window of the tessellation is the convex hull of X, not
the original window of X.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
tess, dirichlet, convexhull.xy, ppp, delaunayDistance, delaunayNetwork
Examples
X <- runifpoint(42)
plot(delaunay(X))
plot(X, add=TRUE)

delaunayDistance

281

delaunayDistance

Distance on Delaunay Triangulation

Description
Computes the graph distance in the Delaunay triangulation of a point pattern.
Usage
delaunayDistance(X)
Arguments
X

Spatial point pattern (object of class "ppp").

Details
The Delaunay triangulation of a spatial point pattern X is defined as follows. First the Dirichlet/Voronoi tessellation of X computed; see dirichlet. Then two points of X are defined to be
Delaunay neighbours if their Dirichlet/Voronoi tiles share a common boundary. Every pair of Delaunay neighbours is joined by a straight line.
The graph distance in the Delaunay triangulation between two points X[i] and X[j] is the minimum number of edges of the Delaunay triangulation that must be traversed to go from X[i] to
X[j].
This command returns a matrix D such that D[i,j] is the graph distance between X[i] and X[j].
Value
A symmetric square matrix with integer entries.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
delaunay, delaunayNetwork
Examples
X <- runifpoint(20)
M <- delaunayDistance(X)
plot(delaunay(X), lty=3)
text(X, labels=M[1, ], cex=2)

282

delaunayNetwork

delaunayNetwork

Linear Network of Delaunay Triangulation or Dirichlet Tessellation

Description
Computes the edges of the Delaunay triangulation or Dirichlet tessellation of a point pattern, and
returns the result as a linear network object.
Usage
delaunayNetwork(X)
dirichletNetwork(X, ...)
Arguments
X

A point pattern (object of class "ppp").

...

Arguments passed to as.linnet.psp

Details
For delaunayNetwork, points of X which are neighbours in the Delaunay triangulation (see delaunay)
will be joined by a straight line. The result will be returned as a linear network (object of class
"linnet").
For dirichletNetwork, the Dirichlet tessellation is computed (see dirichlet) and the edges of the
tiles of the tessellation are extracted. This is converted to a linear network using as.linnet.psp.
Value
Linear network (object of class "linnet") or NULL.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
delaunay, dirichlet, delaunayDistance
Examples
LE <- delaunayNetwork(cells)
LI <- dirichletNetwork(cells)

deletebranch

283

deletebranch

Delete or Extract a Branch of a Tree

Description
Deletes or extracts a given branch of a tree.
Usage
deletebranch(X, ...)
## S3 method for class 'linnet'
deletebranch(X, code, labels, ...)
## S3 method for class 'lpp'
deletebranch(X, code, labels, ...)
extractbranch(X, ...)
## S3 method for class 'linnet'
extractbranch(X, code, labels, ..., which=NULL)
## S3 method for class 'lpp'
extractbranch(X, code, labels, ..., which=NULL)
Arguments
X

Linear network (object of class "linnet") or point pattern on a linear network
(object of class "lpp").

code

Character string. Label of the branch to be deleted or extracted.

labels

Vector of character strings. Branch labels for the vertices of the network, usually
obtained from treebranchlabels.

...

Arguments passed to methods.

which

Logical vector indicating which vertices of the network should be extracted.
Overrides code and labels.

Details
The linear network L <- X or L <- as.linnet(X) must be a tree, that is, it has no loops.
The argument labels should be a character vector giving tree branch labels for each vertex of the
network. It is usually obtained by calling treebranchlabels.
The branch designated by the string code will be deleted or extracted.
The return value is the result of deleting or extracting this branch from X along with any data
associated with this branch (such as points or marks).
Value
Another object of the same type as X obtained by deleting or extracting the specified branch.

284

deltametric

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
treebranchlabels, branchlabelfun, linnet
Examples
# make a simple tree
m <- simplenet$m
m[8,10] <- m[10,8] <- FALSE
L <- linnet(vertices(simplenet), m)
plot(L, main="")
# compute branch labels
tb <- treebranchlabels(L, 1)
tbc <- paste0("[", tb, "]")
text(vertices(L), labels=tbc, cex=2)
# delete branch B
LminusB <- deletebranch(L, "b", tb)
plot(LminusB, add=TRUE, col="green")
# extract branch B
LB <- extractbranch(L, "b", tb)
plot(LB, add=TRUE, col="red")

deltametric

Delta Metric

Description
Computes the discrepancy between two sets A and B according to Baddeley’s delta-metric.
Usage
deltametric(A, B, p = 2, c = Inf, ...)
Arguments
A,B

The two sets which will be compared. Windows (objects of class "owin"),
point patterns (objects of class "ppp") or line segment patterns (objects of class
"psp").

p

Index of the Lp metric. Either a positive numeric value, or Inf.

c

Distance threshold. Either a positive numeric value, or Inf.

...

Arguments passed to as.mask to determine the pixel resolution of the distance
maps computed by distmap.

deltametric

285

Details
Baddeley (1992a, 1992b) defined a distance between two sets A and B contained in a space W by


1
∆(A, B) =
|W |

Z

p

1/p

|min(c, d(x, A)) − min(c, d(x, B))| dx
W

where c ≥ 0 is a distance threshold parameter, 0 < p ≤ ∞ is the exponent parameter, and d(x, A)
denotes the shortest distance from a point x to the set A. Also |W| denotes the area or volume of
the containing space W .
This is defined so that it is a metric, i.e.
• ∆(A, B) = 0 if and only if A = B
• ∆(A, B) = ∆(B, A)
• ∆(A, C) ≤ ∆(A, B) + ∆(B, C)
It is topologically equivalent to the Hausdorff metric (Baddeley, 1992a) but has better stability
properties in practical applications (Baddeley, 1992b).
If p = ∞ and c = ∞ the Delta metric is equal to the Hausdorff metric.
The algorithm uses distmap to compute the distance maps d(x, A) and d(x, B), then approximates
the integral numerically. The accuracy of the computation depends on the pixel resolution which is
controlled through the extra arguments ... passed to as.mask.
Value
A numeric value.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Baddeley, A.J. (1992a) Errors in binary images and an Lp version of the Hausdorff metric. Nieuw
Archief voor Wiskunde 10, 157–183.
Baddeley, A.J. (1992b) An error metric for binary images. In W. Foerstner and S. Ruwiedel (eds)
Robust Computer Vision. Karlsruhe: Wichmann. Pages 59–78.
See Also
distmap
Examples
X <- runifpoint(20)
Y <- runifpoint(10)
deltametric(X, Y, p=1,c=0.1)

286

density.lpp

density.lpp

Kernel Estimate of Intensity on a Linear Network

Description
Estimates the intensity of a point process on a linear network by applying kernel smoothing to the
point pattern data.
Usage
## S3 method for class 'lpp'
density(x, sigma, ...,
weights=NULL,
kernel="gaussian",
continuous=TRUE,
epsilon = 1e-06, verbose = TRUE,
debug = FALSE, savehistory = TRUE,
old=FALSE)
## S3 method for class 'splitppx'
density(x, sigma, ...)
Arguments
x

Point pattern on a linear network (object of class "lpp") to be smoothed.

sigma

Smoothing bandwidth (standard deviation of the kernel) in the same units as the
spatial coordinates of x.

...

Arguments passed to as.mask determining the resolution of the result.

weights

Optional. Numeric vector of weights associated with the points of x. Weights
may be positive, negative or zero.

kernel

Character string specifying the smoothing kernel. See dkernel for possible
options.

continuous

Logical value indicating whether to compute the “equal-split continuous” smoother
(continuous=TRUE, the default) or the “equal-split discontinuous” smoother
(continuous=FALSE).

epsilon

Tolerance value. A tail of the kernel with total mass less than epsilon may be
deleted.

verbose

Logical value indicating whether to print progress reports.

debug

Logical value indicating whether to print debugging information.

savehistory

Logical value indicating whether to save the entire history of the algorithm, for
the purposes of evaluating performance.

old

Logical value indicating whether to use the old, very slow algorithm for the
equal-split continuous estimator.

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287

Details
Kernel smoothing is applied to the points of x using one of the rules described in Okabe and Sugihara (2012) and McSwiggan et al (2016). The result is a pixel image on the linear network (class
"linim") which can be plotted.
If continuous=TRUE (the default), smoothing is performed using the “equal-split continuous” rule
described in Section 9.2.3 of Okabe and Sugihara (2012). The resulting function is continuous on
the linear network.
If continuous=FALSE, smoothing is performed using the “equal-split discontinuous” rule described
in Section 9.2.2 of Okabe and Sugihara (2012). The resulting function is not continuous.
In the default case (where continuous=TRUE and kernel="gaussian" and old=FALSE), computation is performed rapidly by solving the classical heat equation on the network, as described in
McSwiggan et al (2016). Computational time is short, but increases quadratically with sigma. The
arguments epsilon,debug,verbose,savehistory are ignored.
In all other cases, computation is performed by path-tracing as described in Okabe and Sugihara
(2012); computation can be extremely slow, and time increases exponentially with sigma.
There is also a method for split point patterns on a linear network (class "splitppx") which will
return a list of pixel images.
Value
A pixel image on the linear network (object of class "linim").
Author(s)
Adrian Baddeley  and Greg McSwiggan.
References
McSwiggan, G., Baddeley, A. and Nair, G. (2016) Kernel density estimation on a linear network.
Scandinavian Journal of Statistics, In press.
Okabe, A. and Sugihara, K. (2012) Spatial analysis along networks. Wiley.
See Also
lpp, linim
Examples
X <- runiflpp(3, simplenet)
D <- density(X, 0.2, verbose=FALSE)
plot(D, style="w", main="", adjust=2)
Dw <- density(X, 0.2, weights=c(1,2,-1), verbose=FALSE)
De <- density(X, 0.2, kernel="epanechnikov", verbose=FALSE)
Ded <- density(X, 0.2, kernel="epanechnikov", continuous=FALSE, verbose=FALSE)

288

density.ppp

density.ppp

Kernel Smoothed Intensity of Point Pattern

Description
Compute a kernel smoothed intensity function from a point pattern.
Usage
## S3 method for class 'ppp'
density(x, sigma=NULL, ...,
weights=NULL, edge=TRUE, varcov=NULL,
at="pixels", leaveoneout=TRUE,
adjust=1, diggle=FALSE, se=FALSE,
kernel="gaussian",
scalekernel=is.character(kernel),
positive=FALSE, verbose=TRUE)
Arguments
x

Point pattern (object of class "ppp").

sigma

Standard deviation of isotropic smoothing kernel. Either a numerical value, or a
function that computes an appropriate value of sigma.

weights

Optional weights to be attached to the points. A numeric vector, numeric matrix,
an expression, or a pixel image.

...

Additional arguments passed to pixellate.ppp and as.mask to determine the
pixel resolution, or passed to sigma if it is a function.

edge

Logical value indicating whether to apply edge correction.

varcov

Variance-covariance matrix of anisotropic smoothing kernel. Incompatible with
sigma.

at

String specifying whether to compute the intensity values at a grid of pixel locations (at="pixels") or only at the points of x (at="points").

leaveoneout

Logical value indicating whether to compute a leave-one-out estimator. Applicable only when at="points".

adjust

Optional. Adjustment factor for the smoothing parameter.

diggle

Logical. If TRUE, use the Jones-Diggle improved edge correction, which is more
accurate but slower to compute than the default correction.

kernel

The smoothing kernel. A character string specifying the smoothing kernel (current options are "gaussian", "epanechnikov", "quartic" or "disc"), or a
pixel image (object of class "im") containing values of the kernel, or a function(x,y)
which yields values of the kernel.

scalekernel

Logical value. If scalekernel=TRUE, then the kernel will be rescaled to the
bandwidth determined by sigma and varcov: this is the default behaviour when
kernel is a character string. If scalekernel=FALSE, then sigma and varcov
will be ignored: this is the default behaviour when kernel is a function or a
pixel image.

se

Logical value indicating whether to compute standard errors as well.

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289

positive

Logical value indicating whether to force all density values to be positive numbers. Default is FALSE.

verbose

Logical value indicating whether to issue warnings about numerical problems
and conditions.

Details
This is a method for the generic function density.
It computes a fixed-bandwidth kernel estimate (Diggle, 1985) of the intensity function of the point
process that generated the point pattern x.
By default it computes the convolution of the isotropic Gaussian kernel of standard deviation sigma
with point masses at each of the data points in x. Anisotropic Gaussian kernels are also supported.
Each point has unit weight, unless the argument weights is given.
If edge=TRUE, the intensity estimate is corrected for edge effect bias in one of two ways:
• If diggle=FALSE (the default) the intensity estimate is correted by dividing it by the convolution of the Gaussian kernel with the window of observation. This is the approach originally
described in Diggle (1985). Thus the intensity value at a point u is
X
λ̂(u) = e(u)
k(xi − u)wi
i

where k is the Gaussian smoothing kernel, e(u) is an edge correction factor, and wi are the
weights.
• If diggle=TRUE then the code uses the improved edge correction described by Jones (1993)
and Diggle (2010, equation 18.9). This has been shown to have better performance (Jones,
1993) but is slightly slower to compute. The intensity value at a point u is
X
λ̂(u) =
k(xi − u)wi e(xi )
i

where again k is the Gaussian smoothing kernel, e(xi ) is an edge correction factor, and wi are
the weights.
In both cases, the edge correction term e(u) is the reciprocal of the kernel mass inside the window:
Z
1
=
k(v − u) dv
e(u)
W
where W is the observation window.
The smoothing kernel is determined by the arguments sigma, varcov and adjust.
• if sigma is a single numerical value, this is taken as the standard deviation of the isotropic
Gaussian kernel.
• alternatively sigma may be a function that computes an appropriate bandwidth for the isotropic
Gaussian kernel from the data point pattern by calling sigma(x). To perform automatic bandwidth selection using cross-validation, it is recommended to use the functions bw.diggle or
bw.ppl.
• The smoothing kernel may be chosen to be any Gaussian kernel, by giving the variancecovariance matrix varcov. The arguments sigma and varcov are incompatible.
• Alternatively sigma may be a vector of length 2 giving the standard deviations of two independent Gaussian coordinates, thus equivalent to varcov = diag(rep(sigma^2, 2)).

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density.ppp
• if neither sigma nor varcov is specified, an isotropic Gaussian kernel will be used, with a
default value of sigma calculated by a simple rule of thumb that depends only on the size of
the window.
• The argument adjust makes it easy for the user to change the bandwidth specified by any
of the rules above. The value of sigma will be multiplied by the factor adjust. The matrix
varcov will be multiplied by adjust^2. To double the smoothing bandwidth, set adjust=2.
If at="pixels" (the default), intensity values are computed at every location u in a fine grid, and
are returned as a pixel image. The point pattern is first discretised using pixellate.ppp, then the
intensity is computed using the Fast Fourier Transform. Accuracy depends on the pixel resolution
and the discretisation rule. The pixel resolution is controlled by the arguments ... passed to
as.mask (specify the number of pixels by dimyx or the pixel size by eps). The discretisation rule
is controlled by the arguments ... passed to pixellate.ppp (the default rule is that each point
is allocated to the nearest pixel centre; this can be modified using the arguments fractional and
preserve).
If at="points", the intensity values are computed to high accuracy at the points of x only. Computation is performed by directly evaluating and summing the Gaussian kernel contributions without
discretising the data. The result is a numeric vector giving the density values. The intensity value
at a point xi is (if diggle=FALSE)
X
λ̂(xi ) = e(xi )
k(xj − xi )wj
j

or (if diggle=TRUE)
λ̂(xi ) =

X

k(xj − xi )wj e(xj )

j

If leaveoneout=TRUE (the default), then the sum in the equation is taken over all j not equal to i, so
that the intensity value at a data point is the sum of kernel contributions from all other data points.
If leaveoneout=FALSE then the sum is taken over all j, so that the intensity value at a data point
includes a contribution from the same point.
If weights is a matrix with more than one column, then the calculation is effectively repeated for
each column of weights. The result is a list of images (if at="pixels") or a matrix of numerical
values (if at="points").
The argument weights can also be an expression. It will be evaluated in the data frame as.data.frame(x)
to obtain a vector or matrix of weights. The expression may involve the symbols x and y representing the Cartesian coordinates, the symbol marks representing the mark values if there is only one
column of marks, and the names of the columns of marks if there are several columns.
The argument weights can also be a pixel image (object of class "im"). numerical weights for the
data points will be extracted from this image (by looking up the pixel values at the locations of the
data points in x).
To select the bandwidth sigma automatically by cross-validation, use bw.diggle or bw.ppl.
To perform spatial interpolation of values that were observed at the points of a point pattern, use
Smooth.ppp.
For adaptive nonparametric estimation, see adaptive.density. For data sharpening, see sharpen.ppp.
To compute a relative risk surface or probability map for two (or more) types of points, use relrisk.
Value
By default, the result is a pixel image (object of class "im"). Pixel values are estimated intensity
values, expressed in “points per unit area”.

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291

If at="points", the result is a numeric vector of length equal to the number of points in x. Values
are estimated intensity values at the points of x.
In either case, the return value has attributes "sigma" and "varcov" which report the smoothing
bandwidth that was used.
If weights is a matrix with more than one column, then the result is a list of images (if at="pixels")
or a matrix of numerical values (if at="points").
If se=TRUE, the result is a list with two elements named estimate and SE, each of the format
described above.
Negative Values
Negative and zero values of the density estimate are possible when at="pixels" because of numerical errors in finite-precision arithmetic.
By default, density.ppp does not try to repair such errors. This would take more computation time
and is not always needed. (Also it would not be appropriate if weights include negative values.)
To ensure that the resulting density values are always positive, set positive=TRUE.
Note
This function is often misunderstood.
The result of density.ppp is not a spatial smoothing of the marks or weights attached to the point
pattern. To perform spatial interpolation of values that were observed at the points of a point pattern,
use Smooth.ppp.
The result of density.ppp is not a probability density. It is an estimate of the intensity function of
the point process that generated the point pattern data. Intensity is the expected number of random
points per unit area. The units of intensity are “points per unit area”. Intensity is usually a function
of spatial location, and it is this function which is estimated by density.ppp. The integral of the
intensity function over a spatial region gives the expected number of points falling in this region.
Inspecting an estimate of the intensity function is usually the first step in exploring a spatial point
pattern dataset. For more explanation, see Baddeley, Rubak and Turner (2015) or Diggle (2003,
2010).
If you have two (or more) types of points, and you want a probability map or relative risk surface
(the spatially-varying probability of a given type), use relrisk.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
References
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
Diggle, P.J. (1985) A kernel method for smoothing point process data. Applied Statistics (Journal
of the Royal Statistical Society, Series C) 34 (1985) 138–147.
Diggle, P.J. (2003) Statistical analysis of spatial point patterns, Second edition. Arnold.
Diggle, P.J. (2010) Nonparametric methods. Chapter 18, pp. 299–316 in A.E. Gelfand, P.J. Diggle,
M. Fuentes and P. Guttorp (eds.) Handbook of Spatial Statistics, CRC Press, Boca Raton, FL.
Jones, M.C. (1993) Simple boundary corrections for kernel density estimation. Statistics and Computing 3, 135–146.

292

density.psp

See Also
bw.diggle, bw.ppl, Smooth.ppp, sharpen.ppp, adaptive.density, relrisk, ppp.object, im.object
Examples
if(interactive()) {
opa <- par(mfrow=c(1,2))
plot(density(cells, 0.05))
plot(density(cells, 0.05, diggle=TRUE))
par(opa)
v <- diag(c(0.05, 0.07)^2)
plot(density(cells, varcov=v))
}
Z
Z
Z
Z
Z
Z

<<<<<<-

density(cells,
density(cells,
density(cells,
density(cells,
density(cells,
density(cells,

0.05)
0.05, diggle=TRUE)
0.05, se=TRUE)
varcov=diag(c(0.05^2, 0.07^2)))
0.05, weights=data.frame(a=1:42,b=42:1))
0.05, weights=expression(x))

# automatic bandwidth selection
plot(density(cells, sigma=bw.diggle(cells)))
# equivalent:
plot(density(cells, bw.diggle))
# evaluate intensity at points
density(cells, 0.05, at="points")
plot(density(cells, sigma=0.4, kernel="epanechnikov"))
# relative risk calculation by hand (see relrisk.ppp)
lung <- split(chorley)$lung
larynx <- split(chorley)$larynx
D <- density(lung, sigma=2)
plot(density(larynx, sigma=2, weights=1/D))

density.psp

Kernel Smoothing of Line Segment Pattern or Linear Network

Description
Compute a kernel smoothed intensity function from a line segment pattern or a linear network.
Usage
## S3 method for class 'psp'
density(x, sigma, ..., edge=TRUE,
method=c("FFT", "C", "interpreted"),
at=NULL)
## S3 method for class 'linnet'
density(x, ...)

density.psp

293

Arguments
x
sigma
...
edge
method
at

Line segment pattern (object of class "psp") or linear network (object of class
"linnet") to be smoothed.
Standard deviation of isotropic Gaussian smoothing kernel.
Extra arguments, including arguments passed to as.mask to determine the resolution of the resulting image.
Logical flag indicating whether to apply edge correction.
Character string (partially matched) specifying the method of computation. Option "FFT" is the fastest, while "C" is the most accurate.
Optional. An object specifying the locations where density values should be
computed. Either a window (object of class "owin") or a point pattern (object
of class "ppp" or "lpp").

Details
These are methods for the generic function density for the classes "psp" (line segment patterns)
and "linnet" (linear networks). If x is a linear network, it is first converted to a line segment
pattern.
A kernel estimate of the intensity of the line segment pattern is computed. The result is the convolution of the isotropic Gaussian kernel, of standard deviation sigma, with the line segments. The
result is computed as follows:
• if method="FFT" (the default), the line segments are discretised using pixellate.psp, then
the Fast Fourier Transform is used to calculate the convolution. This method is the fastest, but
is slightly less accurate. Accuracy can be improved by increasing pixel resolution.
• if method="C" the exact value of the convolution at the centre of each pixel is computed
analytically using C code;
• if method="interpreted", the exact value of the convolution at the centre of each pixel is
computed analytically using R code. This method is the slowest.
If edge=TRUE this result is adjusted for edge effects by dividing it by the convolution of the same
Gaussian kernel with the observation window.
If the argument at is given, then it specifies the locations where density values should be computed.
• If at is a window, then the window is converted to a binary mask using the arguments ...,
and density values are computed at the centre of each pixel in this mask. The result is a pixel
image.
• If at is a point pattern, then density values are computed at each point location, and the result
is a numeric vector.
Value
A pixel image (object of class "im") or a numeric vector.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
psp.object, im.object, density

294

density.splitppp

Examples
L <- psp(runif(20),runif(20),runif(20),runif(20), window=owin())
D <- density(L, sigma=0.03)
plot(D, main="density(L)")
plot(L, add=TRUE)

density.splitppp

Kernel Smoothed Intensity of Split Point Pattern

Description
Compute a kernel smoothed intensity function for each of the components of a split point pattern,
or each of the point patterns in a list.
Usage
## S3 method for class 'splitppp'
density(x, ..., se=FALSE)
## S3 method for class 'ppplist'
density(x, ..., se=FALSE)
Arguments
x

Split point pattern (object of class "splitppp" created by split.ppp) to be
smoothed. Alternatively a list of point patterns, of class "ppplist".

...

Arguments passed to density.ppp to control the smoothing, pixel resolution,
edge correction etc.

se

Logical value indicating whether to compute standard errors as well.

Details
This is a method for the generic function density.
The argument x should be a list of point patterns, and should belong to one of the classes "ppplist"
or "splitppp".
Typically x is obtained by applying the function split.ppp to a point pattern y by calling split(y).
This splits the points of y into several sub-patterns.
A kernel estimate of the intensity function of each of the point patterns is computed using density.ppp.
The return value is usually a list, each of whose entries is a pixel image (object of class "im"). The
return value also belongs to the class "solist" and can be plotted or printed.
If the argument at="points" is given, the result is a list of numeric vectors giving the intensity
values at the data points.
If se=TRUE, the result is a list with two elements named estimate and SE, each of the format
described above.

deriv.fv

295

Value
A list of pixel images (objects of class "im") which can be plotted or printed; or a list of numeric
vectors giving the values at specified points.
If se=TRUE, the result is a list with two elements named estimate and SE, each of the format
described above.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
ppp.object, im.object
Examples
Z <- density(split(amacrine), 0.05)
plot(Z)

deriv.fv

Calculate Derivative of Function Values

Description
Applies numerical differentiation to the values in selected columns of a function value table.
Usage
## S3 method for class 'fv'
deriv(expr, which = "*", ...,
method=c("spline", "numeric"),
kinks=NULL,
periodic=FALSE,
Dperiodic=periodic)
Arguments
expr
which

...
method
kinks
periodic
Dperiodic

Function values to be differentiated. A function value table (object of class "fv",
see fv.object).
Character vector identifying which columns of the table should be differentiated.
Either a vector containing names of columns, or one of the wildcard strings "*"
or "." explained below.
Extra arguments passed to smooth.spline to control the differentiation algorithm, if method="spline".
Differentiation method. A character string, partially matched to either "spline"
or "numeric".
Optional vector of x values where the derivative is allowed to be discontinuous.
Logical value indicating whether the function expr is periodic.
Logical value indicating whether the resulting derivative should be a periodic
function.

296

detpointprocfamilyfun

Details
This command performs numerical differentiation on the function values in a function value table
(object of class "fv"). The differentiation is performed either by smooth.spline or by a naive
numerical difference algorithm.
The command deriv is generic. This is the method for objects of class "fv".
Differentiation is applied to every column (or to each of the selected columns) of function values in
turn, using the function argument as the x coordinate and the selected column as the y coordinate.
The original function values are then replaced by the corresponding derivatives.
The optional argument which specifies which of the columns of function values in expr will be
differentiated. The default (indicated by the wildcard which="*") is to differentiate all function
values, i.e.\ all columns except the function argument. Alternatively which="." designates the subset of function values that are displayed in the default plot. Alternatively which can be a character
vector containing the names of columns of expr.
If the argument kinks is given, it should be a numeric vector giving the discontinuity points of the
function: the value or values of the function argument at which the function is not differentiable.
Differentiation will be performed separately on intervals between the discontinuity points.
If periodic=TRUE then the function expr is taken to be periodic, with period equal to the range of
the function argument in expr. The resulting derivative is periodic.
If periodic=FALSE but Dperiodic=TRUE, then the derivative is assumed to be periodic. This would
be appropriate if expr is the cumulative distribution function of an angular variable, for example.
Value
Another function value table (object of class "fv") of the same format.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
with.fv, fv.object, smooth.spline
Examples
G <- Gest(cells)
plot(deriv(G, which=".", spar=0.5))
A <- pairorient(redwood, 0.05, 0.15)
DA <- deriv(A, spar=0.6, Dperiodic=TRUE)

detpointprocfamilyfun Construct a New Determinantal Point Process Model Family Function

Description
Function to ease the implementation of a new determinantal point process model family.

detpointprocfamilyfun

297

Usage
detpointprocfamilyfun(kernel = NULL,
specden = NULL, basis = "fourierbasis",
convkernel = NULL, Kfun = NULL, valid = NULL, intensity = NULL,
dim = 2, name = "User-defined", isotropic = TRUE, range = NULL,
parbounds = NULL, specdenrange = NULL, startpar = NULL, ...)
Arguments
kernel

function specifying the kernel. May be set to NULL. See Details.

specden

function specifying the spectral density. May be set to NULL. See Details.

basis

character string giving the name of the basis. Defaults to the Fourier basis. See
Details.

convkernel

function specifying the k-fold auto-convolution of the kernel. May be set to
NULL. See Details.

Kfun

function specifying the K-function. May be set to NULL. See Details.

valid

function determining whether a given set of parameter values yields a valid
model. May be set to NULL. See Examples.

intensity

character string specifying which parameter is the intensity in the model family.
Should be NULL if the model family has no intensity parameter.

dim

character strig specifying which parameter is the dimension of the state space
in this model family (if any). Alternatively a positive integer specifying the
dimension.

name

character string giving the name of the model family used for printing.

isotropic

logical value indicating whether or not the model is isotropic.

range

function determining the interaction range of the model. May be set to NULL.
See Examples.

parbounds

function determining the bounds for each model parameter when all other parameters are fixed. May be set to NULL. See Examples.

specdenrange

function specifying the the range of the spectral density if it is finite (only the
case for very few models). May be set to NULL.

startpar

function determining starting values for parameters in any estimation algorithm.
May be set to NULL. See Examples.

...

Additional arguments for inclusion in the returned model object. These are not
checked in any way.

Details
A determinantal point process family is specified either in terms of a kernel (a positive semi-definite
function, i.e. a covariance function) or a spectral density, or preferably both. One of these can be
NULL if it is unknown, but not both. When both are supplied they must have the same arguments. The
first argument gives the values at which the function should be evaluated. In general the function
should accept an n by d matrix or data.frame specifying n(>= 0) points in dimension d. If the
model is isotropic it only needs to accept a non-negative valued numeric of length n. (In fact there
is currently almost no support for non-isotropic models, so it is recommended not to specify such a
model.) The name of this argument could be chosen freely, but x is recommended. The remaining
arguments are the parameters of the model. If one of these is an intensity parameter the name should

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detpointprocfamilyfun
be mentioned in the argument intensity. If one of these specifies the dimension of the model it
should be mentioned in the argument dim.
The kernel and spectral density is with respect to a specific set of basis functions, which is typically
the Fourier basis. However this can be changed to any user-supplied basis in the argument basis.
If such an alternative is supplied it must be the name of a function expecting the same arguments as
fourierbasis and returning the results in the same form as fourierbasis.
If supplied, the arguments of convkernel must obey the following: first argument should be like
the first argument of kernel and/or specden (see above). The second argument (preferably called k)
should be the positive integer specifying how many times the auto-convolution is done (i.e. the k
in k-fold auto-convolution). The remaining arguments must agree with the arguments of kernel
and/or specden (see above).
If supplied, the arguments of Kfun should be like the arguments of kernel and specden (see above).

Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
Examples
## Example of how to define the Gauss family
exGauss <- detpointprocfamilyfun(
name="Gaussian",
kernel=function(x, lambda, alpha, d){
lambda*exp(-(x/alpha)^2)
},
specden=function(x, lambda, alpha, d){
lambda * (sqrt(pi)*alpha)^d * exp(-(x*alpha*pi)^2)
},
convkernel=function(x, k, lambda, alpha, d){
logres <- k*log(lambda*pi*alpha^2) - log(pi*k*alpha^2) - x^2/(k*alpha^2)
return(exp(logres))
},
Kfun = function(x, lambda, alpha, d){
pi*x^2 - pi*alpha^2/2*(1-exp(-2*x^2/alpha^2))
},
valid=function(lambda, alpha, d){
lambda>0 && alpha>0 && d>=1 && lambda <= (sqrt(pi)*alpha)^(-d)
},
isotropic=TRUE,
intensity="lambda",
dim="d",
range=function(alpha, bound = .99){
if(missing(alpha))
stop("The parameter alpha is missing.")
if(!(is.numeric(bound)&&bound>0&&bound<1))
stop("Argument bound must be a numeric between 0 and 1.")
return(alpha*sqrt(-log(sqrt(1-bound))))
},
parbounds=function(name, lambda, alpha, d){
switch(name,
lambda = c(0, (sqrt(pi)*alpha)^(-d)),
alpha = c(0, lambda^(-1/d)/sqrt(pi)),

dfbetas.ppm

299
stop("Parameter name misspecified")
)

},
startpar=function(model, X){
rslt <- NULL
if("lambda" %in% model$freepar){
lambda <- intensity(X)
rslt <- c(rslt, "lambda" = lambda)
model <- update(model, lambda=lambda)
}
if("alpha" %in% model$freepar){
alpha <- .8*dppparbounds(model, "alpha")[2]
rslt <- c(rslt, "alpha" = alpha)
}
return(rslt)
}
)
exGauss
m <- exGauss(lambda=100, alpha=.05, d=2)
m

dfbetas.ppm

Parameter influence measure

Description
Computes the deletion influence measure for each parameter in a fitted point process model.
Usage
## S3 method for class 'ppm'
dfbetas(model, ..., drop = FALSE, iScore=NULL,
iHessian=NULL, iArgs=NULL)
Arguments
model

Fitted point process model (object of class "ppm").

...

Ignored, except for the arguments dimyx and eps which are passed to as.mask
to control the spatial resolution of the image of the density component.

drop

Logical. Whether to include (drop=FALSE) or exclude (drop=TRUE) contributions from quadrature points that were not used to fit the model.

iScore,iHessian
Components of the score vector and Hessian matrix for the irregular parameters,
if required. See Details.
iArgs

List of extra arguments for the functions iScore, iHessian if required.

Details
Given a fitted spatial point process model, this function computes the influence measure for each
parameter, as described in Baddeley, Chang and Song (2013).
This is a method for the generic function dfbetas.

300

dg.envelope
The influence measure for each parameter θ is a signed measure in two-dimensional space. It
consists of a discrete mass on each data point (i.e. each point in the point pattern to which the
model was originally fitted) and a continuous density at all locations. The mass at a data point
represents the change in the fitted value of the parameter θ that would occur if this data point were
to be deleted. The density at other non-data locations represents the effect (on the fitted value of
θ) of deleting these locations (and their associated covariate values) from the input to the fitting
procedure.
If the point process model trend has irregular parameters that were fitted (using ippm) then the
influence calculation requires the first and second derivatives of the log trend with respect to the
irregular parameters. The argument iScore should be a list, with one entry for each irregular
parameter, of R functions that compute the partial derivatives of the log trend (i.e. log intensity or
log conditional intensity) with respect to each irregular parameter. The argument iHessian should
be a list, with p2 entries where p is the number of irregular parameters, of R functions that compute
the second order partial derivatives of the log trend with respect to each pair of irregular parameters.

Value
An object of class "msr" representing a signed or vector-valued measure.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
References
Baddeley, A. and Chang, Y.M. and Song, Y. (2013) Leverage and influence diagnostics for spatial
point process models. Scandinavian Journal of Statistics 40, 86–104.
See Also
leverage.ppm, influence.ppm, ppmInfluence
Examples
X <- rpoispp(function(x,y) { exp(3+3*x) })
fit <- ppm(X, ~x+y)
plot(dfbetas(fit))
plot(Smooth(dfbetas(fit)))

dg.envelope

Global Envelopes for Dao-Genton Test

Description
Computes the global envelopes corresponding to the Dao-Genton test of goodness-of-fit.

dg.envelope

301

Usage
dg.envelope(X, ...,
nsim = 19, nsimsub=nsim-1, nrank = 1,
alternative=c("two.sided", "less", "greater"),
leaveout=1, interpolate = FALSE,
savefuns=FALSE, savepatterns=FALSE,
verbose = TRUE)
Arguments
X

Either a point pattern dataset (object of class "ppp", "lpp" or "pp3") or a fitted
point process model (object of class "ppm", "kppm" or "slrm").

...

Arguments passed to mad.test or envelope to control the conduct of the test.
Useful arguments include fun to determine the summary function, rinterval
to determine the range of r values used in the test, and verbose=FALSE to turn
off the messages.

nsim

Number of simulated patterns to be generated in the primary experiment.

nsimsub

Number of simulations in each basic test. There will be nsim repetitions of the
basic test, each involving nsimsub simulated realisations, so there will be a total
of nsim * (nsimsub + 1) simulations.

nrank

Integer. Rank of the envelope value amongst the nsim simulated values. A rank
of 1 means that the minimum and maximum simulated values will be used.

alternative

Character string determining whether the envelope corresponds to a two-sided
test (alternative="two.sided", the default) or a one-sided test with a lower
critical boundary (alternative="less") or a one-sided test with an upper critical boundary (alternative="greater").

leaveout

Optional integer 0, 1 or 2 indicating how to calculate the deviation between the
observed summary function and the nominal reference value, when the reference
value must be estimated by simulation. See Details.

interpolate

Logical value indicating whether to interpolate the distribution of the test statistic by kernel smoothing, as described in Dao and Genton (2014, Section 5).

savefuns

Logical flag indicating whether to save the simulated function values (from the
first stage).

savepatterns

Logical flag indicating whether to save the simulated point patterns (from the
first stage).

verbose

Logical value determining whether to print progress reports.

Details
Computes global simulation envelopes corresponding to the Dao-Genton (2014) adjusted Monte
Carlo goodness-of-fit test. The envelopes are described in Baddeley et al (2015).
If X is a point pattern, the null hypothesis is CSR.
If X is a fitted model, the null hypothesis is that model.
Value
An object of class "fv".

302

dg.progress

Author(s)
Adrian Baddeley , Andrew Hardegen, Tom Lawrence, Robin
Milne, Gopalan Nair and Suman Rakshit. Implemented by Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Dao, N.A. and Genton, M. (2014) A Monte Carlo adjusted goodness-of-fit test for parametric models describing spatial point patterns. Journal of Graphical and Computational Statistics 23, 497–
517.
Baddeley, A., Hardegen, A., Lawrence, L., Milne, R.K., Nair, G.M. and Rakshit, S. (2015) Pushing
the envelope: extensions of graphical Monte Carlo tests. Submitted for publication.
See Also
dg.test, mad.test, envelope
Examples
ns <- if(interactive()) 19 else 4
E <- dg.envelope(swedishpines, Lest, nsim=ns)
E
plot(E)
Eo <- dg.envelope(swedishpines, Lest, alternative="less", nsim=ns)
Ei <- dg.envelope(swedishpines, Lest, interpolate=TRUE, nsim=ns)

dg.progress

Progress Plot of Dao-Genton Test of Spatial Pattern

Description
Generates a progress plot (envelope representation) of the Dao-Genton test for a spatial point pattern.
Usage
dg.progress(X, fun = Lest, ...,
exponent = 2, nsim = 19, nsimsub = nsim - 1,
nrank = 1, alpha, leaveout=1, interpolate = FALSE, rmin=0,
savefuns = FALSE, savepatterns = FALSE, verbose=TRUE)
Arguments
X

Either a point pattern (object of class "ppp", "lpp" or other class), a fitted point
process model (object of class "ppm", "kppm" or other class) or an envelope
object (class "envelope").

fun

Function that computes the desired summary statistic for a point pattern.

...

Arguments passed to envelope. Useful arguments include alternative to
specify one-sided or two-sided envelopes.

dg.progress

303

exponent

Positive number. The exponent of the Lp distance. See Details.

nsim

Number of repetitions of the basic test.

nsimsub

Number of simulations in each basic test. There will be nsim repetitions of the
basic test, each involving nsimsub simulated realisations, so there will be a total
of nsim * (nsimsub + 1) simulations.

nrank

Integer. The rank of the critical value of the Monte Carlo test, amongst the nsim
simulated values. A rank of 1 means that the minimum and maximum simulated
values will become the critical values for the test.

alpha

Optional. The significance level of the test. Equivalent to nrank/(nsim+1)
where nsim is the number of simulations.

leaveout

Optional integer 0, 1 or 2 indicating how to calculate the deviation between the
observed summary function and the nominal reference value, when the reference
value must be estimated by simulation. See Details.

interpolate

Logical value indicating how to compute the critical value. If interpolate=FALSE
(the default), a standard Monte Carlo test is performed, and the critical value
is the largest simulated value of the test statistic (if nrank=1) or the nrank-th
largest (if nrank is another number). If interpolate=TRUE, kernel density estimation is applied to the simulated values, and the critical value is the upper
alpha quantile of this estimated distribution.

rmin

Optional. Left endpoint for the interval of r values on which the test statistic is
calculated.

savefuns

Logical value indicating whether to save the simulated function values (from the
first stage).

savepatterns

Logical value indicating whether to save the simulated point patterns (from the
first stage).

verbose

Logical value indicating whether to print progress reports.

Details
The Dao and Genton (2014) test for a spatial point pattern is described in dg.test. This test
depends on the choice of an interval of distance values (the argument rinterval). A progress plot
or envelope representation of the test (Baddeley et al, 2014) is a plot of the test statistic (and the
corresponding critical value) against the length of the interval rinterval.
The command dg.progress effectively performs dg.test on X using all possible intervals of the
form [0, R], and returns the resulting values of the test statistic, and the corresponding critical values
of the test, as a function of R.
The result is an object of class "fv" that can be plotted to obtain the progress plot. The display
shows the test statistic (solid black line) and the test acceptance region (grey shading). If X is an
envelope object, then some of the data stored in X may be re-used:
• If X is an envelope object containing simulated functions, and fun=NULL, then the code will
re-use the simulated functions stored in X.
• If X is an envelope object containing simulated point patterns, then fun will be applied to the
stored point patterns to obtain the simulated functions. If fun is not specified, it defaults to
Lest.
• Otherwise, new simulations will be performed, and fun defaults to Lest.
If the argument rmin is given, it specifies the left endpoint of the interval defining the test statistic:
the tests are performed using intervals [rmin , R] where R ≥ rmin .

304

dg.sigtrace
The argument leaveout specifies how to calculate the discrepancy between the summary function
for the data and the nominal reference value, when the reference value must be estimated by simulation. The values leaveout=0 and leaveout=1 are both algebraically equivalent (Baddeley et al,
2014, Appendix) to computing the difference observed - reference where the reference is the
mean of simulated values. The value leaveout=2 gives the leave-two-out discrepancy proposed by
Dao and Genton (2014).

Value
An object of class "fv" that can be plotted to obtain the progress plot.
Author(s)
Adrian Baddeley , Andrew Hardegen, Tom Lawrence, Robin
Milne, Gopalan Nair and Suman Rakshit. Implemented by Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Baddeley, A., Diggle, P., Hardegen, A., Lawrence, T., Milne, R. and Nair, G. (2014) On tests of
spatial pattern based on simulation envelopes. Ecological Monographs 84 (3) 477–489.
Baddeley, A., Hardegen, A., Lawrence, L., Milne, R.K., Nair, G.M. and Rakshit, S. (2015) Pushing
the envelope: extensions of graphical Monte Carlo tests. Submitted for publication.
Dao, N.A. and Genton, M. (2014) A Monte Carlo adjusted goodness-of-fit test for parametric models describing spatial point patterns. Journal of Graphical and Computational Statistics 23, 497–
517.
See Also
dg.test, dclf.progress
Examples
ns <- if(interactive()) 19 else 5
plot(dg.progress(cells, nsim=ns))

dg.sigtrace

Significance Trace of Dao-Genton Test

Description
Generates a Significance Trace of the Dao and Genton (2014) test for a spatial point pattern.
Usage
dg.sigtrace(X, fun = Lest, ...,
exponent = 2, nsim = 19, nsimsub = nsim - 1,
alternative = c("two.sided", "less", "greater"),
rmin=0, leaveout=1,
interpolate = FALSE, confint = TRUE, alpha = 0.05,
savefuns=FALSE, savepatterns=FALSE, verbose=FALSE)

dg.sigtrace

305

Arguments
X

Either a point pattern (object of class "ppp", "lpp" or other class), a fitted point
process model (object of class "ppm", "kppm" or other class) or an envelope
object (class "envelope").

fun

Function that computes the desired summary statistic for a point pattern.

...

Arguments passed to envelope.

exponent

Positive number. Exponent used in the test statistic. Use exponent=2 for the
Diggle-Cressie-Loosmore-Ford test, and exponent=Inf for the Maximum Absolute Deviation test. See Details.

nsim

Number of repetitions of the basic test.

nsimsub

Number of simulations in each basic test. There will be nsim repetitions of the
basic test, each involving nsimsub simulated realisations, so there will be a total
of nsim * (nsimsub + 1) simulations.

alternative

Character string specifying the alternative hypothesis. The default (alternative="two.sided")
is that the true value of the summary function is not equal to the theoretical
value postulated under the null hypothesis. If alternative="less" the alternative hypothesis is that the true value of the summary function is lower than the
theoretical value.

rmin

Optional. Left endpoint for the interval of r values on which the test statistic is
calculated.

leaveout

Optional integer 0, 1 or 2 indicating how to calculate the deviation between the
observed summary function and the nominal reference value, when the reference
value must be estimated by simulation. See Details.

interpolate

Logical value indicating whether to interpolate the distribution of the test statistic by kernel smoothing, as described in Dao and Genton (2014, Section 5).

confint

Logical value indicating whether to compute a confidence interval for the ‘true’
p-value.

alpha

Significance level to be plotted (this has no effect on the calculation but is simply
plotted as a reference value).

savefuns

Logical flag indicating whether to save the simulated function values (from the
first stage).

savepatterns

Logical flag indicating whether to save the simulated point patterns (from the
first stage).

verbose

Logical flag indicating whether to print progress reports.

Details
The Dao and Genton (2014) test for a spatial point pattern is described in dg.test. This test
depends on the choice of an interval of distance values (the argument rinterval). A significance
trace (Bowman and Azzalini, 1997; Baddeley et al, 2014, 2015) of the test is a plot of the p-value
obtained from the test against the length of the interval rinterval.
The command dg.sigtrace effectively performs dg.test on X using all possible intervals of the
form [0, R], and returns the resulting p-values as a function of R.
The result is an object of class "fv" that can be plotted to obtain the significance trace. The plot
shows the Dao-Genton adjusted p-value (solid black line), the critical value 0.05 (dashed red line),
and a pointwise 95% confidence band (grey shading) for the ‘true’ (Neyman-Pearson) p-value. The
confidence band is based on the Agresti-Coull (1998) confidence interval for a binomial proportion.

306

dg.sigtrace
If X is an envelope object and fun=NULL then the code will re-use the simulated functions stored in
X.
If the argument rmin is given, it specifies the left endpoint of the interval defining the test statistic:
the tests are performed using intervals [rmin , R] where R ≥ rmin .
The argument leaveout specifies how to calculate the discrepancy between the summary function
for the data and the nominal reference value, when the reference value must be estimated by simulation. The values leaveout=0 and leaveout=1 are both algebraically equivalent (Baddeley et al,
2014, Appendix) to computing the difference observed - reference where the reference is the
mean of simulated values. The value leaveout=2 gives the leave-two-out discrepancy proposed by
Dao and Genton (2014).

Value
An object of class "fv" that can be plotted to obtain the significance trace.
Author(s)
Adrian Baddeley , Andrew Hardegen, Tom Lawrence, Robin
Milne, Gopalan Nair and Suman Rakshit. Implemented by Adrian Baddeley ,
Rolf Turner  and Ege Rubak .
References
Agresti, A. and Coull, B.A. (1998) Approximate is better than “Exact” for interval estimation of
binomial proportions. American Statistician 52, 119–126.
Baddeley, A., Diggle, P., Hardegen, A., Lawrence, T., Milne, R. and Nair, G. (2014) On tests of
spatial pattern based on simulation envelopes. Ecological Monographs 84(3) 477–489.
Baddeley, A., Hardegen, A., Lawrence, L., Milne, R.K., Nair, G.M. and Rakshit, S. (2015) Pushing
the envelope: extensions of graphical Monte Carlo tests. Submitted for publication.
Bowman, A.W. and Azzalini, A. (1997) Applied smoothing techniques for data analysis: the kernel
approach with S-Plus illustrations. Oxford University Press, Oxford.
Dao, N.A. and Genton, M. (2014) A Monte Carlo adjusted goodness-of-fit test for parametric models describing spatial point patterns. Journal of Graphical and Computational Statistics 23, 497–
517.

See Also
dg.test for the Dao-Genton test, dclf.sigtrace for significance traces of other tests.
Examples
ns <- if(interactive()) 19 else 5
plot(dg.sigtrace(cells, nsim=ns))

dg.test

dg.test

307

Dao-Genton Adjusted Goodness-Of-Fit Test

Description
Performs the Dao and Genton (2014) adjusted goodness-of-fit test of spatial pattern.
Usage
dg.test(X, ...,
exponent = 2, nsim=19, nsimsub=nsim-1,
alternative=c("two.sided", "less", "greater"),
reuse = TRUE, leaveout=1, interpolate = FALSE,
savefuns=FALSE, savepatterns=FALSE,
verbose = TRUE)
Arguments
X

Either a point pattern dataset (object of class "ppp", "lpp" or "pp3") or a fitted
point process model (object of class "ppm", "kppm", "lppm" or "slrm").

...

Arguments passed to dclf.test or mad.test or envelope to control the conduct of the test. Useful arguments include fun to determine the summary function, rinterval to determine the range of r values used in the test, and use.theory
described under Details.

exponent

Exponent used in the test statistic. Use exponent=2 for the Diggle-CressieLoosmore-Ford test, and exponent=Inf for the Maximum Absolute Deviation
test.

nsim

Number of repetitions of the basic test.

nsimsub

Number of simulations in each basic test. There will be nsim repetitions of the
basic test, each involving nsimsub simulated realisations, so there will be a total
of nsim * (nsimsub + 1) simulations.

alternative

Character string specifying the alternative hypothesis. The default (alternative="two.sided")
is that the true value of the summary function is not equal to the theoretical
value postulated under the null hypothesis. If alternative="less" the alternative hypothesis is that the true value of the summary function is lower than the
theoretical value.

reuse

Logical value indicating whether to re-use the first stage simulations at the second stage, as described by Dao and Genton (2014).

leaveout

Optional integer 0, 1 or 2 indicating how to calculate the deviation between the
observed summary function and the nominal reference value, when the reference
value must be estimated by simulation. See Details.

interpolate

Logical value indicating whether to interpolate the distribution of the test statistic by kernel smoothing, as described in Dao and Genton (2014, Section 5).

savefuns

Logical flag indicating whether to save the simulated function values (from the
first stage).

savepatterns

Logical flag indicating whether to save the simulated point patterns (from the
first stage).

verbose

Logical value indicating whether to print progress reports.

308

dg.test

Details
Performs the Dao-Genton (2014) adjusted Monte Carlo goodness-of-fit test, in the equivalent form
described by Baddeley et al (2014).
If X is a point pattern, the null hypothesis is CSR.
If X is a fitted model, the null hypothesis is that model.
The argument use.theory passed to envelope determines whether to compare the summary function for the data to its theoretical value for CSR (use.theory=TRUE) or to the sample mean of
simulations from CSR (use.theory=FALSE).
The argument leaveout specifies how to calculate the discrepancy between the summary function
for the data and the nominal reference value, when the reference value must be estimated by simulation. The values leaveout=0 and leaveout=1 are both algebraically equivalent (Baddeley et al,
2014, Appendix) to computing the difference observed - reference where the reference is the
mean of simulated values. The value leaveout=2 gives the leave-two-out discrepancy proposed by
Dao and Genton (2014).
The Dao-Genton test is biased when the significance level is very small (small p-values are not
reliable) and we recommend bits.test in this case.
Value
A hypothesis test (object of class "htest" which can be printed to show the outcome of the test.
Author(s)

Adrian Baddeley, Andrew Hardegen, Tom Lawrence, Robin Milne, Gopalan Nair and Suman Rakshit. Implemented by Adrian Baddeley , Rolf Turner .
References
Dao, N.A. and Genton, M. (2014) A Monte Carlo adjusted goodness-of-fit test for parametric models describing spatial point patterns. Journal of Graphical and Computational Statistics 23, 497–
517.
Baddeley, A., Diggle, P.J., Hardegen, A., Lawrence, T., Milne, R.K. and Nair, G. (2014) On tests of
spatial pattern based on simulation envelopes. Ecological Monographs 84 (3) 477–489.
Baddeley, A., Hardegen, A., Lawrence, L., Milne, R.K., Nair, G.M. and Rakshit, S. (2017) On twostage Monte Carlo tests of composite hypotheses. Computational Statistics and Data Analysis, in
press.
See Also
bits.test, dclf.test, mad.test
Examples
ns <- if(interactive()) 19 else 4
dg.test(cells, nsim=ns)
dg.test(cells, alternative="less", nsim=ns)
dg.test(cells, nsim=ns, interpolate=TRUE)

diagnose.ppm

diagnose.ppm

309

Diagnostic Plots for Fitted Point Process Model

Description
Given a point process model fitted to a point pattern, produce diagnostic plots based on residuals.
Usage
diagnose.ppm(object, ..., type="raw", which="all", sigma=NULL,
rbord=reach(object), cumulative=TRUE,
plot.it=TRUE, rv = NULL,
compute.sd=is.poisson(object), compute.cts=TRUE,
envelope=FALSE, nsim=39, nrank=1,
typename, check=TRUE, repair=TRUE,
oldstyle=FALSE, splineargs=list(spar=0.5))
## S3 method for class 'diagppm'
plot(x, ..., which,
plot.neg=c("image", "discrete", "contour", "imagecontour"),
plot.smooth=c("imagecontour", "image", "contour", "persp"),
plot.sd, spacing=0.1, outer=3,
srange=NULL, monochrome=FALSE, main=NULL)
Arguments
object

The fitted point process model (an object of class "ppm") for which diagnostics
should be produced. This object is usually obtained from ppm.

type

String indicating the type of residuals or weights to be used. Current options are
"eem" for the Stoyan-Grabarnik exponential energy weights, "raw" for the raw
residuals, "inverse" for the inverse-lambda residuals, and "pearson" for the
Pearson residuals. A partial match is adequate.

which

Character string or vector indicating the choice(s) of plots to be generated. Options are "all", "marks", "smooth", "x", "y" and "sum". Multiple choices
may be given but must be matched exactly. See Details.

sigma

Bandwidth for kernel smoother in "smooth" option.

rbord

Width of border to avoid edge effects. The diagnostic calculations will be confined to those points of the data pattern which are at least rbord units away from
the edge of the window. (An infinite value of rbord will be ignored.)

cumulative

Logical flag indicating whether the lurking variable plots for the x and y coordinates will be the plots of cumulative sums of marks (cumulative=TRUE) or the
plots of marginal integrals of the smoothed residual field (cumulative=FALSE).

plot.it

Logical value indicating whether plots should be shown. If plot.it=FALSE, the
computed diagnostic quantities are returned without plotting them.

plot.neg

String indicating how the density part of the residual measure should be plotted.

plot.smooth

String indicating how the smoothed residual field should be plotted.

310

diagnose.ppm
compute.sd,plot.sd
Logical values indicating whether error bounds should be computed and added
to the "x" and "y" plots. The default is TRUE for Poisson models and FALSE for
non-Poisson models. See Details.
envelope,nsim,nrank
Arguments passed to lurking in order to plot simulation envelopes for the lurking variable plots.
rv

Usually absent. Advanced use only. If this argument is present, the values of the
residuals will not be calculated from the fitted model object but will instead be
taken directly from rv.

spacing

The spacing between plot panels (when a four-panel plot is generated) expressed
as a fraction of the width of the window of the point pattern.

outer

The distance from the outermost line of text to the nearest plot panel, expressed
as a multiple of the spacing between plot panels.

srange

Vector of length 2 that will be taken as giving the range of values of the smoothed
residual field, when generating an image plot of this field. This is useful if you
want to generate diagnostic plots for two different fitted models using the same
colour map.

monochrome

Flag indicating whether images should be displayed in greyscale (suitable for
publication) or in colour (suitable for the screen). The default is to display in
colour.

check

Logical value indicating whether to check the internal format of object. If
there is any possibility that this object has been restored from a dump file, or
has otherwise lost track of the environment where it was originally computed,
set check=TRUE.

repair

Logical value indicating whether to repair the internal format of object, if it is
found to be damaged.

oldstyle

Logical flag indicating whether error bounds should be plotted using the approximation given in the original paper (oldstyle=TRUE), or using the correct
asymptotic formula (oldstyle=FALSE).

splineargs

Argument passed to lurking to control the smoothing in the lurking variable
plot.

x

The value returned from a previous call to diagnose.ppm. An object of class
"diagppm".

typename

String to be used as the name of the residuals.

main

Main title for the plot.

...

Extra arguments, controlling either the resolution of the smoothed image (passed
from diagnose.ppm to density.ppp) or the appearance of the plots (passed
from diagnose.ppm to plot.diagppm and from plot.diagppm to plot.default).

compute.cts

Advanced use only.

Details
The function diagnose.ppm generates several diagnostic plots for a fitted point process model. The
plots display the residuals from the fitted model (Baddeley et al, 2005) or alternatively the ‘exponential energy marks’ (Stoyan and Grabarnik, 1991). These plots can be used to assess goodness-of-fit,
to identify outliers in the data, and to reveal departures from the fitted model. See also the companion function qqplot.ppm.

diagnose.ppm

311

The argument object must be a fitted point process model (object of class "ppm") typically produced by the maximum pseudolikelihood fitting algorithm ppm).
The argument type selects the type of residual or weight that will be computed. Current options
are:
"eem": exponential energy marks (Stoyan and Grabarnik, 1991) computed by eem. These are positive weights attached to the data points (i.e. the points of the point pattern dataset to which
the model was fitted). If the fitted model is correct, then the sum of these weights for all data
points in a spatial region B has expected value equal to the area of B. See eem for further
explanation.
"raw", "inverse" or "pearson": point process residuals (Baddeley et al, 2005) computed by the
function residuals.ppm. These are residuals attached both to the data points and to some
other points in the window of observation (namely, to the dummy points of the quadrature
scheme used to fit the model). If the fitted model is correct, then the sum of the residuals in a
spatial region B has mean zero. The options are
• "raw": the raw residuals;
• "inverse": the ‘inverse-lambda’ residuals, a counterpart of the exponential energy weights;
• "pearson": the Pearson residuals.
See residuals.ppm for further explanation.
The argument which selects the type of plot that is produced. Options are:
"marks": plot the residual measure. For the exponential energy weights (type="eem") this displays circles centred at the points of the data pattern, with radii proportional to the exponential energy weights. For the residuals (type="raw", type="inverse" or type="pearson")
this again displays circles centred at the points of the data pattern with radii proportional to
the (positive) residuals, while the plotting of the negative residuals depends on the argument
plot.neg. If plot.neg="image" then the negative part of the residual measure, which is a
density, is plotted as a colour image. If plot.neg="discrete" then the discretised negative
residuals (obtained by approximately integrating the negative density using the quadrature
scheme of the fitted model) are plotted as squares centred at the dummy points with side
lengths proportional to the (negative) residuals. [To control the size of the circles and squares,
use the argument maxsize.]
"smooth": plot a kernel-smoothed version of the residual measure. Each data or dummy point is
taken to have a ‘mass’ equal to its residual or exponential energy weight. (Note that residuals
can be negative). This point mass is then replaced by a bivariate isotropic Gaussian density
with standard deviation sigma. The value of the smoothed residual field at any point in the
window is the sum of these weighted densities. If the fitted model is correct, this smoothed
field should be flat, and its height should be close to 0 (for the residuals) or 1 (for the exponential energy weights). The field is plotted either as an image, contour plot or perspective view
of a surface, according to the argument plot.smooth. The range of values of the smoothed
field is printed if the option which="sum" is also selected.
"x": produce a ‘lurking variable’ plot for the x coordinate. This is a plot of h(x) against x (solid
lines) and of E(h(x)) against x (dashed lines), where h(x) is defined below, and E(h(x))
denotes the expectation of h(x) assuming the fitted model is true.
• if cumulative=TRUE then h(x) is the cumulative sum of the weights or residuals for all
points which have X coordinate less than or equal to x. For the residuals E(h(x)) = 0,
and for the exponential energy weights E(h(x)) = area of the subset of the window to
the left of the line X = x.

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• if cumulative=FALSE then h(x) is the marginal integral of the smoothed residual field
(see the case which="smooth" described above) on the x axis. This is approximately the
derivative of the plot for cumulative=TRUE. The value of h(x) is computed by summing
the values of the smoothed residual field over all pixels with the given x coordinate. For
the residuals E(h(x)) = 0, and for the exponential energy weights E(h(x)) = length of
the intersection between the observation window and the line X = x.
If plot.sd = TRUE, then superimposed on the lurking variable plot are the pointwise twostandard-deviation error limits for h(x) calculated for the inhomogeneous Poisson process.
The default is plot.sd = TRUE for Poisson models and plot.sd = FALSE for non-Poisson
models.
"y": produce a similar lurking variable plot for the y coordinate.
"sum": print the sum of the weights or residuals for all points in the window (clipped by a margin
rbord if required) and the area of the same window. If the fitted model is correct the sum
of the exponential energy weights should equal the area of the window, while the sum of the
residuals should equal zero. Also print the range of values of the smoothed field displayed in
the "smooth" case.
"all": All four of the diagnostic plots listed above are plotted together in a two-by-two display.
Top left panel is "marks" plot. Bottom right panel is "smooth" plot. Bottom left panel is "x"
plot. Top right panel is "y" plot, rotated 90 degrees.
The argument rbord ensures there are no edge effects in the computation of the residuals. The
diagnostic calculations will be confined to those points of the data pattern which are at least rbord
units away from the edge of the window. The value of rbord should be greater than or equal to the
range of interaction permitted in the model.
By default, the two-standard-deviation limits are calculated from the exact formula for the asymptotic variance of the residuals under the asymptotic normal approximation, equation (37) of Baddeley et al (2006). However, for compatibility with the original paper of Baddeley et al (2005),
if oldstyle=TRUE, the two-standard-deviation limits are calculated using the innovation variance,
an over-estimate of the true variance of the residuals. (However, see the section about Replicated
Data).
The argument rv would normally be used only by experts. It enables the user to substitute arbitrary
values for the residuals or marks, overriding the usual calculations. If rv is present, then instead
of calculating the residuals from the fitted model, the algorithm takes the residuals from the object
rv, and plots them in the manner appropriate to the type of residual or mark selected by type. If
type ="eem" then rv should be similar to the return value of eem, namely, a numeric vector of
length equal to the number of points in the original data point pattern. Otherwise, rv should be
similar to the return value of residuals.ppm, that is, it should be an object of class "msr" (see
msr) representing a signed measure.
The return value of diagnose.ppm is an object of class "diagppm". The plot method for this class
is documented here. There is also a print method. See the Examples.
In plot.diagppm, if a four-panel diagnostic plot is produced (the default), then the extra arguments
xlab, ylab, rlab determine the text labels for the x and y coordinates and the residuals, respectively. The undocumented arguments col.neg and col.smooth control the colour maps used in the
top left and bottom right panels respectively.
See also the companion functions qqplot.ppm, which produces a Q-Q plot of the residuals, and
lurking, which produces lurking variable plots for any spatial covariate.

Value
An object of class "diagppm" which contains the coordinates needed to reproduce the selected
plots. This object can be plotted using plot.diagppm and printed using print.diagppm.

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313

Replicated Data
Note that if object is a model that was obtained by first fitting a model to replicated point pattern
data using mppm and then using subfits to extract a model for one of the individual point patterns,
then the variance calculations are only implemented for the innovation variance (oldstyle=TRUE)
and this is the default in such cases.
Author(s)
Adrian Baddeley  and Rolf Turner 
References
Baddeley, A., Turner, R., Møller, J. and Hazelton, M. (2005) Residual analysis for spatial point
processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
Baddeley, A., Møller, J. and Pakes, A.G. (2008) Properties of residuals for spatial point processes.
Annals of the Institute of Statistical Mathematics 60, 627–649.
Stoyan, D. and Grabarnik, P. (1991) Second-order characteristics for stochastic structures connected
with Gibbs point processes. Mathematische Nachrichten, 151:95–100.
See Also
residuals.ppm, eem, ppm.object, qqplot.ppm, lurking, ppm
Examples
fit <- ppm(cells ~x, Strauss(r=0.15))
diagnose.ppm(fit)
## Not run:
diagnose.ppm(fit, type="pearson")
## End(Not run)
diagnose.ppm(fit, which="marks")
diagnose.ppm(fit, type="raw", plot.neg="discrete")
diagnose.ppm(fit, type="pearson", which="smooth")
# save the diagnostics and plot them later
u <- diagnose.ppm(fit, rbord=0.15, plot.it=FALSE)
## Not run:
plot(u)
plot(u, which="marks")
## End(Not run)

diameter

Diameter of an Object

Description
Computes the diameter of an object such as a two-dimensional window or three-dimensional box.

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diameter.box3

Usage
diameter(x)
Arguments
x

A window or other object whose diameter will be computed.

Details
This function computes the diameter of an object such as a two-dimensional window or a threedimensional box. The diameter is the maximum distance between any two points in the object.
The function diameter is generic, with methods for the class "owin" (two-dimensional windows),
"box3" (three-dimensional boxes), "boxx" (multi-dimensional boxes) and "linnet" (linear networks).
Value
The numerical value of the diameter of the object.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
diameter.owin, diameter.box3, diameter.boxx, diameter.linnet

diameter.box3

Geometrical Calculations for Three-Dimensional Box

Description
Calculates the volume, diameter, shortest side, side lengths, or eroded volume of a three-dimensional
box.
Usage
## S3 method for class 'box3'
diameter(x)
## S3 method for class 'box3'
volume(x)
shortside(x)
sidelengths(x)
eroded.volumes(x, r)
## S3 method for class 'box3'
shortside(x)

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315

## S3 method for class 'box3'
sidelengths(x)
## S3 method for class 'box3'
eroded.volumes(x, r)
Arguments
x

Three-dimensional box (object of class "box3").

r

Numeric value or vector of numeric values for which eroded volumes should be
calculated.

Details
diameter.box3 computes the diameter of the box. volume.box3 computes the volume of the box.
shortside.box3 finds the shortest of the three side lengths of the box. sidelengths.box3 returns
all three side lengths of the box.
eroded.volumes computes, for each entry r[i], the volume of the smaller box obtained by removing a slab of thickness r[i] from each face of the box. This smaller box is the subset consisting of
points that lie at least r[i] units away from the boundary of the box.
Value
For diameter.box3, shortside.box3 and volume.box3, a single numeric value. For sidelengths.box3,
a vector of three numbers. For eroded.volumes, a numeric vector of the same length as r.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
as.box3
Examples
X <- box3(c(0,10),c(0,10),c(0,5))
diameter(X)
volume(X)
sidelengths(X)
shortside(X)
hd <- shortside(X)/2
eroded.volumes(X, seq(0,hd, length=10))

316

diameter.boxx

diameter.boxx

Geometrical Calculations for Multi-Dimensional Box

Description
Calculates the volume, diameter, shortest side, side lengths, or eroded volume of a multi-dimensional
box.
Usage
## S3 method for class 'boxx'
diameter(x)
## S3 method for class 'boxx'
volume(x)
## S3 method for class 'boxx'
shortside(x)
## S3 method for class 'boxx'
sidelengths(x)
## S3 method for class 'boxx'
eroded.volumes(x, r)
Arguments
x

Multi-dimensional box (object of class "boxx").

r

Numeric value or vector of numeric values for which eroded volumes should be
calculated.

Details
diameter.boxx, volume.boxx and shortside.boxx compute the diameter, volume and shortest
side length of the box. sidelengths.boxx returns the lengths of each side of the box.
eroded.volumes.boxx computes, for each entry r[i], the volume of the smaller box obtained
by removing a slab of thickness r[i] from each face of the box. This smaller box is the subset
consisting of points that lie at least r[i] units away from the boundary of the box.
Value
For diameter.boxx, shortside.boxx and volume.boxx, a single numeric value. For sidelengths.boxx,
a numeric vector of length equal to the number of spatial dimensions. For eroded.volumes.boxx,
a numeric vector of the same length as r.
Author(s)
Adrian Baddeley 
and Rolf Turner 

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317

See Also
boxx
Examples
X <- boxx(c(0,10),c(0,10),c(0,5),c(0,2))
diameter(X)
volume(X)
shortside(X)
sidelengths(X)
hd <- shortside(X)/2
eroded.volumes(X, seq(0,hd, length=10))

diameter.linnet

Diameter and Bounding Radius of a Linear Network

Description
Compute the diameter or bounding radius of a linear network measured using the shortest path
distance.
Usage
## S3 method for class 'linnet'
diameter(x)
## S3 method for class 'linnet'
boundingradius(x, ...)
Arguments
x

Linear network (object of class "linnet").

...

Ignored.

Details
The diameter of a linear network (in the shortest path distance) is the maximum value of the shortestpath distance between any two points u and v on the network.
The bounding radius of a linear network (in the shortest path distance) is the minimum value, over
all points u on the network, of the maximum shortest-path distance from u to another point v on the
network.
The functions boundingradius and diameter are generic; the functions boundingradius.linnet
and diameter.linnet are the methods for objects of class linnet.
Value
A single numeric value.
Author(s)
Adrian Baddeley 

318

diameter.owin

See Also
boundingradius, diameter, linnet
Examples
diameter(simplenet)
boundingradius(simplenet)

diameter.owin

Diameter of a Window

Description
Computes the diameter of a window.
Usage
## S3 method for class 'owin'
diameter(x)
Arguments
x

A window whose diameter will be computed.

Details
This function computes the diameter of a window of arbitrary shape, i.e. the maximum distance
between any two points in the window.
The argument x should be a window (an object of class "owin", see owin.object for details) or
can be given in any format acceptable to as.owin().
The function diameter is generic. This function is the method for the class "owin".
Value
The numerical value of the diameter of the window.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
area.owin, perimeter, edges, owin, as.owin
Examples
w <- owin(c(0,1),c(0,1))
diameter(w)
# returns sqrt(2)
data(letterR)
diameter(letterR)

DiggleGatesStibbard

319

DiggleGatesStibbard

Diggle-Gates-Stibbard Point Process Model

Description
Creates an instance of the Diggle-Gates-Stibbard point process model which can then be fitted to
point pattern data.
Usage
DiggleGatesStibbard(rho)
Arguments
rho

Interaction range

Details
Diggle, Gates and Stibbard (1987) proposed a pairwise interaction point process in which each pair
of points separated by a distance d contributes a factor e(d) to the probability density, where
 
πd
e(d) = sin2
2ρ
for d < ρ, and e(d) is equal to 1 for d ≥ ρ.
The function ppm(), which fits point process models to point pattern data, requires an argument
of class "interact" describing the interpoint interaction structure of the model to be fitted. The
appropriate description of the Diggle-Gates-Stibbard pairwise interaction is yielded by the function
DiggleGatesStibbard(). See the examples below.
Note that this model does not have any regular parameters (as explained in the section on Interaction
Parameters in the help file for ppm). The parameter ρ is not estimated by ppm.
Value
An object of class "interact" describing the interpoint interaction structure of the Diggle-GatesStibbard process with interaction range rho.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
References
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns.
Australian and New Zealand Journal of Statistics 42, 283–322.
Ripley, B.D. (1981) Spatial statistics. John Wiley and Sons.
Diggle, P.J., Gates, D.J., and Stibbard, A. (1987) A nonparametric estimator for pairwise-interaction
point processes. Biometrika 74, 763 – 770. Scandinavian Journal of Statistics 21, 359–373.

320

DiggleGratton

See Also
ppm, pairwise.family, DiggleGratton, rDGS, ppm.object
Examples
DiggleGatesStibbard(0.02)
# prints a sensible description of itself
## Not run:
ppm(cells ~1, DiggleGatesStibbard(0.05))
# fit the stationary D-G-S process to `cells'
## End(Not run)
ppm(cells ~ polynom(x,y,3), DiggleGatesStibbard(0.05))
# fit a nonstationary D-G-S process
# with log-cubic polynomial trend

DiggleGratton

Diggle-Gratton model

Description
Creates an instance of the Diggle-Gratton pairwise interaction point process model, which can then
be fitted to point pattern data.
Usage
DiggleGratton(delta=NA, rho)
Arguments
delta

lower threshold δ

rho

upper threshold ρ

Details
Diggle and Gratton (1984, pages 208-210) introduced the pairwise interaction point process with
pair potential h(t) of the form

κ
t−δ
h(t) =
if δ ≤ t ≤ ρ
ρ−δ
with h(t) = 0 for t < δ and h(t) = 1 for t > ρ. Here δ, ρ and κ are parameters.
Note that we use the symbol κ where Diggle and Gratton (1984) and Diggle, Gates and Stibbard
(1987) use β, since in spatstat we reserve the symbol β for an intensity parameter.
The parameters must all be nonnegative, and must satisfy δ ≤ ρ.
The potential is inhibitory, i.e.\ this model is only appropriate for regular point patterns. The strength
of inhibition increases with κ. For κ = 0 the model is a hard core process with hard core radius δ.
For κ = ∞ the model is a hard core process with hard core radius ρ.

dilated.areas

321

The irregular parameters δ, ρ must be given in the call to DiggleGratton, while the regular parameter κ will be estimated.
If the lower threshold delta is missing or NA, it will be estimated from the data when ppm is called.
The estimated value of delta is the minimum nearest neighbour distance multiplied by n/(n + 1),
where n is the number of data points.
Value
An object of class "interact" describing the interpoint interaction structure of a point process.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
References
Diggle, P.J., Gates, D.J. and Stibbard, A. (1987) A nonparametric estimator for pairwise-interaction
point processes. Biometrika 74, 763 – 770.
Diggle, P.J. and Gratton, R.J. (1984) Monte Carlo methods of inference for implicit statistical models. Journal of the Royal Statistical Society, series B 46, 193 – 212.
See Also
ppm, ppm.object, Pairwise
Examples
ppm(cells ~1, DiggleGratton(0.05, 0.1))

dilated.areas

Areas of Morphological Dilations

Description
Computes the areas of successive morphological dilations.
Usage
dilated.areas(X, r, W=as.owin(X), ..., constrained=TRUE, exact = FALSE)
Arguments
X
r
W
...
constrained
exact

Object to be dilated. A point pattern (object of class "ppp"), a line segment
pattern (object of class "psp"), or a window (object of class "owin").
Numeric vector of radii for the dilations.
Window (object of class "owin") inside which the areas will be computed, if
constrained=TRUE.
Arguments passed to distmap to control the pixel resolution, if exact=FALSE.
Logical flag indicating whether areas should be restricted to the window W.
Logical flag indicating whether areas should be computed using analytic geometry (which is slower but more accurate). Currently available only when X is a
point pattern.

322

dilation

Details
This function computes the areas of the dilations of X by each of the radii r[i]. Areas may also be
computed inside a specified window W.
The morphological dilation of a set X by a distance r > 0 is the subset consisting of all points x
such that the distance from x to X is less than or equal to r.
When X is a point pattern, the dilation by a distance r is the union of discs of radius r centred at the
points of X.
The argument r should be a vector of nonnegative numbers.
If exact=TRUE and if X is a point pattern, then the areas are computed using analytic geometry,
which is slower but much more accurate. Otherwise the computation is performed using distmap.
To compute the dilated object itself, use dilation.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
owin, as.owin, dilation, eroded.areas
Examples
X <- runifpoint(10)
a <- dilated.areas(X, c(0.1,0.2), W=square(1), exact=TRUE)

dilation

Morphological Dilation

Description
Perform morphological dilation of a window, a line segment pattern or a point pattern
Usage
dilation(w, r, ...)
## S3 method for class 'owin'
dilation(w, r, ..., polygonal=NULL, tight=TRUE)
## S3 method for class 'ppp'
dilation(w, r, ..., polygonal=TRUE, tight=TRUE)
## S3 method for class 'psp'
dilation(w, r, ..., polygonal=TRUE, tight=TRUE)

dilation

323

Arguments
w

A window (object of class "owin" or a line segment pattern (object of class
"psp") or a point pattern (object of class "ppp").

r

positive number: the radius of dilation.

...

extra arguments passed to as.mask controlling the pixel resolution, if the pixel
approximation is used; or passed to disc if the polygonal approximation is used.

polygonal

Logical flag indicating whether to compute a polygonal approximation to the
dilation (polygonal=TRUE) or a pixel grid approximation (polygonal=FALSE).

tight

Logical flag indicating whether the bounding frame of the window should be
taken as the smallest rectangle enclosing the dilated region (tight=TRUE), or
should be the dilation of the bounding frame of w (tight=FALSE).

Details
The morphological dilation of a set W by a distance r > 0 is the set consisting of all points lying
at most r units away from W . Effectively, dilation adds a margin of width r onto the set W .
If polygonal=TRUE then a polygonal approximation to the dilation is computed. If polygonal=FALSE
then a pixel approximation to the dilation is computed from the distance map of w. The arguments
"..." are passed to as.mask to control the pixel resolution.
When w is a window, the default (when polygonal=NULL) is to compute a polygonal approximation
if w is a rectangle or polygonal window, and to compute a pixel approximation if w is a window of
type "mask".
Value
If r > 0, an object of class "owin" representing the dilated region. If r=0, the result is identical to
w.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
erosion for the opposite operation.
dilationAny for morphological dilation using any shape.
owin, as.owin
Examples
plot(dilation(letterR, 0.2))
plot(letterR, add=TRUE, lwd=2, border="red")
X <- psp(runif(10), runif(10), runif(10), runif(10), window=owin())
plot(dilation(X, 0.1))
plot(X, add=TRUE, col="red")

324

dimhat

dim.detpointprocfamily
Dimension of Determinantal Point Process Model

Description
Extracts the dimension of a determinantal point process model.
Usage
## S3 method for class 'detpointprocfamily'
dim(x)
Arguments
x

object of class "detpointprocfamily".

Value
A numeric (or NULL if the dimension of the model is unspecified).
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 

dimhat

Estimate Dimension of Central Subspace

Description
Given the kernel matrix that characterises a central subspace, this function estimates the dimension
of the subspace.
Usage
dimhat(M)
Arguments
M

Kernel of subspace. A symmetric, non-negative definite, numeric matrix, typically obtained from sdr.

dirichlet

325

Details
This function computes the maximum descent estimate of the dimension of the central subspace
with a given kernel matrix M.
The matrix M should be the kernel matrix of a central subspace, which can be obtained from sdr. It
must be a symmetric, non-negative-definite, numeric matrix.
The algorithm finds the eigenvalues λ1 ≥ . . . ≥ λn of M , and then determines the index k for
which λk /λk−1 is greatest.
Value
A single integer giving the estimated dimension.
Author(s)
Matlab original by Yongtao Guan, translated to R by Suman Rakshit.
References
Guan, Y. and Wang, H. (2010) Sufficient dimension reduction for spatial point processes directed
by Gaussian random fields. Journal of the Royal Statistical Society, Series B, 72, 367–387.
See Also
sdr, subspaceDistance

dirichlet

Dirichlet Tessellation of Point Pattern

Description
Computes the Dirichlet tessellation of a spatial point pattern. Also known as the Voronoi or Thiessen
tessellation.
Usage
dirichlet(X)
Arguments
X

Spatial point pattern (object of class "ppp").

Details
In a spatial point pattern X, the Dirichlet tile associated with a particular point X[i] is the region
of space that is closer to X[i] than to any other point in X. The Dirichlet tiles divide the twodimensional plane into disjoint regions, forming a tessellation.
The Dirichlet tessellation is also known as the Voronoi or Thiessen tessellation.
This function computes the Dirichlet tessellation (within the original window of X) using the function deldir in the package deldir.

326

dirichletAreas
To ensure that there is a one-to-one correspondence between the points of X and the tiles of dirichlet(X),
duplicated points in X should first be removed by X <- unique(X, rule="deldir").
The tiles of the tessellation will be computed as polygons if the original window is a rectangle or a
polygon. Otherwise the tiles will be computed as binary masks.

Value
A tessellation (object of class "tess").
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
tess, delaunay, ppp, dirichletVertices
Examples
X <- runifpoint(42)
plot(dirichlet(X))
plot(X, add=TRUE)

dirichletAreas

Compute Areas of Tiles in Dirichlet Tessellation

Description
Calculates the area of each tile in the Dirichlet-Voronoi tessellation of a point pattern.
Usage
dirichletAreas(X)
Arguments
X

Point pattern (object of class "ppp").

Details
This is an efficient algorithm to calculate the areas of the tiles in the Dirichlet-Voronoi tessellation.
If the window of X is a binary pixel mask, the tile areas are computed by counting pixels. Otherwise
the areas are computed exactly using analytic geometry.
If any points of X are duplicated, the duplicates will have tile area zero.
Value
Numeric vector with one entry for each point of X.

dirichletVertices

327

Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
dirichlet, dirichletVertices
Examples
aa <- dirichletAreas(cells)

dirichletVertices

Vertices and Edges of Dirichlet Tessellation

Description
Computes the Dirichlet-Voronoi tessellation of a point pattern and extracts the vertices or edges of
the tiles.
Usage
dirichletVertices(X)
dirichletEdges(X)
Arguments
X

Point pattern (object of class "ppp").

Details
These function compute the Dirichlet-Voronoi tessellation of X (see dirichlet) and extract the
vertices or edges of the tiles of the tessellation.
The Dirichlet vertices are the spatial locations which are locally farthest away from X, that is, where
the distance function of X reaches a local maximum.
The Dirichlet edges are the dividing lines equally distant between a pair of points of X.
The Dirichlet tessellation of X is computed using dirichlet. The vertices or edges of all tiles of
the tessellation are extracted.
For dirichletVertices, any vertex which lies on the boundary of the window of X is deleted. The
remaining vertices are returned, as a point pattern, without duplicated entries.
Value
dirichletVertices returns a point pattern (object of class "ppp") in the same window as X.
dirichletEdges returns a line segment pattern (object of class "psp").

328

dirichletWeights

Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
dirichlet, dirichletAreas
Examples
plot(dirichlet(cells))
plot(dirichletVertices(cells), add=TRUE)
ed <- dirichletEdges(cells)

dirichletWeights

Compute Quadrature Weights Based on Dirichlet Tessellation

Description
Computes quadrature weights for a given set of points, using the areas of tiles in the Dirichlet
tessellation.
Usage
dirichletWeights(X, window=NULL, exact=TRUE, ...)
Arguments
X

Data defining a point pattern.

window

Default window for the point pattern

exact

Logical value. If TRUE, compute exact areas using the package deldir. If FALSE,
compute approximate areas using a pixel raster.

...

Ignored.

Details
This function computes a set of quadrature weights for a given pattern of points (typically comprising both “data” and ‘dummy” points). See quad.object for an explanation of quadrature weights
and quadrature schemes.
The weights are computed using the Dirichlet tessellation. First X and (optionally) window are
converted into a point pattern object. Then the Dirichlet tessellation of the points of X is computed.
The weight attached to a point of X is the area of its Dirichlet tile (inside the window Window(X)).
If exact=TRUE the Dirichlet tessellation is computed exactly by the Lee-Schachter algorithm using the package deldir. Otherwise a pixel raster approximation is constructed and the areas are
approximations to the true weights. In all cases the sum of the weights is equal to the area of the
window.

disc

329

Value
Vector of nonnegative weights for each point in X.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
quad.object, gridweights
Examples
Q <- quadscheme(runifpoispp(10))
X <- as.ppp(Q) # data and dummy points together
w <- dirichletWeights(X, exact=FALSE)

disc

Circular Window

Description
Creates a circular window
Usage
disc(radius=1, centre=c(0,0), ..., mask=FALSE, npoly=128, delta=NULL)
Arguments
radius
centre
mask
npoly
delta
...

Radius of the circle.
The centre of the circle.
Logical flag controlling the type of approximation to a perfect circle. See Details.
Number of edges of the polygonal approximation, if mask=FALSE. Incompatible
with delta.
Tolerance of polygonal approximation: the length of arc that will be replaced by
one edge of the polygon. Incompatible with npoly.
Arguments passed to as.mask determining the pixel resolution, if mask=TRUE.

Details
This command creates a window object representing a disc, with the given radius and centre.
By default, the circle is approximated by a polygon with npoly edges.
If mask=TRUE, then the disc is approximated by a binary pixel mask. The resolution of the mask is
controlled by the arguments ... which are passed to as.mask.
The argument radius must be a single positive number. The argument centre specifies the disc
centre: it can be either a numeric vector of length 2 giving the coordinates, or a list(x,y) giving
the coordinates of exactly one point, or a point pattern (object of class "ppp") containing exactly
one point.

330

discpartarea

Value
An object of class "owin" (see owin.object) specifying a window.
Note
This function can also be used to generate regular polygons, by setting npoly to a small integer
value. For example npoly=5 generates a pentagon and npoly=13 a triskaidecagon.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
ellipse, discs, owin.object, owin, as.mask
Examples
# unit disc
W <- disc()
# disc of radius 3 centred at x=10, y=5
W <- disc(3, c(10,5))
#
plot(disc())
plot(disc(mask=TRUE))
# nice smooth circle
plot(disc(npoly=256))
# how to control the resolution of the mask
plot(disc(mask=TRUE, dimyx=256))
# check accuracy of approximation
area(disc())/pi
area(disc(mask=TRUE))/pi

discpartarea

Area of Part of Disc

Description
Compute area of intersection between a disc and a window
Usage
discpartarea(X, r, W=as.owin(X))
Arguments
X

Point pattern (object of class "ppp") specifying the centres of the discs. Alternatively, X may be in any format acceptable to as.ppp.

r

Matrix, vector or numeric value specifying the radii of the discs.

W

Window (object of class "owin") with which the discs should be intersected.

discretise

331

Details
This algorithm computes the exact area of the intersection between a window W and a disc (or each
of several discs). The centres of the discs are specified by the point pattern X, and their radii are
specified by r.
If r is a single numeric value, then the algorithm computes the area of intersection between W and
the disc of radius r centred at each point of X, and returns a one-column matrix containing one entry
for each point of X.
If r is a vector of length m, then the algorithm returns an n * m matrix in which the entry on row i,
column j is the area of the intersection between W and the disc centred at X[i] with radius r[j].
If r is a matrix, it should have one row for each point in X. The algorithm returns a matrix in which
the entry on row i, column j is the area of the intersection between W and the disc centred at X[i]
with radius r[i,j].
Areas are computed by analytic geometry.
Value
Numeric matrix, with one row for each point of X.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
owin, disc
Examples
data(letterR)
X <- runifpoint(3, letterR)
discpartarea(X, 0.2)

discretise

Safely Convert Point Pattern Window to Binary Mask

Description
Given a point pattern, discretise its window by converting it to a binary pixel mask, adjusting the
mask so that it still contains all the points.
Usage
discretise(X, eps = NULL, dimyx = NULL, xy = NULL)
Arguments
X
eps
dimyx
xy

A point pattern (object of class "ppp") to be converted.
(optional) width and height of each pixel
(optional) pixel array dimensions
(optional) pixel coordinates

332

discretise

Details
This function modifies the point pattern X by converting its observation window Window(X) to a
binary pixel image (a window of type "mask"). It ensures that no points of X are deleted by the
discretisation.
The window is first discretised using as.mask. It can happen that points of X that were inside
the original window may fall outside the new mask. The discretise function corrects this by
augmenting the mask (so that the mask includes any pixel that contains a point of the pattern).
The arguments eps, dimyx and xy control the fineness of the pixel array. They are passed to
as.mask.
If eps, dimyx and xy are all absent or NULL, and if the window of X is of type "mask" to start with,
then discretise(X) returns X unchanged.
See as.mask for further details about the arguments eps, dimyx, and xy, and the process of converting a window to one of type mask.
Value
A point pattern (object of class "ppp"), identical to X, except that its observation window has been
converted to one of type mask.
Error checking
Before doing anything, discretise checks that all the points of the pattern are actually inside the
original window. This is guaranteed to be the case if the pattern was constructed using ppp or
as.ppp. However anomalies are possible if the point pattern was created or manipulated inappropriately. These will cause an error.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
as.mask
Examples
data(demopat)
X <- demopat
plot(X, main="original pattern")
Y <- discretise(X, dimyx=50)
plot(Y, main="discretise(X)")
stopifnot(npoints(X) == npoints(Y))
# what happens if we just convert the window to a mask?
W <- Window(X)
M <- as.mask(W, dimyx=50)
plot(M, main="window of X converted to mask")
plot(X, add=TRUE, pch=16)
plot(X[M], add=TRUE, pch=1, cex=1.5)
XM <- X[M]
cat(paste(npoints(X) - npoints(XM), "points of X lie outside M\n"))

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333

discs

Union of Discs

Description
Make a spatial region composed of discs with given centres and radii.
Usage
discs(centres, radii = marks(centres)/2, ...,
separate = FALSE, mask = FALSE, trim = TRUE,
delta = NULL, npoly=NULL)
Arguments
centres

Point pattern giving the locations of centres for the discs.

radii

Vector of radii for each disc, or a single number giving a common radius. (Notice that the default assumes that the marks of X are diameters.)

...

Optional arguments passed to as.mask to determine the pixel resolution, if
mask=TRUE.

separate

Logical. If TRUE, the result is a list containing each disc as a separate entry. If
FALSE (the default), the result is a window obtained by forming the union of the
discs.

mask

Logical. If TRUE, the result is a binary mask window. If FALSE, the result is a
polygonal window. Applies only when separate=FALSE.

trim

Logical value indicating whether to restrict the result to the original window of
the centres. Applies only when separate=FALSE.

delta

Argument passed to disc to determine the tolerance for the polygonal approximation of each disc. Applies only when mask=FALSE. Incompatible with npoly.

npoly

Argument passed to disc to determine the number of edges in the polygonal
approximation of each disc. Applies only when mask=FALSE. Incompatible with
delta.

Details
This command is typically applied to a marked point pattern dataset X in which the marks represent
the sizes of objects. The result is a spatial region representing the space occupied by the objects.
If the marks of X represent the diameters of circular objects, then the result of discs(X) is a spatial region constructed by taking discs, of the specified diameters, centred at the points of X, and
forming the union of these discs. If the marks of X represent the areas of objects, one could take
discs(X, sqrt(marks(X)/pi)) to produce discs of equivalent area.
A fast algorithm is used to compute the result as a binary mask, when mask=TRUE. This option is
recommended unless polygons are really necessary.
If mask=FALSE, the discs will be constructed as polygons by the function disc. To avoid computational problems, by default, the discs will all be constructed using the same physical tolerance
value delta passed to disc. The default is such that the smallest disc will be approximated by
a 16-sided polygon. (The argument npoly should not normally be used, to avoid computational
problems arising with small radii.)

334

distcdf

Value
If separate=FALSE, a window (object of class "owin").
If separate=TRUE, a list of windows.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
disc, union.owin
Examples
plot(discs(anemones, mask=TRUE, eps=0.5))

distcdf

Distribution Function of Interpoint Distance

Description
Computes the cumulative distribution function of the distance between two independent random
points in a given window or windows.
Usage
distcdf(W, V=W, ..., dW=1, dV=dW, nr=1024, regularise=TRUE)
Arguments
W

A window (object of class "owin") containing the first random point.

V

Optional. Another window containing the second random point. Defaults to W.

...

Arguments passed to as.mask to determine the pixel resolution for the calculation.

dV, dW

Optional. Probability densities (not necessarily normalised) for the first and
second random points respectively. Data in any format acceptable to as.im, for
example, a function(x,y) or a pixel image or a numeric value. The default
corresponds to a uniform distribution over the window.

nr

Integer. The number of values of interpoint distance r for which the CDF will
be computed. Should be a large value!

regularise

Logical value indicating whether to smooth the results for very small distances,
to avoid discretisation artefacts.

distfun

335

Details
This command computes the Cumulative Distribution Function CDF (r) = P rob(T ≤ r) of the
Euclidean distance T = kX1 − X2 k between two independent random points X1 and X2 .
In the simplest case, the command distcdf(W), the random points are assumed to be uniformly
distributed in the same window W.
Alternatively the two random points may be uniformly distributed in two different windows W and
V.
In the most general case the first point X1 is random in the window W with a probability density
proportional to dW, and the second point X2 is random in a different window V with probability
density proportional to dV. The values of dW and dV must be finite and nonnegative.
The calculation is performed by numerical integration of the set covariance function setcov for
uniformly distributed points, and by computing the covariance function imcov in the general case.
The accuracy of the result depends on the pixel resolution used to represent the windows: this is
controlled by the arguments ... which are passed to as.mask. For example use eps=0.1 to specify
pixels of size 0.1 units.
The arguments W or V may also be point patterns (objects of class "ppp"). The result is the cumulative distribution function of the distance from a randomly selected point in the point pattern, to a
randomly selected point in the other point pattern or window.
If regularise=TRUE (the default), values of the cumulative distribution function for very short
distances are smoothed to avoid discretisation artefacts. Smoothing is applied to all distances shorter
than the width of 7 pixels.
Value
An object of class "fv", see fv.object, which can be plotted directly using plot.fv.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
setcov, as.mask.
Examples
# The unit disc
B <- disc()
plot(distcdf(B))

distfun

Distance Map as a Function

Description
Compute the distance function of an object, and return it as a function.

336

distfun

Usage
distfun(X, ...)
## S3 method for class 'ppp'
distfun(X, ..., k=1, undef=Inf)
## S3 method for class 'psp'
distfun(X, ...)
## S3 method for class 'owin'
distfun(X, ..., invert=FALSE)
Arguments
X

Any suitable dataset representing a two-dimensional object, such as a point pattern (object of class "ppp"), a window (object of class "owin") or a line segment
pattern (object of class "psp").

...

Extra arguments are ignored.

k

An integer. The distance to the kth nearest point will be computed.

undef

The value that should be returned if the distance is undefined (that is, if X contains fewer than k points).

invert

If TRUE, compute the distance transform of the complement of X.

Details
The “distance function” of a set of points A is the mathematical function f such that, for any twodimensional spatial location (x, y), the function value f(x,y) is the shortest distance from (x, y) to
A.
The command f <- distfun(X) returns a function in the R language, with arguments x,y, that
represents the distance function of X. Evaluating the function f in the form v <- f(x,y), where x
and y are any numeric vectors of equal length containing coordinates of spatial locations, yields the
values of the distance function at these locations. Alternatively x can be a point pattern (object of
class "ppp" or "lpp") of locations at which the distance function should be computed (and then y
should be missing).
This should be contrasted with the related command distmap which computes the distance function
of X on a grid of locations, and returns the distance values in the form of a pixel image.
The result of f <- distfun(X) also belongs to the class "funxy" and to the special class "distfun".
It can be printed and plotted immediately as shown in the Examples.
A distfun object can be converted to a pixel image using as.im.
Value
A function with arguments x,y. The function also belongs to the class "distfun" which has a
method for print. It also belongs to the class "funxy" which has methods for plot, contour and
persp.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .

distfun.lpp

337

See Also
distmap, plot.funxy
Examples
data(letterR)
f <- distfun(letterR)
f
plot(f)
f(0.2, 0.3)
plot(distfun(letterR, invert=TRUE), eps=0.1)
d <- distfun(cells)
d2 <- distfun(cells, k=2)
d(0.5, 0.5)
d2(0.5, 0.5)
z <- d(japanesepines)

distfun.lpp

Distance Map on Linear Network

Description
Compute the distance function of a point pattern on a linear network.
Usage
## S3 method for class 'lpp'
distfun(X, ..., k=1)
Arguments
X

A point pattern on a linear network (object of class "lpp").

k

An integer. The distance to the kth nearest point will be computed.

...

Extra arguments are ignored.

Details
On a linear network L, the “geodesic distance function” of a set of points A in L is the mathematical
function f such that, for any location s on L, the function value f(s) is the shortest-path distance
from s to A.
The command distfun.lpp is a method for the generic command distfun for the class "lpp" of
point patterns on a linear network.
If X is a point pattern on a linear network, f <- distfun(X) returns a function in the R language
that represents the distance function of X. Evaluating the function f in the form v <- f(x,y), where
x and y are any numeric vectors of equal length containing coordinates of spatial locations, yields
the values of the distance function at these locations. More efficiently f can be called in the form
v <- f(x, y, seg, tp) where seg and tp are the local coordinates on the network. It can also
be called as v <- f(x) where x is a point pattern on the same linear network.

338

distmap
The function f obtained from f <- distfun(X) also belongs to the class "linfun". It can be
printed and plotted immediately as shown in the Examples. It can be converted to a pixel image
using as.linim.

Value
A function with arguments x,y and optional arguments seg,tp. It also belongs to the class
"linfun" which has methods for plot, print etc.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
linfun, methods.linfun.
To identify which point is the nearest neighbour, see nnfun.lpp.
Examples
data(letterR)
X <- runiflpp(3, simplenet)
f <- distfun(X)
f
plot(f)
# using a distfun as a covariate in a point process model:
Y <- runiflpp(4, simplenet)
fit <- lppm(Y ~D, covariates=list(D=f))
f(Y)

distmap

Distance Map

Description
Compute the distance map of an object, and return it as a pixel image. Generic.
Usage
distmap(X, ...)
Arguments
X

Any suitable dataset representing a two-dimensional object, such as a point pattern (object of class "ppp"), a window (object of class "owin") or a line segment
pattern (object of class "psp").

...

Arguments passed to as.mask to control pixel resolution.

distmap.owin

339

Details
The “distance map” of a set of points A is the function f whose value f(x) is defined for any
two-dimensional location x as the shortest distance from x to A.
This function computes the distance map of the set X and returns the distance map as a pixel image.
This is generic. Methods are provided for point patterns (distmap.ppp), line segment patterns
(distmap.psp) and windows (distmap.owin).
Value
A pixel image (object of class "im") whose grey scale values are the values of the distance map.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
distmap.ppp, distmap.psp, distmap.owin, distfun
Examples
data(cells)
U <- distmap(cells)
data(letterR)
V <- distmap(letterR)
## Not run:
plot(U)
plot(V)
## End(Not run)

distmap.owin

Distance Map of Window

Description
Computes the distance from each pixel to the nearest point in the given window.
Usage
## S3 method for class 'owin'
distmap(X, ..., discretise=FALSE, invert=FALSE)
Arguments
X

A window (object of class "owin").

...

Arguments passed to as.mask to control pixel resolution.

discretise

Logical flag controlling the choice of algorithm when X is a polygonal window.
See Details.

invert

If TRUE, compute the distance transform of the complement of the window.

340

distmap.owin

Details
The “distance map” of a window W is the function f whose value f(u) is defined for any twodimensional location u as the shortest distance from u to W .
This function computes the distance map of the window X and returns the distance map as a pixel
image. The greyscale value at a pixel u equals the distance from u to the nearest pixel in X.
Additionally, the return value has an attribute "bdry" which is also a pixel image. The grey values
in "bdry" give the distance from each pixel to the bounding rectangle of the image.
If X is a binary pixel mask, the distance values computed are not the usual Euclidean distances.
Instead the distance between two pixels is measured by the length of the shortest path connecting
the two pixels. A path is a series of steps between neighbouring pixels (each pixel has 8 neighbours).
This is the standard ‘distance transform’ algorithm of image processing (Rosenfeld and Kak, 1968;
Borgefors, 1986).
If X is a polygonal window, then exact Euclidean distances will be computed if discretise=FALSE.
If discretise=TRUE then the window will first be converted to a binary pixel mask and the discrete
path distances will be computed.
The arguments ... are passed to as.mask to control the pixel resolution.
This function is a method for the generic distmap.
Value
A pixel image (object of class "im") whose greyscale values are the values of the distance map. The
return value has an attribute "bdry" which is a pixel image.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Borgefors, G. Distance transformations in digital images. Computer Vision, Graphics and Image
Processing 34 (1986) 344–371.
Rosenfeld, A. and Pfalz, J.L. Distance functions on digital pictures. Pattern Recognition 1 (1968)
33-61.
See Also
distmap, distmap.ppp, distmap.psp
Examples
data(letterR)
U <- distmap(letterR)
## Not run:
plot(U)
plot(attr(U, "bdry"))
## End(Not run)

distmap.ppp

distmap.ppp

341

Distance Map of Point Pattern

Description
Computes the distance from each pixel to the nearest point in the given point pattern.
Usage
## S3 method for class 'ppp'
distmap(X, ...)
Arguments
X

A point pattern (object of class "ppp").

...

Arguments passed to as.mask to control pixel resolution.

Details
The “distance map” of a point pattern X is the function f whose value f(u) is defined for any
two-dimensional location u as the shortest distance from u to X.
This function computes the distance map of the point pattern X and returns the distance map as a
pixel image. The greyscale value at a pixel u equals the distance from u to the nearest point of the
pattern X.
Additionally, the return value has two attributes, "index" and "bdry", which are also pixel images.
The grey values in "bdry" give the distance from each pixel to the bounding rectangle of the image.
The grey values in "index" are integers identifying which point of X is closest.
This is a method for the generic function distmap.
Note that this function gives the distance from the centre of each pixel to the nearest data point. To
compute the exact distance from a given spatial location to the nearest data point in X, use distfun
or nncross.
Value
A pixel image (object of class "im") whose greyscale values are the values of the distance map. The
return value has attributes "index" and "bdry" which are also pixel images.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
Generic function distmap and other methods distmap.psp, distmap.owin.
Generic function distfun.
Nearest neighbour distance nncross

342

distmap.psp

Examples
data(cells)
U <- distmap(cells)
## Not run:
plot(U)
plot(attr(U, "bdry"))
plot(attr(U, "index"))
## End(Not run)

distmap.psp

Distance Map of Line Segment Pattern

Description
Computes the distance from each pixel to the nearest line segment in the given line segment pattern.
Usage
## S3 method for class 'psp'
distmap(X, ...)
Arguments
X

A line segment pattern (object of class "psp").

...

Arguments passed to as.mask to control pixel resolution.

Details
The “distance map” of a line segment pattern X is the function f whose value f(u) is defined for
any two-dimensional location u as the shortest distance from u to X.
This function computes the distance map of the line segment pattern X and returns the distance map
as a pixel image. The greyscale value at a pixel u equals the distance from u to the nearest line
segment of the pattern X. Distances are computed using analytic geometry.
Additionally, the return value has two attributes, "index" and "bdry", which are also pixel images.
The grey values in "bdry" give the distance from each pixel to the bounding rectangle of the image.
The grey values in "index" are integers identifying which line segment of X is closest.
This is a method for the generic function distmap.
Note that this function gives the exact distance from the centre of each pixel to the nearest line segment. To compute the exact distance from the points in a point pattern to the nearest line segment,
use distfun or one of the low-level functions nncross or project2segment.
Value
A pixel image (object of class "im") whose greyscale values are the values of the distance map. The
return value has attributes "index" and "bdry" which are also pixel images.
Author(s)
Adrian Baddeley 
and Rolf Turner 

divide.linnet

343

See Also
distmap, distmap.owin, distmap.ppp, distfun, nncross, nearestsegment, project2segment.
Examples
a <- psp(runif(20),runif(20),runif(20),runif(20), window=owin())
Z <- distmap(a)
plot(Z)
plot(a, add=TRUE)

divide.linnet

Divide Linear Network at Cut Points

Description
Make a tessellation of a linear network by dividing it into pieces demarcated by the points of a point
pattern.
Usage
divide.linnet(X)
Arguments
X

Point pattern on a linear network (object of class "lpp").

Details
The points X are interpreted as dividing the linear network L=as.linnet(X) into separate pieces.
Two locations on L belong to the same piece if and only if they can be joined by a path in L that
does not cross any of the points of X.
The result is a tessellation of the network (object of class "lintess") representing the division of
L into pieces.
Value
A tessellation on a linear network (object of class "lintess").
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak  and Greg McSwiggan.
See Also
linnet, lintess.
Examples
X <- runiflpp(5, simplenet)
plot(divide.linnet(X))
plot(X, add=TRUE, pch=16)

344

dkernel

dkernel

Kernel distributions and random generation

Description
Density, distribution function, quantile function and random generation for several distributions
used in kernel estimation for numerical data.
Usage
dkernel(x,
pkernel(q,
qkernel(p,
rkernel(n,

kernel
kernel
kernel
kernel

=
=
=
=

"gaussian",
"gaussian",
"gaussian",
"gaussian",

mean
mean
mean
mean

=
=
=
=

0,
0,
0,
0,

sd
sd
sd
sd

=
=
=
=

1)
1, lower.tail = TRUE)
1, lower.tail = TRUE)
1)

Arguments
x, q

Vector of quantiles.

p

Vector of probabilities.

kernel

String name of the kernel. Options are "gaussian", "rectangular", "triangular",
"epanechnikov", "biweight", "cosine" and "optcosine". (Partial matching
is used).

n

Number of observations.

mean

Mean of distribution.

sd

Standard deviation of distribution.

lower.tail

logical; if TRUE (the default), then probabilities are P (X ≤ x), otherwise,
P (X > x).

Details
These functions give the probability density, cumulative distribution function, quantile function and
random generation for several distributions used in kernel estimation for one-dimensional (numerical) data.
The available kernels are those used in density.default, namely "gaussian", "rectangular",
"triangular", "epanechnikov", "biweight", "cosine" and "optcosine". For more information about these kernels, see density.default.
dkernel gives the probability density, pkernel gives the cumulative distribution function, qkernel
gives the quantile function, and rkernel generates random deviates.
Value
A numeric vector. For dkernel, a vector of the same length as x containing the corresponding
values of the probability density. For pkernel, a vector of the same length as x containing the
corresponding values of the cumulative distribution function. For qkernel, a vector of the same
length as p containing the corresponding quantiles. For rkernel, a vector of length n containing
randomly generated values.

dmixpois

345

Author(s)
Adrian Baddeley  
and Martin Hazelton
See Also
density.default, kernel.factor
Examples
x <- seq(-3,3,length=100)
plot(x, dkernel(x, "epa"), type="l",
main=c("Epanechnikov kernel", "probability density"))
plot(x, pkernel(x, "opt"), type="l",
main=c("OptCosine kernel", "cumulative distribution function"))
p <- seq(0,1, length=256)
plot(p, qkernel(p, "biw"), type="l",
main=c("Biweight kernel", "cumulative distribution function"))
y <- rkernel(100, "tri")
hist(y, main="Random variates from triangular density")
rug(y)

dmixpois

Mixed Poisson Distribution

Description
Density, distribution function, quantile function and random generation for a mixture of Poisson
distributions.
Usage
dmixpois(x,
pmixpois(q,
qmixpois(p,
rmixpois(n,

mu,
mu,
mu,
mu,

sd,
sd,
sd,
sd,

invlink
invlink
invlink
invlink

=
=
=
=

exp, GHorder = 5)
exp, lower.tail = TRUE, GHorder = 5)
exp, lower.tail = TRUE, GHorder = 5)
exp)

Arguments
x
q
p
n
mu
sd
invlink
lower.tail
GHorder

vector of (non-negative integer) quantiles.
vector of quantiles.
vector of probabilities.
number of random values to return.
Mean of the linear predictor. A single numeric value.
Standard deviation of the linear predictor. A single numeric value.
Inverse link function. A function in the R language, used to transform the linear
predictor into the parameter lambda of the Poisson distribution.
Logical. If TRUE (the default), probabilities are P [X ≤ x], otherwise, P [X >
x].
Number of quadrature points in the Gauss-Hermite quadrature approximation.
A small positive integer.

346

domain

Details
These functions are analogous to dpois ppois, qpois and rpois except that they apply to a mixture
of Poisson distributions.
In effect, the Poisson mean parameter lambda is randomised by setting lambda = invlink(Z)
where Z has a Gaussian N (µ, σ 2 ) distribution. The default is invlink=exp which means that
lambda is lognormal. Set invlink=I to assume that lambda is approximately Normal.
For dmixpois, pmixpois and qmixpois, the probability distribution is approximated using GaussHermite quadrature. For rmixpois, the deviates are simulated exactly.
Value
Numeric vector: dmixpois gives probability masses, ppois gives cumulative probabilities, qpois
gives (non-negative integer) quantiles, and rpois generates (non-negative integer) random deviates.
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak 
See Also
dpois, gauss.hermite.
Examples
dmixpois(7, 10, 1, invlink = I)
dpois(7, 10)
pmixpois(7, log(10), 0.2)
ppois(7, 10)
qmixpois(0.95, log(10), 0.2)
qpois(0.95, 10)
x <- rmixpois(100, log(10), log(1.2))
mean(x)
var(x)

domain

Extract the Domain of any Spatial Object

Description
Given a spatial object such as a point pattern, in any number of dimensions, this function extracts
the spatial domain in which the object is defined.

domain

347

Usage
domain(X, ...)
## S3 method for class 'ppp'
domain(X, ...)
## S3 method for class 'psp'
domain(X, ...)
## S3 method for class 'im'
domain(X, ...)
## S3 method for class 'ppx'
domain(X, ...)
## S3 method for class 'pp3'
domain(X, ...)
## S3 method for class 'lpp'
domain(X, ...)
## S3 method for class 'ppm'
domain(X, ..., from=c("points", "covariates"))
## S3 method for class 'kppm'
domain(X, ..., from=c("points", "covariates"))
## S3 method for class 'dppm'
domain(X, ..., from=c("points", "covariates"))
## S3 method for class 'lpp'
domain(X, ...)
## S3 method for class 'lppm'
domain(X, ...)
## S3 method for class 'msr'
domain(X, ...)
## S3 method for class 'quad'
domain(X, ...)
## S3 method for class 'quadratcount'
domain(X, ...)
## S3 method for class 'quadrattest'
domain(X, ...)
## S3 method for class 'tess'
domain(X, ...)
## S3 method for class 'linfun'

348

domain
domain(X, ...)
## S3 method for class 'lintess'
domain(X, ...)
## S3 method for class 'im'
domain(X, ...)
## S3 method for class 'layered'
domain(X, ...)
## S3 method for class 'distfun'
domain(X, ...)
## S3 method for class 'nnfun'
domain(X, ...)
## S3 method for class 'funxy'
domain(X, ...)
## S3 method for class 'rmhmodel'
domain(X, ...)
## S3 method for class 'leverage.ppm'
domain(X, ...)
## S3 method for class 'influence.ppm'
domain(X, ...)

Arguments
X

A spatial object such as a point pattern (in any number of dimensions), line
segment pattern or pixel image.

...

Extra arguments. They are ignored by all the methods listed here.

from

Character string. See Details.

Details
The function domain is generic.
For a spatial object X in any number of dimensions, domain(X) extracts the spatial domain in which
X is defined.
For a two-dimensional object X, typically domain(X) is the same as domain(X).
The exception is that, if X is a point pattern on a linear network (class "lpp") or a point process
model on a linear network (class "lppm"), then domain(X) is the linear network on which the points
lie, while Window(X) is the two-dimensional window containing the linear network.
The argument from applies when X is a fitted point process model (object of class "ppm", "kppm"
or "dppm"). If from="data" (the default), domain extracts the window of the original point pattern
data to which the model was fitted. If from="covariates" then domain returns the window in
which the spatial covariates of the model were provided.

dppapproxkernel

349

Value
A spatial object representing the domain of X. Typically a window (object of class "owin"), a threedimensional box ("box3"), a multidimensional box ("boxx") or a linear network ("linnet").
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
Window, Frame
Examples
domain(cells)
domain(bei.extra$elev)
domain(chicago)

dppapproxkernel

Approximate Determinantal Point Process Kernel

Description
Returns an approximation to the kernel of a determinantal point process, as a function of one argument x.
Usage
dppapproxkernel(model, trunc = 0.99, W = NULL)
Arguments
model

Object of class "detpointprocfamily".

trunc

Numeric specifying how the model truncation is performed. See Details section
of simulate.detpointprocfamily.

W

Optional window – undocumented at the moment.

Value
A function
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 

350

dppBessel

dppapproxpcf

Approximate Pair Correlation Function of Determinantal Point Process Model

Description
Returns an approximation to the theoretical pair correlation function of a determinantal point process model, as a function of one argument x.
Usage
dppapproxpcf(model, trunc = 0.99, W = NULL)
Arguments
model

Object of class "detpointprocfamily".

trunc

Numeric specifying how the model truncation is performed. See Details section
of simulate.detpointprocfamily.

W

Optional window – undocumented at the moment.

Details
This function is usually NOT needed for anything. It only exists for investigative purposes.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
Examples
f <- dppapproxpcf(dppMatern(lambda = 100, alpha=.028, nu=1, d=2))
plot(f, xlim = c(0,0.1))

dppBessel

Bessel Type Determinantal Point Process Model

Description
Function generating an instance of the Bessel-type determinantal point process model.
Usage
dppBessel(...)
Arguments
...

arguments of the form tag=value specifying the model parameters. See Details.

dppCauchy

351

Details
The possible parameters are:
• the intensity lambda as a positive numeric
• the scale parameter alpha as a positive numeric
• the shape parameter sigma as a non-negative numeric
• the dimension d as a positive integer
Value
An object of class "detpointprocfamily".
Author(s)
Frederic Lavancier and Christophe Biscio. Modified by Ege Rubak  , Adrian
Baddeley 
and Rolf Turner 
See Also
dppCauchy, dppGauss, dppMatern, dppPowerExp
Examples
m <- dppBessel(lambda=100, alpha=.05, sigma=0, d=2)

dppCauchy

Generalized Cauchy Determinantal Point Process Model

Description
Function generating an instance of the (generalized) Cauchy determinantal point process model.
Usage
dppCauchy(...)
Arguments
...

arguments of the form tag=value specifying the parameters. See Details.

Details
The (generalized) Cauchy DPP is defined in (Lavancier, Møller and Rubak, 2015) The possible
parameters are:
• the intensity lambda as a positive numeric
• the scale parameter alpha as a positive numeric
• the shape parameter nu as a positive numeric (artificially required to be less than 20 in the
code for numerical stability)
• the dimension d as a positive integer

352

dppeigen

Value
An object of class "detpointprocfamily".
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Lavancier, F. Møller, J. and Rubak, E. (2015) Determinantal point process models and statistical
inference Journal of the Royal Statistical Society, Series B 77, 853–977.
See Also
dppBessel, dppGauss, dppMatern, dppPowerExp
Examples
m <- dppCauchy(lambda=100, alpha=.05, nu=1, d=2)

dppeigen

Internal function calculating eig and index

Description
This function is mainly for internal package use and is usually not called by the user.
Usage
dppeigen(model, trunc, Wscale, stationary = FALSE)
Arguments
model

object of class "detpointprocfamily"

trunc

numeric giving the truncation

Wscale

numeric giving the scale of the window relative to a unit box

stationary

logical indicating whether the stationarity of the model should be used (only
works in dimension 2).

Value
A list
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 

dppGauss

dppGauss

353

Gaussian Determinantal Point Process Model

Description
Function generating an instance of the Gaussian determinantal point process model.
Usage
dppGauss(...)
Arguments
...

arguments of the form tag=value specifying the parameters. See Details.

Details
The Gaussian DPP is defined in (Lavancier, Møller and Rubak, 2015) The possible parameters are:
• the intensity lambda as a positive numeric
• the scale parameter alpha as a positive numeric
• the dimension d as a positive integer
Value
An object of class "detpointprocfamily".
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Lavancier, F. Møller, J. and Rubak, E. (2015) Determinantal point process models and statistical
inference Journal of the Royal Statistical Society, Series B 77, 853–977.
See Also
dppBessel, dppCauchy, dppMatern, dppPowerExp
Examples
m <- dppGauss(lambda=100, alpha=.05, d=2)

354

dppm

dppkernel

Extract Kernel from Determinantal Point Process Model Object

Description
Returns the kernel of a determinantal point process model as a function of one argument x.
Usage
dppkernel(model, ...)
Arguments
model

Model of class "detpointprocfamily".

...

Arguments passed to dppapproxkernel if the exact kernel is unknown

Value
A function
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
Examples
kernelMatern <- dppkernel(dppMatern(lambda = 100, alpha=.01, nu=1, d=2))
plot(kernelMatern, xlim = c(0,0.1))

dppm

Fit Determinantal Point Process Model

Description
Fit a determinantal point process model to a point pattern.
Usage
dppm(formula, family, data=NULL,
...,
startpar = NULL,
method = c("mincon", "clik2", "palm"),
weightfun=NULL,
control=list(),
algorithm="Nelder-Mead",
statistic="K",
statargs=list(),

dppm

355
rmax = NULL,
covfunargs=NULL,
use.gam=FALSE,
nd=NULL, eps=NULL)

Arguments
formula

A formula in the R language specifying the data (on the left side) and the form
of the model to be fitted (on the right side). For a stationary model it suffices to
provide a point pattern without a formula. See Details.

family

Information specifying the family of point processes to be used in the model.
Typically one of the family functions dppGauss, dppMatern, dppCauchy, dppBessel
or dppPowerExp. Alternatively a character string giving the name of a family
function, or the result of calling one of the family functions. See Details.

data

The values of spatial covariates (other than the Cartesian coordinates) required
by the model. A named list of pixel images, functions, windows, tessellations
or numeric constants.

...

Additional arguments. See Details.

startpar

Named vector of starting parameter values for the optimization.

method

The fitting method. Either "mincon" for minimum contrast, "clik2" for second
order composite likelihood, or "palm" for Palm likelihood. Partially matched.

weightfun

Optional weighting function w in the composite likelihood or Palm likelihood.
A function in the R language. See Details.

control

List of control parameters passed to the optimization function optim.

algorithm

Character string determining the mathematical optimisation algorithm to be used
by optim. See the argument method of optim.

statistic

Name of the summary statistic to be used for minimum contrast estimation:
either "K" or "pcf".

statargs

Optional list of arguments to be used when calculating the statistic. See
Details.

rmax
Maximum value of interpoint distance to use in the composite likelihood.
covfunargs,use.gam,nd,eps
Arguments passed to ppm when fitting the intensity.
Details
This function fits a determinantal point process model to a point pattern dataset as described in
Lavancier et al. (2015).
The model to be fitted is specified by the arguments formula and family.
The argument formula should normally be a formula in the R language. The left hand side of
the formula specifies the point pattern dataset to which the model should be fitted. This should be
a single argument which may be a point pattern (object of class "ppp") or a quadrature scheme
(object of class "quad"). The right hand side of the formula is called the trend and specifies the
form of the logarithm of the intensity of the process. Alternatively the argument formula may be a
point pattern or quadrature scheme, and the trend formula is taken to be ~1.
The argument family specifies the family of point processes to be used in the model. It is typically
one of the family functions dppGauss, dppMatern, dppCauchy, dppBessel or dppPowerExp. Alternatively it may be a character string giving the name of a family function, or the result of calling one

356

dppm
of the family functions. A family function belongs to class "detpointprocfamilyfun". The result
of calling a family function is a point process family, which belongs to class "detpointprocfamily".
The algorithm first estimates the intensity function of the point process using ppm. If the trend
formula is ~1 (the default if a point pattern or quadrature scheme is given rather than a "formula")
then the model is homogeneous. The algorithm begins by estimating the intensity as the number of
points divided by the area of the window. Otherwise, the model is inhomogeneous. The algorithm
begins by fitting a Poisson process with log intensity of the form specified by the formula trend.
(See ppm for further explanation).
The interaction parameters of the model are then fitted either by minimum contrast estimation, or
by maximum composite likelihood.
Minimum contrast: If method = "mincon" (the default) interaction parameters of the model will
be fitted by minimum contrast estimation, that is, by matching the theoretical K-function of
the model to the empirical K-function of the data, as explained in mincontrast.
For a homogeneous model ( trend = ~1 ) the empirical K-function of the data is computed using Kest, and the interaction parameters of the model are estimated by the method of
minimum contrast.
For an inhomogeneous model, the inhomogeneous K function is estimated by Kinhom using
the fitted intensity. Then the interaction parameters of the model are estimated by the method
of minimum contrast using the inhomogeneous K function. This two-step estimation procedure is heavily inspired by Waagepetersen (2007).
If statistic="pcf" then instead of using the K-function, the algorithm will use the pair
correlation function pcf for homogeneous models and the inhomogeneous pair correlation
function pcfinhom for inhomogeneous models. In this case, the smoothing parameters of the
pair correlation can be controlled using the argument statargs, as shown in the Examples.
Additional arguments ... will be passed to mincontrast to control the minimum contrast
fitting algorithm.
Composite likelihood: If method = "clik2" the interaction parameters of the model will be
fitted by maximising the second-order composite likelihood (Guan, 2006). The log composite
likelihood is


Z Z
X
X


w(dij ) log ρ(dij ; θ) −
w(dij ) log
w(ku − vk)ρ(ku − vk; θ) du dv
i,j

i,j

D

D

where the sums are taken over all pairs of data points xi , xj separated by a distance dij =
kxi − xj k less than rmax, and the double integral is taken over all pairs of locations u, v in
the spatial window of the data. Here ρ(d; θ) is the pair correlation function of the model with
cluster parameters θ.
The function w in the composite likelihood is a weighting function and may be chosen arbitrarily. It is specified by the argument weightfun. If this is missing or NULL then the default
is a threshold weight function, w(d) = 1(d ≤ R), where R is rmax/2.
Palm likelihood: If method = "palm" the interaction parameters of the model will be fitted by
maximising the Palm loglikelihood (Tanaka et al, 2008)
Z
X
w(xi , xj ) log λP (xj | xi ; θ) −
w(xi , u)λP (u | xi ; θ)du
i,j

D

with the same notation as above. Here λP (u|v; θ is the Palm intensity of the model at location
u given there is a point at v.

dppm

357

In all three methods, the optimisation is performed by the generic optimisation algorithm optim.
The behaviour of this algorithm can be modified using the argument control. Useful control
arguments include trace, maxit and abstol (documented in the help for optim).
Finally, it is also possible to fix any parameters desired before the optimisation by specifying them
as name=value in the call to the family function. See Examples.
Value
An object of class "dppm" representing the fitted model. There are methods for printing, plotting,
predicting and simulating objects of this class.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
References
Lavancier, F. Møller, J. and Rubak, E. (2015) Determinantal point process models and statistical
inference Journal of the Royal Statistical Society, Series B 77, 853–977.
Guan, Y. (2006) A composite likelihood approach in fitting spatial point process models. Journal
of the American Statistical Association 101, 1502–1512.
Tanaka, U. and Ogata, Y. and Stoyan, D. (2008) Parameter estimation and model selection for
Neyman-Scott point processes. Biometrical Journal 50, 43–57.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous NeymanScott processes. Biometrics 63, 252–258.
See Also
methods for dppm objects: plot.dppm, fitted.dppm, predict.dppm, simulate.dppm, methods.dppm,
as.ppm.dppm, Kmodel.dppm, pcfmodel.dppm.
Minimum contrast fitting algorithm: mincontrast.
Deterimantal point process models: dppGauss, dppMatern, dppCauchy, dppBessel, dppPowerExp,
Summary statistics: Kest, Kinhom, pcf, pcfinhom.
See also ppm
Examples
jpines <- residualspaper$Fig1
dppm(jpines ~ 1, dppGauss)
dppm(jpines ~ 1, dppGauss, method="c")
dppm(jpines ~ 1, dppGauss, method="p")
# Fixing the intensity to lambda=2 rather than the Poisson MLE 2.04:
dppm(jpines ~ 1, dppGauss(lambda=2))
if(interactive()) {
# The following is quite slow (using K-function)
dppm(jpines ~ x, dppMatern)

358

dppMatern
}
# much faster using pair correlation function
dppm(jpines ~ x, dppMatern, statistic="pcf", statargs=list(stoyan=0.2))
# Fixing the Matern shape parameter to nu=2 rather than estimating it:
dppm(jpines ~ x, dppMatern(nu=2))

dppMatern

Whittle-Matern Determinantal Point Process Model

Description
Function generating an instance of the Whittle-Matern determinantal point process model
Usage
dppMatern(...)
Arguments
...

arguments of the form tag=value specifying the parameters. See Details.

Details
The Whittle-Matérn DPP is defined in (Lavancier, Møller and Rubak, 2015) The possible parameters are:
• the intensity lambda as a positive numeric
• the scale parameter alpha as a positive numeric
• the shape parameter nu as a positive numeric (artificially required to be less than 20 in the
code for numerical stability)
• the dimension d as a positive integer
Value
An object of class "detpointprocfamily".
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Lavancier, F. Møller, J. and Rubak, E. (2015) Determinantal point process models and statistical
inference Journal of the Royal Statistical Society, Series B 77, 853–977.
See Also
dppBessel, dppCauchy, dppGauss, dppPowerExp

dppparbounds

359

Examples
m <- dppMatern(lambda=100, alpha=.02, nu=1, d=2)

dppparbounds

Parameter Bound for a Determinantal Point Process Model

Description
Returns the lower and upper bound for a specific parameter of a determinantal point process model
when all other parameters are fixed.
Usage
dppparbounds(model, name, ...)
Arguments
model

Model of class "detpointprocfamily".

name

name of the parameter for which the bound should be computed.

...

Additional arguments passed to the parbounds function of the given model

Value
A data.frame containing lower and upper bounds.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
Examples
model <- dppMatern(lambda=100, alpha=.01, nu=1, d=2)
dppparbounds(model, "lambda")

dppPowerExp

Power Exponential Spectral Determinantal Point Process Model

Description
Function generating an instance of the Power Exponential Spectral determinantal point process
model.
Usage
dppPowerExp(...)

360

dppspecden

Arguments
...

arguments of the form tag=value specifying the parameters. See Details.

Details
The Power Exponential Spectral DPP is defined in (Lavancier, Møller and Rubak, 2015) The possible parameters are:
• the intensity lambda as a positive numeric
• the scale parameter alpha as a positive numeric
• the shape parameter nu as a positive numeric (artificially required to be less than 20 in the
code for numerical stability)
• the dimension d as a positive integer
Value
An object of class "detpointprocfamily".
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Lavancier, F. Møller, J. and Rubak, E. (2015) Determinantal point process models and statistical
inference Journal of the Royal Statistical Society, Series B 77, 853–977.
See Also
dppBessel, dppCauchy, dppGauss, dppMatern
Examples
m <- dppPowerExp(lambda=100, alpha=.01, nu=1, d=2)

dppspecden

Extract Spectral Density from Determinantal Point Process Model Object

Description
Returns the spectral density of a determinantal point process model as a function of one argument
x.
Usage
dppspecden(model)

dppspecdenrange

361

Arguments
model

Model of class "detpointprocfamily".

Value
A function
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
dppspecdenrange
Examples
model <- dppMatern(lambda = 100, alpha=.01, nu=1, d=2)
dppspecden(model)

dppspecdenrange

Range of Spectral Density of a Determinantal Point Process Model

Description
Computes the range of the spectral density of a determinantal point process model.
Usage
dppspecdenrange(model)
Arguments
model

Model of class "detpointprocfamily".

Value
Numeric value (possibly Inf).
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
dppspecden

362

dummify

Examples
m <- dppBessel(lambda=100, alpha=0.05, sigma=1, d=2)
dppspecdenrange(m)

dummify

Convert Data to Numeric Values by Constructing Dummy Variables

Description
Converts data of any kind to numeric values. A factor is expanded to a set of dummy variables.
Usage
dummify(x)
Arguments
x

Vector, factor, matrix or data frame to be converted.

Details
This function converts data (such as a factor) to numeric values in order that the user may calculate,
for example, the mean, variance, covariance and correlation of the data.
If x is a numeric vector or integer vector, it is returned unchanged.
If x is a logical vector, it is converted to a 0-1 matrix with 2 columns. The first column contains a 1
if the logical value is FALSE, and the second column contains a 1 if the logical value is TRUE.
If x is a complex vector, it is converted to a matrix with 2 columns, containing the real and imaginary
parts.
If x is a factor, the result is a matrix of 0-1 dummy variables. The matrix has one column for each
possible level of the factor. The (i,j) entry is equal to 1 when the ith factor value equals the jth
level, and is equal to 0 otherwise.
If x is a matrix or data frame, the appropriate conversion is applied to each column of x.
Note that, unlike model.matrix, this command converts a factor into a full set of dummy variables
(one column for each level of the factor).
Value
A numeric matrix.
Author(s)
Adrian Baddeley 
Examples
chara <- sample(letters[1:3], 8, replace=TRUE)
logi <- (runif(8) < 0.3)
comp <- round(4*runif(8) + 3*runif(8) * 1i, 1)
nume <- 8:1 + 0.1
df <- data.frame(nume, chara, logi, comp)
df
dummify(df)

dummy.ppm

dummy.ppm

363

Extract Dummy Points Used to Fit a Point Process Model

Description
Given a fitted point process model, this function extracts the ‘dummy points’ of the quadrature
scheme used to fit the model.
Usage
dummy.ppm(object, drop=FALSE)
Arguments
object

fitted point process model (an object of class "ppm").

drop

Logical value determining whether to delete dummy points that were not used
to fit the model.

Details
An object of class "ppm" represents a point process model that has been fitted to data. It is typically
produced by the model-fitting algorithm ppm.
The maximum pseudolikelihood algorithm in ppm approximates the pseudolikelihood integral by
a sum over a finite set of quadrature points, which is constructed by augmenting the original data
point pattern by a set of “dummy” points. The fitted model object returned by ppm contains complete
information about this quadrature scheme. See ppm or ppm.object for further information.
This function dummy.ppm extracts the dummy points of the quadrature scheme. A typical use of this
function would be to count the number of dummy points, to gauge the accuracy of the approximation
to the exact pseudolikelihood.
It may happen that some dummy points are not actually used in fitting the model (typically because
the value of a covariate is NA at these points). The argument drop specifies whether these unused
dummy points shall be deleted (drop=TRUE) or retained (drop=FALSE) in the return value.
See ppm.object for a list of all operations that can be performed on objects of class "ppm".
Value
A point pattern (object of class "ppp").
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
ppm.object, ppp.object, ppm

364

duplicated.ppp

Examples
data(cells)
fit <- ppm(cells, ~1, Strauss(r=0.1))
X <- dummy.ppm(fit)
npoints(X)
# this is the number of dummy points in the quadrature scheme

duplicated.ppp

Determine Duplicated Points in a Spatial Point Pattern

Description
Determines which points in a spatial point pattern are duplicates of previous points, and returns a
logical vector.
Usage
## S3 method for class 'ppp'
duplicated(x, ..., rule=c("spatstat", "deldir", "unmark"))
## S3 method for class 'ppx'
duplicated(x, ...)
## S3 method for class 'ppp'
anyDuplicated(x, ...)
## S3 method for class 'ppx'
anyDuplicated(x, ...)
Arguments
x

A spatial point pattern (object of class "ppp" or "ppx").

...

Ignored.

rule

Character string. The rule for determining duplicated points.

Details
These are methods for the generic functions duplicated and anyDuplicated for point pattern
datasets (of class "ppp", see ppp.object, or class "ppx").
anyDuplicated(x) is a faster version of any(duplicated(x)).
Two points in a point pattern are deemed to be identical if their x, y coordinates are the same, and
their marks are also the same (if they carry marks). The Examples section illustrates how it is
possible for a point pattern to contain a pair of identical points.
This function determines which points in x duplicate other points that appeared earlier in the sequence. It returns a logical vector with entries that are TRUE for duplicated points and FALSE for
unique (non-duplicated) points.
If rule="spatstat" (the default), two points are deemed identical if their coordinates are equal
according to ==, and their marks are equal according to ==. This is the most stringent possible
test. If rule="unmark", duplicated points are determined by testing equality of their coordinates

edge.Ripley

365

only, using ==. If rule="deldir", duplicated points are determined by testing equality of their
coordinates only, using the function duplicatedxy in the package deldir, which currently uses
duplicated.data.frame. Setting rule="deldir" will ensure consistency with functions in the
deldir package.
Value
duplicated(x) returns a logical vector of length equal to the number of points in x.
anyDuplicated(x) is a number equal to 0 if there are no duplicated points, and otherwise is equal
to the index of the first duplicated point.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
ppp.object, unique.ppp, multiplicity.ppp
Examples
X <- ppp(c(1,1,0.5), c(2,2,1), window=square(3))
duplicated(X)
duplicated(X, rule="deldir")

edge.Ripley

Ripley’s Isotropic Edge Correction

Description
Computes Ripley’s isotropic edge correction weights for a point pattern.
Usage
edge.Ripley(X, r, W = Window(X), method = "C", maxweight = 100)
rmax.Ripley(W)
Arguments
X

Point pattern (object of class "ppp").

W

Window for which the edge correction is required.

r

Vector or matrix of interpoint distances for which the edge correction should be
computed.

method

Choice of algorithm. Either "interpreted" or "C". This is needed only for
debugging purposes.

maxweight

Maximum permitted value of the edge correction weight.

366

edge.Ripley

Details
The function edge.Ripley computes Ripley’s (1977) isotropic edge correction weight, which is
used in estimating the K function and in many other contexts.
The function rmax.Ripley computes the maximum value of distance r for which the isotropic edge
correction estimate of K(r) is valid.
For a single point x in a window W , and a distance r > 0, the isotropic edge correction weight is
e(u, r) =

2πr
length(c(u, r) ∩ W )

where c(u, r) is the circle of radius r centred at the point u. The denominator is the length of the
overlap between this circle and the window W .
The function edge.Ripley computes this edge correction weight for each point in the point pattern
X and for each corresponding distance value in the vector or matrix r.
If r is a vector, with one entry for each point in X, then the result is a vector containing the edge
correction weights e(X[i], r[i]) for each i.
If r is a matrix, with one row for each point in X, then the result is a matrix whose i,j entry gives
the edge correction weight e(X[i], r[i,j]). For example edge.Ripley(X, pairdist(X))
computes all the edge corrections required for the K-function.
If any value of the edge correction weight exceeds maxwt, it is set to maxwt.
The function rmax.Ripley computes the smallest distance r such that it is possible to draw a circle
of radius r, centred at a point of W, such that the circle does not intersect the interior of W.
Value
A numeric vector or matrix.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Ripley, B.D. (1977) Modelling spatial patterns (with discussion). Journal of the Royal Statistical
Society, Series B, 39, 172 – 212.
See Also
edge.Trans, rmax.Trans, Kest
Examples
v <- edge.Ripley(cells, pairdist(cells))
rmax.Ripley(Window(cells))

edge.Trans

edge.Trans

367

Translation Edge Correction

Description
Computes Ohser and Stoyan’s translation edge correction weights for a point pattern.
Usage
edge.Trans(X, Y = X, W = Window(X),
exact = FALSE, paired = FALSE,
...,
trim = spatstat.options("maxedgewt"),
dx=NULL, dy=NULL,
give.rmax=FALSE, gW=NULL)
rmax.Trans(W, g=setcov(W))
Arguments
X,Y

Point patterns (objects of class "ppp").

W

Window for which the edge correction is required.

exact

Logical. If TRUE, a slow algorithm will be used to compute the exact value. If
FALSE, a fast algorithm will be used to compute the approximate value.

paired

Logical value indicating whether X and Y are paired. If TRUE, compute the edge
correction for corresponding points X[i], Y[i] for all i. If FALSE, compute
the edge correction for each possible pair of points X[i], Y[j] for all i and j.

...

Ignored.

trim

Maximum permitted value of the edge correction weight.

dx,dy

Alternative data giving the x and y coordinates of the vector differences between
the points. Incompatible with X and Y. See Details.

give.rmax

Logical. If TRUE, also compute the value of rmax.Trans(W) and return it as an
attribute of the result.

g, gW

Optional. Set covariance of W, if it has already been computed. Not required if
W is a rectangle.

Details
The function edge.Trans computes Ohser and Stoyan’s translation edge correction weight, which
is used in estimating the K function and in many other contexts.
The function rmax.Trans computes the maximum value of distance r for which the translation
edge correction estimate of K(r) is valid.
For a pair of points x and y in a window W , the translation edge correction weight is
e(u, r) =

area(W )
area(W ∩ (W + y − x))

where W + y − x is the result of shifting the window W by the vector y − x. The denominator is
the area of the overlap between this shifted window and the original window.

368

edge.Trans
The function edge.Trans computes this edge correction weight. If paired=TRUE, then X and Y
should contain the same number of points. The result is a vector containing the edge correction
weights e(X[i], Y[i]) for each i.
If paired=FALSE, then the result is a matrix whose i,j entry gives the edge correction weight
e(X[i], Y[j]).
Computation is exact if the window is a rectangle. Otherwise,
• if exact=TRUE, the edge correction weights are computed exactly using overlap.owin, which
can be quite slow.
• if exact=FALSE (the default), the weights are computed rapidly by evaluating the set covariance function setcov using the Fast Fourier Transform.
If any value of the edge correction weight exceeds trim, it is set to trim.
The arguments dx and dy can be provided as an alternative to X and Y. If paired=TRUE then dx,dy
should be vectors of equal length such that the vector difference of the ith pair is c(dx[i], dy[i]).
If paired=FALSE then dx,dy should be matrices of the same dimensions, such that the vector difference between X[i] and Y[j] is c(dx[i,j], dy[i,j]). The argument W is needed.
The value of rmax.Trans is the shortest distance from the origin (0, 0) to the boundary of the
support of the set covariance function of W. It is computed by pixel approximation using setcov,
unless W is a rectangle, when rmax.Trans(W) is the length of the shortest side of the rectangle.

Value
Numeric vector or matrix.

Author(s)
Adrian Baddeley  and Rolf Turner .

References
Ohser, J. (1983) On estimators for the reduced second moment measure of point processes. Mathematische Operationsforschung und Statistik, series Statistics, 14, 63 – 71.

See Also
rmax.Trans, edge.Ripley, setcov, Kest

Examples
v <- edge.Trans(cells)
rmax.Trans(Window(cells))

edges

369

edges

Extract Boundary Edges of a Window.

Description
Extracts the boundary edges of a window and returns them as a line segment pattern.

Usage
edges(x, ..., window = NULL, check = FALSE)
Arguments
x

A window (object of class "owin"), or data acceptable to as.owin, specifying
the window whose boundary is to be extracted.

...

Ignored.

window

Window to contain the resulting line segments. Defaults to as.rectangle(x).

check

Logical. Whether to check the validity of the resulting segment pattern.

Details
The boundary edges of the window x will be extracted as a line segment pattern.
Value
A line segment pattern (object of class "psp").
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
perimeter for calculating the total length of the boundary.
Examples
edges(square(1))
edges(letterR)

370

edges2triangles

edges2triangles

List Triangles in a Graph

Description
Given a list of edges between vertices, compile a list of all triangles formed by these edges.
Usage
edges2triangles(iedge, jedge, nvert=max(iedge, jedge), ...,
check=TRUE, friendly=rep(TRUE, nvert))
Arguments
iedge,jedge

Integer vectors, of equal length, specifying the edges.

nvert

Number of vertices in the network.

...

Ignored

check

Logical. Whether to check validity of input data.

friendly

Optional. For advanced use. See Details.

Details
This low level function finds all the triangles (cliques of size 3) in a finite graph with nvert vertices
and with edges specified by iedge, jedge.
The interpretation of iedge, jedge is that each successive pair of entries specifies an edge in the
graph. The kth edge joins vertex iedge[k] to vertex jedge[k]. Entries of iedge and jedge must
be integers from 1 to nvert.
To improve efficiency in some applications, the optional argument friendly can be used. It should
be a logical vector of length nvert specifying a labelling of the vertices, such that two vertices j,k
which are not friendly (friendly[j] = friendly[k] = FALSE) are never connected by an edge.
Value
A 3-column matrix of integers, in which each row represents a triangle.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
edges2vees
Examples
i <- c(1, 2, 5, 5, 1, 4, 2)
j <- c(2, 3, 3, 1, 3, 2, 5)
edges2triangles(i, j)

edges2vees

edges2vees

371

List Dihedral Triples in a Graph

Description
Given a list of edges between vertices, compile a list of all ‘vees’ or dihedral triples formed by these
edges.
Usage
edges2vees(iedge, jedge, nvert=max(iedge, jedge), ...,
check=TRUE)
Arguments
iedge,jedge

Integer vectors, of equal length, specifying the edges.

nvert

Number of vertices in the network.

...

Ignored

check

Logical. Whether to check validity of input data.

Details
Given a finite graph with nvert vertices and with edges specified by iedge, jedge, this low-level
function finds all ‘vees’ or ‘dihedral triples’ in the graph, that is, all triples of vertices (i,j,k)
where i and j are joined by an edge and i and k are joined by an edge.
The interpretation of iedge, jedge is that each successive pair of entries specifies an edge in the
graph. The kth edge joins vertex iedge[k] to vertex jedge[k]. Entries of iedge and jedge must
be integers from 1 to nvert.
Value
A 3-column matrix of integers, in which each row represents a triple of vertices, with the first vertex
joined to the other two vertices.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
edges2triangles
Examples
i <- c(1, 2, 5, 5, 1, 4, 2)
j <- c(2, 3, 3, 1, 3, 2, 5)
edges2vees(i, j)

372

edit.hyperframe

edit.hyperframe

Invoke Text Editor on Hyperframe

Description
Invokes a text editor allowing the user to inspect and change entries in a hyperframe.
Usage
## S3 method for class 'hyperframe'
edit(name, ...)
Arguments
name

A hyperframe (object of class "hyperframe").

...

Other arguments passed to edit.data.frame.

Details
The function edit is generic. This function is the methods for objects of class "hyperframe".
The hyperframe name is converted to a data frame or array, and the text editor is invoked. The user
can change entries in the columns of data, and create new columns of data.
Only the columns of atomic data (numbers, characters, factor values etc) can be edited.
Note that the original object name is not changed; the function returns the edited dataset.
Value
Another hyperframe.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
edit.data.frame, edit.ppp
Examples
if(interactive()) Z <- edit(flu)

edit.ppp

edit.ppp

373

Invoke Text Editor on Spatial Data

Description
Invokes a text editor allowing the user to inspect and change entries in a spatial dataset.
Usage
## S3 method for class 'ppp'
edit(name, ...)
## S3 method for class 'psp'
edit(name, ...)
## S3 method for class 'im'
edit(name, ...)
Arguments
name

A spatial dataset (object of class "ppp", "psp" or "im").

...

Other arguments passed to edit.data.frame.

Details
The function edit is generic. These functions are methods for spatial objects of class "ppp", "psp"
and "im".
The spatial dataset name is converted to a data frame or array, and the text editor is invoked. The
user can change the values of spatial coordinates or marks of the points in a point pattern, or the
coordinates or marks of the segments in a segment pattern, or the pixel values in an image. The
names of the columns of marks can also be edited.
If name is a pixel image, it is converted to a matrix and displayed in the same spatial orientation as
if the image had been plotted.
Note that the original object name is not changed; the function returns the edited dataset.
Value
Object of the same kind as name containing the edited data.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
edit.data.frame, edit.hyperframe

374

eem

Examples
if(interactive()) Z <- edit(cells)

eem

Exponential Energy Marks

Description
Given a point process model fitted to a point pattern, compute the Stoyan-Grabarnik diagnostic
“exponential energy marks” for the data points.
Usage
eem(fit, check=TRUE)
Arguments
fit

The fitted point process model. An object of class "ppm".

check

Logical value indicating whether to check the internal format of fit. If there
is any possibility that this object has been restored from a dump file, or has
otherwise lost track of the environment where it was originally computed, set
check=TRUE.

Details
Stoyan and Grabarnik (1991) proposed a diagnostic tool for point process models fitted to spatial
point pattern data. Each point xi of the data pattern X is given a ‘mark’ or ‘weight’
mi =

1
λ̂(xi , X)

where λ̂(xi , X) is the conditional intensity of the fitted model. If the fitted model is correct, then
the sum of these marks for all points in a region B has expected value equal to the area of B.
The argument fit must be a fitted point process model (object of class "ppm"). Such objects
are produced by the maximum pseudolikelihood fitting algorithm ppm). This fitted model object
contains complete information about the original data pattern and the model that was fitted to it.
The value returned by eem is the vector of weights m[i] associated with the points x[i] of the original
data pattern. The original data pattern (in corresponding order) can be extracted from fit using
data.ppm.
The function diagnose.ppm produces a set of sensible diagnostic plots based on these weights.
Value
A vector containing the values of the exponential energy mark for each point in the pattern.
Author(s)
Adrian Baddeley  and Rolf Turner 

effectfun

375

References
Stoyan, D. and Grabarnik, P. (1991) Second-order characteristics for stochastic structures connected
with Gibbs point processes. Mathematische Nachrichten, 151:95–100.
See Also
diagnose.ppm, ppm.object, data.ppm, residuals.ppm, ppm
Examples
data(cells)
fit <- ppm(cells, ~x, Strauss(r=0.15))
ee <- eem(fit)
sum(ee)/area(Window(cells)) # should be about 1 if model is correct
Y <- setmarks(cells, ee)
plot(Y, main="Cells data\n Exponential energy marks")

effectfun

Compute Fitted Effect of a Spatial Covariate in a Point Process Model

Description
Compute the trend or intensity of a fitted point process model as a function of one of its covariates.
Usage
effectfun(model, covname, ..., se.fit=FALSE)
Arguments
model

A fitted point process model (object of class "ppm", "kppm", "lppm", "dppm",
"rppm" or "profilepl").

covname

The name of the covariate. A character string. (Needed only if the model has
more than one covariate.)

...

The fixed values of other covariates (in the form name=value) if required.

se.fit

Logical. If TRUE, asymptotic standard errors of the estimates will be computed,
together with a 95% confidence interval.

Details
The object model should be an object of class "ppm", "kppm", "lppm", "dppm", "rppm" or "profilepl"
representing a point process model fitted to point pattern data.
The model’s trend formula should involve a spatial covariate named covname. This could be "x"
or "y" representing one of the Cartesian coordinates. More commonly the covariate is another,
external variable that was supplied when fitting the model.
The command effectfun computes the fitted trend of the point process model as a function of the
covariate named covname. The return value can be plotted immediately, giving a plot of the fitted
trend against the value of the covariate.
If the model also involves covariates other than covname, then these covariates will be held fixed.
Values for these other covariates must be provided as arguments to effectfun in the form name=value.

376

effectfun
If se.fit=TRUE, the algorithm also calculates the asymptotic standard error of the fitted trend, and
a (pointwise) asymptotic 95% confidence interval for the true trend.
This command is just a wrapper for the prediction method predict.ppm. For more complicated
computations about the fitted intensity, use predict.ppm.

Value
A data frame containing a column of values of the covariate and a column of values of the fitted
trend. If se.fit=TRUE, there are 3 additional columns containing the standard error and the upper
and lower limits of a confidence interval.
If the covariate named covname is numeric (rather than a factor or logical variable), the return value
is also of class "fv" so that it can be plotted immediately.

Trend and intensity
For a Poisson point process model, the trend is the same as the intensity of the point process. For
a more general Gibbs model, the trend is the first order potential in the model (the first order term
in the Gibbs representation). In Poisson or Gibbs models fitted by ppm, the trend is the only part of
the model that depends on the covariates.

Determinantal point process models with fixed intensity
The function dppm which fits a determinantal point process model allows the user to specify the
intensity lambda. In such cases the effect function is undefined, and effectfun stops with an error
message.

Author(s)
Adrian Baddeley  and Rolf Turner .

See Also
ppm, predict.ppm, fv.object

Examples
X <- copper$SouthPoints
D <- distfun(copper$SouthLines)
fit <- ppm(X ~ polynom(D, 5))
effectfun(fit)
plot(effectfun(fit, se.fit=TRUE))
fitx <- ppm(X ~ x + polynom(D, 5))
plot(effectfun(fitx, "D", x=20))

ellipse

377

ellipse

Elliptical Window.

Description
Create an elliptical window.
Usage
ellipse(a, b, centre=c(0,0), phi=0, ..., mask=FALSE, npoly = 128)
Arguments
a,b

The half-lengths of the axes of the ellipse.

centre

The centre of the ellipse.

phi

The (anti-clockwise) angle through which the ellipse should be rotated (about
its centre) starting from an orientation in which the axis of half-length a is horizontal.

mask

Logical value controlling the type of approximation to a perfect ellipse. See
Details.

...

Arguments passed to as.mask to determine the pixel resolution, if mask is TRUE.

npoly

The number of edges in the polygonal approximation to the ellipse.

Details
This command creates a window object representing an ellipse with the given centre and axes.
By default, the ellipse is approximated by a polygon with npoly edges.
If mask=TRUE, then the ellipse is approximated by a binary pixel mask. The resolution of the mask
is controlled by the arguments ... which are passed to as.mask.
The arguments a and b must be single positive numbers. The argument centre specifies the ellipse
centre: it can be either a numeric vector of length 2 giving the coordinates, or a list(x,y) giving
the coordinates of exactly one point, or a point pattern (object of class "ppp") containing exactly
one point.
Value
An object of class owin (either of type “polygonal” or of type “mask”) specifying an elliptical
window.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
disc, owin.object, owin, as.mask

378

Emark

Examples
W <- ellipse(a=5,b=2,centre=c(5,1),phi=pi/6)
plot(W,lwd=2,border="red")
WM <- ellipse(a=5,b=2,centre=c(5,1),phi=pi/6,mask=TRUE,dimyx=512)
plot(WM,add=TRUE,box=FALSE)

Emark

Diagnostics for random marking

Description
Estimate the summary functions E(r) and V (r) for a marked point pattern, proposed by Schlather
et al (2004) as diagnostics for dependence between the points and the marks.
Usage
Emark(X, r=NULL,
correction=c("isotropic", "Ripley", "translate"),
method="density", ..., normalise=FALSE)
Vmark(X, r=NULL,
correction=c("isotropic", "Ripley", "translate"),
method="density", ..., normalise=FALSE)
Arguments
X

The observed point pattern. An object of class "ppp" or something acceptable
to as.ppp. The pattern should have numeric marks.

r

Optional. Numeric vector. The values of the argument r at which the function
E(r) or V (r) should be evaluated. There is a sensible default.

correction

A character vector containing any selection of the options "isotropic", "Ripley"
or "translate". It specifies the edge correction(s) to be applied.

method

A character vector indicating the user’s choice of density estimation technique
to be used. Options are "density", "loess", "sm" and "smrep".

...

Arguments passed to the density estimation routine (density, loess or sm.density)
selected by method.

normalise

IfTRUE, normalise the estimate of E(r) or V (r) so that it would have value equal
to 1 if the marks are independent of the points.

Details
For a marked point process, Schlather et al (2004) defined the functions E(r) and V (r) to be the
conditional mean and conditional variance of the mark attached to a typical random point, given
that there exists another random point at a distance r away from it.
More formally,
E(r) = E0u [M (0)]
and
V (r) = E0u [(M (0) − E(u))2 ]

Emark

379

where E0u denotes the conditional expectation given that there are points of the process at the
locations 0 and u separated by a distance r, and where M (0) denotes the mark attached to the point
0.
These functions may serve as diagnostics for dependence between the points and the marks. If the
points and marks are independent, then E(r) and V (r) should be constant (not depending on r).
See Schlather et al (2004).
The argument X must be a point pattern (object of class "ppp") or any data that are acceptable to
as.ppp. It must be a marked point pattern with numeric marks.
The argument r is the vector of values for the distance r at which kf (r) is estimated.
This algorithm assumes that X can be treated as a realisation of a stationary (spatially homogeneous)
random spatial point process in the plane, observed through a bounded window. The window (which
is specified in X as Window(X)) may have arbitrary shape.
Biases due to edge effects are treated in the same manner as in Kest. The edge corrections implemented here are
isotropic/Ripley Ripley’s isotropic correction (see Ripley, 1988; Ohser, 1983). This is implemented only for rectangular and polygonal windows (not for binary masks).
translate Translation correction (Ohser, 1983). Implemented for all window geometries, but slow
for complex windows.
Note that the estimator assumes the process is stationary (spatially homogeneous).
The numerator and denominator of the mark correlation function (in the expression above) are
estimated using density estimation techniques. The user can choose between
"density" which uses the standard kernel density estimation routine density, and works only for
evenly-spaced r values;
"loess" which uses the function loess in the package modreg;
"sm" which uses the function sm.density in the package sm and is extremely slow;
"smrep" which uses the function sm.density in the package sm and is relatively fast, but may
require manual control of the smoothing parameter hmult.
Value
If marks(X) is a numeric vector, the result is an object of class "fv" (see fv.object). If marks(X)
is a data frame, the result is a list of objects of class "fv", one for each column of marks.
An object of class "fv" is essentially a data frame containing numeric columns
r

the values of the argument r at which the function E(r) or V (r) has been estimated

theo

the theoretical, constant value of E(r) or V (r) when the marks attached to different points are independent

together with a column or columns named "iso" and/or "trans", according to the selected edge
corrections. These columns contain estimates of the function E(r) or V (r) obtained by the edge
corrections named.
Author(s)
Adrian Baddeley  and Rolf Turner 

380

emend

References
Schlather, M. and Ribeiro, P. and Diggle, P. (2004) Detecting dependence between marks and locations of marked point processes. Journal of the Royal Statistical Society, series B 66 (2004) 79-83.
See Also
Mark correlation markcorr, mark variogram markvario for numeric marks.
Mark connection function markconnect and multitype K-functions Kcross, Kdot for factor-valued
marks.
Examples
plot(Emark(spruces))
E <- Emark(spruces, method="density", kernel="epanechnikov")
plot(Vmark(spruces))

emend

Force Model to be Valid

Description
Check whether a model is valid, and if not, find the nearest model which is valid.
Usage
emend(object, ...)
Arguments
object

A statistical model, belonging to some class.

...

Arguments passed to methods.

Details
The function emend is generic, and has methods for several classes of statistical models in the
spatstat package (mostly point process models). Its purpose is to check whether a given model is
valid (for example, that none of the model parameters are NA) and, if not, to find the nearest model
which is valid.
See the methods for more information.
Value
Another model of the same kind.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 

emend.ppm

381

See Also
emend.ppm, emend.lppm, valid.

emend.ppm

Force Point Process Model to be Valid

Description
Ensures that a fitted point process model satisfies the integrability conditions for existence of the
point process.
Usage
project.ppm(object, ..., fatal=FALSE, trace=FALSE)
## S3 method for class 'ppm'
emend(object, ..., fatal=FALSE, trace=FALSE)
Arguments
object

Fitted point process model (object of class "ppm").

...

Ignored.

fatal

Logical value indicating whether to generate an error if the model cannot be
projected to a valid model.

trace

Logical value indicating whether to print a trace of the decision process.

Details
The functions emend.ppm and project.ppm are identical: emend.ppm is a method for the generic
emend, while project.ppm is an older name for the same function.
The purpose of the function is to ensure that a fitted model is valid.
The model-fitting function ppm fits Gibbs point process models to point pattern data. By default, the
fitted model returned by ppm may not actually exist as a point process.
First, some of the fitted coefficients of the model may be NA or infinite values. This usually occurs
when the data are insufficient to estimate all the parameters. The model is said to be unidentifiable
or confounded.
Second, unlike a regression model, which is well-defined for any finite values of the fitted regression
coefficients, a Gibbs point process model is only well-defined if the fitted interaction parameters
satisfy some constraints. A famous example is the Strauss process (see Strauss) which exists only
when the interaction parameter γ is less than or equal to 1. For values γ > 1, the probability density
is not integrable and the process does not exist (and cannot be simulated).
By default, ppm does not enforce the constraint that a fitted Strauss process (for example) must
satisfy γ ≤ 1. This is because a fitted parameter value of γ > 1 could be useful information for data
analysis, as it indicates that the Strauss model is not appropriate, and suggests a clustered model
should be fitted.
The function emend.ppm or project.ppm modifies the model object so that the model is valid.
It identifies the terms in the model object that are associated with illegal parameter values (i.e.
parameter values which are either NA, infinite, or outside their permitted range). It considers all

382

endpoints.psp
possible sub-models of object obtained by deleting one or more of these terms. It identifies which
of these submodels are valid, and chooses the valid submodel with the largest pseudolikelihood.
The result of emend.ppm or project.ppm is the true maximum pseudolikelihood fit to the data.
For large datasets or complex models, the algorithm used in emend.ppm or project.ppm may be
time-consuming, because it takes time to compute all the sub-models. A faster, approximate algorithm can be applied by setting spatstat.options(project.fast=TRUE). This produces a valid
submodel, which may not be the maximum pseudolikelihood submodel.
Use the function valid.ppm to check whether a fitted model object specifies a well-defined point
process.
Use the expression all(is.finite(coef(object))) to determine whether all parameters are
identifiable.

Value
Another point process model (object of class "ppm").
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
ppm, valid.ppm, emend, spatstat.options
Examples
fit <- ppm(redwood, ~1, Strauss(0.1))
coef(fit)
fit2 <- emend(fit)
coef(fit2)

endpoints.psp

Endpoints of Line Segment Pattern

Description
Extracts the endpoints of each line segment in a line segment pattern.
Usage
endpoints.psp(x, which="both")
Arguments
x

A line segment pattern (object of class "psp").

which

String specifying which endpoint or endpoints should be returned. See Details.

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383

Details
This function extracts one endpoint, or both endpoints, from each of the line segments in x, and
returns these points as a point pattern object.
The argument which determines which endpoint or endpoints of each line segment should be returned:
which="both" (the default): both endpoints of each line segment are returned. The result is a point
pattern with twice as many points as there are line segments in x.
which="first" select the first endpoint of each line segment (returns the points with coordinates
x$ends$x0, x$ends$y0).
which="second" select the second endpoint of each line segment (returns the points with coordinates x$ends$x1, x$ends$y1).
which="left" select the left-most endpoint (the endpoint with the smaller x coordinate) of each
line segment.
which="right" select the right-most endpoint (the endpoint with the greater x coordinate) of each
line segment.
which="lower" select the lower endpoint (the endpoint with the smaller y coordinate) of each line
segment.
which="upper" select the upper endpoint (the endpoint with the greater y coordinate) of each line
segment.
The result is a point pattern. It also has an attribute "id" which is an integer vector identifying the
segment which contributed each point.
Value
Point pattern (object of class "ppp").
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
psp.object, ppp.object, midpoints.psp
Examples
a <- psp(runif(10), runif(10), runif(10), runif(10), window=owin())
plot(a)
b <- endpoints.psp(a, "left")
plot(b, add=TRUE)

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envelope

Simulation Envelopes of Summary Function

Description
Computes simulation envelopes of a summary function.
Usage
envelope(Y, fun, ...)
## S3 method for class 'ppp'
envelope(Y, fun=Kest, nsim=99, nrank=1, ...,
funargs=list(), funYargs=funargs,
simulate=NULL, fix.n=FALSE, fix.marks=FALSE,
verbose=TRUE, clipdata=TRUE,
transform=NULL, global=FALSE, ginterval=NULL, use.theory=NULL,
alternative=c("two.sided", "less", "greater"),
scale=NULL, clamp=FALSE,
savefuns=FALSE, savepatterns=FALSE,
nsim2=nsim, VARIANCE=FALSE, nSD=2, Yname=NULL, maxnerr=nsim,
do.pwrong=FALSE, envir.simul=NULL)
## S3 method for class 'ppm'
envelope(Y, fun=Kest, nsim=99, nrank=1, ...,
funargs=list(), funYargs=funargs,
simulate=NULL, fix.n=FALSE, fix.marks=FALSE,
verbose=TRUE, clipdata=TRUE,
start=NULL, control=update(default.rmhcontrol(Y), nrep=nrep), nrep=1e5,
transform=NULL, global=FALSE, ginterval=NULL, use.theory=NULL,
alternative=c("two.sided", "less", "greater"),
scale=NULL, clamp=FALSE,
savefuns=FALSE, savepatterns=FALSE,
nsim2=nsim, VARIANCE=FALSE, nSD=2, Yname=NULL, maxnerr=nsim,
do.pwrong=FALSE, envir.simul=NULL)
## S3 method for class 'kppm'
envelope(Y, fun=Kest, nsim=99, nrank=1, ...,
funargs=list(), funYargs=funargs,
simulate=NULL,
verbose=TRUE, clipdata=TRUE,
transform=NULL, global=FALSE, ginterval=NULL, use.theory=NULL,
alternative=c("two.sided", "less", "greater"),
scale=NULL, clamp=FALSE,
savefuns=FALSE, savepatterns=FALSE,
nsim2=nsim, VARIANCE=FALSE, nSD=2, Yname=NULL, maxnerr=nsim,
do.pwrong=FALSE, envir.simul=NULL)
Arguments
Y

Object containing point pattern data. A point pattern (object of class "ppp") or
a fitted point process model (object of class "ppm" or "kppm").

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385

fun

Function that computes the desired summary statistic for a point pattern.

nsim

Number of simulated point patterns to be generated when computing the envelopes.

nrank

Integer. Rank of the envelope value amongst the nsim simulated values. A rank
of 1 means that the minimum and maximum simulated values will be used.

...

Extra arguments passed to fun.

funargs

A list, containing extra arguments to be passed to fun.

funYargs

Optional. A list, containing extra arguments to be passed to fun when applied
to the original data Y only.

simulate

Optional. Specifies how to generate the simulated point patterns. If simulate
is an expression in the R language, then this expression will be evaluated nsim
times, to obtain nsim point patterns which are taken as the simulated patterns
from which the envelopes are computed. If simulate is a list of point patterns,
then the entries in this list will be treated as the simulated patterns from which
the envelopes are computed. Alternatively simulate may be an object produced
by the envelope command: see Details.

fix.n

Logical. If TRUE, simulated patterns will have the same number of points as the
original data pattern. This option is currently not available for envelope.kppm.

fix.marks

Logical. If TRUE, simulated patterns will have the same number of points and the
same marks as the original data pattern. In a multitype point pattern this means
that the simulated patterns will have the same number of points of each type as
the original data. This option is currently not available for envelope.kppm.

verbose

Logical flag indicating whether to print progress reports during the simulations.

clipdata

Logical flag indicating whether the data point pattern should be clipped to the
same window as the simulated patterns, before the summary function for the data
is computed. This should usually be TRUE to ensure that the data and simulations
are properly comparable.

start,control

Optional. These specify the arguments start and control of rmh, giving complete control over the simulation algorithm. Applicable only when Y is a fitted
model of class "ppm".

nrep

Number of iterations in the Metropolis-Hastings simulation algorithm. Applicable only when Y is a fitted model of class "ppm".

transform

Optional. A transformation to be applied to the function values, before the envelopes are computed. An expression object (see Details).

global

Logical flag indicating whether envelopes should be pointwise (global=FALSE)
or simultaneous (global=TRUE).

ginterval

Optional. A vector of length 2 specifying the interval of r values for the simultaneous critical envelopes. Only relevant if global=TRUE.

use.theory

Logical value indicating whether to use the theoretical value, computed by fun,
as the reference value for simultaneous envelopes. Applicable only when global=TRUE.
Default is use.theory=TRUE if Y is a point pattern, or a point process model
equivalent to Complete Spatial Randomness, and use.theory=FALSE otherwise.

alternative

Character string determining whether the envelope corresponds to a two-sided
test (side="two.sided", the default) or a one-sided test with a lower critical
boundary (side="less") or a one-sided test with an upper critical boundary
(side="greater").

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scale

clamp

savefuns
savepatterns
nsim2

VARIANCE
nSD
Yname
maxnerr

do.pwrong

envir.simul

Optional. Scaling function for global envelopes. A function in the R language
which determines the relative scale of deviations, as a function of distance r,
when computing the global envelopes. Applicable only when global=TRUE.
Summary function values for distance r will be divided by scale(r) before
the maximum deviation is computed. The resulting global envelopes will have
width proportional to scale(r).
Logical value indicating how to compute envelopes when alternative="less"
or alternative="greater". Deviations of the observed summary function
from the theoretical summary function are initially evaluated as signed real
numbers, with large positive values indicating consistency with the alternative
hypothesis. If clamp=FALSE (the default), these values are not changed. If
clamp=TRUE, any negative values are replaced by zero.
Logical flag indicating whether to save all the simulated function values.
Logical flag indicating whether to save all the simulated point patterns.
Number of extra simulated point patterns to be generated if it is necessary to
use simulation to estimate the theoretical mean of the summary function. Only
relevant when global=TRUE and the simulations are not based on CSR.
Logical. If TRUE, critical envelopes will be calculated as sample mean plus or
minus nSD times sample standard deviation.
Number of estimated standard deviations used to determine the critical envelopes,
if VARIANCE=TRUE.
Character string that should be used as the name of the data point pattern Y when
printing or plotting the results.
Maximum number of rejected patterns. If fun yields an error when applied to
a simulated point pattern (for example, because the pattern is empty and fun
requires at least one point), the pattern will be rejected and a new random point
pattern will be generated. If this happens more than maxnerr times, the algorithm will give up.
Logical. If TRUE, the algorithm will also estimate the true significance level of
the “wrong” test (the test that declares the summary function for the data to be
significant if it lies outside the pointwise critical boundary at any point). This
estimate is printed when the result is printed.
Environment in which to evaluate the expression simulate, if not the current
environment.

Details
The envelope command performs simulations and computes envelopes of a summary statistic
based on the simulations. The result is an object that can be plotted to display the envelopes.
The envelopes can be used to assess the goodness-of-fit of a point process model to point pattern
data.
For the most basic use, if you have a point pattern X and you want to test Complete Spatial Randomness (CSR), type plot(envelope(X, Kest,nsim=39)) to see the K function for X plotted together
with the envelopes of the K function for 39 simulations of CSR.
The envelope function is generic, with methods for the classes "ppp", "ppm" and "kppm" described here. There are also methods for the classes "pp3", "lpp" and "lppm" which are described
separately under envelope.pp3 and envelope.lpp. Envelopes can also be computed from other
envelopes, using envelope.envelope.
To create simulation envelopes, the command envelope(Y, ...) first generates nsim random
point patterns in one of the following ways.

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387

• If Y is a point pattern (an object of class "ppp") and simulate=NULL, then we generate nsim
simulations of Complete Spatial Randomness (i.e. nsim simulated point patterns each being
a realisation of the uniform Poisson point process) with the same intensity as the pattern
Y. (If Y is a multitype point pattern, then the simulated patterns are also given independent
random marks; the probability distribution of the random marks is determined by the relative
frequencies of marks in Y.)
• If Y is a fitted point process model (an object of class "ppm" or "kppm") and simulate=NULL,
then this routine generates nsim simulated realisations of that model.
• If simulate is supplied, then it determines how the simulated point patterns are generated. It
may be either
– an expression in the R language, typically containing a call to a random generator. This
expression will be evaluated nsim times to yield nsim point patterns. For example if
simulate=expression(runifpoint(100)) then each simulated pattern consists of exactly 100 independent uniform random points.
– a list of point patterns. The entries in this list will be taken as the simulated patterns.
– an object of class "envelope". This should have been produced by calling envelope
with the argument savepatterns=TRUE. The simulated point patterns that were saved in
this object will be extracted and used as the simulated patterns for the new envelope computation. This makes it possible to plot envelopes for two different summary functions
based on exactly the same set of simulated point patterns.
The summary statistic fun is applied to each of these simulated patterns. Typically fun is one of
the functions Kest, Gest, Fest, Jest, pcf, Kcross, Kdot, Gcross, Gdot, Jcross, Jdot, Kmulti,
Gmulti, Jmulti or Kinhom. It may also be a character string containing the name of one of these
functions.
The statistic fun can also be a user-supplied function; if so, then it must have arguments X and r
like those in the functions listed above, and it must return an object of class "fv".
Upper and lower critical envelopes are computed in one of the following ways:
pointwise: by default, envelopes are calculated pointwise (i.e. for each value of the distance argument r), by sorting the nsim simulated values, and taking the m-th lowest and m-th highest
values, where m = nrank. For example if nrank=1, the upper and lower envelopes are the
pointwise maximum and minimum of the simulated values.
The pointwise envelopes are not “confidence bands” for the true value of the function! Rather,
they specify the critical points for a Monte Carlo test (Ripley, 1981). The test is constructed
by choosing a fixed value of r, and rejecting the null hypothesis if the observed function
value lies outside the envelope at this value of r. This test has exact significance level
alpha = 2 * nrank/(1 + nsim).
simultaneous: if global=TRUE, then the envelopes are determined as follows. First we calculate
the theoretical mean value of the summary statistic (if we are testing CSR, the theoretical value
is supplied by fun; otherwise we perform a separate set of nsim2 simulations, compute the
average of all these simulated values, and take this average as an estimate of the theoretical
mean value). Then, for each simulation, we compare the simulated curve to the theoretical
curve, and compute the maximum absolute difference between them (over the interval of r
values specified by ginterval). This gives a deviation value di for each of the nsim simulations. Finally we take the m-th largest of the deviation values, where m=nrank, and call
this dcrit. Then the simultaneous envelopes are of the form lo = expected - dcrit and
hi = expected + dcrit where expected is either the theoretical mean value theo (if we
are testing CSR) or the estimated theoretical value mmean (if we are testing another model).
The simultaneous critical envelopes have constant width 2 * dcrit.

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envelope
The simultaneous critical envelopes allow us to perform a different Monte Carlo test (Ripley,
1981). The test rejects the null hypothesis if the graph of the observed function lies outside the
envelope at any value of r. This test has exact significance level alpha = nrank/(1 + nsim).
This test can also be performed using mad.test.
based on sample moments: if VARIANCE=TRUE, the algorithm calculates the (pointwise) sample
mean and sample variance of the simulated functions. Then the envelopes are computed as
mean plus or minus nSD standard deviations. These envelopes do not have an exact significance interpretation. They are a naive approximation to the critical points of the NeymanPearson test assuming the summary statistic is approximately Normally distributed.
The return value is an object of class "fv" containing the summary function for the data point
pattern, the upper and lower simulation envelopes, and the theoretical expected value (exact or estimated) of the summary function for the model being tested. It can be plotted using plot.envelope.
If VARIANCE=TRUE then the return value also includes the sample mean, sample variance and other
quantities.
Arguments can be passed to the function fun through .... This means that you simply specify
these arguments in the call to envelope, and they will be passed to fun. In particular, the argument
correction determines the edge correction to be used to calculate the summary statistic. See the
section on Edge Corrections, and the Examples.
Arguments can also be passed to the function fun through the list funargs. This mechanism is typically used if an argument of fun has the same name as an argument of envelope. The list funargs
should contain entries of the form name=value, where each name is the name of an argument of
fun.
There is also an option, rarely used, in which different function arguments are used when computing
the summary function for the data Y and for the simulated patterns. If funYargs is given, it will
be used when the summary function for the data Y is computed, while funargs will be used when
computing the summary function for the simulated patterns. This option is only needed in rare
cases: usually the basic principle requires that the data and simulated patterns must be treated
equally, so that funargs and funYargs should be identical.
If Y is a fitted cluster point process model (object of class "kppm"), and simulate=NULL, then the
model is simulated directly using simulate.kppm.
If Y is a fitted Gibbs point process model (object of class "ppm"), and simulate=NULL, then the
model is simulated by running the Metropolis-Hastings algorithm rmh. Complete control over this
algorithm is provided by the arguments start and control which are passed to rmh.
For simultaneous critical envelopes (global=TRUE) the following options are also useful:
ginterval determines the interval of r values over which the deviation between curves is calculated. It should be a numeric vector of length 2. There is a sensible default (namely, the
recommended plotting interval for fun(X), or the range of r values if r is explicitly specified).
transform specifies a transformation of the summary function fun that will be carried out before
the deviations are computed. Such transforms are useful if global=TRUE or VARIANCE=TRUE.
The transform must be an expression object using the symbol . to represent the function
value (and possibly other symbols recognised by with.fv). For example, the conventional
way to normalise the K function (Ripley, 1981) is to transform it to the L function L(r) =
p
K(r)/π and this is implemented by setting transform=expression(sqrt(./pi)).
It is also possible to extract the summary functions for each of the individual simulated point patterns, by setting savefuns=TRUE. Then the return value also has an attribute "simfuns" containing
all the summary functions for the individual simulated patterns. It is an "fv" object containing
functions named sim1, sim2, ... representing the nsim summary functions.

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389

It is also possible to save the simulated point patterns themselves, by setting savepatterns=TRUE.
Then the return value also has an attribute "simpatterns" which is a list of length nsim containing
all the simulated point patterns.
See plot.envelope and plot.fv for information about how to plot the envelopes.
Different envelopes can be recomputed from the same data using envelope.envelope. Envelopes
can be combined using pool.envelope.
Value
An object of class "envelope" and "fv", see fv.object, which can be printed and plotted directly.
Essentially a data frame containing columns
r

the vector of values of the argument r at which the summary function fun has
been estimated

obs

values of the summary function for the data point pattern

lo

lower envelope of simulations

hi

upper envelope of simulations

and either
theo

theoretical value of the summary function under CSR (Complete Spatial Randomness, a uniform Poisson point process) if the simulations were generated
according to CSR

mmean

estimated theoretical value of the summary function, computed by averaging
simulated values, if the simulations were not generated according to CSR.

Additionally, if savepatterns=TRUE, the return value has an attribute "simpatterns" which is a
list containing the nsim simulated patterns. If savefuns=TRUE, the return value has an attribute
"simfuns" which is an object of class "fv" containing the summary functions computed for each
of the nsim simulated patterns.
Errors and warnings
An error may be generated if one of the simulations produces a point pattern that is empty, or is
otherwise unacceptable to the function fun.
The upper envelope may be NA (plotted as plus or minus infinity) if some of the function values
computed for the simulated point patterns are NA. Whether this occurs will depend on the function
fun, but it usually happens when the simulated point pattern does not contain enough points to
compute a meaningful value.
Confidence intervals
Simulation envelopes do not compute confidence intervals; they generate significance bands. If you
really need a confidence interval for the true summary function of the point process, use lohboot.
See also varblock.
Edge corrections
It is common to apply a correction for edge effects when calculating a summary function such as
the K function. Typically the user has a choice between several possible edge corrections. In a
call to envelope, the user can specify the edge correction to be applied in fun, using the argument
correction. See the Examples below.

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Summary functions in spatstat Summary functions that are available in spatstat, such as Kest,
Gest and pcf, have a standard argument called correction which specifies the name of one
or more edge corrections.
The list of available edge corrections is different for each summary function, and may also
depend on the kind of window in which the point pattern is recorded. In the case of Kest (the
default and most frequently used value of fun) the best edge correction is Ripley’s isotropic
correction if the window is rectangular or polygonal, and the translation correction if the
window is a binary mask. See the help files for the individual functions for more information.
All the summary functions in spatstat recognise the option correction="best" which gives
the “best” (most accurate) available edge correction for that function.
In a call to envelope, if fun is one of the summary functions provided in spatstat, then the
default is correction="best". This means that by default, the envelope will be computed
using the “best” available edge correction.
The user can override this default by specifying the argument correction. For example the
computation can be accelerated by choosing another edge correction which is less accurate
than the “best” one, but faster to compute.
User-written summary functions If fun is a function written by the user, then envelope has to
guess what to do.
If fun has an argument called correction, or has ... arguments, then envelope assumes
that the function can handle a correction argument. To compute the envelope, fun will be
called with a correction argument. The default is correction="best", unless overridden
in the call to envelope.
Otherwise, if fun does not have an argument called correction and does not have ... arguments, then envelope assumes that the function cannot handle a correction argument. To
compute the envelope, fun is called without a correction argument.

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
References
Baddeley, A., Diggle, P.J., Hardegen, A., Lawrence, T., Milne, R.K. and Nair, G. (2014) On tests of
spatial pattern based on simulation envelopes. Ecological Monographs 84 (3) 477–489.
Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.
Diggle, P.J. Statistical analysis of spatial point patterns. Arnold, 2003.
Ripley, B.D. (1981) Spatial statistics. John Wiley and Sons.
Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.
Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical
statistics. John Wiley and Sons.
See Also
dclf.test, mad.test for envelope-based tests.
fv.object, plot.envelope, plot.fv, envelope.envelope, pool.envelope for handling envelopes. There are also methods for print and summary.
Kest, Gest, Fest, Jest, pcf, ppp, ppm, default.expand

envelope
Examples
X <- simdat
# Envelope of K function under CSR
## Not run:
plot(envelope(X))
## End(Not run)
# Translation edge correction (this is also FASTER):
## Not run:
plot(envelope(X, correction="translate"))
## End(Not run)
# Global envelopes
## Not run:
plot(envelope(X, Lest, global=TRUE))
plot(envelope(X, Kest, global=TRUE, scale=function(r) { r }))
## End(Not run)
# Envelope of K function for simulations from Gibbs model
## Not run:
fit <- ppm(cells ~1, Strauss(0.05))
plot(envelope(fit))
plot(envelope(fit), global=TRUE)
## End(Not run)
# Envelope of K function for simulations from cluster model
fit <- kppm(redwood ~1, "Thomas")
## Not run:
plot(envelope(fit, Gest))
plot(envelope(fit, Gest, global=TRUE))
## End(Not run)
# Envelope of G function under CSR
## Not run:
plot(envelope(X, Gest))
## End(Not run)
# Envelope of L function under CSR
# L(r) = sqrt(K(r)/pi)
## Not run:
E <- envelope(X, Kest)
plot(E, sqrt(./pi) ~ r)

391

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## End(Not run)
# Simultaneous critical envelope for L function
# (alternatively, use Lest)
## Not run:
plot(envelope(X, Kest, transform=expression(sqrt(./pi)), global=TRUE))
## End(Not run)
## One-sided envelope
## Not run:
plot(envelope(X, Lest, alternative="less"))
## End(Not run)
# How to pass arguments needed to compute the summary functions:
# We want envelopes for Jcross(X, "A", "B")
# where "A" and "B" are types of points in the dataset 'demopat'
data(demopat)
## Not run:
plot(envelope(demopat, Jcross, i="A", j="B"))
## End(Not run)
# Use of `simulate'
## Not run:
plot(envelope(cells, Gest, simulate=expression(runifpoint(42))))
plot(envelope(cells, Gest, simulate=expression(rMaternI(100,0.02))))
## End(Not run)
# Envelope under random toroidal shifts
data(amacrine)
## Not run:
plot(envelope(amacrine, Kcross, i="on", j="off",
simulate=expression(rshift(amacrine, radius=0.25))))
## End(Not run)
# Envelope under random shifts with erosion
## Not run:
plot(envelope(amacrine, Kcross, i="on", j="off",
simulate=expression(rshift(amacrine, radius=0.1, edge="erode"))))
## End(Not run)
# Envelope of INHOMOGENEOUS K-function with fitted trend
# The following is valid.
# Setting lambda=fit means that the fitted model is re-fitted to
# each simulated pattern to obtain the intensity estimates for Kinhom.

envelope.envelope

393

# (lambda=NULL would also be valid)
fit <- kppm(redwood ~1, clusters="MatClust")
## Not run:
plot(envelope(fit, Kinhom, lambda=fit, nsim=19))
## End(Not run)
#
#
#
#
#
#
#
#
#
#
#

Note that the principle of symmetry, essential to the validity of
simulation envelopes, requires that both the observed and
simulated patterns be subjected to the same method of intensity
estimation. In the following example it would be incorrect to set the
argument 'lambda=red.dens' in the envelope command, because this
would mean that the inhomogeneous K functions of the simulated
patterns would be computed using the intensity function estimated
from the original redwood data, violating the symmetry. There is
still a concern about the fact that the simulations are generated
from a model that was fitted to the data; this is only a problem in
small datasets.

## Not run:
red.dens <- density(redwood, sigma=bw.diggle)
plot(envelope(redwood, Kinhom, sigma=bw.diggle,
simulate=expression(rpoispp(red.dens))))
## End(Not run)
# Precomputed list of point patterns
## Not run:
nX <- npoints(X)
PatList <- list()
for(i in 1:19) PatList[[i]] <- runifpoint(nX)
E <- envelope(X, Kest, nsim=19, simulate=PatList)
## End(Not run)
# re-using the same point patterns
## Not run:
EK <- envelope(X, Kest, savepatterns=TRUE)
EG <- envelope(X, Gest, simulate=EK)
## End(Not run)

envelope.envelope

Recompute Envelopes

Description
Given a simulation envelope (object of class "envelope"), compute another envelope from the
same simulation data using different parameters.

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envelope.envelope

Usage
## S3 method for class 'envelope'
envelope(Y, fun = NULL, ...,
transform=NULL, global=FALSE, VARIANCE=FALSE)
Arguments
Y

A simulation envelope (object of class "envelope").

fun
Optional. Summary function to be applied to the simulated point patterns.
...,transform,global,VARIANCE
Parameters controlling the type of envelope that is re-computed. See envelope.
Details
This function can be used to re-compute a simulation envelope from previously simulated data,
using different parameter settings for the envelope: for example, a different significance level, or a
global envelope instead of a pointwise envelope.
The function envelope is generic. This is the method for the class "envelope".
The argument Y should be a simulation envelope (object of class "envelope") produced by any of
the methods for envelope. Additionally, Y must contain either
• the simulated point patterns that were used to create the original envelope (so Y should have
been created by calling envelope with savepatterns=TRUE);
• the summary functions of the simulated point patterns that were used to create the original
envelope (so Y should have been created by calling envelope with savefuns=TRUE).
If the argument fun is given, it should be a summary function that can be applied to the simulated
point patterns that were used to create Y. The envelope of the summary function fun for these point
patterns will be computed using the parameters specified in ....
If fun is not given, then:
• If Y contains the summary functions that were used to compute the original envelope, then the
new envelope will be computed from these original summary functions.
• Otherwise, if Y contains the simulated point patterns. then the K function Kest will be applied to each of these simulated point patterns, and the new envelope will be based on the K
functions.
The new envelope will be computed using the parameters specified in ....
See envelope for a full list of envelope parameters. Frequently-used parameters include nrank and
nsim (to change the number of simulations used and the significance level of the envelope), global
(to change from pointwise to global envelopes) and VARIANCE (to compute the envelopes from the
sample moments instead of the ranks).
Value
An envelope (object of class "envelope".
Author(s)
Adrian Baddeley 
and Rolf Turner 

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395

See Also
envelope
Examples
E <- envelope(cells, Kest, nsim=19, savefuns=TRUE, savepatterns=TRUE)
E2 <- envelope(E, nrank=2)
Eg <- envelope(E, global=TRUE)
EG <- envelope(E, Gest)
EL <- envelope(E, transform=expression(sqrt(./pi)))

envelope.lpp

Envelope for Point Patterns on Linear Network

Description
Enables envelopes to be computed for point patterns on a linear network.
Usage
## S3 method for class 'lpp'
envelope(Y, fun=linearK, nsim=99, nrank=1, ...,
funargs=list(), funYargs=funargs,
simulate=NULL, fix.n=FALSE, fix.marks=FALSE, verbose=TRUE,
transform=NULL,global=FALSE,ginterval=NULL,use.theory=NULL,
alternative=c("two.sided", "less", "greater"),
scale=NULL, clamp=FALSE,
savefuns=FALSE, savepatterns=FALSE,
nsim2=nsim, VARIANCE=FALSE, nSD=2, Yname=NULL,
do.pwrong=FALSE, envir.simul=NULL)
## S3 method for class 'lppm'
envelope(Y, fun=linearK, nsim=99, nrank=1, ...,
funargs=list(), funYargs=funargs,
simulate=NULL, fix.n=FALSE, fix.marks=FALSE, verbose=TRUE,
transform=NULL,global=FALSE,ginterval=NULL,use.theory=NULL,
alternative=c("two.sided", "less", "greater"),
scale=NULL, clamp=FALSE,
savefuns=FALSE, savepatterns=FALSE,
nsim2=nsim, VARIANCE=FALSE, nSD=2, Yname=NULL,
do.pwrong=FALSE, envir.simul=NULL)
Arguments
Y

A point pattern on a linear network (object of class "lpp") or a fitted point
process model on a linear network (object of class "lppm").

fun

Function that is to be computed for each simulated pattern.

nsim

Number of simulations to perform.

nrank

Integer. Rank of the envelope value amongst the nsim simulated values. A rank
of 1 means that the minimum and maximum simulated values will be used.

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envelope.lpp
...

Extra arguments passed to fun.

funargs

A list, containing extra arguments to be passed to fun.

funYargs

Optional. A list, containing extra arguments to be passed to fun when applied
to the original data Y only.

simulate

Optional. Specifies how to generate the simulated point patterns. If simulate
is an expression in the R language, then this expression will be evaluated nsim
times, to obtain nsim point patterns which are taken as the simulated patterns
from which the envelopes are computed. If simulate is a list of point patterns,
then the entries in this list will be treated as the simulated patterns from which
the envelopes are computed. Alternatively simulate may be an object produced
by the envelope command: see Details.

fix.n

Logical. If TRUE, simulated patterns will have the same number of points as the
original data pattern.

fix.marks

Logical. If TRUE, simulated patterns will have the same number of points and
the same marks as the original data pattern. In a multitype point pattern this
means that the simulated patterns will have the same number of points of each
type as the original data.

verbose

Logical flag indicating whether to print progress reports during the simulations.

transform

Optional. A transformation to be applied to the function values, before the envelopes are computed. An expression object (see Details).

global

Logical flag indicating whether envelopes should be pointwise (global=FALSE)
or simultaneous (global=TRUE).

ginterval

Optional. A vector of length 2 specifying the interval of r values for the simultaneous critical envelopes. Only relevant if global=TRUE.

use.theory

Logical value indicating whether to use the theoretical value, computed by fun,
as the reference value for simultaneous envelopes. Applicable only when global=TRUE.

alternative

Character string determining whether the envelope corresponds to a two-sided
test (side="two.sided", the default) or a one-sided test with a lower critical
boundary (side="less") or a one-sided test with an upper critical boundary
(side="greater").

scale

Optional. Scaling function for global envelopes. A function in the R language
which determines the relative scale of deviations, as a function of distance r,
when computing the global envelopes. Applicable only when global=TRUE.
Summary function values for distance r will be divided by scale(r) before
the maximum deviation is computed. The resulting global envelopes will have
width proportional to scale(r).

clamp

Logical value indicating how to compute envelopes when alternative="less"
or alternative="greater". Deviations of the observed summary function
from the theoretical summary function are initially evaluated as signed real
numbers, with large positive values indicating consistency with the alternative
hypothesis. If clamp=FALSE (the default), these values are not changed. If
clamp=TRUE, any negative values are replaced by zero.

savefuns

Logical flag indicating whether to save all the simulated function values.

savepatterns

Logical flag indicating whether to save all the simulated point patterns.

nsim2

Number of extra simulated point patterns to be generated if it is necessary to
use simulation to estimate the theoretical mean of the summary function. Only
relevant when global=TRUE and the simulations are not based on CSR.

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397

VARIANCE

Logical. If TRUE, critical envelopes will be calculated as sample mean plus or
minus nSD times sample standard deviation.

nSD

Number of estimated standard deviations used to determine the critical envelopes,
if VARIANCE=TRUE.

Yname

Character string that should be used as the name of the data point pattern Y when
printing or plotting the results.

do.pwrong

Logical. If TRUE, the algorithm will also estimate the true significance level of
the “wrong” test (the test that declares the summary function for the data to be
significant if it lies outside the pointwise critical boundary at any point). This
estimate is printed when the result is printed.

envir.simul

Environment in which to evaluate the expression simulate, if not the current
environment.

Details
This is a method for the generic function envelope applicable to point patterns on a linear network.
The argument Y can be either a point pattern on a linear network, or a fitted point process model on
a linear network. The function fun will be evaluated for the data and also for nsim simulated point
patterns on the same linear network. The upper and lower envelopes of these evaluated functions
will be computed as described in envelope.
The type of simulation is determined as follows.
• if Y is a point pattern (object of class "lpp") and simulate is missing or NULL, then random
point patterns will be generated according to a Poisson point process on the linear network on
which Y is defined, with intensity estimated from Y.
• if Y is a fitted point process model (object of class "lppm") and simulate is missing or NULL,
then random point patterns will be generated by simulating from the fitted model.
• If simulate is present, it should be an expression that can be evaluated to yield random point
patterns on the same linear network as Y.
The function fun should accept as its first argument a point pattern on a linear network (object of
class "lpp") and should have another argument called r or a ... argument.
Value
Function value table (object of class "fv") with additional information, as described in envelope.
Author(s)
Ang Qi Wei  and Adrian Baddeley 
References
Ang, Q.W. (2010) Statistical methodology for events on a network. Master’s thesis, School of
Mathematics and Statistics, University of Western Australia.
Ang, Q.W., Baddeley, A. and Nair, G. (2012) Geometrically corrected second-order analysis of
events on a linear network, with applications to ecology and criminology. Scandinavian Journal of
Statistics 39, 591–617.
Okabe, A. and Yamada, I. (2001) The K-function method on a network and its computational implementation. Geographical Analysis 33, 271-290.

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envelope.pp3

See Also
envelope, linearK
Examples
if(interactive()) {
ns <- 39
np <- 40
} else { ns <- np <- 3 }
X <- runiflpp(np, simplenet)
# uniform Poisson: random numbers of points
envelope(X, nsim=ns)
# uniform Poisson: conditional on observed number of points
envelope(X, fix.n=TRUE, nsim=ns)
# nonuniform Poisson
fit <- lppm(X ~x)
envelope(fit, nsim=ns)
#multitype
marks(X) <- sample(letters[1:2], np, replace=TRUE)
envelope(X, nsim=ns)

envelope.pp3

Simulation Envelopes of Summary Function for 3D Point Pattern

Description
Computes simulation envelopes of a summary function for a three-dimensional point pattern.
Usage
## S3 method for class 'pp3'
envelope(Y, fun=K3est, nsim=99, nrank=1, ...,
funargs=list(), funYargs=funargs, simulate=NULL, verbose=TRUE,
transform=NULL,global=FALSE,ginterval=NULL,use.theory=NULL,
alternative=c("two.sided", "less", "greater"),
scale=NULL, clamp=FALSE,
savefuns=FALSE, savepatterns=FALSE,
nsim2=nsim, VARIANCE=FALSE, nSD=2, Yname=NULL, maxnerr=nsim,
do.pwrong=FALSE, envir.simul=NULL)
Arguments
Y

A three-dimensional point pattern (object of class "pp3").

fun

Function that computes the desired summary statistic for a 3D point pattern.

nsim

Number of simulated point patterns to be generated when computing the envelopes.

nrank

Integer. Rank of the envelope value amongst the nsim simulated values. A rank
of 1 means that the minimum and maximum simulated values will be used.

envelope.pp3

399

...

Extra arguments passed to fun.

funargs

A list, containing extra arguments to be passed to fun.

funYargs

Optional. A list, containing extra arguments to be passed to fun when applied
to the original data Y only.

simulate

Optional. Specifies how to generate the simulated point patterns. If simulate
is an expression in the R language, then this expression will be evaluated nsim
times, to obtain nsim point patterns which are taken as the simulated patterns
from which the envelopes are computed. If simulate is a list of point patterns,
then the entries in this list will be treated as the simulated patterns from which
the envelopes are computed. Alternatively simulate may be an object produced
by the envelope command: see Details.

verbose

Logical flag indicating whether to print progress reports during the simulations.

transform

Optional. A transformation to be applied to the function values, before the envelopes are computed. An expression object (see Details).

global

Logical flag indicating whether envelopes should be pointwise (global=FALSE)
or simultaneous (global=TRUE).

ginterval

Optional. A vector of length 2 specifying the interval of r values for the simultaneous critical envelopes. Only relevant if global=TRUE.

use.theory

Logical value indicating whether to use the theoretical value, computed by fun,
as the reference value for simultaneous envelopes. Applicable only when global=TRUE.

alternative

Character string determining whether the envelope corresponds to a two-sided
test (side="two.sided", the default) or a one-sided test with a lower critical
boundary (side="less") or a one-sided test with an upper critical boundary
(side="greater").

scale

Optional. Scaling function for global envelopes. A function in the R language
which determines the relative scale of deviations, as a function of distance r,
when computing the global envelopes. Applicable only when global=TRUE.
Summary function values for distance r will be divided by scale(r) before
the maximum deviation is computed. The resulting global envelopes will have
width proportional to scale(r).

clamp

Logical value indicating how to compute envelopes when alternative="less"
or alternative="greater". Deviations of the observed summary function
from the theoretical summary function are initially evaluated as signed real
numbers, with large positive values indicating consistency with the alternative
hypothesis. If clamp=FALSE (the default), these values are not changed. If
clamp=TRUE, any negative values are replaced by zero.

savefuns

Logical flag indicating whether to save all the simulated function values.

savepatterns

Logical flag indicating whether to save all the simulated point patterns.

nsim2

Number of extra simulated point patterns to be generated if it is necessary to
use simulation to estimate the theoretical mean of the summary function. Only
relevant when global=TRUE and the simulations are not based on CSR.

VARIANCE

Logical. If TRUE, critical envelopes will be calculated as sample mean plus or
minus nSD times sample standard deviation.

nSD

Number of estimated standard deviations used to determine the critical envelopes,
if VARIANCE=TRUE.

Yname

Character string that should be used as the name of the data point pattern Y when
printing or plotting the results.

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envelope.pp3
maxnerr

do.pwrong

envir.simul

Maximum number of rejected patterns. If fun yields an error when applied to
a simulated point pattern (for example, because the pattern is empty and fun
requires at least one point), the pattern will be rejected and a new random point
pattern will be generated. If this happens more than maxnerr times, the algorithm will give up.
Logical. If TRUE, the algorithm will also estimate the true significance level of
the “wrong” test (the test that declares the summary function for the data to be
significant if it lies outside the pointwise critical boundary at any point). This
estimate is printed when the result is printed.
Environment in which to evaluate the expression simulate, if not the current
environment.

Details
The envelope command performs simulations and computes envelopes of a summary statistic
based on the simulations. The result is an object that can be plotted to display the envelopes.
The envelopes can be used to assess the goodness-of-fit of a point process model to point pattern
data.
The envelope function is generic, with methods for the classes "ppp", "ppm" and "kppm" described
in the help file for envelope. This function envelope.pp3 is the method for three-dimensional
point patterns (objects of class "pp3").
For the most basic use, if you have a 3D point pattern X and you want to test Complete Spatial
Randomness (CSR), type plot(envelope(X, K3est,nsim=39)) to see the three-dimensional K
function for X plotted together with the envelopes of the three-dimensional K function for 39 simulations of CSR.
To create simulation envelopes, the command envelope(Y, ...) first generates nsim random
point patterns in one of the following ways.
• If simulate=NULL, then we generate nsim simulations of Complete Spatial Randomness (i.e.
nsim simulated point patterns each being a realisation of the uniform Poisson point process)
with the same intensity as the pattern Y.
• If simulate is supplied, then it determines how the simulated point patterns are generated.
See envelope for details.
The summary statistic fun is applied to each of these simulated patterns. Typically fun is one of the
functions K3est, G3est, F3est or pcf3est. It may also be a character string containing the name
of one of these functions.
For further information, see the documentation for envelope.
Value
A function value table (object of class "fv") which can be plotted directly. See envelope for further
details.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Baddeley, A.J, Moyeed, R.A., Howard, C.V. and Boyde, A. (1993) Analysis of a three-dimensional
point pattern with replication. Applied Statistics 42, 641–668.

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401

See Also
pp3, rpoispp3, K3est, G3est, F3est, pcf3est.
Examples
X <- rpoispp3(20, box3())
## Not run:
plot(envelope(X, nsim=39))
## End(Not run)

envelopeArray

Array of Simulation Envelopes of Summary Function

Description
Compute an array of simulation envelopes using a summary function that returns an array of curves.
Usage
envelopeArray(X, fun, ..., dataname = NULL, verb = FALSE, reuse = TRUE)
Arguments
X

fun
...
dataname
verb
reuse

Object containing point pattern data. A point pattern (object of class "ppp",
"lpp", "pp3" or "ppx") or a fitted point process model (object of class "ppm",
"kppm" or "lppm").
Function that computes the desired summary statistic for a point pattern. The
result of fun should be a function array (object of class "fasp").
Arguments passed to envelope to control the simulations, or passed to fun
when evaluating the function.
Optional character string name for the data.
Logical value indicating whether to print progress reports.
Logical value indicating whether the envelopes in each panel should be based
on the same set of simulated patterns (reuse=TRUE, the default) or on different,
independent sets of simulated patterns (reuse=FALSE).

Details
This command is the counterpart of envelope when the function fun that is evaluated on each
simulated point pattern will return an object of class "fasp" representing an array of summary
functions.
Simulated point patterns are generated according to the rules described for envelope. In brief, if X
is a point pattern, the algorithm generates simulated point patterns of the same kind, according to
complete spatial randomness. If X is a fitted model, the algorithm generates simulated point patterns
according to this model.
For each simulated point pattern Y, the function fun is invoked. The result Z <- fun(Y, ...)
should be an object of class "fasp" representing an array of summary functions. The dimensions
of the array Z should be the same for each simulated pattern Y.
This algorithm finds the simulation envelope of the summary functions in each cell of the array.

402

eroded.areas

Value
An object of class "fasp" representing an array of envelopes.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
envelope, alltypes.
Examples
A <- envelopeArray(finpines, markcrosscorr, nsim=9)
plot(A)

eroded.areas

Areas of Morphological Erosions

Description
Computes the areas of successive morphological erosions of a window.
Usage
eroded.areas(w, r, subset=NULL)
Arguments
w

A window.

r

Numeric vector of radii at which erosions will be performed.

subset

Optional window inside which the areas should be computed.

Details
This function computes the areas of the erosions of the window w by each of the radii r[i].
The morphological erosion of a set W by a distance r > 0 is the subset consisting of all points
x ∈ W such that the distance from x to the boundary of W is greater than or equal to r. In other
words it is the result of trimming a margin of width r off the set W .
The argument r should be a vector of positive numbers. The argument w should be a window (an
object of class "owin", see owin.object for details) or can be given in any format acceptable to
as.owin().
Unless w is a rectangle, the computation is performed using a pixel raster approximation.
To compute the eroded window itself, use erosion.
Value
Numeric vector, of the same length as r, giving the areas of the successive erosions.

erosion

403

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
owin, as.owin, erosion
Examples
w <- owin(c(0,1),c(0,1))
a <- eroded.areas(w, seq(0.01,0.49,by=0.01))

erosion

Morphological Erosion by a Disc

Description
Perform morphological erosion of a window, a line segment pattern or a point pattern by a disc.
Usage
erosion(w, r, ...)
## S3 method for class 'owin'
erosion(w, r, shrink.frame=TRUE, ...,
strict=FALSE, polygonal=NULL)
## S3 method for class 'ppp'
erosion(w, r,...)
## S3 method for class 'psp'
erosion(w, r,...)
Arguments
w

A window (object of class "owin" or a line segment pattern (object of class
"psp") or a point pattern (object of class "ppp").

r

positive number: the radius of erosion.

shrink.frame

logical: if TRUE, erode the bounding rectangle as well.

...

extra arguments to as.mask controlling the pixel resolution, if pixel approximation is used.

strict

Logical flag determining the fate of boundary pixels, if pixel approximation is
used. See details.

polygonal

Logical flag indicating whether to compute a polygonal approximation to the
erosion (polygonal=TRUE) or a pixel grid approximation (polygonal=FALSE).

404

erosionAny

Details
The morphological erosion of a set W by a distance r > 0 is the subset consisting of all points
x ∈ W such that the distance from x to the boundary of W is greater than or equal to r. In other
words it is the result of trimming a margin of width r off the set W .
If polygonal=TRUE then a polygonal approximation to the erosion is computed. If polygonal=FALSE
then a pixel approximation to the erosion is computed from the distance map of w. The arguments
"..." are passed to as.mask to control the pixel resolution. The erosion consists of all pixels
whose distance from the boundary of w is strictly greater than r (if strict=TRUE) or is greater than
or equal to r (if strict=FALSE).
When w is a window, the default (when polygonal=NULL) is to compute a polygonal approximation
if w is a rectangle or polygonal window, and to compute a pixel approximation if w is a window of
type "mask".
If shrink.frame is false, the resulting window is given the same outer, bounding rectangle as the
original window w. If shrink.frame is true, the original bounding rectangle is also eroded by the
same distance r.
To simply compute the area of the eroded window, use eroded.areas.
Value
If r > 0, an object of class "owin" representing the eroded region (or NULL if this region is empty).
If r=0, the result is identical to w.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
dilation for the opposite operation.
erosionAny for morphological erosion using any shape.
owin, as.owin, eroded.areas
Examples
plot(letterR, main="erosion(letterR, 0.2)")
plot(erosion(letterR, 0.2), add=TRUE, col="red")

erosionAny

Morphological Erosion of Windows

Description
Compute the morphological erosion of one spatial window by another.
Usage
erosionAny(A, B)
A %(-)% B

eval.fasp

405

Arguments
A,B

Windows (objects of class "owin").

Details
The operator A %(-)% B and function erosionAny(A,B) are synonymous: they both compute the
morphological erosion of the window A by the window B.
The morphological erosion A B of region A by region B is the spatial region consisting of all
vectors z such that, when B is shifted by the vector z, the result is a subset of A.
Equivalently
A

B = ((Ac ⊕ (−B))c

where ⊕ is the Minkowski sum, Ac denotes the set complement, and (−B) is the reflection of B
through the origin, consisting of all vectors −b where b is a point in B.
If B is a disc of radius r, then erosionAny(A, B) is equivalent to erosion(A, r). See erosion.
The algorithm currently computes the result as a polygonal window using the polyclip library. It
will be quite slow if applied to binary mask windows.
Value
Another window (object of class "owin").
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
erosion, MinkowskiSum
Examples
B <- square(c(-0.1, 0.1))
RminusB <- letterR %(-)% B
FR <- grow.rectangle(Frame(letterR), 0.3)
plot(FR, main="", type="n")
plot(letterR, add=TRUE, lwd=2, hatch=TRUE, box=FALSE)
plot(RminusB, add=TRUE, col="blue", box=FALSE)
plot(shift(B, vec=c(3.49, 2.98)),
add=TRUE, border="red", lwd=2)

eval.fasp

Evaluate Expression Involving Function Arrays

Description
Evaluates any expression involving one or more function arrays (fasp objects) and returns another
function array.

406

eval.fasp

Usage
eval.fasp(expr, envir, dotonly=TRUE)
Arguments
expr

An expression involving the names of objects of class "fasp".

envir

Optional. The environment in which to evaluate the expression, or a named list
containing "fasp" objects to be used in the expression.

dotonly

Logical. Passed to eval.fv.

Details
This is a wrapper to make it easier to perform pointwise calculations with the arrays of summary
functions used in spatial statistics.
A function array (object of class "fasp") can be regarded as a matrix whose entries are functions.
Objects of this kind are returned by the command alltypes.
Suppose X is an object of class "fasp". Then eval.fasp(X+3) effectively adds 3 to the value of
every function in the array X, and returns the resulting object.
Suppose X and Y are two objects of class "fasp" which are compatible (for example the arrays must
have the same dimensions). Then eval.fasp(X + Y) will add the corresponding functions in each
cell of the arrays X and Y, and return the resulting array of functions.
Suppose X is an object of class "fasp" and f is an object of class "fv". Then eval.fasp(X + f)
will add the function f to the functions in each cell of the array X, and return the resulting array of
functions.
In general, expr can be any expression involving (a) the names of objects of class "fasp" or "fv",
(b) scalar constants, and (c) functions which are vectorised. See the Examples.
First eval.fasp determines which of the variable names in the expression expr refer to objects of
class "fasp". The expression is then evaluated for each cell of the array using eval.fv.
The expression expr must be vectorised. There must be at least one object of class "fasp" in the
expression. All such objects must be compatible.

Value
Another object of class "fasp".
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
fasp.object, Kest

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407

Examples
# manipulating the K function
K <- alltypes(amacrine, "K")
# expressions involving a fasp object
eval.fasp(K + 3)
L <- eval.fasp(sqrt(K/pi))
# expression involving two fasp objects
D <- eval.fasp(K - L)
# subtracting the unmarked K function from the cross-type K functions
K0 <- Kest(unmark(amacrine))
DK <- eval.fasp(K - K0)
## Use of 'envir'
S <- eval.fasp(1-G, list(G=alltypes(amacrine, "G")))

eval.fv

Evaluate Expression Involving Functions

Description
Evaluates any expression involving one or more function value (fv) objects, and returns another
object of the same kind.
Usage
eval.fv(expr, envir, dotonly=TRUE, equiv=NULL, relabel=TRUE)
Arguments
expr

An expression.

envir

Optional. The environment in which to evaluate the expression, or a named list
containing "fv" objects to be used in the expression.

dotonly

Logical. See Details.

equiv

Mapping between column names of different objects that are deemed to be
equivalent. See Details.

relabel

Logical value indicating whether to compute appropriate labels for the resulting
function. This should normally be TRUE (the default). See Details.

Details
This is a wrapper to make it easier to perform pointwise calculations with the summary functions
used in spatial statistics.
An object of class "fv" is essentially a data frame containing several different statistical estimates
of the same function. Such objects are returned by Kest and its relatives.
For example, suppose X is an object of class "fv" containing several different estimates of the
Ripley’s K function K(r), evaluated at a sequence of values of r. Then eval.fv(X+3) effectively
adds 3 to each function estimate in X, and returns the resulting object.

408

eval.fv
Suppose X and Y are two objects of class "fv" which are compatible (in particular they have the
same vector of r values). Then eval.im(X + Y) will add the corresponding function values in X
and Y, and return the resulting function.
In general, expr can be any expression involving (a) the names of objects of class "fv", (b) scalar
constants, and (c) functions which are vectorised. See the Examples.
First eval.fv determines which of the variable names in the expression expr refer to objects of
class "fv". Each such name is replaced by a vector containing the function values. The expression
is then evaluated. The result should be a vector; it is taken as the new vector of function values.
The expression expr must be vectorised. There must be at least one object of class "fv" in the
expression. If the objects are not compatible, they will be made compatible by harmonise.fv.
If dotonly=TRUE (the default), the expression will be evaluated only for those columns of an "fv"
object that contain values of the function itself (rather than values of the derivative of the function,
the hazard rate, etc). If dotonly=FALSE, the expression will be evaluated for all columns.
For example the result of Fest includes several columns containing estimates of the empty space
function F (r), but also includes an estimate of the hazard h(r) of F (r). Transformations that are
valid for F may not be valid for h. Accordingly, h would normally be omitted from the calculation.
The columns of an object x that represent the function itself are identified by its “dot” names,
fvnames(x, "."). They are the columns normally plotted by plot.fv and identified by the symbol
"." in plot formulas in plot.fv.
The argument equiv can be used to specify that two different column names in different function
objects are mathematically equivalent or cognate. It should be a list of name=value pairs, or a
named vector of character strings, indicating the pairing of equivalent names. (Without this argument, these columns would be discarded.) See the Examples.
The argument relabel should normally be TRUE (the default). It determines whether to compute
appropriate mathematical labels and descriptions for the resulting function object (used when the
object is printed or plotted). If relabel=FALSE then this does not occur, and the mathematical labels
and descriptions in the result are taken from the function object that appears first in the expression.
This reduces computation time slightly (for advanced use only).

Value
Another object of class "fv".
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
fv.object, Kest
Examples
# manipulating the K function
X <- rpoispp(42)
Ks <- Kest(X)
eval.fv(Ks + 3)
Ls <- eval.fv(sqrt(Ks/pi))

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409

# manipulating two K functions
Y <- rpoispp(20)
Kr <- Kest(Y)
Kdif <- eval.fv(Ks - Kr)
Z <- eval.fv(sqrt(Ks/pi) - sqrt(Kr/pi))
## Use of 'envir'
U <- eval.fv(sqrt(K), list(K=Kest(cells)))
## Use of 'equiv'
Fc <- Fest(cells)
Gc <- Gest(cells)
# Hanisch and Chiu-Stoyan estimators are cognate
Dc <- eval.fv(Fc - Gc, equiv=list(cs="han"))

eval.im

Evaluate Expression Involving Pixel Images

Description
Evaluates any expression involving one or more pixel images, and returns a pixel image.
Usage
eval.im(expr, envir, harmonize=TRUE)
Arguments
expr

An expression.

envir

Optional. The environment in which to evaluate the expression, or a named list
containing pixel images to be used in the expression.

harmonize

Logical. Whether to resolve inconsistencies between the pixel grids.

Details
This function is a wrapper to make it easier to perform pixel-by-pixel calculations in an image.
Pixel images in spatstat are represented by objects of class "im" (see im.object). These are
essentially matrices of pixel values, with extra attributes recording the pixel dimensions, etc.
Suppose X is a pixel image. Then eval.im(X+3) will add 3 to the value of every pixel in X, and
return the resulting pixel image.
Suppose X and Y are two pixel images with compatible dimensions: they have the same number of
pixels, the same physical size of pixels, and the same bounding box. Then eval.im(X + Y) will
add the corresponding pixel values in X and Y, and return the resulting pixel image.
In general, expr can be any expression in the R language involving (a) the names of pixel images,
(b) scalar constants, and (c) functions which are vectorised. See the Examples.
First eval.im determines which of the variable names in the expression expr refer to pixel images. Each such name is replaced by a matrix containing the pixel values. The expression is then
evaluated. The result should be a matrix; it is taken as the matrix of pixel values.
The expression expr must be vectorised. There must be at least one pixel image in the expression.

410

eval.linim
All images must have compatible dimensions. If harmonize=TRUE, images that have incompatible
dimensions will be resampled so that they are compatible. If harmonize=FALSE, images that are
incompatible will cause an error.

Value
An image object of class "im".
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
as.im, compatible.im, harmonise.im, im.object
Examples
# test images
X <- as.im(function(x,y) { x^2 - y^2 }, unit.square())
Y <- as.im(function(x,y) { 3 * x + y }, unit.square())
eval.im(X + 3)
eval.im(X - Y)
eval.im(abs(X - Y))
Z <- eval.im(sin(X * pi) + Y)
## Use of 'envir'
W <- eval.im(sin(U), list(U=density(cells)))

eval.linim

Evaluate Expression Involving Pixel Images on Linear Network

Description
Evaluates any expression involving one or more pixel images on a linear network, and returns a
pixel image on the same linear network.
Usage
eval.linim(expr, envir, harmonize=TRUE)
Arguments
expr

An expression in the R language, involving the names of objects of class "linim".

envir

Optional. The environment in which to evaluate the expression.

harmonize

Logical. Whether to resolve inconsistencies between the pixel grids.

eval.linim

411

Details
This function a wrapper to make it easier to perform pixel-by-pixel calculations. It is one of several
functions whose names begin with eval which work on objects of different types. This particular
function is designed to work with objects of class "linim" which represent pixel images on a linear
network.
Suppose X is a pixel image on a linear network (object of class "linim". Then eval.linim(X+3)
will add 3 to the value of every pixel in X, and return the resulting pixel image on the same linear
network.
Suppose X and Y are two pixel images on the same linear network, with compatible pixel dimensions.
Then eval.linim(X + Y) will add the corresponding pixel values in X and Y, and return the
resulting pixel image on the same linear network.
In general, expr can be any expression in the R language involving (a) the names of pixel images,
(b) scalar constants, and (c) functions which are vectorised. See the Examples.
First eval.linim determines which of the variable names in the expression expr refer to pixel
images. Each such name is replaced by a matrix containing the pixel values. The expression is then
evaluated. The result should be a matrix; it is taken as the matrix of pixel values.
The expression expr must be vectorised. There must be at least one linear pixel image in the
expression.
All images must have compatible dimensions. If harmonize=TRUE, images that have incompatible
dimensions will be resampled so that they are compatible. If harmonize=FALSE, images that are
incompatible will cause an error.
Value
An image object of class "linim".
Author(s)
Adrian Baddeley  and Rolf Turner 
See Also
eval.im, linim
Examples
M <- as.mask.psp(as.psp(simplenet))
Z <- as.im(function(x,y) {x-y}, W=M)
X <- linim(simplenet, Z)
X
Y <- linfun(function(x,y,seg,tp){y^2+x}, simplenet)
Y <- as.linim(Y)
eval.linim(X + 3)
eval.linim(X - Y)
eval.linim(abs(X - Y))
Z <- eval.linim(sin(X * pi) + Y)

412

ewcdf

ewcdf

Weighted Empirical Cumulative Distribution Function

Description
Compute a weighted version of the empirical cumulative distribution function.
Usage
ewcdf(x, weights = rep(1/length(x), length(x)))
Arguments
x

Numeric vector of observations.

weights

Numeric vector of non-negative weights for x.

Details
This is a modification of the standard function ecdf allowing the observations x to have weights.
The weighted e.c.d.f. (empirical cumulative distribution function) Fn is defined so that, for any real
number y, the value of Fn(y) is equal to the total weight of all entries of x that are less than or equal
to y. That is Fn(y) = sum(weights[x <= y]).
Thus Fn is a step function which jumps at the values of x. The height of the jump at a point y is the
total weight of all entries in x number of tied observations at that value. Missing values are ignored.
If weights is omitted, the default is equivalent to ecdf(x) except for the class membership.
The result of ewcdf is a function, of class "ewcdf", inheriting from the classes "ecdf" and "stepfun".
The class ewcdf has methods for print and quantile. The inherited class ecdf has methods for
plot and summary.
Value
A function, of class "ewcdf", inheriting from "ecdf" and "stepfun".
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
ecdf.
quantile.ewcdf
Examples
x <- rnorm(100)
w <- runif(100)
plot(ewcdf(x,w))

exactMPLEstrauss

exactMPLEstrauss

413

Exact Maximum Pseudolikelihood Estimate for Stationary Strauss
Process

Description
Computes, to very high accuracy, the Maximum Pseudolikelihood Estimates of the parameters of a
stationary Strauss point process.
Usage
exactMPLEstrauss(X, R, ngrid = 2048, plotit = FALSE, project=TRUE)
Arguments
X

Data to which the Strauss process will be fitted. A point pattern dataset (object
of class "ppp").

R

Interaction radius of the Strauss process. A non-negative number.

ngrid

Grid size for calculation of integrals. An integer, giving the number of grid
points in the x and y directions.

plotit

Logical. If TRUE, the log pseudolikelihood is plotted on the current device.

project

Logical. If TRUE (the default), the parameter γ is constrained to lie in the interval
[0, 1]. If FALSE, this constraint is not applied.

Details
This function is intended mainly for technical investigation of algorithm performance. Its practical
use is quite limited.
It fits the stationary Strauss point process model to the point pattern dataset X by maximum pseudolikelihood (with the border edge correction) using an algorithm with very high accuracy. This
algorithm is more accurate than the default behaviour of the model-fitting function ppm because the
discretisation is much finer.
Ripley (1988) and Baddeley and Turner (2000) derived the log pseudolikelihood for the stationary
Strauss process, and eliminated the parameter β, obtaining an exact formula for the partial log
pseudolikelihood as a function of the interaction parameter γ only. The algorithm evaluates this
expression to a high degree of accuracy, using numerical integration on a ngrid * ngrid lattice,
uses optim to maximise the log pseudolikelihood with respect to γ, and finally recovers β.
The result is a vector of length 2, containing the fitted coefficients log β and log γ. These values
correspond to the entries that would be obtained with coef(ppm(X, ~1, Strauss(R))). The fitted
coefficients are typically accurate to within 10−6 as shown in Baddeley and Turner (2013).
Note however that (by default) exactMPLEstrauss constrains the parameter γ to lie in the interval [0, 1] in which the point process is well defined (Kelly and Ripley, 1976) whereas ppm does
not constrain the value of γ (by default). This behaviour is controlled by the argument project
to ppm and exactMPLEstrauss. The default for ppm is project=FALSE, while the default for
exactMPLEstrauss is project=TRUE.
Value
Vector of length 2.

414

expand.owin

Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns.
Australian and New Zealand Journal of Statistics 42, 283–322.
Baddeley, A. and Turner, R. (2013) Bias correction for parameter estimates of spatial point process
models. Journal of Statistical Computation and Simulation 2012. doi: 10.1080/00949655.2012.755976
Kelly, F.P. and Ripley, B.D. (1976) On Strauss’s model for clustering. Biometrika 63, 357–360.
Ripley, B.D. (1988) Statistical inference for spatial processes. Cambridge University Press.
See Also
ppm
Examples
if(interactive()) {
exactMPLEstrauss(cells, 0.1)
coef(ppm(cells, ~1, Strauss(0.1)))
coef(ppm(cells, ~1, Strauss(0.1), nd=128))
exactMPLEstrauss(redwood, 0.04)
exactMPLEstrauss(redwood, 0.04, project=FALSE)
coef(ppm(redwood, ~1, Strauss(0.04)))
}

expand.owin

Apply Expansion Rule

Description
Applies an expansion rule to a window.
Usage
expand.owin(W, ...)
Arguments
W

A window.

...

Arguments passed to rmhexpand to determine an expansion rule.

Extract.anylist

415

Details
The argument W should be a window (an object of class "owin").
This command applies the expansion rule specified by the arguments ... to the window W, yielding
another window.
The arguments ... are passed to rmhexpand to determine the expansion rule.
For other transformations of the scale, location and orientation of a window, see shift, affine and
rotate.
Value
A window (object of class "owin").
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
rmhexpand about expansion rules.
shift, rotate, affine for other types of manipulation.
Examples
expand.owin(square(1), 9)
expand.owin(square(1), distance=0.5)
expand.owin(letterR, length=2)
expand.owin(letterR, distance=0.1)

Extract.anylist

Extract or Replace Subset of a List of Things

Description
Extract or replace a subset of a list of things.
Usage
## S3 method for class 'anylist'
x[i, ...]
## S3 replacement method for class 'anylist'
x[i] <- value
Arguments
x

An object of class "anylist" representing a list of things.

i

Subset index. Any valid subset index in the usual R sense.

value

Replacement value for the subset.

...

Ignored.

416

Extract.fasp

Details
These are the methods for extracting and replacing subsets for the class "anylist".
The argument x should be an object of class "anylist" representing a list of things. See anylist.
The method replaces a designated subset of x, and returns an object of class "anylist".
Value
Another object of class "anylist".
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
anylist, plot.anylist, summary.anylist
Examples
x <- anylist(A=runif(10), B=runif(10), C=runif(10))
x[1] <- list(A=rnorm(10))

Extract.fasp

Extract Subset of Function Array

Description
Extract a subset of a function array (an object of class "fasp").
Usage
## S3 method for class 'fasp'
x[I, J, drop=TRUE,...]
Arguments
x

A function array. An object of class "fasp".

I

any valid expression for a subset of the row indices of the array.

J

any valid expression for a subset of the column indices of the array.

drop

Logical. When the selected subset consists of only one cell of the array, if
drop=FALSE the result is still returned as a 1×1 array of functions (class "fasp")
while if drop=TRUE it is returned as a function (class "fv").

...

Ignored.

Details
A function array can be regarded as a matrix whose entries are functions. See fasp.object for an
explanation of function arrays.
This routine extracts a sub-array according to the usual conventions for matrix indexing.

Extract.fv

417

Value
A function array (of class "fasp"). Exceptionally, if the array has only one cell, and if drop=TRUE,
then the result is a function value table (class "fv").
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
fasp.object
Examples
# Lansing woods data - multitype points with 6 types
woods <- lansing
# compute 6 x 6 array of all cross-type K functions
a <- alltypes(woods, "K")
# extract first three marks only
b <- a[1:3,1:3]
## Not run: plot(b)
# subset of array pertaining to hickories
h <- a[levels(marks(woods)) == "hickory", ]
## Not run: plot(h)

Extract.fv

Extract or Replace Subset of Function Values

Description
Extract or replace a subset of an object of class "fv".
Usage
## S3 method for class 'fv'
x[i, j, ..., drop=FALSE]
## S3 replacement method for class 'fv'
x[i, j] <- value
## S3 replacement method for class 'fv'
x$name <- value
Arguments
x

a function value object, of class "fv" (see fv.object). Essentially a data frame.

i

any appropriate subset index. Selects a subset of the rows of the data frame, i.e.
a subset of the domain of the function(s) represented by x.

418

Extract.fv
j
name
...
drop

value

any appropriate subset index for the columns of the data frame. Selects some of
the functions present in x.
the name of a column of the data frame.
Ignored.
Logical. If TRUE, the result is a data frame or vector containing the selected rows
and columns of data. If FALSE (the default), the result is another object of class
"fv".
Replacement value for the column or columns selected by name or j.

Details
These functions extract a designated subset of an object of class "fv", or replace the designated
subset with other data, or delete the designated subset.
The subset is specified by the row index i and column index j, or by the column name name. Either
i or j may be missing, or both may be missing.
The function [.fv is a method for the generic operator [ for the class "fv". It extracts the designated
subset of x, and returns it as another object of class "fv" (if drop=FALSE) or as a data frame or vector
(if drop=TRUE).
The function [<-.fv is a method for the generic operator [<- for the class "fv". If value is NULL,
the designated subset of x will be deleted from x. Otherwise, the designated subset of x will be
replaced by the data contained in value. The return value is the modified object x.
The function $<-.fv is a method for the generic operator $<- for the class "fv". If value is NULL,
the designated column of x will be deleted from x. Otherwise, the designated column of x will be
replaced by the data contained in value. The return value is the modified object x.
Value
The result of [.fv with drop=TRUE is a data frame or vector.
Otherwise, the result is another object of class "fv".
Author(s)
Adrian Baddeley  and Rolf Turner 
See Also
fv.object
Examples
K <- Kest(cells)
# discard the estimates of K(r) for r
Ksub <- K[K$r <= 0.1, ]
# extract the border method estimates
bor <- K[ , "border", drop=TRUE]
# or equivalently
bor <- K$border
# remove the border-method estimates
K$border <- NULL
K

> 0.1

Extract.hyperframe

419

Extract.hyperframe

Extract or Replace Subset of Hyperframe

Description
Extract or replace a subset of a hyperframe.
Usage
## S3 method for class 'hyperframe'
x[i, j, drop, strip=drop, ...]
## S3 replacement method for class 'hyperframe'
x[i, j] <- value
## S3 method for class 'hyperframe'
x$name
## S3 replacement method for class 'hyperframe'
x$name <- value
Arguments
x

A hyperframe (object of class "hyperframe").

i,j

Row and column indices.

drop,strip

Logical values indicating what to do when the hyperframe has only one row or
column. See Details.

...

Ignored.

name

Name of a column of the hyperframe.

value

Replacement value for the subset. A hyperframe or (if the subset is a single
column) a list or an atomic vector.

Details
These functions extract a designated subset of a hyperframe, or replace the designated subset with
another hyperframe.
The function [.hyperframe is a method for the subset operator [ for the class "hyperframe". It
extracts the subset of x specified by the row index i and column index j.
The argument drop determines whether the array structure will be discarded if possible. The argument strip determines whether the list structure in a row or column or cell will be discarded
if possible. If drop=FALSE (the default), the return value is always a hyperframe or data frame. If
drop=TRUE, and if the selected subset has only one row, or only one column, or both, then
• if strip=FALSE, the result is a list, with one entry for each array cell that was selected.
• if strip=TRUE,
– if the subset has one row containing several columns, the result is a list or (if possible) an
atomic vector;
– if the subset has one column containing several rows, the result is a list or (if possible) an
atomic vector;
– if the subset has exactly one row and exactly one column, the result is the object (or
atomic value) contained in this row and column.

420

Extract.im
The function [<-.hyperframe is a method for the subset replacement operator [<- for the class
"hyperframe". It replaces the designated subset with the hyperframe value. The subset of x to
be replaced is designated by the arguments i and j as above. The replacement value should be a
hyperframe with the appropriate dimensions, or (if the specified subset is a single column) a list of
the appropriate length.
The function $.hyperframe is a method for $ for hyperframes. It extracts the relevant column of the
hyperframe. The result is always a list (i.e. equivalent to using [.hyperframe with strip=FALSE).
The function $<-.hyperframe is a method for $<- for hyperframes. It replaces the relevant column
of the hyperframe. The replacement value should be a list of the appropriate length.

Value
A hyperframe (of class "hyperframe").
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
hyperframe
Examples
h <- hyperframe(X=list(square(1), square(2)), Y=list(sin, cos))
h
h[1, ]
h[1, ,drop=TRUE]
h[ , 1]
h[ , 1, drop=TRUE]
h[1,1]
h[1,1,drop=TRUE]
h[1,1,drop=TRUE,strip=FALSE]
h[1,1] <- list(square(3))
# extract column
h$X
# replace existing column
h$Y <- list(cells, cells)
# add new column
h$Z <- list(cells, cells)

Extract.im

Extract Subset of Image

Description
Extract a subset or subregion of a pixel image.
Usage
## S3 method for class 'im'
x[i, j, ..., drop=TRUE, tight=FALSE,
raster=NULL, rescue=is.owin(i)]

Extract.im

421

Arguments
x

A two-dimensional pixel image. An object of class "im".

i

Object defining the subregion or subset to be extracted. Either a spatial window
(an object of class "owin"), or a pixel image with logical values, or a linear
network (object of class "linnet") or a point pattern (an object of class "ppp"),
or any type of index that applies to a matrix, or something that can be converted
to a point pattern by as.ppp (using the window of x).

j

An integer or logical vector serving as the column index if matrix indexing is
being used. Ignored if i is a spatial object.

...

Ignored.

drop

Logical value. Locations in w that lie outside the spatial domain of the image x
return a pixel value of NA if drop=FALSE, and are omitted if drop=TRUE.

tight

Logical value. If tight=TRUE, and if the result of the subset operation is an
image, the image will be trimmed to the smallest possible rectangle.

raster

Optional. An object of class "owin" or "im" determining a pixel grid.

rescue

Logical value indicating whether rectangular blocks of data should always be
returned as pixel images.

Details
This function extracts a subset of the pixel values in a pixel image. (To reassign the pixel values,
see [<-.im).
The image x must be an object of class "im" representing a pixel image defined inside a rectangle
in two-dimensional space (see im.object).
The subset to be extracted is determined by the arguments i,j according to the following rules
(which are checked in this order):
1. i is a spatial object such as a window, a pixel image with logical values, a linear network, or
a point pattern; or
2. i,j are indices for the matrix as.matrix(x); or
3. i can be converted to a point pattern by as.ppp(i, W=Window(x)), and i is not a matrix.
If i is a spatial window (an object of class "owin"), the values of the image inside this window are
extracted (after first clipping the window to the spatial domain of the image if necessary).
If i is a linear network (object of class "linnet"), the values of the image on this network are
extracted.
If i is a pixel image with logical values, it is interpreted as a spatial window (with TRUE values
inside the window and FALSE outside).
If i is a point pattern (an object of class "ppp"), then the values of the pixel image at the points of
this pattern are extracted. This is a simple way to read the pixel values at a given spatial location.
At locations outside the spatial domain of the image, the pixel value is undefined, and is taken to be
NA. The logical argument drop determines whether such NA values will be returned or omitted. It
also influences the format of the return value.
If i is a point pattern (or something that can be converted to a point pattern), then X[i, drop=FALSE]
is a numeric vector containing the pixel values at each of the points of the pattern. Its length is equal
to the number of points in the pattern i. It may contain NAs corresponding to points which lie outside the spatial domain of the image x. By contrast, X[i] or X[i, drop=TRUE] contains only those
pixel values which are not NA. It may be shorter.

422

Extract.im
If i is a spatial window then X[i, drop=FALSE] is another pixel image of the same dimensions as
X obtained by setting all pixels outside the window i to have value NA. When the result is displayed
by plot.im the effect is that the pixel image x is clipped to the window i.
If i is a linear network (object of class "linnet") then X[i, drop=FALSE] is another pixel image
of the same dimensions as X obtained by restricting the pixel image X to the linear network. The
result also belongs to the class "linim" (pixel image on a linear network).
If i is a spatial window then X[i, drop=TRUE] is either:
• a numeric vector containing the pixel values for all pixels that lie inside the window i. This
happens if i is not a rectangle (i.e. i$type != "rectangle") or if rescue=FALSE.
• a pixel image. This happens only if i is a rectangle (i$type = "rectangle") and rescue=TRUE
(the default).
If the optional argument raster is given, then it should be a binary image mask or a pixel image.
Then x will first be converted to an image defined on the pixel grid implied by raster, before the
subset operation is carried out. In particular, x[i, raster=i, drop=FALSE] will return an image
defined on the same pixel array as the object i.
If i does not satisfy any of the conditions above, then the algorithm attempts to interpret i and j as
indices for the matrix as.matrix(x). Either i or j may be missing or blank. The result is usually
a vector or matrix of pixel values. Exceptionally the result is a pixel image if i,j determines a
rectangular subset of the pixel grid, and if the user specifies rescue=TRUE.
Finally, if none of the above conditions is met, the object i may also be a data frame or list of
x,y coordinates which will be converted to a point pattern, taking the observation window to be
Window(x). Then the pixel values at these points will be extracted as a vector.

Value
Either a pixel image or a vector of pixel values. See Details.
Warnings
If you have a 2-column matrix containing the x, y coordinates of point locations, then to prevent
this being interpreted as an array index, you should convert it to a data.frame or to a point pattern.
If W is a window or a pixel image, then x[W, drop=FALSE] will return an image defined on the
same pixel array as the original image x. If you want to obtain an image whose pixel dimensions
agree with those of W, use the raster argument, x[W, raster=W, drop=FALSE].
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
im.object, [<-.im, ppp.object, as.ppp, owin.object, plot.im
Examples
# make up an image
X <- setcov(unit.square())
plot(X)
# a rectangular subset

Extract.influence.ppm

423

W <- owin(c(0,0.5),c(0.2,0.8))
Y <- X[W]
plot(Y)
# a polygonal subset
R <- affine(letterR, diag(c(1,1)/2), c(-2,-0.7))
plot(X[R, drop=FALSE])
plot(X[R, drop=FALSE, tight=TRUE])
# a point pattern
P <- rpoispp(20)
Y <- X[P]
# look up a specified location
X[list(x=0.1,y=0.2)]
# 10 x 10 pixel array
X <- as.im(function(x,y) { x + y }, owin(c(-1,1),c(-1,1)), dimyx=10)
# 100 x 100
W <- as.mask(disc(1, c(0,0)), dimyx=100)
# 10 x 10 raster
X[W,drop=FALSE]
# 100 x 100 raster
X[W, raster=W, drop=FALSE]

Extract.influence.ppm Extract Subset of Influence Object

Description
Extract a subset of an influence object, or extract the influence values at specified locations.
Usage
## S3 method for class 'influence.ppm'
x[i, ...]
Arguments
x

A influence object (of class "influence.ppm") computed by influence.ppm.

i

Subset index (passed to [.ppp). Either a spatial window (object of class "owin")
or an integer index.

...

Ignored.

Details
An object of class "influence.ppm" contains the values of the likelihood influence for a point
process model, computed by influence.ppm. This is effectively a marked point pattern obtained
by marking each of the original data points with its likelihood influence.
This function extracts a designated subset of the influence values, either as another influence object,
or as a vector of numeric values.

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Extract.layered
The function [.influence.ppm is a method for [ for the class "influence.ppm". The argument i
should be an index applicable to a point pattern. It may be either a spatial window (object of class
"owin") or a sequence index. The result will be another influence object (of class influence.ppm).
To extract the influence values as a numeric vector, use marks(as.ppp(x)).

Value
Another object of class "influence.ppm".
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
influence.ppm.
Examples
fit <- ppm(cells, ~x)
infl <- influence(fit)
b <- owin(c(0.1, 0.3), c(0.2, 0.4))
infl[b]
infl[1:5]
marks(as.ppp(infl))[1:3]

Extract.layered

Extract or Replace Subset of a Layered Object

Description
Extract or replace some or all of the layers of a layered object, or extract a spatial subset of each
layer.
Usage
## S3 method for class 'layered'
x[i, j, drop=FALSE, ...]
## S3 replacement method for class 'layered'
x[i] <- value
## S3 replacement method for class 'layered'
x[[i]] <- value

Extract.layered

425

Arguments
x

A layered object (class "layered").

i

Subset index for the list of layers. A logical vector, integer vector or character
vector specifying which layers are to be extracted or replaced.

j

Subset index to be applied to the data in each layer. Typically a spatial window
(class "owin").

drop

Logical. If i specifies only a single layer and drop=TRUE, then the contents of
this layer will be returned.

...

Additional arguments, passed to other subset methods if the subset index is a
window.

value

List of objects which shall replace the designated subset, or an object which
shall replace the designated element.

Details
A layered object represents data that should be plotted in successive layers, for example, a background and a foreground. See layered.
The function [.layered extracts a designated subset of a layered object. It is a method for [ for
the class "layered".
The functions [<-.layered and [[<-.layered replace a designated subset or designated entry of
the object by new values. They are methods for [<- and [[<- for the "layered" class.
The index i specifies which layers will be retained. It should be a valid subset index for the list of
layers.
The index j will be applied to each layer. It is typically a spatial window (class "owin") so that
each of the layers will be restricted to the same spatial region. Alternatively j may be any subset
index which is permissible for the "[" method for each of the layers.
Value
Usually an object of class "layered".
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
layered
Examples
D <- distmap(cells)
L <- layered(D, cells,
plotargs=list(list(ribbon=FALSE), list(pch=16)))
L[-2]
L[, square(0.5)]
L[[3]] <- japanesepines
L

426

Extract.leverage.ppm

Extract.leverage.ppm

Extract Subset of Leverage Object

Description
Extract a subset of a leverage map, or extract the leverage values at specified locations.
Usage
## S3 method for class 'leverage.ppm'
x[i, ..., update=TRUE]
Arguments
x

A leverage object (of class "leverage.ppm") computed by leverage.ppm.

i

Subset index (passed to [.im). Either a spatial window (object of class "owin")
or a spatial point pattern (object of class "ppp").

...

Further arguments passed to [.im, especially the argument drop.

update

Logical value indicating whether to update the internally-stored value of the
mean leverage, by averaging over the specified subset.

Details
An object of class "leverage.ppm" contains the values of the leverage function for a point process
model, computed by leverage.ppm.
This function extracts a designated subset of the leverage values, either as another leverage object,
or as a vector of numeric values.
The function [.leverage.ppm is a method for [ for the class "leverage.ppm". The argument i
should be either
• a spatial window (object of class "owin") determining a region where the leverage map is
required. The result will typically be another leverage map (object of class leverage.ppm).
• a spatial point pattern (object of class "ppp") specifying locations at which the leverage values
are required. The result will be a numeric vector.
The subset operator for images, [.im, is applied to the leverage map. If this yields a pixel image,
then the result of [.leverage.ppm is another leverage object. Otherwise, a vector containing the
numeric values of leverage is returned.
Value
Another object of class "leverage.ppm", or a vector of numeric values of leverage.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
leverage.ppm.

Extract.linim

427

Examples
fit <- ppm(cells ~x)
lev <- leverage(fit)
b <- owin(c(0.1, 0.3), c(0.2, 0.4))
lev[b]
lev[cells]

Extract.linim

Extract Subset of Pixel Image on Linear Network

Description
Extract a subset of a pixel image on a linear network.
Usage
## S3 method for class 'linim'
x[i, ..., drop=TRUE]
Arguments
x

A pixel image on a linear network (object of class "linim").

i

Spatial window defining the subregion. Either a spatial window (an object of
class "owin"), or a logical-valued pixel image, or any type of index that applies
to a matrix, or a point pattern (an object of class "lpp" or "ppp"), or something
that can be converted to a point pattern by as.lpp (using the network on which
x is defined).

...

Additional arguments passed to [.im.

drop

Logical value indicating whether NA values should be omitted from the result.

Details
This function is a method for the subset operator "[" for pixel images on linear networks (objects
of class "linim").
The pixel image x will be restricted to the domain specified by i.
Pixels outside the domain of x are assigned the value NA; if drop=TRUE (the default) such NA values
are deleted from the result; if drop=FALSE, then NA values are retained.
If i is a window (or a logical-valued pixel image) then x[i] is another pixel image of class "linim",
representing the restriction of x to the spatial domain specified by i.
If i is a point pattern, then x[i] is the vector of pixel values of x at the locations specified by i.
Value
Another pixel image on a linear network (object of class "linim") or a vector of pixel values.
Author(s)
Adrian Baddeley 

428

Extract.linnet

Examples
M <- as.mask.psp(as.psp(simplenet))
Z <- as.im(function(x,y){x}, W=M)
Y <- linim(simplenet, Z)
X <- runiflpp(4, simplenet)
Y[X]
Y[square(c(0.3, 0.6))]

Extract.linnet

Extract Subset of Linear Network

Description
Extract a subset of a linear network.
Usage
## S3 method for class 'linnet'
x[i, ..., snip=TRUE]
Arguments
x

A linear network (object of class "linnet").

i

Spatial window defining the subregion. An object of class "owin".

snip

Logical. If TRUE (the default), segments of x which cross the boundary of i will
be cut by the boundary. If FALSE, these segments will be deleted.

...

Ignored.

Details
This function computes the intersection between the linear network x and the domain specified by
i.
This function is a method for the subset operator "[" for linear networks (objects of class "linnet").
It is provided mainly for completeness.
The index i should be a window.
The argument snip specifies what to do with segments of x which cross the boundary of i. If
snip=FALSE, such segments are simply deleted. If snip=TRUE (the default), such segments are cut
into pieces by the boundary of i, and those pieces which lie inside the window i are included in the
resulting network.
Value
Another linear network (object of class "linnet").
Author(s)
Adrian Baddeley , Rolf Turner ,
Ege Rubak  and Suman Rakshit.

Extract.listof

429

Examples
p <- par(mfrow=c(1,2), mar=0.2+c(0,0,1,0))
B <- owin(c(0.1,0.7),c(0.19,0.6))
plot(simplenet, main="x[w, snip=TRUE]")
plot(simplenet[B], add=TRUE, col="green", lwd=3)
plot(B, add=TRUE, border="red", lty=3)
plot(simplenet, main="x[w, snip=FALSE]")
plot(simplenet[B, snip=FALSE], add=TRUE, col="green", lwd=3)
plot(B, add=TRUE, border="red", lty=3)
par(p)

Extract.listof

Extract or Replace Subset of a List of Things

Description
Replace a subset of a list of things.
Usage
## S3 replacement method for class 'listof'
x[i] <- value
Arguments
x

An object of class "listof" representing a list of things which all belong to one
class.

i

Subset index. Any valid subset index in the usual R sense.

value

Replacement value for the subset.

Details
This is a subset replacement method for the class "listof".
The argument x should be an object of class "listof" representing a list of things that all belong
to one class.
The method replaces a designated subset of x, and returns an object of class "listof".
Value
Another object of class "listof".
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
plot.listof, summary.listof

430

Extract.lpp

Examples
x <- list(A=runif(10), B=runif(10), C=runif(10))
class(x) <- c("listof", class(x))
x[1] <- list(A=rnorm(10))

Extract.lpp

Extract Subset of Point Pattern on Linear Network

Description
Extract a subset of a point pattern on a linear network.
Usage
## S3 method for class 'lpp'
x[i, j, drop=FALSE, ..., snip=TRUE]
Arguments
x

A point pattern on a linear network (object of class "lpp").

i

Subset index. A valid subset index in the usual R sense, indicating which points
should be retained.

j

Spatial window (object of class "owin") delineating the region that should be
retained.

drop

Logical value indicating whether to remove unused levels of the marks, if the
marks are a factor.

snip

Logical. If TRUE (the default), segments of the network which cross the boundary
of the window j will be cut by the boundary. If FALSE, these segments will be
deleted.

...

Ignored.

Details
This function extracts a designated subset of a point pattern on a linear network.
The function [.lpp is a method for [ for the class "lpp". It extracts a designated subset of a point
pattern. The argument i should be a subset index in the usual R sense: either a numeric vector
of positive indices (identifying the points to be retained), a numeric vector of negative indices
(identifying the points to be deleted) or a logical vector of length equal to the number of points in
the point pattern x. In the latter case, the points (x$x[i], x$y[i]) for which subset[i]=TRUE
will be retained, and the others will be deleted.
The argument j, if present, should be a spatial window. The pattern inside the region will be
retained. Line segments that cross the boundary of the window are deleted in the current implementation.
The argument drop determines whether to remove unused levels of a factor, if the point pattern is
multitype (i.e. the marks are a factor) or if the marks are a data frame or hyperframe in which some
of the columns are factors.
The argument snip specifies what to do with segments of the network which cross the boundary of
the window j. If snip=FALSE, such segments are simply deleted. If snip=TRUE (the default), such

Extract.msr

431

segments are cut into pieces by the boundary of j, and those pieces which lie inside the window ji
are included in the resulting network.
Use unmark to remove all the marks in a marked point pattern, and subset.lpp to remove only
some columns of marks.
Value
A point pattern on a linear network (of class "lpp").
Author(s)
Adrian Baddeley  and Rolf Turner 
See Also
lpp, subset.lpp
Examples
# Chicago crimes data - remove cases of assault
chicago[marks(chicago) != "assault"]
# equivalent to subset(chicago, select=-assault)
# spatial window subset
B <- owin(c(350, 700), c(600, 1000))
plot(chicago)
plot(B, add=TRUE, lty=2, border="red", lwd=3)
op <- par(mfrow=c(1,2), mar=0.6+c(0,0,1,0))
plot(B, main="chicago[B, snip=FALSE]", lty=3, border="red")
plot(chicago[, B, snip=FALSE], add=TRUE)
plot(B, main="chicago[B, snip=TRUE]", lty=3, border="red")
plot(chicago[, B, snip=TRUE], add=TRUE)
par(op)

Extract.msr

Extract Subset of Signed or Vector Measure

Description
Extract a subset of a signed measure or vector-valued measure.
Usage
## S3 method for class 'msr'
x[i, j, ...]
Arguments
x

A signed or vector measure. An object of class "msr" (see msr).

i

Object defining the subregion or subset to be extracted. Either a spatial window
(an object of class "owin"), or a pixel image with logical values, or any type of
index that applies to a matrix.

432

Extract.owin
j

Subset index selecting the vector coordinates to be extracted, if x is a vectorvalued measure.

...

Ignored.

Details
This operator extracts a subset of the data which determines the signed measure or vector-valued
measure x. The result is another measure.
Value
An object of class "msr".
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
msr
Examples
X <- rpoispp(function(x,y) { exp(3+3*x) })
fit <- ppm(X ~x+y)
rp <- residuals(fit, type="pearson")
rs <- residuals(fit, type="score")
rp[square(0.5)]
rs[ , 2:3]

Extract.owin

Extract Subset of Window

Description
Extract a subset of a window.
Usage
## S3 method for class 'owin'
x[i, ...]
Arguments
x

A spatial window (object of class "owin").

i

Object defining the subregion. Either a spatial window, or a pixel image with
logical values.

...

Ignored.

Extract.ppp

433

Details
This function computes the intersection between the window x and the domain specified by i, using
intersect.owin.
This function is a method for the subset operator "[" for spatial windows (objects of class "owin").
It is provided mainly for completeness.
The index i may be either a window, or a pixel image with logical values (the TRUE values of the
image specify the spatial domain).
Value
Another spatial window (object of class "owin").
Author(s)
Adrian Baddeley  and Rolf Turner 
See Also
intersect.owin
Examples
W <- owin(c(2.5, 3.2), c(1.4, 2.9))
plot(letterR)
plot(letterR[W], add=TRUE, col="red")

Extract.ppp

Extract or Replace Subset of Point Pattern

Description
Extract or replace a subset of a point pattern. Extraction of a subset has the effect of thinning the
points and/or trimming the window.
Usage
##
x[i,
##
x[i,

S3
j,
S3
j]

method for class 'ppp'
drop=FALSE, ..., clip=FALSE]
replacement method for class 'ppp'
<- value

Arguments
x

A two-dimensional point pattern. An object of class "ppp".

i

Subset index. Either a valid subset index in the usual R sense, indicating which
points should be retained, or a window (an object of class "owin") delineating a
subset of the original observation window, or a pixel image with logical values
defining a subset of the original observation window.

value

Replacement value for the subset. A point pattern.

j

Redundant. Included for backward compatibility.

434

Extract.ppp
drop

Logical value indicating whether to remove unused levels of the marks, if the
marks are a factor.

clip

Logical value indicating how to form the window of the resulting point pattern,
when i is a window. If clip=FALSE (the default), the result has window equal to
i. If clip=TRUE, the resulting window is the intersection between the window
of x and the window i.

...

Ignored. This argument is required for compatibility with the generic function.

Details
These functions extract a designated subset of a point pattern, or replace the designated subset with
another point pattern.
The function [.ppp is a method for [ for the class "ppp". It extracts a designated subset of a
point pattern, either by “thinning” (retaining/deleting some points of a point pattern) or “trimming”
(reducing the window of observation to a smaller subregion and retaining only those points which
lie in the subregion) or both.
The pattern will be “thinned” if i is a subset index in the usual R sense: either a numeric vector
of positive indices (identifying the points to be retained), a numeric vector of negative indices
(identifying the points to be deleted) or a logical vector of length equal to the number of points in
the point pattern x. In the latter case, the points (x$x[i], x$y[i]) for which subset[i]=TRUE
will be retained, and the others will be deleted.
The pattern will be “trimmed” if i is an object of class "owin" specifying a window of observation.
The points of x lying inside the new window i will be retained. Alternatively i may be a pixel
image (object of class "im") with logical values; the pixels with the value TRUE will be interpreted
as a window.
The argument drop determines whether to remove unused levels of a factor, if the point pattern is
multitype (i.e. the marks are a factor) or if the marks are a data frame in which some of the columns
are factors.
The function [<-.ppp is a method for [<- for the class "ppp". It replaces the designated subset
with the point pattern value. The subset of x to be replaced is designated by the argument i as
above.
The replacement point pattern value must lie inside the window of the original pattern x. The
ordering of points in x will be preserved if the replacement pattern value has the same number of
points as the subset to be replaced. Otherwise the ordering is unpredictable.
If the original pattern x has marks, then the replacement pattern value must also have marks, of the
same type.
Use the function unmark to remove marks from a marked point pattern.
Use the function split.ppp to select those points in a marked point pattern which have a specified
mark.
Value
A point pattern (of class "ppp").
Warnings
The function does not check whether i is a subset of Window(x). Nor does it check whether value
lies inside Window(x).

Extract.ppp

435

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
subset.ppp.
ppp.object, owin.object, unmark, split.ppp, cut.ppp
Examples
# Longleaf pines data
lon <- longleaf
## Not run:
plot(lon)
## End(Not run)
# adult trees defined to have diameter at least 30 cm
longadult <- subset(lon, marks >= 30)
## Not run:
plot(longadult)
## End(Not run)
# note that the marks are still retained.
# Use unmark(longadult) to remove the marks
# New Zealand trees data
## Not run:
plot(nztrees)
# plot shows a line of trees at the far right
abline(v=148, lty=2)
# cut along this line
## End(Not run)
nzw <- owin(c(0,148),c(0,95)) # the subwindow
# trim dataset to this subwindow
nzsub <- nztrees[nzw]
## Not run:
plot(nzsub)
## End(Not run)
# Redwood data
## Not run:
plot(redwood)
## End(Not run)
# Random thinning: delete 60% of data
retain <- (runif(npoints(redwood)) < 0.4)
thinred <- redwood[retain]
## Not run:
plot(thinred)
## End(Not run)
# Scramble 60% of data

436

Extract.ppx
X <- redwood
modif <- (runif(npoints(X)) < 0.6)
X[modif] <- runifpoint(ex=X[modif])
# Lansing woods data - multitype points
lan <- lansing
# Hickory trees
hicks <- split(lansing)$hickory
# Trees in subwindow
win <- owin(c(0.3, 0.6),c(0.2, 0.5))
lsub <- lan[win]
# Scramble the locations of trees in subwindow, retaining their marks
lan[win] <- runifpoint(ex=lsub) %mark% marks(lsub)
# Extract oaks only
oaknames <- c("redoak", "whiteoak", "blackoak")
oak <- lan[marks(lan) %in% oaknames, drop=TRUE]
oak <- subset(lan, marks %in% oaknames, drop=TRUE)
# To clip or not to clip
X <- runifpoint(25, letterR)
B <- owin(c(2.2, 3.9), c(2, 3.5))
opa <- par(mfrow=c(1,2))
plot(X, main="X[B]")
plot(X[B], border="red", cols="red", add=TRUE, show.all=TRUE, main="")
plot(X, main="X[B, clip=TRUE]")
plot(B, add=TRUE, lty=2)
plot(X[B, clip=TRUE], border="blue", cols="blue", add=TRUE,
show.all=TRUE, main="")
par(opa)

Extract.ppx

Extract Subset of Multidimensional Point Pattern

Description
Extract a subset of a multidimensional point pattern.
Usage
## S3 method for class 'ppx'
x[i, drop=FALSE, ...]
Arguments
x
i
drop
...

A multidimensional point pattern (object of class "ppx").
Subset index. A valid subset index in the usual R sense, indicating which points
should be retained; or a spatial domain of class "boxx" or "box3".
Logical value indicating whether to remove unused levels of the marks, if the
marks are a factor.
Ignored.

Extract.psp

437

Details
This function extracts a designated subset of a multidimensional point pattern.
The function [.ppx is a method for [ for the class "ppx". It extracts a designated subset of a point
pattern. The argument i may be either
• a subset index in the usual R sense: either a numeric vector of positive indices (identifying
the points to be retained), a numeric vector of negative indices (identifying the points to be
deleted) or a logical vector of length equal to the number of points in the point pattern x. In
the latter case, the points (x$x[i], x$y[i]) for which subset[i]=TRUE will be retained,
and the others will be deleted.
• a spatial domain of class "boxx" or "box3". Points falling inside this region will be retained.
The argument drop determines whether to remove unused levels of a factor, if the point pattern is
multitype (i.e. the marks are a factor) or if the marks are a data frame or hyperframe in which some
of the columns are factors.
Use the function unmark to remove marks from a marked point pattern.
Value
A multidimensional point pattern (of class "ppx").
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
ppx
Examples
df <- data.frame(x=runif(4),y=runif(4),z=runif(4))
X <- ppx(data=df, coord.type=c("s","s","t"))
X[-2]

Extract.psp

Extract Subset of Line Segment Pattern

Description
Extract a subset of a line segment pattern.
Usage
## S3 method for class 'psp'
x[i, j, drop, ..., fragments=TRUE]

438

Extract.psp

Arguments
x

A two-dimensional line segment pattern. An object of class "psp".

i

Subset index. Either a valid subset index in the usual R sense, indicating which
segments should be retained, or a window (an object of class "owin") delineating a subset of the original observation window.

j

Redundant - included for backward compatibility.

drop

Ignored. Required for compatibility with generic function.

...

Ignored.

fragments

Logical value indicating whether to retain all pieces of line segments that intersect the new window (fragments=TRUE, the default) or to retain only those line
segments that lie entirely inside the new window (fragments=FALSE).

Details
These functions extract a designated subset of a line segment pattern.
The function [.psp is a method for [ for the class "psp". It extracts a designated subset of a
line segment pattern, either by “thinning” (retaining/deleting some line segments of a line segment
pattern) or “trimming” (reducing the window of observation to a smaller subregion and clipping the
line segments to this boundary) or both.
The pattern will be “thinned” if subset is specified. The line segments designated by subset will
be retained. Here subset can be a numeric vector of positive indices (identifying the line segments
to be retained), a numeric vector of negative indices (identifying the line segments to be deleted)
or a logical vector of length equal to the number of line segments in the line segment pattern x. In
the latter case, the line segments for which subset[i]=TRUE will be retained, and the others will be
deleted.
The pattern will be “trimmed” if window is specified. This should be an object of class owin specifying a window of observation to which the line segment pattern x will be trimmed. Line segments
of x lying inside the new window will be retained unchanged. Line segments lying partially inside
the new window and partially outside it will, by default, be clipped so that they lie entirely inside
the window; but if fragments=FALSE, such segments will be removed.
Both “thinning” and “trimming” can be performed together.
Value
A line segment pattern (of class "psp").
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
psp.object, owin.object
Examples
a <- psp(runif(20),runif(20),runif(20),runif(20), window=owin())
plot(a)
# thinning

Extract.quad

439

id <- sample(c(TRUE, FALSE), 20, replace=TRUE)
b <- a[id]
plot(b, add=TRUE, lwd=3)
# trimming
plot(a)
w <- owin(c(0.1,0.7), c(0.2, 0.8))
b <- a[w]
plot(b, add=TRUE, col="red", lwd=2)
plot(w, add=TRUE)
u <- a[w, fragments=FALSE]
plot(u, add=TRUE, col="blue", lwd=3)

Extract.quad

Subset of Quadrature Scheme

Description
Extract a subset of a quadrature scheme.
Usage
## S3 method for class 'quad'
x[...]
Arguments
x

A quadrature scheme (object of class "quad").

...

Arguments passed to [.ppp to determine the subset.

Details
This function extracts a designated subset of a quadrature scheme.
The function [.quad is a method for [ for the class "quad". It extracts a designated subset of a
quadrature scheme.
The subset to be extracted is determined by the arguments ... which are interpreted by [.ppp. Thus
it is possible to take the subset consisting of all quadrature points that lie inside a given region, or a
subset of quadrature points identified by numeric indices.
Value
A quadrature scheme (object of class "quad").
Author(s)
Adrian Baddeley  and Rolf Turner 
See Also
quad.object, [.ppp.

440

Extract.solist

Examples
Q <- quadscheme(nztrees)
W <- owin(c(0,148),c(0,95)) # a subwindow
Q[W]

Extract.solist

Extract or Replace Subset of a List of Spatial Objects

Description
Extract or replace some entries in a list of spatial objects, or extract a designated sub-region in each
object.
Usage
## S3 method for class 'solist'
x[i, ...]
## S3 replacement method for class 'solist'
x[i] <- value
Arguments
x

An object of class "solist" representing a list of two-dimensional spatial objects.

i

Subset index. Any valid subset index for vectors in the usual R sense, or a
window (object of class "owin").

value

Replacement value for the subset.

...

Ignored.

Details
These are methods for extracting and replacing subsets for the class "solist".
The argument x should be an object of class "solist" representing a list of two-dimensional spatial
objects. See solist.
For the subset method, the subset index i can be either a vector index (specifying some elements of
the list) or a spatial window (specifying a spatial sub-region).
For the replacement method, i must be a vector index: the designated elements will be replaced.
Value
Another object of the same class as x.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 

Extract.splitppp

441

See Also
solist, plot.solist, summary.solist
Examples
x <- solist(japanesepines, cells, redwood)
x[2:3]
x[square(0.5)]
x[1] <- list(finpines)

Extract.splitppp

Extract or Replace Sub-Patterns

Description
Extract or replace some of the sub-patterns in a split point pattern.
Usage
## S3 method for class 'splitppp'
x[...]
## S3 replacement method for class 'splitppp'
x[...] <- value
Arguments
x

An object of class "splitppp", representing a point pattern separated into a list
of sub-patterns.

...

Subset index. Any valid subset index in the usual R sense.

value

Replacement value for the subset. A list of point patterns.

Details
These are subset methods for the class "splitppp".
The argument x should be an object of class "splitppp", representing a point pattern that has been
separated into a list of sub-patterns. It is created by split.ppp.
The methods extract or replace a designated subset of the list x, and return an object of class
"splitppp".
Value
Another object of class "splitppp".
Author(s)
Adrian Baddeley  and Rolf Turner 
See Also
split.ppp, plot.splitppp, summary.splitppp

442

Extract.tess

Examples
data(amacrine) # multitype point pattern
y <- split(amacrine)
y[1]
y["off"]
y[1] <- list(runifpoint(42, Window(amacrine)))

Extract.tess

Extract or Replace Subset of Tessellation

Description
Extract, change or delete a subset of the tiles of a tessellation, to make a new tessellation.
Usage
##
x[i,
##
x[i,

S3 method for class 'tess'
...]
S3 replacement method for class 'tess'
...] <- value

Arguments
x

A tessellation (object of class "tess").

i

Subset index for the tiles of the tessellation. Alternatively a window (object of
class "owin").

...

One argument that specifies the subset to be extracted or changed. Any valid
format for the subset index in a list.

value

Replacement value for the selected tiles of the tessellation. A list of windows
(objects of class "owin") or NULL.

Details
A tessellation (object of class "tess", see tess) is effectively a list of tiles (spatial regions) that
cover a spatial region. The subset operator [.tess extracts some of these tiles and forms a new
tessellation, which of course covers a smaller region than the original.
For [.tess only, the subset index can also be a window (object of class "owin"). The tessellation
x is then intersected with the window.
The replacement operator changes the selected tiles. The replacement value may be either NULL
(which causes the selected tiles to be removed from x) or a list of the same length as the selected
subset. The entries of value may be windows (objects of class "owin") or NULL to indicate that the
corresponding tile should be deleted.
Generally it does not make sense to replace a tile in a tessellation with a completely different tile,
because the tiles are expected to fit together. However this facility is sometimes useful for making
small adjustments to polygonal tiles.
Value
A tessellation (object of class "tess").

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443

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
tess, tiles, intersect.tess.
Examples
A <- tess(xgrid=0:4, ygrid=0:3)
B <- A[c(1, 3, 7)]
E <- A[-1]
A[c(2, 5, 11)] <- NULL

F3est

Empty Space Function of a Three-Dimensional Point Pattern

Description
Estimates the empty space function F3 (r) from a three-dimensional point pattern.
Usage
F3est(X, ..., rmax = NULL, nrval = 128, vside = NULL,
correction = c("rs", "km", "cs"),
sphere = c("fudge", "ideal", "digital"))
Arguments
X

Three-dimensional point pattern (object of class "pp3").

...

Ignored.

rmax

Optional. Maximum value of argument r for which F3 (r) will be estimated.

nrval

Optional. Number of values of r for which F3 (r) will be estimated. A large
value of nrval is required to avoid discretisation effects.

vside

Optional. Side length of the voxels in the discrete approximation.

correction

Optional. Character vector specifying the edge correction(s) to be applied. See
Details.

sphere

Optional. Character string specifying how to calculate the theoretical value of
F3 (r) for a Poisson process. See Details.

444

F3est

Details
For a stationary point process Φ in three-dimensional space, the empty space function is
F3 (r) = P (d(0, Φ) ≤ r)
where d(0, Φ) denotes the distance from a fixed origin 0 to the nearest point of Φ.
The three-dimensional point pattern X is assumed to be a partial realisation of a stationary point
process Φ. The empty space function of Φ can then be estimated using techniques described in the
References.
The box containing the point pattern is discretised into cubic voxels of side length vside. The
distance function d(u, Φ) is computed for every voxel centre point u using a three-dimensional version of the distance transform algorithm (Borgefors, 1986). The empirical cumulative distribution
function of these values, with appropriate edge corrections, is the estimate of F3 (r).
The available edge corrections are:
"rs": the reduced sample (aka minus sampling, border correction) estimator (Baddeley et al, 1993)
"km": the three-dimensional version of the Kaplan-Meier estimator (Baddeley and Gill, 1997)
"cs": the three-dimensional generalisation of the Chiu-Stoyan or Hanisch estimator (Chiu and
Stoyan, 1998).
Alternatively correction="all" selects all options.
The result includes a column theo giving the theoretical value of F3 (r) for a uniform Poisson
process (Complete Spatial Randomness). This value depends on the volume of the sphere of radius
r measured in the discretised distance metric. The argument sphere determines how this will be
calculated.
• If sphere="ideal" the calculation will use the volume of an ideal sphere of radius r namely
(4/3)πr3 . This is not recommended because the theoretical values of F3 (r) are inaccurate.
• If sphere="fudge" then the volume of the ideal sphere will be multiplied by 0.78, which
gives the approximate volume of the sphere in the discretised distance metric.
• If sphere="digital" then the volume of the sphere in the discretised distance metric is computed exactly using another distance transform. This takes longer to compute, but is exact.
Value
A function value table (object of class "fv") that can be plotted, printed or coerced to a data frame
containing the function values.
Warnings
A small value of vside and a large value of nrval are required for reasonable accuracy.
The default value of vside ensures that the total number of voxels is 2^22 or about 4 million. To
change the default number of voxels, see spatstat.options("nvoxel").
Author(s)
Adrian Baddeley 
and Rana Moyeed.

fardist

445

References
Baddeley, A.J, Moyeed, R.A., Howard, C.V. and Boyde, A. Analysis of a three-dimensional point
pattern with replication. Applied Statistics 42 (1993) 641–668.
Baddeley, A.J. and Gill, R.D. (1997) Kaplan-Meier estimators of interpoint distance distributions
for spatial point processes. Annals of Statistics 25, 263–292.
Borgefors, G. (1986) Distance transformations in digital images. Computer Vision, Graphics and
Image Processing 34, 344–371.
Chiu, S.N. and Stoyan, D. (1998) Estimators of distance distributions for spatial patterns. Statistica
Neerlandica 52, 239–246.
See Also
G3est, K3est, pcf3est.
Examples
X <- rpoispp3(42)
Z <- F3est(X)
if(interactive()) plot(Z)

fardist

Farthest Distance to Boundary of Window

Description
Computes the farthest distance from each pixel, or each data point, to the boundary of the window.
Usage
fardist(X, ...)
## S3 method for class 'owin'
fardist(X, ..., squared=FALSE)
## S3 method for class 'ppp'
fardist(X, ..., squared=FALSE)
Arguments
X
...
squared

A spatial object such as a window or point pattern.
Arguments passed to as.mask to determine the pixel resolution, if required.
Logical. If TRUE, the squared distances will be returned.

Details
The function fardist is generic, with methods for the classes owin and ppp.
For a window W, the command fardist(W) returns a pixel image in which the value at each pixel
is the largest distance from that pixel to the boundary of W.
For a point pattern X, with window W, the command fardist(X) returns a numeric vector with one
entry for each point of X, giving the largest distance from that data point to the boundary of W.

446

fasp.object

Value
For fardist.owin, a pixel image (object of class "im").
For fardist.ppp, a numeric vector.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
Examples
fardist(cells)
plot(FR <- fardist(letterR))

fasp.object

Function Arrays for Spatial Patterns

Description
A class "fasp" to represent a “matrix” of functions, amenable to plotting as a matrix of plot panels.
Details
An object of this class is a convenient way of storing (and later plotting, editing, etc) a set of
functions fi,j (r) of a real argument r, defined for each possible pair (i, j) of indices 1 ≤ i, j ≤ n.
We may think of this as a matrix or array of functions fi,j .
Function arrays are particularly useful in the analysis of a multitype point pattern (a point pattern in
which the points are identified as belonging to separate types). We may want to compute a summary
function for the points of type i only, for each of the possible types i. This produces a 1 × m array
of functions. Alternatively we may compute a summary function for each possible pair of types
(i, j). This produces an m × m array of functions.
For multitype point patterns the command alltypes will compute arrays of summary functions for
each possible type or for each possible pair of types. The function alltypes returns an object of
class "fasp".
An object of class "fasp" is a list containing at least the following components:
fns A list of data frames, each representing one of the functions.
which A matrix representing the spatial arrangement of the functions. If which[i,j] = k then
the function represented by fns[[k]] should be plotted in the panel at position (i, j). If
which[i,j] = NA then nothing is plotted in that position.
titles A list of character strings, providing suitable plotting titles for the functions.
default.formulae A list of default formulae for plotting each of the functions.
title A character string, giving a default title for the array when it is plotted.

Fest

447

Functions available
There are methods for plot, print and "[" for this class.
The plot method displays the entire array of functions. The method [.fasp selects a sub-array
using the natural indices i,j.
The command eval.fasp can be used to apply a transformation to each function in the array, and
to combine two arrays.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
alltypes, plot.fasp, [.fasp, eval.fasp
Examples
# multitype point pattern
data(amacrine)
GG <- alltypes(amacrine, "G")
plot(GG)
# select the row corresponding to cells of type "on"
Gon <- GG["on", ]
plot(Gon)
# extract the G function for i = "on", j = "off"
Gonoff <- GG["on", "off", drop=TRUE]
# Fisher variance stabilising transformation
GGfish <- eval.fasp(asin(sqrt(GG)))
plot(GGfish)

Fest

Estimate the Empty Space Function or its Hazard Rate

Description
Estimates the empty space function F (r) or its hazard rate h(r) from a point pattern in a window
of arbitrary shape.
Usage
Fest(X, ..., eps, r=NULL, breaks=NULL,
correction=c("rs", "km", "cs"),
domain=NULL)
Fhazard(X, ...)

448

Fest

Arguments
X

The observed point pattern, from which an estimate of F (r) will be computed.
An object of class ppp, or data in any format acceptable to as.ppp().

...

Extra arguments, passed from Fhazard to Fest. Extra arguments to Fest are
ignored.

eps

Optional. A positive number. The resolution of the discrete approximation to
Euclidean distance (see below). There is a sensible default.

r

Optional. Numeric vector. The values of the argument r at which F (r) should
be evaluated. There is a sensible default. First-time users are strongly advised
not to specify this argument. See below for important conditions on r.

breaks

This argument is for internal use only.

correction

Optional. The edge correction(s) to be used to estimate F (r). A vector of character strings selected from "none", "rs", "km", "cs" and "best". Alternatively
correction="all" selects all options.

domain

Optional. Calculations will be restricted to this subset of the window. See Details.

Details
Fest computes an estimate of the empty space function F (r), and Fhazard computes an estimate
of its hazard rate h(r).
The empty space function (also called the “spherical contact distribution” or the “point-to-nearestevent” distribution) of a stationary point process X is the cumulative distribution function F of the
distance from a fixed point in space to the nearest point of X.
An estimate of F derived from a spatial point pattern dataset can be used in exploratory data analysis
and formal inference about the pattern (Cressie, 1991; Diggle, 1983; Ripley, 1988). In exploratory
analyses, the estimate of F is a useful statistic summarising the sizes of gaps in the pattern. For
inferential purposes, the estimate of F is usually compared to the true value of F for a completely
random (Poisson) point process, which is
F (r) = 1 − e−λπr

2

where λ is the intensity (expected number of points per unit area). Deviations between the empirical
and theoretical F curves may suggest spatial clustering or spatial regularity.
This algorithm estimates the empty space function F from the point pattern X. It assumes that X can
be treated as a realisation of a stationary (spatially homogeneous) random spatial point process in
the plane, observed through a bounded window. The window (which is specified in X) may have
arbitrary shape.
The argument X is interpreted as a point pattern object (of class "ppp", see ppp.object) and can be
supplied in any of the formats recognised by as.ppp.
The algorithm uses two discrete approximations which are controlled by the parameter eps and by
the spacing of values of r respectively. (See below for details.) First-time users are strongly advised
not to specify these arguments.
The estimation of F is hampered by edge effects arising from the unobservability of points of
the random pattern outside the window. An edge correction is needed to reduce bias (Baddeley,
1998; Ripley, 1988). The edge corrections implemented here are the border method or "reduced
sample" estimator, the spatial Kaplan-Meier estimator (Baddeley and Gill, 1997) and the ChiuStoyan estimator (Chiu and Stoyan, 1998).

Fest

449
Our implementation makes essential use of the distance transform algorithm of image processing
(Borgefors, 1986). A fine grid of pixels is created in the observation window. The Euclidean
distance between two pixels is approximated by the length of the shortest path joining them in
the grid, where a path is a sequence √
of steps between adjacent pixels, and horizontal, vertical and
diagonal steps have length 1, 1 and 2 respectively in pixel units. If the pixel grid is sufficiently
fine then this is an accurate approximation.
The parameter eps is the pixel width of the rectangular raster used to compute the distance transform
(see below). It must not be too large: the absolute error in distance values due to discretisation is
bounded by eps.
If eps is not specified, the function checks whether the window Window(X) contains pixel raster
information. If so, then eps is set equal to the pixel width of the raster; otherwise, eps defaults to
1/100 of the width of the observation window.
The argument r is the vector of values for the distance r at which F (r) should be evaluated. It
is also used to determine the breakpoints (in the sense of hist) for the computation of histograms
of distances. The estimators are computed from histogram counts. This introduces a discretisation
error which is controlled by the fineness of the breakpoints.
First-time users would be strongly advised not to specify r. However, if it is specified, r must
satisfy r[1] = 0, and max(r) must be larger than the radius of the largest disc contained in the
window. Furthermore, the spacing of successive r values must be very fine (ideally not greater than
eps/4).
The algorithm also returns an estimate of the hazard rate function, h(r) of F (r). The hazard rate is
defined by
d
h(r) = − log(1 − F (r))
dr
The hazard rate of F has been proposed as a useful exploratory statistic (Baddeley and Gill, 1994).
The estimate of h(r) given here is a discrete approximation to the hazard rate of the Kaplan-Meier
estimator of F . Note that F is absolutely continuous (for any stationary point process X), so the
hazard function always exists (Baddeley and Gill, 1997).
If the argument domain is given, the estimate of F (r) will be based only on the empty space distances measured from locations inside domain (although their nearest data points may lie outside
domain). This is useful in bootstrap techniques. The argument domain should be a window (object
of class "owin") or something acceptable to as.owin. It must be a subset of the window of the
point pattern X.
The naive empirical distribution of distances from each location in the window to the nearest point
of the data pattern, is a biased estimate of F . However this is also returned by the algorithm (if
correction="none"), as it is sometimes useful in other contexts. Care should be taken not to use
the uncorrected empirical F as if it were an unbiased estimator of F .

Value
An object of class "fv", see fv.object, which can be plotted directly using plot.fv.
The result of Fest is essentially a data frame containing up to seven columns:
r

the values of the argument r at which the function F (r) has been estimated

rs

the “reduced sample” or “border correction” estimator of F (r)

km

the spatial Kaplan-Meier estimator of F (r)

hazard

the hazard rate λ(r) of F (r) by the spatial Kaplan-Meier method

cs

the Chiu-Stoyan estimator of F (r)

450

Fest
raw
theo

the uncorrected estimate of F (r), i.e. the empirical distribution of the distance
from a random point in the window to the nearest point of the data pattern X
the theoretical value of F (r) for a stationary Poisson process of the same estimated intensity.

The result of Fhazard contains only three columns
r
hazard
theo

the values of the argument r at which the hazard rate h(r) has been estimated
the spatial Kaplan-Meier estimate of the hazard rate h(r)
the theoretical value of h(r) for a stationary Poisson process of the same estimated intensity.

Warnings
The reduced sample (border method) estimator of F is pointwise approximately unbiased, but need
not be a valid distribution function; it may not be a nondecreasing function of r. Its range is always
within [0, 1].
The spatial Kaplan-Meier estimator of F is always nondecreasing but its maximum value may be
less than 1.
The estimate of hazard rate h(r) returned by the algorithm is an approximately unbiased estimate
for the integral of h() over the corresponding histogram cell. It may exhibit oscillations due to
discretisation effects. We recommend modest smoothing, such as kernel smoothing with kernel
width equal to the width of a histogram cell, using Smooth.fv.
Note
Sizeable amounts of memory may be needed during the calculation.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Baddeley, A.J. Spatial sampling and censoring. In O.E. Barndorff-Nielsen, W.S. Kendall and
M.N.M. van Lieshout (eds) Stochastic Geometry: Likelihood and Computation. Chapman and
Hall, 1998. Chapter 2, pages 37-78.
Baddeley, A.J. and Gill, R.D. The empty space hazard of a spatial pattern. Research Report 1994/3,
Department of Mathematics, University of Western Australia, May 1994.
Baddeley, A.J. and Gill, R.D. Kaplan-Meier estimators of interpoint distance distributions for spatial
point processes. Annals of Statistics 25 (1997) 263-292.
Borgefors, G. Distance transformations in digital images. Computer Vision, Graphics and Image
Processing 34 (1986) 344-371.
Chiu, S.N. and Stoyan, D. (1998) Estimators of distance distributions for spatial patterns. Statistica
Neerlandica 52, 239–246.
Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.
Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 1983.
Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.
Stoyan, D, Kendall, W.S. and Mecke, J. Stochastic geometry and its applications. 2nd edition.
Springer Verlag, 1995.

Fiksel

451

See Also
Gest, Jest, Kest, km.rs, reduced.sample, kaplan.meier
Examples
Fc <- Fest(cells, 0.01)
# Tip: don't use F for the left hand side!
# That's an abbreviation for FALSE
plot(Fc)
# P-P style plot
plot(Fc, cbind(km, theo) ~ theo)
# The empirical F is above the Poisson F
# indicating an inhibited pattern
## Not run:
plot(Fc, . ~ theo)
plot(Fc, asin(sqrt(.)) ~ asin(sqrt(theo)))
## End(Not run)

Fiksel

The Fiksel Interaction

Description
Creates an instance of Fiksel’s double exponential pairwise interaction point process model, which
can then be fitted to point pattern data.
Usage
Fiksel(r, hc=NA, kappa)
Arguments
r
hc
kappa

The interaction radius of the Fiksel model
The hard core distance
The rate parameter

Details
Fiksel (1984) introduced a pairwise interaction point process with the following interaction function
c. For two points u and v separated by a distance d = ||u − v||, the interaction c(u, v) is equal to 0
if d < h, equal to 1 if d > r, and equal to
exp(a exp(−κd))
if h ≤ d ≤ r, where h, r, κ, a are parameters.
A graph of this interaction function is shown in the Examples. The interpretation of the parameters
is as follows.

452

Fiksel
• h is the hard core distance: distinct points are not permitted to come closer than a distance h
apart.
• r is the interaction range: points further than this distance do not interact.
• κ is the rate or slope parameter, controlling the decay of the interaction as distance increases.
• a is the interaction strength parameter, controlling the strength and type of interaction. If a
is zero, the process is Poisson. If a is positive, the process is clustered. If a is negative, the
process is inhibited (regular).
The function ppm(), which fits point process models to point pattern data, requires an argument
of class "interact" describing the interpoint interaction structure of the model to be fitted. The
appropriate description of the Fiksel pairwise interaction is yielded by the function Fiksel(). See
the examples below.
The parameters h, r and κ must be fixed and given in the call to Fiksel, while the canonical
parameter a is estimated by ppm().
To estimate h, r andκ it is possible to use profilepl. The maximum likelihood estimator ofh is
the minimum interpoint distance.
If the hard core distance argument hc is missing or NA, it will be estimated from the data when
ppm is called. The estimated value of hc is the minimum nearest neighbour distance multiplied by
n/(n + 1), where n is the number of data points.
See also Stoyan, Kendall and Mecke (1987) page 161.

Value
An object of class "interact" describing the interpoint interaction structure of the Fiksel process
with interaction radius r, hard core distance hc and rate parameter kappa.
Author(s)
Adrian Baddeley  and Rolf Turner 
References
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns.
Australian and New Zealand Journal of Statistics 42, 283–322.
Fiksel, T. (1984) Estimation of parameterized pair potentials of marked and non-marked Gibbsian
point processes. Electronische Informationsverabeitung und Kybernetika 20, 270–278.
Stoyan, D, Kendall, W.S. and Mecke, J. (1987) Stochastic geometry and its applications. Wiley.
See Also
ppm, pairwise.family, ppm.object, StraussHard
Examples
Fiksel(r=1,hc=0.02, kappa=2)
# prints a sensible description of itself
data(spruces)
X <- unmark(spruces)
fit <- ppm(X ~ 1, Fiksel(r=3.5, kappa=1))
plot(fitin(fit))

Finhom

Finhom

453

Inhomogeneous Empty Space Function

Description
Estimates the inhomogeneous empty space function of a non-stationary point pattern.
Usage
Finhom(X, lambda = NULL, lmin = NULL, ...,
sigma = NULL, varcov = NULL,
r = NULL, breaks = NULL, ratio = FALSE, update = TRUE)
Arguments
X

lambda

lmin
sigma,varcov
...
r
breaks
ratio
update

The observed data point pattern, from which an estimate of the inhomogeneous
F function will be computed. An object of class "ppp" or in a format recognised
by as.ppp()
Optional. Values of the estimated intensity function. Either a vector giving the
intensity values at the points of the pattern X, a pixel image (object of class "im")
giving the intensity values at all locations, a fitted point process model (object of
class "ppm") or a function(x,y) which can be evaluated to give the intensity
value at any location.
Optional. The minimum possible value of the intensity over the spatial domain.
A positive numerical value.
Optional arguments passed to density.ppp to control the smoothing bandwidth, when lambda is estimated by kernel smoothing.
Extra arguments passed to as.mask to control the pixel resolution, or passed to
density.ppp to control the smoothing bandwidth.
vector of values for the argument r at which the inhomogeneous K function
should be evaluated. Not normally given by the user; there is a sensible default.
This argument is for internal use only.
Logical. If TRUE, the numerator and denominator of the estimate will also be
saved, for use in analysing replicated point patterns.
Logical. If lambda is a fitted model (class "ppm" or "kppm") and update=TRUE
(the default), the model will first be refitted to the data X (using update.ppm
or update.kppm) before the fitted intensity is computed. If update=FALSE, the
fitted intensity of the model will be computed without fitting it to X.

Details
This command computes estimates of the inhomogeneous F -function (van Lieshout, 2010) of a
point pattern. It is the counterpart, for inhomogeneous spatial point patterns, of the empty space
function F for homogeneous point patterns computed by Fest.
The argument X should be a point pattern (object of class "ppp").
The inhomogeneous F function is computed using the border correction, equation (6) in Van
Lieshout (2010).
The argument lambda should supply the (estimated) values of the intensity function λ of the point
process. It may be either

454

Finhom
a numeric vector containing the values of the intensity function at the points of the pattern X.
a pixel image (object of class "im") assumed to contain the values of the intensity function at all
locations in the window.
a fitted point process model (object of class "ppm" or "kppm") whose fitted trend can be used as
the fitted intensity. (If update=TRUE the model will first be refitted to the data X before the
trend is computed.)
a function which can be evaluated to give values of the intensity at any locations.
omitted: if lambda is omitted, then it will be estimated using a ‘leave-one-out’ kernel smoother.
If lambda is a numeric vector, then its length should be equal to the number of points in the pattern
X. The value lambda[i] is assumed to be the the (estimated) value of the intensity λ(xi ) for the
point xi of the pattern X. Each value must be a positive number; NA’s are not allowed.
If lambda is a pixel image, the domain of the image should cover the entire window of the point
pattern. If it does not (which may occur near the boundary because of discretisation error), then
the missing pixel values will be obtained by applying a Gaussian blur to lambda using blur, then
looking up the values of this blurred image for the missing locations. (A warning will be issued in
this case.)
If lambda is a function, then it will be evaluated in the form lambda(x,y) where x and y are vectors
of coordinates of the points of X. It should return a numeric vector with length equal to the number
of points in X.
If lambda is omitted, then it will be estimated using a ‘leave-one-out’ kernel smoother, as described
in Baddeley, Møller and Waagepetersen (2000). The estimate lambda[i] for the point X[i] is computed by removing X[i] from the point pattern, applying kernel smoothing to the remaining points
using density.ppp, and evaluating the smoothed intensity at the point X[i]. The smoothing kernel
bandwidth is controlled by the arguments sigma and varcov, which are passed to density.ppp
along with any extra arguments.

Value
An object of class "fv", see fv.object, which can be plotted directly using plot.fv.
Author(s)
Original code by Marie-Colette van Lieshout. C implementation and R adaptation by Adrian Baddeley 
and Ege Rubak .
References
Van Lieshout, M.N.M. and Baddeley, A.J. (1996) A nonparametric measure of spatial interaction in
point patterns. Statistica Neerlandica 50, 344–361.
Van Lieshout, M.N.M. (2010) A J-function for inhomogeneous point processes. Statistica Neerlandica 65, 183–201.
See Also
Ginhom, Jinhom, Fest

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455

Examples
## Not run:
plot(Finhom(swedishpines, sigma=bw.diggle, adjust=2))
## End(Not run)
plot(Finhom(swedishpines, sigma=10))

fitin.ppm

Extract the Interaction from a Fitted Point Process Model

Description
Given a point process model that has been fitted to point pattern data, this function extracts the
interpoint interaction part of the model as a separate object.
Usage
fitin(object)
## S3 method for class 'ppm'
fitin(object)
## S3 method for class 'profilepl'
fitin(object)
Arguments
object

A fitted point process model (object of class "ppm" or "profilepl").

Details
An object of class "ppm" describes a fitted point process model. It contains information about the
original data to which the model was fitted, the spatial trend that was fitted, the interpoint interaction
that was fitted, and other data. See ppm.object) for details of this class.
The function fitin extracts from this model the information about the fitted interpoint interaction
only. The information is organised as an object of class "fii" (fitted interpoint interaction). This
object can be printed or plotted.
Users may find this a convenient way to plot the fitted interpoint interaction term, as shown in the
Examples.
For a pairwise interaction, the plot of the fitted interaction shows the pair interaction function (the
contribution to the probability density from a pair of points as a function of the distance between
them). For a higher-order interaction, the plot shows the strongest interaction (the value most different from 1) that could ever arise at the given distance.
The fitted interaction coefficients can also be extracted from this object using coef.
Value
An object of class "fii" representing the fitted interpoint interaction. This object can be printed
and plotted.

456

fitted.lppm

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
Methods for handling fitted interactions: methods.fii, reach.fii, as.interact.fii.
Background: ppm, ppm.object.
Examples
# unmarked
model <- ppm(swedishpines ~1, PairPiece(seq(3,19,by=4)))
f <- fitin(model)
f
plot(f)
# extract fitted interaction coefficients
coef(f)
# multitype
# fit the stationary multitype Strauss process to `amacrine'
r <- 0.02 * matrix(c(1,2,2,1), nrow=2,ncol=2)
model <- ppm(amacrine ~1, MultiStrauss(r))
f <- fitin(model)
f
plot(f)

fitted.lppm

Fitted Intensity for Point Process on Linear Network

Description
Given a point process model fitted to a point pattern on a linear network, compute the fitted intensity
of the model at the points of the pattern, or at the points of the quadrature scheme used to fit the
model.
Usage
## S3 method for class 'lppm'
fitted(object, ...,
dataonly = FALSE, new.coef = NULL,
leaveoneout = FALSE)
Arguments
object

Fitted point process model on a linear network (object of class "lppm").

...

Ignored.

dataonly

Logical value indicating whether to computed fitted intensities at the points of
the original point pattern dataset (dataonly=TRUE) or at all the quadrature points
of the quadrature scheme used to fit the model (dataonly=FALSE, the default).

fitted.mppm

457

new.coef

Numeric vector of parameter values to replace the fitted model parameters coef(object).

leaveoneout

Logical. If TRUE the fitted value at each data point will be computed using a
leave-one-out method. See Details.

Details
This is a method for the generic function fitted for the class "lppm" of fitted point process models
on a linear network.
The locations u at which the fitted conditional intensity/trend is evaluated, are the points of the
quadrature scheme used to fit the model in ppm. They include the data points (the points of the
original point pattern dataset x) and other “dummy” points in the window of observation.
If leaveoneout=TRUE, fitted values will be computed for the data points only, using a ‘leave-oneout’ rule: the fitted value at X[i] is effectively computed by deleting this point from the data and
re-fitting the model to the reduced pattern X[-i], then predicting the value at X[i]. (Instead of
literally performing this calculation, we apply a Taylor approximation using the influence function
computed in dfbetas.ppm.
Value
A vector containing the values of the fitted spatial trend.
Entries in this vector correspond to the quadrature points (data or dummy points) used to fit the
model. The quadrature points can be extracted from object by union.quad(quad.ppm(object)).
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
lppm, predict.lppm
Examples
fit <- lppm(spiders~x+y)
a <- fitted(fit)
b <- fitted(fit, dataonly=TRUE)

fitted.mppm

Fitted Conditional Intensity for Multiple Point Process Model

Description
Given a point process model fitted to multiple point patterns, compute the fitted conditional intensity
of the model at the points of each data pattern, or at the points of the quadrature schemes used to fit
the model.

458

fitted.mppm

Usage
## S3 method for class 'mppm'
fitted(object, ..., type = "lambda", dataonly = FALSE)
Arguments
object

The fitted model. An object of class "mppm" obtained from mppm.

...

Ignored.

type

Type of fitted values: either "trend" for the spatial trend, or "lambda" or "cif"
for the conditional intensity.

dataonly

If TRUE, fitted values are computed only for the points of the data point patterns.
If FALSE, fitted values are computed for the points of the quadrature schemes
used to fit the model.

Details
ˆ of the fitted point
This function evaluates the conditional intensity λ̂(u, x) or spatial trend b(u)
process model for certain locations u, for each of the original point patterns x to which the model
was fitted.
The locations u at which the fitted conditional intensity/trend is evaluated, are the points of the
quadrature schemes used to fit the model in mppm. They include the data points (the points of the
original point pattern datasets) and other “dummy” points in the window of observation.
Use predict.mppm to compute the fitted conditional intensity at other locations or with other values
of the explanatory variables.
Value
A list of vectors (one for each row of the original hyperframe, i.e. one vector for each of the original
point patterns) containing the values of the fitted conditional intensity or (if type="trend") the
fitted spatial trend.
Entries in these vector correspond to the quadrature points (data or dummy points) used to fit the
model. The quadrature points can be extracted from object by quad.mppm(object).
Author(s)
Adrian Baddeley , Ida-Maria Sintorn and Leanne Bischoff.
Implemented by Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. London: Chapman and Hall/CRC Press.
See Also
mppm, predict.mppm

fitted.ppm

459

Examples
model <- mppm(Bugs ~ x, data=hyperframe(Bugs=waterstriders),
interaction=Strauss(7))
cifs <- fitted(model)

fitted.ppm

Fitted Conditional Intensity for Point Process Model

Description
Given a point process model fitted to a point pattern, compute the fitted conditional intensity or
fitted trend of the model at the points of the pattern, or at the points of the quadrature scheme used
to fit the model.
Usage
## S3 method for class 'ppm'
fitted(object, ..., type="lambda", dataonly=FALSE,
new.coef=NULL, leaveoneout=FALSE, drop=FALSE, check=TRUE, repair=TRUE,
dropcoef=FALSE)
Arguments
object

The fitted point process model (an object of class "ppm")

...

Ignored.

type

String (partially matched) indicating whether the fitted value is the conditional
intensity ("lambda" or "cif") or the first order trend ("trend") or the logarithm
of conditional intensity ("link").

dataonly

Logical. If TRUE, then values will only be computed at the points of the data
point pattern. If FALSE, then values will be computed at all the points of the
quadrature scheme used to fit the model, including the points of the data point
pattern.

new.coef

Numeric vector of parameter values to replace the fitted model parameters coef(object).

leaveoneout

Logical. If TRUE the fitted value at each data point will be computed using a
leave-one-out method. See Details.

drop

Logical value determining whether to delete quadrature points that were not
used to fit the model.

check

Logical value indicating whether to check the internal format of object. If
there is any possibility that this object has been restored from a dump file, or
has otherwise lost track of the environment where it was originally computed,
set check=TRUE.

repair

Logical value indicating whether to repair the internal format of object, if it is
found to be damaged.

dropcoef

Internal use only.

460

fitted.ppm

Details
The argument object must be a fitted point process model (object of class "ppm"). Such objects
are produced by the model-fitting algorithm ppm).
This function evaluates the conditional intensity λ̂(u, x) or spatial trend b̂(u) of the fitted point
process model for certain locations u, where x is the original point pattern dataset to which the
model was fitted.
The locations u at which the fitted conditional intensity/trend is evaluated, are the points of the
quadrature scheme used to fit the model in ppm. They include the data points (the points of the
original point pattern dataset x) and other “dummy” points in the window of observation.
If leaveoneout=TRUE, fitted values will be computed for the data points only, using a ‘leave-oneout’ rule: the fitted value at X[i] is effectively computed by deleting this point from the data and
re-fitting the model to the reduced pattern X[-i], then predicting the value at X[i]. (Instead of
literally performing this calculation, we apply a Taylor approximation using the influence function
computed in dfbetas.ppm.
The argument drop is explained in quad.ppm.
Use predict.ppm to compute the fitted conditional intensity at other locations or with other values
of the explanatory variables.
Value
A vector containing the values of the fitted conditional intensity, fitted spatial trend, or logarithm of
the fitted conditional intensity.
Entries in this vector correspond to the quadrature points (data or dummy points) used to fit the
model. The quadrature points can be extracted from object by union.quad(quad.ppm(object)).
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
References
Baddeley, A., Turner, R., Møller, J. and Hazelton, M. (2005). Residual analysis for spatial point
processes (with discussion). Journal of the Royal Statistical Society, Series B 67, 617–666.
See Also
ppm.object, ppm, predict.ppm
Examples
str <- ppm(cells ~x, Strauss(r=0.1))
lambda <- fitted(str)
# extract quadrature points in corresponding order
quadpoints <- union.quad(quad.ppm(str))
# plot conditional intensity values
# as circles centred on the quadrature points
quadmarked <- setmarks(quadpoints, lambda)
plot(quadmarked)

fitted.slrm

461

if(!interactive()) str <- ppm(cells ~ x)
lambdaX <- fitted(str, leaveoneout=TRUE)

fitted.slrm

Fitted Probabilities for Spatial Logistic Regression

Description
Given a fitted Spatial Logistic Regression model, this function computes the fitted probabilities for
each pixel.
Usage
## S3 method for class 'slrm'
fitted(object, ...)
Arguments
object

a fitted spatial logistic regression model. An object of class "slrm".

...

Ignored.

Details
This is a method for the generic function fitted for spatial logistic regression models (objects of
class "slrm", usually obtained from the function slrm).
The algorithm computes the fitted probabilities of the presence of a random point in each pixel.
Value
A pixel image (object of class "im") containing the fitted probability for each pixel.
Author(s)
Adrian Baddeley  
and Rolf Turner 
See Also
slrm, fitted
Examples
X <- rpoispp(42)
fit <- slrm(X ~ x+y)
plot(fitted(fit))

462

fixef.mppm

fixef.mppm

Extract Fixed Effects from Point Process Model

Description
Given a point process model fitted to a list of point patterns, extract the fixed effects of the model.
A method for fixef.
Usage
## S3 method for class 'mppm'
fixef(object, ...)
Arguments
object

A fitted point process model (an object of class "mppm").

...

Ignored.

Details
This is a method for the generic function fixef.
The argument object must be a fitted point process model (object of class "mppm") produced by
the fitting algorithm mppm). This represents a point process model that has been fitted to a list of
several point pattern datasets. See mppm for information.
This function extracts the coefficients of the fixed effects of the model.
Value
A numeric vector of coefficients.
Author(s)
Adrian Baddeley, Ida-Maria Sintorn and Leanne Bischoff. Implemented in spatstat by Adrian
Baddeley , Rolf Turner  and
Ege Rubak .
References
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. London: Chapman and Hall/CRC Press.
See Also
coef.mppm

flipxy

463

Examples
H <- hyperframe(Y = waterstriders)
# Tweak data to exaggerate differences
H$Y[[1]] <- rthin(H$Y[[1]], 0.3)
m1 <- mppm(Y ~ id, data=H, Strauss(7))
fixef(m1)
m2 <- mppm(Y ~ 1, random=~1|id, data=H, Strauss(7))
fixef(m2)

flipxy

Exchange X and Y Coordinates

Description
Exchanges the x and y coordinates in a spatial dataset.
Usage
flipxy(X)
## S3 method
flipxy(X)
## S3 method
flipxy(X)
## S3 method
flipxy(X)
## S3 method
flipxy(X)

for class 'owin'
for class 'ppp'
for class 'psp'
for class 'im'

Arguments
X

Spatial dataset. An object of class "owin", "ppp", "psp" or "im".

Details
This function swaps the x and y coordinates of a spatial dataset. This could also be performed using
the command affine, but flipxy is faster.
The function flipxy is generic, with methods for the classes of objects listed above.
Value
Another object of the same type, representing the result of swapping the x and y coordinates.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
affine, reflect, rotate, shift

464

FmultiInhom

Examples
data(cells)
X <- flipxy(cells)

FmultiInhom

Inhomogeneous Marked F-Function

Description
For a marked point pattern, estimate the inhomogeneous version of the multitype F function, effectively the cumulative distribution function of the distance from a fixed point to the nearest point in
subset J, adjusted for spatially varying intensity.
Usage
FmultiInhom(X, J,
lambda = NULL, lambdaJ = NULL, lambdamin = NULL,
...,
r = NULL)
Arguments
X

A spatial point pattern (object of class "ppp".

J

A subset index specifying the subset of points to which distances are measured.
Any kind of subset index acceptable to [.ppp.

lambda

Intensity estimates for each point of X. A numeric vector of length equal to
npoints(X). Incompatible with lambdaJ.

lambdaJ

Intensity estimates for each point of X[J]. A numeric vector of length equal to
npoints(X[J]). Incompatible with lambda.

lambdamin

A lower bound for the intensity, or at least a lower bound for the values in
lambdaJ or lambda[J].

...

Ignored.

r

Vector of distance values at which the inhomogeneous G function should be
estimated. There is a sensible default.

Details
See Cronie and Van Lieshout (2015).
Value
Object of class "fv" containing the estimate of the inhomogeneous multitype F function.
Author(s)

Ottmar Cronie and Marie-Colette van Lieshout. Rewritten for spatstat by Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
beginner
Examples
foo

formula.fv

Extract or Change the Plot Formula for a Function Value Table

Description
Extract or change the default plotting formula for an object of class "fv" (function value table).
Usage
## S3 method for class 'fv'
formula(x, ...)
formula(x, ...) <- value
## S3 replacement method for class 'fv'
formula(x, ...) <- value
Arguments
x

An object of class "fv", containing the values of several estimates of a function.

...

Arguments passed to other methods.

value

New value of the formula. Either a formula or a character string.

Details
A function value table (object of class "fv", see fv.object) is a convenient way of storing and
plotting several different estimates of the same function.
The default behaviour of plot(x) for a function value table x is determined by a formula associated
with x called its plot formula. See plot.fv for explanation about these formulae.
The function formula.fv is a method for the generic command formula. It extracts the plot formula associated with the object.
The function formula<- is generic. It changes the formula associated with an object.
The function formula<-.fv is the method for formula<- for the class "fv". It changes the plot
formula associated with the object.

formula.ppm

467

Value
The result of formula.fv is a character string containing the plot formula. The result of formula<-.fv
is a new object of class "fv".
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
fv, plot.fv, formula.
Examples
K <- Kest(cells)
formula(K)
formula(K) <- (iso ~ r)

formula.ppm

Model Formulae for Gibbs Point Process Models

Description
Extract the trend formula, or the terms in the trend formula, in a fitted Gibbs point process model.
Usage
## S3 method for class 'ppm'
formula(x, ...)
## S3 method for class 'ppm'
terms(x, ...)
Arguments
x

An object of class "ppm", representing a fitted point process model.

...

Arguments passed to other methods.

Details
These functions are methods for the generic commands formula and terms for the class "ppm".
An object of class "ppm" represents a fitted Poisson or Gibbs point process model. It is obtained
from the model-fitting function ppm.
The method formula.ppm extracts the trend formula from the fitted model x (the formula originally
specified as the argument trend to ppm). The method terms.ppm extracts the individual terms in
the trend formula.
Value
See the help files for the corresponding generic functions.

468

fourierbasis

Author(s)
Adrian Baddeley 
See Also
ppm, as.owin, coef.ppm, extractAIC.ppm, fitted.ppm, logLik.ppm, model.frame.ppm, model.matrix.ppm,
plot.ppm, predict.ppm, residuals.ppm, simulate.ppm, summary.ppm, update.ppm, vcov.ppm.
Examples
data(cells)
fit <- ppm(cells, ~x)
formula(fit)
terms(fit)

fourierbasis

Fourier Basis Functions

Description
Evaluates the Fourier basis functions on a d-dimensional box with d-dimensional frequencies ki at
the d-dimensional coordinates xj .
Usage
fourierbasis(x, k, win = boxx(rep(list(0:1), ncol(k))))
fourierbasisraw(x, k, boxlengths)
Arguments
x

Coordinates. A data.frame or matrix with n rows and d columns giving the
d-dimensional coordinates.

k

Frequencies. A data.frame or matrix with m rows and d columns giving the
frequencies of the Fourier-functions.

win

window (of class "owin", "box3" or "boxx") giving the d-dimensional box domain of the Fourier functions.

boxlengths

numeric giving the side lengths of the box domain of the Fourier functions.

Details
The result is an m by n matrix where the (i, j)’th entry is the d-dimensional Fourier basis function
with frequency ki evaluated at the point xj , i.e.,
X
1
p
exp(2πi
l = 1d ki,l xj,l /Ll )
|W |
where Ll , l = 1, ..., d are the box side lengths and |W | is the volume of the domain (window/box).
Note that the algorithm does not check whether the coordinates given in x are contained in the
given box. Actually the box is only used to determine the side lengths and volume of the domain
for normalization.
The stripped down faster version fourierbasisraw doesn’t do checking or conversion of arguments and requires x and k to be matrices.

Frame

469

Value
An m by n matrix of complex values.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
Examples
## 27 rows of three dimensional Fourier frequencies:
k <- expand.grid(-1:1,-1:1, -1:1)
## Two random points in the three dimensional unit box:
x <- rbind(runif(3),runif(3))
## 27 by 2 resulting matrix:
v <- fourierbasis(x, k)
head(v)

Frame

Extract or Change the Containing Rectangle of a Spatial Object

Description
Given a spatial object (such as a point pattern or pixel image) in two dimensions, these functions
extract or change the containing rectangle inside which the object is defined.
Usage
Frame(X)
## Default S3 method:
Frame(X)
Frame(X) <- value
## S3 replacement method for class 'owin'
Frame(X) <- value
## S3 replacement method for class 'ppp'
Frame(X) <- value
## S3 replacement method for class 'im'
Frame(X) <- value
## Default S3 replacement method:
Frame(X) <- value

470

fryplot

Arguments
X

A spatial object such as a point pattern, line segment pattern or pixel image.

value

A rectangular window (object of class "owin" of type "rectangle") to be used
as the new containing rectangle for X.

Details
The functions Frame and Frame<- are generic.
Frame(X) extracts the rectangle inside which X is defined.
Frame(X) <- R changes the rectangle inside which X is defined to the new rectangle R.
Value
The result of Frame is a rectangular window (object of class "owin" of type "rectangle").
The result of Frame<- is the updated object X, of the same class as X.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
Window
Examples
Frame(cells)
X <- demopat
Frame(X)
Frame(X) <- owin(c(0, 11000), c(400, 8000))

fryplot

Fry Plot of Point Pattern

Description
Displays the Fry plot (Patterson plot) of a spatial point pattern.
Usage
fryplot(X, ..., width=NULL, from=NULL, to=NULL, axes=FALSE)
frypoints(X, from=NULL, to=NULL, dmax=Inf)

fryplot

471

Arguments
X

A point pattern (object of class "ppp") or something acceptable to as.ppp.

...

Optional arguments to control the appearance of the plot.

width

Optional parameter indicating the width of a box for a zoomed-in view of the
Fry plot near the origin.

from,to

Optional. Subset indices specifying which points of X will be considered when
forming the vectors (drawn from each point of from, to each point of to.)

axes

Logical value indicating whether to draw axes, crossing at the origin.

dmax

Maximum distance between points. Pairs at greater distances do not contribute
to the result. The default means there is no maximum distance.

Details
The function fryplot generates a Fry plot (or Patterson plot); frypoints returns the points of the
Fry plot as a point pattern dataset.
Fry (1979) and Hanna and Fry (1979) introduced a manual graphical method for investigating features of a spatial point pattern of mineral deposits. A transparent sheet, marked with an origin or
centre point, is placed over the point pattern. The transparent sheet is shifted so that the origin
lies over one of the data points, and the positions of all the other data points are copied onto the
transparent sheet. This procedure is repeated for each data point in turn. The resulting plot (the Fry
plot) is a pattern of n(n − 1) points, where n is the original number of data points. This procedure
was previously proposed by Patterson (1934, 1935) for studying inter-atomic distances in crystals,
and is also known as a Patterson plot.
The function fryplot generates the Fry/Patterson plot. Standard graphical parameters such as
main, pch, lwd, col, bg, cex can be used to control the appearance of the plot. To zoom in (to view
only a subset of the Fry plot at higher magnification), use the argument width to specify the width
of a rectangular field of view centred at the origin, or the standard graphical arguments xlim and
ylim to specify another rectangular field of view. (The actual field of view may be slightly larger,
depending on the graphics device.)
The function frypoints returns the points of the Fry plot as a point pattern object. There may be a
large number of points in this pattern, so this function should be used only if further analysis of the
Fry plot is required.
Fry plots are particularly useful for recognising anisotropy in regular point patterns. A void around
the origin in the Fry plot suggests regularity (inhibition between points) and the shape of the void
gives a clue to anisotropy in the pattern. Fry plots are also useful for detecting periodicity or
rounding of the spatial coordinates.
In mathematical terms, the Fry plot of a point pattern X is simply a plot of the vectors X[i] - X[j]
connecting all pairs of distinct points in X.
The Fry plot is related to the K function (see Kest) and the reduced second moment measure (see
Kmeasure). For example, the number of points in the Fry plot lying within a circle of given radius is
an unnormalised and uncorrected version of the K function. The Fry plot has a similar appearance
to the plot of the reduced second moment measure Kmeasure when the smoothing parameter sigma
is very small.
The Fry plot does not adjust for the effect of the size and shape of the sampling window. The density
of points in the Fry plot tapers off near the edges of the plot. This is an edge effect, a consequence
of the bounded sampling window. In geological applications this is usually not important, because
interest is focused on the behaviour near the origin where edge effects can be ignored. To correct
for the edge effect, use Kmeasure or Kest or its relatives.

472

funxy

Value
fryplot returns NULL. frypoints returns a point pattern (object of class "ppp").
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Fry, N. (1979) Random point distributions and strain measurement in rocks. Tectonophysics 60,
89–105.
Hanna, S.S. and Fry, N. (1979) A comparison of methods of strain determination in rocks from
southwest Dyfed (Pembrokeshire) and adjacent areas. Journal of Structural Geology 1, 155–162.
Patterson, A.L. (1934) A Fourier series method for the determination of the component of interatomic distances in crystals. Physics Reviews 46, 372–376.
Patterson, A.L. (1935) A direct method for the determination of the components of inter-atomic
distances in crystals. Zeitschrift fuer Krystallographie 90, 517–554.
See Also
Kmeasure, Kest
Examples
## unmarked data
fryplot(cells)
Y <- frypoints(cells)
## numerical marks
fryplot(longleaf, width=4, axes=TRUE)
## multitype points
fryplot(amacrine, width=0.2,
from=(marks(amacrine) == "on"),
chars=c(3,16), cols=2:3,
main="Fry plot centred at an On-cell")
points(0,0)

funxy

Spatial Function Class

Description
A simple class of functions of spatial location
Usage
funxy(f, W)

fv

473

Arguments
f

A function in the R language with arguments x,y (at least)

W

Window (object of class "owin") inside which the function is well-defined.

Details
This creates an object of class "funxy". This is a simple mechanism for handling a function of
spatial location f (x, y) to make it easier to display and manipulate.
f should be a function in the R language. The first two arguments of f must be named x and y
respectively.
W should be a window (object of class "owin") inside which the function f is well-defined.
The function f should be vectorised: that is, if x and y are numeric vectors of the same length n,
then v <- f(x,y) should be a vector of length n.
The resulting function g <- funxy(f, W) has the same formal arguments as f. It accepts numeric
vectors x,y as described above, but if y is missing, then x may be a point pattern (object of class
"ppp" or "lpp") from which the coordinates should be extracted.
Value
A function with the same arguments as f, which also belongs to the class "funxy". This class has
methods for print, plot, contour and persp.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
plot.funxy
Examples
f <- function(x,y) { x^2 + y^2 - 1}
g <- funxy(f, square(2))
g(0.2, 0.3)
g
g(cells[1:4])

fv

Create a Function Value Table

Description
Advanced Use Only. This low-level function creates an object of class "fv" from raw numerical
data.
Usage
fv(x, argu = "r", ylab = NULL, valu, fmla = NULL, alim = NULL,
labl = names(x), desc = NULL, unitname = NULL, fname = NULL, yexp = ylab)

474

fv

Arguments
x

A data frame with at least 2 columns containing the values of the function argument and the corresponding values of (one or more versions of) the function.

argu

String. The name of the column of x that contains the values of the function
argument.

ylab

Either NULL, or an R language expression representing the mathematical name
of the function. See Details.

valu

String. The name of the column of x that should be taken as containing the
function values, in cases where a single column is required.

fmla

Either NULL, or a formula specifying the default plotting behaviour. See Details.

alim

Optional. The default range of values of the function argument for which the
function will be plotted. Numeric vector of length 2.

labl

Optional. Plot labels for the columns of x. A vector of strings, with one entry
for each column of x.

desc

Optional. Descriptions of the columns of x. A vector of strings, with one entry
for each column of x.

unitname

Optional. Name of the unit (usually a unit of length) in which the function argument is expressed. Either a single character string, or a vector of two character
strings giving the singular and plural forms, respectively.

fname

Optional. The name of the function itself. A character string.

yexp

Optional. Alternative form of ylab more suitable for annotating an axis of the
plot. See Details.

Details
This documentation is provided for experienced programmers who want to modify the internal
behaviour of spatstat. Other users please see fv.object.
The low-level function fv is used to create an object of class "fv" from raw numerical data.
The data frame x contains the numerical data. It should have one column (typically but not necessarily named "r") giving the values of the function argument for which the function has been
evaluated; and at least one other column, containing the corresponding values of the function.
Typically there is more than one column of function values. These columns typically give the values
of different versions or estimates of the same function, for example, different estimates of the K
function obtained using different edge corrections. However they may also contain the values of
related functions such as the derivative or hazard rate.
argu specifies the name of the column of x that contains the values of the function argument (typically argu="r" but this is not compulsory).
valu specifies the name of another column that contains the ‘recommended’ estimate of the function. It will be used to provide function values in those situations where a single column of data is
required. For example, envelope computes its simulation envelopes using the recommended value
of the summary function.
fmla specifies the default plotting behaviour. It should be a formula, or a string that can be converted
to a formula. Variables in the formula are names of columns of x. See plot.fv for the interpretation
of this formula.
alim specifies the recommended range of the function argument. This is used in situations where
statistical theory or statistical practice indicates that the computed estimates of the function are not

fv

475
trustworthy outside a certain range of values of the function argument. By default, plot.fv will
restrict the plot to this range.
fname is a string giving the name of the function itself. For example, the K function would have
fname="K".
ylab is a mathematical expression for the function value, used when labelling an axis of the plot,
or when printing a description of the function. It should be an R language object. For example the
K function’s mathematical name K(r) is rendered by ylab=quote(K(r)).
If yexp is present, then ylab will be used only for printing, and yexp will be used for annotating axes
in a plot. (Otherwise yexp defaults to ylab). For example the cross-type K function K1,2 (r) is rendered by something like ylab=quote(Kcross[1,2](r)) and yexp=quote(Kcross[list(1,2)](r))
to get the most satisfactory behaviour.
(A useful tip: use substitute instead of quote to insert values of variables into an expression, e.g.
substitute(Kcross[i,j](r), list(i=42,j=97)) yields the same as quote(Kcross[42, 97](r)).)
labl is a character vector specifying plot labels for each column of x. These labels will appear on
the plot axes (in non-default plots), legends and printed output. Entries in labl may contain the
string "%s" which will be replaced by fname. For example the border-corrected estimate of the K
function has label "%s[bord](r)" which becomes "K[bord](r)".
desc is a character vector containing intelligible explanations of each column of x. Entries in desc
may contain the string "%s" which will be replaced by ylab. For example the border correction
estimate of the K function has description "border correction estimate of %s".

Value
An object of class "fv", see fv.object.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
See plot.fv for plotting an "fv" object.
See as.function.fv to convert an "fv" object to an R function.
Use cbind.fv to combine several "fv" objects. Use bind.fv to glue additional columns onto an
existing "fv" object.
Use range.fv to compute the range of y values for a function, and with.fv for more complicated
calculations.
The functions fvnames, fvnames<- allow the user to use standard abbreviations to refer to columns
of an "fv" object.
Undocumented functions for modifying an "fv" object include tweak.fv.entry and rebadge.fv.
Examples
df <- data.frame(r=seq(0,5,by=0.1))
df <- transform(df, a=pi*r^2, b=3*r^2)
X <- fv(df, "r", quote(A(r)),
"a", cbind(a, b) ~ r,
alim=c(0,4),
labl=c("r", "%s[true](r)", "%s[approx](r)"),

476

fv.object
desc=c("radius of circle",
"true area %s",
"rough area %s"),
fname="A")

X

fv.object

Function Value Table

Description
A class "fv" to support the convenient plotting of several estimates of the same function.
Details
An object of this class is a convenient way of storing and plotting several different estimates of the
same function.
It is a data frame with extra attributes indicating the recommended way of plotting the function, and
other information.
There are methods for print and plot for this class.
Objects of class "fv" are returned by Fest, Gest,Jest, and Kest along with many other functions.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
Objects of class "fv" are returned by Fest, Gest,Jest, and Kest along with many other functions.
See plot.fv for plotting an "fv" object.
See as.function.fv to convert an "fv" object to an R function.
Use cbind.fv to combine several "fv" objects. Use bind.fv to glue additional columns onto an
existing "fv" object.
Undocumented functions for modifying an "fv" object include fvnames, fvnames<-, tweak.fv.entry
and rebadge.fv.
Examples
data(cells)
K <- Kest(cells)
class(K)
K

# prints a sensible summary

plot(K)

fvnames

477

fvnames

Abbreviations for Groups of Columns in Function Value Table

Description
Groups of columns in a function value table (object of class "fv") identified by standard abbreviations.
Usage
fvnames(X, a = ".")
fvnames(X, a = ".") <- value

Arguments
X

Function value table (object of class "fv"). See fv.object.

a

One of the standard abbreviations listed below.

value

Character vector containing names of columns of X.

Details
An object of class "fv" represents a table of values of a function, usually a summary function
for spatial data such as the K-function, for which several different statistical estimators may be
available. The different estimates are stored as columns of the table.
Auxiliary information carried in the object X specifies some columns or groups of columns of this
table that should be used for particular purposes. For convenience these groups can be referred to
by standard abbreviations which are recognised by various functions in the spatstat package, such
as plot.fv.
These abbreviations are:
".x"
".y"
"."
".s"
".a"

the function argument
the recommended value of the function
all function values to be plotted by default
(in order of plotting)
the upper and lower limits of shading
(for envelopes and confidence intervals)
all function values

The command fvnames(X, a) expands the abbreviation a and returns a character vector containing
the names of the columns.
The assignment fvnames(X, a) <- value changes the definition of the abbreviation a to the
character vector value. It does not change the labels of any columns.
Note that fvnames(x, ".") lists the columns of values that will be plotted by default, in the order
that they would be plotted, not in order of the column position. The order in which curves are
plotted affects the colours and line styles associated with the curves.

478

G3est

Value
For fvnames, a character vector.
For fvnames<-, the updated object.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
fv.object, plot.fv
Examples
K <- Kest(cells)
fvnames(K, ".y")
fvnames(K, ".y") <- "trans"

G3est

Nearest Neighbour Distance Distribution Function of a ThreeDimensional Point Pattern

Description
Estimates the nearest-neighbour distance distribution function G3 (r) from a three-dimensional
point pattern.
Usage
G3est(X, ..., rmax = NULL, nrval = 128, correction = c("rs", "km", "Hanisch"))
Arguments
X

Three-dimensional point pattern (object of class "pp3").

...

Ignored.

rmax

Optional. Maximum value of argument r for which G3 (r) will be estimated.

nrval

Optional. Number of values of r for which G3 (r) will be estimated. A large
value of nrval is required to avoid discretisation effects.

correction

Optional. Character vector specifying the edge correction(s) to be applied. See
Details.

G3est

479

Details
For a stationary point process Φ in three-dimensional space, the nearest-neighbour function is
G3 (r) = P (d∗ (x, Φ) ≤ r | x ∈ Φ)
the cumulative distribution function of the distance d∗ (x, Φ) from a typical point x in Φ to its nearest
neighbour, i.e. to the nearest other point of Φ.
The three-dimensional point pattern X is assumed to be a partial realisation of a stationary point
process Φ. The nearest neighbour function of Φ can then be estimated using techniques described
in the References. For each data point, the distance to the nearest neighbour is computed. The
empirical cumulative distribution function of these values, with appropriate edge corrections, is the
estimate of G3 (r).
The available edge corrections are:
"rs": the reduced sample (aka minus sampling, border correction) estimator (Baddeley et al, 1993)
"km": the three-dimensional version of the Kaplan-Meier estimator (Baddeley and Gill, 1997)
"Hanisch": the three-dimensional generalisation of the Hanisch estimator (Hanisch, 1984).
Alternatively correction="all" selects all options.
Value
A function value table (object of class "fv") that can be plotted, printed or coerced to a data frame
containing the function values.
Warnings
A large value of nrval is required in order to avoid discretisation effects (due to the use of histograms in the calculation).
Author(s)
Adrian Baddeley 
and Rana Moyeed.
References
Baddeley, A.J, Moyeed, R.A., Howard, C.V. and Boyde, A. (1993) Analysis of a three-dimensional
point pattern with replication. Applied Statistics 42, 641–668.
Baddeley, A.J. and Gill, R.D. (1997) Kaplan-Meier estimators of interpoint distance distributions
for spatial point processes. Annals of Statistics 25, 263–292.
Hanisch, K.-H. (1984) Some remarks on estimators of the distribution function of nearest neighbour
distance in stationary spatial point patterns. Mathematische Operationsforschung und Statistik,
series Statistics 15, 409–412.
See Also
F3est, K3est, pcf3est
Examples
X <- rpoispp3(42)
Z <- G3est(X)
if(interactive()) plot(Z)

480

gauss.hermite

gauss.hermite

Gauss-Hermite Quadrature Approximation to Expectation for Normal
Distribution

Description
Calculates an approximation to the expected value of any function of a normally-distributed random
variable, using Gauss-Hermite quadrature.
Usage
gauss.hermite(f, mu = 0, sd = 1, ..., order = 5)
Arguments
f

The function whose moment should be approximated.

mu

Mean of the normal distribution.

sd

Standard deviation of the normal distribution.

...

Additional arguments passed to f.

order

Number of quadrature points in the Gauss-Hermite quadrature approximation.
A small positive integer.

Details
This algorithm calculates the approximate expected value of f(Z) when Z is a normally-distributed
random variable with mean mu and standard deviation sd. The expected value is an integral with
respect to the Gaussian density; this integral is approximated using Gauss-Hermite quadrature.
The argument f should be a function in the R language whose first argument is the variable Z.
Additional arguments may be passed through .... The value returned by f may be a single numeric
value, a vector, or a matrix. The values returned by f for different values of Z must have compatible
dimensions.
The result is a weighted average of several values of f.
Value
Numeric value, vector or matrix.
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak .
Examples
gauss.hermite(function(x) x^2, 3, 1)

Gcom

Gcom

481

Model Compensator of Nearest Neighbour Function

Description
Given a point process model fitted to a point pattern dataset, this function computes the compensator of the nearest neighbour distance distribution function G based on the fitted model (as well
as the usual nonparametric estimates of G based on the data alone). Comparison between the nonparametric and model-compensated G functions serves as a diagnostic for the model.
Usage
Gcom(object, r = NULL, breaks = NULL, ...,
correction = c("border", "Hanisch"),
conditional = !is.poisson(object),
restrict=FALSE,
model=NULL,
trend = ~1, interaction = Poisson(),
rbord = reach(interaction),
ppmcorrection="border",
truecoef = NULL, hi.res = NULL)
Arguments
object

Object to be analysed. Either a fitted point process model (object of class "ppm")
or a point pattern (object of class "ppp") or quadrature scheme (object of class
"quad").

r

Optional. Vector of values of the argument r at which the function G(r) should
be computed. This argument is usually not specified. There is a sensible default.

breaks

This argument is for internal use only.

correction

Edge correction(s) to be employed in calculating the compensator. Options are
"border", "Hanisch" and "best". Alternatively correction="all" selects
all options.

conditional

Optional. Logical value indicating whether to compute the estimates for the
conditional case. See Details.

restrict

Logical value indicating whether to compute the restriction estimator (restrict=TRUE)
or the reweighting estimator (restrict=FALSE, the default). Applies only if
conditional=TRUE. See Details.

model

Optional. A fitted point process model (object of class "ppm") to be re-fitted to
the data using update.ppm, if object is a point pattern. Overrides the arguments trend,interaction,rbord,ppmcorrection.
trend,interaction,rbord
Optional. Arguments passed to ppm to fit a point process model to the data, if
object is a point pattern. See ppm for details.
...

Extra arguments passed to ppm.

ppmcorrection

The correction argument to ppm.

truecoef

Optional. Numeric vector. If present, this will be treated as if it were the true
coefficient vector of the point process model, in calculating the diagnostic. Incompatible with hi.res.

482

Gcom
hi.res

Optional. List of parameters passed to quadscheme. If this argument is present,
the model will be re-fitted at high resolution as specified by these parameters.
The coefficients of the resulting fitted model will be taken as the true coefficients. Then the diagnostic will be computed for the default quadrature scheme,
but using the high resolution coefficients.

Details
This command provides a diagnostic for the goodness-of-fit of a point process model fitted to a
point pattern dataset. It computes different estimates of the nearest neighbour distance distribution
function G of the dataset, which should be approximately equal if the model is a good fit to the data.
The first argument, object, is usually a fitted point process model (object of class "ppm"), obtained
from the model-fitting function ppm.
For convenience, object can also be a point pattern (object of class "ppp"). In that case, a point
process model will be fitted to it, by calling ppm using the arguments trend (for the first order
trend), interaction (for the interpoint interaction) and rbord (for the erosion distance in the border correction for the pseudolikelihood). See ppm for details of these arguments.
The algorithm first extracts the original point pattern dataset (to which the model was fitted) and
computes the standard nonparametric estimates of the G function. It then also computes the modelcompensated G function. The different functions are returned as columns in a data frame (of class
"fv"). The interpretation of the columns is as follows (ignoring edge corrections):
bord: the nonparametric border-correction estimate of G(r),
P
I{di ≤ r}I{bi > r}
Ĝ(r) = i P
i I{bi > r}
where di is the distance from the i-th data point to its nearest neighbour, and bi is the distance
from the i-th data point to the boundary of the window W .
bcom: the model compensator of the border-correction estimate
R
λ(u, x)I{b(u) > r}I{d(u, x) ≤ r}
P
C Ĝ(r) =
1 + i I{bi > r}
where λ(u, x) denotes the conditional intensity of the model at the location u, and d(u, x)
denotes the distance from u to the nearest point in x, while b(u) denotes the distance from u
to the boundary of the windowW .
han: the nonparametric Hanisch estimate of G(r)
Ĝ(r) =
where
D(r) =
in which W

r

D(r)
D(∞)

X I{xi ∈ W d }I{di ≤ r}
i
area(W
di )
i

denotes the erosion of the window W by a distance r.

hcom: the corresponding model-compensated function
Z
λ(u, x)I(u ∈ W d(u) )I(d(u) ≤ r)
C G(r) =
D̂(∞)area(W d(u) ) + 1
W
where d(u) = d(u, x) is the (‘empty space’) distance from location u to the nearest point of
x.

Gcom

483

If the fitted model is a Poisson point process, then the formulae above are exactly what is computed.
If the fitted model is not Poisson, the formulae above are modified slightly to handle edge effects.
The modification is determined by the arguments conditional and restrict. The value of
conditional defaults to FALSE for Poisson models and TRUE for non-Poisson models. If conditional=FALSE
then the formulae above are not modified. If conditional=TRUE, then the algorithm calculates the
restriction estimator if restrict=TRUE, and calculates the reweighting estimator if restrict=FALSE.
See Appendix E of Baddeley, Rubak and Møller (2011). See also spatstat.options('eroded.intensity').
Thus, by default, the reweighting estimator is computed for non-Poisson models.
The border-corrected and Hanisch-corrected estimates of G(r) are approximately unbiased estimates of the G-function, assuming the point process is stationary. The model-compensated functions are unbiased estimates of the mean value of the corresponding nonparametric estimate, assuming the model is true. Thus, if the model is a good fit, the mean value of the difference between
the nonparametric and model-compensated estimates is approximately zero.
To compute the difference between the nonparametric and model-compensated functions, use Gres.
Value
A function value table (object of class "fv"), essentially a data frame of function values. There is a
plot method for this class. See fv.object.
Author(s)
Adrian Baddeley 
Ege Rubak  and Jesper Møller.
References
Baddeley, A., Rubak, E. and Møller, J. (2011) Score, pseudo-score and residual diagnostics for
spatial point process models. Statistical Science 26, 613–646.
See Also
Related functions: Gest, Gres.
Alternative functions: Kcom, psstA, psstG, psst.
Model fitting: ppm.
Examples
data(cells)
fit0 <- ppm(cells, ~1) # uniform Poisson
G0 <- Gcom(fit0)
G0
plot(G0)
# uniform Poisson is clearly not correct
# Hanisch estimates only
plot(Gcom(fit0), cbind(han, hcom) ~ r)
fit1 <- ppm(cells, ~1, Strauss(0.08))
plot(Gcom(fit1), cbind(han, hcom) ~ r)
# Try adjusting interaction distance

484

Gcross
fit2 <- update(fit1, Strauss(0.10))
plot(Gcom(fit2), cbind(han, hcom) ~ r)
G3 <- Gcom(cells, interaction=Strauss(0.12))
plot(G3, cbind(han, hcom) ~ r)

Gcross

Multitype Nearest Neighbour Distance Function (i-to-j)

Description
For a multitype point pattern, estimate the distribution of the distance from a point of type i to the
nearest point of type j.
Usage
Gcross(X, i, j, r=NULL, breaks=NULL, ..., correction=c("rs", "km", "han"))
Arguments
X

The observed point pattern, from which an estimate of the cross type distance
distribution function Gij (r) will be computed. It must be a multitype point
pattern (a marked point pattern whose marks are a factor). See under Details.

i

The type (mark value) of the points in X from which distances are measured.
A character string (or something that will be converted to a character string).
Defaults to the first level of marks(X).

j

The type (mark value) of the points in X to which distances are measured. A
character string (or something that will be converted to a character string). Defaults to the second level of marks(X).

r

Optional. Numeric vector. The values of the argument r at which the distribution
function Gij (r) should be evaluated. There is a sensible default. First-time
users are strongly advised not to specify this argument. See below for important
conditions on r.

breaks

This argument is for internal use only.

...

Ignored.

correction

Optional. Character string specifying the edge correction(s) to be used. Options
are "none", "rs", "km", "hanisch" and "best". Alternatively correction="all"
selects all options.

Details
This function Gcross and its companions Gdot and Gmulti are generalisations of the function Gest
to multitype point patterns.
A multitype point pattern is a spatial pattern of points classified into a finite number of possible
“colours” or “types”. In the spatstat package, a multitype pattern is represented as a single point
pattern object in which the points carry marks, and the mark value attached to each point determines
the type of that point.
The argument X must be a point pattern (object of class "ppp") or any data that are acceptable to
as.ppp. It must be a marked point pattern, and the mark vector X$marks must be a factor. The

Gcross

485

arguments i and j will be interpreted as levels of the factor X$marks. (Warning: this means that an
integer value i=3 will be interpreted as the number 3, not the 3rd smallest level).
The “cross-type” (type i to type j) nearest neighbour distance distribution function of a multitype
point process is the cumulative distribution function Gij (r) of the distance from a typical random
point of the process with type i the nearest point of type j.
An estimate of Gij (r) is a useful summary statistic in exploratory data analysis of a multitype point
pattern. If the process of type i points were independent of the process of type j points, then Gij (r)
would equal Fj (r), the empty space function of the type j points. For a multitype Poisson point
process where the type i points have intensity λi , we have
Gij (r) = 1 − e−λj πr

2

Deviations between the empirical and theoretical Gij curves may suggest dependence between the
points of types i and j.
This algorithm estimates the distribution function Gij (r) from the point pattern X. It assumes that
X can be treated as a realisation of a stationary (spatially homogeneous) random spatial point process in the plane, observed through a bounded window. The window (which is specified in X as
Window(X)) may have arbitrary shape. Biases due to edge effects are treated in the same manner as
in Gest.
The argument r is the vector of values for the distance r at which Gij (r) should be evaluated. It is
also used to determine the breakpoints (in the sense of hist) for the computation of histograms of
distances. The reduced-sample and Kaplan-Meier estimators are computed from histogram counts.
In the case of the Kaplan-Meier estimator this introduces a discretisation error which is controlled
by the fineness of the breakpoints.
First-time users would be strongly advised not to specify r. However, if it is specified, r must
satisfy r[1] = 0, and max(r) must be larger than the radius of the largest disc contained in the
window. Furthermore, the successive entries of r must be finely spaced.
The algorithm also returns an estimate of the hazard rate function, λ(r), of Gij (r). This estimate
should be used with caution as Gij (r) is not necessarily differentiable.
The naive empirical distribution of distances from each point of the pattern X to the nearest other
point of the pattern, is a biased estimate of Gij . However this is also returned by the algorithm, as
it is sometimes useful in other contexts. Care should be taken not to use the uncorrected empirical
Gij as if it were an unbiased estimator of Gij .
Value
An object of class "fv" (see fv.object).
Essentially a data frame containing six numeric columns
r

the values of the argument r at which the function Gij (r) has been estimated

rs

the “reduced sample” or “border correction” estimator of Gij (r)

han

the Hanisch-style estimator of Gij (r)

km

the spatial Kaplan-Meier estimator of Gij (r)

hazard

the hazard rate λ(r) of Gij (r) by the spatial Kaplan-Meier method

raw

the uncorrected estimate of Gij (r), i.e. the empirical distribution of the distances from each point of type i to the nearest point of type j

theo

the theoretical value of Gij (r) for a marked Poisson process with the same estimated intensity (see below).

486

Gcross

Warnings
The arguments i and j are always interpreted as levels of the factor X$marks. They are converted
to character strings if they are not already character strings. The value i=1 does not refer to the first
level of the factor.
The function Gij does not necessarily have a density.
The reduced sample estimator of Gij is pointwise approximately unbiased, but need not be a valid
distribution function; it may not be a nondecreasing function of r. Its range is always within [0, 1].
The spatial Kaplan-Meier estimator of Gij is always nondecreasing but its maximum value may be
less than 1.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.
Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 1983.
Diggle, P. J. (1986). Displaced amacrine cells in the retina of a rabbit : analysis of a bivariate spatial
point pattern. J. Neurosci. Meth. 18, 115–125.
Harkness, R.D and Isham, V. (1983) A bivariate spatial point pattern of ants’ nests. Applied Statistics 32, 293–303
Lotwick, H. W. and Silverman, B. W. (1982). Methods for analysing spatial processes of several
types of points. J. Royal Statist. Soc. Ser. B 44, 406–413.
Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.
Stoyan, D, Kendall, W.S. and Mecke, J. Stochastic geometry and its applications. 2nd edition.
Springer Verlag, 1995.
Van Lieshout, M.N.M. and Baddeley, A.J. (1999) Indices of dependence between types in multivariate point patterns. Scandinavian Journal of Statistics 26, 511–532.
See Also
Gdot, Gest, Gmulti
Examples
# amacrine cells data
G01 <- Gcross(amacrine)
# equivalent to:
## Not run:
G01 <- Gcross(amacrine, "off", "on")
## End(Not run)
plot(G01)
# empty space function of `on' points
## Not run:

Gdot

487
F1 <- Fest(split(amacrine)$on, r = G01$r)
lines(F1$r, F1$km, lty=3)

## End(Not run)
# synthetic example
pp <- runifpoispp(30)
pp <- pp %mark% factor(sample(0:1, npoints(pp), replace=TRUE))
G <- Gcross(pp, "0", "1") # note: "0" not 0

Gdot

Multitype Nearest Neighbour Distance Function (i-to-any)

Description
For a multitype point pattern, estimate the distribution of the distance from a point of type i to the
nearest other point of any type.
Usage
Gdot(X, i, r=NULL, breaks=NULL, ..., correction=c("km", "rs", "han"))
Arguments
X

The observed point pattern, from which an estimate of the distance distribution
function Gi• (r) will be computed. It must be a multitype point pattern (a marked
point pattern whose marks are a factor). See under Details.

i

The type (mark value) of the points in X from which distances are measured.
A character string (or something that will be converted to a character string).
Defaults to the first level of marks(X).

r

Optional. Numeric vector. The values of the argument r at which the distribution
function Gi• (r) should be evaluated. There is a sensible default. First-time
users are strongly advised not to specify this argument. See below for important
conditions on r.

breaks

This argument is for internal use only.

...

Ignored.

correction

Optional. Character string specifying the edge correction(s) to be used. Options
are "none", "rs", "km", "hanisch" and "best". Alternatively correction="all"
selects all options.

Details
This function Gdot and its companions Gcross and Gmulti are generalisations of the function Gest
to multitype point patterns.
A multitype point pattern is a spatial pattern of points classified into a finite number of possible
“colours” or “types”. In the spatstat package, a multitype pattern is represented as a single point
pattern object in which the points carry marks, and the mark value attached to each point determines
the type of that point.
The argument X must be a point pattern (object of class "ppp") or any data that are acceptable to
as.ppp. It must be a marked point pattern, and the mark vector X$marks must be a factor. The

488

Gdot
argument will be interpreted as a level of the factor X$marks. (Warning: this means that an integer
value i=3 will be interpreted as the number 3, not the 3rd smallest level.)
The “dot-type” (type i to any type) nearest neighbour distance distribution function of a multitype
point process is the cumulative distribution function Gi• (r) of the distance from a typical random
point of the process with type i the nearest other point of the process, regardless of type.
An estimate of Gi• (r) is a useful summary statistic in exploratory data analysis of a multitype point
pattern. If the type i points were independent of all other points, then Gi• (r) would equal Gii (r),
the nearest neighbour distance distribution function of the type i points alone. For a multitype
Poisson point process with total intensity λ, we have
Gi• (r) = 1 − e−λπr

2

Deviations between the empirical and theoretical Gi• curves may suggest dependence of the type i
points on the other points.
This algorithm estimates the distribution function Gi• (r) from the point pattern X. It assumes that
X can be treated as a realisation of a stationary (spatially homogeneous) random spatial point process in the plane, observed through a bounded window. The window (which is specified in X as
Window(X)) may have arbitrary shape. Biases due to edge effects are treated in the same manner as
in Gest.
The argument r is the vector of values for the distance r at which Gi• (r) should be evaluated. It is
also used to determine the breakpoints (in the sense of hist) for the computation of histograms of
distances. The reduced-sample and Kaplan-Meier estimators are computed from histogram counts.
In the case of the Kaplan-Meier estimator this introduces a discretisation error which is controlled
by the fineness of the breakpoints.
First-time users would be strongly advised not to specify r. However, if it is specified, r must
satisfy r[1] = 0, and max(r) must be larger than the radius of the largest disc contained in the
window. Furthermore, the successive entries of r must be finely spaced.
The algorithm also returns an estimate of the hazard rate function, λ(r), of Gi• (r). This estimate
should be used with caution as Gi• (r) is not necessarily differentiable.
The naive empirical distribution of distances from each point of the pattern X to the nearest other
point of the pattern, is a biased estimate of Gi• . However this is also returned by the algorithm, as
it is sometimes useful in other contexts. Care should be taken not to use the uncorrected empirical
Gi• as if it were an unbiased estimator of Gi• .
Value
An object of class "fv" (see fv.object).
Essentially a data frame containing six numeric columns
r

the values of the argument r at which the function Gi• (r) has been estimated

rs

the “reduced sample” or “border correction” estimator of Gi• (r)

han

the Hanisch-style estimator of Gi• (r)

km

the spatial Kaplan-Meier estimator of Gi• (r)

hazard

the hazard rate λ(r) of Gi• (r) by the spatial Kaplan-Meier method

raw

the uncorrected estimate of Gi• (r), i.e. the empirical distribution of the distances from each point of type i to the nearest other point of any type.

theo

the theoretical value of Gi• (r) for a marked Poisson process with the same estimated intensity (see below).

Gdot

489

Warnings
The argument i is interpreted as a level of the factor X$marks. It is converted to a character string
if it is not already a character string. The value i=1 does not refer to the first level of the factor.
The function Gi• does not necessarily have a density.
The reduced sample estimator of Gi• is pointwise approximately unbiased, but need not be a valid
distribution function; it may not be a nondecreasing function of r. Its range is always within [0, 1].
The spatial Kaplan-Meier estimator of Gi• is always nondecreasing but its maximum value may be
less than 1.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.
Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 1983.
Diggle, P. J. (1986). Displaced amacrine cells in the retina of a rabbit : analysis of a bivariate spatial
point pattern. J. Neurosci. Meth. 18, 115–125.
Harkness, R.D and Isham, V. (1983) A bivariate spatial point pattern of ants’ nests. Applied Statistics 32, 293–303
Lotwick, H. W. and Silverman, B. W. (1982). Methods for analysing spatial processes of several
types of points. J. Royal Statist. Soc. Ser. B 44, 406–413.
Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.
Stoyan, D, Kendall, W.S. and Mecke, J. Stochastic geometry and its applications. 2nd edition.
Springer Verlag, 1995.
Van Lieshout, M.N.M. and Baddeley, A.J. (1999) Indices of dependence between types in multivariate point patterns. Scandinavian Journal of Statistics 26, 511–532.
See Also
Gcross, Gest, Gmulti
Examples
# amacrine cells data
G0. <- Gdot(amacrine, "off")
plot(G0.)
# synthetic example
pp <- runifpoispp(30)
pp <- pp %mark% factor(sample(0:1, npoints(pp), replace=TRUE))
G <- Gdot(pp, "0")
G <- Gdot(pp, 0) # equivalent

490

Gest

Gest

Nearest Neighbour Distance Function G

Description
Estimates the nearest neighbour distance distribution function G(r) from a point pattern in a window of arbitrary shape.
Usage
Gest(X, r=NULL, breaks=NULL, ...,
correction=c("rs", "km", "han"),
domain=NULL)
Arguments
X

The observed point pattern, from which an estimate of G(r) will be computed.
An object of class ppp, or data in any format acceptable to as.ppp().

r

Optional. Numeric vector. The values of the argument r at which G(r) should
be evaluated. There is a sensible default. First-time users are strongly advised
not to specify this argument. See below for important conditions on r.

breaks

This argument is for internal use only.

...

Ignored.

correction

Optional. The edge correction(s) to be used to estimate G(r). A vector of
character strings selected from "none", "rs", "km", "Hanisch" and "best".
Alternatively correction="all" selects all options.

domain

Optional. Calculations will be restricted to this subset of the window. See Details.

Details
The nearest neighbour distance distribution function (also called the “event-to-event” or “interevent” distribution) of a point process X is the cumulative distribution function G of the distance
from a typical random point of X to the nearest other point of X.
An estimate of G derived from a spatial point pattern dataset can be used in exploratory data analysis
and formal inference about the pattern (Cressie, 1991; Diggle, 1983; Ripley, 1988). In exploratory
analyses, the estimate of G is a useful statistic summarising one aspect of the “clustering” of points.
For inferential purposes, the estimate of G is usually compared to the true value of G for a completely random (Poisson) point process, which is
G(r) = 1 − e−λπr

2

where λ is the intensity (expected number of points per unit area). Deviations between the empirical
and theoretical G curves may suggest spatial clustering or spatial regularity.
This algorithm estimates the nearest neighbour distance distribution function G from the point pattern X. It assumes that X can be treated as a realisation of a stationary (spatially homogeneous) random spatial point process in the plane, observed through a bounded window. The window (which
is specified in X as Window(X)) may have arbitrary shape.

Gest

491

The argument X is interpreted as a point pattern object (of class "ppp", see ppp.object) and can be
supplied in any of the formats recognised by as.ppp().
The estimation of G is hampered by edge effects arising from the unobservability of points of the
random pattern outside the window. An edge correction is needed to reduce bias (Baddeley, 1998;
Ripley, 1988). The edge corrections implemented here are the border method or “reduced sample”
estimator, the spatial Kaplan-Meier estimator (Baddeley and Gill, 1997) and the Hanisch estimator
(Hanisch, 1984).
The argument r is the vector of values for the distance r at which G(r) should be evaluated. It
is also used to determine the breakpoints (in the sense of hist) for the computation of histograms
of distances. The estimators are computed from histogram counts. This introduces a discretisation
error which is controlled by the fineness of the breakpoints.
First-time users would be strongly advised not to specify r. However, if it is specified, r must
satisfy r[1] = 0, and max(r) must be larger than the radius of the largest disc contained in the
window. Furthermore, the successive entries of r must be finely spaced.
The algorithm also returns an estimate of the hazard rate function, λ(r), of G(r). The hazard rate
is defined as the derivative
d
λ(r) = − log(1 − G(r))
dr
This estimate should be used with caution as G is not necessarily differentiable.
If the argument domain is given, the estimate of G(r) will be based only on the nearest neighbour
distances measured from points falling inside domain (although their nearest neighbours may lie
outside domain). This is useful in bootstrap techniques. The argument domain should be a window
(object of class "owin") or something acceptable to as.owin. It must be a subset of the window of
the point pattern X.
The naive empirical distribution of distances from each point of the pattern X to the nearest other
point of the pattern, is a biased estimate of G. However it is sometimes useful. It can be returned by
the algorithm, by selecting correction="none". Care should be taken not to use the uncorrected
empirical G as if it were an unbiased estimator of G.
To simply compute the nearest neighbour distance for each point in the pattern, use nndist. To
determine which point is the nearest neighbour of a given point, use nnwhich.
Value
An object of class "fv", see fv.object, which can be plotted directly using plot.fv.
Essentially a data frame containing some or all of the following columns:
r

the values of the argument r at which the function G(r) has been estimated

rs

the “reduced sample” or “border correction” estimator of G(r)

km

the spatial Kaplan-Meier estimator of G(r)

hazard

the hazard rate λ(r) of G(r) by the spatial Kaplan-Meier method

raw

the uncorrected estimate of G(r), i.e. the empirical distribution of the distances
from each point in the pattern X to the nearest other point of the pattern

han

the Hanisch correction estimator of G(r)

theo

the theoretical value of G(r) for a stationary Poisson process of the same estimated intensity.

492

Gest

Warnings
The function G does not necessarily have a density. Any valid c.d.f. may appear as the nearest
neighbour distance distribution function of a stationary point process.
The reduced sample estimator of G is pointwise approximately unbiased, but need not be a valid
distribution function; it may not be a nondecreasing function of r. Its range is always within [0, 1].
The spatial Kaplan-Meier estimator of G is always nondecreasing but its maximum value may be
less than 1.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Baddeley, A.J. Spatial sampling and censoring. In O.E. Barndorff-Nielsen, W.S. Kendall and
M.N.M. van Lieshout (eds) Stochastic Geometry: Likelihood and Computation. Chapman and
Hall, 1998. Chapter 2, pages 37-78.
Baddeley, A.J. and Gill, R.D. Kaplan-Meier estimators of interpoint distance distributions for spatial
point processes. Annals of Statistics 25 (1997) 263-292.
Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.
Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 1983.
Hanisch, K.-H. (1984) Some remarks on estimators of the distribution function of nearest-neighbour
distance in stationary spatial point patterns. Mathematische Operationsforschung und Statistik,
series Statistics 15, 409–412.
Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.
Stoyan, D, Kendall, W.S. and Mecke, J. Stochastic geometry and its applications. 2nd edition.
Springer Verlag, 1995.
See Also
nndist, nnwhich, Fest, Jest, Kest, km.rs, reduced.sample, kaplan.meier
Examples
data(cells)
G <- Gest(cells)
plot(G)
# P-P style plot
plot(G, cbind(km,theo) ~ theo)
# the empirical G is below the Poisson G,
# indicating an inhibited pattern
## Not run:
plot(G, . ~ r)
plot(G, . ~ theo)
plot(G, asin(sqrt(.)) ~ asin(sqrt(theo)))
## End(Not run)

Geyer

493

Geyer

Geyer’s Saturation Point Process Model

Description
Creates an instance of Geyer’s saturation point process model which can then be fitted to point
pattern data.
Usage
Geyer(r,sat)
Arguments
r

Interaction radius. A positive real number.

sat

Saturation threshold. A non-negative real number.

Details
Geyer (1999) introduced the “saturation process”, a modification of the Strauss process (see Strauss)
in which the total contribution to the potential from each point (from its pairwise interaction with
all other points) is trimmed to a maximum value s. The interaction structure of this model is implemented in the function Geyer().
The saturation point process with interaction radius r, saturation threshold s, and parameters β and
γ, is the point process in which each point xi in the pattern X contributes a factor
βγ min(s,t(xi ,X))
to the probability density of the point pattern, where t(xi , X) denotes the number of ‘close neighbours’ of xi in the pattern X. A close neighbour of xi is a point xj with j 6= i such that the distance
between xi and xj is less than or equal to r.
If the saturation threshold s is set to infinity, this model reduces to the Strauss process (see Strauss)
with interaction parameter γ 2 . If s = 0, the model reduces to the Poisson point process. If s is a
finite positive number, then the interaction parameter γ may take any positive value (unlike the case
of the Strauss process), with values γ < 1 describing an ‘ordered’ or ‘inhibitive’ pattern, and values
γ > 1 describing a ‘clustered’ or ‘attractive’ pattern.
The nonstationary saturation process is similar except that the value β is replaced by a function
β(xi ) of location.
The function ppm(), which fits point process models to point pattern data, requires an argument
of class "interact" describing the interpoint interaction structure of the model to be fitted. The
appropriate description of the saturation process interaction is yielded by Geyer(r, sat) where
the arguments r and sat specify the Strauss interaction radius r and the saturation threshold s,
respectively. See the examples below.
Note the only arguments are the interaction radius r and the saturation threshold sat. When r and
sat are fixed, the model becomes an exponential family. The canonical parameters log(β) and
log(γ) are estimated by ppm(), not fixed in Geyer().
Value
An object of class "interact" describing the interpoint interaction structure of Geyer’s saturation
point process with interaction radius r and saturation threshold sat.

494

Gfox

Zero saturation
The value sat=0 is permitted by Geyer, but this is not very useful. For technical reasons, when ppm
fits a Geyer model with sat=0, the default behaviour is to return an “invalid” fitted model in which
the estimate of γ is NA. In order to get a Poisson process model returned when sat=0, you would
need to set emend=TRUE in the call to ppm.
Author(s)
Adrian Baddeley  and Rolf Turner 
References
Geyer, C.J. (1999) Likelihood Inference for Spatial Point Processes. Chapter 3 in O.E. BarndorffNielsen, W.S. Kendall and M.N.M. Van Lieshout (eds) Stochastic Geometry: Likelihood and Computation, Chapman and Hall / CRC, Monographs on Statistics and Applied Probability, number 80.
Pages 79–140.
See Also
ppm, pairwise.family, ppm.object, Strauss, SatPiece
Examples
ppm(cells, ~1, Geyer(r=0.07, sat=2))
# fit the stationary saturation process to `cells'

Gfox

Foxall’s Distance Functions

Description
Given a point pattern X and a spatial object Y, compute estimates of Foxall’s G and J functions.
Usage
Gfox(X, Y, r = NULL, breaks = NULL, correction = c("km", "rs", "han"), ...)
Jfox(X, Y, r = NULL, breaks = NULL, correction = c("km", "rs", "han"), ...)
Arguments
X
Y
r

breaks
correction

...

A point pattern (object of class "ppp") from which distances will be measured.
An object of class "ppp", "psp" or "owin" to which distances will be measured.
Optional. Numeric vector. The values of the argument r at which Gf ox(r) or
Jf ox(r) should be evaluated. There is a sensible default. First-time users are
strongly advised not to specify this argument. See below for important conditions on r.
This argument is for internal use only.
Optional. The edge correction(s) to be used to estimate Gf ox(r) or Jf ox(r). A
vector of character strings selected from "none", "rs", "km", "cs" and "best".
Alternatively correction="all" selects all options.
Extra arguments affecting the discretisation of distances. These arguments are
ignored by Gfox, but Jfox passes them to Hest to determine the discretisation
of the spatial domain.

Gfox

495

Details
Given a point pattern X and another spatial object Y, these functions compute two nonparametric
measures of association between X and Y, introduced by Foxall (Foxall and Baddeley, 2002).
Let the random variable R be the distance from a typical point of X to the object Y. Foxall’s Gfunction is the cumulative distribution function of R:
G(r) = P (R ≤ r)
Let the random variable S be the distance from a fixed point in space to the object Y. The cumulative
distribution function of S is the (unconditional) spherical contact distribution function
H(r) = P (S ≤ r)
which is computed by Hest.
Foxall’s J-function is the ratio
J(r) =

1 − G(r)
1 − H(r)

For further interpretation, see Foxall and Baddeley (2002).
Accuracy of Jfox depends on the pixel resolution, which is controlled by the arguments eps, dimyx
and xy passed to as.mask. For example, use eps=0.1 to specify square pixels of side 0.1 units, and
dimyx=256 to specify a 256 by 256 grid of pixels.
Value
A function value table (object of class "fv") which can be printed, plotted, or converted to a data
frame of values.
Author(s)
Rob Foxall and Adrian Baddeley 
References
Foxall, R. and Baddeley, A. (2002) Nonparametric measures of association between a spatial point
process and a random set, with geological applications. Applied Statistics 51, 165–182.
See Also
Gest, Hest, Jest, Fest
Examples
data(copper)
X <- copper$SouthPoints
Y <- copper$SouthLines
G <- Gfox(X,Y)
J <- Jfox(X,Y, correction="km")
## Not run:
J <- Jfox(X,Y, correction="km", eps=0.25)
## End(Not run)

496

Ginhom

Ginhom

Inhomogeneous Nearest Neighbour Function

Description
Estimates the inhomogeneous nearest neighbour function G of a non-stationary point pattern.
Usage
Ginhom(X, lambda = NULL, lmin = NULL, ...,
sigma = NULL, varcov = NULL,
r = NULL, breaks = NULL, ratio = FALSE, update = TRUE)
Arguments
X

lambda

lmin
sigma,varcov
...
r
breaks
ratio
update

The observed data point pattern, from which an estimate of the inhomogeneous
G function will be computed. An object of class "ppp" or in a format recognised
by as.ppp()
Optional. Values of the estimated intensity function. Either a vector giving the
intensity values at the points of the pattern X, a pixel image (object of class "im")
giving the intensity values at all locations, a fitted point process model (object of
class "ppm") or a function(x,y) which can be evaluated to give the intensity
value at any location.
Optional. The minimum possible value of the intensity over the spatial domain.
A positive numerical value.
Optional arguments passed to density.ppp to control the smoothing bandwidth, when lambda is estimated by kernel smoothing.
Extra arguments passed to as.mask to control the pixel resolution, or passed to
density.ppp to control the smoothing bandwidth.
vector of values for the argument r at which the inhomogeneous K function
should be evaluated. Not normally given by the user; there is a sensible default.
This argument is for internal use only.
Logical. If TRUE, the numerator and denominator of the estimate will also be
saved, for use in analysing replicated point patterns.
Logical. If lambda is a fitted model (class "ppm" or "kppm") and update=TRUE
(the default), the model will first be refitted to the data X (using update.ppm
or update.kppm) before the fitted intensity is computed. If update=FALSE, the
fitted intensity of the model will be computed without fitting it to X.

Details
This command computes estimates of the inhomogeneous G-function (van Lieshout, 2010) of
a point pattern. It is the counterpart, for inhomogeneous spatial point patterns, of the nearestneighbour distance distribution function G for homogeneous point patterns computed by Gest.
The argument X should be a point pattern (object of class "ppp").
The inhomogeneous G function is computed using the border correction, equation (7) in Van
Lieshout (2010).
The argument lambda should supply the (estimated) values of the intensity function λ of the point
process. It may be either

Ginhom

497

a numeric vector containing the values of the intensity function at the points of the pattern X.
a pixel image (object of class "im") assumed to contain the values of the intensity function at all
locations in the window.
a fitted point process model (object of class "ppm" or "kppm") whose fitted trend can be used as
the fitted intensity. (If update=TRUE the model will first be refitted to the data X before the
trend is computed.)
a function which can be evaluated to give values of the intensity at any locations.
omitted: if lambda is omitted, then it will be estimated using a ‘leave-one-out’ kernel smoother.
If lambda is a numeric vector, then its length should be equal to the number of points in the pattern
X. The value lambda[i] is assumed to be the the (estimated) value of the intensity λ(xi ) for the
point xi of the pattern X. Each value must be a positive number; NA’s are not allowed.
If lambda is a pixel image, the domain of the image should cover the entire window of the point
pattern. If it does not (which may occur near the boundary because of discretisation error), then
the missing pixel values will be obtained by applying a Gaussian blur to lambda using blur, then
looking up the values of this blurred image for the missing locations. (A warning will be issued in
this case.)
If lambda is a function, then it will be evaluated in the form lambda(x,y) where x and y are vectors
of coordinates of the points of X. It should return a numeric vector with length equal to the number
of points in X.
If lambda is omitted, then it will be estimated using a ‘leave-one-out’ kernel smoother, as described
in Baddeley, Møller and Waagepetersen (2000). The estimate lambda[i] for the point X[i] is computed by removing X[i] from the point pattern, applying kernel smoothing to the remaining points
using density.ppp, and evaluating the smoothed intensity at the point X[i]. The smoothing kernel
bandwidth is controlled by the arguments sigma and varcov, which are passed to density.ppp
along with any extra arguments.
Value
An object of class "fv", see fv.object, which can be plotted directly using plot.fv.
Author(s)
Original code by Marie-Colette van Lieshout. C implementation and R adaptation by Adrian Baddeley 
and Ege Rubak .
References
Baddeley, A., Møller, J. and Waagepetersen, R. (2000) Non- and semiparametric estimation of
interaction in inhomogeneous point patterns. Statistica Neerlandica 54, 329–350.
Van Lieshout, M.N.M. and Baddeley, A.J. (1996) A nonparametric measure of spatial interaction in
point patterns. Statistica Neerlandica 50, 344–361.
Van Lieshout, M.N.M. (2010) A J-function for inhomogeneous point processes. Statistica Neerlandica 65, 183–201.
See Also
Finhom, Jinhom, Gest

498

Gmulti

Examples
## Not run:
plot(Ginhom(swedishpines, sigma=bw.diggle, adjust=2))
## End(Not run)
plot(Ginhom(swedishpines, sigma=10))

Gmulti

Marked Nearest Neighbour Distance Function

Description
For a marked point pattern, estimate the distribution of the distance from a typical point in subset I
to the nearest point of subset J.
Usage
Gmulti(X, I, J, r=NULL, breaks=NULL, ...,
disjoint=NULL, correction=c("rs", "km", "han"))
Arguments
X

The observed point pattern, from which an estimate of the multitype distance
distribution function GIJ (r) will be computed. It must be a marked point pattern. See under Details.

I

Subset of points of X from which distances are measured.

J

Subset of points in X to which distances are measured.

r

Optional. Numeric vector. The values of the argument r at which the distribution
function GIJ (r) should be evaluated. There is a sensible default. First-time
users are strongly advised not to specify this argument. See below for important
conditions on r.

breaks

This argument is for internal use only.

...

Ignored.

disjoint

Optional flag indicating whether the subsets I and J are disjoint. If missing, this
value will be computed by inspecting the vectors I and J.

correction

Optional. Character string specifying the edge correction(s) to be used. Options
are "none", "rs", "km", "hanisch" and "best". Alternatively correction="all"
selects all options.

Details
The function Gmulti generalises Gest (for unmarked point patterns) and Gdot and Gcross (for
multitype point patterns) to arbitrary marked point patterns.
Suppose XI , XJ are subsets, possibly overlapping, of a marked point process. This function computes an estimate of the cumulative distribution function GIJ (r) of the distance from a typical point
of XI to the nearest distinct point of XJ .
The argument X must be a point pattern (object of class "ppp") or any data that are acceptable to
as.ppp.

Gmulti

499

The arguments I and J specify two subsets of the point pattern. They may be any type of subset
indices, for example, logical vectors of length equal to npoints(X), or integer vectors with entries
in the range 1 to npoints(X), or negative integer vectors.
Alternatively, I and J may be functions that will be applied to the point pattern X to obtain index
vectors. If I is a function, then evaluating I(X) should yield a valid subset index. This option is
useful when generating simulation envelopes using envelope.
This algorithm estimates the distribution function GIJ (r) from the point pattern X. It assumes that
X can be treated as a realisation of a stationary (spatially homogeneous) random spatial point process in the plane, observed through a bounded window. The window (which is specified in X as
Window(X)) may have arbitrary shape. Biases due to edge effects are treated in the same manner as
in Gest.
The argument r is the vector of values for the distance r at which GIJ (r) should be evaluated. It is
also used to determine the breakpoints (in the sense of hist) for the computation of histograms of
distances. The reduced-sample and Kaplan-Meier estimators are computed from histogram counts.
In the case of the Kaplan-Meier estimator this introduces a discretisation error which is controlled
by the fineness of the breakpoints.
First-time users would be strongly advised not to specify r. However, if it is specified, r must
satisfy r[1] = 0, and max(r) must be larger than the radius of the largest disc contained in the
window. Furthermore, the successive entries of r must be finely spaced.
The algorithm also returns an estimate of the hazard rate function, λ(r), of GIJ (r). This estimate
should be used with caution as GIJ (r) is not necessarily differentiable.
The naive empirical distribution of distances from each point of the pattern X to the nearest other
point of the pattern, is a biased estimate of GIJ . However this is also returned by the algorithm, as
it is sometimes useful in other contexts. Care should be taken not to use the uncorrected empirical
GIJ as if it were an unbiased estimator of GIJ .
Value
An object of class "fv" (see fv.object).
Essentially a data frame containing six numeric columns
r

the values of the argument r at which the function GIJ (r) has been estimated

rs

the “reduced sample” or “border correction” estimator of GIJ (r)

han

the Hanisch-style estimator of GIJ (r)

km

the spatial Kaplan-Meier estimator of GIJ (r)

hazard

the hazard rate λ(r) of GIJ (r) by the spatial Kaplan-Meier method

raw

the uncorrected estimate of GIJ (r), i.e. the empirical distribution of the distances from each point of type i to the nearest point of type j

theo

the theoretical value of GIJ (r) for a marked Poisson process with the same
estimated intensity

Warnings
The function GIJ does not necessarily have a density.
The reduced sample estimator of GIJ is pointwise approximately unbiased, but need not be a valid
distribution function; it may not be a nondecreasing function of r. Its range is always within [0, 1].
The spatial Kaplan-Meier estimator of GIJ is always nondecreasing but its maximum value may be
less than 1.

500

GmultiInhom

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
References
Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.
Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 1983.
Diggle, P. J. (1986). Displaced amacrine cells in the retina of a rabbit : analysis of a bivariate spatial
point pattern. J. Neurosci. Meth. 18, 115–125.
Harkness, R.D and Isham, V. (1983) A bivariate spatial point pattern of ants’ nests. Applied Statistics 32, 293–303
Lotwick, H. W. and Silverman, B. W. (1982). Methods for analysing spatial processes of several
types of points. J. Royal Statist. Soc. Ser. B 44, 406–413.
Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.
Stoyan, D, Kendall, W.S. and Mecke, J. Stochastic geometry and its applications. 2nd edition.
Springer Verlag, 1995.
Van Lieshout, M.N.M. and Baddeley, A.J. (1999) Indices of dependence between types in multivariate point patterns. Scandinavian Journal of Statistics 26, 511–532.
See Also
Gcross, Gdot, Gest
Examples
trees <- longleaf
# Longleaf Pine data: marks represent diameter
Gm <- Gmulti(trees, marks(trees) <= 15, marks(trees) >= 25)
plot(Gm)

GmultiInhom

Inhomogeneous Marked G-Function

Description
For a marked point pattern, estimate the inhomogeneous version of the multitype G function, effectively the cumulative distribution function of the distance from a point in subset I to the nearest
point in subset J, adjusted for spatially varying intensity.
Usage
GmultiInhom(X, I, J,
lambda = NULL, lambdaI = NULL, lambdaJ = NULL,
lambdamin = NULL, ...,
r = NULL,
ReferenceMeasureMarkSetI = NULL,
ratio = FALSE)

GmultiInhom

501

Arguments
X

A spatial point pattern (object of class "ppp".

I

A subset index specifying the subset of points from which distances are measured. Any kind of subset index acceptable to [.ppp.

J

A subset index specifying the subset of points to which distances are measured.
Any kind of subset index acceptable to [.ppp.

lambda

Intensity estimates for each point of X. A numeric vector of length equal to
npoints(X). Incompatible with lambdaI,lambdaJ.

lambdaI

Intensity estimates for each point of X[I]. A numeric vector of length equal to
npoints(X[I]). Incompatible with lambda.

lambdaJ

Intensity estimates for each point of X[J]. A numeric vector of length equal to
npoints(X[J]). Incompatible with lambda.

lambdamin

A lower bound for the intensity, or at least a lower bound for the values in
lambdaJ or lambda[J].

...

Ignored.

r

Vector of distance values at which the inhomogeneous G function should be
estimated. There is a sensible default.
ReferenceMeasureMarkSetI
Optional. The total measure of the mark set. A positive number.
ratio

Logical value indicating whether to save ratio information.

Details
See Cronie and Van Lieshout (2015).
Value
Object of class "fv" containing the estimate of the inhomogeneous multitype G function.
Author(s)

Ottmar Cronie and Marie-Colette van Lieshout. Rewritten for spatstat by Adrian Baddeley 
Ege Rubak  and Jesper Møller.
References
Baddeley, A., Rubak, E. and Møller, J. (2011) Score, pseudo-score and residual diagnostics for
spatial point process models. Statistical Science 26, 613–646.
See Also
Related functions: Gcom, Gest.
Alternative functions: Kres, psstA, psstG, psst.
Model-fitting: ppm.

gridcentres

503

Examples
data(cells)
fit0 <- ppm(cells, ~1) # uniform Poisson
G0 <- Gres(fit0)
plot(G0)
# Hanisch correction estimate
plot(G0, hres ~ r)
# uniform Poisson is clearly not correct
fit1 <- ppm(cells, ~1, Strauss(0.08))
plot(Gres(fit1), hres ~ r)
# fit looks approximately OK; try adjusting interaction distance
plot(Gres(cells, interaction=Strauss(0.12)))
# How to make envelopes
## Not run:
E <- envelope(fit1, Gres, model=fit1, nsim=39)
plot(E)
## End(Not run)
# For computational efficiency
Gc <- Gcom(fit1)
G1 <- Gres(Gc)

gridcentres

Rectangular grid of points

Description
Generates a rectangular grid of points in a window
Usage
gridcentres(window, nx, ny)
Arguments
window

A window. An object of class owin, or data in any format acceptable to as.owin().

nx

Number of points in each row of the rectangular grid.

ny

Number of points in each column of the rectangular grid.

Details
This function creates a rectangular grid of points in the window.
The bounding rectangle of the window is divided into a regular nx × ny grid of rectangular tiles.
The function returns the x, y coordinates of the centres of these tiles.
Note that some of these grid points may lie outside the window, if window is not of type "rectangle".
The function inside.owin can be used to select those grid points which do lie inside the window.
See the examples.
This function is useful in creating dummy points for quadrature schemes (see quadscheme) and for
other miscellaneous purposes.

504

gridweights

Value
A list with two components x and y, which are numeric vectors giving the coordinates of the points
of the rectangular grid.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
quad.object, quadscheme, inside.owin, stratrand
Examples
w <- unit.square()
xy <- gridcentres(w, 10,15)
## Not run:
plot(w)
points(xy)
## End(Not run)
bdry <- list(x=c(0.1,0.3,0.7,0.4,0.2),
y=c(0.1,0.1,0.5,0.7,0.3))
w <- owin(c(0,1), c(0,1), poly=bdry)
xy <- gridcentres(w, 30, 30)
ok <- inside.owin(xy$x, xy$y, w)
## Not run:
plot(w)
points(xy$x[ok], xy$y[ok])
## End(Not run)

gridweights

Compute Quadrature Weights Based on Grid Counts

Description
Computes quadrature weights for a given set of points, using the “counting weights” for a grid of
rectangular tiles.
Usage
gridweights(X, ntile, ..., window=NULL, verbose=FALSE, npix=NULL, areas=NULL)
Arguments
X

Data defining a point pattern.

ntile

Number of tiles in each row and column of the rectangular grid. An integer
vector of length 1 or 2.

...

Ignored.

grow.boxx

505

window

Default window for the point pattern

verbose

Logical flag. If TRUE, information will be printed about the computation of the
grid weights.

npix

Dimensions of pixel grid to use when computing a digital approximation to the
tile areas.

areas

Vector of areas of the tiles, if they are already known.

Details
This function computes a set of quadrature weights for a given pattern of points (typically comprising both “data” and ‘dummy” points). See quad.object for an explanation of quadrature weights
and quadrature schemes.
The weights are computed by the “counting weights” rule based on a regular grid of rectangular
tiles. First X and (optionally) window are converted into a point pattern object. Then the bounding
rectangle of the window of the point pattern is divided into a regular ntile[1] * ntile[2] grid of
rectangular tiles. The weight attached to a point of X is the area of the tile in which it lies, divided
by the number of points of X lying in that tile.
For non-rectangular windows the tile areas are currently calculated by approximating the window
as a binary mask. The accuracy of this approximation is controlled by npix, which becomes the
argument dimyx of as.mask.
Value
Vector of nonnegative weights for each point in X.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
quad.object, dirichletWeights
Examples
Q
X
w
w

<<<<-

quadscheme(runifpoispp(10))
as.ppp(Q) # data and dummy points together
gridweights(X, 10)
gridweights(X, c(10, 10))

grow.boxx

Add margins to box in any dimension

Description
Adds a margin to a box of class boxx.

506

grow.rectangle

Usage
grow.boxx(W, left, right = left)
grow.box3(W, left, right = left)
Arguments
W

A box (object of class "boxx" or "box3").

left

Width of margin to be added to left endpoint of box side in every dimension. A
single nonnegative number, or a vector of same length as the dimension of the
box to add different left margin in each dimension.

right

Width of margin to be added to right endpoint of box side in every dimension.
A single nonnegative number, or a vector of same length as the dimension of the
box to add different right margin in each dimension.

Value
Another object of the same class "boxx" or "box3" representing the window after margins are
added.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
grow.rectangle, boxx, box3
Examples
w <- boxx(c(0,10), c(0,10), c(0,10), c(0,10))
# add a margin of size 1 on both sides in all four dimensions
b12 <- grow.boxx(w, 1)
# add margin of size 2 at left, and margin of size 3 at right,
# in each dimension.
v <- grow.boxx(w, 2, 3)

grow.rectangle

Add margins to rectangle

Description
Adds a margin to a rectangle.
Usage
grow.rectangle(W, xmargin=0, ymargin=xmargin, fraction=NULL)

Hardcore

507

Arguments
W

A window (object of class "owin"). Must be of type "rectangle".

xmargin

Width of horizontal margin to be added. A single nonnegative number, or a
vector of length 2 indicating margins of unequal width at left and right.

ymargin

Height of vertical margin to be added. A single nonnegative number, or a vector
of length 2 indicating margins of unequal width at bottom and top.

fraction

Fraction of width and height to be added. A number greater than zero, or a
numeric vector of length 2 indicating different fractions of width and of height,
respectively. Incompatible with specifying xmargin and ymargin.

Details
This is a simple convenience function to add a margin of specified width and height on each side of
a rectangular window. Unequal margins can also be added.
Value
Another object of class "owin" representing the window after margins are added.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
trim.rectangle, dilation, erosion, owin.object
Examples
w <- square(10)
# add a margin of width 1 on all four sides
square12 <- grow.rectangle(w, 1)
# add margin of width 3 on the right side
# and margin of height 4 on top.
v <- grow.rectangle(w, c(0,3), c(0,4))
# grow by 5 percent on all sides
grow.rectangle(w, fraction=0.05)

Hardcore

The Hard Core Point Process Model

Description
Creates an instance of the hard core point process model which can then be fitted to point pattern
data.
Usage
Hardcore(hc=NA)

508

Hardcore

Arguments
hc

The hard core distance

Details
A hard core process with hard core distance h and abundance parameter β is a pairwise interaction
point process in which distinct points are not allowed to come closer than a distance h apart.
The probability density is zero if any pair of points is closer than h units apart, and otherwise equals
f (x1 , . . . , xn ) = αβ n(x)
where x1 , . . . , xn represent the points of the pattern, n(x) is the number of points in the pattern,
and α is the normalising constant.
The function ppm(), which fits point process models to point pattern data, requires an argument
of class "interact" describing the interpoint interaction structure of the model to be fitted. The
appropriate description of the hard core process pairwise interaction is yielded by the function
Hardcore(). See the examples below.
If the hard core distance argument hc is missing or NA, it will be estimated from the data when
ppm is called. The estimated value of hc is the minimum nearest neighbour distance multiplied by
n/(n + 1), where n is the number of data points.
Value
An object of class "interact" describing the interpoint interaction structure of the hard core process with hard core distance hc.
Author(s)
Adrian Baddeley  and Rolf Turner 
References
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns.
Australian and New Zealand Journal of Statistics 42, 283–322.
Ripley, B.D. (1981) Spatial statistics. John Wiley and Sons.
See Also
Strauss, StraussHard, MultiHard, ppm, pairwise.family, ppm.object
Examples
Hardcore(0.02)
# prints a sensible description of itself
## Not run:
ppm(cells, ~1, Hardcore(0.05))
# fit the stationary hard core process to `cells'
## End(Not run)
# estimate hard core radius from data
ppm(cells, ~1, Hardcore())

harmonic

509

ppm(cells, ~1, Hardcore)
ppm(cells, ~ polynom(x,y,3), Hardcore(0.05))
# fit a nonstationary hard core process
# with log-cubic polynomial trend

harmonic

Basis for Harmonic Functions

Description
Evaluates a basis for the harmonic polynomials in x and y of degree less than or equal to n.
Usage
harmonic(x, y, n)
Arguments
x

Vector of x coordinates

y

Vector of y coordinates

n

Maximum degree of polynomial

Details
This function computes a basis for the harmonic polynomials in two variables x and y up to a
given degree n and evaluates them at given x, y locations. It can be used in model formulas (for
example in the model-fitting functions lm,glm,gam and ppm) to specify a linear predictor which is
a harmonic function.
A function f (x, y) is harmonic if
∂2
∂2
f
+
f = 0.
∂x2
∂y 2
The harmonic polynomials of degree less than or equal to n have a basis consisting of 2n functions.
This function was implemented on a suggestion of P. McCullagh for fitting nonstationary spatial
trend to point process models.
Value
A data frame with 2 * n columns giving the values of the basis functions at the coordinates. Each
column is labelled by an algebraic expression for the corresponding basis function.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
ppm, polynom

510

harmonise

Examples
# inhomogeneous point pattern
X <- unmark(longleaf)
# fit Poisson point process with log-cubic intensity
fit.3 <- ppm(X ~ polynom(x,y,3), Poisson())
# fit Poisson process with log-cubic-harmonic intensity
fit.h <- ppm(X ~ harmonic(x,y,3), Poisson())
# Likelihood ratio test
lrts <- 2 * (logLik(fit.3) - logLik(fit.h))
df <- with(coords(X),
ncol(polynom(x,y,3)) - ncol(harmonic(x,y,3)))
pval <- 1 - pchisq(lrts, df=df)

harmonise

Make Objects Compatible

Description
Converts several objects of the same class to a common format so that they can be combined or
compared.
Usage
harmonise(...)
harmonize(...)
Arguments
...

Any number of objects of the same class.

Details
This generic command takes any number of objects of the same class, and attempts to make them
compatible in the sense of compatible so that they can be combined or compared.
There are methods for the classes "fv" (harmonise.fv) and "im" (harmonise.im).
All arguments ... must be objects of the same class. The result will be a list, of length equal to the
number of arguments ..., containing new versions of each of these objects, converted to a common
format. If the arguments were named (name=value) then the return value also carries these names.
Value
A list, of length equal to the number of arguments ..., whose entries are objects of the same class.
If the arguments were named (name=value) then the return value also carries these names.
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak .

harmonise.fv

511

See Also
compatible, harmonise.fv, harmonise.im

harmonise.fv

Make Function Tables Compatible

Description
Convert several objects of class "fv" to the same values of the function argument.
Usage
## S3 method for class 'fv'
harmonise(..., strict=FALSE)
## S3 method for class 'fv'
harmonize(..., strict=FALSE)
Arguments
...

Any number of function tables (objects of class "fv").

strict

Logical. If TRUE, a column of data will be deleted if columns of the same name
do not appear in every object.

Details
A function value table (object of class "fv") is essentially a data frame giving the values of a
function f (x) (or several alternative estimates of this value) at equally-spaced values of the function
argument x.
The command harmonise is generic. This is the method for objects of class "fv".
This command makes any number of "fv" objects compatible, in the loose sense that they have the
same sequence of values of x. They can then be combined by cbind.fv, but not necessarily by
eval.fv.
All arguments ... must be function value tables (objects of class "fv"). The result will be a list, of
length equal to the number of arguments ..., containing new versions of each of these functions,
converted to a common sequence of x values. If the arguments were named (name=value) then the
return value also carries these names.
The range of x values in the resulting functions will be the intersection of the ranges of x values in
the original functions. The spacing of x values in the resulting functions will be the finest (narrowest) of the spacings of the x values in the original functions. Function values are interpolated using
approxfun.
If strict=TRUE, each column of data will be retained only if a column of the same name appears in
all of the arguments .... This ensures that the resulting objects are strictly compatible in the sense
of compatible.fv, and can be combined using eval.fv or collapse.fv.
If strict=FALSE (the default), this does not occur, and then the resulting objects are not guaranteed
to be compatible in the sense of compatible.fv.

512

harmonise.im

Value
A list, of length equal to the number of arguments ..., whose entries are objects of class "fv". If
the arguments were named (name=value) then the return value also carries these names.

Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak .
See Also
fv.object, cbind.fv, eval.fv, compatible.fv
Examples
H <- harmonise(K=Kest(cells), G=Gest(cells))
H
## Not run:
## generates a warning about duplicated columns
try(cbind(H$K, H$G))
## End(Not run)

harmonise.im

Make Pixel Images Compatible

Description
Convert several pixel images to a common pixel raster.

Usage
## S3 method for class 'im'
harmonise(...)
## S3 method for class 'im'
harmonize(...)
Arguments
...

Any number of pixel images (objects of class "im") or data which can be converted to pixel images by as.im.

harmonise.msr

513

Details
This function makes any number of pixel images compatible, by converting them all to a common
pixel grid.
The command harmonise is generic. This is the method for objects of class "im".
At least one of the arguments ... must be a pixel image. Some arguments may be windows (objects
of class "owin"), functions (function(x,y)) or numerical constants. These will be converted to
images using as.im.
The common pixel grid is determined by inspecting all the pixel images in the argument list, computing the bounding box of all the images, then finding the image with the highest spatial resolution,
and extending its pixel grid to cover the bounding box.
The return value is a list with entries corresponding to the input arguments. If the arguments were
named (name=value) then the return value also carries these names.
If you just want to determine the appropriate pixel resolution, without converting the images, use
commonGrid.
Value
A list, of length equal to the number of arguments ..., whose entries are pixel images.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
commonGrid, compatible.im, as.im
Examples
A <- setcov(square(1))
B <- function(x,y) { x }
G <- density(runifpoint(42))
harmonise(X=A, Y=B, Z=G)

harmonise.msr

Make Measures Compatible

Description
Convert several measures to a common quadrature scheme
Usage
## S3 method for class 'msr'
harmonise(...)
Arguments
...

Any number of measures (objects of class "msr").

514

harmonise.owin

Details
This function makes any number of measures compatible, by converting them all to a common
quadrature scheme.
The command harmonise is generic. This is the method for objects of class "msr".
Value
A list, of length equal to the number of arguments ..., whose entries are measures.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
harmonise, msr
Examples
fit1 <- ppm(cells ~ x)
fit2 <- ppm(rpoispp(ex=cells) ~ x)
m1 <- residuals(fit1)
m2 <- residuals(fit2)
harmonise(m1, m2)
s1 <- residuals(fit1, type="score")
s2 <- residuals(fit2, type="score")
harmonise(s1, s2)

harmonise.owin

Make Windows Compatible

Description
Convert several windows to a common pixel raster.
Usage
## S3 method for class 'owin'
harmonise(...)
## S3 method for class 'owin'
harmonize(...)
Arguments
...

Any number of windows (objects of class "owin") or data which can be converted to windows by as.owin.

has.close

515

Details
This function makes any number of windows compatible, by converting them all to a common pixel
grid.
This only has an effect if one of the windows is a binary mask. If all the windows are rectangular
or polygonal, they are returned unchanged.
The command harmonise is generic. This is the method for objects of class "owin".
Each argument must be a window (object of class "owin"), or data that can be converted to a
window by as.owin.
The common pixel grid is determined by inspecting all the windows in the argument list, computing
the bounding box of all the windows, then finding the binary mask with the finest spatial resolution,
and extending its pixel grid to cover the bounding box.
The return value is a list with entries corresponding to the input arguments. If the arguments were
named (name=value) then the return value also carries these names.
If you just want to determine the appropriate pixel resolution, without converting the windows, use
commonGrid.
Value
A list of windows, of length equal to the number of arguments .... The list belongs to the class
"solist".
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
commonGrid, harmonise.im, as.owin
Examples
harmonise(X=letterR,
Y=grow.rectangle(Frame(letterR), 0.2),
Z=as.mask(letterR, eps=0.1),
V=as.mask(letterR, eps=0.07))

has.close

Check Whether Points Have Close Neighbours

Description
For each point in a point pattern, determine whether the point has a close neighbour in the same
pattern.

516

has.close

Usage
has.close(X, r, Y=NULL, ...)
## Default S3 method:
has.close(X,r, Y=NULL, ..., periodic=FALSE)
## S3 method for class 'ppp'
has.close(X,r, Y=NULL, ..., periodic=FALSE, sorted=FALSE)
## S3 method for class 'pp3'
has.close(X,r, Y=NULL, ..., periodic=FALSE, sorted=FALSE)
Arguments
X,Y

Point patterns of class "ppp" or "pp3" or "lpp".

r

Threshold distance: a number greater than zero.

periodic

Logical value indicating whether to measure distances in the periodic sense, so
that opposite sides of the (rectangular) window are treated as identical.

sorted

Logical value, indicating whether the points of X (and Y, if given) are already
sorted into increasing order of the x coordinates.

...

Other arguments are ignored.

Details
This is simply a faster version of (nndist(X) <= r) or (nncross(X,Y,what="dist") <= r).
has.close(X,r) determines, for each point in the pattern X, whether or not this point has a neighbour in the same pattern X which lies at a distance less than or equal to r.
has.close(X,r,Y) determines, for each point in the pattern X, whether or not this point has a
neighbour in the other pattern Y which lies at a distance less than or equal to r.
The function has.close is generic, with methods for "ppp" and "pp3" and a default method.
Value
A logical vector, with one entry for each point of X.
Author(s)
Adrian Baddeley .
See Also
nndist
Examples
has.close(redwood, 0.05)
with(split(amacrine), has.close(on, 0.05, off))

headtail

headtail

517

First or Last Part of a Spatial Pattern

Description
Returns the first few elements (head) or the last few elements (tail) of a spatial pattern.
Usage
## S3 method for class 'ppp'
head(x, n = 6L, ...)
## S3 method for class 'ppx'
head(x, n = 6L, ...)
## S3 method for class 'psp'
head(x, n = 6L, ...)
## S3 method for class 'tess'
head(x, n = 6L, ...)
## S3 method for class 'ppp'
tail(x, n = 6L, ...)
## S3 method for class 'ppx'
tail(x, n = 6L, ...)
## S3 method for class 'psp'
tail(x, n = 6L, ...)
## S3 method for class 'tess'
tail(x, n = 6L, ...)
Arguments
x

A spatial pattern of geometrical figures, such as a spatial pattern of points (an
object of class "ppp", "pp3", "ppx" or "lpp") or a spatial pattern of line segments (an object of class "psp") or a tessellation (object of class "tess").

n

Integer. The number of elements of the pattern that should be extracted.

...

Ignored.

Details
These are methods for the generic functions head and tail. They extract the first or last n elements
from x and return them as an object of the same kind as x.
To inspect the spatial coordinates themselves, use View(x) or head(as.data.frame(x)).
Value
An object of the same class as x.

518

Hest

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
View, edit.
Conversion to data frame: as.data.frame.ppp, as.data.frame.ppx, as.data.frame.psp
Examples
head(cells)
tail(as.psp(spiders), 10)
head(dirichlet(cells), 4)

Hest

Spherical Contact Distribution Function

Description
Estimates the spherical contact distribution function of a random set.
Usage
Hest(X, r=NULL, breaks=NULL, ...,
W,
correction=c("km", "rs", "han"),
conditional=TRUE)
Arguments
X

The observed random set. An object of class "ppp", "psp" or "owin". Alternatively a pixel image (class "im") with logical values.

r

Optional. Vector of values for the argument r at which H(r) should be evaluated. Users are advised not to specify this argument; there is a sensible default.

breaks

This argument is for internal use only.

...

Arguments passed to as.mask to control the discretisation.

W

Optional. A window (object of class "owin") to be taken as the window of
observation. The contact distribution function will be estimated from values of
the contact distance inside W.

correction

Optional. The edge correction(s) to be used to estimate H(r). A vector of
character strings selected from "none", "rs", "km", "han" and "best". Alternatively correction="all" selects all options.

conditional

Logical value indicating whether to compute the conditional or unconditional
distribution. See Details.

Hest

519

Details
The spherical contact distribution function of a stationary random set X is the cumulative distribution function H of the distance from a fixed point in space to the nearest point of X, given that the
point lies outside X. That is, H(r) equals the probability that X lies closer than r units away from
the fixed point x, given that X does not cover x.
Let D = d(x, X) be the shortest distance from an arbitrary point x to the set X. Then the spherical
contact distribution function is
H(r) = P (D ≤ r | D > 0)
For a point process, the spherical contact distribution function is the same as the empty space
function F discussed in Fest.
The argument X may be a point pattern (object of class "ppp"), a line segment pattern (object of
class "psp") or a window (object of class "owin"). It is assumed to be a realisation of a stationary
random set.
The algorithm first calls distmap to compute the distance transform of X, then computes the KaplanMeier and reduced-sample estimates of the cumulative distribution following Hansen et al (1999). If
conditional=TRUE (the default) the algorithm returns an estimate of the spherical contact function
H(r) as defined above. If conditional=FALSE, it instead returns an estimate of the cumulative
distribution function H ∗ (r) = P (D ≤ r) which includes a jump at r = 0 if X has nonzero area.
Accuracy depends on the pixel resolution, which is controlled by the arguments eps, dimyx and
xy passed to as.mask. For example, use eps=0.1 to specify square pixels of side 0.1 units, and
dimyx=256 to specify a 256 by 256 grid of pixels.
Value
An object of class "fv", see fv.object, which can be plotted directly using plot.fv.
Essentially a data frame containing up to six columns:
r

the values of the argument r at which the function H(r) has been estimated

rs

the “reduced sample” or “border correction” estimator of H(r)

km

the spatial Kaplan-Meier estimator of H(r)

hazard

the hazard rate λ(r) of H(r) by the spatial Kaplan-Meier method

han

the spatial Hanisch-Chiu-Stoyan estimator of H(r)

raw

the uncorrected estimate of H(r), i.e. the empirical distribution of the distance
from a fixed point in the window to the nearest point of X

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak  with contributions from Kassel Hingee.
References
Baddeley, A.J. Spatial sampling and censoring. In O.E. Barndorff-Nielsen, W.S. Kendall and
M.N.M. van Lieshout (eds) Stochastic Geometry: Likelihood and Computation. Chapman and
Hall, 1998. Chapter 2, pages 37-78.
Baddeley, A.J. and Gill, R.D. The empty space hazard of a spatial pattern. Research Report 1994/3,
Department of Mathematics, University of Western Australia, May 1994.

520

hextess
Hansen, M.B., Baddeley, A.J. and Gill, R.D. First contact distributions for spatial patterns: regularity and estimation. Advances in Applied Probability 31 (1999) 15-33.
Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.
Stoyan, D, Kendall, W.S. and Mecke, J. Stochastic geometry and its applications. 2nd edition.
Springer Verlag, 1995.

See Also
Fest
Examples
X <- runifpoint(42)
H <- Hest(X)
Y <- rpoisline(10)
H <- Hest(Y)
H <- Hest(Y, dimyx=256)
X <- heather$coarse
plot(Hest(X))
H <- Hest(X, conditional=FALSE)
P <- owin(poly=list(x=c(5.3, 8.5, 8.3, 3.7, 1.3, 3.7),
y=c(9.7, 10.0, 13.6, 14.4, 10.7, 7.2)))
plot(X)
plot(P, add=TRUE, col="red")
H <- Hest(X, W=P)
Z <- as.im(FALSE, Frame(X))
Z[X] <- TRUE
Z <- Z[P, drop=FALSE]
plot(Z)
H <- Hest(Z)

hextess

Hexagonal Grid or Tessellation

Description
Construct a hexagonal grid of points, or a hexagonal tessellation.
Usage
hexgrid(W, s, offset = c(0, 0), origin=NULL, trim = TRUE)
hextess(W, s, offset = c(0, 0), origin=NULL, trim = TRUE)
Arguments
W

Window in which to construct the hexagonal grid or tessellation. An object of
class "owin".

s

Side length of hexagons. A positive number.

offset

Numeric vector of length 2 specifying a shift of the hexagonal grid. See Details.

hextess

521

origin

Numeric vector of length 2 specifying the initial origin of the hexagonal grid,
before the offset is applied. See Details.

trim

Logical value indicating whether to restrict the result to the window W. See Details.

Details
hexgrid constructs a hexagonal grid of points on the window W. If trim=TRUE (the default), the
grid is intersected with W so that all points lie inside W. If trim=FALSE, then we retain all grid points
which are the centres of hexagons that intersect W.
hextess constructs a tessellation of hexagons on the window W. If trim=TRUE (the default), the
tessellation is restricted to the interior of W, so that there will be some fragmentary hexagons near
the boundary of W. If trim=FALSE, the tessellation consists of all hexagons which intersect W.
The points of hexgrid(...) are the centres of the tiles of hextess(...) in the same order.
In the initial position of the grid or tessellation, one of the grid points (tile centres) is placed at
the origin, which defaults to the midpoint of the bounding rectangle of W. The grid can be shifted
relative to this origin by specifing the offset.
Value
The value of hexgrid is a point pattern (object of class "ppp").
The value of hextess is a tessellation (object of class "tess").
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
tess
hexagon
Examples
if(interactive()) {
W <- Window(chorley)
s <- 0.7
} else {
W <- letterR
s <- 0.3
}
plot(hextess(W, s))
plot(hexgrid(W, s), add=TRUE)

522

HierHard

HierHard

The Hierarchical Hard Core Point Process Model

Description
Creates an instance of the hierarchical hard core point process model which can then be fitted to
point pattern data.
Usage
HierHard(hradii=NULL, types=NULL, archy=NULL)
Arguments
hradii

Optional matrix of hard core distances

types

Optional; vector of all possible types (i.e. the possible levels of the marks variable in the data)

archy

Optional: the hierarchical order. See Details.

Details
This is a hierarchical point process model for a multitype point pattern (Högmander and Särkkä,
1999; Grabarnik and Särkkä, 2009). It is appropriate for analysing multitype point pattern data in
which the types are ordered so that the points of type j depend on the points of type 1, 2, . . . , j − 1.
The hierarchical version of the (stationary) hard core process with m types, with hard core distances
hij and parameters βj , is a point process in which each point of type j contributes a factor βj to
the probability density of the point pattern. If any pair of points of types i and j lies closer than hij
units apart, the configuration of points is impossible (probability density zero).
The nonstationary hierarchical hard core process is similar except that the contribution of each
individual point xi is a function β(xi ) of location and type, rather than a constant beta.
The function ppm(), which fits point process models to point pattern data, requires an argument
of class "interact" describing the interpoint interaction structure of the model to be fitted. The
appropriate description of the hierarchical hard core process pairwise interaction is yielded by the
function HierHard(). See the examples below.
The argument types need not be specified in normal use. It will be determined automatically from
the point pattern data set to which the HierHard interaction is applied, when the user calls ppm.
However, the user should be confident that the ordering of types in the dataset corresponds to the
ordering of rows and columns in the matrix radii.
The argument archy can be used to specify a hierarchical ordering of the types. It can be either
a vector of integers or a character vector matching the possible types. The default is the sequence
1, 2, . . . , m meaning that type j depends on types 1, 2, . . . , j − 1.
The matrix iradii must be square, with entries which are either positive numbers, or zero or
NA. A value of zero or NA indicates that no hard core interaction term should be included for this
combination of types.
Note that only the hard core distances are specified in HierHard. The canonical parameters log(βj )
are estimated by ppm(), not fixed in HierHard().

hierpair.family

523

Value
An object of class "interact" describing the interpoint interaction structure of the hierarchical
hard core process with hard core distances hradii[i, j].
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak .
References
Grabarnik, P. and Särkkä, A. (2009) Modelling the spatial structure of forest stands by multivariate
point processes with hierarchical interactions. Ecological Modelling 220, 1232–1240.
Högmander, H. and Särkkä, A. (1999) Multitype spatial point patterns with hierarchical interactions.
Biometrics 55, 1051–1058.
See Also
MultiHard for the corresponding symmetrical interaction.
HierStrauss, HierStraussHard.
Examples
h <- matrix(c(4, NA, 10, 15), 2, 2)
HierHard(h)
# prints a sensible description of itself
ppm(ants ~1, HierHard(h))
# fit the stationary hierarchical hard core process to ants data

hierpair.family

Hierarchical Pairwise Interaction Process Family

Description
An object describing the family of all hierarchical pairwise interaction Gibbs point processes.
Details
Advanced Use Only!
This structure would not normally be touched by the user. It describes the hierarchical pairwise
interaction family of point process models.
Anyway, hierpair.family is an object of class "isf" containing a function hierpair.family$eval
for evaluating the sufficient statistics of any hierarchical pairwise interaction point process model
taking an exponential family form.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak .

524

HierStrauss

See Also
Other families: pairwise.family, pairsat.family, ord.family, inforder.family.
Hierarchical Strauss interaction: HierStrauss.

HierStrauss

The Hierarchical Strauss Point Process Model

Description
Creates an instance of the hierarchical Strauss point process model which can then be fitted to point
pattern data.
Usage
HierStrauss(radii, types=NULL, archy=NULL)
Arguments
radii

Matrix of interaction radii

types

Optional; vector of all possible types (i.e. the possible levels of the marks variable in the data)

archy

Optional: the hierarchical order. See Details.

Details
This is a hierarchical point process model for a multitype point pattern (Högmander and Särkkä,
1999; Grabarnik and Särkkä, 2009). It is appropriate for analysing multitype point pattern data in
which the types are ordered so that the points of type j depend on the points of type 1, 2, . . . , j − 1.
The hierarchical version of the (stationary) Strauss process with m types, with interaction radii rij
and parameters βj and γij is a point process in which each point of type j contributes a factor βj to
the probability density of the point pattern, and a pair of points of types i and j closer than rij units
apart contributes a factor γij to the density provided i ≤ j.
The nonstationary hierarchical Strauss process is similar except that the contribution of each individual point xi is a function β(xi ) of location and type, rather than a constant beta.
The function ppm(), which fits point process models to point pattern data, requires an argument
of class "interact" describing the interpoint interaction structure of the model to be fitted. The
appropriate description of the hierarchical Strauss process pairwise interaction is yielded by the
function HierStrauss(). See the examples below.
The argument types need not be specified in normal use. It will be determined automatically from
the point pattern data set to which the HierStrauss interaction is applied, when the user calls ppm.
However, the user should be confident that the ordering of types in the dataset corresponds to the
ordering of rows and columns in the matrix radii.
The argument archy can be used to specify a hierarchical ordering of the types. It can be either
a vector of integers or a character vector matching the possible types. The default is the sequence
1, 2, . . . , m meaning that type j depends on types 1, 2, . . . , j − 1.
The matrix radii must be symmetric, with entries which are either positive numbers or NA. A value
of NA indicates that no interaction term should be included for this combination of types.
Note that only the interaction radii are specified in HierStrauss. The canonical parameters log(βj )
and log(γij ) are estimated by ppm(), not fixed in HierStrauss().

HierStraussHard

525

Value
An object of class "interact" describing the interpoint interaction structure of the hierarchical
Strauss process with interaction radii radii[i, j].
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak .
References
Grabarnik, P. and Särkkä, A. (2009) Modelling the spatial structure of forest stands by multivariate
point processes with hierarchical interactions. Ecological Modelling 220, 1232–1240.
Högmander, H. and Särkkä, A. (1999) Multitype spatial point patterns with hierarchical interactions.
Biometrics 55, 1051–1058.
See Also
MultiStrauss for the corresponding symmetrical interaction.
HierHard, HierStraussHard.
Examples
r <- matrix(10 * c(3,4,4,3), nrow=2,ncol=2)
HierStrauss(r)
# prints a sensible description of itself
ppm(ants ~1, HierStrauss(r, , c("Messor", "Cataglyphis")))
# fit the stationary hierarchical Strauss process to ants data

HierStraussHard

The Hierarchical Strauss Hard Core Point Process Model

Description
Creates an instance of the hierarchical Strauss-hard core point process model which can then be
fitted to point pattern data.
Usage
HierStraussHard(iradii, hradii=NULL, types=NULL, archy=NULL)
Arguments
iradii

Matrix of interaction radii

hradii

Optional matrix of hard core distances

types

Optional; vector of all possible types (i.e. the possible levels of the marks variable in the data)

archy

Optional: the hierarchical order. See Details.

526

HierStraussHard

Details
This is a hierarchical point process model for a multitype point pattern (Högmander and Särkkä,
1999; Grabarnik and Särkkä, 2009). It is appropriate for analysing multitype point pattern data in
which the types are ordered so that the points of type j depend on the points of type 1, 2, . . . , j − 1.
The hierarchical version of the (stationary) Strauss hard core process with m types, with interaction
radii rij , hard core distances hij and parameters βj and γij is a point process in which each point
of type j contributes a factor βj to the probability density of the point pattern, and a pair of points
of types i and j closer than rij units apart contributes a factor γij to the density provided i ≤ j.
If any pair of points of types i and j lies closer than hij units apart, the configuration of points is
impossible (probability density zero).
The nonstationary hierarchical Strauss hard core process is similar except that the contribution of
each individual point xi is a function β(xi ) of location and type, rather than a constant beta.
The function ppm(), which fits point process models to point pattern data, requires an argument
of class "interact" describing the interpoint interaction structure of the model to be fitted. The
appropriate description of the hierarchical Strauss hard core process pairwise interaction is yielded
by the function HierStraussHard(). See the examples below.
The argument types need not be specified in normal use. It will be determined automatically from
the point pattern data set to which the HierStraussHard interaction is applied, when the user calls
ppm. However, the user should be confident that the ordering of types in the dataset corresponds to
the ordering of rows and columns in the matrix radii.
The argument archy can be used to specify a hierarchical ordering of the types. It can be either
a vector of integers or a character vector matching the possible types. The default is the sequence
1, 2, . . . , m meaning that type j depends on types 1, 2, . . . , j − 1.
The matrices iradii and hradii must be square, with entries which are either positive numbers
or zero or NA. A value of zero or NA indicates that no interaction term should be included for this
combination of types.
Note that only the interaction radii and hard core distances are specified in HierStraussHard. The
canonical parameters log(βj ) and log(γij ) are estimated by ppm(), not fixed in HierStraussHard().
Value
An object of class "interact" describing the interpoint interaction structure of the hierarchical
Strauss-hard core process with interaction radii iradii[i, j] and hard core distances hradii[i, j].
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak .
References
Grabarnik, P. and Särkkä, A. (2009) Modelling the spatial structure of forest stands by multivariate
point processes with hierarchical interactions. Ecological Modelling 220, 1232–1240.
Högmander, H. and Särkkä, A. (1999) Multitype spatial point patterns with hierarchical interactions.
Biometrics 55, 1051–1058.

hist.funxy

527

See Also
MultiStraussHard for the corresponding symmetrical interaction.
HierHard, HierStrauss.
Examples
r <- matrix(c(30, NA, 40, 30), nrow=2,ncol=2)
h <- matrix(c(4, NA, 10, 15), 2, 2)
HierStraussHard(r, h)
# prints a sensible description of itself
ppm(ants ~1, HierStraussHard(r, h))
# fit the stationary hierarchical Strauss-hard core process to ants data

hist.funxy

Histogram of Values of a Spatial Function

Description
Computes and displays a histogram of the values of a spatial function of class "funxy".
Usage
## S3 method for class 'funxy'
hist(x, ..., xname)
Arguments
x

A pixel image (object of class "funxy").

...

Arguments passed to as.im or hist.im.

xname

Optional. Character string to be used as the name of the dataset x.

Details
This function computes and (by default) displays a histogram of the values of the function x.
An object of class "funxy" describes a function of spatial location. It is a function(x,y,..) in
the R language, with additional attributes.
The function hist.funxy is a method for the generic function hist for the class "funxy".
The function is first converted to a pixel image using as.im, then hist.im is called to produce the
histogram.
Any arguments in ... are passed to as.im to determine the pixel resolution, or to hist.im to
determine the histogram breaks and to control or suppress plotting. Useful arguments include W for
the spatial domain, eps,dimyx for pixel resolution, main for the main title.
Value
An object of class "histogram" as returned by hist.default. This object can be plotted.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .

528

hist.im

See Also
spatialcdf for the cumulative distribution function of an image or function.
hist, hist.default.
For other statistical graphics such as Q-Q plots, use as.im(X)[] to extract the pixel values of image
X, and apply the usual statistical graphics commands.
Examples
f <- funxy(function(x,y) {x^2}, unit.square())
hist(f)

hist.im

Histogram of Pixel Values in an Image

Description
Computes and displays a histogram of the pixel values in a pixel image. The hist method for class
"im".
Usage
## S3 method for class 'im'
hist(x, ..., probability=FALSE, xname)
Arguments
x

A pixel image (object of class "im").

...

Arguments passed to hist.default or barplot.

probability

Logical. If TRUE, the histogram will be normalised to give probabilities or probability densities.

xname

Optional. Character string to be used as the name of the dataset x.

Details
This function computes and (by default) displays a histogram of the pixel values in the image x.
An object of class "im" describes a pixel image. See im.object) for details of this class.
The function hist.im is a method for the generic function hist for the class "im".
Any arguments in ... are passed to hist.default (for numeric valued images) or barplot (for
factor or logical images). For example, such arguments control the axes, and may be used to suppress the plotting.
Value
For numeric-valued images, an object of class "histogram" as returned by hist.default. This
object can be plotted.
For factor-valued or logical images, an object of class "barplotdata", which can be plotted. This
is a list with components called counts (contingency table of counts of the numbers of pixels taking
each possible value), probs (corresponding relative frequencies) and mids (graphical x-coordinates
of the midpoints of the bars in the barplot).

hopskel

529

Author(s)
Adrian Baddeley  and Rolf Turner 
See Also
spatialcdf for the cumulative distribution function of an image.
hist, hist.default, barplot.
For other statistical graphics such as Q-Q plots, use X[] to extract the pixel values of image X, and
apply the usual statistical graphics commands.
For information about pixel images see im.object, summary.im.
Examples
X <- as.im(function(x,y) {x^2}, unit.square())
hist(X)
hist(cut(X,3))

hopskel

Hopkins-Skellam Test

Description
Perform the Hopkins-Skellam test of Complete Spatial Randomness, or simply calculate the test
statistic.
Usage
hopskel(X)
hopskel.test(X, ...,
alternative=c("two.sided", "less", "greater",
"clustered", "regular"),
method=c("asymptotic", "MonteCarlo"),
nsim=999)
Arguments
X

Point pattern (object of class "ppp").

alternative

String indicating the type of alternative for the hypothesis test. Partially matched.

method

Method of performing the test. Partially matched.

nsim

Number of Monte Carlo simulations to perform, if a Monte Carlo p-value is
required.

...

Ignored.

530

hopskel

Details
Hopkins and Skellam (1954) proposed a test of Complete Spatial Randomness based on comparing
nearest-neighbour distances with point-event distances.
If the point pattern X contains n points, we first compute the nearest-neighbour distances P1 , . . . , Pn
so that Pi is the distance from the ith data point to the nearest other data point. Then we generate
another completely random pattern U with the same number n of points, and compute for each point
of U the distance to the nearest point of X, giving distances I1 , . . . , In . The test statistic is
P 2
P
A = Pi i2
i Ii
The null distribution of A is roughly an F distribution with shape parameters (2n, 2n). (This is
equivalent to using the test statistic H = A/(1 + A) and referring H to the Beta distribution with
parameters (n, n)).
The function hopskel calculates the Hopkins-Skellam test statistic A, and returns its numeric value.
This can be used as a simple summary of spatial pattern: the value H = 1 is consistent with
Complete Spatial Randomness, while values H < 1 are consistent with spatial clustering, and
values H > 1 are consistent with spatial regularity.
The function hopskel.test performs the test. If method="asymptotic" (the default), the test
statistic H is referred to the F distribution. If method="MonteCarlo", a Monte Carlo test is performed using nsim simulated point patterns.
Value
The value of hopskel is a single number.
The value of hopskel.test is an object of class "htest" representing the outcome of the test. It
can be printed.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Hopkins, B. and Skellam, J.G. (1954) A new method of determining the type of distribution of plant
individuals. Annals of Botany 18, 213–227.
See Also
clarkevans, clarkevans.test, nndist, nncross
Examples
hopskel(redwood)
hopskel(redwood)
hopskel.test(redwood, alternative="clustered")

Hybrid

Hybrid

531

Hybrid Interaction Point Process Model

Description
Creates an instance of a hybrid point process model which can then be fitted to point pattern data.
Usage
Hybrid(...)
Arguments
...

Two or more interactions (objects of class "interact") or objects which can be
converted to interactions. See Details.

Details
A hybrid (Baddeley, Turner, Mateu and Bevan, 2013) is a point process model created by combining
two or more point process models, or an interpoint interaction created by combining two or more
interpoint interactions.
The hybrid of two point processes, with probability densities f (x) and g(x) respectively, is the
point process with probability density
h(x) = c f (x) g(x)
where c is a normalising constant.
Equivalently, the hybrid of two point processes with conditional intensities λ(u, x) and κ(u, x) is
the point process with conditional intensity
φ(u, x) = λ(u, x) κ(u, x).
The hybrid of m > 3 point processes is defined in a similar way.
The function ppm, which fits point process models to point pattern data, requires an argument of
class "interact" describing the interpoint interaction structure of the model to be fitted. The
appropriate description of a hybrid interaction is yielded by the function Hybrid().
The arguments ... will be interpreted as interpoint interactions (objects of class "interact")
and the result will be the hybrid of these interactions. Each argument must either be an interpoint
interaction (object of class "interact"), or a point process model (object of class "ppm") from
which the interpoint interaction will be extracted.
The arguments ... may also be given in the form name=value. This is purely cosmetic: it can be
used to attach simple mnemonic names to the component interactions, and makes the printed output
from print.ppm neater.
Value
An object of class "interact" describing an interpoint interaction structure.
Author(s)
Adrian Baddeley 
and Rolf Turner 

532

hybrid.family

References
Baddeley, A., Turner, R., Mateu, J. and Bevan, A. (2013) Hybrids of Gibbs point process models
and their implementation. Journal of Statistical Software 55:11, 1–43. http://www.jstatsoft.
org/v55/i11/
See Also
ppm
Examples
Hybrid(Strauss(0.1), Geyer(0.2, 3))
Hybrid(Ha=Hardcore(0.05), St=Strauss(0.1), Ge=Geyer(0.2, 3))
fit <- ppm(redwood, ~1, Hybrid(A=Strauss(0.02), B=Geyer(0.1, 2)))
fit
ctr <- rmhcontrol(nrep=5e4, expand=1)
plot(simulate(fit, control=ctr))
# hybrid components can be models (including hybrid models)
Hybrid(fit, S=Softcore(0.5))
# plot.fii only works if every component is a pairwise interaction
data(swedishpines)
fit2 <- ppm(swedishpines, ~1, Hybrid(DG=DiggleGratton(2,10), S=Strauss(5)))
plot(fitin(fit2))
plot(fitin(fit2), separate=TRUE, mar.panel=rep(4,4))

hybrid.family

Hybrid Interaction Family

Description
An object describing the family of all hybrid interactions.
Details
Advanced Use Only!
This structure would not normally be touched by the user. It describes the family of all hybrid point
process models.
If you need to create a specific hybrid interaction model for use in modelling, use the function
Hybrid.
Anyway, hybrid.family is an object of class "isf" containing a function hybrid.family$eval
for evaluating the sufficient statistics of any hybrid interaction point process model.
Author(s)
Adrian Baddeley 
and Rolf Turner 

hyperframe

533

See Also
Use Hybrid to make hybrid interactions.
Other families: pairwise.family, pairsat.family, ord.family, inforder.family.

hyperframe

Hyper Data Frame

Description
Create a hyperframe: a two-dimensional array in which each column consists of values of the same
atomic type (like the columns of a data frame) or objects of the same class.
Usage
hyperframe(...,
row.names=NULL, check.rows=FALSE, check.names=TRUE,
stringsAsFactors=default.stringsAsFactors())
Arguments
...

Arguments of the form value or tag=value. Each value is either an atomic
vector, or a list of objects of the same class, or a single atomic value, or a single
object. Each value will become a column of the array. The tag determines the
name of the column. See Details.
row.names,check.rows,check.names,stringsAsFactors
Arguments passed to data.frame controlling the names of the rows, whether to
check that rows are consistent, whether to check validity of the column names,
and whether to convert character columns to factors.
Details
A hyperframe is like a data frame, except that its entries can be objects of any kind.
A hyperframe is a two-dimensional array in which each column consists of values of one atomic
type (as in a data frame) or consists of objects of one class.
The arguments ... are any number of arguments of the form value or tag=value. Each value
will become a column of the array. The tag determines the name of the column.
Each value can be either
• an atomic vector or factor (i.e. numeric vector, integer vector, character vector, logical vector,
complex vector or factor)
• a list of objects which are all of the same class
• one atomic value, which will be replicated to make an atomic vector or factor
• one object, which will be replicated to make a list of objects.
All columns (vectors, factors and lists) must be of the same length, if their length is greater than 1.
Value
An object of class "hyperframe".

534

identify.ppp

Methods for Hyperframes
There are methods for print, plot, summary, with, split, [, [<,$, $<-, names, as.data.frame
as.list, cbind and rbind for the class of hyperframes. There is also is.hyperframe and as.hyperframe.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
as.hyperframe, as.hyperframe.ppx, plot.hyperframe, [.hyperframe, with.hyperframe, split.hyperframe,
as.data.frame.hyperframe, cbind.hyperframe, rbind.hyperframe
Examples
# equivalent to a data frame
hyperframe(X=1:10, Y=3)
# list of functions
hyperframe(f=list(sin, cos, tan))
# table of functions and matching expressions
hyperframe(f=list(sin, cos, tan),
e=list(expression(sin(x)), expression(cos(x)), expression(tan(x))))
hyperframe(X=1:10, Y=letters[1:10], Z=factor(letters[1:10]),
stringsAsFactors=FALSE)
lambda <- runif(4, min=50, max=100)
X <- lapply(as.list(lambda), function(x) { rpoispp(x) })
h <- hyperframe(lambda=lambda, X=X)
h
h$lambda2 <- lambda^2
h[, "lambda3"] <- lambda^3
h[, "Y"] <- X

identify.ppp

Identify Points in a Point Pattern

Description
If a point pattern is plotted in the graphics window, this function will find the point of the pattern
which is nearest to the mouse position, and print its mark value (or its serial number if there is no
mark).
Usage
## S3 method for class 'ppp'
identify(x, ...)
## S3 method for class 'lpp'
identify(x, ...)

identify.psp

535

Arguments
x
...

A point pattern (object of class "ppp" or "lpp").
Arguments passed to identify.default.

Details
This is a method for the generic function identify for point pattern objects.
The point pattern x should first be plotted using plot.ppp or plot.lpp as appropriate. Then
identify(x) reads the position of the graphics pointer each time the left mouse button is pressed. It
then finds the point of the pattern x closest to the mouse position. If this closest point is sufficiently
close to the mouse pointer, its index (and its mark if any) will be returned as part of the value of the
call.
Each time a point of the pattern is identified, text will be displayed next to the point, showing its
serial number (if x is unmarked) or its mark value (if x is marked).
Value
If x is unmarked, the result is a vector containing the serial numbers of the points in the pattern x
that were identified. If x is marked, the result is a 2-column matrix, the first column containing the
serial numbers and the second containing the marks for these points.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
identify, clickppp
identify.psp

Identify Segments in a Line Segment Pattern

Description
If a line segment pattern is plotted in the graphics window, this function will find the segment which
is nearest to the mouse position, and print its serial number.
Usage
## S3 method for class 'psp'
identify(x, ..., labels=seq_len(nsegments(x)), n=nsegments(x), plot=TRUE)
Arguments
x
labels

n
plot
...

A line segment pattern (object of class "psp").
Labels associated with the segments, to be plotted when the segments are identified. A character vector or numeric vector of length equal to the number of
segments in x.
Maximum number of segments to be identified.
Logical. Whether to plot the labels when a segment is identified.
Arguments passed to text.default controlling the plotting of the labels.

536

idw

Details
This is a method for the generic function identify for line segment pattern objects.
The line segment pattern x should first be plotted using plot.psp. Then identify(x) reads the
position of the graphics pointer each time the left mouse button is pressed. It then finds the segment
in the pattern x that is closest to the mouse position. This segment’s index will be returned as part
of the value of the call.
Each time a segment is identified, text will be displayed next to the point, showing its serial number
(or the relevant entry of labels).
Value
Vector containing the serial numbers of the segments in the pattern x that were identified.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
identify, identify.ppp.

idw

Inverse-distance weighted smoothing of observations at irregular
points

Description
Performs spatial smoothing of numeric values observed at a set of irregular locations using inversedistance weighting.
Usage
idw(X, power=2, at="pixels", ...)
Arguments
X

A marked point pattern (object of class "ppp").

power

Numeric. Power of distance used in the weighting.

at

String specifying whether to compute the intensity values at a grid of pixel locations (at="pixels") or only at the points of X (at="points").

...

Arguments passed to as.mask to control the pixel resolution of the result.

idw

537

Details
This function performs spatial smoothing of numeric values observed at a set of irregular locations.
Smoothing is performed by inverse distance weighting. If the observed values are v1 , . . . , vn at
locations x1 , . . . , xn respectively, then the smoothed value at a location u is
P
wi v i
g(u) = Pi
i wi
where the weights are the inverse p-th powers of distance,
wi =

1
d(u, xi )p

where d(u, xi ) = ||u − xi || is the Euclidean distance from u to xi .
The argument X must be a marked point pattern (object of class "ppp", see ppp.object). The points
of the pattern are taken to be the observation locations xi , and the marks of the pattern are taken to
be the numeric values vi observed at these locations.
The marks are allowed to be a data frame. Then the smoothing procedure is applied to each column
of marks.
If at="pixels" (the default), the smoothed mark value is calculated at a grid of pixels, and the
result is a pixel image. The arguments ... control the pixel resolution. See as.mask.
If at="points", the smoothed mark values are calculated at the data points only, using a leave-oneout rule (the mark value at a data point is excluded when calculating the smoothed value for that
point).
An alternative to inverse-distance weighting is kernel smoothing, which is performed by Smooth.ppp.
Value
If X has a single column of marks:
• If at="pixels" (the default), the result is a pixel image (object of class "im"). Pixel values
are values of the interpolated function.
• If at="points", the result is a numeric vector of length equal to the number of points in X.
Entries are values of the interpolated function at the points of X.
If X has a data frame of marks:
• If at="pixels" (the default), the result is a named list of pixel images (object of class "im").
There is one image for each column of marks. This list also belongs to the class "solist",
for which there is a plot method.
• If at="points", the result is a data frame with one row for each point of X, and one column
for each column of marks. Entries are values of the interpolated function at the points of X.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
density.ppp, ppp.object, im.object.
See Smooth.ppp for kernel smoothing and nnmark for nearest-neighbour interpolation.
To perform other kinds of interpolation, see also the akima package.

538

Iest

Examples
# data frame of marks: trees marked by diameter and height
data(finpines)
plot(idw(finpines))
idw(finpines, at="points")[1:5,]

Iest

Estimate the I-function

Description
Estimates the summary function I(r) for a multitype point pattern.
Usage
Iest(X, ..., eps=NULL, r=NULL, breaks=NULL, correction=NULL)
Arguments
X

The observed point pattern, from which an estimate of I(r) will be computed.
An object of class "ppp", or data in any format acceptable to as.ppp().

...

Ignored.

eps

the resolution of the discrete approximation to Euclidean distance (see below).
There is a sensible default.

r

Optional. Numeric vector of values for the argument r at which I(r) should be
evaluated. There is a sensible default. First-time users are strongly advised not
to specify this argument. See below for important conditions on r.

breaks

This argument is for internal use only.

correction

Optional. Vector of character strings specifying the edge correction(s) to be used
by Jest.

Details
The I function summarises the dependence between types in a multitype point process (Van Lieshout
and Baddeley, 1999) It is based on the concept of the J function for an unmarked point process (Van
Lieshout and Baddeley, 1996). See Jest for information about the J function.
The I function is defined as
I(r) =

m
X

pi Jii (r) − J•• (r)

i=1

where J•• is the J function for the entire point process ignoring the marks, while Jii is the J
function for the process consisting of points of type i only, and pi is the proportion of points which
are of type i.
The I function is designed to measure dependence between points of different types, even if the
points are not Poisson. Let X be a stationary multitype point process, and write Xi for the process of
points of type i. If the processes Xi are independent of each other, then the I-function is identically
equal to 0. Deviations I(r) < 1 or I(r) > 1 typically indicate negative and positive association,
respectively, between types. See Van Lieshout and Baddeley (1999) for further information.

Iest

539
An estimate of I derived from a multitype spatial point pattern dataset can be used in exploratory
data analysis and formal inference about the pattern. The estimate of I(r) is compared against the
constant function 0. Deviations I(r) < 1 or I(r) > 1 may suggest negative and positive association,
respectively.
This algorithm estimates the I-function from the multitype point pattern X. It assumes that X can be
treated as a realisation of a stationary (spatially homogeneous) random spatial marked point process
in the plane, observed through a bounded window.
The argument X is interpreted as a point pattern object (of class "ppp", see ppp.object) and can
be supplied in any of the formats recognised by as.ppp(). It must be a multitype point pattern (it
must have a marks vector which is a factor).
The function Jest is called to compute estimates of the J functions in the formula above. In fact
three different estimates are computed using different edge corrections. See Jest for information.

Value
An object of class "fv", see fv.object, which can be plotted directly using plot.fv.
Essentially a data frame containing
r

the vector of values of the argument r at which the function I has been estimated

rs

the “reduced sample” or “border correction” estimator of I(r) computed from
the border-corrected estimates of J functions

km

the spatial Kaplan-Meier estimator of I(r) computed from the Kaplan-Meier
estimates of J functions

han

the Hanisch-style estimator of I(r) computed from the Hanisch-style estimates
of J functions

un

the uncorrected estimate of I(r) computed from the uncorrected estimates of J

theo

the theoretical value of I(r) for a stationary Poisson process: identically equal
to 0

Note
Sizeable amounts of memory may be needed during the calculation.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Van Lieshout, M.N.M. and Baddeley, A.J. (1996) A nonparametric measure of spatial interaction in
point patterns. Statistica Neerlandica 50, 344–361.
Van Lieshout, M.N.M. and Baddeley, A.J. (1999) Indices of dependence between types in multivariate point patterns. Scandinavian Journal of Statistics 26, 511–532.
See Also
Jest

540

im

Examples
data(amacrine)
Ic <- Iest(amacrine)
plot(Ic, main="Amacrine Cells data")
# values are below I= 0, suggesting negative association
# between 'on' and 'off' cells.

im

Create a Pixel Image Object

Description
Creates an object of class "im" representing a two-dimensional pixel image.
Usage
im(mat, xcol=seq_len(ncol(mat)), yrow=seq_len(nrow(mat)),
xrange=NULL, yrange=NULL,
unitname=NULL)
Arguments
mat

matrix or vector containing the pixel values of the image.

xcol

vector of x coordinates for the pixel grid

yrow

vector of y coordinates for the pixel grid

xrange,yrange

Optional. Vectors of length 2 giving the x and y limits of the enclosing rectangle.
(Ignored if xcol, yrow are present.)

unitname

Optional. Name of unit of length. Either a single character string, or a vector of
two character strings giving the singular and plural forms, respectively.

Details
This function creates an object of class "im" representing a ‘pixel image’ or two-dimensional array
of values.
The pixel grid is rectangular and occupies a rectangular window in the spatial coordinate system.
The pixel values are scalars: they can be real numbers, integers, complex numbers, single characters
or strings, logical values, or categorical values. A pixel’s value can also be NA, meaning that no
value is defined at that location, and effectively that pixel is ‘outside’ the window. Although the
pixel values must be scalar, photographic colour images (i.e., with red, green, and blue brightness
channels) can be represented as character-valued images in spatstat, using R’s standard encoding
of colours as character strings.
The matrix mat contains the ‘greyscale’ values for a rectangular grid of pixels. Note carefully that
the entry mat[i,j] gives the pixel value at the location (xcol[j],yrow[i]). That is, the row
index of the matrix mat corresponds to increasing y coordinate, while the column index of mat
corresponds to increasing x coordinate. Thus yrow has one entry for each row of mat and xcol has
one entry for each column of mat. Under the usual convention in R, a correct display of the image
would be obtained by transposing the matrix, e.g. image.default(xcol, yrow, t(mat)), if you
wanted to do it by hand.

im

541
The entries of mat may be numeric (real or integer), complex, logical, character, or factor values.
If mat is not a matrix, it will be converted into a matrix with nrow(mat) = length(yrow) and
ncol(mat) = length(xcol).
To make a factor-valued image, note that R has a quirky way of handling matrices with factor-valued
entries. The command matrix cannot be used directly, because it destroys factor information. To
make a factor-valued image, do one of the following:
• Create a factor containing the pixel values, say mat <- factor(.....), and then assign
matrix dimensions to it by dim(mat) <- c(nr, nc) where nr, nc are the numbers of rows
and columns. The resulting object mat is both a factor and a vector.
• Supply mat as a one-dimensional factor and specify the arguments xcol and yrow to determine
the dimensions of the image.
• Use the functions cut.im or eval.im to make factor-valued images from other images).
For a description of the methods available for pixel image objects, see im.object.
To convert other kinds of data to a pixel image (for example, functions or windows), use as.im.

Warnings
The internal representation of images is likely to change in future releases of spatstat. The safe
way to extract pixel values from an image object is to use as.matrix.im or [.im.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
im.object for details of the class.
as.im for converting other kinds of data to an image.
as.matrix.im, [.im, eval.im for manipulating images.
Examples
vec <- rnorm(1200)
mat <- matrix(vec, nrow=30, ncol=40)
whitenoise <- im(mat)
whitenoise <- im(mat, xrange=c(0,1), yrange=c(0,1))
whitenoise <- im(mat, xcol=seq(0,1,length=40), yrow=seq(0,1,length=30))
whitenoise <- im(vec, xcol=seq(0,1,length=40), yrow=seq(0,1,length=30))
plot(whitenoise)
# Factor-valued images:
f <- factor(letters[1:12])
dim(f) <- c(3,4)
Z <- im(f)
# Factor image from other image:
cutwhite <- cut(whitenoise, 3)
plot(cutwhite)
# Factor image from raw data

542

im.apply
cutmat <- cut(mat, 3)
dim(cutmat) <- c(30,40)
cutwhite <- im(cutmat)
plot(cutwhite)

im.apply

Apply Function Pixelwise to List of Images

Description
Returns a pixel image obtained by applying a function to the values of corresponding pixels in
several pixel images.
Usage
im.apply(X, FUN, ...)
Arguments
X

A list of pixel images (objects of class "im").

FUN

A function that can be applied to vectors, or a character string giving the name
of such a function.

...

Additional arguments to FUN.

Details
The argument X should be a list of pixel images (objects of class "im"). If the images do not have
identical pixel grids, they will be converted to a common grid using harmonise.im.
At each pixel location, the values of the images in X at that pixel will be extracted as a vector. The
function FUN will be applied to this vector. The result (which should be a single value) becomes the
pixel value of the resulting image.
Value
A pixel image (object of class "im").
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
eval.im for algebraic operations with images.
Examples
DA <- density(split(amacrine))
DA
im.apply(DA, max)

im.object

im.object

543

Class of Images

Description
A class "im" to represent a two-dimensional pixel image.
Details
An object of this class represents a two-dimensional pixel image. It specifies
• the dimensions of the rectangular array of pixels
• x and y coordinates for the pixels
• a numeric value (“grey value”) at each pixel
If X is an object of type im, it contains the following elements:
v
dim
xrange
yrange
xstep
ystep
xcol
yrow

matrix of values
dimensions of matrix v
range of x coordinates of image window
range of y coordinates of image window
width of one pixel
height of one pixel
vector of x coordinates of centres of pixels
vector of y coordinates of centres of pixels

Users are strongly advised not to manipulate these entries directly.
Objects of class "im" may be created by the functions im and as.im. Image objects are also returned
by various functions including distmap, Kmeasure, setcov, eval.im and cut.im.
Image objects may be displayed using the methods plot.im, image.im, persp.im and contour.im.
There are also methods print.im for printing information about an image, summary.im for summarising an image, mean.im for calculating the average pixel value, hist.im for plotting a histogram of pixel values, quantile.im for calculating quantiles of pixel values, and cut.im for
dividing the range of pixel values into categories.
Pixel values in an image may be extracted using the subset operator [.im. To extract all pixel values
from an image object, use as.matrix.im. The levels of a factor-valued image can be extracted and
changed with levels and levels<-.
Calculations involving one or more images (for example, squaring all the pixel values in an image,
converting numbers to factor levels, or subtracting one image from another) can often be done easily
using eval.im. To find all pixels satisfying a certain constraint, use solutionset.
Note carefully that the entry v[i,j] gives the pixel value at the location (xcol[j],yrow[i]. That
is, the row index of the matrix v corresponds to increasing y coordinate, while the column index of
mat corresponds to increasing x coordinate. Thus yrow has one entry for each row of v and xcol
has one entry for each column of v. Under the usual convention in R, a correct display of the image
would be obtained by transposing the matrix, e.g. image.default(xcol, yrow, t(v)), if you
wanted to do it by hand.

544

imcov

Warnings
The internal representation of images is likely to change in future releases of spatstat. Do not
address the entries in an image directly. To extract all pixel values from an image object, use
as.matrix.im.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
im, as.im, plot.im, persp.im, eval.im, [.im

imcov

Spatial Covariance of a Pixel Image

Description
Computes the unnormalised spatial covariance function of a pixel image.
Usage
imcov(X, Y=X)
Arguments
X

A pixel image (object of class "im".

Y

Optional. Another pixel image.

Details
The (uncentred, unnormalised) spatial covariance function of a pixel image X in the plane is the
function C(v) defined for each vector v as
Z
C(v) = X(u)X(u − v) du
where the integral is over all spatial locations u, and where X(u) denotes the pixel value at location
u.
This command computes a discretised approximation to the spatial covariance function, using the
Fast Fourier Transform. The return value is another pixel image (object of class "im") whose
greyscale values are values of the spatial covariance function.
If the argument Y is present, then imcov(X,Y) computes the set cross-covariance function C(u)
defined as
Z
C(v) = X(u)Y (u − v) du.
Note that imcov(X,Y) is equivalent to convolve.im(X,Y,reflectY=TRUE).

improve.kppm

545

Value
A pixel image (an object of class "im") representing the spatial covariance function of X, or the
cross-covariance of X and Y.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
setcov, convolve.im, owin, as.owin, erosion
Examples
X <- as.im(square(1))
v <- imcov(X)
plot(v)

improve.kppm

Improve Intensity Estimate of Fitted Cluster Point Process Model

Description
Update the fitted intensity of a fitted cluster point process model.
Usage
improve.kppm(object, type=c("quasi", "wclik1", "clik1"), rmax = NULL,
eps.rmax = 0.01, dimyx = 50, maxIter = 100, tolerance = 1e-06,
fast = TRUE, vcov = FALSE, fast.vcov = FALSE, verbose = FALSE,
save.internals = FALSE)
Arguments
object

Fitted cluster point process model (object of class "kppm").

type

A character string indicating the method of estimation. Current options are
"clik1", "wclik1" and "quasi" for, respectively, first order composite (Poisson) likelihood, weighted first order composite likelihood and quasi-likelihood.

rmax

Optional. The dependence range. Not usually specified by the user.

eps.rmax

Numeric. A small positive number which is used to determine rmax from the
tail behaviour of the pair correlation function. Namely rmax is the smallest value
of r at which (g(r) − 1)/(g(0) − 1) falls below eps.rmax. Ignored if rmax is
provided.

dimyx

Pixel array dimensions. See Details.

maxIter

Integer. Maximum number of iterations of iterative weighted least squares (Fisher
scoring).

tolerance

Numeric. Tolerance value specifying when to stop iterative weighted least squares
(Fisher scoring).

546

improve.kppm
fast

Logical value indicating whether tapering should be used to make the computations faster (requires the package Matrix).

vcov

Logical value indicating whether to calculate the asymptotic variance covariance/matrix.

fast.vcov

Logical value indicating whether tapering should be used for the variance/covariance
matrix to make the computations faster (requires the package Matrix). Caution:
This is expected to underestimate the true asymptotic variances/covariances.

verbose

A logical indicating whether the details of computations should be printed.

save.internals A logical indicating whether internal quantities should be saved in the returned
object (mostly for development purposes).
Details
This function reestimates the intensity parameters in a fitted "kppm" object. If type="clik1"
estimates are based on the first order composite (Poisson) likelihood, which ignores dependence
between the points. Note that type="clik1" is mainly included for testing purposes and is not recommended for the typical user; instead the more efficient kppm with improve.type="none" should
be used.
When type="quasi" or type="wclik1" the dependence structure between the points is incorporated in the estimation procedure by using the estimated pair correlation function in the estimating
equation.
In all cases the estimating equation is based on dividing the observation window into small subregions and count the number of points in each subregion. To do this the observation window is first
converted into a digital mask by as.mask where the resolution is controlled by the argument dimyx.
The computational time grows with the cube of the number of subregions, so fine grids may take
very long to compute (or even run out of memory).
Value
A fitted cluster point process model of class "kppm".
Author(s)
Abdollah Jalilian 

and Rasmus Waagepetersen  adapted for spatstat by Adrian Baddeley 
References
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous NeymanScott processes, Biometrics, 63, 252-258.
Guan, Y. and Shen, Y. (2010) A weighted estimating equation approach to inference for inhomogeneous spatial point processes, Biometrika, 97, 867-880.
Guan, Y., Jalilian, A. and Waagepetersen, R. (2015) Quasi-likelihood for spatial point processes.
Journal of the Royal Statistical Society, Series B 77, 677–697.
See Also
ppm, kppm, improve.kppm

incircle

547

Examples
# fit a Thomas process using minimum contrast estimation method
# to model interaction between points of the pattern
fit0 <- kppm(bei ~ elev + grad, data = bei.extra)
# fit the log-linear intensity model with quasi-likelihood method
fit1 <- improve.kppm(fit0, type="quasi")
# compare
coef(fit0)
coef(fit1)

incircle

Find Largest Circle Inside Window

Description
Find the largest circle contained in a given window.
Usage
incircle(W)
inradius(W)
Arguments
W

A window (object of class "owin").

Details
Given a window W of any type and shape, the function incircle determines the largest circle that
is contained inside W, while inradius computes its radius only.
For non-rectangular windows, the incircle is computed approximately by finding the maximum of
the distance map (see distmap) of the complement of the window.
Value
The result of incircle is a list with entries x,y,r giving the location (x,y) and radius r of the
incircle.
The result of inradius is the numerical value of radius.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
centroid.owin

548

increment.fv

Examples
W <- square(1)
Wc <- incircle(W)
plot(W)
plot(disc(Wc$r, c(Wc$x, Wc$y)), add=TRUE)
plot(letterR)
Rc <- incircle(letterR)
plot(disc(Rc$r, c(Rc$x, Rc$y)), add=TRUE)
W <- as.mask(letterR)
plot(W)
Rc <- incircle(W)
plot(disc(Rc$r, c(Rc$x, Rc$y)), add=TRUE)

increment.fv

Increments of a Function

Description
Compute the change in the value of a function f when the function argument increases by delta.
Usage
increment.fv(f, delta)
Arguments
f

Object of class "fv" representing a function.

delta

Numeric. The increase in the value of the function argument.

Details
This command computes the new function
g(x) = f (x + h) − f (x − h)
where h = delta/2. The value of g(x) is the change in the value of f over an interval of length
delta centred at x.
Value
Another object of class "fv" compatible with X.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
fv.object, deriv.fv

infline

549

Examples
plot(increment.fv(Kest(cells), 0.05))

infline

Infinite Straight Lines

Description
Define the coordinates of one or more straight lines in the plane
Usage
infline(a = NULL, b = NULL, h = NULL, v = NULL, p = NULL, theta = NULL)
## S3 method for class 'infline'
print(x, ...)
## S3 method for class 'infline'
plot(x, ...)
Arguments
a,b

Numeric vectors of equal length giving the intercepts a and slopes b of the lines.
Incompatible with h,v,p,theta

h

Numeric vector giving the positions of horizontal lines when they cross the y
axis. Incompatible with a,b,v,p,theta

v

Numeric vector giving the positions of vertical lines when they cross the x axis.
Incompatible with a,b,h,p,theta

p,theta

Numeric vectors of equal length giving the polar coordinates of the line. Incompatible with a,b,h,v

x

An object of class "infline"

...

Extra arguments passed to print for printing or abline for plotting

Details
The class infline is a convenient way to handle infinite straight lines in the plane.
The position of a line can be specified in several ways:
• its intercept a and slope b in the equation y = a + bx can be used unless the line is vertical.
• for vertical lines we can use the position v where the line crosses the y axis
• for horizontal lines we can use the position h where the line crosses the x axis
• the polar coordinates p and θ can be used for any line. The line equation is
y cos θ + x sin θ = p
The command infline will accept line coordinates in any of these formats. The arguments a,b,h,v
have the same interpretation as they do in the line-plotting function abline.
The command infline converts between different coordinate systems (e.g. from a,b to p,theta)
and returns an object of class "infline" that contains a representation of the lines in each appropriate coordinate system. This object can be printed and plotted.

550

influence.ppm

Value
The value of infline is an object of class "infline" which is basically a data frame with columns
a,b,h,v,p,theta. Each row of the data frame represents one line. Entries may be NA if a coordinate is not applicable to a particular line.

Author(s)
Adrian Baddeley  and Rolf Turner 
See Also
rotate.infline, clip.infline, chop.tess, whichhalfplane
Examples
infline(a=10:13,b=1)
infline(p=1:3, theta=pi/4)
plot(c(-1,1),c(-1,1),type="n",xlab="",ylab="", asp=1)
plot(infline(p=0.4, theta=seq(0,pi,length=20)))

influence.ppm

Influence Measure for Spatial Point Process Model

Description
Computes the influence measure for a fitted spatial point process model.

Usage
## S3 method for class 'ppm'
influence(model, ..., drop = FALSE, iScore=NULL, iHessian=NULL, iArgs=NULL)
Arguments
model

Fitted point process model (object of class "ppm").

...

Ignored.

drop

Logical. Whether to include (drop=FALSE) or exclude (drop=TRUE) contributions from quadrature points that were not used to fit the model.

iScore,iHessian
Components of the score vector and Hessian matrix for the irregular parameters,
if required. See Details.
iArgs

List of extra arguments for the functions iScore, iHessian if required.

influence.ppm

551

Details
Given a fitted spatial point process model model, this function computes the influence measure
described in Baddeley, Chang and Song (2013).
The function influence is generic, and influence.ppm is the method for objects of class "ppm"
representing point process models.
The influence of a point process model is a value attached to each data point (i.e. each point of the
point pattern to which the model was fitted). The influence value s(xi ) at a data point xi represents
the change in the maximised log (pseudo)likelihood that occurs when the point xi is deleted. A
relatively large value of s(xi ) indicates a data point with a large influence on the fitted model.
If the point process model trend has irregular parameters that were fitted (using ippm) then the
influence calculation requires the first and second derivatives of the log trend with respect to the
irregular parameters. The argument iScore should be a list, with one entry for each irregular
parameter, of R functions that compute the partial derivatives of the log trend (i.e. log intensity or
log conditional intensity) with respect to each irregular parameter. The argument iHessian should
be a list, with p2 entries where p is the number of irregular parameters, of R functions that compute
the second order partial derivatives of the log trend with respect to each pair of irregular parameters.
The result of influence.ppm is an object of class "influence.ppm". It can be plotted (by plot.influence.ppm),
or converted to a marked point pattern by as.ppp (see as.ppp.influence.ppm).
Value
An object of class "influence.ppm" that can be plotted (by plot.influence.ppm). There are also
methods for print, [, as.ppp and as.owin.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
References
Baddeley, A. and Chang, Y.M. and Song, Y. (2013) Leverage and influence diagnostics for spatial
point process models. Scandinavian Journal of Statistics 40, 86–104.

See Also
leverage.ppm, dfbetas.ppm, ppmInfluence, plot.influence.ppm
Examples
X <- rpoispp(function(x,y) { exp(3+3*x) })
fit <- ppm(X ~x+y)
plot(influence(fit))

552

insertVertices

inforder.family

Infinite Order Interaction Family

Description
An object describing the family of all Gibbs point processes with infinite interaction order.
Details
Advanced Use Only!
This structure would not normally be touched by the user. It describes the interaction structure of
Gibbs point processes which have infinite order of interaction, such as the area-interaction process
AreaInter.
Anyway, inforder.family is an object of class "isf" containing a function inforder.family$eval
for evaluating the sufficient statistics of a Gibbs point process model taking an exponential family
form.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns.
Australian and New Zealand Journal of Statistics 42, 283–322.
See Also
AreaInter to create the area interaction process structure.
Other families: pairwise.family, pairsat.family, ord.family.

insertVertices

Insert New Vertices in a Linear Network

Description
Adds new vertices to a linear network at specified locations along the network.
Usage
insertVertices(L, ...)
Arguments
L

Linear network (object of class "linnet") or point pattern on a linear network
(object of class "lpp").

...

Additional arguments passed to as.lpp specifying the positions of the new vertices along the network.

insertVertices

553

Details
This function adds new vertices at locations along an existing linear network.
The argument L can be either a linear network (class "linnet") or some other object that includes
a linear network.
The new vertex locations can be specified either as a point pattern (class "lpp" or "ppp") or using
coordinate vectors x,y or seg,tp or x,y,seg,tp as explained in the help for as.lpp.
This function breaks the existing line segments of L into pieces at the locations specified by the
coordinates seg,tp and creates new vertices at these locations.
The result is the modified object, with an attribute "id" such that the ith added vertex has become
the id[i]th vertex of the new network.
Value
An object of the same class as L representing the result of adding the new vertices. The result also
has an attribute "id" as described in Details.
Author(s)
Adrian Baddeley
See Also
as.lpp
Examples
opa <- par(mfrow=c(1,3), mar=rep(0,4))
simplenet
plot(simplenet, main="")
plot(vertices(simplenet), add=TRUE)
# add two new vertices at specified local coordinates
L <- insertVertices(simplenet, seg=c(3,7), tp=c(0.2, 0.5))
L
plot(L, main="")
plot(vertices(L), add=TRUE)
id <- attr(L, "id")
id
plot(vertices(L)[id], add=TRUE, pch=16)
# add new vertices at three randomly-generated points
X <- runiflpp(3, simplenet)
LL <- insertVertices(simplenet, X)
plot(LL, main="")
plot(vertices(LL), add=TRUE)
ii <- attr(LL, "id")
plot(vertices(LL)[ii], add=TRUE, pch=16)
par(opa)

554

inside.boxx

inside.boxx

Test Whether Points Are Inside A Multidimensional Box

Description
Test whether points lie inside or outside a given multidimensional box.
Usage
inside.boxx(..., w)
Arguments
...

Coordinates of points to be tested. One vector for each dimension (all of same
length). (Alternatively, a single point pattern object of class "ppx" or its coordinates as a "hyperframe")

w

A window. This should be an object of class boxx, or can be given in any format
acceptable to as.boxx().

Details
This function tests whether each of the points (x[i],y[i]) lies inside or outside the window w and
returns TRUE if it is inside.
The boundary of the window is treated as being inside.
Normally each argument provided (except w) must be numeric vectors of equal length (length zero
is allowed) containing the coordinates of points. Alternatively a single point pattern (object of class
"ppx") can be given; then the coordinates of the point pattern are extracted.
Value
Logical vector whose ith entry is TRUE if the corresponding point is inside w.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
boxx, as.boxx
Examples
# Random points in box with side [0,2]
w <- boxx(c(0,2), c(0,2), c(0,2))
#
x
y
z

Random points in box
<- runif(30, min=-1,
<- runif(30, min=-1,
<- runif(30, min=-1,

with side [-1,3]
max=3)
max=3)
max=3)

inside.owin

555

# Points falling in smaller box
ok <- inside.boxx(x, y, z, w=w)
# Same using a point pattern as argument:
X <- ppx(data = cbind(x, y, z), domain = boxx(c(0,3), c(0,3), c(0,3)))
ok2 <- inside.boxx(X, w=w)

inside.owin

Test Whether Points Are Inside A Window

Description
Test whether points lie inside or outside a given window.
Usage
inside.owin(x, y, w)
Arguments
x

Vector of x coordinates of points to be tested. (Alternatively, a point pattern
object providing both x and y coordinates.)

y

Vector of y coordinates of points to be tested.

w

A window. This should be an object of class owin, or can be given in any format
acceptable to as.owin().

Details
This function tests whether each of the points (x[i],y[i]) lies inside or outside the window w and
returns TRUE if it is inside.
The boundary of the window is treated as being inside.
If w is of type "rectangle" or "polygonal", the algorithm uses analytic geometry (the discrete
Stokes theorem). Computation time is linear in the number of points and (for polygonal windows)
in the number of vertices of the boundary polygon. Boundary cases are correct to single precision
accuracy.
If w is of type "mask" then the pixel closest to (x[i],y[i]) is tested. The results may be incorrect
for points lying within one pixel diameter of the window boundary.
Normally x and y must be numeric vectors of equal length (length zero is allowed) containing the
coordinates of points. Alternatively x can be a point pattern (object of class "ppp") while y is
missing; then the coordinates of the point pattern are extracted.
Value
Logical vector whose ith entry is TRUE if the corresponding point (x[i],y[i]) is inside w.
Author(s)
Adrian Baddeley 
and Rolf Turner 

556

integral.im

See Also
owin.object, as.owin
Examples
# hexagonal window
k <- 6
theta <- 2 * pi * (0:(k-1))/k
co <- cos(theta)
si <- sin(theta)
mas <- owin(c(-1,1), c(-1,1), poly=list(x=co, y=si))
## Not run:
plot(mas)
## End(Not run)
# random points in rectangle
x <- runif(30,min=-1, max=1)
y <- runif(30,min=-1, max=1)
ok <- inside.owin(x, y, mas)
## Not run:
points(x[ok], y[ok])
points(x[!ok], y[!ok], pch="x")
## End(Not run)

integral.im

Integral of a Pixel Image

Description
Computes the integral of a pixel image.
Usage
integral(f, domain=NULL, ...)
## S3 method for class 'im'
integral(f, domain=NULL, ...)
Arguments
f

A pixel image (object of class "im") with pixel values that can be treated as
numeric or complex values.

domain

Optional. Window specifying the domain of integration. Alternatively a tessellation.

...

Ignored.

integral.linim

557

Details
The function integral is generic, with methods for "im", "msr", "linim" and "linfun".
The method integral.im treats the pixel image f as a function of the spatial coordinates, and
computes its integral. The integral is calculated by summing the pixel values and multiplying by
the area of one pixel.
The pixel values of f may be numeric, integer, logical or complex. They cannot be factor or character values.
The logical values TRUE and FALSE are converted to 1 and 0 respectively, so that the integral of a
logical image is the total area of the TRUE pixels, in the same units as unitname(x).
If domain is a window (class "owin") then the integration will be restricted to this window. If
domain is a tessellation (class "tess") then the integral of f in each tile of domain will be computed.
Value
A single numeric or complex value (or a vector of such values if domain is a tessellation).
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
eval.im, [.im
Examples
# approximate integral of f(x,y) dx dy
f <- function(x,y){3*x^2 + 2*y}
Z <- as.im(f, square(1))
integral.im(Z)
# correct answer is 2
D <- density(cells)
integral.im(D)
# should be approximately equal to number of points = 42
# integrate over the subset [0.1,0.9] x [0.2,0.8]
W <- owin(c(0.1,0.9), c(0.2,0.8))
integral.im(D, W)

integral.linim

Integral on a Linear Network

Description
Computes the integral (total value) of a function or pixel image over a linear network.

558

integral.msr

Usage
## S3 method for class 'linim'
integral(f, domain=NULL, ...)
## S3 method for class 'linfun'
integral(f, domain=NULL, ..., delta)
Arguments
f

A pixel image on a linear network (class "linim") or a function on a linear
network (class "linfun").

domain

Optional window specifying the domain of integration.

...

Ignored.

delta

Optional. The step length (in coordinate units) for computing the approximate
integral. A single positive number.

Details
The integral (total value of the function over the network) is calculated.
Value
A numeric value.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
linim, integral.im
Examples
# make some data
xcoord <- linfun(function(x,y,seg,tp) { x }, simplenet)
integral(xcoord)
X <- as.linim(xcoord)
integral(X)

integral.msr

Integral of a Measure

Description
Computes the integral (total value) of a measure over its domain.
Usage
## S3 method for class 'msr'
integral(f, domain=NULL, ...)

integral.msr

559

Arguments
f

A signed measure or vector-valued measure (object of class "msr").

domain

Optional window specifying the domain of integration. Alternatively a tessellation.

...

Ignored.

Details
The integral (total value of the measure over its domain) is calculated.
If domain is a window (class "owin") then the integration will be restricted to this window. If
domain is a tessellation (class "tess") then the integral of f in each tile of domain will be computed.
For a multitype measure m, use split.msr to separate the contributions for each type of point, as
shown in the Examples.

Value
A numeric value (for a signed measure) or a vector of values (for a vector-valued measure).

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
msr, integral
Examples
fit <- ppm(cells ~ x)
rr <- residuals(fit)
integral(rr)
# vector-valued measure
rs <- residuals(fit, type="score")
integral(rs)
# multitype
fitA <- ppm(amacrine ~ x)
rrA <- residuals(fitA)
sapply(split(rrA), integral)
# multitype and vector-valued
rsA <- residuals(fitA, type="score")
sapply(split(rsA), integral)

560

intensity

intensity

Intensity of a Dataset or a Model

Description
Generic function for computing the intensity of a spatial dataset or spatial point process model.

Usage
intensity(X, ...)
Arguments
X

A spatial dataset or a spatial point process model.

...

Further arguments depending on the class of X.

Details
This is a generic function for computing the intensity of a spatial dataset or spatial point process
model. There are methods for point patterns (objects of class "ppp") and fitted point process models
(objects of class "ppm").
The empirical intensity of a dataset is the average density (the average amount of ‘stuff’ per unit area
or volume). The empirical intensity of a point pattern is computed by the method intensity.ppp.
The theoretical intensity of a stochastic model is the expected density (expected amount of ‘stuff’
per unit area or volume). The theoretical intensity of a fitted point process model is computed by
the method intensity.ppm.
Value
Usually a numeric value or vector.

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
intensity.ppp, intensity.ppm.

intensity.dppm

intensity.dppm

561

Intensity of Determinantal Point Process Model

Description
Extracts the intensity of a determinantal point process model.
Usage
## S3 method for class 'detpointprocfamily'
intensity(X, ...)
## S3 method for class 'dppm'
intensity(X, ...)
Arguments
X

A determinantal point process model (object of class "detpointprocfamily"
or "dppm").

...

Ignored.

Value
A numeric value (if the model is stationary), a pixel image (if the model is non-stationary) or NA if
the intensity is unknown for the model.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 

intensity.lpp

Empirical Intensity of Point Pattern on Linear Network

Description
Computes the average number of points per unit length in a point pattern on a linear network.
Usage
## S3 method for class 'lpp'
intensity(X, ...)
Arguments
X

A point pattern on a linear network (object of class "lpp").

...

Ignored.

562

intensity.ppm

Details
This is a method for the generic function intensity It computes the empirical intensity of a point
pattern on a linear network (object of class "lpp"), i.e. the average density of points per unit length.
If the point pattern is multitype, the intensities of the different types are computed separately.
Value
A numeric value (giving the intensity) or numeric vector (giving the intensity for each possible
type).
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
intensity, intensity.ppp
Examples
intensity(chicago)

intensity.ppm

Intensity of Fitted Point Process Model

Description
Computes the intensity of a fitted point process model.
Usage
## S3 method for class 'ppm'
intensity(X, ...)
Arguments
X

A fitted point process model (object of class "ppm").

...

Arguments passed to predict.ppm in some cases. See Details.

Details
This is a method for the generic function intensity for fitted point process models (class "ppm").
The intensity of a point process model is the expected number of random points per unit area.
If X is a Poisson point process model, the intensity of the process is computed exactly. The result is
a numerical value if X is a stationary Poisson point process, and a pixel image if X is non-stationary.
(In the latter case, the resolution of the pixel image is controlled by the arguments ... which are
passed to predict.ppm.)
If X is another Gibbs point process model, the intensity is computed approximately using the
Poisson-saddlepoint approximation (Baddeley and Nair, 2012a, 2012b, 2016; Anderssen et al,

intensity.ppm

563

2014). The approximation is currently available for pairwise-interaction models (Baddeley and
Nair, 2012a, 2012b) and for the area-interaction model and Geyer saturation model (Baddeley and
Nair, 2016).
For a non-stationary Gibbs model, the pseudostationary solution (Baddeley and Nair, 2012b; Anderssen et al, 2014) is used. The result is a pixel image, whose resolution is controlled by the
arguments ... which are passed to predict.ppm.

Value
A numeric value (if the model is stationary) or a pixel image.

Author(s)
Adrian Baddeley  and Gopalan Nair.

References
Anderssen, R.S., Baddeley, A., DeHoog, F.R. and Nair, G.M. (2014) Solution of an integral equation
arising in spatial point process theory. Journal of Integral Equations and Applications 26 (4) 437–
453.
Baddeley, A. and Nair, G. (2012a) Fast approximation of the intensity of Gibbs point processes.
Electronic Journal of Statistics 6 1155–1169.
Baddeley, A. and Nair, G. (2012b) Approximating the moments of a spatial point process. Stat 1,
1, 18–30. doi: 10.1002/sta4.5
Baddeley, A. and Nair, G. (2016) Poisson-saddlepoint approximation for spatial point processes
with infinite order interaction. Submitted for publication.

See Also
intensity, intensity.ppp

Examples
fitP <- ppm(swedishpines ~ 1)
intensity(fitP)
fitS <- ppm(swedishpines ~ 1, Strauss(9))
intensity(fitS)
fitSx <- ppm(swedishpines ~ x, Strauss(9))
lamSx <- intensity(fitSx)
fitG <- ppm(swedishpines ~ 1, Geyer(9, 1))
lamG <- intensity(fitG)
fitA <- ppm(swedishpines ~ 1, AreaInter(7))
lamA <- intensity(fitA)

564

intensity.ppp

intensity.ppp

Empirical Intensity of Point Pattern

Description
Computes the average number of points per unit area in a point pattern dataset.
Usage
## S3 method for class 'ppp'
intensity(X, ..., weights=NULL)
## S3 method for class 'splitppp'
intensity(X, ..., weights=NULL)
Arguments
X

A point pattern (object of class "ppp").

weights

Optional. Numeric vector of weights attached to the points of X. Alternatively,
an expression which can be evaluated to give a vector of weights.

...

Ignored.

Details
This is a method for the generic function intensity. It computes the empirical intensity of a point
pattern (object of class "ppp"), i.e. the average density of points per unit area.
If the point pattern is multitype, the intensities of the different types are computed separately.
Note that the intensity will be computed as the number of points per square unit, based on the unit
of length for X, given by unitname(X). If the unit of length is a strange multiple of a standard unit,
like 5.7 metres, then it can be converted to the standard unit using rescale. See the Examples.
If weights are given, then the intensity is computed as the total weight per square unit. The argument weights should be a numeric vector of weights for each point of X (weights may be negative
or zero).
Alternatively weights can be an expression which will be evaluated for the dataset to yield a
vector of weights. The expression may involve the Cartesian coordinates x, y of the points, and the
marks of the points, if any. Variable names permitted in the expression include x and y, the name
marks if X has a single column of marks, the names of any columns of marks if X has a data frame
of marks, and the names of constants or functions that exist in the global environment. See the
Examples.
Value
A numeric value (giving the intensity) or numeric vector (giving the intensity for each possible
type).
Author(s)
Adrian Baddeley 
and Rolf Turner 

intensity.ppx

565

See Also
intensity, intensity.ppm
Examples
japanesepines
intensity(japanesepines)
unitname(japanesepines)
intensity(rescale(japanesepines))
intensity(amacrine)
intensity(split(amacrine))
# numeric vector of weights
volumes <- with(marks(finpines), (pi/4) * height * diameter^2)
intensity(finpines, weights=volumes)
# expression for weights
intensity(finpines, weights=expression((pi/4) * height * diameter^2))

intensity.ppx

Intensity of a Multidimensional Space-Time Point Pattern

Description
Calculates the intensity of points in a multi-dimensional point pattern of class "ppx" or "pp3".
Usage
## S3 method for class 'ppx'
intensity(X, ...)
Arguments
X
...

Point pattern of class "ppx" or "pp3".
Ignored.

Details
This is a method for the generic function intensity. It computes the empirical intensity of a multidimensional point pattern (object of class "ppx" including "pp3"), i.e. the average density of points
per unit volume.
If the point pattern is multitype, the intensities of the different types are computed separately.
Value
A single number or a numeric vector.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 

566

intensity.psp

Examples
X <- osteo$pts[[1]]
intensity(X)
marks(X) <- factor(sample(letters[1:3], npoints(X), replace=TRUE))
intensity(X)

intensity.psp

Empirical Intensity of Line Segment Pattern

Description
Computes the average total length of segments per unit area in a spatial pattern of line segments.
Usage
## S3 method for class 'psp'
intensity(X, ..., weights=NULL)
Arguments
X

A line segment pattern (object of class "psp").

weights

Optional. Numeric vector of weights attached to the segments of X. Alternatively, an expression which can be evaluated to give a vector of weights.

...

Ignored.

Details
This is a method for the generic function intensity. It computes the empirical intensity of a line
segment pattern (object of class "psp"), i.e. the average total segment length per unit area.
If the segment pattern is multitype, the intensities of the different types are computed separately.
Note that the intensity will be computed as the length per area in units per square unit, based on
the unit of length for X, given by unitname(X). If the unit of length is a strange multiple of a
standard unit, like 5.7 metres, then it can be converted to the standard unit using rescale. See
the Examples.
If weights are given, then the intensity is computed as the total weight times length per square unit.
The argument weights should be a numeric vector of weights for each point of X (weights may be
negative or zero).
Alternatively weights can be an expression which will be evaluated for the dataset to yield a
vector of weights. The expression may involve the Cartesian coordinates x, y of the points, and the
marks of the points, if any. Variable names permitted in the expression include x0, x1, y0, y1 for the
coordinates of the segment endpoint, the name marks if X has a single column of marks, the names
of any columns of marks if X has a data frame of marks, and the names of constants or functions
that exist in the global environment. See the Examples.
Value
A numeric value (giving the intensity) or numeric vector (giving the intensity for each possible
type).

intensity.quadratcount

567

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
intensity
Examples
S <- as.psp(simplenet)
intensity(S)
intensity(S, weights=runif(nsegments(S)))
intensity(S, weights=expression((x0+x1)/2))

intensity.quadratcount
Intensity Estimates Using Quadrat Counts

Description
Uses quadrat count data to estimate the intensity of a point pattern in each tile of a tessellation,
assuming the intensity is constant in each tile.
Usage
## S3 method for class 'quadratcount'
intensity(X, ..., image=FALSE)
Arguments
X

An object of class "quadratcount".

image

Logical value specifying whether to return a table of estimated intensities (the
default) or a pixel image of the estimated intensity (image=TRUE).

...

Arguments passed to as.mask to determine the resolution of the pixel image, if
image=TRUE.

Details
This is a method for the generic function intensity. It computes an estimate of the intensity of a
point pattern from its quadrat counts.
The argument X should be an object of class "quadratcount". It would have been obtained by
applying the function quadratcount to a point pattern (object of class "ppp"). It contains the
counts of the numbers of points of the point pattern falling in each tile of a tessellation.
Using this information, intensity.quadratcount divides the quadrat counts by the tile areas,
yielding the average density of points per unit area in each tile of the tessellation.
If image=FALSE (the default), these intensity values are returned in a contingency table. Cells of the
contingency table correspond to tiles of the tessellation.
If image=TRUE, the estimated intensity function is returned as a pixel image. For each pixel, the
pixel value is the estimated intensity in the tile which contains that pixel.

568

interp.colourmap

Value
If image=FALSE (the default), a contingency table. If image=TRUE, a pixel image (object of class
"im").
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
intensity, quadratcount
Examples
qa <- quadratcount(swedishpines, 4,3)
qa
intensity(qa)
plot(intensity(qa, image=TRUE))

interp.colourmap

Interpolate smoothly between specified colours

Description
Given a colourmap object which maps numbers to colours, this function interpolates smoothly
between the colours, yielding a new colour map.
Usage
interp.colourmap(m, n = 512)
Arguments
m

A colour map (object of class "colourmap").

n

Number of colour steps to be created in the new colour map.

Details
Given a colourmap object m, which maps numerical values to colours, this function interpolates the
mapping, yielding a new colour map.
This makes it easy to build a colour map that has smooth gradation between different colours or
shades. First specify a small vector of numbers x which should be mapped to specific colours y.
Use m <- colourmap(y, inputs=x) to create a colourmap that represents this simple mapping.
Then apply interp.colourmap(m) to obtain a smooth transition between these points.
Value
Another colour map (object of class "colourmap").

interp.im

569

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
colourmap, tweak.colourmap, colourtools.
Examples
co <- colourmap(inputs=c(0, 0.5, 1), c("black", "red", "white"))
plot(interp.colourmap(co))

interp.im

Interpolate a Pixel Image

Description
Interpolates the values of a pixel image at any desired location in the frame.
Usage
interp.im(Z, x, y=NULL)
Arguments
Z

Pixel image (object of class "im") with numeric or integer values.

x,y

Vectors of Cartesian coordinates. Alternatively x can be a point pattern and y
can be missing.

Details
A value at each location (x[i],y[i]) will be interpolated using the pixel values of Z at the four
surrounding pixel centres, by simple bilinear interpolation.
At the boundary (where (x[i],y[i]) is not surrounded by four pixel centres) the value at the
nearest pixel is taken.
The arguments x,y can be anything acceptable to xy.coords.
Value
Vector of interpolated values, with NA for points that lie outside the domain of the image.
Author(s)
Adrian Baddeley 
and Rolf Turner 

570

intersect.owin

Examples
opa <- par(mfrow=c(1,2))
# coarse image
V <- as.im(function(x,y) { x^2 + y }, owin(), dimyx=10)
plot(V, main="coarse image", col=terrain.colors(256))
# lookup value at location (0.5,0.5)
V[list(x=0.5,y=0.5)]
# interpolated value at location (0.5,0.5)
interp.im(V, 0.5, 0.5)
# true value is 0.75
# how to obtain an interpolated image at a desired resolution
U <- as.im(interp.im, W=owin(), Z=V, dimyx=256)
plot(U, main="interpolated image", col=terrain.colors(256))
par(opa)

intersect.owin

Intersection, Union or Set Subtraction of Windows

Description
Yields the intersection, union or set subtraction of windows.
Usage
intersect.owin(..., fatal=TRUE, p)
union.owin(..., p)
setminus.owin(A, B, ..., p)
Arguments
A,B
...
fatal
p

Windows (objects of class "owin").
Windows, or arguments passed to as.mask to control the discretisation.
Logical. Determines what happens if the intersection is empty.
Optional list of parameters passed to polyclip to control the accuracy of polygon geometry.

Details
The function intersect.owin computes the intersection between the windows given in ..., while
union.owin computes their union. The function setminus.owin computes the intersection of A
with the complement of B.
For intersect.owin and union.owin, the arguments ... must be either
•
•
•
•

window objects of class "owin",
data that can be coerced to this class by as.owin),
lists of windows, of class "solist",
named arguments of as.mask to control the discretisation if required.

For setminus.owin, the arguments ... must be named arguments of as.mask.
If the intersection is empty, then if fatal=FALSE the result is NULL, while if fatal=TRUE an error
occurs.

intersect.owin
Value
A window (object of class "owin") or possibly NULL.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
is.subset.owin, overlap.owin, boundingbox, owin.object
Examples
# rectangles
u <- unit.square()
v <- owin(c(0.5,3.5), c(0.4,2.5))
# polygon
data(letterR)
# mask
m <- as.mask(letterR)
# two rectangles
intersect.owin(u, v)
union.owin(u,v)
setminus.owin(u,v)
# polygon and rectangle
intersect.owin(letterR, v)
union.owin(letterR,v)
setminus.owin(letterR,v)
# mask and rectangle
intersect.owin(m, v)
union.owin(m,v)
setminus.owin(m,v)
# mask and polygon
p <- rotate(v, 0.2)
intersect.owin(m, p)
union.owin(m,p)
setminus.owin(m,p)
# two polygons
A <- letterR
B <- rotate(letterR, 0.2)
plot(boundingbox(A,B), main="intersection")
w <- intersect.owin(A, B)
plot(w, add=TRUE, col="lightblue")
plot(A, add=TRUE)
plot(B, add=TRUE)
plot(boundingbox(A,B), main="union")
w <- union.owin(A,B)

571

572

intersect.tess
plot(w, add=TRUE, col="lightblue")
plot(A, add=TRUE)
plot(B, add=TRUE)
plot(boundingbox(A,B), main="set minus")
w <- setminus.owin(A,B)
plot(w, add=TRUE, col="lightblue")
plot(A, add=TRUE)
plot(B, add=TRUE)
# intersection and union of three windows
C <- shift(B, c(0.2, 0.3))
plot(union.owin(A,B,C))
plot(intersect.owin(A,B,C))

intersect.tess

Intersection of Two Tessellations

Description
Yields the intersection of two tessellations, or the intersection of a tessellation with a window.
Usage
intersect.tess(X, Y, ..., keepmarks=FALSE)
Arguments
X,Y

Two tessellations (objects of class "tess"), or windows (objects of class "tess"),
or other data that can be converted to tessellations by as.tess.

...

Optional arguments passed to as.mask to control the discretisation, if required.

keepmarks

Logical value. If TRUE, the marks attached to the tiles of X and Y will be retained
as marks of the intersection tiles.

Details
A tessellation is a collection of disjoint spatial regions (called tiles) that fit together to form a larger
spatial region. See tess.
If X and Y are not tessellations, they are first converted into tessellations by as.tess.
The function intersect.tess then computes the intersection between the two tessellations. This
is another tessellation, each of whose tiles is the intersection of a tile from X and a tile from Y.
One possible use of this function is to slice a window W into subwindows determined by a tessellation. See the Examples.
Value
A tessellation (object of class "tess").
Author(s)
Adrian Baddeley 
and Rolf Turner 

invoke.symbolmap

573

See Also
tess, as.tess, intersect.owin
Examples
opa <- par(mfrow=c(1,3))
# polygon
data(letterR)
plot(letterR)
# tessellation of rectangles
X <- tess(xgrid=seq(2, 4, length=10), ygrid=seq(0, 3.5, length=8))
plot(X)
plot(intersect.tess(X, letterR))
A <- runifpoint(10)
B <- runifpoint(10)
plot(DA <- dirichlet(A))
plot(DB <- dirichlet(B))
plot(intersect.tess(DA, DB))
par(opa)
marks(DA) <- 1:10
marks(DB) <- 1:10
plot(Z <- intersect.tess(DA,DB, keepmarks=TRUE))
mZ <- marks(Z)
tZ <- tiles(Z)
for(i in which(mZ[,1] == 3)) plot(tZ[[i]], add=TRUE, col="pink")

invoke.symbolmap

Plot Data Using Graphics Symbol Map

Description
Apply a graphics symbol map to a vector of data values and plot the resulting symbols.
Usage
invoke.symbolmap(map, values, x=NULL, y = NULL, ..., add = FALSE,
do.plot = TRUE, started = add && do.plot)
Arguments
map

Graphics symbol map (object of class "symbolmap").

values

Vector of data that can be mapped by the symbol map.

x,y

Coordinate vectors for the spatial locations of the symbols to be plotted.

...

Additional graphics parameters.

add

Logical value indicating whether to add the symbols to an existing plot (add=TRUE)
or to initialise a new plot (add=FALSE, the default).

do.plot

Logical value indicating whether to actually perform the plotting.

started

Logical value indicating whether the plot has already been initialised.

574

iplot

Details
A symbol map is an association between data values and graphical symbols.
This command applies the symbol map map to the data values and plots the resulting symbols at
the locations given by xy.coords(x,y).
Value
(Invisibly) the maximum diameter of the symbols, in user coordinate units.
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak .
See Also
plot.symbolmap to plot the graphics map itself.
symbolmap to create a graphics map.
Examples
g <- symbolmap(range=c(-1,1),
shape=function(x) ifelse(x > 0, "circles", "squares"),
size=function(x) sqrt(ifelse(x > 0, x/pi, -x))/15,
bg=function(x) ifelse(x > 0, "green", "red"))
plot(square(1), main="")
a <- invoke.symbolmap(g, runif(10, -1, 1), runifpoint(10), add=TRUE)
a

iplot

Point and Click Interface for Displaying Spatial Data

Description
Plot spatial data with interactive (point-and-click) control over the plot.
Usage
iplot(x, ...)
## S3 method for class 'ppp'
iplot(x, ..., xname)
## S3 method for class 'linnet'
iplot(x, ..., xname)
## S3 method for class 'lpp'
iplot(x, ..., xname)
## S3 method for class 'layered'

iplot

575

iplot(x, ..., xname, visible)
## Default S3 method:
iplot(x, ..., xname)
Arguments
x

The spatial object to be plotted. An object of class "ppp", "psp", "im", "owin",
"linnet", "lpp" or "layered".

...

Ignored.

xname

Optional. Character string to use as the title of the dataset.

visible

Optional. Logical vector indicating which layers of x should initially be turned
on (visible).

Details
The function iplot generates a plot of the spatial dataset x and allows interactive control over the
appearance of the plot using a point-and-click interface.
The function iplot is generic, with methods for for point patterns (iplot.ppp), layered objects
(iplot.layered) and a default method. The default method will handle objects of class "psp",
"im" and "owin" at least.
A new popup window is launched. The spatial dataset x is displayed in the middle of the window
using the appropriate plot method.
The left side of the window contains buttons and sliders allowing the user to change the plot parameters.
The right side of the window contains navigation controls for zooming (changing magnification),
panning (shifting the field of view relative to the data), redrawing and exiting.
If the user clicks in the area where the point pattern is displayed, the field of view will be re-centred
at the point that was clicked.
Value
NULL.
Package Dependence
This function requires the package rpanel to be loaded.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
istat

576

ippm

Examples
if(interactive() && require(rpanel)) {
iplot(cells)
iplot(amacrine)
iplot(lansing)
L <- layered(D=distmap(cells), P=cells,
plotargs=list(list(ribbon=FALSE), list(pch=16)))
iplot(L)
}

ippm

Fit Point Process Model Involving Irregular Trend Parameters

Description
Experimental extension to ppm which finds optimal values of the irregular trend parameters in a
point process model.

Usage
ippm(Q, ...,
iScore=NULL,
start=list(),
covfunargs=start,
nlm.args=list(stepmax=1/2),
silent=FALSE,
warn.unused=TRUE)
Arguments
Q,...

Arguments passed to ppm to fit the point process model.

iScore

Optional. A named list of R functions that compute the partial derivatives of the
logarithm of the trend, with respect to each irregular parameter. See Details.

start

Named list containing initial values of the irregular parameters over which to
optimise.

covfunargs

Argument passed to ppm. A named list containing values for all irregular parameters required by the covariates in the model. Must include all the parameters
named in start.

nlm.args

Optional list of arguments passed to nlm to control the optimization algorithm.

silent

Logical. Whether to print warnings if the optimization algorithm fails to converge.

warn.unused

Logical. Whether to print a warning if some of the parameters in start are not
used in the model.

ippm

577

Details
This function is an experimental extension to the point process model fitting command ppm. The
extension allows the trend of the model to include irregular parameters, which will be maximised
by a Newton-type iterative method, using nlm.
For the sake of explanation, consider a Poisson point process with intensity function λ(u) at location
u. Assume that
λ(u) = exp(α + βZ(u)) f (u, γ)
where α, β, γ are parameters to be estimated, Z(u) is a spatial covariate function, and f is some
known function. Then the parameters α, β are called regular because they appear in a loglinear
form; the parameter γ is called irregular.
To fit this model using ippm, we specify the intensity using the trend formula in the same way as
usual for ppm. The trend formula is a representation of the log intensity. In the above example the
log intensity is
log λ(u) = α + βZ(u) + log f (u, γ)
So the model above would be encoded with the trend formula ~Z + offset(log(f)). Note that
the irregular part of the model is an offset term, which means that it is included in the log trend as it
is, without being multiplied by another regular parameter.
The optimisation runs faster if we specify the derivative of log f (u, γ) with respect to γ. We call this
the irregular score. To specify this, the user must write an R function that computes the irregular
score for any value of γ at any location (x,y).
Thus, to code such a problem,
1. The argument trend should define the log intensity, with the irregular part as an offset;
2. The argument start should be a list containing initial values of each of the irregular parameters;
3. The argument iScore, if provided, must be a list (with one entry for each entry of start)
of functions with arguments x,y,..., that evaluate the partial derivatives of log f (u, γ) with
respect to each irregular parameter.
The coded example below illustrates the model with two irregular parameters γ, δ and irregular term
f ((x, y), (γ, δ)) = 1 + exp(γ − δx3 )
Arguments ... passed to ppm may also include interaction. In this case the model is not a
Poisson point process but a more general Gibbs point process; the trend formula trend determines
the first-order trend of the model (the first order component of the conditional intensity), not the
intensity.
Value
A fitted point process model (object of class "ppm").
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
ppm, profilepl

578

is.connected

Examples
nd <- 32
gamma0 <- 3
delta0 <- 5
POW <- 3
# Terms in intensity
Z <- function(x,y) { -2*y }
f <- function(x,y,gamma,delta) { 1 + exp(gamma - delta * x^POW) }
# True intensity
lamb <- function(x,y,gamma,delta) { 200 * exp(Z(x,y)) * f(x,y,gamma,delta) }
# Simulate realisation
lmax <- max(lamb(0,0,gamma0,delta0), lamb(1,1,gamma0,delta0))
set.seed(42)
X <- rpoispp(lamb, lmax=lmax, win=owin(), gamma=gamma0, delta=delta0)
# Partial derivatives of log f
DlogfDgamma <- function(x,y, gamma, delta) {
topbit <- exp(gamma - delta * x^POW)
topbit/(1 + topbit)
}
DlogfDdelta <- function(x,y, gamma, delta) {
topbit <- exp(gamma - delta * x^POW)
- (x^POW) * topbit/(1 + topbit)
}
# irregular score
Dlogf <- list(gamma=DlogfDgamma, delta=DlogfDdelta)
# fit model
ippm(X ~Z + offset(log(f)),
covariates=list(Z=Z, f=f),
iScore=Dlogf,
start=list(gamma=1, delta=1),
nlm.args=list(stepmax=1),
nd=nd)

is.connected

Determine Whether an Object is Connected

Description
Determine whether an object is topologically connected.
Usage
is.connected(X, ...)
## Default S3 method:
is.connected(X, ...)
## S3 method for class 'linnet'
is.connected(X, ...)

is.connected.ppp

579

Arguments
X
...

A spatial object such as a pixel image (object of class "im"), a window (object
of class "owin") or a linear network (object of class "linnet").
Arguments passed to connected to determine the connected components.

Details
The command is.connected(X) returns TRUE if the object X consists of a single, topologicallyconnected piece, and returns FALSE if X consists of several pieces which are not joined together.
The function is.connected is generic. The default method is.connected.default works for
many classes of objects, including windows (class "owin") and images (class "im"). There is a
method for linear networks, is.connected.linnet, described here, and a method for point patterns
described in is.connected.ppp.
Value
A logical value.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
connected, is.connected.ppp.
Examples
d <- distmap(cells, dimyx=256)
X <- levelset(d, 0.07)
plot(X)
is.connected(X)

is.connected.ppp

Determine Whether a Point Pattern is Connected

Description
Determine whether a point pattern is topologically connected when all pairs of points closer than a
threshold distance are joined.
Usage
## S3 method for class 'ppp'
is.connected(X, R, ...)
Arguments
X
R
...

A point pattern (object of class "ppp").
Threshold distance. Pairs of points closer than R units apart will be joined together.
Ignored.

580

is.convex

Details
The function is.connected is generic. This is the method for point patterns (objects of class
"ppp").
The point pattern X is first converted into an abstract graph by joining every pair of points that lie
closer than R units apart. Then the algorithm determines whether this graph is connected.
That is, the result of is.connected(X) is TRUE if any point in X can be reached from any other
point, by a series of steps between points of X, each step being shorter than R units in length.
Value
A logical value.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
is.connected, connected.ppp.
Examples
is.connected(redwoodfull, 0.1)
is.connected(redwoodfull, 0.2)

is.convex

Test Whether a Window is Convex

Description
Determines whether a window is convex.
Usage
is.convex(x)
Arguments
x

Window (object of class "owin").

Details
If x is a rectangle, the result is TRUE.
If x is polygonal, the result is TRUE if x consists of a single polygon and this polygon is equal to
the minimal convex hull of its vertices computed by chull.
If x is a mask, the algorithm first extracts all boundary pixels of x using vertices. Then it computes
the (polygonal) convex hull K of the boundary pixels. The result is TRUE if every boundary pixel
lies within one pixel diameter of an edge of K.

is.dppm

581

Value
Logical value, equal to TRUE if x is convex.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
owin, convexhull.xy, vertices

is.dppm

Recognise Fitted Determinantal Point Process Models

Description
Check that an object inherits the class dppm
Usage
is.dppm(x)
Arguments
x

Any object.

Value
A single logical value.
Author(s)
Ege Rubak  , Adrian Baddeley 
 and Rolf Turner  

is.empty

Test Whether An Object Is Empty

Description
Checks whether the argument is an empty window, an empty point pattern, etc.

582

is.hybrid

Usage
is.empty(x)
## S3 method for class 'owin'
is.empty(x)
## S3 method for class 'ppp'
is.empty(x)
## S3 method for class 'psp'
is.empty(x)
## Default S3 method:
is.empty(x)
Arguments
x

A window (object of class "owin"), a point pattern (object of class "ppp"), or a
line segment pattern (object of class "psp").

Details
This function tests whether the object x represents an empty spatial object, such as an empty window, a point pattern with zero points, or a line segment pattern with zero line segments.
An empty window can be obtained as the output of intersect.owin, erosion, opening, complement.owin
and some other operations.
An empty point pattern or line segment pattern can be obtained as the result of simulation.
Value
Logical value.
Author(s)
Adrian Baddeley 
and Rolf Turner 

is.hybrid

Test Whether Object is a Hybrid

Description
Tests where a point process model or point process interaction is a hybrid of several interactions.
Usage
is.hybrid(x)
## S3 method for class 'ppm'
is.hybrid(x)
## S3 method for class 'interact'
is.hybrid(x)

is.hybrid

583

Arguments
x

A point process model (object of class "ppm") or a point process interaction
structure (object of class "interact").

Details
A hybrid (Baddeley, Turner, Mateu and Bevan, 2012) is a point process model created by combining
two or more point process models, or an interpoint interaction created by combining two or more
interpoint interactions.
The function is.hybrid is generic, with methods for point process models (objects of class "ppm")
and point process interactions (objects of class "interact"). These functions return TRUE if the
object x is a hybrid, and FALSE if it is not a hybrid.
Hybrids of two or more interpoint interactions are created by the function Hybrid. Such a hybrid
interaction can then be fitted to point pattern data using ppm.
Value
TRUE if the object is a hybrid, and FALSE otherwise.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Baddeley, A., Turner, R., Mateu, J. and Bevan, A. (2013) Hybrids of Gibbs point process models
and their implementation. Journal of Statistical Software 55:11, 1–43. http://www.jstatsoft.
org/v55/i11/
See Also
Hybrid
Examples
S <- Strauss(0.1)
is.hybrid(S)
H <- Hybrid(Strauss(0.1), Geyer(0.2, 3))
is.hybrid(H)
data(redwood)
fit <- ppm(redwood, ~1, H)
is.hybrid(fit)

584

is.lpp

is.im

Test Whether An Object Is A Pixel Image

Description
Tests whether its argument is a pixel image (object of class "im").
Usage
is.im(x)
Arguments
x

Any object.

Details
This function tests whether the argument x is a pixel image object of class "im". For details of this
class, see im.object.
The object is determined to be an image if it inherits from class "im".
Value
TRUE if x is a pixel image, otherwise FALSE.
Author(s)
Adrian Baddeley 
and Rolf Turner 

is.lpp

Test Whether An Object Is A Point Pattern on a Linear Network

Description
Checks whether its argument is a point pattern on a linear network (object of class "lpp").
Usage
is.lpp(x)
Arguments
x

Any object.

Details
This function tests whether the object x is a point pattern object of class "lpp".

is.marked

585

Value
TRUE if x is a point pattern of class "lpp", otherwise FALSE.
Author(s)
Adrian Baddeley 
and Rolf Turner 

is.marked

Test Whether Marks Are Present

Description
Generic function to test whether a given object (usually a point pattern or something related to a
point pattern) has “marks” attached to the points.
Usage
is.marked(X, ...)
Arguments
X

Object to be inspected

...

Other arguments.

Details
“Marks” are observations attached to each point of a point pattern. For example the longleaf
dataset contains the locations of trees, each tree being marked by its diameter; the amacrine dataset
gives the locations of cells of two types (on/off) and the type of cell may be regarded as a mark
attached to the location of the cell.
Other objects related to point patterns, such as point process models, may involve marked points.
This function tests whether the object X contains or involves marked points. It is generic; methods
are provided for point patterns (objects of class "ppp") and point process models (objects of class
"ppm").
Value
Logical value, equal to TRUE if X is marked.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
is.marked.ppp, is.marked.ppm

586

is.marked.ppm

is.marked.ppm

Test Whether A Point Process Model is Marked

Description
Tests whether a fitted point process model involves “marks” attached to the points.
Usage
## S3 method for class 'ppm'
is.marked(X, ...)
## S3 method for class 'lppm'
is.marked(X, ...)
Arguments
X

Fitted point process model (object of class "ppm") usually obtained from ppm.
Alternatively, a model of class "lppm".

...

Ignored.

Details
“Marks” are observations attached to each point of a point pattern. For example the longleaf
dataset contains the locations of trees, each tree being marked by its diameter; the amacrine dataset
gives the locations of cells of two types (on/off) and the type of cell may be regarded as a mark
attached to the location of the cell.
The argument X is a fitted point process model (an object of class "ppm") typically obtained by
fitting a model to point pattern data using ppm.
This function returns TRUE if the original data (to which the model X was fitted) were a marked
point pattern.
Note that this is not the same as testing whether the model involves terms that depend on the marks
(i.e. whether the fitted model ignores the marks in the data). Currently we have not implemented a
test for this.
If this function returns TRUE, the implications are (for example) that any simulation of this model
will require simulation of random marks as well as random point locations.
Value
Logical value, equal to TRUE if X is a model that was fitted to a marked point pattern dataset.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
is.marked, is.marked.ppp

is.marked.ppp

587

Examples
X <- lansing
# Multitype point pattern --- trees marked by species

fit1 <- ppm(X, ~ marks, Poisson())
is.marked(fit1)
# TRUE
fit2 <- ppm(X, ~ 1, Poisson())
is.marked(fit2)
# TRUE
# Unmarked point pattern
fit3 <- ppm(cells, ~ 1, Poisson())
is.marked(fit3)
# FALSE

is.marked.ppp

Test Whether A Point Pattern is Marked

Description
Tests whether a point pattern has “marks” attached to the points.
Usage
## S3 method for class 'ppp'
is.marked(X, na.action="warn", ...)
Arguments
X

Point pattern (object of class "ppp")

na.action

String indicating what to do if NA values are encountered amongst the marks.
Options are "warn", "fatal" and "ignore".

...

Ignored.

Details
“Marks” are observations attached to each point of a point pattern. For example the longleaf
dataset contains the locations of trees, each tree being marked by its diameter; the amacrine dataset
gives the locations of cells of two types (on/off) and the type of cell may be regarded as a mark
attached to the location of the cell.
This function tests whether the point pattern X contains or involves marked points. It is a method
for the generic function is.marked.
The argument na.action determines what action will be taken if the point pattern has a vector of
marks but some or all of the marks are NA. Options are "fatal" to cause a fatal error; "warn" to
issue a warning and then return TRUE; and "ignore" to take no action except returning TRUE.

588

is.multitype

Value
Logical value, equal to TRUE if X is a marked point pattern.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
is.marked, is.marked.ppm
Examples
data(cells)
is.marked(cells) #FALSE
data(longleaf)
is.marked(longleaf) #TRUE

is.multitype

Test whether Object is Multitype

Description
Generic function to test whether a given object (usually a point pattern or something related to a
point pattern) has “marks” attached to the points which classify the points into several types.
Usage
is.multitype(X, ...)
Arguments
X

Object to be inspected

...

Other arguments.

Details
“Marks” are observations attached to each point of a point pattern. For example the longleaf
dataset contains the locations of trees, each tree being marked by its diameter; the amacrine dataset
gives the locations of cells of two types (on/off) and the type of cell may be regarded as a mark
attached to the location of the cell. Other objects related to point patterns, such as point process
models, may involve marked points.
This function tests whether the object X contains or involves marked points, and that the marks are
a factor.
For example, the amacrine dataset is multitype (there are two types of cells, on and off), but the
longleaf dataset is not multitype (the marks are real numbers).
This function is generic; methods are provided for point patterns (objects of class "ppp") and point
process models (objects of class "ppm").

is.multitype.ppm

589

Value
Logical value, equal to TRUE if X is multitype.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
is.multitype.ppp, is.multitype.ppm

is.multitype.ppm

Test Whether A Point Process Model is Multitype

Description
Tests whether a fitted point process model involves “marks” attached to the points that classify the
points into several types.
Usage
## S3 method for class 'ppm'
is.multitype(X, ...)
## S3 method for class 'lppm'
is.multitype(X, ...)
Arguments
X

Fitted point process model (object of class "ppm") usually obtained from ppm.
Alternatively a model of class "lppm".

...

Ignored.

Details
“Marks” are observations attached to each point of a point pattern. For example the longleaf
dataset contains the locations of trees, each tree being marked by its diameter; the amacrine dataset
gives the locations of cells of two types (on/off) and the type of cell may be regarded as a mark
attached to the location of the cell.
The argument X is a fitted point process model (an object of class "ppm") typically obtained by
fitting a model to point pattern data using ppm.
This function returns TRUE if the original data (to which the model X was fitted) were a multitype
point pattern.
Note that this is not the same as testing whether the model involves terms that depend on the marks
(i.e. whether the fitted model ignores the marks in the data). Currently we have not implemented a
test for this.
If this function returns TRUE, the implications are (for example) that any simulation of this model
will require simulation of random marks as well as random point locations.

590

is.multitype.ppp

Value
Logical value, equal to TRUE if X is a model that was fitted to a multitype point pattern dataset.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
is.multitype, is.multitype.ppp
Examples
X <- lansing
# Multitype point pattern --- trees marked by species

fit1 <- ppm(X, ~ marks, Poisson())
is.multitype(fit1)
# TRUE
fit2 <- ppm(X, ~ 1, Poisson())
is.multitype(fit2)
# TRUE
# Unmarked point pattern
fit3 <- ppm(cells, ~ 1, Poisson())
is.multitype(fit3)
# FALSE

is.multitype.ppp

Test Whether A Point Pattern is Multitype

Description
Tests whether a point pattern has “marks” attached to the points which classify the points into
several types.
Usage
## S3 method for class 'ppp'
is.multitype(X, na.action="warn", ...)
## S3 method for class 'lpp'
is.multitype(X, na.action="warn", ...)

is.owin

591

Arguments
X

Point pattern (object of class "ppp" or "lpp")

na.action

String indicating what to do if NA values are encountered amongst the marks.
Options are "warn", "fatal" and "ignore".

...

Ignored.

Details
“Marks” are observations attached to each point of a point pattern. For example the longleaf
dataset contains the locations of trees, each tree being marked by its diameter; the amacrine dataset
gives the locations of cells of two types (on/off) and the type of cell may be regarded as a mark
attached to the location of the cell.
This function tests whether the point pattern X contains or involves marked points, and that the
marks are a factor. It is a method for the generic function is.multitype.
For example, the amacrine dataset is multitype (there are two types of cells, on and off), but the
longleaf dataset is not multitype (the marks are real numbers).
The argument na.action determines what action will be taken if the point pattern has a vector of
marks but some or all of the marks are NA. Options are "fatal" to cause a fatal error; "warn" to
issue a warning and then return TRUE; and "ignore" to take no action except returning TRUE.
Value
Logical value, equal to TRUE if X is a multitype point pattern.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
is.multitype, is.multitype.ppm
Examples
is.multitype(cells) #FALSE - no marks
is.multitype(longleaf) #FALSE - real valued marks
is.multitype(amacrine) #TRUE

is.owin

Test Whether An Object Is A Window

Description
Checks whether its argument is a window (object of class "owin").
Usage
is.owin(x)

592

is.ppm

Arguments
x

Any object.

Details
This function tests whether the object x is a window object of class "owin". See owin.object for
details of this class.
The result is determined to be TRUE if x inherits from "owin", i.e. if x has "owin" amongst its
classes.
Value
TRUE if x is a point pattern, otherwise FALSE.
Author(s)
Adrian Baddeley 
and Rolf Turner 
is.ppm

Test Whether An Object Is A Fitted Point Process Model

Description
Checks whether its argument is a fitted point process model (object of class "ppm", "kppm", "lppm"
or "slrm").
Usage
is.ppm(x)
is.kppm(x)
is.lppm(x)
is.slrm(x)
Arguments
x

Any object.

Details
These functions test whether the object x is a fitted point process model object of the specified class.
The result of is.ppm(x) is TRUE if x has "ppm" amongst its classes, and otherwise FALSE. Similarly
for the other functions.
Value
A single logical value.
Author(s)
Adrian Baddeley 
and Rolf Turner 

is.ppp

593

is.ppp

Test Whether An Object Is A Point Pattern

Description
Checks whether its argument is a point pattern (object of class "ppp").
Usage
is.ppp(x)
Arguments
x

Any object.

Details
This function tests whether the object x is a point pattern object of class "ppp". See ppm.object
for details of this class.
The result is determined to be TRUE if x inherits from "ppp", i.e. if x has "ppp" amongst its classes.
Value
TRUE if x is a point pattern, otherwise FALSE.
Author(s)
Adrian Baddeley 
and Rolf Turner 

is.rectangle

Determine Type of Window

Description
Determine whether a window is a rectangle, a polygonal region, or a binary mask.
Usage
is.rectangle(w)
is.polygonal(w)
is.mask(w)
Arguments
w

Window to be inspected. An object of class "owin".

594

is.stationary

Details
These simple functions determine whether a window w (object of class "owin") is a rectangle
(is.rectangle(w) = TRUE), a domain with polygonal boundary (is.polygonal(w) = TRUE),
or a binary pixel mask (is.mask(w) = TRUE).
Value
Logical value, equal to TRUE if w is a window of the specified type.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
owin

is.stationary

Recognise Stationary and Poisson Point Process Models

Description
Given a point process model that has been fitted to data, determine whether the model is a stationary
point process, and whether it is a Poisson point process.
Usage
is.stationary(x)
## S3 method for
is.stationary(x)
## S3 method for
is.stationary(x)
## S3 method for
is.stationary(x)
## S3 method for
is.stationary(x)
## S3 method for
is.stationary(x)
## S3 method for
is.stationary(x)
## S3 method for
is.stationary(x)

class 'ppm'
class 'kppm'
class 'lppm'
class 'slrm'
class 'rmhmodel'
class 'dppm'
class 'detpointprocfamily'

is.poisson(x)
## S3 method for class 'ppm'
is.poisson(x)
## S3 method for class 'kppm'
is.poisson(x)
## S3 method for class 'lppm'
is.poisson(x)

is.stationary

595

## S3 method for class 'slrm'
is.poisson(x)
## S3 method for class 'rmhmodel'
is.poisson(x)
## S3 method for class 'interact'
is.poisson(x)
Arguments
x

A fitted spatial point process model (object of class "ppm", "kppm", "lppm",
"dppm" or "slrm") or similar object.

Details
The argument x represents a fitted spatial point process model or a similar object.
is.stationary(x) returns TRUE if x represents a stationary point process, and FALSE if not.
is.poisson(x) returns TRUE if x represents a Poisson point process, and FALSE if not.
The functions is.stationary and is.poisson are generic, with methods for the classes "ppm"
(Gibbs point process models), "kppm" (cluster or Cox point process models), "slrm" (spatial logistic regression models) and "rmhmodel" (model specifications for the Metropolis-Hastings algorithm). Additionally is.stationary has a method for classes "detpointprocfamily" and "dppm"
(both determinantal point processes) and is.poisson has a method for class "interact" (interaction structures for Gibbs models).
is.poisson.kppm will return FALSE, unless the model x is degenerate: either x has zero intensity
so that its realisations are empty with probability 1, or it is a log-Gaussian Cox process where the
log intensity has zero variance.
is.poisson.slrm will always return TRUE, by convention.
Value
A logical value.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
is.marked to determine whether a model is a marked point process.
summary.ppm for detailed information.
Model-fitting functions ppm, dppm, kppm, lppm, slrm.
Examples
data(cells)
data(redwood)
fit <- ppm(cells ~ x)
is.stationary(fit)

596

is.subset.owin
is.poisson(fit)
fut <- kppm(redwood ~ 1, "MatClust")
is.stationary(fut)
is.poisson(fut)
fot <- slrm(cells ~ x)
is.stationary(fot)
is.poisson(fot)

is.subset.owin

Determine Whether One Window is Contained In Another

Description
Tests whether window A is a subset of window B.
Usage
is.subset.owin(A, B)
Arguments
A

A window object (see Details).

B

A window object (see Details).

Details
This function tests whether the window A is a subset of the window B.
The arguments A and B must be window objects (either objects of class "owin", or data that can be
coerced to this class by as.owin).
Various algorithms are used, depending on the geometrical type of the two windows.
Note that if B is not rectangular, the algorithm proceeds by discretising A, converting it to a pixel
mask using as.mask. In this case the resulting answer is only “approximately correct”. The accuracy of the approximation can be controlled: see as.mask.
Value
Logical scalar; TRUE if A is a sub-window of B, otherwise FALSE.
Author(s)
Adrian Baddeley 
and Rolf Turner 
Examples
w1 <- as.owin(c(0,1,0,1))
w2 <- as.owin(c(-1,2,-1,2))
is.subset.owin(w1,w2) # Returns TRUE.
is.subset.owin(w2,w1) # Returns FALSE.

istat

597

istat

Point and Click Interface for Exploratory Analysis of Point Pattern

Description
Compute various summary functions for a point pattern using a point-and-click interface.
Usage
istat(x, xname)
Arguments
x
xname

The spatial point pattern to be analysed. An object of class "ppp".
Optional. Character string to use as the title of the dataset.

Details
This command launches an interactive (point-and-click) interface which offers a choice of spatial
summary functions that can be applied to the point pattern x.
The selected summary function is computed for the point pattern x and plotted in a popup window.
The selection of functions includes Kest, Lest, pcf, Fest ,Gest and Jest. For the function pcf it
is possible to control the bandwidth parameter bw.
There is also an option to show simulation envelopes of the summary function.
Value
NULL.
Note
Before adjusting the bandwidth parameter bw, it is advisable to select No simulation envelopes to
save a lot of computation time.
Package Dependence
This function requires the package rpanel to be loaded.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
iplot
Examples
if(interactive() && require(rpanel)) {
istat(swedishpines)
}

598

Jcross

Jcross

Multitype J Function (i-to-j)

Description
For a multitype point pattern, estimate the multitype J function summarising the interpoint dependence between points of type i and of type j.
Usage
Jcross(X, i, j, eps=NULL, r=NULL, breaks=NULL, ..., correction=NULL)
Arguments
X

The observed point pattern, from which an estimate of the multitype J function
Jij (r) will be computed. It must be a multitype point pattern (a marked point
pattern whose marks are a factor). See under Details.

i

The type (mark value) of the points in X from which distances are measured.
A character string (or something that will be converted to a character string).
Defaults to the first level of marks(X).

j

The type (mark value) of the points in X to which distances are measured. A
character string (or something that will be converted to a character string). Defaults to the second level of marks(X).

eps

A positive number. The resolution of the discrete approximation to Euclidean
distance (see below). There is a sensible default.

r

Optional. Numeric vector. The values of the argument r at which the function
Jij (r) should be evaluated. There is a sensible default. First-time users are
strongly advised not to specify this argument. See below for important conditions on r.

breaks

This argument is for internal use only.

...

Ignored.

correction

Optional. Character string specifying the edge correction(s) to be used. Options
are "none", "rs", "km", "Hanisch" and "best". Alternatively correction="all"
selects all options.

Details
This function Jcross and its companions Jdot and Jmulti are generalisations of the function Jest
to multitype point patterns.
A multitype point pattern is a spatial pattern of points classified into a finite number of possible
“colours” or “types”. In the spatstat package, a multitype pattern is represented as a single point
pattern object in which the points carry marks, and the mark value attached to each point determines
the type of that point.
The argument X must be a point pattern (object of class "ppp") or any data that are acceptable
to as.ppp. It must be a marked point pattern, and the mark vector X$marks must be a factor. The
argument i will be interpreted as a level of the factor X$marks. (Warning: this means that an integer
value i=3 will be interpreted as the number 3, not the 3rd smallest level).

Jcross

599

The “type i to type j” multitype J function of a stationary multitype point process X was introduced
by Van lieshout and Baddeley (1999). It is defined by
Jij (r) =

1 − Gij (r)
1 − Fj (r)

where Gij (r) is the distribution function of the distance from a type i point to the nearest point
of type j, and Fj (r) is the distribution function of the distance from a fixed point in space to the
nearest point of type j in the pattern.
An estimate of Jij (r) is a useful summary statistic in exploratory data analysis of a multitype point
pattern. If the subprocess of type i points is independent of the subprocess of points of type j,
then Jij (r) ≡ 1. Hence deviations of the empirical estimate of Jij from the value 1 may suggest
dependence between types.
This algorithm estimates Jij (r) from the point pattern X. It assumes that X can be treated as a realisation of a stationary (spatially homogeneous) random spatial point process in the plane, observed
through a bounded window. The window (which is specified in X as Window(X)) may have arbitrary
shape. Biases due to edge effects are treated in the same manner as in Jest, using the Kaplan-Meier
and border corrections. The main work is done by Gmulti and Fest.
The argument r is the vector of values for the distance r at which Jij (r) should be evaluated. The
values of r must be increasing nonnegative numbers and the maximum r value must exceed the
radius of the largest disc contained in the window.
Value
An object of class "fv" (see fv.object).
Essentially a data frame containing six numeric columns
J

the recommended estimator of Jij (r), currently the Kaplan-Meier estimator.

r

the values of the argument r at which the function Jij (r) has been estimated

km

the Kaplan-Meier estimator of Jij (r)

rs

the “reduced sample” or “border correction” estimator of Jij (r)

han

the Hanisch-style estimator of Jij (r)

un

the “uncorrected” estimator of Jij (r) formed by taking the ratio of uncorrected
empirical estimators of 1 − Gij (r) and 1 − Fj (r), see Gdot and Fest.

theo

the theoretical value of Jij (r) for a marked Poisson process, namely 1.

The result also has two attributes "G" and "F" which are respectively the outputs of Gcross and
Fest for the point pattern.
Warnings
The arguments i and j are always interpreted as levels of the factor X$marks. They are converted
to character strings if they are not already character strings. The value i=1 does not refer to the first
level of the factor.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .

600

Jdot

References
Van Lieshout, M.N.M. and Baddeley, A.J. (1996) A nonparametric measure of spatial interaction in
point patterns. Statistica Neerlandica 50, 344–361.
Van Lieshout, M.N.M. and Baddeley, A.J. (1999) Indices of dependence between types in multivariate point patterns. Scandinavian Journal of Statistics 26, 511–532.
See Also
Jdot, Jest, Jmulti
Examples
# Lansing woods data: 6 types of trees
woods <- lansing
Jhm <- Jcross(woods, "hickory", "maple")
# diagnostic plot for independence between hickories and maples
plot(Jhm)
# synthetic example with two types "a" and "b"
pp <- runifpoint(30) %mark% factor(sample(c("a","b"), 30, replace=TRUE))
J <- Jcross(pp)

Jdot

Multitype J Function (i-to-any)

Description
For a multitype point pattern, estimate the multitype J function summarising the interpoint dependence between the type i points and the points of any type.
Usage
Jdot(X, i, eps=NULL, r=NULL, breaks=NULL, ..., correction=NULL)
Arguments
X

The observed point pattern, from which an estimate of the multitype J function
Ji• (r) will be computed. It must be a multitype point pattern (a marked point
pattern whose marks are a factor). See under Details.

i

The type (mark value) of the points in X from which distances are measured.
A character string (or something that will be converted to a character string).
Defaults to the first level of marks(X).

eps

A positive number. The resolution of the discrete approximation to Euclidean
distance (see below). There is a sensible default.

r

numeric vector. The values of the argument r at which the function Ji• (r)
should be evaluated. There is a sensible default. First-time users are strongly
advised not to specify this argument. See below for important conditions on r.

breaks

This argument is for internal use only.

...

Ignored.

Jdot

601
correction

Optional. Character string specifying the edge correction(s) to be used. Options
are "none", "rs", "km", "Hanisch" and "best". Alternatively correction="all"
selects all options.

Details
This function Jdot and its companions Jcross and Jmulti are generalisations of the function Jest
to multitype point patterns.
A multitype point pattern is a spatial pattern of points classified into a finite number of possible
“colours” or “types”. In the spatstat package, a multitype pattern is represented as a single point
pattern object in which the points carry marks, and the mark value attached to each point determines
the type of that point.
The argument X must be a point pattern (object of class "ppp") or any data that are acceptable
to as.ppp. It must be a marked point pattern, and the mark vector X$marks must be a factor. The
argument i will be interpreted as a level of the factor X$marks. (Warning: this means that an integer
value i=3 will be interpreted as the number 3, not the 3rd smallest level.)
The “type i to any type” multitype J function of a stationary multitype point process X was introduced by Van lieshout and Baddeley (1999). It is defined by
Ji• (r) =

1 − Gi• (r)
1 − F• (r)

where Gi• (r) is the distribution function of the distance from a type i point to the nearest other
point of the pattern, and F• (r) is the distribution function of the distance from a fixed point in space
to the nearest point of the pattern.
An estimate of Ji• (r) is a useful summary statistic in exploratory data analysis of a multitype point
pattern. If the pattern is a marked Poisson point process, then Ji• (r) ≡ 1. If the subprocess of type
i points is independent of the subprocess of points of all types not equal to i, then Ji• (r) equals
Jii (r), the ordinary J function (see Jest and Van Lieshout and Baddeley (1996)) of the points of
type i. Hence deviations from zero of the empirical estimate of Ji• − Jii may suggest dependence
between types.
This algorithm estimates Ji• (r) from the point pattern X. It assumes that X can be treated as a realisation of a stationary (spatially homogeneous) random spatial point process in the plane, observed
through a bounded window. The window (which is specified in X as Window(X)) may have arbitrary
shape. Biases due to edge effects are treated in the same manner as in Jest, using the Kaplan-Meier
and border corrections. The main work is done by Gmulti and Fest.
The argument r is the vector of values for the distance r at which Ji• (r) should be evaluated. The
values of r must be increasing nonnegative numbers and the maximum r value must exceed the
radius of the largest disc contained in the window.
Value
An object of class "fv" (see fv.object).
Essentially a data frame containing six numeric columns
J

the recommended estimator of Ji• (r), currently the Kaplan-Meier estimator.

r

the values of the argument r at which the function Ji• (r) has been estimated

km

the Kaplan-Meier estimator of Ji• (r)

rs

the “reduced sample” or “border correction” estimator of Ji• (r)

han

the Hanisch-style estimator of Ji• (r)

602

Jdot
un

the “uncorrected” estimator of Ji• (r) formed by taking the ratio of uncorrected
empirical estimators of 1 − Gi• (r) and 1 − F• (r), see Gdot and Fest.

theo

the theoretical value of Ji• (r) for a marked Poisson process, namely 1.

The result also has two attributes "G" and "F" which are respectively the outputs of Gdot and Fest
for the point pattern.

Warnings
The argument i is interpreted as a level of the factor X$marks. It is converted to a character string
if it is not already a character string. The value i=1 does not refer to the first level of the factor.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Van Lieshout, M.N.M. and Baddeley, A.J. (1996) A nonparametric measure of spatial interaction in
point patterns. Statistica Neerlandica 50, 344–361.
Van Lieshout, M.N.M. and Baddeley, A.J. (1999) Indices of dependence between types in multivariate point patterns. Scandinavian Journal of Statistics 26, 511–532.

See Also
Jcross, Jest, Jmulti
Examples
# Lansing woods data: 6 types of trees
woods <- lansing
Jh. <- Jdot(woods, "hickory")
plot(Jh.)
# diagnostic plot for independence between hickories and other trees
Jhh <- Jest(split(woods)$hickory)
plot(Jhh, add=TRUE, legendpos="bottom")
## Not run:
# synthetic example with two marks "a" and "b"
pp <- runifpoint(30) %mark% factor(sample(c("a","b"), 30, replace=TRUE))
J <- Jdot(pp, "a")
## End(Not run)

Jest

603

Jest

Estimate the J-function

Description
Estimates the summary function J(r) for a point pattern in a window of arbitrary shape.
Usage
Jest(X, ..., eps=NULL, r=NULL, breaks=NULL, correction=NULL)
Arguments
X

The observed point pattern, from which an estimate of J(r) will be computed.
An object of class "ppp", or data in any format acceptable to as.ppp().

...

Ignored.

eps

the resolution of the discrete approximation to Euclidean distance (see below).
There is a sensible default.

r

vector of values for the argument r at which J(r) should be evaluated. There
is a sensible default. First-time users are strongly advised not to specify this
argument. See below for important conditions on r.

breaks

This argument is for internal use only.

correction

Optional. Character string specifying the choice of edge correction(s) in Fest
and Gest. See Details.

Details
The J function (Van Lieshout and Baddeley, 1996) of a stationary point process is defined as
J(r) =

1 − G(r)
1 − F (r)

where G(r) is the nearest neighbour distance distribution function of the point process (see Gest)
and F (r) is its empty space function (see Fest).
For a completely random (uniform Poisson) point process, the J-function is identically equal to
1. Deviations J(r) < 1 or J(r) > 1 typically indicate spatial clustering or spatial regularity,
respectively. The J-function is one of the few characteristics that can be computed explicitly for
a wide range of point processes. See Van Lieshout and Baddeley (1996), Baddeley et al (2000),
Thonnes and Van Lieshout (1999) for further information.
An estimate of J derived from a spatial point pattern dataset can be used in exploratory data analysis
and formal inference about the pattern. The estimate of J(r) is compared against the constant
function 1. Deviations J(r) < 1 or J(r) > 1 may suggest spatial clustering or spatial regularity,
respectively.
This algorithm estimates the J-function from the point pattern X. It assumes that X can be treated
as a realisation of a stationary (spatially homogeneous) random spatial point process in the plane,
observed through a bounded window. The window (which is specified in X as Window(X)) may have
arbitrary shape.
The argument X is interpreted as a point pattern object (of class "ppp", see ppp.object) and can be
supplied in any of the formats recognised by as.ppp().

604

Jest
The functions Fest and Gest are called to compute estimates of F (r) and G(r) respectively. These
estimates are then combined by simply taking the ratio J(r) = (1 − G(r))/(1 − F (r)).
In fact several different estimates are computed using different edge corrections (Baddeley, 1998).
The Kaplan-Meier estimate (returned as km) is the ratio J = (1-G)/(1-F) of the Kaplan-Meier
estimates of 1 − F and 1 − G computed by Fest and Gest respectively. This is computed if
correction=NULL or if correction includes "km".
The Hanisch-style estimate (returned as han) is the ratio J = (1-G)/(1-F) where F is the ChiuStoyan estimate of F and G is the Hanisch estimate of G. This is computed if correction=NULL or
if correction includes "cs" or "han".
The reduced-sample or border corrected estimate (returned as rs) is the same ratio J = (1-G)/(1-F)
of the border corrected estimates. This is computed if correction=NULL or if correction includes
"rs" or "border".
These edge-corrected estimators are slightly biased for J, since they are ratios of approximately
unbiased estimators. The logarithm of the Kaplan-Meier estimate is exactly unbiased for log J.
The uncorrected estimate (returned as un and computed only if correction includes "none") is the
ratio J = (1-G)/(1-F) of the uncorrected (“raw”) estimates of the survival functions of F and G,
which are the empirical distribution functions of the empty space distances Fest(X,...)$raw and
of the nearest neighbour distances Gest(X,...)$raw. The uncorrected estimates of F and G are
severely biased. However the uncorrected estimate of J is approximately unbiased (if the process is
close to Poisson); it is insensitive to edge effects, and should be used when edge effects are severe
(see Baddeley et al, 2000).
The algorithm for Fest uses two discrete approximations which are controlled by the parameter eps
and by the spacing of values of r respectively. See Fest for details. First-time users are strongly
advised not to specify these arguments.
Note that the value returned by Jest includes the output of Fest and Gest as attributes (see the last
example below). If the user is intending to compute the F,G and J functions for the point pattern, it
is only necessary to call Jest.

Value
An object of class "fv", see fv.object, which can be plotted directly using plot.fv.
Essentially a data frame containing
r

the vector of values of the argument r at which the function J has been estimated

rs

the “reduced sample” or “border correction” estimator of J(r) computed from
the border-corrected estimates of F and G

km

the spatial Kaplan-Meier estimator of J(r) computed from the Kaplan-Meier
estimates of F and G

han

the Hanisch-style estimator of J(r) computed from the Hanisch estimate of G
and the Chiu-Stoyan estimate of F

un

the uncorrected estimate of J(r) computed from the uncorrected estimates of F
and G

theo

the theoretical value of J(r) for a stationary Poisson process: identically equal
to 1

The data frame also has attributes
F

the output of Fest for this point pattern, containing three estimates of the empty
space function F (r) and an estimate of its hazard function

Jest

605
G

the output of Gest for this point pattern, containing three estimates of the nearest neighbour distance distribution function G(r) and an estimate of its hazard
function

Note
Sizeable amounts of memory may be needed during the calculation.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Baddeley, A.J. Spatial sampling and censoring. In O.E. Barndorff-Nielsen, W.S. Kendall and
M.N.M. van Lieshout (eds) Stochastic Geometry: Likelihood and Computation. Chapman and
Hall, 1998. Chapter 2, pages 37–78.
Baddeley, A.J. and Gill, R.D. The empty space hazard of a spatial pattern. Research Report 1994/3,
Department of Mathematics, University of Western Australia, May 1994.
Baddeley, A.J. and Gill, R.D. Kaplan-Meier estimators of interpoint distance distributions for spatial
point processes. Annals of Statistics 25 (1997) 263–292.
Baddeley, A., Kerscher, M., Schladitz, K. and Scott, B.T. Estimating the J function without edge
correction. Statistica Neerlandica 54 (2000) 315–328.
Borgefors, G. Distance transformations in digital images. Computer Vision, Graphics and Image
Processing 34 (1986) 344–371.
Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.
Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 1983.
Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.
Stoyan, D, Kendall, W.S. and Mecke, J. Stochastic geometry and its applications. 2nd edition.
Springer Verlag, 1995.
Thonnes, E. and Van Lieshout, M.N.M, A comparative study on the power of Van Lieshout and
Baddeley’s J-function. Biometrical Journal 41 (1999) 721–734.
Van Lieshout, M.N.M. and Baddeley, A.J. A nonparametric measure of spatial interaction in point
patterns. Statistica Neerlandica 50 (1996) 344–361.
See Also
Jinhom, Fest, Gest, Kest, km.rs, reduced.sample, kaplan.meier
Examples
data(cells)
J <- Jest(cells, 0.01)
plot(J, main="cells data")
# values are far above J = 1, indicating regular pattern
data(redwood)
J <- Jest(redwood, 0.01, legendpos="center")
plot(J, main="redwood data")
# values are below J = 1, indicating clustered pattern

606

Jinhom

Jinhom

Inhomogeneous J-function

Description
Estimates the inhomogeneous J function of a non-stationary point pattern.
Usage
Jinhom(X, lambda = NULL, lmin = NULL, ...,
sigma = NULL, varcov = NULL,
r = NULL, breaks = NULL, update = TRUE)
Arguments
X

The observed data point pattern, from which an estimate of the inhomogeneous
J function will be computed. An object of class "ppp" or in a format recognised
by as.ppp()

lambda

Optional. Values of the estimated intensity function. Either a vector giving the
intensity values at the points of the pattern X, a pixel image (object of class "im")
giving the intensity values at all locations, a fitted point process model (object
of class "ppm" or "kppm") or a function(x,y) which can be evaluated to give
the intensity value at any location.

lmin

Optional. The minimum possible value of the intensity over the spatial domain.
A positive numerical value.

sigma,varcov

Optional arguments passed to density.ppp to control the smoothing bandwidth, when lambda is estimated by kernel smoothing.

...

Extra arguments passed to as.mask to control the pixel resolution, or passed to
density.ppp to control the smoothing bandwidth.

r

vector of values for the argument r at which the inhomogeneous K function
should be evaluated. Not normally given by the user; there is a sensible default.

breaks

This argument is for internal use only.

update

Logical. If lambda is a fitted model (class "ppm" or "kppm") and update=TRUE
(the default), the model will first be refitted to the data X (using update.ppm
or update.kppm) before the fitted intensity is computed. If update=FALSE, the
fitted intensity of the model will be computed without fitting it to X.

Details
This command computes estimates of the inhomogeneous J-function (Van Lieshout, 2010) of a
point pattern. It is the counterpart, for inhomogeneous spatial point patterns, of the J function for
homogeneous point patterns computed by Jest.
The argument X should be a point pattern (object of class "ppp").
The inhomogeneous J function is computed as Jinhom(r) = (1−Ginhom(r))/(1−F inhom(r))
where Ginhom, F inhom are the inhomogeneous G and F functions computed using the border
correction (equations (7) and (6) respectively in Van Lieshout, 2010).
The argument lambda should supply the (estimated) values of the intensity function λ of the point
process. It may be either

Jinhom

607

a numeric vector containing the values of the intensity function at the points of the pattern X.
a pixel image (object of class "im") assumed to contain the values of the intensity function at all
locations in the window.
a fitted point process model (object of class "ppm" or "kppm") whose fitted trend can be used as
the fitted intensity. (If update=TRUE the model will first be refitted to the data X before the
trend is computed.)
a function which can be evaluated to give values of the intensity at any locations.
omitted: if lambda is omitted, then it will be estimated using a ‘leave-one-out’ kernel smoother.
If lambda is a numeric vector, then its length should be equal to the number of points in the pattern
X. The value lambda[i] is assumed to be the the (estimated) value of the intensity λ(xi ) for the
point xi of the pattern X. Each value must be a positive number; NA’s are not allowed.
If lambda is a pixel image, the domain of the image should cover the entire window of the point
pattern. If it does not (which may occur near the boundary because of discretisation error), then
the missing pixel values will be obtained by applying a Gaussian blur to lambda using blur, then
looking up the values of this blurred image for the missing locations. (A warning will be issued in
this case.)
If lambda is a function, then it will be evaluated in the form lambda(x,y) where x and y are vectors
of coordinates of the points of X. It should return a numeric vector with length equal to the number
of points in X.
If lambda is omitted, then it will be estimated using a ‘leave-one-out’ kernel smoother, as described
in Baddeley, Møller and Waagepetersen (2000). The estimate lambda[i] for the point X[i] is computed by removing X[i] from the point pattern, applying kernel smoothing to the remaining points
using density.ppp, and evaluating the smoothed intensity at the point X[i]. The smoothing kernel
bandwidth is controlled by the arguments sigma and varcov, which are passed to density.ppp
along with any extra arguments.
Value
An object of class "fv", see fv.object, which can be plotted directly using plot.fv.
Author(s)
Original code by Marie-Colette van Lieshout. C implementation and R adaptation by Adrian Baddeley 
and Ege Rubak .
References
Baddeley, A., Møller, J. and Waagepetersen, R. (2000) Non- and semiparametric estimation of
interaction in inhomogeneous point patterns. Statistica Neerlandica 54, 329–350.
van Lieshout, M.N.M. and Baddeley, A.J. (1996) A nonparametric measure of spatial interaction in
point patterns. Statistica Neerlandica 50, 344–361.
van Lieshout, M.N.M. (2010) A J-function for inhomogeneous point processes. Statistica Neerlandica 65, 183–201.
See Also
Ginhom, Finhom, Jest

608

Jmulti

Examples
## Not run:
plot(Jinhom(swedishpines, sigma=bw.diggle, adjust=2))
## End(Not run)
plot(Jinhom(swedishpines, sigma=10))

Jmulti

Marked J Function

Description
For a marked point pattern, estimate the multitype J function summarising dependence between the
points in subset I and those in subset J.
Usage
Jmulti(X, I, J, eps=NULL, r=NULL, breaks=NULL, ..., disjoint=NULL,
correction=NULL)
Arguments
X

The observed point pattern, from which an estimate of the multitype distance
distribution function JIJ (r) will be computed. It must be a marked point pattern.
See under Details.

I

Subset of points of X from which distances are measured. See Details.

J

Subset of points in X to which distances are measured. See Details.

eps

A positive number. The pixel resolution of the discrete approximation to Euclidean distance (see Jest). There is a sensible default.

r

numeric vector. The values of the argument r at which the distribution function
JIJ (r) should be evaluated. There is a sensible default. First-time users are
strongly advised not to specify this argument. See below for important conditions on r.

breaks

This argument is for internal use only.

...

Ignored.

disjoint

Optional flag indicating whether the subsets I and J are disjoint. If missing, this
value will be computed by inspecting the vectors I and J.

correction

Optional. Character string specifying the edge correction(s) to be used. Options
are "none", "rs", "km", "Hanisch" and "best". Alternatively correction="all"
selects all options.

Details
The function Jmulti generalises Jest (for unmarked point patterns) and Jdot and Jcross (for
multitype point patterns) to arbitrary marked point patterns.
Suppose XI , XJ are subsets, possibly overlapping, of a marked point process. Define
JIJ (r) =

1 − GIJ (r)
1 − FJ (r)

Jmulti

609

where FJ (r) is the cumulative distribution function of the distance from a fixed location to the
nearest point of XJ , and GIJ (r) is the distribution function of the distance from a typical point of
XI to the nearest distinct point of XJ .
The argument X must be a point pattern (object of class "ppp") or any data that are acceptable to
as.ppp.
The arguments I and J specify two subsets of the point pattern. They may be any type of subset
indices, for example, logical vectors of length equal to npoints(X), or integer vectors with entries
in the range 1 to npoints(X), or negative integer vectors.
Alternatively, I and J may be functions that will be applied to the point pattern X to obtain index
vectors. If I is a function, then evaluating I(X) should yield a valid subset index. This option is
useful when generating simulation envelopes using envelope.
It is assumed that X can be treated as a realisation of a stationary (spatially homogeneous) random
spatial point process in the plane, observed through a bounded window. The window (which is
specified in X as Window(X)) may have arbitrary shape. Biases due to edge effects are treated in the
same manner as in Jest.
The argument r is the vector of values for the distance r at which JIJ (r) should be evaluated. It is
also used to determine the breakpoints (in the sense of hist) for the computation of histograms of
distances. The reduced-sample and Kaplan-Meier estimators are computed from histogram counts.
In the case of the Kaplan-Meier estimator this introduces a discretisation error which is controlled
by the fineness of the breakpoints.
First-time users would be strongly advised not to specify r. However, if it is specified, r must
satisfy r[1] = 0, and max(r) must be larger than the radius of the largest disc contained in the
window. Furthermore, the successive entries of r must be finely spaced.
Value
An object of class "fv" (see fv.object).
Essentially a data frame containing six numeric columns
r
rs
km
han
un
theo

the values of the argument r at which the function JIJ (r) has been estimated
the “reduced sample” or “border correction” estimator of JIJ (r)
the spatial Kaplan-Meier estimator of JIJ (r)
the Hanisch-style estimator of JIJ (r)
the uncorrected estimate of JIJ (r), formed by taking the ratio of uncorrected
empirical estimators of 1 − GIJ (r) and 1 − FJ (r), see Gdot and Fest.
the theoretical value of JIJ (r) for a marked Poisson process with the same estimated intensity, namely 1.

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
References
Van Lieshout, M.N.M. and Baddeley, A.J. (1999) Indices of dependence between types in multivariate point patterns. Scandinavian Journal of Statistics 26, 511–532.
See Also
Jcross, Jdot, Jest

610

K3est

Examples
trees <- longleaf
# Longleaf Pine data: marks represent diameter
Jm <- Jmulti(trees, marks(trees) <= 15, marks(trees) >= 25)
plot(Jm)

K3est

K-function of a Three-Dimensional Point Pattern

Description
Estimates the K-function from a three-dimensional point pattern.
Usage
K3est(X, ...,
rmax = NULL, nrval = 128,
correction = c("translation", "isotropic"),
ratio=FALSE)
Arguments
X

Three-dimensional point pattern (object of class "pp3").

...

Ignored.

rmax

Optional. Maximum value of argument r for which K3 (r) will be estimated.

nrval

Optional. Number of values of r for which K3 (r) will be estimated. A large
value of nrval is required to avoid discretisation effects.

correction

Optional. Character vector specifying the edge correction(s) to be applied. See
Details.

ratio

Logical. If TRUE, the numerator and denominator of each edge-corrected estimate will also be saved, for use in analysing replicated point patterns.

Details
For a stationary point process Φ in three-dimensional space, the three-dimensional K function is
K3 (r) =

1
E(N (Φ, x, r) | x ∈ Φ)
λ

where λ is the intensity of the process (the expected number of points per unit volume) and N (Φ, x, r)
is the number of points of Φ, other than x itself, which fall within a distance r of x. This is the
three-dimensional generalisation of Ripley’s K function for two-dimensional point processes (Ripley, 1977).
The three-dimensional point pattern X is assumed to be a partial realisation of a stationary point
process Φ. The distance between each pair of distinct points is computed. The empirical cumulative
distribution function of these values, with appropriate edge corrections, is renormalised to give the
estimate of K3 (r).
The available edge corrections are:

kaplan.meier

611

"translation": the Ohser translation correction estimator (Ohser, 1983; Baddeley et al, 1993)
"isotropic": the three-dimensional counterpart of Ripley’s isotropic edge correction (Ripley,
1977; Baddeley et al, 1993).
Alternatively correction="all" selects all options.
Value
A function value table (object of class "fv") that can be plotted, printed or coerced to a data frame
containing the function values.
Author(s)
Adrian Baddeley 
and Rana Moyeed.
References
Baddeley, A.J, Moyeed, R.A., Howard, C.V. and Boyde, A. (1993) Analysis of a three-dimensional
point pattern with replication. Applied Statistics 42, 641–668.
Ohser, J. (1983) On estimators for the reduced second moment measure of point processes. Mathematische Operationsforschung und Statistik, series Statistics, 14, 63 – 71.
Ripley, B.D. (1977) Modelling spatial patterns (with discussion). Journal of the Royal Statistical
Society, Series B, 39, 172 – 212.
See Also
F3est, G3est, pcf3est
Examples
X <- rpoispp3(42)
Z <- K3est(X)
if(interactive()) plot(Z)

kaplan.meier

Kaplan-Meier Estimator using Histogram Data

Description
Compute the Kaplan-Meier estimator of a survival time distribution function, from histogram data
Usage
kaplan.meier(obs, nco, breaks, upperobs=0)

612

kaplan.meier

Arguments
obs

vector of n integers giving the histogram of all observations (censored or uncensored survival times)

nco

vector of n integers giving the histogram of uncensored observations (those survival times that are less than or equal to the censoring time)

breaks

Vector of n + 1 breakpoints which were used to form both histograms.

upperobs

Number of observations beyond the rightmost breakpoint, if any.

Details
This function is needed mainly for internal use in spatstat, but may be useful in other applications
where you want to form the Kaplan-Meier estimator from a huge dataset.
Suppose Ti are the survival times of individuals i = 1, . . . , M with unknown distribution function
F (t) which we wish to estimate. Suppose these times are right-censored by random censoring
times Ci . Thus the observations consist of right-censored survival times T̃i = min(Ti , Ci ) and
non-censoring indicators Di = 1{Ti ≤ Ci } for each i.
If the number of observations M is large, it is efficient to use histograms. Form the histogram
obs of all observed times T̃i . That is, obs[k] counts the number of values T̃i in the interval
(breaks[k],breaks[k+1]] for k > 1 and [breaks[1],breaks[2]] for k = 1. Also form the
histogram nco of all uncensored times, i.e. those T̃i such that Di = 1. These two histograms are
the arguments passed to kaplan.meier.
The vectors km and lambda returned by kaplan.meier are (histogram approximations to) the
Kaplan-Meier estimator of F (t) and its hazard rate λ(t). Specifically, km[k] is an estimate of
F(breaks[k+1]), and lambda[k] is an estimate of the average of λ(t) over the interval (breaks[k],breaks[k+1]).
The histogram breaks must include 0. If the histogram breaks do not span the range of the observations, it is important to count how many survival times T̃i exceed the rightmost breakpoint, and
give this as the value upperobs.
Value
A list with two elements:
km

Kaplan-Meier estimate of the survival time c.d.f. F (t)

lambda

corresponding Nelson-Aalen estimate of the hazard rate λ(t)

These are numeric vectors of length n.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
reduced.sample, km.rs

Kcom

Kcom

613

Model Compensator of K Function

Description
Given a point process model fitted to a point pattern dataset, this function computes the compensator
of the K function based on the fitted model (as well as the usual nonparametric estimates of K based
on the data alone). Comparison between the nonparametric and model-compensated K functions
serves as a diagnostic for the model.
Usage
Kcom(object, r = NULL, breaks = NULL, ...,
correction = c("border", "isotropic", "translate"),
conditional = !is.poisson(object),
restrict = FALSE,
model = NULL,
trend = ~1, interaction = Poisson(), rbord = reach(interaction),
compute.var = TRUE,
truecoef = NULL, hi.res = NULL)
Arguments
object

Object to be analysed. Either a fitted point process model (object of class "ppm")
or a point pattern (object of class "ppp") or quadrature scheme (object of class
"quad").

r

Optional. Vector of values of the argument r at which the function K(r) should
be computed. This argument is usually not specified. There is a sensible default.

breaks

This argument is for advanced use only.

...

Ignored.

correction

Optional vector of character strings specifying the edge correction(s) to be used.
See Kest for options.

conditional

Optional. Logical value indicating whether to compute the estimates for the
conditional case. See Details.

restrict

Logical value indicating whether to compute the restriction estimator (restrict=TRUE)
or the reweighting estimator (restrict=FALSE, the default). Applies only if
conditional=TRUE. See Details.

model

Optional. A fitted point process model (object of class "ppm") to be re-fitted to
the data using update.ppm, if object is a point pattern. Overrides the arguments trend,interaction,rbord.
trend,interaction,rbord
Optional. Arguments passed to ppm to fit a point process model to the data, if
object is a point pattern. See ppm for details.
compute.var

Logical value indicating whether to compute the Poincare variance bound for
the residual K function (calculation is only implemented for the isotropic correction).

truecoef

Optional. Numeric vector. If present, this will be treated as if it were the true
coefficient vector of the point process model, in calculating the diagnostic. Incompatible with hi.res.

614

Kcom
hi.res

Optional. List of parameters passed to quadscheme. If this argument is present,
the model will be re-fitted at high resolution as specified by these parameters.
The coefficients of the resulting fitted model will be taken as the true coefficients. Then the diagnostic will be computed for the default quadrature scheme,
but using the high resolution coefficients.

Details
This command provides a diagnostic for the goodness-of-fit of a point process model fitted to a
point pattern dataset. It computes an estimate of the K function of the dataset, together with a
model compensator of the K function, which should be approximately equal if the model is a good
fit to the data.
The first argument, object, is usually a fitted point process model (object of class "ppm"), obtained
from the model-fitting function ppm.
For convenience, object can also be a point pattern (object of class "ppp"). In that case, a point
process model will be fitted to it, by calling ppm using the arguments trend (for the first order
trend), interaction (for the interpoint interaction) and rbord (for the erosion distance in the border correction for the pseudolikelihood). See ppm for details of these arguments.
The algorithm first extracts the original point pattern dataset (to which the model was fitted) and
computes the standard nonparametric estimates of the K function. It then also computes the model
compensator of the K function. The different function estimates are returned as columns in a data
frame (of class "fv").
The argument correction determines the edge correction(s) to be applied. See Kest for explanation of the principle of edge corrections. The following table gives the options for the correction
argument, and the corresponding column names in the result:
correction
"isotropic"
"translate"
"border"

description of correction
Ripley isotropic correction
Ohser-Stoyan translation correction
border correction

nonparametric
iso
trans
border

compensator
icom
tcom
bcom

The nonparametric estimates can all be expressed in the form
XX
K̂(r) =
e(xi , xj , r, x)I{d(xi , xj ) ≤ r}
i

j
Ege Rubak  and Jesper Møller.
References
Baddeley, A., Rubak, E. and Møller, J. (2011) Score, pseudo-score and residual diagnostics for
spatial point process models. Statistical Science 26, 613–646.
See Also
Related functions: Kres, Kest.
Alternative functions: Gcom, psstG, psstA, psst.
Point process models: ppm.
Examples
fit0 <- ppm(cells, ~1) # uniform Poisson
if(interactive()) {
plot(Kcom(fit0))
# compare the isotropic-correction estimates
plot(Kcom(fit0), cbind(iso, icom) ~ r)
# uniform Poisson is clearly not correct
}
fit1 <- ppm(cells, ~1, Strauss(0.08))
K1 <- Kcom(fit1)
K1
if(interactive()) {
plot(K1)
plot(K1, cbind(iso, icom) ~ r)
plot(K1, cbind(trans, tcom) ~ r)
# how to plot the difference between nonparametric estimates and compensators
plot(K1, iso - icom ~ r)
# fit looks approximately OK; try adjusting interaction distance
}

616

Kcross
fit2 <- ppm(cells, ~1, Strauss(0.12))
K2 <- Kcom(fit2)
if(interactive()) {
plot(K2)
plot(K2, cbind(iso, icom) ~ r)
plot(K2, iso - icom ~ r)
}

Kcross

Multitype K Function (Cross-type)

Description
For a multitype point pattern, estimate the multitype K function which counts the expected number
of points of type j within a given distance of a point of type i.
Usage
Kcross(X, i, j, r=NULL, breaks=NULL, correction,
..., ratio=FALSE, from, to )
Arguments
X

The observed point pattern, from which an estimate of the cross type K function
Kij (r) will be computed. It must be a multitype point pattern (a marked point
pattern whose marks are a factor). See under Details.

i

The type (mark value) of the points in X from which distances are measured.
A character string (or something that will be converted to a character string).
Defaults to the first level of marks(X).

j

The type (mark value) of the points in X to which distances are measured. A
character string (or something that will be converted to a character string). Defaults to the second level of marks(X).

r

numeric vector. The values of the argument r at which the distribution function
Kij (r) should be evaluated. There is a sensible default. First-time users are
strongly advised not to specify this argument. See below for important conditions on r.

breaks

This argument is for internal use only.

correction

A character vector containing any selection of the options "border", "bord.modif",
"isotropic", "Ripley", "translate", "translation", "none" or "best". It
specifies the edge correction(s) to be applied. Alternatively correction="all"
selects all options.

...

Ignored.

ratio

Logical. If TRUE, the numerator and denominator of each edge-corrected estimate will also be saved, for use in analysing replicated point patterns.

from,to

An alternative way to specify i and j respectively.

Kcross

617

Details
This function Kcross and its companions Kdot and Kmulti are generalisations of the function Kest
to multitype point patterns.
A multitype point pattern is a spatial pattern of points classified into a finite number of possible
“colours” or “types”. In the spatstat package, a multitype pattern is represented as a single point
pattern object in which the points carry marks, and the mark value attached to each point determines
the type of that point.
The argument X must be a point pattern (object of class "ppp") or any data that are acceptable to
as.ppp. It must be a marked point pattern, and the mark vector X$marks must be a factor.
The arguments i and j will be interpreted as levels of the factor X$marks. If i and j are missing,
they default to the first and second level of the marks factor, respectively.
The “cross-type” (type i to type j) K function of a stationary multitype point process X is defined
so that λj Kij (r) equals the expected number of additional random points of type j within a distance
r of a typical point of type i in the process X. Here λj is the intensity of the type j points, i.e. the
expected number of points of type j per unit area. The function Kij is determined by the second
order moment properties of X.
An estimate of Kij (r) is a useful summary statistic in exploratory data analysis of a multitype point
pattern. If the process of type i points were independent of the process of type j points, then Kij (r)
would equal πr2 . Deviations between the empirical Kij curve and the theoretical curve πr2 may
suggest dependence between the points of types i and j.
This algorithm estimates the distribution function Kij (r) from the point pattern X. It assumes that
X can be treated as a realisation of a stationary (spatially homogeneous) random spatial point process in the plane, observed through a bounded window. The window (which is specified in X as
Window(X)) may have arbitrary shape. Biases due to edge effects are treated in the same manner as
in Kest, using the border correction.
The argument r is the vector of values for the distance r at which Kij (r) should be evaluated. The
values of r must be increasing nonnegative numbers and the maximum r value must not exceed the
radius of the largest disc contained in the window.
The pair correlation function can also be applied to the result of Kcross; see pcf.
Value
An object of class "fv" (see fv.object).
Essentially a data frame containing numeric columns
r

the values of the argument r at which the function Kij (r) has been estimated

theo

the theoretical value of Kij (r) for a marked Poisson process, namely πr2

together with a column or columns named "border", "bord.modif", "iso" and/or "trans", according to the selected edge corrections. These columns contain estimates of the function Kij (r)
obtained by the edge corrections named.
If ratio=TRUE then the return value also has two attributes called "numerator" and "denominator"
which are "fv" objects containing the numerators and denominators of each estimate of K(r).
Warnings
The arguments i and j are always interpreted as levels of the factor X$marks. They are converted
to character strings if they are not already character strings. The value i=1 does not refer to the first
level of the factor.

618

Kcross.inhom

Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.
Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 1983.
Harkness, R.D and Isham, V. (1983) A bivariate spatial point pattern of ants’ nests. Applied Statistics 32, 293–303
Lotwick, H. W. and Silverman, B. W. (1982). Methods for analysing spatial processes of several
types of points. J. Royal Statist. Soc. Ser. B 44, 406–413.
Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.
Stoyan, D, Kendall, W.S. and Mecke, J. Stochastic geometry and its applications. 2nd edition.
Springer Verlag, 1995.
See Also
Kdot, Kest, Kmulti, pcf
Examples
# amacrine cells data
K01 <- Kcross(amacrine, "off", "on")
plot(K01)
## Not run:
K10 <- Kcross(amacrine, "on", "off")
# synthetic example: point pattern with marks 0 and 1
pp <- runifpoispp(50)
pp <- pp %mark% factor(sample(0:1, npoints(pp), replace=TRUE))
K <- Kcross(pp, "0", "1")
K <- Kcross(pp, 0, 1) # equivalent
## End(Not run)

Kcross.inhom

Inhomogeneous Cross K Function

Description
For a multitype point pattern, estimate the inhomogeneous version of the cross K function, which
counts the expected number of points of type j within a given distance of a point of type i, adjusted
for spatially varying intensity.

Kcross.inhom

619

Usage
Kcross.inhom(X, i, j, lambdaI=NULL, lambdaJ=NULL, ..., r=NULL, breaks=NULL,
correction = c("border", "isotropic", "Ripley", "translate"),
sigma=NULL, varcov=NULL,
lambdaIJ=NULL,
lambdaX=NULL, update=TRUE, leaveoneout=TRUE)
Arguments
X

i

j

lambdaI

lambdaJ

r

breaks
correction

...
sigma
varcov

lambdaIJ
lambdaX

The observed point pattern, from which an estimate of the inhomogeneous cross
type K function Kij (r) will be computed. It must be a multitype point pattern
(a marked point pattern whose marks are a factor). See under Details.
The type (mark value) of the points in X from which distances are measured.
A character string (or something that will be converted to a character string).
Defaults to the first level of marks(X).
The type (mark value) of the points in X to which distances are measured. A
character string (or something that will be converted to a character string). Defaults to the second level of marks(X).
Optional. Values of the the estimated intensity of the sub-process of points of
type i. Either a pixel image (object of class "im"), a numeric vector containing
the intensity values at each of the type i points in X, a fitted point process model
(object of class "ppm" or "kppm" or "dppm"), or a function(x,y) which can be
evaluated to give the intensity value at any location.
Optional. Values of the the estimated intensity of the sub-process of points of
type j. Either a pixel image (object of class "im"), a numeric vector containing
the intensity values at each of the type j points in X, a fitted point process model
(object of class "ppm" or "kppm" or "dppm"), or a function(x,y) which can be
evaluated to give the intensity value at any location.
Optional. Numeric vector giving the values of the argument r at which the cross
K function Kij (r) should be evaluated. There is a sensible default. First-time
users are strongly advised not to specify this argument. See below for important
conditions on r.
This argument is for advanced use only.
A character vector containing any selection of the options "border", "bord.modif",
"isotropic", "Ripley" ,"translate", "translation", "none" or "best". It
specifies the edge correction(s) to be applied. Alternatively correction="all"
selects all options.
Ignored.
Standard deviation of isotropic Gaussian smoothing kernel, used in computing
leave-one-out kernel estimates of lambdaI, lambdaJ if they are omitted.
Variance-covariance matrix of anisotropic Gaussian kernel, used in computing
leave-one-out kernel estimates of lambdaI, lambdaJ if they are omitted. Incompatible with sigma.
Optional. A matrix containing estimates of the product of the intensities lambdaI
and lambdaJ for each pair of points of types i and j respectively.
Optional. Values of the intensity for all points of X. Either a pixel image (object
of class "im"), a numeric vector containing the intensity values at each of the
points in X, a fitted point process model (object of class "ppm" or "kppm" or
"dppm"), or a function(x,y) which can be evaluated to give the intensity value
at any location. If present, this argument overrides both lambdaI and lambdaJ.

620

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update

Logical value indicating what to do when lambdaI, lambdaJ or lambdaX is a
fitted point process model (class "ppm", "kppm" or "dppm"). If update=TRUE
(the default), the model will first be refitted to the data X (using update.ppm
or update.kppm) before the fitted intensity is computed. If update=FALSE, the
fitted intensity of the model will be computed without re-fitting it to X.

leaveoneout

Logical value (passed to density.ppp or fitted.ppm) specifying whether to
use a leave-one-out rule when calculating the intensity.

Details
This is a generalisation of the function Kcross to include an adjustment for spatially inhomogeneous
intensity, in a manner similar to the function Kinhom.
The inhomogeneous cross-type K function is described by Møller and Waagepetersen (2003, pages
48-49 and 51-53).
Briefly, given a multitype point process, suppose the sub-process of points of type j has intensity
function λj (u) at spatial locations u. Suppose we place a mass of 1/λj (ζ) at each point ζ of type
j. Then the expected total mass per unit area is 1. The inhomogeneous “cross-type” K function
inhom (r) equals the expected total mass within a radius r of a point of the process of type i.
Kij
inhom (r)
If the process of type i points were independent of the process of type j points, then Kij
would equal πr2 . Deviations between the empirical Kij curve and the theoretical curve πr2 suggest
dependence between the points of types i and j.
The argument X must be a point pattern (object of class "ppp") or any data that are acceptable to
as.ppp. It must be a marked point pattern, and the mark vector X$marks must be a factor.
The arguments i and j will be interpreted as levels of the factor X$marks. (Warning: this means
that an integer value i=3 will be interpreted as the number 3, not the 3rd smallest level). If i and j
are missing, they default to the first and second level of the marks factor, respectively.
The argument lambdaI supplies the values of the intensity of the sub-process of points of type i. It
may be either
a pixel image (object of class "im") which gives the values of the type i intensity at all locations
in the window containing X;
a numeric vector containing the values of the type i intensity evaluated only at the data points of
type i. The length of this vector must equal the number of type i points in X.
a function which can be evaluated to give values of the intensity at any locations.
a fitted point process model (object of class "ppm", "kppm" or "dppm") whose fitted trend can be
used as the fitted intensity. (If update=TRUE the model will first be refitted to the data X before
the trend is computed.)
omitted: if lambdaI is omitted then it will be estimated using a leave-one-out kernel smoother.
If lambdaI is omitted, then it will be estimated using a ‘leave-one-out’ kernel smoother, as described
in Baddeley, Møller and Waagepetersen (2000). The estimate of lambdaI for a given point is
computed by removing the point from the point pattern, applying kernel smoothing to the remaining
points using density.ppp, and evaluating the smoothed intensity at the point in question. The
smoothing kernel bandwidth is controlled by the arguments sigma and varcov, which are passed to
density.ppp along with any extra arguments.
Similarly lambdaJ should contain estimated values of the intensity of the sub-process of points of
type j. It may be either a pixel image, a function, a numeric vector, or omitted.
Alternatively if the argument lambdaX is given, then it specifies the intensity values for all points of
X, and the arguments lambdaI, lambdaJ will be ignored.

Kcross.inhom

621

The optional argument lambdaIJ is for advanced use only. It is a matrix containing estimated values
of the products of these two intensities for each pair of data points of types i and j respectively.
The argument r is the vector of values for the distance r at which Kij (r) should be evaluated. The
values of r must be increasing nonnegative numbers and the maximum r value must exceed the
radius of the largest disc contained in the window.
The argument correction chooses the edge correction as explained e.g. in Kest.
The pair correlation function can also be applied to the result of Kcross.inhom; see pcf.
Value
An object of class "fv" (see fv.object).
Essentially a data frame containing numeric columns
r

the values of the argument r at which the function Kij (r) has been estimated

theo

the theoretical value of Kij (r) for a marked Poisson process, namely πr2

together with a column or columns named "border", "bord.modif", "iso" and/or "trans", according to the selected edge corrections. These columns contain estimates of the function Kij (r)
obtained by the edge corrections named.
Warnings
The arguments i and j are always interpreted as levels of the factor X$marks. They are converted
to character strings if they are not already character strings. The value i=1 does not refer to the first
level of the factor.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
References
Baddeley, A., Møller, J. and Waagepetersen, R. (2000) Non- and semiparametric estimation of
interaction in inhomogeneous point patterns. Statistica Neerlandica 54, 329–350.
Møller, J. and Waagepetersen, R. Statistical Inference and Simulation for Spatial Point Processes
Chapman and Hall/CRC Boca Raton, 2003.
See Also
Kcross, Kinhom, Kdot.inhom, Kmulti.inhom, pcf
Examples
# Lansing Woods data
woods <- lansing
ma <- split(woods)$maple
wh <- split(woods)$whiteoak
# method (1): estimate intensities by nonparametric smoothing
lambdaM <- density.ppp(ma, sigma=0.15, at="points")
lambdaW <- density.ppp(wh, sigma=0.15, at="points")

622

Kdot
K <- Kcross.inhom(woods, "whiteoak", "maple", lambdaW, lambdaM)
# method (2): leave-one-out
K <- Kcross.inhom(woods, "whiteoak", "maple", sigma=0.15)
# method (3): fit parametric intensity model
fit <- ppm(woods ~marks * polynom(x,y,2))
# alternative (a): use fitted model as 'lambda' argument
K <- Kcross.inhom(woods, "whiteoak", "maple",
lambdaI=fit, lambdaJ=fit, update=FALSE)
K <- Kcross.inhom(woods, "whiteoak", "maple",
lambdaX=fit, update=FALSE)
# alternative (b): evaluate fitted intensities at data points
# (these are the intensities of the sub-processes of each type)
inten <- fitted(fit, dataonly=TRUE)
# split according to types of points
lambda <- split(inten, marks(woods))
K <- Kcross.inhom(woods, "whiteoak", "maple",
lambda$whiteoak, lambda$maple)
# synthetic example: type A points have intensity 50,
#
type B points have intensity 100 * x
lamB <- as.im(function(x,y){50 + 100 * x}, owin())
X <- superimpose(A=runifpoispp(50), B=rpoispp(lamB))
K <- Kcross.inhom(X, "A", "B",
lambdaI=as.im(50, Window(X)), lambdaJ=lamB)

Kdot

Multitype K Function (i-to-any)

Description
For a multitype point pattern, estimate the multitype K function which counts the expected number
of other points of the process within a given distance of a point of type i.
Usage
Kdot(X, i, r=NULL, breaks=NULL, correction, ..., ratio=FALSE, from)
Arguments
X

The observed point pattern, from which an estimate of the multitype K function
Ki• (r) will be computed. It must be a multitype point pattern (a marked point
pattern whose marks are a factor). See under Details.

i

The type (mark value) of the points in X from which distances are measured.
A character string (or something that will be converted to a character string).
Defaults to the first level of marks(X).

r

numeric vector. The values of the argument r at which the distribution function
Ki• (r) should be evaluated. There is a sensible default. First-time users are
strongly advised not to specify this argument. See below for important conditions on r.

breaks

This argument is for internal use only.

Kdot

623

correction

A character vector containing any selection of the options "border", "bord.modif",
"isotropic", "Ripley", "translate", "translation", "none" or "best". It
specifies the edge correction(s) to be applied. Alternatively correction="all"
selects all options.

...

Ignored.

ratio

Logical. If TRUE, the numerator and denominator of each edge-corrected estimate will also be saved, for use in analysing replicated point patterns.

from

An alternative way to specify i.

Details
This function Kdot and its companions Kcross and Kmulti are generalisations of the function Kest
to multitype point patterns.
A multitype point pattern is a spatial pattern of points classified into a finite number of possible
“colours” or “types”. In the spatstat package, a multitype pattern is represented as a single point
pattern object in which the points carry marks, and the mark value attached to each point determines
the type of that point.
The argument X must be a point pattern (object of class "ppp") or any data that are acceptable to
as.ppp. It must be a marked point pattern, and the mark vector X$marks must be a factor.
The argument i will be interpreted as a level of the factor X$marks. If i is missing, it defaults to the
first level of the marks factor, i = levels(X$marks)[1].
The “type i to any type” multitype K function of a stationary multitype point process X is defined
so that λKi• (r) equals the expected number of additional random points within a distance r of a
typical point of type i in the process X. Here λ is the intensity of the process, i.e. the expected
number of points of X per unit area. The function Ki• is determined by the second order moment
properties of X.
An estimate of Ki• (r) is a useful summary statistic in exploratory data analysis of a multitype point
pattern. If the subprocess of type i points were independent of the subprocess of points of all types
not equal to i, then Ki• (r) would equal πr2 . Deviations between the empirical Ki• curve and the
theoretical curve πr2 may suggest dependence between types.
This algorithm estimates the distribution function Ki• (r) from the point pattern X. It assumes that
X can be treated as a realisation of a stationary (spatially homogeneous) random spatial point process in the plane, observed through a bounded window. The window (which is specified in X as
Window(X)) may have arbitrary shape. Biases due to edge effects are treated in the same manner as
in Kest, using the border correction.
The argument r is the vector of values for the distance r at which Ki• (r) should be evaluated. The
values of r must be increasing nonnegative numbers and the maximum r value must exceed the
radius of the largest disc contained in the window.
The pair correlation function can also be applied to the result of Kdot; see pcf.
Value
An object of class "fv" (see fv.object).
Essentially a data frame containing numeric columns
r

the values of the argument r at which the function Ki• (r) has been estimated

theo

the theoretical value of Ki• (r) for a marked Poisson process, namely πr2

624

Kdot
together with a column or columns named "border", "bord.modif", "iso" and/or "trans", according to the selected edge corrections. These columns contain estimates of the function Ki• (r)
obtained by the edge corrections named.
If ratio=TRUE then the return value also has two attributes called "numerator" and "denominator"
which are "fv" objects containing the numerators and denominators of each estimate of K(r).

Warnings
The argument i is interpreted as a level of the factor X$marks. It is converted to a character string
if it is not already a character string. The value i=1 does not refer to the first level of the factor.
The reduced sample estimator of Ki• is pointwise approximately unbiased, but need not be a valid
distribution function; it may not be a nondecreasing function of r. Its range is always within [0, 1].
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.
Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 1983.
Harkness, R.D and Isham, V. (1983) A bivariate spatial point pattern of ants’ nests. Applied Statistics 32, 293–303
Lotwick, H. W. and Silverman, B. W. (1982). Methods for analysing spatial processes of several
types of points. J. Royal Statist. Soc. Ser. B 44, 406–413.
Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.
Stoyan, D, Kendall, W.S. and Mecke, J. Stochastic geometry and its applications. 2nd edition.
Springer Verlag, 1995.
See Also
Kdot, Kest, Kmulti, pcf
Examples
# Lansing woods data: 6 types of trees
woods <- lansing

Kh. <- Kdot(woods, "hickory")
# diagnostic plot for independence between hickories and other trees
plot(Kh.)
## Not run:
# synthetic example with two marks "a" and "b"
pp <- runifpoispp(50)
pp <- pp %mark% factor(sample(c("a","b"), npoints(pp), replace=TRUE))
K <- Kdot(pp, "a")
## End(Not run)

Kdot.inhom

Kdot.inhom

625

Inhomogeneous Multitype K Dot Function

Description
For a multitype point pattern, estimate the inhomogeneous version of the dot K function, which
counts the expected number of points of any type within a given distance of a point of type i,
adjusted for spatially varying intensity.
Usage
Kdot.inhom(X, i, lambdaI=NULL, lambdadot=NULL, ..., r=NULL, breaks=NULL,
correction = c("border", "isotropic", "Ripley", "translate"),
sigma=NULL, varcov=NULL, lambdaIdot=NULL,
lambdaX=NULL, update=TRUE, leaveoneout=TRUE)
Arguments
X

The observed point pattern, from which an estimate of the inhomogeneous cross
type K function Ki• (r) will be computed. It must be a multitype point pattern
(a marked point pattern whose marks are a factor). See under Details.

i

The type (mark value) of the points in X from which distances are measured.
A character string (or something that will be converted to a character string).
Defaults to the first level of marks(X).

lambdaI

Optional. Values of the estimated intensity of the sub-process of points of type
i. Either a pixel image (object of class "im"), a numeric vector containing the
intensity values at each of the type i points in X, a fitted point process model
(object of class "ppm" or "kppm" or "dppm"), or a function(x,y) which can be
evaluated to give the intensity value at any location.

lambdadot

Optional. Values of the estimated intensity of the entire point process, Either
a pixel image (object of class "im"), a numeric vector containing the intensity
values at each of the points in X, a fitted point process model (object of class
"ppm" or "kppm" or "dppm"), or a function(x,y) which can be evaluated to
give the intensity value at any location.

...

Ignored.

r

Optional. Numeric vector giving the values of the argument r at which the cross
K function Kij (r) should be evaluated. There is a sensible default. First-time
users are strongly advised not to specify this argument. See below for important
conditions on r.

breaks

This argument is for internal use only.

correction

A character vector containing any selection of the options "border", "bord.modif",
"isotropic", "Ripley", "translate", "translation", "none" or "best". It
specifies the edge correction(s) to be applied. Alternatively correction="all"
selects all options.

sigma

Standard deviation of isotropic Gaussian smoothing kernel, used in computing
leave-one-out kernel estimates of lambdaI, lambdadot if they are omitted.

varcov

Variance-covariance matrix of anisotropic Gaussian kernel, used in computing
leave-one-out kernel estimates of lambdaI, lambdadot if they are omitted. Incompatible with sigma.

626

Kdot.inhom
lambdaIdot

Optional. A matrix containing estimates of the product of the intensities lambdaI
and lambdadot for each pair of points, the first point of type i and the second
of any type.

lambdaX

Optional. Values of the intensity for all points of X. Either a pixel image (object of class "im"), a numeric vector containing the intensity values at each of
the points in X, a fitted point process model (object of class "ppm" or "kppm"
or "dppm"), or a function(x,y) which can be evaluated to give the intensity
value at any location. If present, this argument overrides both lambdaI and
lambdadot.

update

Logical value indicating what to do when lambdaI, lambdadot or lambdaX is
a fitted point process model (class "ppm", "kppm" or "dppm"). If update=TRUE
(the default), the model will first be refitted to the data X (using update.ppm
or update.kppm) before the fitted intensity is computed. If update=FALSE, the
fitted intensity of the model will be computed without re-fitting it to X.

leaveoneout

Logical value (passed to density.ppp or fitted.ppm) specifying whether to
use a leave-one-out rule when calculating the intensity.

Details
This is a generalisation of the function Kdot to include an adjustment for spatially inhomogeneous
intensity, in a manner similar to the function Kinhom.
Briefly, given a multitype point process, consider the points without their types, and suppose this
unmarked point process has intensity function λ(u) at spatial locations u. Suppose we place a mass
of 1/λ(ζ) at each point ζ of the process. Then the expected total mass per unit area is 1. The
inhom (r) equals the expected total mass within a radius
inhomogeneous “dot-type” K function Ki•
r of a point of the process of type i, discounting this point itself.
inhom (r)
If the process of type i points were independent of the points of other types, then Ki•
2
would equal πr . Deviations between the empirical Ki• curve and the theoretical curve πr2 suggest
dependence between the points of types i and j for j 6= i.
The argument X must be a point pattern (object of class "ppp") or any data that are acceptable to
as.ppp. It must be a marked point pattern, and the mark vector X$marks must be a factor.
The argument i will be interpreted as a level of the factor X$marks. (Warning: this means that an
integer value i=3 will be interpreted as the number 3, not the 3rd smallest level). If i is missing, it
defaults to the first level of the marks factor, i = levels(X$marks)[1].
The argument lambdaI supplies the values of the intensity of the sub-process of points of type i. It
may be either
a pixel image (object of class "im") which gives the values of the type i intensity at all locations
in the window containing X;
a numeric vector containing the values of the type i intensity evaluated only at the data points of
type i. The length of this vector must equal the number of type i points in X.
a function of the form function(x,y) which can be evaluated to give values of the intensity at
any locations.
a fitted point process model (object of class "ppm", "kppm" or "dppm") whose fitted trend can be
used as the fitted intensity. (If update=TRUE the model will first be refitted to the data X before
the trend is computed.)
omitted: if lambdaI is omitted then it will be estimated using a leave-one-out kernel smoother.

Kdot.inhom

627

If lambdaI is omitted, then it will be estimated using a ‘leave-one-out’ kernel smoother, as described
in Baddeley, Møller and Waagepetersen (2000). The estimate of lambdaI for a given point is
computed by removing the point from the point pattern, applying kernel smoothing to the remaining
points using density.ppp, and evaluating the smoothed intensity at the point in question. The
smoothing kernel bandwidth is controlled by the arguments sigma and varcov, which are passed to
density.ppp along with any extra arguments.
Similarly the argument lambdadot should contain estimated values of the intensity of the entire
point process. It may be either a pixel image, a numeric vector of length equal to the number of
points in X, a function, or omitted.
Alternatively if the argument lambdaX is given, then it specifies the intensity values for all points
of X, and the arguments lambdaI, lambdadot will be ignored. (The two arguments lambdaI,
lambdadot allow the user to specify two different methods for calculating the intensities of the
two kinds of points, while lambdaX ensures that the same method is used for both kinds of points.)
For advanced use only, the optional argument lambdaIdot is a matrix containing estimated values
of the products of these two intensities for each pair of points, the first point of type i and the second
of any type.
The argument r is the vector of values for the distance r at which Ki• (r) should be evaluated. The
values of r must be increasing nonnegative numbers and the maximum r value must exceed the
radius of the largest disc contained in the window.
The argument correction chooses the edge correction as explained e.g. in Kest.
The pair correlation function can also be applied to the result of Kcross.inhom; see pcf.
Value
An object of class "fv" (see fv.object).
Essentially a data frame containing numeric columns
r

the values of the argument r at which the function Ki• (r) has been estimated

theo

the theoretical value of Ki• (r) for a marked Poisson process, namely πr2

together with a column or columns named "border", "bord.modif", "iso" and/or "trans", according to the selected edge corrections. These columns contain estimates of the function Ki• (r)
obtained by the edge corrections named.
Warnings
The argument i is interpreted as a level of the factor X$marks. It is converted to a character string
if it is not already a character string. The value i=1 does not refer to the first level of the factor.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
References
Møller, J. and Waagepetersen, R. Statistical Inference and Simulation for Spatial Point Processes
Chapman and Hall/CRC Boca Raton, 2003.
See Also
Kdot, Kinhom, Kcross.inhom, Kmulti.inhom, pcf

628

kernel.factor

Examples
# Lansing Woods data
woods <- lansing
woods <- woods[seq(1,npoints(woods), by=10)]
ma <- split(woods)$maple
lg <- unmark(woods)
# Estimate intensities by nonparametric smoothing
lambdaM <- density.ppp(ma, sigma=0.15, at="points")
lambdadot <- density.ppp(lg, sigma=0.15, at="points")
K <- Kdot.inhom(woods, "maple", lambdaI=lambdaM,
lambdadot=lambdadot)
# Equivalent
K <- Kdot.inhom(woods, "maple", sigma=0.15)
# Fit model
fit <- ppm(woods ~ marks * polynom(x,y,2))
K <- Kdot.inhom(woods, "maple", lambdaX=fit, update=FALSE)
# synthetic example: type A points have intensity 50,
#
type B points have intensity 50 + 100 * x
lamB <- as.im(function(x,y){50 + 100 * x}, owin())
lamdot <- as.im(function(x,y) { 100 + 100 * x}, owin())
X <- superimpose(A=runifpoispp(50), B=rpoispp(lamB))
K <- Kdot.inhom(X, "B", lambdaI=lamB,
lambdadot=lamdot)

kernel.factor

Scale factor for density kernel

Description
Returns a scale factor for the kernels used in density estimation for numerical data.
Usage
kernel.factor(kernel = "gaussian")
Arguments
kernel

String name of the kernel. Options are "gaussian", "rectangular", "triangular",
"epanechnikov", "biweight", "cosine" and "optcosine". (Partial matching
is used).

Details
Kernel estimation of a probability density in one dimension is performed by density.default
using a kernel function selected from the list above.
This function computes a scale constant for the kernel. For the Gaussian kernel, this constant is
equal to 1. Otherwise, the constant c is such that the kernel with standard deviation 1 is supported
on the interval [−c, c].
For more information about these kernels, see density.default.

kernel.moment

629

Value
A single number.
Author(s)
Adrian Baddeley  and Martin Hazelton
See Also
density.default, dkernel, kernel.moment, kernel.squint
Examples
kernel.factor("rect")
# bandwidth for Epanechnikov kernel with half-width h=1
h <- 1
bw <- h/kernel.factor("epa")

kernel.moment

Moment of Smoothing Kernel

Description
Computes the complete or incomplete mth moment of a smoothing kernel.
Usage
kernel.moment(m, r, kernel = "gaussian")
Arguments
m

Exponent (order of moment). An integer.

r

Upper limit of integration for the incomplete moment. A numeric value or numeric vector. Set r=Inf to obtain the complete moment.

kernel

String name of the kernel. Options are "gaussian", "rectangular", "triangular",
"epanechnikov", "biweight", "cosine" and "optcosine". (Partial matching
is used).

Details
Kernel estimation of a probability density in one dimension is performed by density.default
using a kernel function selected from the list above. For more information about these kernels, see
density.default.
The function kernel.moment computes the partial integral
Z r
tm k(t)dt
−∞

where k(t) is the selected kernel, r is the upper limit of integration, and m is the exponent or order.

630

kernel.squint

Value
A single number, or a numeric vector of the same length as r.
Author(s)
Adrian Baddeley  and Martin Hazelton.
See Also
density.default, dkernel, kernel.factor,
Examples
kernel.moment(1, 0.1, "epa")
curve(kernel.moment(2, x, "epa"), from=-1, to=1)

kernel.squint

Integral of Squared Kernel

Description
Computes the integral of the squared kernel, for the kernels used in density estimation for numerical
data.
Usage
kernel.squint(kernel = "gaussian", bw=1)
Arguments
kernel

String name of the kernel. Options are "gaussian", "rectangular", "triangular",
"epanechnikov", "biweight", "cosine" and "optcosine". (Partial matching
is used).

bw

Bandwidth (standard deviation) of the kernel.

Details
Kernel estimation of a probability density in one dimension is performed by density.default
using a kernel function selected from the list above.
This function computes the integral of the squared kernel,
Z ∞
R=
k(x)2 dx
−∞

where k(x) is the kernel with bandwidth bw.
Value
A single number.

Kest

631

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak  and Martin Hazelton
See Also
density.default, dkernel, kernel.moment, kernel.factor
Examples
kernel.squint("gaussian", 3)
# integral of squared Epanechnikov kernel with half-width h=1
h <- 1
bw <- h/kernel.factor("epa")
kernel.squint("epa", bw)

Kest

K-function

Description
Estimates Ripley’s reduced second moment function K(r) from a point pattern in a window of
arbitrary shape.
Usage
Kest(X, ..., r=NULL, rmax=NULL, breaks=NULL,
correction=c("border", "isotropic", "Ripley", "translate"),
nlarge=3000, domain=NULL, var.approx=FALSE, ratio=FALSE)
Arguments
X

The observed point pattern, from which an estimate of K(r) will be computed.
An object of class "ppp", or data in any format acceptable to as.ppp().

...

Ignored.

r

Optional. Vector of values for the argument r at which K(r) should be evaluated. Users are advised not to specify this argument; there is a sensible default.
If necessary, specify rmax.

rmax

Optional. Maximum desired value of the argument r.

breaks

This argument is for internal use only.

correction

Optional. A character vector containing any selection of the options "none",
"border", "bord.modif", "isotropic", "Ripley", "translate", "translation",
"rigid", "none", "good" or "best". It specifies the edge correction(s) to be
applied. Alternatively correction="all" selects all options.

nlarge

Optional. Efficiency threshold. If the number of points exceeds nlarge, then
only the border correction will be computed (by default), using a fast algorithm.

domain

Optional. Calculations will be restricted to this subset of the window. See Details.

632

Kest
var.approx

Logical. If TRUE, the approximate variance of K̂(r) under CSR will also be
computed.

ratio

Logical. If TRUE, the numerator and denominator of each edge-corrected estimate will also be saved, for use in analysing replicated point patterns.

Details
The K function (variously called “Ripley’s K-function” and the “reduced second moment function”) of a stationary point process X is defined so that λK(r) equals the expected number of
additional random points within a distance r of a typical random point of X. Here λ is the intensity
of the process, i.e. the expected number of points of X per unit area. The K function is determined
by the second order moment properties of X.
An estimate of K derived from a spatial point pattern dataset can be used in exploratory data analysis and formal inference about the pattern (Cressie, 1991; Diggle, 1983; Ripley, 1977, 1988). In
exploratory analyses, the estimate of K is a useful statistic summarising aspects of inter-point “dependence” and “clustering”. For inferential purposes, the estimate of K is usually compared to the
true value of K for a completely random (Poisson) point process, which is K(r) = πr2 . Deviations
between the empirical and theoretical K curves may suggest spatial clustering or spatial regularity.
This routine Kest estimates the K function of a stationary point process, given observation of the
process inside a known, bounded window. The argument X is interpreted as a point pattern object (of
class "ppp", see ppp.object) and can be supplied in any of the formats recognised by as.ppp().
The estimation of K is hampered by edge effects arising from the unobservability of points of the
random pattern outside the window. An edge correction is needed to reduce bias (Baddeley, 1998;
Ripley, 1988). The corrections implemented here are
border the border method or “reduced sample” estimator (see Ripley, 1988). This is the least
efficient (statistically) and the fastest to compute. It can be computed for a window of arbitrary
shape.
isotropic/Ripley Ripley’s isotropic correction (see Ripley, 1988; Ohser, 1983). This is implemented for rectangular and polygonal windows (not for binary masks).
translate/translation Translation correction (Ohser, 1983). Implemented for all window geometries, but slow for complex windows.
rigid Rigid motion correction (Ohser and Stoyan, 1981). Implemented for all window geometries,
but slow for complex windows.
none Uncorrected estimate. An estimate of the K function without edge correction. (i.e. setting
eij = 1 in the equation below. This estimate is biased and should not be used for data analysis,
unless you have an extremely large point pattern (more than 100,000 points).
best Selects the best edge correction that is available for the geometry of the window. Currently
this is Ripley’s isotropic correction for a rectangular or polygonal window, and the translation
correction for masks.
good Selects the best edge correction that can be computed in a reasonable time. This is the same
as "best" for datasets with fewer than 3000 points; otherwise the selected edge correction is
"border", unless there are more than 100,000 points, when it is "none".
The estimates of K(r) are of the form
K̂(r) =

XX
a
I(dij ≤ r)eij
n(n − 1) i j

where a is the area of the window, n is the number of data points, and the sum is taken over all
ordered pairs of points i and j in X. Here dij is the distance between the two points, and I(dij ≤ r)

Kest

633

is the indicator that equals 1 if the distance is less than or equal to r. The term eij is the edge
correction weight (which depends on the choice of edge correction listed above).
Note that this estimator assumes the process is stationary (spatially homogeneous). For inhomogeneous point patterns, see Kinhom.
If the point pattern X contains more than about 3000 points, the isotropic and translation edge corrections can be computationally prohibitive. The computations for the border method are much faster,
and are statistically efficient when there are large numbers of points. Accordingly, if the number of
points in X exceeds the threshold nlarge, then only the border correction will be computed. Setting nlarge=Inf or correction="best" will prevent this from happening. Setting nlarge=0 is
equivalent to selecting only the border correction with correction="border".
If X contains more than about 100,000 points, even the border correction is time-consuming. You
may want to consider setting correction="none" in this case. There is an even faster algorithm
for the uncorrected estimate.
Approximations to the variance of K̂(r) are available, for the case of the isotropic edge correction
estimator, assuming complete spatial randomness (Ripley, 1988; Lotwick and Silverman, 1982;
Diggle, 2003, pp 51-53). If var.approx=TRUE, then the result of Kest also has a column named
rip giving values of Ripley’s (1988) approximation to var(K̂(r)), and (if the window is a rectangle)
a column named ls giving values of Lotwick and Silverman’s (1982) approximation.
If the argument domain is given, the calculations will be restricted to a subset of the data. In
the formula for K(r) above, the first point i will be restricted to lie inside domain. The result is an
approximately unbiased estimate of K(r) based on pairs of points in which the first point lies inside
domain and the second point is unrestricted. This is useful in bootstrap techniques. The argument
domain should be a window (object of class "owin") or something acceptable to as.owin. It must
be a subset of the window of the point pattern X.
The estimator Kest ignores marks. Its counterparts for multitype point patterns are Kcross, Kdot,
and for general marked point patterns see Kmulti.
Some writers, particularly Stoyan (1994, 1995) advocate the use of the “pair correlation function”
g(r) =

K 0 (r)
2πr

where K 0 (r) is the derivative of K(r). See pcf on how to estimate this function.
Value
An object of class "fv", see fv.object, which can be plotted directly using plot.fv.
Essentially a data frame containing columns
r

the vector of values of the argument r at which the function K has been estimated

theo

the theoretical value K(r) = πr2 for a stationary Poisson process

together with columns named "border", "bord.modif", "iso" and/or "trans", according to the
selected edge corrections. These columns contain estimates of the function K(r) obtained by the
edge corrections named.
If var.approx=TRUE then the return value also has columns rip and ls containing approximations
to the variance of K̂(r) under CSR.
If ratio=TRUE then the return value also has two attributes called "numerator" and "denominator"
which are "fv" objects containing the numerators and denominators of each estimate of K(r).

634

Kest

Envelopes, significance bands and confidence intervals
To compute simulation envelopes for the K-function under CSR, use envelope.
To compute a confidence interval for the true K-function, use varblock or lohboot.
Warnings
The estimator of K(r) is approximately unbiased for each fixed r. Bias increases with r and depends on the window geometry. For a rectangular window it is prudent to restrict the r values to a
maximum of 1/4 of the smaller side length of the rectangle. Bias may become appreciable for point
patterns consisting of fewer than 15 points.
While K(r) is always a non-decreasing function, the estimator of K is not guaranteed to be nondecreasing. This is rarely a problem in practice.
Author(s)
Adrian Baddeley  and Rolf Turner 
References
Baddeley, A.J. Spatial sampling and censoring. In O.E. Barndorff-Nielsen, W.S. Kendall and
M.N.M. van Lieshout (eds) Stochastic Geometry: Likelihood and Computation. Chapman and
Hall, 1998. Chapter 2, pages 37–78.
Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.
Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 1983.
Ohser, J. (1983) On estimators for the reduced second moment measure of point processes. Mathematische Operationsforschung und Statistik, series Statistics, 14, 63 – 71.
Ohser, J. and Stoyan, D. (1981) On the second-order and orientation analysis of planar stationary
point processes. Biometrical Journal 23, 523–533.
Ripley, B.D. (1977) Modelling spatial patterns (with discussion). Journal of the Royal Statistical
Society, Series B, 39, 172 – 212.
Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.
Stoyan, D, Kendall, W.S. and Mecke, J. (1995) Stochastic geometry and its applications. 2nd
edition. Springer Verlag.
Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical
statistics. John Wiley and Sons.
See Also
localK to extract individual summands in the K function.
pcf for the pair correlation.
Fest, Gest, Jest for alternative summary functions.
Kcross, Kdot, Kinhom, Kmulti for counterparts of the K function for multitype point patterns.
reduced.sample for the calculation of reduced sample estimators.

Kest.fft

635

Examples
X <- runifpoint(50)
K <- Kest(X)
K <- Kest(cells, correction="isotropic")
plot(K)
plot(K, main="K function for cells")
# plot the L function
plot(K, sqrt(iso/pi) ~ r)
plot(K, sqrt(./pi) ~ r, ylab="L(r)", main="L function for cells")

Kest.fft

K-function using FFT

Description
Estimates the reduced second moment function K(r) from a point pattern in a window of arbitrary
shape, using the Fast Fourier Transform.
Usage
Kest.fft(X, sigma, r=NULL, ..., breaks=NULL)
Arguments
X
sigma
r
...
breaks

The observed point pattern, from which an estimate of K(r) will be computed.
An object of class "ppp", or data in any format acceptable to as.ppp().
Standard deviation of the isotropic Gaussian smoothing kernel.
Optional. Vector of values for the argument r at which K(r) should be evaluated. There is a sensible default.
Arguments passed to as.mask determining the spatial resolution for the FFT
calculation.
This argument is for internal use only.

Details
This is an alternative to the function Kest for estimating the K function. It may be useful for very
large patterns of points.
Whereas Kest computes the distance between each pair of points analytically, this function discretises the point pattern onto a rectangular pixel raster and applies Fast Fourier Transform techniques
to estimate K(t). The hard work is done by the function Kmeasure.
The result is an approximation whose accuracy depends on the resolution of the pixel raster. The resolution is controlled by the arguments ..., or by setting the parameter npixel in spatstat.options.
Value
An object of class "fv" (see fv.object).
Essentially a data frame containing columns
r
border
theo

the vector of values of the argument r at which the function K has been estimated
the estimates of K(r) for these values of r
the theoretical value K(r) = πr2 for a stationary Poisson process

636

Kinhom

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
References
Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.
Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 1983.
Ohser, J. (1983) On estimators for the reduced second moment measure of point processes. Mathematische Operationsforschung und Statistik, series Statistics, 14, 63 – 71.
Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.
Stoyan, D, Kendall, W.S. and Mecke, J. (1995) Stochastic geometry and its applications. 2nd
edition. Springer Verlag.
Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical
statistics. John Wiley and Sons.
See Also
Kest, Kmeasure, spatstat.options
Examples
pp <- runifpoint(10000)
Kpp <- Kest.fft(pp, 0.01)
plot(Kpp)

Kinhom

Inhomogeneous K-function

Description
Estimates the inhomogeneous K function of a non-stationary point pattern.
Usage
Kinhom(X, lambda=NULL, ..., r = NULL, breaks = NULL,
correction=c("border", "bord.modif", "isotropic", "translate"),
renormalise=TRUE,
normpower=1,
update=TRUE,
leaveoneout=TRUE,
nlarge = 1000,
lambda2=NULL, reciplambda=NULL, reciplambda2=NULL,
diagonal=TRUE,
sigma=NULL, varcov=NULL,
ratio=FALSE)

Kinhom

637

Arguments
X

The observed data point pattern, from which an estimate of the inhomogeneous
K function will be computed. An object of class "ppp" or in a format recognised
by as.ppp()

lambda

Optional. Values of the estimated intensity function. Either a vector giving the
intensity values at the points of the pattern X, a pixel image (object of class "im")
giving the intensity values at all locations, a fitted point process model (object
of class "ppm" or "kppm") or a function(x,y) which can be evaluated to give
the intensity value at any location.

...

Extra arguments. Ignored if lambda is present. Passed to density.ppp if
lambda is omitted.

r

vector of values for the argument r at which the inhomogeneous K function
should be evaluated. Not normally given by the user; there is a sensible default.

breaks

This argument is for internal use only.

correction

A character vector containing any selection of the options "border", "bord.modif",
"isotropic", "Ripley", "translate", "translation", "none" or "best". It
specifies the edge correction(s) to be applied. Alternatively correction="all"
selects all options.

renormalise

Logical. Whether to renormalise the estimate. See Details.

normpower

Integer (usually either 1 or 2). Normalisation power. See Details.

update

Logical value indicating what to do when lambda is a fitted model (class "ppm",
"kppm" or "dppm"). If update=TRUE (the default), the model will first be refitted
to the data X (using update.ppm or update.kppm) before the fitted intensity is
computed. If update=FALSE, the fitted intensity of the model will be computed
without re-fitting it to X.

leaveoneout

Logical value (passed to density.ppp or fitted.ppm) specifying whether to
use a leave-one-out rule when calculating the intensity.

nlarge

Optional. Efficiency threshold. If the number of points exceeds nlarge, then
only the border correction will be computed, using a fast algorithm.

lambda2

Advanced use only. Matrix containing estimates of the products λ(xi )λ(xj ) of
the intensities at each pair of data points xi and xj .

reciplambda

Alternative to lambda. Values of the estimated reciprocal 1/λ of the intensity
function. Either a vector giving the reciprocal intensity values at the points of
the pattern X, a pixel image (object of class "im") giving the reciprocal intensity
values at all locations, or a function(x,y) which can be evaluated to give the
reciprocal intensity value at any location.

reciplambda2

Advanced use only. Alternative to lambda2. A matrix giving values of the
estimated reciprocal products 1/λ(xi )λ(xj ) of the intensities at each pair of
data points xi and xj .

diagonal

Do not use this argument.

sigma,varcov

Optional arguments passed to density.ppp to control the smoothing bandwidth, when lambda is estimated by kernel smoothing.

ratio

Logical. If TRUE, the numerator and denominator of each edge-corrected estimate will also be saved, for use in analysing replicated point patterns.

638

Kinhom

Details
This computes a generalisation of the K function for inhomogeneous point patterns, proposed by
Baddeley, Møller and Waagepetersen (2000).
The “ordinary” K function (variously known as the reduced second order moment function and
Ripley’s K function), is described under Kest. It is defined only for stationary point processes.
The inhomogeneous K function Kinhom (r) is a direct generalisation to nonstationary point processes. Suppose x is a point process with non-constant intensity λ(u) at each location u. Define
Kinhom (r) to be the expected value, given that u is a point of x, of the sum of all terms 1/λ(xj ) over
all points xj in the process separated from u by a distance less than r. This reduces to the ordinary
K function if λ() is constant. If x is an inhomogeneous Poisson process with intensity function
λ(u), then Kinhom (r) = πr2 .
Given a point pattern dataset, the inhomogeneous K function can be estimated essentially by summing the values 1/(λ(xi )λ(xj )) for all pairs of points xi , xj separated by a distance less than r.
This allows us to inspect a point pattern for evidence of interpoint interactions after allowing for
spatial inhomogeneity of the pattern. Values Kinhom (r) > πr2 are suggestive of clustering.
The argument lambda should supply the (estimated) values of the intensity function λ. It may be
either
a numeric vector containing the values of the intensity function at the points of the pattern X.
a pixel image (object of class "im") assumed to contain the values of the intensity function at all
locations in the window.
a fitted point process model (object of class "ppm", "kppm" or "dppm") whose fitted trend can be
used as the fitted intensity. (If update=TRUE the model will first be refitted to the data X before
the trend is computed.)
a function which can be evaluated to give values of the intensity at any locations.
omitted: if lambda is omitted, then it will be estimated using a ‘leave-one-out’ kernel smoother.
If lambda is a numeric vector, then its length should be equal to the number of points in the pattern
X. The value lambda[i] is assumed to be the the (estimated) value of the intensity λ(xi ) for the
point xi of the pattern X. Each value must be a positive number; NA’s are not allowed.
If lambda is a pixel image, the domain of the image should cover the entire window of the point
pattern. If it does not (which may occur near the boundary because of discretisation error), then
the missing pixel values will be obtained by applying a Gaussian blur to lambda using blur, then
looking up the values of this blurred image for the missing locations. (A warning will be issued in
this case.)
If lambda is a function, then it will be evaluated in the form lambda(x,y) where x and y are vectors
of coordinates of the points of X. It should return a numeric vector with length equal to the number
of points in X.
If lambda is omitted, then it will be estimated using a ‘leave-one-out’ kernel smoother, as described
in Baddeley, Møller and Waagepetersen (2000). The estimate lambda[i] for the point X[i] is computed by removing X[i] from the point pattern, applying kernel smoothing to the remaining points
using density.ppp, and evaluating the smoothed intensity at the point X[i]. The smoothing kernel
bandwidth is controlled by the arguments sigma and varcov, which are passed to density.ppp
along with any extra arguments.
Edge corrections are used to correct bias in the estimation of Kinhom . Each edge-corrected estimate
of Kinhom (r) is of the form
b inhom (r) = (1/A)
K

X X 1{dij ≤ r}e(xi , xj , r)
λ(xi )λ(xj )
i
j

Kinhom

639

where A is a constant denominator, dij is the distance between points xi and xj , and e(xi , xj , r) is
an edge correction factor. For the ‘border’ correction,
1(bi > r)
j 1(bj > r)/λ(xj )

e(xi , xj , r) = P

where bi is the distance from xi to the boundary of the window. For the ‘modified border’ correction,
e(xi , xj , r) =

1(bi > r)
area(W r)

where W r is the eroded window obtained by trimming a margin of width r from the border of
the original window. For the ‘translation’ correction,
e(xi , xj , r) =

1
area(W ∩ (W + (xj − xi )))

and for the ‘isotropic’ correction,
e(xi , xj , r) =

1
area(W )g(xi , xj )

where g(xi , xj ) is the fraction of the circumference of the circle with centre xi and radius ||xi −xj ||
which lies inside the window.
If renormalise=TRUE P
(the default), then the estimates described above are multiplied by cnormpower
where c = area(W )/ (1/λ(xi )). This rescaling reduces the variability and bias of the estimate
in small samples and in cases of very strong inhomogeneity. The default value of normpower is 1
(for consistency with previous versions of spatstat)
P but the most sensible value is 2, which would
correspond to rescaling the lambda values so that (1/λ(xi )) = area(W ).
If the point pattern X contains more than about 1000 points, the isotropic and translation edge corrections can be computationally prohibitive. The computations for the border method are much faster,
and are statistically efficient when there are large numbers of points. Accordingly, if the number of
points in X exceeds the threshold nlarge, then only the border correction will be computed. Setting nlarge=Inf or correction="best" will prevent this from happening. Setting nlarge=0 is
equivalent to selecting only the border correction with correction="border".
The pair correlation function can also be applied to the result of Kinhom; see pcf.
Value
An object of class "fv" (see fv.object).
Essentially a data frame containing at least the following columns,
r

the vector of values of the argument r at which Kinhom (r) has been estimated

theo

vector of values of πr2 , the theoretical value of Kinhom (r) for an inhomogeneous
Poisson process

and containing additional columns according to the choice specified in the correction argument. The additional columns are named border, trans and iso and give the estimated values
of Kinhom (r) using the border correction, translation correction, and Ripley isotropic correction,
respectively.
If ratio=TRUE then the return value also has two attributes called "numerator" and "denominator"
which are "fv" objects containing the numerators and denominators of each estimate of Kinhom (r).

640

Kinhom

Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Baddeley, A., Møller, J. and Waagepetersen, R. (2000) Non- and semiparametric estimation of
interaction in inhomogeneous point patterns. Statistica Neerlandica 54, 329–350.
See Also
Kest, pcf
Examples
# inhomogeneous pattern of maples
X <- unmark(split(lansing)$maple)
# (1) intensity function estimated by model-fitting
# Fit spatial trend: polynomial in x and y coordinates
fit <- ppm(X, ~ polynom(x,y,2), Poisson())
# (a) predict intensity values at points themselves,
#
obtaining a vector of lambda values
lambda <- predict(fit, locations=X, type="trend")
# inhomogeneous K function
Ki <- Kinhom(X, lambda)
plot(Ki)
# (b) predict intensity at all locations,
#
obtaining a pixel image
lambda <- predict(fit, type="trend")
Ki <- Kinhom(X, lambda)
plot(Ki)
# (2) intensity function estimated by heavy smoothing
Ki <- Kinhom(X, sigma=0.1)
plot(Ki)
# (3) simulated data: known intensity function
lamfun <- function(x,y) { 50 + 100 * x }
# inhomogeneous Poisson process
Y <- rpoispp(lamfun, 150, owin())
# inhomogeneous K function
Ki <- Kinhom(Y, lamfun)
plot(Ki)
# How to make simulation envelopes:
#
Example shows method (2)
## Not run:
smo <- density.ppp(X, sigma=0.1)
Ken <- envelope(X, Kinhom, nsim=99,
simulate=expression(rpoispp(smo)),
sigma=0.1, correction="trans")
plot(Ken)
## End(Not run)

km.rs

641

km.rs

Kaplan-Meier and Reduced Sample Estimator using Histograms

Description
Compute the Kaplan-Meier and Reduced Sample estimators of a survival time distribution function,
using histogram techniques
Usage
km.rs(o, cc, d, breaks)
Arguments
o
cc
d
breaks

vector of observed survival times
vector of censoring times
vector of non-censoring indicators
Vector of breakpoints to be used to form histograms.

Details
This function is needed mainly for internal use in spatstat, but may be useful in other applications
where you want to form the Kaplan-Meier estimator from a huge dataset.
Suppose Ti are the survival times of individuals i = 1, . . . , M with unknown distribution function
F (t) which we wish to estimate. Suppose these times are right-censored by random censoring
times Ci . Thus the observations consist of right-censored survival times T̃i = min(Ti , Ci ) and
non-censoring indicators Di = 1{Ti ≤ Ci } for each i.
The arguments to this function are vectors o, cc, d of observed values of T̃i , Ci and Di respectively.
The function computes histograms and forms the reduced-sample and Kaplan-Meier estimates of
F (t) by invoking the functions kaplan.meier and reduced.sample. This is efficient if the lengths
of o, cc, d (i.e. the number of observations) is large.
The vectors km and hazard returned by kaplan.meier are (histogram approximations to) the
Kaplan-Meier estimator of F (t) and its hazard rate λ(t). Specifically, km[k] is an estimate of
F(breaks[k+1]), and lambda[k] is an estimate of the average of λ(t) over the interval (breaks[k],breaks[k+1]).
This approximation is exact only if the survival times are discrete and the histogram breaks are fine
enough to ensure that each interval (breaks[k],breaks[k+1]) contains only one possible value
of the survival time.
The vector rs is the reduced-sample estimator, rs[k] being the reduced sample estimate of F(breaks[k+1]).
This value is exact, i.e. the use of histograms does not introduce any approximation error in the
reduced-sample estimator.
Value
A list with five elements
rs
km
hazard
r
breaks

Reduced-sample estimate of the survival time c.d.f. F (t)
Kaplan-Meier estimate of the survival time c.d.f. F (t)
corresponding Nelson-Aalen estimate of the hazard rate λ(t)
values of t for which F (t) is estimated
the breakpoints vector

642

Kmark

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
reduced.sample, kaplan.meier

Kmark

Mark-Weighted K Function

Description
Estimates the mark-weighted K function of a marked point pattern.
Usage
Kmark(X, f = NULL, r = NULL,
correction = c("isotropic", "Ripley", "translate"), ...,
f1 = NULL, normalise = TRUE, returnL = FALSE, fargs = NULL)
markcorrint(X, f = NULL, r = NULL,
correction = c("isotropic", "Ripley", "translate"), ...,
f1 = NULL, normalise = TRUE, returnL = FALSE, fargs = NULL)
Arguments
X

The observed point pattern. An object of class "ppp" or something acceptable
to as.ppp.

f

Optional. Test function f used in the definition of the mark correlation function.
An R function with at least two arguments. There is a sensible default.

r

Optional. Numeric vector. The values of the argument r at which the mark
correlation function kf (r) should be evaluated. There is a sensible default.

correction

A character vector containing any selection of the options "isotropic", "Ripley"
or "translate". It specifies the edge correction(s) to be applied. Alternatively
correction="all" selects all options.

...

Ignored.

f1

An alternative to f. If this argument is given, then f is assumed to take the form
f (u, v) = f1 (u)f1 (v).

normalise

If normalise=FALSE, compute only the numerator of the expression for the
mark correlation.

returnL

Compute the analogue of the K-function if returnL=FALSE or the analogue of
the L-function if returnL=TRUE.

fargs

Optional. A list of extra arguments to be passed to the function f or f1.

Kmark

643

Details
The functions Kmark and markcorrint are identical. (Eventually markcorrint will be deprecated.)
The mark-weighted K function Kf (r) of a marked point process (Penttinen et al, 1992) is a generalisation of Ripley’s K function, in which the contribution from each pair of points is weighted by
a function of their marks. If the marks of the two points are m1 , m2 then the weight is proportional
to f (m1 , m2 ) where f is a specified test function.
The mark-weighted K function is defined so that
Cf (r)
E[f (M1 , M2 )]

λKf (r) =
where
"
Cf (r) = E

#
X

f (m(u), m(x))10 < ||u − x|| ≤ r u ∈ X

x∈X

for any spatial location u taken to be a typical point of the point process X. Here ||u − x|| is
the euclidean distance between u and x, so that the sum is taken over all random points x that
lie within a distance r of the point u. The function Cf (r) is the unnormalised mark-weighted K
function. To obtain Kf (r) we standardise Cf (r) by dividing by E[f (M1 , M2 )], the expected value
of f (M1 , M2 ) when M1 and M2 are independent random marks with the same distribution as the
marks in the point process.
Under the hypothesis of random labelling, the mark-weighted K function is equal to Ripley’s K
function, Kf (r) = K(r).
The mark-weighted K function is sometimes called the mark correlation integral because it is
related to the mark correlation function kf (r) and the pair correlation function g(r) by
Z
Kf (r) = 2π

r

skf (s) g(s) ds
0

See markcorr for a definition of the mark correlation function.
Given a marked point pattern X, this command computes edge-corrected estimates of the markweighted K function. If returnL=FALSE then the estimated function Kf (r) is returned; otherwise
the function
q
Lf (r) =

Kf (r)/π

is returned.
Value
An object of class "fv" (see fv.object).
Essentially a data frame containing numeric columns
r

the values of the argument r at which the mark correlation integral Kf (r) has
been estimated

theo

the theoretical value of Kf (r) when the marks attached to different points are
independent, namely πr2

together with a column or columns named "iso" and/or "trans", according to the selected edge
corrections. These columns contain estimates of the mark-weighted K function Kf (r) obtained by
the edge corrections named (if returnL=FALSE).

644

Kmeasure

Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Penttinen, A., Stoyan, D. and Henttonen, H. M. (1992) Marked point processes in forest statistics.
Forest Science 38 (1992) 806-824.
Illian, J., Penttinen, A., Stoyan, H. and Stoyan, D. (2008) Statistical analysis and modelling of
spatial point patterns. Chichester: John Wiley.
See Also
markcorr to estimate the mark correlation function.
Examples
# CONTINUOUS-VALUED MARKS:
# (1) Spruces
# marks represent tree diameter
# mark correlation function
ms <- Kmark(spruces)
plot(ms)
# (2) simulated data with independent marks
X <- rpoispp(100)
X <- X %mark% runif(npoints(X))
Xc <- Kmark(X)
plot(Xc)
#
#
#
M

MULTITYPE DATA:
Hughes' amacrine data
Cells marked as 'on'/'off'
<- Kmark(amacrine, function(m1,m2) {m1==m2},
correction="translate")
plot(M)

Kmeasure

Reduced Second Moment Measure

Description
Estimates the reduced second moment measure κ from a point pattern in a window of arbitrary
shape.
Usage
Kmeasure(X, sigma, edge=TRUE, ..., varcov=NULL)

Kmeasure

645

Arguments
X

The observed point pattern, from which an estimate of κ will be computed. An
object of class "ppp", or data in any format acceptable to as.ppp().

sigma

Standard deviation σ of the Gaussian smoothing kernel. Incompatible with
varcov.

edge

Logical value indicating whether an edge correction should be applied.

...

Arguments passed to as.mask controlling the pixel resolution.

varcov

Variance-covariance matrix of the Gaussian smoothing kernel. Incompatible
with sigma.

Details
Given a point pattern dataset, this command computes an estimate of the reduced second moment
measure κ of the point process. The result is a pixel image whose pixel values are estimates of the
density of the reduced second moment measure.
The reduced second moment measure κ can be regarded as a generalisation of the more familiar
K-function. An estimate of κ derived from a spatial point pattern dataset can be useful in exploratory data analysis. Its advantage over the K-function is that it is also sensitive to anisotropy
and directional effects.
In a nutshell, the command Kmeasure computes a smoothed version of the Fry plot. As explained
under fryplot, the Fry plot is a scatterplot of the vectors joining all pairs of points in the pattern.
The reduced second moment measure is (essentially) defined as the average of the Fry plot over
different realisations of the point process. The command Kmeasure effectively smooths the Fry plot
of a dataset to obtain an estimate of the reduced second moment measure.
In formal terms, the reduced second moment measure κ of a stationary point process X is a measure
defined on the two-dimensional plane such that, for a ‘typical’ point x of the process, the expected
number of other points y of the process such that the vector y − x lies in a region A, equals λκ(A).
Here λ is the intensity of the process, i.e. the expected number of points of X per unit area.
The K-function is a special case. The function value K(t) is the value of the reduced second
moment measure for the disc of radius t centred at the origin; that is, K(t) = κ(b(0, t)).
The command Kmeasure computes an estimate of κ from a point pattern dataset X, which is assumed
to be a realisation of a stationary point process, observed inside a known, bounded window. Marks
are ignored.
The algorithm approximates the point pattern and its window by binary pixel images, introduces a
Gaussian smoothing kernel and uses the Fast Fourier Transform fft to form a density estimate of
κ. The calculation corresponds to the edge correction known as the “translation correction”.
The Gaussian smoothing kernel may be specified by either of the arguments sigma or varcov. If
sigma is a single number, this specifies an isotropic Gaussian kernel with standard deviation sigma
on each coordinate axis. If sigma is a vector of two numbers, this specifies a Gaussian kernel with
standard deviation sigma[1] on the x axis, standard deviation sigma[2] on the y axis, and zero
correlation between the x and y axes. If varcov is given, this specifies the variance-covariance
matrix of the Gaussian kernel. There do not seem to be any well-established rules for selecting the
smoothing kernel in this context.
The density estimate of κ is returned in the form of a real-valued pixel image. Pixel values are
estimates of the normalised second moment density at the centre of the pixel. (The uniform Poisson
process would have values identically equal to 1.) The image x and y coordinates are on the same
scale as vector displacements in the original point pattern window. The point x=0, y=0 corresponds
to the ‘typical point’. A peak in the image near (0,0) suggests clustering; a dip in the image near
(0,0) suggests inhibition; peaks or dips at other positions suggest possible periodicity.

646

Kmeasure
If desired, the value of κ(A) for a region A can be estimated by computing the integral of the
pixel image over the domain A, i.e.\ summing the pixel values and multiplying by pixel area, using
integral.im. One possible application is to compute anisotropic counterparts of the K-function
(in which the disc of radius t is replaced by another shape). See Examples.

Value
A real-valued pixel image (an object of class "im", see im.object) whose pixel values are estimates
of the density of the reduced second moment measure at each location.
Warning
Some writers use the term reduced second moment measure when they mean the K-function. This
has caused confusion.
As originally defined, the reduced second moment measure is a measure, obtained by modifying the
second moment measure, while the K-function is a function obtained by evaluating this measure
for discs of increasing radius. In spatstat, the K-function is computed by Kest and the reduced
second moment measure is computed by Kmeasure.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Stoyan, D, Kendall, W.S. and Mecke, J. (1995) Stochastic geometry and its applications. 2nd
edition. Springer Verlag.
Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical
statistics. John Wiley and Sons.
See Also
Kest, fryplot, spatstat.options, integral.im, im.object
Examples
data(cells)
plot(Kmeasure(cells, 0.05))
# shows pronounced dip around origin consistent with strong inhibition
data(redwood)
plot(Kmeasure(redwood, 0.03), col=grey(seq(1,0,length=32)))
# shows peaks at several places, reflecting clustering and ?periodicity
M <- Kmeasure(cells, 0.05)
# evaluate measure on a sector
W <- Window(M)
ang <- as.im(atan2, W)
rad <- as.im(function(x,y){sqrt(x^2+y^2)}, W)
sector <- solutionset(ang > 0 & ang < 1 & rad < 0.6)
integral.im(M[sector, drop=FALSE])

Kmodel

647

Kmodel

K Function or Pair Correlation Function of a Point Process Model

Description
Returns the theoretical K function or the pair correlation function of a point process model.
Usage
Kmodel(model, ...)
pcfmodel(model, ...)
Arguments
model

A fitted point process model of some kind.

...

Ignored.

Details
For certain types of point process models, it is possible to write down a mathematical expression
for the K function or the pair correlation function of the model.
The functions Kmodel and pcfmodel give the theoretical K-function and the theoretical pair correlation function for a point process model that has been fitted to data.
The functions Kmodel and pcfmodel are generic, with methods for the classes "kppm" (cluster
processes and Cox processes) and "ppm" (Gibbs processes).
The return value is a function in the R language, which takes one argument r. Evaluation of this
function, on a numeric vector r, yields values of the desired K function or pair correlation function
at these distance values.
Value
A function in the R language, which takes one argument r.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
Kest or pcf to estimate the K function or pair correlation function nonparametrically from data.
Kmodel.kppm for the method for cluster processes and Cox processes.
Kmodel.ppm for the method for Gibbs processes.

648

Kmodel.dppm

Kmodel.dppm

K-function or Pair Correlation Function of a Determinantal Point Process Model

Description
Returns the theoretical K-function or theoretical pair correlation function of a determinantal point
process model as a function of one argument r.

Usage
## S3 method for class 'dppm'
Kmodel(model, ...)
## S3 method for class 'dppm'
pcfmodel(model, ...)
## S3 method for class 'detpointprocfamily'
Kmodel(model, ...)
## S3 method for class 'detpointprocfamily'
pcfmodel(model, ...)

Arguments
model

Model of class "detpointprocfamily" or "dppm".

...

Ignored (not quite true – there is some undocumented internal use)

Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 

Examples
model <- dppMatern(lambda=100, alpha=.01, nu=1, d=2)
KMatern <- Kmodel(model)
pcfMatern <- pcfmodel(model)
plot(KMatern, xlim = c(0,0.05))
plot(pcfMatern, xlim = c(0,0.05))

Kmodel.kppm

649

Kmodel.kppm

K Function or Pair Correlation Function of Cluster Model or Cox
model

Description
Returns the theoretical K function or the pair correlation function of a cluster point process model
or Cox point process model.
Usage
## S3 method for class 'kppm'
Kmodel(model, ...)
## S3 method for class 'kppm'
pcfmodel(model, ...)
Arguments
model

A fitted cluster point process model (object of class "kppm") typically obtained
from the model-fitting algorithm kppm.

...

Ignored.

Details
For certain types of point process models, it is possible to write down a mathematical expression
for the K function or the pair correlation function of the model. In particular this is possible for a
fitted cluster point process model (object of class "kppm" obtained from kppm).
The functions Kmodel and pcfmodel are generic. The functions documented here are the methods
for the class "kppm".
The return value is a function in the R language, which takes one argument r. Evaluation of this
function, on a numeric vector r, yields values of the desired K function or pair correlation function
at these distance values.
Value
A function in the R language, which takes one argument r.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
Kest or pcf to estimate the K function or pair correlation function nonparametrically from data.
kppm to fit cluster models.
Kmodel for the generic functions.
Kmodel.ppm for the method for Gibbs processes.

650

Kmodel.ppm

Examples
data(redwood)
fit <- kppm(redwood, ~x, "MatClust")
K <- Kmodel(fit)
K(c(0.1, 0.2))
curve(K(x), from=0, to=0.25)

Kmodel.ppm

K Function or Pair Correlation Function of Gibbs Point Process model

Description
Returns the theoretical K function or the pair correlation function of a fitted Gibbs point process
model.
Usage
## S3 method for class 'ppm'
Kmodel(model, ...)
## S3 method for class 'ppm'
pcfmodel(model, ...)
Arguments
model

A fitted Poisson or Gibbs point process model (object of class "ppm") typically
obtained from the model-fitting algorithm ppm.

...

Ignored.

Details
This function computes an approximation to the K function or the pair correlation function of a
Gibbs point process.
The functions Kmodel and pcfmodel are generic. The functions documented here are the methods
for the class "ppm".
The approximation is only available for stationary pairwise-interaction models. It uses the second
order Poisson-saddlepoint approximation (Baddeley and Nair, 2012b) which is a combination of
the Poisson-Boltzmann-Emden and Percus-Yevick approximations.
The return value is a function in the R language, which takes one argument r. Evaluation of this
function, on a numeric vector r, yields values of the desired K function or pair correlation function
at these distance values.
Value
A function in the R language, which takes one argument r.
Author(s)
Adrian Baddeley 
and Gopalan Nair.

Kmulti

651

References
Baddeley, A. and Nair, G. (2012a) Fast approximation of the intensity of Gibbs point processes.
Electronic Journal of Statistics 6 1155–1169.
Baddeley, A. and Nair, G. (2012b) Approximating the moments of a spatial point process. Stat 1,
1, 18–30. doi: 10.1002/sta4.5
See Also
Kest or pcf to estimate the K function or pair correlation function nonparametrically from data.
ppm to fit Gibbs models.
Kmodel for the generic functions.
Kmodel.kppm for the method for cluster/Cox processes.
Examples
fit <- ppm(swedishpines, ~1, Strauss(8))
p <- pcfmodel(fit)
K <- Kmodel(fit)
p(6)
K(8)
curve(K(x), from=0, to=15)

Kmulti

Marked K-Function

Description
For a marked point pattern, estimate the multitype K function which counts the expected number
of points of subset J within a given distance from a typical point in subset I.
Usage
Kmulti(X, I, J, r=NULL, breaks=NULL, correction, ..., ratio=FALSE)
Arguments
X

The observed point pattern, from which an estimate of the multitype K function
KIJ (r) will be computed. It must be a marked point pattern. See under Details.

I

Subset index specifying the points of X from which distances are measured. See
Details.

J

Subset index specifying the points in X to which distances are measured. See
Details.

r

numeric vector. The values of the argument r at which the multitype K function
KIJ (r) should be evaluated. There is a sensible default. First-time users are
strongly advised not to specify this argument. See below for important conditions on r.

breaks

This argument is for internal use only.

652

Kmulti
correction

A character vector containing any selection of the options "border", "bord.modif",
"isotropic", "Ripley", "translate", "translation", "none" or "best". It
specifies the edge correction(s) to be applied. Alternatively correction="all"
selects all options.

...

Ignored.

ratio

Logical. If TRUE, the numerator and denominator of each edge-corrected estimate will also be saved, for use in analysing replicated point patterns.

Details
The function Kmulti generalises Kest (for unmarked point patterns) and Kdot and Kcross (for
multitype point patterns) to arbitrary marked point patterns.
Suppose XI , XJ are subsets, possibly overlapping, of a marked point process. The multitype K
function is defined so that λJ KIJ (r) equals the expected number of additional random points of
XJ within a distance r of a typical point of XI . Here λJ is the intensity of XJ i.e. the expected
number of points of XJ per unit area. The function KIJ is determined by the second order moment
properties of X.
The argument X must be a point pattern (object of class "ppp") or any data that are acceptable to
as.ppp.
The arguments I and J specify two subsets of the point pattern. They may be any type of subset
indices, for example, logical vectors of length equal to npoints(X), or integer vectors with entries
in the range 1 to npoints(X), or negative integer vectors.
Alternatively, I and J may be functions that will be applied to the point pattern X to obtain index
vectors. If I is a function, then evaluating I(X) should yield a valid subset index. This option is
useful when generating simulation envelopes using envelope.
The argument r is the vector of values for the distance r at which KIJ (r) should be evaluated. It is
also used to determine the breakpoints (in the sense of hist) for the computation of histograms of
distances.
First-time users would be strongly advised not to specify r. However, if it is specified, r must
satisfy r[1] = 0, and max(r) must be larger than the radius of the largest disc contained in the
window.
This algorithm assumes that X can be treated as a realisation of a stationary (spatially homogeneous)
random spatial point process in the plane, observed through a bounded window. The window (which
is specified in X as Window(X)) may have arbitrary shape.
Biases due to edge effects are treated in the same manner as in Kest. The edge corrections implemented here are
border the border method or “reduced sample” estimator (see Ripley, 1988). This is the least
efficient (statistically) and the fastest to compute. It can be computed for a window of arbitrary
shape.
isotropic/Ripley Ripley’s isotropic correction (see Ripley, 1988; Ohser, 1983). This is currently
implemented only for rectangular and polygonal windows.
translate Translation correction (Ohser, 1983). Implemented for all window geometries.
The pair correlation function pcf can also be applied to the result of Kmulti.
Value
An object of class "fv" (see fv.object).
Essentially a data frame containing numeric columns

Kmulti

653

r
theo

the values of the argument r at which the function KIJ (r) has been estimated
the theoretical value of KIJ (r) for a marked Poisson process, namely πr2

together with a column or columns named "border", "bord.modif", "iso" and/or "trans", according to the selected edge corrections. These columns contain estimates of the function KIJ (r)
obtained by the edge corrections named.
If ratio=TRUE then the return value also has two attributes called "numerator" and "denominator"
which are "fv" objects containing the numerators and denominators of each estimate of K(r).
Warnings
The function KIJ is not necessarily differentiable.
The border correction (reduced sample) estimator of KIJ used here is pointwise approximately
unbiased, but need not be a nondecreasing function of r, while the true KIJ must be nondecreasing.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
References
Cressie, N.A.C. Statistics for spatial data. John Wiley and Sons, 1991.
Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 1983.
Diggle, P. J. (1986). Displaced amacrine cells in the retina of a rabbit : analysis of a bivariate spatial
point pattern. J. Neurosci. Meth. 18, 115–125.
Harkness, R.D and Isham, V. (1983) A bivariate spatial point pattern of ants’ nests. Applied Statistics 32, 293–303
Lotwick, H. W. and Silverman, B. W. (1982). Methods for analysing spatial processes of several
types of points. J. Royal Statist. Soc. Ser. B 44, 406–413.
Ripley, B.D. Statistical inference for spatial processes. Cambridge University Press, 1988.
Stoyan, D, Kendall, W.S. and Mecke, J. Stochastic geometry and its applications. 2nd edition.
Springer Verlag, 1995.
Van Lieshout, M.N.M. and Baddeley, A.J. (1999) Indices of dependence between types in multivariate point patterns. Scandinavian Journal of Statistics 26, 511–532.
See Also
Kcross, Kdot, Kest, pcf
Examples
# Longleaf Pine data: marks represent diameter
trees <- longleaf
K <- Kmulti(trees, marks(trees) <= 15, marks(trees) >= 25)
plot(K)
# functions determining subsets
f1 <- function(X) { marks(X) <= 15 }
f2 <- function(X) { marks(X) >= 15 }
K <- Kmulti(trees, f1, f2)

654

Kmulti.inhom

Kmulti.inhom

Inhomogeneous Marked K-Function

Description
For a marked point pattern, estimate the inhomogeneous version of the multitype K function which
counts the expected number of points of subset J within a given distance from a typical point in
subset I, adjusted for spatially varying intensity.
Usage
Kmulti.inhom(X, I, J, lambdaI=NULL, lambdaJ=NULL,
...,
r=NULL, breaks=NULL,
correction=c("border", "isotropic", "Ripley", "translate"),
lambdaIJ=NULL,
sigma=NULL, varcov=NULL,
lambdaX=NULL, update=TRUE, leaveoneout=TRUE)
Arguments
X

The observed point pattern, from which an estimate of the inhomogeneous multitype K function KIJ (r) will be computed. It must be a marked point pattern.
See under Details.

I

Subset index specifying the points of X from which distances are measured. See
Details.

J

Subset index specifying the points in X to which distances are measured. See
Details.

lambdaI

Optional. Values of the estimated intensity of the sub-process X[I]. Either a
pixel image (object of class "im"), a numeric vector containing the intensity
values at each of the points in X[I], a fitted point process model (object of class
"ppm" or "kppm" or "dppm"), or a function(x,y) which can be evaluated to
give the intensity value at any location,

lambdaJ

Optional. Values of the estimated intensity of the sub-process X[J]. Either a
pixel image (object of class "im"), a numeric vector containing the intensity
values at each of the points in X[J], a fitted point process model (object of class
"ppm" or "kppm" or "dppm"), or a function(x,y) which can be evaluated to
give the intensity value at any location.

...

Ignored.

r

Optional. Numeric vector. The values of the argument r at which the multitype
K function KIJ (r) should be evaluated. There is a sensible default. First-time
users are strongly advised not to specify this argument. See below for important
conditions on r.

breaks

This argument is for internal use only.

correction

A character vector containing any selection of the options "border", "bord.modif",
"isotropic", "Ripley", "translate", "none" or "best". It specifies the
edge correction(s) to be applied. Alternatively correction="all" selects all
options.

Kmulti.inhom

655

lambdaIJ

Optional. A matrix containing estimates of the product of the intensities lambdaI
and lambdaJ for each pair of points, the first point belonging to subset I and the
second point to subset J.

sigma,varcov

Optional arguments passed to density.ppp to control the smoothing bandwidth, when lambda is estimated by kernel smoothing.

lambdaX

Optional. Values of the intensity for all points of X. Either a pixel image (object
of class "im"), a numeric vector containing the intensity values at each of the
points in X, a fitted point process model (object of class "ppm" or "kppm" or
"dppm"), or a function(x,y) which can be evaluated to give the intensity value
at any location. If present, this argument overrides both lambdaI and lambdaJ.

update

Logical value indicating what to do when lambdaI, lambdaJ or lambdaX is a
fitted point process model (class "ppm", "kppm" or "dppm"). If update=TRUE
(the default), the model will first be refitted to the data X (using update.ppm
or update.kppm) before the fitted intensity is computed. If update=FALSE, the
fitted intensity of the model will be computed without re-fitting it to X.

leaveoneout

Logical value (passed to density.ppp or fitted.ppm) specifying whether to
use a leave-one-out rule when calculating the intensity.

Details
The function Kmulti.inhom is the counterpart, for spatially-inhomogeneous marked point patterns,
of the multitype K function Kmulti.
Suppose X is a marked point process, with marks of any kind. Suppose XI , XJ are two subprocesses, possibly overlapping. Typically XI would consist of those points of X whose marks
lie in a specified range of mark values, and similarly for XJ . Suppose that λI (u), λJ (u) are the
spatially-varying intensity functions of XI and XJ respectively. Consider all the pairs of points
(u, v) in the point process X such that the first point u belongs to XI , the second point v belongs
to XJ , and the distance between u and v is less than a specified distance r. Give this pair (u, v) the
numerical weight 1/(λI (u)λJ (u)). Calculate the sum of these weights over all pairs of points as
described. This sum (after appropriate edge-correction and normalisation) is the estimated inhomogeneous multitype K function.
The argument X must be a point pattern (object of class "ppp") or any data that are acceptable to
as.ppp.
The arguments I and J specify two subsets of the point pattern. They may be any type of subset
indices, for example, logical vectors of length equal to npoints(X), or integer vectors with entries
in the range 1 to npoints(X), or negative integer vectors.
Alternatively, I and J may be functions that will be applied to the point pattern X to obtain index
vectors. If I is a function, then evaluating I(X) should yield a valid subset index. This option is
useful when generating simulation envelopes using envelope.
The argument lambdaI supplies the values of the intensity of the sub-process identified by index I.
It may be either
a pixel image (object of class "im") which gives the values of the intensity of X[I] at all locations
in the window containing X;
a numeric vector containing the values of the intensity of X[I] evaluated only at the data points
of X[I]. The length of this vector must equal the number of points in X[I].
a function of the form function(x,y) which can be evaluated to give values of the intensity at
any locations.

656

Kmulti.inhom
a fitted point process model (object of class "ppm", "kppm" or "dppm") whose fitted trend can be
used as the fitted intensity. (If update=TRUE the model will first be refitted to the data X before
the trend is computed.)
omitted: if lambdaI is omitted then it will be estimated using a leave-one-out kernel smoother.
If lambdaI is omitted, then it will be estimated using a ‘leave-one-out’ kernel smoother, as described
in Baddeley, Møller and Waagepetersen (2000). The estimate of lambdaI for a given point is
computed by removing the point from the point pattern, applying kernel smoothing to the remaining
points using density.ppp, and evaluating the smoothed intensity at the point in question. The
smoothing kernel bandwidth is controlled by the arguments sigma and varcov, which are passed to
density.ppp along with any extra arguments.
Similarly lambdaJ supplies the values of the intensity of the sub-process identified by index J.
Alternatively if the argument lambdaX is given, then it specifies the intensity values for all points of
X, and the arguments lambdaI, lambdaJ will be ignored.
The argument r is the vector of values for the distance r at which KIJ (r) should be evaluated. It is
also used to determine the breakpoints (in the sense of hist) for the computation of histograms of
distances.
First-time users would be strongly advised not to specify r. However, if it is specified, r must
satisfy r[1] = 0, and max(r) must be larger than the radius of the largest disc contained in the
window.
Biases due to edge effects are treated in the same manner as in Kinhom. The edge corrections
implemented here are
border the border method or “reduced sample” estimator (see Ripley, 1988). This is the least
efficient (statistically) and the fastest to compute. It can be computed for a window of arbitrary
shape.
isotropic/Ripley Ripley’s isotropic correction (see Ripley, 1988; Ohser, 1983). This is currently
implemented only for rectangular windows.
translate Translation correction (Ohser, 1983). Implemented for all window geometries.
The pair correlation function pcf can also be applied to the result of Kmulti.inhom.

Value
An object of class "fv" (see fv.object).
Essentially a data frame containing numeric columns
r
theo

the values of the argument r at which the function KIJ (r) has been estimated
the theoretical value of KIJ (r) for a marked Poisson process, namely πr2

together with a column or columns named "border", "bord.modif", "iso" and/or "trans", according to the selected edge corrections. These columns contain estimates of the function KIJ (r)
obtained by the edge corrections named.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Baddeley, A., Møller, J. and Waagepetersen, R. (2000) Non- and semiparametric estimation of
interaction in inhomogeneous point patterns. Statistica Neerlandica 54, 329–350.

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657

See Also
Kmulti, Kdot.inhom, Kcross.inhom, pcf
Examples
# Finnish Pines data: marked by diameter and height
plot(finpines, which.marks="height")
II <- (marks(finpines)$height <= 2)
JJ <- (marks(finpines)$height > 3)
K <- Kmulti.inhom(finpines, II, JJ)
plot(K)
# functions determining subsets
f1 <- function(X) { marks(X)$height <= 2 }
f2 <- function(X) { marks(X)$height > 3 }
K <- Kmulti.inhom(finpines, f1, f2)

kppm

Fit Cluster or Cox Point Process Model

Description
Fit a homogeneous or inhomogeneous cluster process or Cox point process model to a point pattern.
Usage
kppm(X, ...)
## S3 method for class 'formula'
kppm(X,
clusters = c("Thomas","MatClust","Cauchy","VarGamma","LGCP"),
...,
data=NULL)
## S3 method for class 'ppp'
kppm(X,
trend = ~1,
clusters = c("Thomas","MatClust","Cauchy","VarGamma","LGCP"),
data = NULL,
...,
covariates=data,
subset,
method = c("mincon", "clik2", "palm"),
improve.type = c("none", "clik1", "wclik1", "quasi"),
improve.args = list(),
weightfun=NULL,
control=list(),
algorithm="Nelder-Mead",
statistic="K",
statargs=list(),
rmax = NULL,
covfunargs=NULL,

658

kppm
use.gam=FALSE,
nd=NULL, eps=NULL)
## S3 method for class 'quad'
kppm(X,
trend = ~1,
clusters = c("Thomas","MatClust","Cauchy","VarGamma","LGCP"),
data = NULL,
...,
covariates=data,
subset,
method = c("mincon", "clik2", "palm"),
improve.type = c("none", "clik1", "wclik1", "quasi"),
improve.args = list(),
weightfun=NULL,
control=list(),
algorithm="Nelder-Mead",
statistic="K",
statargs=list(),
rmax = NULL,
covfunargs=NULL,
use.gam=FALSE,
nd=NULL, eps=NULL)

Arguments
X

A point pattern dataset (object of class "ppp" or "quad") to which the model
should be fitted, or a formula in the R language defining the model. See Details.

trend

An R formula, with no left hand side, specifying the form of the log intensity.

clusters

Character string determining the cluster model. Partially matched. Options are
"Thomas", "MatClust", "Cauchy", "VarGamma" and "LGCP".

data,covariates

The values of spatial covariates (other than the Cartesian coordinates) required
by the model. A named list of pixel images, functions, windows, tessellations
or numeric constants.
...

Additional arguments. See Details.

subset

Optional. A subset of the spatial domain, to which the model-fitting should be
restricted. A window (object of class "owin") or a logical-valued pixel image
(object of class "im"), or an expression (possibly involving the names of entries
in data) which can be evaluated to yield a window or pixel image.

method

The fitting method. Either "mincon" for minimum contrast, "clik2" for second
order composite likelihood, or "palm" for Palm likelihood. Partially matched.

improve.type

Method for updating the initial estimate of the trend. Initially the trend is estimated as if the process is an inhomogeneous Poisson process. The default,
improve.type = "none", is to use this initial estimate. Otherwise, the trend
estimate is updated by improve.kppm, using information about the pair correlation function. Options are "clik1" (first order composite likelihood, essentially
equivalent to "none"), "wclik1" (weighted first order composite likelihood)
and "quasi" (quasi likelihood).

improve.args

Additional arguments passed to improve.kppm when improve.type != "none".
See Details.

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659

weightfun

Optional weighting function w in the composite likelihood or Palm likelihood.
A function in the R language. See Details.

control

List of control parameters passed to the optimization function optim.

algorithm

Character string determining the mathematical optimisation algorithm to be used
by optim. See the argument method of optim.

statistic

Name of the summary statistic to be used for minimum contrast estimation:
either "K" or "pcf".

statargs

Optional list of arguments to be used when calculating the statistic. See
Details.

rmax
Maximum value of interpoint distance to use in the composite likelihood.
covfunargs,use.gam,nd,eps
Arguments passed to ppm when fitting the intensity.
Details
This function fits a clustered point process model to the point pattern dataset X.
The model may be either a Neyman-Scott cluster process or another Cox process. The type of model
is determined by the argument clusters. Currently the options are clusters="Thomas" for the
Thomas process, clusters="MatClust" for the Matern cluster process, clusters="Cauchy" for
the Neyman-Scott cluster process with Cauchy kernel, clusters="VarGamma" for the NeymanScott cluster process with Variance Gamma kernel (requires an additional argument nu to be passed
through the dots; see rVarGamma for details), and clusters="LGCP" for the log-Gaussian Cox
process (may require additional arguments passed through ...; see rLGCP for details on argument
names). The first four models are Neyman-Scott cluster processes.
The algorithm first estimates the intensity function of the point process using ppm. The argument
X may be a point pattern (object of class "ppp") or a quadrature scheme (object of class "quad").
The intensity is specified by the trend argument. If the trend formula is ~1 (the default) then the
model is homogeneous. The algorithm begins by estimating the intensity as the number of points
divided by the area of the window. Otherwise, the model is inhomogeneous. The algorithm begins
by fitting a Poisson process with log intensity of the form specified by the formula trend. (See ppm
for further explanation).
The argument X may also be a formula in the R language. The right hand side of the formula gives
the trend as described above. The left hand side of the formula gives the point pattern dataset to
which the model should be fitted.
If improve.type="none" this is the final estimate of the intensity. Otherwise, the intensity estimate
is updated, as explained in improve.kppm. Additional arguments to improve.kppm are passed as a
named list in improve.args.
The clustering parameters of the model are then fitted either by minimum contrast estimation, or by
maximum composite likelihood.
Minimum contrast: If method = "mincon" (the default) clustering parameters of the model will
be fitted by minimum contrast estimation, that is, by matching the theoretical K-function of
the model to the empirical K-function of the data, as explained in mincontrast.
For a homogeneous model ( trend = ~1 ) the empirical K-function of the data is computed
using Kest, and the parameters of the cluster model are estimated by the method of minimum
contrast.
For an inhomogeneous model, the inhomogeneous K function is estimated by Kinhom using
the fitted intensity. Then the parameters of the cluster model are estimated by the method of
minimum contrast using the inhomogeneous K function. This two-step estimation procedure
is due to Waagepetersen (2007).

660

kppm
If statistic="pcf" then instead of using the K-function, the algorithm will use the pair
correlation function pcf for homogeneous models and the inhomogeneous pair correlation
function pcfinhom for inhomogeneous models. In this case, the smoothing parameters of the
pair correlation can be controlled using the argument statargs, as shown in the Examples.
Additional arguments ... will be passed to mincontrast to control the minimum contrast
fitting algorithm.
Composite likelihood: If method = "clik2" the clustering parameters of the model will be fitted by maximising the second-order composite likelihood (Guan, 2006). The log composite
likelihood is


Z Z
X
X
w(dij ) log ρ(dij ; θ) − 
w(dij ) log
w(ku − vk)ρ(ku − vk; θ) du dv
i,j

i,j

D

D

where the sums are taken over all pairs of data points xi , xj separated by a distance dij =
kxi − xj k less than rmax, and the double integral is taken over all pairs of locations u, v in
the spatial window of the data. Here ρ(d; θ) is the pair correlation function of the model with
cluster parameters θ.
The function w in the composite likelihood is a weighting function and may be chosen arbitrarily. It is specified by the argument weightfun. If this is missing or NULL then the default
is a threshold weight function, w(d) = 1(d ≤ R), where R is rmax/2.
Palm likelihood: If method = "palm" the clustering parameters of the model will be fitted by
maximising the Palm loglikelihood (Tanaka et al, 2008)
Z
X
w(xi , xj ) log λP (xj | xi ; θ) −
w(xi , u)λP (u | xi ; θ)du
i,j

D

with the same notation as above. Here λP (u|v; θ is the Palm intensity of the model at location
u given there is a point at v.
In all three methods, the optimisation is performed by the generic optimisation algorithm optim.
The behaviour of this algorithm can be modified using the argument control. Useful control
arguments include trace, maxit and abstol (documented in the help for optim).
Fitting the LGCP model requires the RandomFields package, except in the default case where the
exponential covariance is assumed.
Value
An object of class "kppm" representing the fitted model. There are methods for printing, plotting,
predicting, simulating and updating objects of this class.
Log-Gaussian Cox Models
To fit a log-Gaussian Cox model with non-exponential covariance, specify clusters="LGCP" and
use additional arguments to specify the covariance structure. These additional arguments can be
given individually in the call to kppm, or they can be collected together in a list called covmodel.

For example a Matern model with parameter ν = 0.5 could be specified either by kppm(X, clusters="LGCP", model="m
or by kppm(X, clusters="LGCP", covmodel=list(model="matern", nu=0.5)).
The argument model specifies the type of covariance model: the default is model="exp" for an exponential covariance. Alternatives include "matern", "cauchy" and "spheric". Model names correspond to functions beginning with RM in the RandomFields package: for example model="matern"
corresponds to the function RMmatern in the RandomFields package.

kppm

661

Additional arguments are passed to the relevant function in the RandomFields package: for example if model="matern" then the additional argument nu is required, and is passed to the function
RMmatern in the RandomFields package.
Note that it is not possible to use anisotropic covariance models because the kppm technique assumes
the pair correlation function is isotropic.
Error and warning messages
See ppm.ppp for a list of common error messages and warnings originating from the first stage of
model-fitting.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak , with contributions from Abdollah Jalilian and Rasmus
Waagepetersen.
References
Guan, Y. (2006) A composite likelihood approach in fitting spatial point process models. Journal
of the American Statistical Association 101, 1502–1512.
Jalilian, A., Guan, Y. and Waagepetersen, R. (2012) Decomposition of variance for spatial Cox
processes. Scandinavian Journal of Statistics 40, 119–137.
Tanaka, U. and Ogata, Y. and Stoyan, D. (2008) Parameter estimation and model selection for
Neyman-Scott point processes. Biometrical Journal 50, 43–57.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous NeymanScott processes. Biometrics 63, 252–258.
See Also
Methods for kppm objects: plot.kppm, fitted.kppm, predict.kppm, simulate.kppm, update.kppm,
vcov.kppm, methods.kppm, as.ppm.kppm, Kmodel.kppm, pcfmodel.kppm.
Minimum contrast fitting algorithm: mincontrast.
Alternative fitting algorithms: thomas.estK, matclust.estK, lgcp.estK, cauchy.estK, vargamma.estK,
thomas.estpcf, matclust.estpcf, lgcp.estpcf, cauchy.estpcf, vargamma.estpcf,
Summary statistics: Kest, Kinhom, pcf, pcfinhom.
See also ppm
Examples
# method for point patterns
kppm(redwood, ~1, "Thomas")
# method for formulas
kppm(redwood ~ 1, "Thomas")
kppm(redwood ~ 1, "Thomas", method="c")
kppm(redwood ~ 1, "Thomas", method="p")
kppm(redwood ~ x, "MatClust")
kppm(redwood ~ x, "MatClust", statistic="pcf", statargs=list(stoyan=0.2))
kppm(redwood ~ x, cluster="Cauchy", statistic="K")
kppm(redwood, cluster="VarGamma", nu = 0.5, statistic="pcf")

662

Kres

# LGCP models
kppm(redwood ~ 1, "LGCP", statistic="pcf")
if(require("RandomFields")) {
kppm(redwood ~ x, "LGCP", statistic="pcf",
model="matern", nu=0.3,
control=list(maxit=10))
}
# fit with composite likelihood method
kppm(redwood ~ x, "VarGamma", method="clik2", nu.ker=-3/8)
# fit intensity with quasi-likelihood method
kppm(redwood ~ x, "Thomas", improve.type = "quasi")

Kres

Residual K Function

Description
Given a point process model fitted to a point pattern dataset, this function computes the residual K
function, which serves as a diagnostic for goodness-of-fit of the model.
Usage
Kres(object, ...)
Arguments
object

Object to be analysed. Either a fitted point process model (object of class
"ppm"), a point pattern (object of class "ppp"), a quadrature scheme (object
of class "quad"), or the value returned by a previous call to Kcom.

...

Arguments passed to Kcom.

Details
This command provides a diagnostic for the goodness-of-fit of a point process model fitted to a
point pattern dataset. It computes a residual version of the K function of the dataset, which should
be approximately zero if the model is a good fit to the data.
In normal use, object is a fitted point process model or a point pattern. Then Kres first calls Kcom
to compute both the nonparametric estimate of the K function and its model compensator. Then
Kres computes the difference between them, which is the residual K-function.
Alternatively, object may be a function value table (object of class "fv") that was returned by a
previous call to Kcom. Then Kres computes the residual from this object.
Value
A function value table (object of class "fv"), essentially a data frame of function values. There is a
plot method for this class. See fv.object.

Kres

663

Author(s)
Adrian Baddeley 
Ege Rubak  and Jesper Møller.

References
Baddeley, A., Rubak, E. and Møller, J. (2011) Score, pseudo-score and residual diagnostics for
spatial point process models. Statistical Science 26, 613–646.

See Also
Related functions: Kcom, Kest.
Alternative functions: Gres, psstG, psstA, psst.
Point process models: ppm.

Examples
data(cells)
fit0 <- ppm(cells, ~1) # uniform Poisson
K0 <- Kres(fit0)
K0
plot(K0)
# isotropic-correction estimate
plot(K0, ires ~ r)
# uniform Poisson is clearly not correct
fit1 <- ppm(cells, ~1, Strauss(0.08))
K1 <- Kres(fit1)
if(interactive()) {
plot(K1, ires ~ r)
# fit looks approximately OK; try adjusting interaction distance
plot(Kres(cells, interaction=Strauss(0.12)))
}
# How to make envelopes
## Not run:
E <- envelope(fit1, Kres, model=fit1, nsim=19)
plot(E)
## End(Not run)
# For computational efficiency
Kc <- Kcom(fit1)
K1 <- Kres(Kc)

664

Kscaled

Kscaled

Locally Scaled K-function

Description
Estimates the locally-rescaled K-function of a point process.
Usage
Kscaled(X, lambda=NULL, ..., r = NULL, breaks = NULL,
rmax = 2.5,
correction=c("border", "isotropic", "translate"),
renormalise=FALSE, normpower=1,
sigma=NULL, varcov=NULL)
Lscaled(...)
Arguments
X

The observed data point pattern, from which an estimate of the locally scaled K
function will be computed. An object of class "ppp" or in a format recognised
by as.ppp().

lambda

Optional. Values of the estimated intensity function. Either a vector giving the
intensity values at the points of the pattern X, a pixel image (object of class
"im") giving the intensity values at all locations, a function(x,y) which can
be evaluated to give the intensity value at any location, or a fitted point process
model (object of class "ppm").

...

Arguments passed from Lscaled to Kscaled and from Kscaled to density.ppp
if lambda is omitted.

r

vector of values for the argument r at which the locally scaled K function should
be evaluated. (These are rescaled distances.) Not normally given by the user;
there is a sensible default.

breaks

This argument is for internal use only.

rmax

maximum value of the argument r that should be used. (This is the rescaled
distance).

correction

A character vector containing any selection of the options "border", "isotropic",
"Ripley", "translate", "translation", "none" or "best". It specifies the
edge correction(s) to be applied. Alternatively correction="all" selects all
options.

renormalise

Logical. Whether to renormalise the estimate. See Details.

normpower

Integer (usually either 1 or 2). Normalisation power. See Details.

sigma,varcov

Optional arguments passed to density.ppp to control the smoothing bandwidth, when lambda is estimated by kernel smoothing.

Kscaled

665

Details
Kscaled computes an estimate of the K function
p for a locally scaled point process. Lscaled
computes the corresponding L function L(r) = K(r)/π.
Locally scaled point processes are a class of models for inhomogeneous point patterns, introduced
by Hahn et al (2003). They include inhomogeneous Poisson processes, and many other models.
The template K function of a locally-scaled process is a counterpart of the “ordinary” Ripley K
function, in which the distances between points of the process are measured on a spatially-varying
scale (such that the locally rescaled process has unit intensity).
The template K function is an indicator of interaction between the points. For an inhomogeneous
Poisson process, the theoretical template K function is approximately equal to K(r) = πr2 . Values
Kscaled (r) > πr2 are suggestive of clustering.
Kscaled computes an estimate
of the template K function and Lscaled computes the correspondp
ing L function L(r) = K(r)/π.
The locally scaled interpoint distances are computed using an approximation proposed by Hahn
(2007). The Euclidean distance between two points is multiplied by the average of the square roots
of the intensity values at the two points.
The argument lambda should supply the (estimated) values of the intensity function λ. It may be
either
a numeric vector containing the values of the intensity function at the points of the pattern X.
a pixel image (object of class "im") assumed to contain the values of the intensity function at all
locations in the window.
a function which can be evaluated to give values of the intensity at any locations.
omitted: if lambda is omitted, then it will be estimated using a ‘leave-one-out’ kernel smoother.
If lambda is a numeric vector, then its length should be equal to the number of points in the pattern
X. The value lambda[i] is assumed to be the the (estimated) value of the intensity λ(xi ) for the
point xi of the pattern X. Each value must be a positive number; NA’s are not allowed.
If lambda is a pixel image, the domain of the image should cover the entire window of the point
pattern. If it does not (which may occur near the boundary because of discretisation error), then
the missing pixel values will be obtained by applying a Gaussian blur to lambda using blur, then
looking up the values of this blurred image for the missing locations. (A warning will be issued in
this case.)
If lambda is a function, then it will be evaluated in the form lambda(x,y) where x and y are vectors
of coordinates of the points of X. It should return a numeric vector with length equal to the number
of points in X.
If lambda is omitted, then it will be estimated using a ‘leave-one-out’ kernel smoother, as described
in Baddeley, Møller and Waagepetersen (2000). The estimate lambda[i] for the point X[i] is computed by removing X[i] from the point pattern, applying kernel smoothing to the remaining points
using density.ppp, and evaluating the smoothed intensity at the point X[i]. The smoothing kernel
bandwidth is controlled by the arguments sigma and varcov, which are passed to density.ppp
along with any extra arguments.
If renormalise=TRUE, the estimated intensity lambda is multiplied by c( normpower/2) before
performing other calculations, where c = area(W )/sum[i](1/lambda(x[i])). This renormalisation has about the same effect as in Kinhom, reducing the variability and bias of the estimate in small
samples and in cases of very strong inhomogeneity.
Edge corrections are used to correct bias in the estimation of Kscaled . First the interpoint distances
are rescaled, and then edge corrections are applied as in Kest. See Kest for details of the edge
corrections and the options for the argument correction.

666

Kscaled
The pair correlation function can also be applied to the result of Kscaled; see pcf and pcf.fv.

Value
An object of class "fv" (see fv.object).
Essentially a data frame containing at least the following columns,
r

the vector of values of the argument r at which the pair correlation function g(r)
has been estimated

theo

vector of values of πr2 , the theoretical value of Kscaled (r) for an inhomogeneous Poisson process

and containing additional columns according to the choice specified in the correction argument. The additional columns are named border, trans and iso and give the estimated values
of Kscaled (r) using the border correction, translation correction, and Ripley isotropic correction,
respectively.
Author(s)
Ute Hahn, Adrian Baddeley 
and Rolf Turner 
References
Baddeley, A., Møller, J. and Waagepetersen, R. (2000) Non- and semiparametric estimation of
interaction in inhomogeneous point patterns. Statistica Neerlandica 54, 329–350.
Hahn, U. (2007) Global and Local Scaling in the Statistics of Spatial Point Processes. Habilitationsschrift, Universitaet Augsburg.
Hahn, U., Jensen, E.B.V., van Lieshout, M.N.M. and Nielsen, L.S. (2003) Inhomogeneous spatial
point processes by location-dependent scaling. Advances in Applied Probability 35, 319–336.
Prokes̆ová, M., Hahn, U. and Vedel Jensen, E.B. (2006) Statistics for locally scaled point patterns.
In A. Baddeley, P. Gregori, J. Mateu, R. Stoica and D. Stoyan (eds.) Case Studies in Spatial Point
Pattern Modelling. Lecture Notes in Statistics 185. New York: Springer Verlag. Pages 99–123.
See Also
Kest, pcf
Examples
data(bronzefilter)
X <- unmark(bronzefilter)
K <- Kscaled(X)
fit <- ppm(X, ~x)
lam <- predict(fit)
K <- Kscaled(X, lam)

Ksector

Ksector

667

Sector K-function

Description
A directional counterpart of Ripley’s K function, in which pairs of points are counted only when
the vector joining the pair happens to lie in a particular range of angles.
Usage
Ksector(X, begin = 0, end = 360, ...,
units = c("degrees", "radians"),
r = NULL, breaks = NULL,
correction = c("border", "isotropic", "Ripley", "translate"),
domain=NULL, ratio = FALSE, verbose=TRUE)
Arguments
X

The observed point pattern, from which an estimate of K(r) will be computed.
An object of class "ppp", or data in any format acceptable to as.ppp().

begin,end

Numeric values giving the range of angles inside which points will be counted.
Angles are measured in degrees (if units="degrees", the default) or radians (if
units="radians") anti-clockwise from the positive x-axis.

...

Ignored.

units

Units in which the angles begin and end are expressed.

r

Optional. Vector of values for the argument r at which K(r) should be evaluated. Users are advised not to specify this argument; there is a sensible default.

breaks

This argument is for internal use only.

correction

Optional. A character vector containing any selection of the options "none",
"border", "bord.modif", "isotropic", "Ripley", "translate", "translation",
"none", "good" or "best". It specifies the edge correction(s) to be applied. Alternatively correction="all" selects all options.

domain

Optional window. The first point xi of each pair of points will be constrained to
lie in domain.

ratio

Logical. If TRUE, the numerator and denominator of each edge-corrected estimate will also be saved, for use in analysing replicated point patterns.

verbose

Logical value indicating whether to print progress reports and warnings.

Details
This is a directional counterpart of Ripley’s K function (see Kest) in which, instead of counting all
pairs of points within a specified distance r, we count only the pairs (xi , xj ) for which the vector
xj − xi falls in a particular range of angles.
This can be used to evaluate evidence for anisotropy in the point pattern X.
Value
An object of class "fv" containing the estimated function.

668

LambertW

Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
Kest
Examples
K <- Ksector(swedishpines, 0, 90)
plot(K)

LambertW

Lambert’s W Function

Description
Computes Lambert’s W-function.
Usage
LambertW(x)
Arguments
x

Vector of nonnegative numbers.

Details
Lambert’s W-function is the inverse function of f (y) = yey . That is, W is the function such that
W (x)eW (x) = x
This command LambertW computes W (x) for each entry in the argument x. If the library gsl has
been installed, then the function lambert_W0 in that library is invoked. Otherwise, values of the
W-function are computed by root-finding, using the function uniroot.
Computation using gsl is about 100 times faster.
If any entries of x are infinite or NA, the corresponding results are NA.
Value
Numeric vector.
Author(s)
Adrian Baddeley 
and Rolf Turner 

laslett

669

References
Corless, R, Gonnet, G, Hare, D, Jeffrey, D and Knuth, D (1996), On the Lambert W function.
Computational Mathematics, 5, 325–359.
Roy, R and Olver, F (2010), Lambert W function. In Olver, F, Lozier, D and Boisvert, R (eds.),
NIST Handbook of Mathematical Functions, Cambridge University Press.
Examples
LambertW(exp(1))

laslett

Laslett’s Transform

Description
Apply Laslett’s Transform to a spatial region, returning the original and transformed regions, and
the original and transformed positions of the lower tangent points. This is a diagnostic for the
Boolean model.
Usage
laslett(X, ..., verbose = FALSE, plotit = TRUE, discretise = FALSE,
type=c("lower", "upper", "left", "right"))
Arguments
X

Spatial region to be transformed. A window (object of class "owin") or a
logical-valued pixel image (object of class "im").

...

Graphics arguments to control the plot (passed to plot.laslett when plotit=TRUE)
or arguments determining the pixel resolution (passed to as.mask).

verbose

Logical value indicating whether to print progress reports.

plotit

Logical value indicating whether to plot the result.

discretise

Logical value indicating whether polygonal windows should first be converted
to pixel masks before the Laslett transform is computed. This should be set to
TRUE for very complicated polygons.

type

Type of tangent points to be detected. This also determines the direction of
contraction in the set transformation. Default is type="lower".

Details
This function finds the lower tangent points of the spatial region X, then applies Laslett’s Transform
to the space, and records the transformed positions of the lower tangent points.
Laslett’s transform is a diagnostic for the Boolean Model. A test of the Boolean model can be
performed by applying a test of CSR to the transformed tangent points. See the Examples.
The rationale is that, if the region X was generated by a Boolean model with convex grains, then the
lower tangent points of X, when subjected to Laslett’s transform, become a Poisson point process
(Cressie, 1993, section 9.3.5; Molchanov, 1997; Barbour and Schmidt, 2001).

670

laslett
Intuitively, Laslett’s transform is a way to account for the fact that tangent points of X cannot occur
inside X. It treats the interior of X as empty space, and collapses this empty space so that only the
exterior of X remains. In this collapsed space, the tangent points are completely random.
Formally, Laslett’s transform is a random (i.e. data-dependent) spatial transformation which maps
each spatial location (x, y) to a new location (x0 , y) at the same height y. The transformation is
defined so that x0 is the total uncovered length of the line segment from (0, y) to (x, y), that is, the
total length of the parts of this segment that fall outside the region X.
In more colourful terms, suppose we use an abacus to display a pixellated version of X. Each wire
of the abacus represents one horizontal line in the pixel image. Each pixel lying outside the region
X is represented by a bead of the abacus; pixels inside X are represented by the absence of a bead.
Next we find any beads which are lower tangent points of X, and paint them green. Then Laslett’s
Transform is applied by pushing all beads to the left, as far as possible. The final locations of all
the beads provide a new spatial region, inside which is the point pattern of tangent points (marked
by the green-painted beads).
If plotit=TRUE (the default), a before-and-after plot is generated, showing the region X and the tangent points before and after the transformation. This plot can also be generated by calling plot(a)
where a is the object returned by the function laslett.
If the argument type is given, then this determines the type of tangents that will be detected, and also
the direction of contraction in Laslett’s transform. The computation is performed by first rotating X,
applying Laslett’s transform for lower tangent points, then rotating back.
There are separate algorithms for polygonal windows and pixellated windows (binary masks).
The polygonal algorithm may be slow for very complicated polygons. If this happens, setting
discretise=TRUE will convert the polygonal window to a binary mask and invoke the pixel raster
algorithm.

Value
A list, which also belongs to the class "laslett" so that it can immediately be printed and plotted.
The list elements are:
oldX: the original dataset X;
TanOld: a point pattern, whose window is Frame(X), containing the lower tangent points of X;
TanNew: a point pattern, whose window is the Laslett transform of Frame(X), and which contains
the Laslett-transformed positions of the tangent points;
Rect: a rectangular window, which is the largest rectangle lying inside the transformed set;
df: a data frame giving the locations of the tangent points before and after transformation.
type: character string specifying the type of tangents.
Author(s)
Kassel Hingee and Adrian Baddeley .
References
Barbour, A.D. and Schmidt, V. (2001) On Laslett’s Transform for the Boolean Model. Advances in
Applied Probability 33(1), 1–5.
Cressie, N.A.C. (1993) Statistics for spatial data, second edition. John Wiley and Sons.
Molchanov, I. (1997) Statistics of the Boolean Model for Practitioners and Mathematicians. Wiley.

latest.news

671

See Also
plot.laslett
Examples
a <- laslett(heather$coarse)
with(a, clarkevans.test(TanNew[Rect], correction="D", nsim=39))
X <- discs(runifpoint(15) %mark% 0.2, npoly=16)
b <- laslett(X)

latest.news

Print News About Latest Version of Package

Description
Prints the news documentation for the current version of spatstat or another specified package.
Usage
latest.news(package = "spatstat", doBrowse=FALSE)
Arguments
package

Name of package for which the latest news should be printed.

doBrowse

Logical value indicating whether to display the results in a browser window
instead of printing them.

Details
By default, this function prints the news documentation about changes in the current installed version of the spatstat package. The function can be called simply by typing its name without parentheses (see the Examples).
If package is given, then the function reads the news for the specified package from its NEWS file (if
it has one) and prints only the entries that refer to the current version of the package.
To see the news for all previous versions as well as the current version, use the R utility news. See
the Examples.
Value
Null.
Author(s)
Adrian Baddeley  and Rolf Turner 
See Also
news, bugfixes

672

layered

Examples
if(interactive()) {
# current news
latest.news
# all news
news(package="spatstat")
}

layered

Create List of Plotting Layers

Description
Given several objects which are capable of being plotted, create a list containing these objects as if
they were successive layers of a plot. The list can then be plotted in different ways.
Usage
layered(..., plotargs = NULL, LayerList=NULL)
Arguments
...

Objects which can be plotted by plot.

plotargs

Default values of the plotting arguments for each of the objects. A list of lists of
arguments of the form name=value.

LayerList

A list of objects. Incompatible with ....

Details
Layering is a simple mechanism for controlling a high-level plot that is composed of several successive plots, for example, a background and a foreground plot. The layering mechanism makes it
easier to issue the plot command, to switch on or off the plotting of each individual layer, to control
the plotting arguments that are passed to each layer, and to zoom in.
Each individual layer in the plot should be saved as an object that can be plotted using plot. It will
typically belong to some class, which has a method for the generic function plot.
The command layered simply saves the objects ... as a list of class "layered". This list can
then be plotted by the method plot.layered. Thus, you only need to type a single plot command to produce the multi-layered plot. Individual layers of the plot can be switched on or off, or
manipulated, using arguments to plot.layered.
The argument plotargs contains default values of the plotting arguments for each layer. It should
be a list, with one entry for each object in .... Each entry of plotargs should be a list of arguments
in the form name=value, which are recognised by the plot method for the relevant layer.
The plotargs can also include an argument named .plot specifying (the name of) a function to
perform the plotting instead of the generic plot.
The length of plotargs should either be equal to the number of layers, or equal to 1. In the latter
case it will be replicated to the appropriate length.

layerplotargs

673

Value
A list, belonging to the class "layered". There are methods for plot, "[", "shift", "affine",
"rotate" and "rescale".
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
plot.layered, methods.layered, as.layered, [.layered, layerplotargs.
Examples
D <- distmap(cells)
L <- layered(D, cells)
L
L <- layered(D, cells,
plotargs=list(list(ribbon=FALSE), list(pch=16)))
plot(L)
layerplotargs(L)[[1]] <- list(.plot="contour")
plot(L)

layerplotargs

Extract or Replace the Plot Arguments of a Layered Object

Description
Extracts or replaces the plot arguments of a layered object.
Usage
layerplotargs(L)
layerplotargs(L) <- value
Arguments
L

An object of class "layered" created by the function layered.

value

Replacement value. A list, with the same length as L, whose elements are lists
of plot arguments.

Details
These commands extract or replace the plotargs in a layered object. See layered.
The replacement value should normally have the same length as the current value. However, it can
also be a list with one element which is a list of parameters. This will be replicated to the required
length.
For the assignment function layerplotargs<-, the argument L can be any spatial object; it will be
converted to a layered object with a single layer.

674

layout.boxes

Value
layerplotargs returns a list of lists of plot arguments.
"layerplotargs<-" returns the updated object of class "layered".
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
layered, methods.layered, [.layered.
Examples
W <- square(2)
L <- layered(W=W, X=cells)
## The following are equivalent
layerplotargs(L) <- list(list(), list(pch=16))
layerplotargs(L)[[2]] <- list(pch=16)
layerplotargs(L)$X <- list(pch=16)
## The following are equivalent
layerplotargs(L) <- list(list(cex=2), list(cex=2))
layerplotargs(L) <- list(list(cex=2))

layout.boxes

Generate a Row or Column Arrangement of Rectangles.

Description
A simple utility to generate a row or column of boxes (rectangles) for use in point-and-click panels.
Usage
layout.boxes(B, n, horizontal = FALSE, aspect = 0.5, usefrac = 0.9)
Arguments
B
n
horizontal
aspect
usefrac

Bounding rectangle for the boxes. An object of class "owin".
Integer. The number of boxes.
Logical. If TRUE, arrange the boxes in a horizontal row. If FALSE (the default),
arrange them in a vertical column.
Aspect ratio (height/width) of each box.
Number between 0 and 1. The fraction of height or width of B that should be
occupied by boxes.

Details
This simple utility generates a list of boxes (rectangles) inside the bounding box B arranged in a
regular row or column. It is useful for generating the positions of the panel buttons in the function
simplepanel.

Lcross

675

Value
A list of rectangles.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
simplepanel
Examples
B <- owin(c(0,10),c(0,1))
boxes <- layout.boxes(B, 5, horizontal=TRUE)
plot(B, main="", col="blue")
niets <- lapply(boxes, plot, add=TRUE, col="grey")

Lcross

Multitype L-function (cross-type)

Description
Calculates an estimate of the cross-type L-function for a multitype point pattern.
Usage
Lcross(X, i, j, ..., from, to)
Arguments
X

The observed point pattern, from which an estimate of the cross-type L function
Lij (r) will be computed. It must be a multitype point pattern (a marked point
pattern whose marks are a factor). See under Details.

i

The type (mark value) of the points in X from which distances are measured.
A character string (or something that will be converted to a character string).
Defaults to the first level of marks(X).

j

The type (mark value) of the points in X to which distances are measured. A
character string (or something that will be converted to a character string). Defaults to the second level of marks(X).

...

Arguments passed to Kcross.

from,to

An alternative way to specify i and j respectively.

676

Lcross

Details
The cross-type L-function is a transformation of the cross-type K-function,
r
Lij (r) =

Kij (r)
π

where Kij (r) is the cross-type K-function from type i to type j. See Kcross for information about
the cross-type K-function.
The command Lcross first calls Kcross to compute the estimate of the cross-type K-function, and
then applies the square root transformation.
For a marked point pattern in which the points of type i are independent of the points of type j, the
theoretical value of the L-function is Lij (r) = r. The square root also has the effect of stabilising
the variance of the estimator, so that Lij is more appropriate for use in simulation envelopes and
hypothesis tests.

Value
An object of class "fv", see fv.object, which can be plotted directly using plot.fv.
Essentially a data frame containing columns
r

the vector of values of the argument r at which the function Lij has been estimated

theo

the theoretical value Lij (r) = r for a stationary Poisson process

together with columns named "border", "bord.modif", "iso" and/or "trans", according to the
selected edge corrections. These columns contain estimates of the function Lij obtained by the
edge corrections named.

Author(s)
Adrian Baddeley 
and Rolf Turner 

See Also
Kcross, Ldot, Lest

Examples
data(amacrine)
L <- Lcross(amacrine, "off", "on")
plot(L)

Lcross.inhom

Lcross.inhom

677

Inhomogeneous Cross Type L Function

Description
For a multitype point pattern, estimate the inhomogeneous version of the cross-type L function.
Usage
Lcross.inhom(X, i, j, ...)
Arguments
X

The observed point pattern, from which an estimate of the inhomogeneous cross
type L function Lij (r) will be computed. It must be a multitype point pattern (a
marked point pattern whose marks are a factor). See under Details.

i

The type (mark value) of the points in X from which distances are measured.
A character string (or something that will be converted to a character string).
Defaults to the first level of marks(X).

j

The type (mark value) of the points in X to which distances are measured. A
character string (or something that will be converted to a character string). Defaults to the second level of marks(X).

...

Other arguments passed to Kcross.inhom.

Details
This is a generalisation of the function Lcross to include an adjustment for spatially inhomogeneous
intensity, in a manner similar to the function Linhom.
All the arguments are passed to Kcross.inhom, which estimates the inhomogeneous multitype K
function
Kij (r) for the point pattern. The resulting values are then transformed by taking L(r) =
p
K(r)/π.
Value
An object of class "fv" (see fv.object).
Essentially a data frame containing numeric columns
r

the values of the argument r at which the function Lij (r) has been estimated

theo

the theoretical value of Lij (r) for a marked Poisson process, identically equal
to r

together with a column or columns named "border", "bord.modif", "iso" and/or "trans", according to the selected edge corrections. These columns contain estimates of the function Lij (r)
obtained by the edge corrections named.
Warnings
The arguments i and j are always interpreted as levels of the factor X$marks. They are converted
to character strings if they are not already character strings. The value i=1 does not refer to the first
level of the factor.

678

Ldot

Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Møller, J. and Waagepetersen, R. Statistical Inference and Simulation for Spatial Point Processes
Chapman and Hall/CRC Boca Raton, 2003.
See Also
Lcross, Linhom, Kcross.inhom
Examples
# Lansing Woods data
woods <- lansing
ma <- split(woods)$maple
wh <- split(woods)$whiteoak
# method (1): estimate intensities by nonparametric smoothing
lambdaM <- density.ppp(ma, sigma=0.15, at="points")
lambdaW <- density.ppp(wh, sigma=0.15, at="points")
L <- Lcross.inhom(woods, "whiteoak", "maple", lambdaW, lambdaM)
# method (2): fit parametric intensity model
fit <- ppm(woods ~marks * polynom(x,y,2))
# evaluate fitted intensities at data points
# (these are the intensities of the sub-processes of each type)
inten <- fitted(fit, dataonly=TRUE)
# split according to types of points
lambda <- split(inten, marks(woods))
L <- Lcross.inhom(woods, "whiteoak", "maple",
lambda$whiteoak, lambda$maple)
# synthetic example: type A points have intensity 50,
#
type B points have intensity 100 * x
lamB <- as.im(function(x,y){50 + 100 * x}, owin())
X <- superimpose(A=runifpoispp(50), B=rpoispp(lamB))
L <- Lcross.inhom(X, "A", "B",
lambdaI=as.im(50, Window(X)), lambdaJ=lamB)

Ldot

Multitype L-function (i-to-any)

Description
Calculates an estimate of the multitype L-function (from type i to any type) for a multitype point
pattern.
Usage
Ldot(X, i, ..., from)

Ldot

679

Arguments
X

The observed point pattern, from which an estimate of the dot-type L function
Lij (r) will be computed. It must be a multitype point pattern (a marked point
pattern whose marks are a factor). See under Details.

i

The type (mark value) of the points in X from which distances are measured.
A character string (or something that will be converted to a character string).
Defaults to the first level of marks(X).

...

Arguments passed to Kdot.

from

An alternative way to specify i.

Details
This command computes
r

Ki• (r)
π
where Ki• (r) is the multitype K-function from points of type i to points of any type. See Kdot for
information about Ki• (r).
Li• (r) =

The command Ldot first calls Kdot to compute the estimate of the i-to-any K-function, and then
applies the square root transformation.
For a marked Poisson point process, the theoretical value of the L-function is Li• (r) = r. The
square root also has the effect of stabilising the variance of the estimator, so that Li• is more
appropriate for use in simulation envelopes and hypothesis tests.
Value
An object of class "fv", see fv.object, which can be plotted directly using plot.fv.
Essentially a data frame containing columns
r

the vector of values of the argument r at which the function Li• has been estimated

theo

the theoretical value Li• (r) = r for a stationary Poisson process

together with columns named "border", "bord.modif", "iso" and/or "trans", according to the
selected edge corrections. These columns contain estimates of the function Li• obtained by the
edge corrections named.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
Kdot, Lcross, Lest
Examples
data(amacrine)
L <- Ldot(amacrine, "off")
plot(L)

680

Ldot.inhom

Ldot.inhom

Inhomogeneous Multitype L Dot Function

Description
For a multitype point pattern, estimate the inhomogeneous version of the dot L function.
Usage
Ldot.inhom(X, i, ...)
Arguments
X

The observed point pattern, from which an estimate of the inhomogeneous cross
type L function Li• (r) will be computed. It must be a multitype point pattern (a
marked point pattern whose marks are a factor). See under Details.

i

The type (mark value) of the points in X from which distances are measured.
A character string (or something that will be converted to a character string).
Defaults to the first level of marks(X).

...

Other arguments passed to Kdot.inhom.

Details
This a generalisation of the function Ldot to include an adjustment for spatially inhomogeneous
intensity, in a manner similar to the function Linhom.
All the arguments are passed to Kdot.inhom, which estimates the inhomogeneous multitype K
function
Ki• (r) for the point pattern. The resulting values are then transformed by taking L(r) =
p
K(r)/π.
Value
An object of class "fv" (see fv.object).
Essentially a data frame containing numeric columns
r

the values of the argument r at which the function Li• (r) has been estimated

theo

the theoretical value of Li• (r) for a marked Poisson process, identical to r.

together with a column or columns named "border", "bord.modif", "iso" and/or "trans", according to the selected edge corrections. These columns contain estimates of the function Li• (r)
obtained by the edge corrections named.
Warnings
The argument i is interpreted as a level of the factor X$marks. It is converted to a character string
if it is not already a character string. The value i=1 does not refer to the first level of the factor.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 

lengths.psp

681

References
Møller, J. and Waagepetersen, R. Statistical Inference and Simulation for Spatial Point Processes
Chapman and Hall/CRC Boca Raton, 2003.
See Also
Ldot, Linhom, Kdot.inhom, Lcross.inhom.
Examples
# Lansing Woods data
lan <- lansing
lan <- lan[seq(1,npoints(lan), by=10)]
ma <- split(lan)$maple
lg <- unmark(lan)
# Estimate intensities by nonparametric smoothing
lambdaM <- density.ppp(ma, sigma=0.15, at="points")
lambdadot <- density.ppp(lg, sigma=0.15, at="points")
L <- Ldot.inhom(lan, "maple", lambdaI=lambdaM,
lambdadot=lambdadot)
# synthetic example: type A points have intensity 50,
#
type B points have intensity 50 + 100 * x
lamB <- as.im(function(x,y){50 + 100 * x}, owin())
lamdot <- as.im(function(x,y) { 100 + 100 * x}, owin())
X <- superimpose(A=runifpoispp(50), B=rpoispp(lamB))
L <- Ldot.inhom(X, "B", lambdaI=lamB,
lambdadot=lamdot)

lengths.psp

Lengths of Line Segments

Description
Computes the length of each line segment in a line segment pattern.
Usage
lengths.psp(x, squared=FALSE)
Arguments
x

A line segment pattern (object of class "psp").

squared

Logical value indicating whether to return the squared lengths (squared=TRUE)
or the lengths themselves (squared=FALSE, the default).

Details
The length of each line segment is computed and the lengths are returned as a numeric vector.
Using squared lengths may be more efficient for some purposes, for example, to find the length of
the shortest segment, sqrt(min(lengths.psp(x, squared=TRUE))) is faster than min(lengths.psp(x)).

682

LennardJones

Value
Numeric vector.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
summary.psp, midpoints.psp, angles.psp
Examples
a <- psp(runif(10), runif(10), runif(10), runif(10), window=owin())
b <- lengths.psp(a)

LennardJones

The Lennard-Jones Potential

Description
Creates the Lennard-Jones pairwise interaction structure which can then be fitted to point pattern
data.
Usage
LennardJones(sigma0=NA)
Arguments
sigma0

Optional. Initial estimate of the parameter σ. A positive number.

Details
In a pairwise interaction point process with the Lennard-Jones pair potential (Lennard-Jones, 1924)
each pair of points in the point pattern, a distance d apart, contributes a factor

 

σ 12  σ 6
v(d) = exp −4
−
d
d
to the probability density, where σ and  are positive parameters to be estimated.
See Examples for a plot of this expression.
This potential causes very strong inhibition between points at short range, and attraction between
points at medium range. The parameter σ is called the characteristic diameter and controls the scale
of interaction. The parameter  is called the well depth and determines the strength of attraction. The
potential switches from inhibition to attraction at d = σ. The maximum value of the pair potential
is exp() occuring at distance d = 21/6 σ. Interaction is usually considered to be negligible for
distances d > 2.5σ max{1, 1/6 }.
This potential is used to model interactions between uncharged molecules in statistical physics.

LennardJones

683

The function ppm(), which fits point process models to point pattern data, requires an argument of
class "interact" describing the interpoint interaction structure of the model to be fitted. The appropriate description of the Lennard-Jones pairwise interaction is yielded by the function LennardJones().
See the examples below.
Value
An object of class "interact" describing the Lennard-Jones interpoint interaction structure.
Rescaling
To avoid numerical instability, the interpoint distances d are rescaled when fitting the model.
Distances are rescaled by dividing by sigma0. In the formula for v(d) above, the interpoint distance
d will be replaced by d/sigma0.
The rescaling happens automatically by default. If the argument sigma0 is missing or NA (the
default), then sigma0 is taken to be the minimum nearest-neighbour distance in the data point
pattern (in the call to ppm).
If the argument sigma0 is given, it should be a positive number, and it should be a rough estimate
of the parameter σ.
The “canonical regular parameters” estimated by ppm are θ1 = 4(σ/σ0 )12 and θ2 = 4(σ/σ0 )6 .
Warnings and Errors
Fitting the Lennard-Jones model is extremely unstable, because of the strong dependence between
the functions d−12 and d−6 . The fitting algorithm often fails to converge. Try increasing the number
of iterations of the GLM fitting algorithm, by setting gcontrol=list(maxit=1e3) in the call to
ppm.
Errors are likely to occur if this model is fitted to a point pattern dataset which does not exhibit both
short-range inhibition and medium-range attraction between points. The values of the parameters σ
and  may be NA (because the fitted canonical parameters have opposite sign, which usually occurs
when the pattern is completely random).
An absence of warnings does not mean that the fitted model is sensible. A negative value of  may
be obtained (usually when the pattern is strongly clustered); this does not correspond to a valid
point process model, but the software does not issue a warning.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Lennard-Jones, J.E. (1924) On the determination of molecular fields. Proc Royal Soc London A
106, 463–477.
See Also
ppm, pairwise.family, ppm.object

684

Lest

Examples
fit <- ppm(cells ~1, LennardJones(), rbord=0.1)
fit
plot(fitin(fit))

Lest

L-function

Description
Calculates an estimate of the L-function (Besag’s transformation of Ripley’s K-function) for a
spatial point pattern.
Usage
Lest(X, ...)
Arguments
X
...

The observed point pattern, from which an estimate of L(r) will be computed.
An object of class "ppp", or data in any format acceptable to as.ppp().
Other arguments passed to Kest to control the estimation procedure.

Details
This command computes an estimate of the L-function for the spatial point pattern X. The Lfunction is a transformation of Ripley’s K-function,
r
K(r)
L(r) =
π
where K(r) is the K-function.
See Kest for information about Ripley’s K-function. The transformation to L was proposed by
Besag (1977).
The command Lest first calls Kest to compute the estimate of the K-function, and then applies the
square root transformation.
For a completely random (uniform Poisson) point pattern, the theoretical value of the L-function is
L(r) = r. The square root also has the effect of stabilising the variance of the estimator, so that
L(r) is more appropriate for use in simulation envelopes and hypothesis tests.
See Kest for the list of arguments.
Value
An object of class "fv", see fv.object, which can be plotted directly using plot.fv.
Essentially a data frame containing columns
r
theo

the vector of values of the argument r at which the function L has been estimated
the theoretical value L(r) = r for a stationary Poisson process

together with columns named "border", "bord.modif", "iso" and/or "trans", according to the
selected edge corrections. These columns contain estimates of the function L(r) obtained by the
edge corrections named.

levelset

685

Variance approximations
If the argument var.approx=TRUE is given, the return value includes columns rip and ls containing approximations to the variance of L̂(r) under CSR. These are obtained by the delta method
from the variance approximations described in Kest.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Besag, J. (1977) Discussion of Dr Ripley’s paper. Journal of the Royal Statistical Society, Series B,
39, 193–195.
See Also
Kest, pcf
Examples
data(cells)
L <- Lest(cells)
plot(L, main="L function for cells")

levelset

Level Set of a Pixel Image

Description
Given a pixel image, find all pixels which have values less than a specified threshold value (or
greater than a threshold, etc), and assemble these pixels into a window.
Usage
levelset(X, thresh, compare="<=")
Arguments
X

A pixel image (object of class "im").

thresh

Threshold value. A single number or value compatible with the pixel values in
X.

compare

Character string specifying one of the comparison operators "<", ">", "==", "<=", ">=", "!=".

686

leverage.ppm

Details
If X is a pixel image with numeric values, then levelset(X, thresh) finds the region of space
where the pixel values are less than or equal to the threshold value thresh. This region is returned
as a spatial window.
The argument compare specifies how the pixel values should be compared with the threshold value.
Instead of requiring pixel values to be less than or equal to thresh, you can specify that they must
be less than (<), greater than (>), equal to (==), greater than or equal to (>=), or not equal to (!=) the
threshold value thresh.
If X has non-numeric pixel values (for example, logical or factor values) it is advisable to use only
the comparisons == and !=, unless you really know what you are doing.
For more complicated logical comparisons, see solutionset.
Value
A spatial window (object of class "owin", see owin.object) containing the pixels satisfying the
constraint.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
im.object, as.owin, solutionset.
Examples
# test image
X <- as.im(function(x,y) { x^2 - y^2 }, unit.square())
W <- levelset(X, 0.2)
W <- levelset(X, -0.3, ">")
# compute area of level set
area(levelset(X, 0.1))

leverage.ppm

Leverage Measure for Spatial Point Process Model

Description
Computes the leverage measure for a fitted spatial point process model.
Usage
leverage(model, ...)
## S3 method for class 'ppm'
leverage(model, ..., drop = FALSE, iScore=NULL, iHessian=NULL, iArgs=NULL)

leverage.ppm

687

Arguments
model

Fitted point process model (object of class "ppm").

...

Ignored, except for the arguments dimyx and eps which are passed to as.mask
to control the spatial resolution of the result.

drop

Logical. Whether to include (drop=FALSE) or exclude (drop=TRUE) contributions from quadrature points that were not used to fit the model.

iScore,iHessian
Components of the score vector and Hessian matrix for the irregular parameters,
if required. See Details.
iArgs

List of extra arguments for the functions iScore, iHessian if required.

Details
The function leverage is generic, and leverage.ppm is the method for objects of class "ppm".
Given a fitted spatial point process model model, the function leverage.ppm computes the leverage
of the model, described in Baddeley, Chang and Song (2013).
The leverage of a spatial point process model is a function of spatial location, and is typically
displayed as a colour pixel image. The leverage value h(u) at a spatial location u represents the
change in the fitted trend of the fitted point process model that would have occurred if a data point
were to have occurred at the location u. A relatively large value of h() indicates a part of the space
where the data have a potentially strong effect on the fitted model (specifically, a strong effect on
the intensity or trend of the fitted model) due to the values of the covariates.
If the point process model trend has irregular parameters that were fitted (using ippm) then the
leverage calculation requires the first and second derivatives of the log trend with respect to the
irregular parameters. The argument iScore should be a list, with one entry for each irregular
parameter, of R functions that compute the partial derivatives of the log trend (i.e. log intensity or
log conditional intensity) with respect to each irregular parameter. The argument iHessian should
be a list, with p2 entries where p is the number of irregular parameters, of R functions that compute
the second order partial derivatives of the log trend with respect to each pair of irregular parameters.
The result of leverage.ppm is an object of class "leverage.ppm". It can be plotted (by plot.leverage.ppm)
or converted to a pixel image by as.im (see as.im.leverage.ppm).
Value
An object of class "leverage.ppm" that can be plotted (by plot.leverage.ppm). There are also
methods for persp, print, [, as.im, as.function and as.owin.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
References
Baddeley, A., Chang, Y.M. and Song, Y. (2013) Leverage and influence diagnostics for spatial point
process models. Scandinavian Journal of Statistics 40, 86–104.
See Also
influence.ppm, dfbetas.ppm, ppmInfluence, plot.leverage.ppm as.function.leverage.ppm

688

lgcp.estK

Examples
X <- rpoispp(function(x,y) { exp(3+3*x) })
fit <- ppm(X ~x+y)
plot(leverage(fit))

lgcp.estK

Fit a Log-Gaussian Cox Point Process by Minimum Contrast

Description
Fits a log-Gaussian Cox point process model to a point pattern dataset by the Method of Minimum
Contrast.
Usage
lgcp.estK(X, startpar=c(var=1,scale=1),
covmodel=list(model="exponential"),
lambda=NULL,
q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...)
Arguments
X

Data to which the model will be fitted. Either a point pattern or a summary
statistic. See Details.

startpar

Vector of starting values for the parameters of the log-Gaussian Cox process
model.

covmodel

Specification of the covariance model for the log-Gaussian field. See Details.

lambda

Optional. An estimate of the intensity of the point process.

q,p

Optional. Exponents for the contrast criterion.

rmin, rmax

Optional. The interval of r values for the contrast criterion.

...

Optional arguments passed to optim to control the optimisation algorithm. See
Details.

Details
This algorithm fits a log-Gaussian Cox point process (LGCP) model to a point pattern dataset by
the Method of Minimum Contrast, using the K function of the point pattern.
The shape of the covariance of the LGCP must be specified: the default is the exponential covariance
function, but other covariance models can be selected.
The argument X can be either
a point pattern: An object of class "ppp" representing a point pattern dataset. The K function of
the point pattern will be computed using Kest, and the method of minimum contrast will be
applied to this.
a summary statistic: An object of class "fv" containing the values of a summary statistic, computed for a point pattern dataset. The summary statistic should be the K function, and this
object should have been obtained by a call to Kest or one of its relatives.

lgcp.estK

689

The algorithm fits a log-Gaussian Cox point process (LGCP) model to X, by finding the parameters
of the LGCP model which give the closest match between the theoretical K function of the LGCP
model and the observed K function. For a more detailed explanation of the Method of Minimum
Contrast, see mincontrast.
The model fitted is a stationary, isotropic log-Gaussian Cox process (Møller and Waagepetersen,
2003, pp. 72-76). To define this process we start with a stationary Gaussian random field Z in the
two-dimensional plane, with constant mean µ and covariance function C(r). Given Z, we generate
a Poisson point process Y with intensity function λ(u) = exp(Z(u)) at location u. Then Y is a
log-Gaussian Cox process.
The K-function of the LGCP is
Z
K(r) =

r

2πs exp(C(s)) ds.
0

The intensity of the LGCP is
λ = exp(µ +

C(0)
).
2

The covariance function C(r) is parametrised in the form
C(r) = σ 2 c(r/α)
where σ 2 and α are parameters controlling the strength and the scale of autocorrelation, respectively,
and c(r) is a known covariance function determining the shape of the covariance. The strength
and scale parameters σ 2 and α will be estimated by the algorithm as the values var and scale
respectively. The template covariance function c(r) must be specified as explained below.
In this algorithm, the Method of Minimum Contrast is first used to find optimal values of the parameters σ 2 and α. Then the remaining parameter µ is inferred from the estimated intensity λ.
The template covariance function c(r) is specified using the argument covmodel. This should be of
the form list(model="modelname", ...) where modelname is a string identifying the template
model as explained below, and ... are optional arguments of the form tag=value giving the values
of parameters controlling the shape of the template model. The default is the exponential covariance
c(r) = e−r so that the scaled covariance is
C(r) = σ 2 e−r/α .
To determine the template model, the string "modelname" will be prefixed by "RM" and the code
will search for a function of this name in the RandomFields package. For a list of available models
see RMmodel in the RandomFields package. For example the Matern covariance with exponent
ν = 0.3 is specified by covmodel=list(model="matern", nu=0.3) corresponding to the function
RMmatern in the RandomFields package.
If the argument lambda is provided, then this is used as the value of λ. Otherwise, if X is a point
pattern, then λ will be estimated from X. If X is a summary statistic and lambda is missing, then the
intensity λ cannot be estimated, and the parameter µ will be returned as NA.
The remaining arguments rmin,rmax,q,p control the method of minimum contrast; see mincontrast.
The optimisation algorithm can be controlled through the additional arguments "..." which are
passed to the optimisation function optim. For example, to constrain the parameter values to a
certain range, use the argument method="L-BFGS-B" to select an optimisation algorithm that respects box constraints, and use the arguments lower and upper to specify (vectors of) minimum
and maximum values for each parameter.

690

lgcp.estK

Value
An object of class "minconfit". There are methods for printing and plotting this object. It contains
the following main components:
par

Vector of fitted parameter values.

fit

Function value table (object of class "fv") containing the observed values of the
summary statistic (observed) and the theoretical values of the summary statistic
computed from the fitted model parameters.

Note
This function is considerably slower than lgcp.estpcf because of the computation time required
for the integral in the K-function.
Computation can be accelerated, at the cost of less accurate results, by setting spatstat.options(fastK.lgcp=TRUE).
Author(s)

Rasmus Waagepetersen . Adapted for spatstat by Adrian Baddeley .
References
Møller, J, Syversveen, A. and Waagepetersen, R. (1998) Log Gaussian Cox Processes. Scandinavian Journal of Statistics 25, 451–482.
Møller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point
Processes. Chapman and Hall/CRC, Boca Raton.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous NeymanScott processes. Biometrics 63, 252–258.
See Also
lgcp.estpcf for alternative method of fitting LGCP.
matclust.estK, thomas.estK for other models.
mincontrast for the generic minimum contrast fitting algorithm, including important parameters
that affect the accuracy of the fit.
RMmodel in the RandomFields package, for covariance function models.
Kest for the K function.
Examples
if(interactive()) {
u <- lgcp.estK(redwood)
} else {
# slightly faster - better starting point
u <- lgcp.estK(redwood, c(var=1, scale=0.1))
}
u
plot(u)
if(FALSE) {

lgcp.estpcf

}

691
## takes several minutes!
lgcp.estK(redwood, covmodel=list(model="matern", nu=0.3))

lgcp.estpcf

Fit a Log-Gaussian Cox Point Process by Minimum Contrast

Description
Fits a log-Gaussian Cox point process model to a point pattern dataset by the Method of Minimum
Contrast using the pair correlation function.
Usage
lgcp.estpcf(X,
startpar=c(var=1,scale=1),
covmodel=list(model="exponential"),
lambda=NULL,
q = 1/4, p = 2, rmin = NULL, rmax = NULL, ..., pcfargs=list())
Arguments
X

Data to which the model will be fitted. Either a point pattern or a summary
statistic. See Details.

startpar

Vector of starting values for the parameters of the log-Gaussian Cox process
model.

covmodel

Specification of the covariance model for the log-Gaussian field. See Details.

lambda

Optional. An estimate of the intensity of the point process.

q,p

Optional. Exponents for the contrast criterion.

rmin, rmax

Optional. The interval of r values for the contrast criterion.

...

Optional arguments passed to optim to control the optimisation algorithm. See
Details.

pcfargs

Optional list containing arguments passed to pcf.ppp to control the smoothing
in the estimation of the pair correlation function.

Details
This algorithm fits a log-Gaussian Cox point process (LGCP) model to a point pattern dataset by
the Method of Minimum Contrast, using the estimated pair correlation function of the point pattern.
The shape of the covariance of the LGCP must be specified: the default is the exponential covariance
function, but other covariance models can be selected.
The argument X can be either
a point pattern: An object of class "ppp" representing a point pattern dataset. The pair correlation
function of the point pattern will be computed using pcf, and the method of minimum contrast
will be applied to this.
a summary statistic: An object of class "fv" containing the values of a summary statistic, computed for a point pattern dataset. The summary statistic should be the pair correlation function,
and this object should have been obtained by a call to pcf or one of its relatives.

692

lgcp.estpcf
The algorithm fits a log-Gaussian Cox point process (LGCP) model to X, by finding the parameters
of the LGCP model which give the closest match between the theoretical pair correlation function
of the LGCP model and the observed pair correlation function. For a more detailed explanation of
the Method of Minimum Contrast, see mincontrast.
The model fitted is a stationary, isotropic log-Gaussian Cox process (Møller and Waagepetersen,
2003, pp. 72-76). To define this process we start with a stationary Gaussian random field Z in the
two-dimensional plane, with constant mean µ and covariance function C(r). Given Z, we generate
a Poisson point process Y with intensity function λ(u) = exp(Z(u)) at location u. Then Y is a
log-Gaussian Cox process.
The theoretical pair correlation function of the LGCP is
g(r) = exp(C(s))
The intensity of the LGCP is
λ = exp(µ +

C(0)
).
2

The covariance function C(r) takes the form
C(r) = σ 2 c(r/α)
where σ 2 and α are parameters controlling the strength and the scale of autocorrelation, respectively,
and c(r) is a known covariance function determining the shape of the covariance. The strength and
scale parameters σ 2 and α will be estimated by the algorithm. The template covariance function
c(r) must be specified as explained below.
In this algorithm, the Method of Minimum Contrast is first used to find optimal values of the parameters σ 2 and α. Then the remaining parameter µ is inferred from the estimated intensity λ.
The template covariance function c(r) is specified using the argument covmodel. This should be of
the form list(model="modelname", ...) where modelname is a string identifying the template
model as explained below, and ... are optional arguments of the form tag=value giving the values
of parameters controlling the shape of the template model. The default is the exponential covariance
c(r) = e−r so that the scaled covariance is
C(r) = σ 2 e−r/α .
To determine the template model, the string "modelname" will be prefixed by "RM" and the code
will search for a function of this name in the RandomFields package. For a list of available models
see RMmodel in the RandomFields package. For example the Matern covariance with exponent
ν = 0.3 is specified by covmodel=list(model="matern", nu=0.3) corresponding to the function
RMmatern in the RandomFields package.
If the argument lambda is provided, then this is used as the value of λ. Otherwise, if X is a point
pattern, then λ will be estimated from X. If X is a summary statistic and lambda is missing, then the
intensity λ cannot be estimated, and the parameter µ will be returned as NA.
The remaining arguments rmin,rmax,q,p control the method of minimum contrast; see mincontrast.
The optimisation algorithm can be controlled through the additional arguments "..." which are
passed to the optimisation function optim. For example, to constrain the parameter values to a
certain range, use the argument method="L-BFGS-B" to select an optimisation algorithm that respects box constraints, and use the arguments lower and upper to specify (vectors of) minimum
and maximum values for each parameter.

lgcp.estpcf

693

Value
An object of class "minconfit". There are methods for printing and plotting this object. It contains
the following main components:
par

Vector of fitted parameter values.

fit

Function value table (object of class "fv") containing the observed values of the
summary statistic (observed) and the theoretical values of the summary statistic
computed from the fitted model parameters.

Author(s)
Adrian Baddeley  with modifications by Shen Guochun and
Rasmus Waagepetersen  and Ege Rubak .

References
Møller, J., Syversveen, A. and Waagepetersen, R. (1998) Log Gaussian Cox Processes. Scandinavian Journal of Statistics 25, 451–482.
Møller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point
Processes. Chapman and Hall/CRC, Boca Raton.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous NeymanScott processes. Biometrics 63, 252–258.

See Also
lgcp.estK for alternative method of fitting LGCP.
matclust.estpcf, thomas.estpcf for other models.
mincontrast for the generic minimum contrast fitting algorithm, including important parameters
that affect the accuracy of the fit.
RMmodel in the RandomFields package, for covariance function models.
pcf for the pair correlation function.

Examples
data(redwood)
u <- lgcp.estpcf(redwood, c(var=1, scale=0.1))
u
plot(u)
if(require(RandomFields)) {
lgcp.estpcf(redwood, covmodel=list(model="matern", nu=0.3))
}

694

lineardirichlet

lineardirichlet

Dirichlet Tessellation on a Linear Network

Description
Given a point pattern on a linear network, compute the Dirichlet (or Voronoi or Thiessen) tessellation induced by the points.
Usage
lineardirichlet(X)
Arguments
X

Point pattern on a linear network (object of class "lpp").

Details
The Dirichlet tessellation induced by a point pattern X on a linear network L is a partition of L into
subsets. The subset L[i] associated with the data point X[i] is the part of L lying closer to X[i]
than to any other data point X[j], where distance is measured by the shortest path.
Value
A tessellation on a linear network (object of class "lintess").
Missing tiles
If the linear network is not connected, and if one of the connected components contains no data
points, then the Dirichlet tessellation is mathematically undefined inside this component. The resulting tessellation object includes a tile with label NA, which contains this component of the network. A plot of the tessellation will not show this tile.
Author(s)
Adrian Baddeley .
See Also
lintess
Examples
X <- runiflpp(5, simplenet)
plot(lineardirichlet(X), lwd=3)
points(X)

lineardisc

lineardisc

695

Compute Disc of Given Radius in Linear Network

Description
Computes the ‘disc’ of given radius and centre in a linear network.
Usage
lineardisc(L, x = locator(1), r, plotit = TRUE,
cols=c("blue", "red","green"))
countends(L, x = locator(1), r, toler=NULL)
Arguments
L

Linear network (object of class "linnet").

x

Location of centre of disc. Either a point pattern (object of class "ppp") containing exactly 1 point, or a numeric vector of length 2.

r

Radius of disc.

plotit

Logical. Whether to plot the disc.

cols

Colours for plotting the disc. A numeric or character vector of length 3 specifying the colours of the disc centre, disc lines and disc endpoints respectively.

toler

Optional. Distance threshold for countends. See Details. There is a sensible
default.

Details
The ‘disc’ B(u, r) of centre x and radius r in a linear network L is the set of all points u in L such
that the shortest path distance from x to u is less than or equal to r. This is a union of line segments
contained in L.
The relative boundary of the disc B(u, r) is the set of points v such that the shortest path distance
from x to u is equal to r.
The function lineardisc computes the disc of radius r and its relative boundary, optionally plots
them, and returns them. The faster function countends simply counts the number of points in the
relative boundary.
The optional threshold toler is used to suppress numerical errors in countends. If the distance
from u to a network vertex v is between r-toler and r+toler, the vertex will be treated as lying
on the relative boundary.
Value
The value of lineardisc is a list with two entries:
lines

Line segment pattern (object of class "psp") representing the interior disc

endpoints

Point pattern (object of class "ppp") representing the relative boundary of the
disc.

The value of countends is an integer giving the number of points in the relative boundary.

696

linearK

Author(s)
Ang Qi Wei  and Adrian Baddeley 
References
Ang, Q.W. (2010) Statistical methodology for events on a network. Master’s thesis, School of
Mathematics and Statistics, University of Western Australia.
Ang, Q.W., Baddeley, A. and Nair, G. (2012) Geometrically corrected second-order analysis of
events on a linear network, with applications to ecology and criminology. Scandinavian Journal of
Statistics 39, 591–617.
See Also
linnet
Examples
# letter 'A'
v <- ppp(x=(-2):2, y=3*c(0,1,2,1,0), c(-3,3), c(-1,7))
edg <- cbind(1:4, 2:5)
edg <- rbind(edg, c(2,4))
letterA <- linnet(v, edges=edg)
lineardisc(letterA, c(0,3), 1.6)
# count the endpoints
countends(letterA, c(0,3), 1.6)
# cross-check (slower)
en <- lineardisc(letterA, c(0,3), 1.6, plotit=FALSE)$endpoints
npoints(en)

linearK

Linear K Function

Description
Computes an estimate of the linear K function for a point pattern on a linear network.
Usage
linearK(X, r=NULL, ..., correction="Ang", ratio=FALSE)
Arguments
X

Point pattern on linear network (object of class "lpp").

r

Optional. Numeric vector of values of the function argument r. There is a
sensible default.

...

Ignored.

correction

Geometry correction. Either "none" or "Ang". See Details.

ratio

Logical. If TRUE, the numerator and denominator of the estimate will also be
saved, for use in analysing replicated point patterns.

linearKcross

697

Details
This command computes the linear K function from point pattern data on a linear network.
If correction="none", the calculations do not include any correction for the geometry of the linear
network. The result is the network K function as defined by Okabe and Yamada (2001).
If correction="Ang", the pair counts are weighted using Ang’s correction (Ang, 2010; Ang et al,
2012).
Value
Function value table (object of class "fv").
Author(s)
Ang Qi Wei  and Adrian Baddeley .
References
Ang, Q.W. (2010) Statistical methodology for spatial point patterns on a linear network. MSc thesis,
University of Western Australia.
Ang, Q.W., Baddeley, A. and Nair, G. (2012) Geometrically corrected second-order analysis of
events on a linear network, with applications to ecology and criminology. Scandinavian Journal of
Statistics 39, 591–617.
Okabe, A. and Yamada, I. (2001) The K-function method on a network and its computational implementation. Geographical Analysis 33, 271-290.
See Also
compileK, lpp
Examples
data(simplenet)
X <- rpoislpp(5, simplenet)
linearK(X)
linearK(X, correction="none")

linearKcross

Multitype K Function (Cross-type) for Linear Point Pattern

Description
For a multitype point pattern on a linear network, estimate the multitype K function which counts
the expected number of points of type j within a given distance of a point of type i.
Usage
linearKcross(X, i, j, r=NULL, ..., correction="Ang")

698

linearKcross

Arguments
X

The observed point pattern, from which an estimate of the cross type K function
Kij (r) will be computed. An object of class "lpp" which must be a multitype
point pattern (a marked point pattern whose marks are a factor).

i

Number or character string identifying the type (mark value) of the points in X
from which distances are measured. Defaults to the first level of marks(X).

j

Number or character string identifying the type (mark value) of the points in X
to which distances are measured. Defaults to the second level of marks(X).

r

numeric vector. The values of the argument r at which the K-function Kij (r)
should be evaluated. There is a sensible default. First-time users are strongly
advised not to specify this argument. See below for important conditions on r.

correction

Geometry correction. Either "none" or "Ang". See Details.

...

Ignored.

Details
This is a counterpart of the function Kcross for a point pattern on a linear network (object of class
"lpp").
The arguments i and j will be interpreted as levels of the factor marks(X). If i and j are missing,
they default to the first and second level of the marks factor, respectively.
The argument r is the vector of values for the distance r at which Kij (r) should be evaluated. The
values of r must be increasing nonnegative numbers and the maximum r value must not exceed the
radius of the largest disc contained in the window.
Value
An object of class "fv" (see fv.object).
Warnings
The arguments i and j are interpreted as levels of the factor marks(X). Beware of the usual trap
with factors: numerical values are not interpreted in the same way as character values.
Author(s)
Adrian Baddeley 
References
Baddeley, A, Jammalamadaka, A. and Nair, G. (to appear) Multitype point process analysis of
spines on the dendrite network of a neuron. Applied Statistics (Journal of the Royal Statistical
Society, Series C), In press.
See Also
linearKdot, linearK.
Examples
data(chicago)
K <- linearKcross(chicago, "assault", "robbery")

linearKcross.inhom

699

linearKcross.inhom

Inhomogeneous multitype K Function (Cross-type) for Linear Point
Pattern

Description
For a multitype point pattern on a linear network, estimate the inhomogeneous multitype K function
which counts the expected number of points of type j within a given distance of a point of type i.
Usage
linearKcross.inhom(X, i, j, lambdaI, lambdaJ,
r=NULL, ..., correction="Ang", normalise=TRUE)
Arguments
X

The observed point pattern, from which an estimate of the cross type K function
Kij (r) will be computed. An object of class "lpp" which must be a multitype
point pattern (a marked point pattern whose marks are a factor).

i

Number or character string identifying the type (mark value) of the points in X
from which distances are measured. Defaults to the first level of marks(X).

j

Number or character string identifying the type (mark value) of the points in X
to which distances are measured. Defaults to the second level of marks(X).

lambdaI

Intensity values for the points of type i. Either a numeric vector, a function,
a pixel image (object of class "im" or "linim") or a fitted point process model
(object of class "ppm" or "lppm").

lambdaJ

Intensity values for the points of type j. Either a numeric vector, a function,
a pixel image (object of class "im" or "linim") or a fitted point process model
(object of class "ppm" or "lppm").

r

numeric vector. The values of the argument r at which the K-function Kij (r)
should be evaluated. There is a sensible default. First-time users are strongly
advised not to specify this argument. See below for important conditions on r.

correction

Geometry correction. Either "none" or "Ang". See Details.

...

Arguments passed to lambdaI and lambdaJ if they are functions.

normalise

Logical. If TRUE (the default), the denominator of the estimator is data-dependent
(equal to the sum of the reciprocal intensities at the points of type i), which reduces the sampling variability. If FALSE, the denominator is the length of the
network.

Details
This is a counterpart of the function Kcross.inhom for a point pattern on a linear network (object
of class "lpp").
The arguments i and j will be interpreted as levels of the factor marks(X). If i and j are missing,
they default to the first and second level of the marks factor, respectively.
The argument r is the vector of values for the distance r at which Kij (r) should be evaluated. The
values of r must be increasing nonnegative numbers and the maximum r value must not exceed the
radius of the largest disc contained in the window.

700

linearKdot
If lambdaI or lambdaJ is a fitted point process model, the default behaviour is to update the model
by re-fitting it to the data, before computing the fitted intensity. This can be disabled by setting
update=FALSE.

Value
An object of class "fv" (see fv.object).
Warnings
The arguments i and j are interpreted as levels of the factor marks(X). Beware of the usual trap
with factors: numerical values are not interpreted in the same way as character values.
Author(s)
Adrian Baddeley 
References
Baddeley, A, Jammalamadaka, A. and Nair, G. (to appear) Multitype point process analysis of
spines on the dendrite network of a neuron. Applied Statistics (Journal of the Royal Statistical
Society, Series C), In press.
See Also
linearKdot, linearK.
Examples
lam <- table(marks(chicago))/(summary(chicago)$totlength)
lamI <- function(x,y,const=lam[["assault"]]){ rep(const, length(x)) }
lamJ <- function(x,y,const=lam[["robbery"]]){ rep(const, length(x)) }
K <- linearKcross.inhom(chicago, "assault", "robbery", lamI, lamJ)
## Not run:
fit <- lppm(chicago, ~marks + x)
linearKcross.inhom(chicago, "assault", "robbery", fit, fit)
## End(Not run)

linearKdot

Multitype K Function (Dot-type) for Linear Point Pattern

Description
For a multitype point pattern on a linear network, estimate the multitype K function which counts
the expected number of points (of any type) within a given distance of a point of type i.
Usage
linearKdot(X, i, r=NULL, ..., correction="Ang")

linearKdot

701

Arguments
X

The observed point pattern, from which an estimate of the dot type K function
Ki• (r) will be computed. An object of class "lpp" which must be a multitype
point pattern (a marked point pattern whose marks are a factor).

i

Number or character string identifying the type (mark value) of the points in X
from which distances are measured. Defaults to the first level of marks(X).

r

numeric vector. The values of the argument r at which the K-function Ki• (r)
should be evaluated. There is a sensible default. First-time users are strongly
advised not to specify this argument. See below for important conditions on r.

correction

Geometry correction. Either "none" or "Ang". See Details.

...

Ignored.

Details
This is a counterpart of the function Kdot for a point pattern on a linear network (object of class
"lpp").
The argument i will be interpreted as levels of the factor marks(X). If i is missing, it defaults to
the first level of the marks factor.
The argument r is the vector of values for the distance r at which Ki• (r) should be evaluated. The
values of r must be increasing nonnegative numbers and the maximum r value must not exceed the
radius of the largest disc contained in the window.
Value
An object of class "fv" (see fv.object).
Warnings
The argument i is interpreted as a level of the factor marks(X). Beware of the usual trap with
factors: numerical values are not interpreted in the same way as character values.
Author(s)
Adrian Baddeley 
References
Baddeley, A, Jammalamadaka, A. and Nair, G. (to appear) Multitype point process analysis of
spines on the dendrite network of a neuron. Applied Statistics (Journal of the Royal Statistical
Society, Series C), In press.
See Also
Kdot, linearKcross, linearK.
Examples
data(chicago)
K <- linearKdot(chicago, "assault")

702

linearKdot.inhom

linearKdot.inhom

Inhomogeneous multitype K Function (Dot-type) for Linear Point Pattern

Description
For a multitype point pattern on a linear network, estimate the inhomogeneous multitype K function
which counts the expected number of points (of any type) within a given distance of a point of type
i.
Usage
linearKdot.inhom(X, i, lambdaI, lambdadot, r=NULL, ...,
correction="Ang", normalise=TRUE)
Arguments
X

i
lambdaI

lambdadot

r

correction
...
normalise

The observed point pattern, from which an estimate of the dot type K function
Ki• (r) will be computed. An object of class "lpp" which must be a multitype
point pattern (a marked point pattern whose marks are a factor).
Number or character string identifying the type (mark value) of the points in X
from which distances are measured. Defaults to the first level of marks(X).
Intensity values for the points of type i. Either a numeric vector, a function,
a pixel image (object of class "im" or "linim") or a fitted point process model
(object of class "ppm" or "lppm").
Intensity values for all points of X. Either a numeric vector, a function, a pixel
image (object of class "im" or "linim") or a fitted point process model (object
of class "ppm" or "lppm").
numeric vector. The values of the argument r at which the K-function Ki• (r)
should be evaluated. There is a sensible default. First-time users are strongly
advised not to specify this argument. See below for important conditions on r.
Geometry correction. Either "none" or "Ang". See Details.
Arguments passed to lambdaI and lambdadot if they are functions.
Logical. If TRUE (the default), the denominator of the estimator is data-dependent
(equal to the sum of the reciprocal intensities at the points of type i), which reduces the sampling variability. If FALSE, the denominator is the length of the
network.

Details
This is a counterpart of the function Kdot.inhom for a point pattern on a linear network (object of
class "lpp").
The argument i will be interpreted as levels of the factor marks(X). If i is missing, it defaults to
the first level of the marks factor.
The argument r is the vector of values for the distance r at which Ki• (r) should be evaluated. The
values of r must be increasing nonnegative numbers and the maximum r value must not exceed the
radius of the largest disc contained in the window.
If lambdaI or lambdadot is a fitted point process model, the default behaviour is to update the
model by re-fitting it to the data, before computing the fitted intensity. This can be disabled by
setting update=FALSE.

linearKinhom

703

Value
An object of class "fv" (see fv.object).
Warnings
The argument i is interpreted as a level of the factor marks(X). Beware of the usual trap with
factors: numerical values are not interpreted in the same way as character values.
Author(s)
Adrian Baddeley 
References
Baddeley, A, Jammalamadaka, A. and Nair, G. (to appear) Multitype point process analysis of
spines on the dendrite network of a neuron. Applied Statistics (Journal of the Royal Statistical
Society, Series C), In press.
See Also
linearKdot, linearK.
Examples
lam <- table(marks(chicago))/(summary(chicago)$totlength)
lamI <- function(x,y,const=lam[["assault"]]){ rep(const, length(x)) }
lam. <- function(x,y,const=sum(lam)){ rep(const, length(x)) }
K <- linearKdot.inhom(chicago, "assault", lamI, lam.)
## Not run:
fit <- lppm(chicago, ~marks + x)
linearKdot.inhom(chicago, "assault", fit, fit)
## End(Not run)

linearKinhom

Inhomogeneous Linear K Function

Description
Computes an estimate of the inhomogeneous linear K function for a point pattern on a linear network.
Usage
linearKinhom(X, lambda=NULL, r=NULL, ..., correction="Ang",
normalise=TRUE, normpower=1,
update=TRUE, leaveoneout=TRUE, ratio=FALSE)

704

linearKinhom

Arguments
X

Point pattern on linear network (object of class "lpp").

lambda

Intensity values for the point pattern. Either a numeric vector, a function, a
pixel image (object of class "im" or "linim") or a fitted point process model
(object of class "ppm" or "lppm").

r

Optional. Numeric vector of values of the function argument r. There is a
sensible default.

...

Ignored.

correction

Geometry correction. Either "none" or "Ang". See Details.

normalise

Logical. If TRUE (the default), the denominator of the estimator is data-dependent
(equal to the sum of the reciprocal intensities at the data points, raised to normpower),
which reduces the sampling variability. If FALSE, the denominator is the length
of the network.

normpower

Integer (usually either 1 or 2). Normalisation power. See Details.

update

Logical value indicating what to do when lambda is a fitted model (class "lppm"
or "ppm"). If update=TRUE (the default), the model will first be refitted to the
data X (using update.lppm or update.ppm) before the fitted intensity is computed. If update=FALSE, the fitted intensity of the model will be computed
without re-fitting it to X.

leaveoneout

Logical value (passed to fitted.lppm or fitted.ppm) specifying whether to
use a leave-one-out rule when calculating the intensity, when lambda is a fitted
model. Supported only when update=TRUE.

ratio

Logical. If TRUE, the numerator and denominator of the estimate will also be
saved, for use in analysing replicated point patterns.

Details
This command computes the inhomogeneous version of the linear K function from point pattern
data on a linear network.
If lambda = NULL the result is equivalent to the homogeneous K function linearK. If lambda is
given, then it is expected to provide estimated values of the intensity of the point process at each
point of X. The argument lambda may be a numeric vector (of length equal to the number of points
in X), or a function(x,y) that will be evaluated at the points of X to yield numeric values, or a
pixel image (object of class "im") or a fitted point process model (object of class "ppm" or "lppm").
If lambda is a fitted point process model, the default behaviour is to update the model by re-fitting
it to the data, before computing the fitted intensity. This can be disabled by setting update=FALSE.
If correction="none", the calculations do not include any correction for the geometry of the linear
network. If correction="Ang", the pair counts are weighted using Ang’s correction (Ang, 2010).
Each estimate is initially computed as
b inhom (r) =
K

X X 1{dij ≤ r}e(xi , xj )
1
length(L) i j
λ(xi )λ(xj )

where L is the linear network, dij is the distance between points xi and xj , and e(xi , xj ) is a
weight. If correction="none" then this weight is equal to 1, while if correction="Ang" the
weight is e(xi , xj , r) = 1/m(xi , dij ) where m(u, t) is the number of locations on the network that
lie exactly t units distant from location u by the shortest path.

linearmarkconnect

705

If normalise=TRUE (thePdefault), then the estimates described above are multiplied by cnormpower
where c = length(L)/ (1/λ(xi )). This rescaling reduces the variability and bias of the estimate
in small samples and in cases of very strong inhomogeneity. The default value of normpower is 1
(for consistency with previous versions of spatstat)
P but the most sensible value is 2, which would
correspond to rescaling the lambda values so that (1/λ(xi )) = area(W ).
Value
Function value table (object of class "fv").
Author(s)
Ang Qi Wei  and Adrian Baddeley 
References
Ang, Q.W. (2010) Statistical methodology for spatial point patterns on a linear network. MSc thesis,
University of Western Australia.
Ang, Q.W., Baddeley, A. and Nair, G. (2012) Geometrically corrected second-order analysis of
events on a linear network, with applications to ecology and criminology. Scandinavian Journal of
Statistics 39, 591–617.
See Also
lpp
Examples
data(simplenet)
X <- rpoislpp(5, simplenet)
fit <- lppm(X ~x)
K <- linearKinhom(X, lambda=fit)
plot(K)

linearmarkconnect

Mark Connection Function for Multitype Point Pattern on Linear Network

Description
For a multitype point pattern on a linear network, estimate the mark connection function from points
of type i to points of type j.
Usage
linearmarkconnect(X, i, j, r=NULL, ...)

706

linearmarkconnect

Arguments
X

i
j
r

...

The observed point pattern, from which an estimate of the mark connection
function pij (r) will be computed. An object of class "lpp" which must be a
multitype point pattern (a marked point pattern whose marks are a factor).
Number or character string identifying the type (mark value) of the points in X
from which distances are measured. Defaults to the first level of marks(X).
Number or character string identifying the type (mark value) of the points in X
to which distances are measured. Defaults to the second level of marks(X).
numeric vector. The values of the argument r at which the function pij (r) should
be evaluated. There is a sensible default. First-time users are strongly advised
not to specify this argument. See below for important conditions on r.
Arguments passed to linearpcfcross and linearpcf.

Details
This is a counterpart of the function markconnect for a point pattern on a linear network (object of
class "lpp").
The argument i will be interpreted as levels of the factor marks(X). If i is missing, it defaults to
the first level of the marks factor.
The argument r is the vector of values for the distance r at which pij (r) should be evaluated. The
values of r must be increasing nonnegative numbers and the maximum r value must not exceed the
radius of the largest disc contained in the window.
Value
An object of class "fv" (see fv.object).
Warnings
The argument i is interpreted as a level of the factor marks(X). Beware of the usual trap with
factors: numerical values are not interpreted in the same way as character values.
Author(s)
Adrian Baddeley 
References
Baddeley, A, Jammalamadaka, A. and Nair, G. (to appear) Multitype point process analysis of
spines on the dendrite network of a neuron. Applied Statistics (Journal of the Royal Statistical
Society, Series C), In press.
See Also
linearpcfcross, linearpcf, linearmarkequal, markconnect.
Examples
pab <- linearmarkconnect(chicago, "assault", "burglary")
## Not run:
plot(alltypes(chicago, linearmarkconnect))
## End(Not run)

linearmarkequal

linearmarkequal

707

Mark Connection Function for Multitype Point Pattern on Linear Network

Description
For a multitype point pattern on a linear network, estimate the mark connection function from points
of type i to points of type j.
Usage
linearmarkequal(X, r=NULL, ...)
Arguments
X

The observed point pattern, from which an estimate of the mark connection
function pij (r) will be computed. An object of class "lpp" which must be a
multitype point pattern (a marked point pattern whose marks are a factor).

r

numeric vector. The values of the argument r at which the function pij (r) should
be evaluated. There is a sensible default. First-time users are strongly advised
not to specify this argument. See below for important conditions on r.

...

Arguments passed to linearpcfcross and linearpcf.

Details
This is the mark equality function for a point pattern on a linear network (object of class "lpp").
The argument r is the vector of values for the distance r at which pij (r) should be evaluated. The
values of r must be increasing nonnegative numbers and the maximum r value must not exceed the
radius of the largest disc contained in the window.
Value
An object of class "fv" (see fv.object).
Author(s)
Adrian Baddeley 
References
Baddeley, A, Jammalamadaka, A. and Nair, G. (to appear) Multitype point process analysis of
spines on the dendrite network of a neuron. Applied Statistics (Journal of the Royal Statistical
Society, Series C), In press.
See Also
linearpcfcross, linearpcf, linearmarkconnect, markconnect.

708

linearpcf

Examples
if(interactive()) {
X <- chicago
} else {
X <- runiflpp(20, simplenet) %mark% sample(c("A","B"), 20,
replace=TRUE)
}
p <- linearmarkequal(X)

linearpcf

Linear Pair Correlation Function

Description
Computes an estimate of the linear pair correlation function for a point pattern on a linear network.
Usage
linearpcf(X, r=NULL, ..., correction="Ang", ratio=FALSE)
Arguments
X

Point pattern on linear network (object of class "lpp").

r

Optional. Numeric vector of values of the function argument r. There is a
sensible default.

...

Arguments passed to density.default to control the smoothing.

correction

Geometry correction. Either "none" or "Ang". See Details.

ratio

Logical. If TRUE, the numerator and denominator of each estimate will also be
saved, for use in analysing replicated point patterns.

Details
This command computes the linear pair correlation function from point pattern data on a linear
network.
The pair correlation function is estimated from the shortest-path distances between each pair of
data points, using the fixed-bandwidth kernel smoother density.default, with a bias correction
at each end of the interval of r values. To switch off the bias correction, set endcorrect=FALSE.
The bandwidth for smoothing the pairwise distances is determined by arguments ... passed to
density.default, mainly the arguments bw and adjust. The default is to choose the bandwidth
by Silverman’s rule of thumb bw="nrd0" explained in density.default.
If correction="none", the calculations do not include any correction for the geometry of the linear
network. The result is an estimate of the first derivative of the network K function defined by Okabe
and Yamada (2001).
If correction="Ang", the pair counts are weighted using Ang’s correction (Ang, 2010). The result
is an estimate of the pair correlation function in the linear network.

linearpcfcross

709

Value
Function value table (object of class "fv").
If ratio=TRUE then the return value also has two attributes called "numerator" and "denominator"
which are "fv" objects containing the numerators and denominators of each estimate of g(r).
Author(s)
Ang Qi Wei  and Adrian Baddeley .
References
Ang, Q.W. (2010) Statistical methodology for spatial point patterns on a linear network. MSc thesis,
University of Western Australia.
Ang, Q.W., Baddeley, A. and Nair, G. (2012) Geometrically corrected second-order analysis of
events on a linear network, with applications to ecology and criminology. Scandinavian Journal of
Statistics 39, 591–617.
Okabe, A. and Yamada, I. (2001) The K-function method on a network and its computational implementation. Geographical Analysis 33, 271-290.
See Also
linearK, linearpcfinhom, lpp
Examples
data(simplenet)
X <- rpoislpp(5, simplenet)
linearpcf(X)
linearpcf(X, correction="none")

linearpcfcross

Multitype Pair Correlation Function (Cross-type) for Linear Point Pattern

Description
For a multitype point pattern on a linear network, estimate the multitype pair correlation function
from points of type i to points of type j.
Usage
linearpcfcross(X, i, j, r=NULL, ..., correction="Ang")
Arguments
X

The observed point pattern, from which an estimate of the i-to-any pair correlation function gij (r) will be computed. An object of class "lpp" which must be
a multitype point pattern (a marked point pattern whose marks are a factor).

i

Number or character string identifying the type (mark value) of the points in X
from which distances are measured. Defaults to the first level of marks(X).

710

linearpcfcross
j

Number or character string identifying the type (mark value) of the points in X
to which distances are measured. Defaults to the second level of marks(X).

r

numeric vector. The values of the argument r at which the function gij (r) should
be evaluated. There is a sensible default. First-time users are strongly advised
not to specify this argument. See below for important conditions on r.

correction

Geometry correction. Either "none" or "Ang". See Details.

...

Arguments passed to density.default to control the kernel smoothing.

Details
This is a counterpart of the function pcfcross for a point pattern on a linear network (object of
class "lpp").
The argument i will be interpreted as levels of the factor marks(X). If i is missing, it defaults to
the first level of the marks factor.
The argument r is the vector of values for the distance r at which gij (r) should be evaluated. The
values of r must be increasing nonnegative numbers and the maximum r value must not exceed the
radius of the largest disc contained in the window.
Value
An object of class "fv" (see fv.object).
Warnings
The argument i is interpreted as a level of the factor marks(X). Beware of the usual trap with
factors: numerical values are not interpreted in the same way as character values.
Author(s)
Adrian Baddeley 
References
Baddeley, A, Jammalamadaka, A. and Nair, G. (to appear) Multitype point process analysis of
spines on the dendrite network of a neuron. Applied Statistics (Journal of the Royal Statistical
Society, Series C), In press.
See Also
linearpcfdot, linearpcf, pcfcross.
Examples
data(chicago)
g <- linearpcfcross(chicago, "assault")

linearpcfcross.inhom

711

linearpcfcross.inhom

Inhomogeneous Multitype Pair Correlation Function (Cross-type) for
Linear Point Pattern

Description
For a multitype point pattern on a linear network, estimate the inhomogeneous multitype pair correlation function from points of type i to points of type j.
Usage
linearpcfcross.inhom(X, i, j, lambdaI, lambdaJ, r=NULL, ...,
correction="Ang", normalise=TRUE)
Arguments
X

The observed point pattern, from which an estimate of the i-to-any pair correlation function gij (r) will be computed. An object of class "lpp" which must be
a multitype point pattern (a marked point pattern whose marks are a factor).

i

Number or character string identifying the type (mark value) of the points in X
from which distances are measured. Defaults to the first level of marks(X).

j

Number or character string identifying the type (mark value) of the points in X
to which distances are measured. Defaults to the second level of marks(X).

lambdaI

Intensity values for the points of type i. Either a numeric vector, a function,
a pixel image (object of class "im" or "linim") or a fitted point process model
(object of class "ppm" or "lppm").

lambdaJ

Intensity values for the points of type j. Either a numeric vector, a function,
a pixel image (object of class "im" or "linim") or a fitted point process model
(object of class "ppm" or "lppm").

r

numeric vector. The values of the argument r at which the function gij (r) should
be evaluated. There is a sensible default. First-time users are strongly advised
not to specify this argument. See below for important conditions on r.

correction

Geometry correction. Either "none" or "Ang". See Details.

...

Arguments passed to density.default to control the kernel smoothing.

normalise

Logical. If TRUE (the default), the denominator of the estimator is data-dependent
(equal to the sum of the reciprocal intensities at the points of type i), which reduces the sampling variability. If FALSE, the denominator is the length of the
network.

Details
This is a counterpart of the function pcfcross.inhom for a point pattern on a linear network (object
of class "lpp").
The argument i will be interpreted as levels of the factor marks(X). If i is missing, it defaults to
the first level of the marks factor.
The argument r is the vector of values for the distance r at which gij (r) should be evaluated. The
values of r must be increasing nonnegative numbers and the maximum r value must not exceed the
radius of the largest disc contained in the window.

712

linearpcfdot
If lambdaI or lambdaJ is a fitted point process model, the default behaviour is to update the model
by re-fitting it to the data, before computing the fitted intensity. This can be disabled by setting
update=FALSE.

Value
An object of class "fv" (see fv.object).
Warnings
The argument i is interpreted as a level of the factor marks(X). Beware of the usual trap with
factors: numerical values are not interpreted in the same way as character values.
Author(s)
Adrian Baddeley 
References
Baddeley, A, Jammalamadaka, A. and Nair, G. (to appear) Multitype point process analysis of
spines on the dendrite network of a neuron. Applied Statistics (Journal of the Royal Statistical
Society, Series C), In press.
See Also
linearpcfdot, linearpcf, pcfcross.inhom.
Examples
lam <- table(marks(chicago))/(summary(chicago)$totlength)
lamI <- function(x,y,const=lam[["assault"]]){ rep(const, length(x)) }
lamJ <- function(x,y,const=lam[["robbery"]]){ rep(const, length(x)) }
g <- linearpcfcross.inhom(chicago, "assault", "robbery", lamI, lamJ)
## Not run:
fit <- lppm(chicago, ~marks + x)
linearpcfcross.inhom(chicago, "assault", "robbery", fit, fit)
## End(Not run)

linearpcfdot

Multitype Pair Correlation Function (Dot-type) for Linear Point Pattern

Description
For a multitype point pattern on a linear network, estimate the multitype pair correlation function
from points of type i to points of any type.
Usage
linearpcfdot(X, i, r=NULL, ..., correction="Ang")

linearpcfdot

713

Arguments
X

The observed point pattern, from which an estimate of the i-to-any pair correlation function gi• (r) will be computed. An object of class "lpp" which must be
a multitype point pattern (a marked point pattern whose marks are a factor).

i

Number or character string identifying the type (mark value) of the points in X
from which distances are measured. Defaults to the first level of marks(X).

r

numeric vector. The values of the argument r at which the function gi• (r) should
be evaluated. There is a sensible default. First-time users are strongly advised
not to specify this argument. See below for important conditions on r.

correction

Geometry correction. Either "none" or "Ang". See Details.

...

Arguments passed to density.default to control the kernel smoothing.

Details
This is a counterpart of the function pcfdot for a point pattern on a linear network (object of class
"lpp").
The argument i will be interpreted as levels of the factor marks(X). If i is missing, it defaults to
the first level of the marks factor.
The argument r is the vector of values for the distance r at which gi• (r) should be evaluated. The
values of r must be increasing nonnegative numbers and the maximum r value must not exceed the
radius of the largest disc contained in the window.
Value
An object of class "fv" (see fv.object).
Warnings
The argument i is interpreted as a level of the factor marks(X). Beware of the usual trap with
factors: numerical values are not interpreted in the same way as character values.
Author(s)
Adrian Baddeley 
References
Baddeley, A, Jammalamadaka, A. and Nair, G. (to appear) Multitype point process analysis of
spines on the dendrite network of a neuron. Applied Statistics (Journal of the Royal Statistical
Society, Series C), In press.
See Also
linearpcfcross, linearpcf. pcfcross.
Examples
data(chicago)
g <- linearpcfdot(chicago, "assault")

714

linearpcfdot.inhom

linearpcfdot.inhom

Inhomogeneous Multitype Pair Correlation Function (Dot-type) for
Linear Point Pattern

Description
For a multitype point pattern on a linear network, estimate the inhomogeneous multitype pair correlation function from points of type i to points of any type.
Usage
linearpcfdot.inhom(X, i, lambdaI, lambdadot, r=NULL, ...,
correction="Ang", normalise=TRUE)
Arguments
X

The observed point pattern, from which an estimate of the i-to-any pair correlation function gi• (r) will be computed. An object of class "lpp" which must be
a multitype point pattern (a marked point pattern whose marks are a factor).

i

Number or character string identifying the type (mark value) of the points in X
from which distances are measured. Defaults to the first level of marks(X).

lambdaI

Intensity values for the points of type i. Either a numeric vector, a function,
a pixel image (object of class "im" or "linim") or a fitted point process model
(object of class "ppm" or "lppm").

lambdadot

Intensity values for all points of X. Either a numeric vector, a function, a pixel
image (object of class "im" or "linim") or a fitted point process model (object
of class "ppm" or "lppm").

r

numeric vector. The values of the argument r at which the function gi• (r) should
be evaluated. There is a sensible default. First-time users are strongly advised
not to specify this argument. See below for important conditions on r.

correction

Geometry correction. Either "none" or "Ang". See Details.

...

Arguments passed to density.default to control the kernel smoothing.

normalise

Logical. If TRUE (the default), the denominator of the estimator is data-dependent
(equal to the sum of the reciprocal intensities at the points of type i), which reduces the sampling variability. If FALSE, the denominator is the length of the
network.

Details
This is a counterpart of the function pcfdot.inhom for a point pattern on a linear network (object
of class "lpp").
The argument i will be interpreted as levels of the factor marks(X). If i is missing, it defaults to
the first level of the marks factor.
The argument r is the vector of values for the distance r at which gi• (r) should be evaluated. The
values of r must be increasing nonnegative numbers and the maximum r value must not exceed the
radius of the largest disc contained in the window.
If lambdaI or lambdadot is a fitted point process model, the default behaviour is to update the
model by re-fitting it to the data, before computing the fitted intensity. This can be disabled by
setting update=FALSE.

linearpcfinhom

715

Value
An object of class "fv" (see fv.object).
Warnings
The argument i is interpreted as a level of the factor marks(X). Beware of the usual trap with
factors: numerical values are not interpreted in the same way as character values.
Author(s)
Adrian Baddeley 
References
Baddeley, A, Jammalamadaka, A. and Nair, G. (to appear) Multitype point process analysis of
spines on the dendrite network of a neuron. Applied Statistics (Journal of the Royal Statistical
Society, Series C), In press.
See Also
linearpcfcross.inhom, linearpcfcross, pcfcross.inhom.
Examples
lam <- table(marks(chicago))/(summary(chicago)$totlength)
lamI <- function(x,y,const=lam[["assault"]]){ rep(const, length(x)) }
lam. <- function(x,y,const=sum(lam)){ rep(const, length(x)) }
g <- linearpcfdot.inhom(chicago, "assault", lamI, lam.)
## Not run:
fit <- lppm(chicago, ~marks + x)
linearpcfdot.inhom(chicago, "assault", fit, fit)
## End(Not run)

linearpcfinhom

Inhomogeneous Linear Pair Correlation Function

Description
Computes an estimate of the inhomogeneous linear pair correlation function for a point pattern on
a linear network.
Usage
linearpcfinhom(X, lambda=NULL, r=NULL, ..., correction="Ang",
normalise=TRUE, normpower=1,
update = TRUE, leaveoneout = TRUE,
ratio = FALSE)

716

linearpcfinhom

Arguments
X

Point pattern on linear network (object of class "lpp").

lambda

Intensity values for the point pattern. Either a numeric vector, a function, a
pixel image (object of class "im") or a fitted point process model (object of
class "ppm" or "lppm").

r

Optional. Numeric vector of values of the function argument r. There is a
sensible default.

...

Arguments passed to density.default to control the smoothing.

correction

Geometry correction. Either "none" or "Ang". See Details.

normalise

Logical. If TRUE (the default), the denominator of the estimator is data-dependent
(equal to the sum of the reciprocal intensities at the data points, raised to normpower),
which reduces the sampling variability. If FALSE, the denominator is the length
of the network.

normpower

Integer (usually either 1 or 2). Normalisation power. See explanation in linearKinhom.

update

Logical value indicating what to do when lambda is a fitted model (class "lppm"
or "ppm"). If update=TRUE (the default), the model will first be refitted to the
data X (using update.lppm or update.ppm) before the fitted intensity is computed. If update=FALSE, the fitted intensity of the model will be computed
without re-fitting it to X.

leaveoneout

Logical value (passed to fitted.lppm or fitted.ppm) specifying whether to
use a leave-one-out rule when calculating the intensity, when lambda is a fitted
model. Supported only when update=TRUE.

ratio

Logical. If TRUE, the numerator and denominator of each estimate will also be
saved, for use in analysing replicated point patterns.

Details
This command computes the inhomogeneous version of the linear pair correlation function from
point pattern data on a linear network.
If lambda = NULL the result is equivalent to the homogeneous pair correlation function linearpcf.
If lambda is given, then it is expected to provide estimated values of the intensity of the point
process at each point of X. The argument lambda may be a numeric vector (of length equal to the
number of points in X), or a function(x,y) that will be evaluated at the points of X to yield numeric
values, or a pixel image (object of class "im") or a fitted point process model (object of class "ppm"
or "lppm").
If lambda is a fitted point process model, the default behaviour is to update the model by re-fitting
it to the data, before computing the fitted intensity. This can be disabled by setting update=FALSE.
If correction="none", the calculations do not include any correction for the geometry of the linear
network. If correction="Ang", the pair counts are weighted using Ang’s correction (Ang, 2010).
The bandwidth for smoothing the pairwise distances is determined by arguments ... passed to
density.default, mainly the arguments bw and adjust. The default is to choose the bandwidth
by Silverman’s rule of thumb bw="nrd0" explained in density.default.
Value
Function value table (object of class "fv").
If ratio=TRUE then the return value also has two attributes called "numerator" and "denominator"
which are "fv" objects containing the numerators and denominators of each estimate of g(r).

linequad

717

Author(s)
Ang Qi Wei  and Adrian Baddeley .
References
Ang, Q.W. (2010) Statistical methodology for spatial point patterns on a linear network. MSc thesis,
University of Western Australia.
Ang, Q.W., Baddeley, A. and Nair, G. (2012) Geometrically corrected second-order analysis of
events on a linear network, with applications to ecology and criminology. Scandinavian Journal of
Statistics 39, 591–617.
Okabe, A. and Yamada, I. (2001) The K-function method on a network and its computational implementation. Geographical Analysis 33, 271-290.
See Also
linearpcf, linearKinhom, lpp
Examples
data(simplenet)
X <- rpoislpp(5, simplenet)
fit <- lppm(X ~x)
K <- linearpcfinhom(X, lambda=fit)
plot(K)

linequad

Quadrature Scheme on a Linear Network

Description
Generates a quadrature scheme (an object of class "quad") on a linear network.
Usage
linequad(X, Y, ..., eps = NULL, nd = 1000, random = FALSE)
Arguments
X

Data points. An object of class "lpp" or "ppp".

Y

Line segments on which the points of X lie. An object of class "psp". Required
only when X is a "ppp" object.

...

Ignored.

eps

Optional. Spacing between successive dummy points along each segment.

nd

Optional. Total number of dummy points to be generated.

random

Logical value indicating whether the sequence of dummy points should start at
a randomly-chosen position along each segment.

718

linfun

Details
This command generates a quadrature scheme (object of class "quad") from a pattern of points on
a linear network.
Normally the user does not need to call linequad explicitly. It is invoked by spatstat functions
when needed. A quadrature scheme is required by lppm in order to fit point process models to
point pattern data on a linear network. A quadrature scheme is also used by rhohat.lpp and other
functions.
In order to create the quadrature scheme, dummy points are placed along each line segment of the
network. The dummy points are evenly-spaced with spacing eps. The default is eps = totlen/nd
where totlen is the total length of all line segments in the network.
Value
A quadrature scheme (object of class "quad").
Author(s)
Adrian Baddeley , Greg McSwiggan and Suman Rakshit.
See Also
lppm

linfun

Function on a Linear Network

Description
Create a function on a linear network.
Usage
linfun(f, L)
Arguments
f

A function in the R language.

L

A linear network (object of class "linnet") on which f is defined.

Details
This creates an object of class "linfun". This is a simple mechanism for handling a function
defined on a linear network, to make it easier to display and manipulate.
f should be a function in the R language, with formal arguments f(x,y,seg,tp) or f(x,y,seg,tp, ...)
where x,y are Cartesian coordinates of locations on the linear network, seg, tp are the local coordinates, and ... are optional additional arguments.
The function f should be vectorised: that is, if x,y,seg,tp are numeric vectors of the same length
n, then v <- f(x,y,seg,tp) should be a vector of length n.
L should be a linear network (object of class "linnet") inside which the function f is well-defined.

Linhom

719

The result is a function g in the R language which belongs to the special class "linfun". This
function can be called as g(X) where X is an "lpp" object, or called as g(x,y) or g(x,y,seg,tp)
where x,y,seg,tp are coordinates. There are several methods for this class including print, plot
and as.linim.
Value
A function in the R\ language. It also belongs to the class "linfun" which has methods for plot,
print etc.
Author(s)
Adrian Baddeley  and Rolf Turner 
See Also
methods.linfun for methods applicable to "linfun" objects.
distfun.lpp, nnfun.lpp.
Examples
f <- linfun(function(x,y,seg,tp) { x+y }, simplenet)
plot(f)
X <- runiflpp(3, simplenet)
plot(X, add=TRUE, cex=2)
f(X)

Linhom

L-function

Description
Calculates an estimate of the inhomogeneous version of the L-function (Besag’s transformation of
Ripley’s K-function) for a spatial point pattern.
Usage
Linhom(...)
Arguments
...

Arguments passed to Kinhom to estimate the inhomogeneous K-function.

Details
This command computes an estimate of the inhomogeneous version of the L-function for a spatial
point pattern
The original L-function is a transformation (proposed by Besag) of Ripley’s K-function,
r
K(r)
L(r) =
π
where K(r) is the Ripley K-function of a spatially homogeneous point pattern, estimated by Kest.

720

linim
The inhomogeneous L-function is the corresponding transformation of the inhomogeneous Kfunction, estimated by Kinhom. It is appropriate when the point pattern clearly does not have a
homogeneous intensity of points. It was proposed by Baddeley, Møller and Waagepetersen (2000).
The command Linhom first calls Kinhom to compute the estimate of the inhomogeneous K-function,
and then applies the square root transformation.
For a Poisson point pattern (homogeneous or inhomogeneous), the theoretical value of the inhomogeneous L-function is L(r) = r. The square root also has the effect of stabilising the variance of
the estimator, so that L is more appropriate for use in simulation envelopes and hypothesis tests.

Value
An object of class "fv", see fv.object, which can be plotted directly using plot.fv.
Essentially a data frame containing columns
r

the vector of values of the argument r at which the function L has been estimated

theo

the theoretical value L(r) = r for a stationary Poisson process

together with columns named "border", "bord.modif", "iso" and/or "trans", according to the
selected edge corrections. These columns contain estimates of the function L(r) obtained by the
edge corrections named.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Baddeley, A., Møller, J. and Waagepetersen, R. (2000) Non- and semiparametric estimation of
interaction in inhomogeneous point patterns. Statistica Neerlandica 54, 329–350.
See Also
Kest, Lest, Kinhom, pcf
Examples
data(japanesepines)
X <- japanesepines
L <- Linhom(X, sigma=0.1)
plot(L, main="Inhomogeneous L function for Japanese Pines")

linim

Create Pixel Image on Linear Network

Description
Creates an object of class "linim" that represents a pixel image on a linear network.
Usage
linim(L, Z, ..., restrict=TRUE, df=NULL)

linim

721

Arguments
L

Linear network (object of class "linnet").

Z

Pixel image (object of class "im").

...

Ignored.

restrict

Advanced use only. Logical value indicating whether to ensure that all pixels in
Z which do not lie on the network L have pixel value NA. This condition must
be satisfied, but if you set restrict=FALSE it will not be checked, and the code
will run faster.

df

Advanced use only. Data frame giving full details of the mapping between the
pixels of Z and the lines of L. See Details.

Details
This command creates an object of class "linim" that represents a pixel image defined on a linear
network. Typically such objects are used to represent the result of smoothing or model-fitting on
the network. Most users will not need to call linim directly.
The argument L is a linear network (object of class "linnet"). It gives the exact spatial locations
of the line segments of the network, and their connectivity.
The argument Z is a pixel image object of class "im" that gives a pixellated approximation of the
function values.
For increased efficiency, advanced users may specify the optional argument df. This is a data frame
giving the precomputed mapping between the pixels of Z and the line segments of L. It should
have columns named xc, yc containing the coordinates of the pixel centres, x,y containing the
projections of these pixel centres onto the linear network, mapXY identifying the line segment on
which each projected point lies, and tp giving the parametric position of (x,y) along the segment.
Value
Object of class "linim" that also inherits the class "im". There is a special method for plotting this
class.
Author(s)
Adrian Baddeley 
References
Ang, Q.W. (2010) Statistical methodology for events on a network. Master’s thesis, School of
Mathematics and Statistics, University of Western Australia.
Ang, Q.W., Baddeley, A. and Nair, G. (2012) Geometrically corrected second-order analysis of
events on a linear network, with applications to ecology and criminology. Scandinavian Journal of
Statistics 39, 591–617.
McSwiggan, G., Nair, M.G. and Baddeley, A. (2012) Fitting Poisson point process models to events
on a linear network. Manuscript in preparation.
See Also
plot.linim, linnet, eval.linim, Math.linim, im.

722

linnet

Examples
Z <- as.im(function(x,y) {x-y}, Frame(simplenet))
X <- linim(simplenet, Z)
X

linnet

Create a Linear Network

Description
Creates an object of class "linnet" representing a network of line segments.
Usage
linnet(vertices, m, edges, sparse=FALSE, warn=TRUE)
Arguments
vertices

Point pattern (object of class "ppp") specifying the vertices of the network.

m

Adjacency matrix. A matrix or sparse matrix of logical values equal to TRUE
when the corresponding vertices are joined by a line. (Specify either m or edges.)

edges

Edge list. A two-column matrix of integers, specifying all pairs of vertices that
should be joined by an edge. (Specify either m or edges.)

sparse

Optional. Logical value indicating whether to use a sparse matrix representation
of the network. See Details.

warn

Logical value indicating whether to issue a warning if the resulting network is
not connected.

Details
An object of class "linnet" represents a network of straight line segments in two dimensions. The
function linnet creates such an object from the minimal information: the spatial location of each
vertex (endpoint, crossing point or meeting point of lines) and information about which vertices are
joined by an edge.
If sparse=FALSE (the default), the algorithm will compute and store various properties of the network, including the adjacency matrix m and a matrix giving the shortest-path distances between each
pair of vertices in the network. This is more efficient for small datasets. However it can require large
amounts of memory and can take a long time to execute.
If sparse=TRUE, then the shortest-path distances will not be computed, and the network adjacency
matrix m will be stored as a sparse matrix. This saves a lot of time and memory when creating the
linear network.
If the argument edges is given, then it will also determine the ordering of the line segments when
they are stored or extracted. For example, edges[i,] corresponds to as.psp(L)[i].
Value
Object of class "linnet" representing the linear network.

lintess

723

Author(s)
Ang Qi Wei  and Adrian Baddeley 
See Also
simplenet for an example of a linear network.
methods.linnet for methods applicable to linnet objects.
Special tools: thinNetwork, insertVertices, connected.linnet, lixellate.
delaunayNetwork for the Delaunay triangulation as a network.
ppp, psp.
Examples
# letter 'A' specified by adjacency matrix
v <- ppp(x=(-2):2, y=3*c(0,1,2,1,0), c(-3,3), c(-1,7))
m <- matrix(FALSE, 5,5)
for(i in 1:4) m[i,i+1] <- TRUE
m[2,4] <- TRUE
m <- m | t(m)
letterA <- linnet(v, m)
plot(letterA)
# letter 'A' specified by edge list
edg <- cbind(1:4, 2:5)
edg <- rbind(edg, c(2,4))
letterA <- linnet(v, edges=edg)

lintess

Tessellation on a Linear Network

Description
Create a tessellation on a linear network.
Usage
lintess(L, df)
Arguments
L

Linear network (object of class "linnet").

df

Data frame of coordinates of endpoints of the tiles of the tessellation.

Details
A tessellation on a linear network L is a partition of the network into non-overlapping pieces (tiles).
Each tile consists of one or more line segments which are subsets of the line segments making up
the network. A tile can consist of several disjoint pieces.
The data frame df should have columns named seg, t0, t1 and tile.
Each row of the data frame specifies one sub-segment of the network and allocates it to a particular
tile.

724

lixellate
The seg column specifies which line segment of the network contains the sub-segment. Values of
seg are integer indices for the segments in as.psp(L).
The t0 and t1 columns specify the start and end points of the sub-segment. They should be numeric
values between 0 and 1 inclusive, where the values 0 and 1 representing the network vertices that
are joined by this network segment.
The tile column specifies which tile of the tessellation includes this sub-segment. It will be coerced
to a factor and its levels will be the names of the tiles.
If df is missing or NULL, the result is a tessellation with only one tile, consisting of the entire network
L.

Value
An object of class "lintess". There are methods for print, plot and summary for this object.
Author(s)
Adrian Baddeley  and Greg McSwiggan.
See Also
linnet for linear networks.
plot.lintess for plotting.
divide.linnet to make a tessellation demarcated by given points.
lineardirichlet to create the Dirichlet-Voronoi tessellation from a point pattern on a linear network.
as.linfun.lintess, as.linnet.lintess and as.linim to convert to other classes.
tile.lengths to compute the length of each tile in the tessellation.
The undocumented methods Window.lintess and as.owin.lintess extract the spatial window.
Examples
# tessellation consisting of one tile for each existing segment
ns <- nsegments(simplenet)
df <- data.frame(seg=1:ns, t0=0, t1=1, tile=letters[1:ns])
u <- lintess(simplenet, df)
u
plot(u)

lixellate

Subdivide Segments of a Network

Description
Each line segment of a linear network will be divided into several shorter segments (line elements
or lixels).
Usage
lixellate(X, ..., nsplit, eps, sparse = TRUE)

lixellate

725

Arguments
X

A linear network (object of class "linnet") or a point pattern on a linear network (object of class "lpp").

...

Ignored.

nsplit

Number of pieces into which each line segment of X should be divided. Either
a single integer, or an integer vector with one entry for each line segment in X.
Incompatible with eps.

eps

Maximum length of the resulting pieces of line segment. A single numeric value.
Incompatible with nsplit.

sparse

Optional. Logical value specifying whether the resulting linear network should
be represented using a sparse matrix. If sparse=NULL, then the representation
will be the same as in X.

Details
Each line segment in X will be subdivided into equal pieces. The result is an object of the same kind
as X, representing the same data as X except that the segments have been subdivided.
Splitting is controlled by the arguments nsplit and eps, exactly one of which should be given.
If nsplit is given, it specifies the number of pieces into which each line segment of X should be
divided. It should be either a single integer, or an integer vector of length equal to the number of
line segments in X.
If eps is given, it specifies the maximum length of any resulting piece of line segment.
It is strongly advisable to use sparse=TRUE (the default) to limit the computation time.
If X is a point pattern (class "lpp") then the spatial coordinates and marks of each data point are
unchanged, but the local coordinates will change, because they are adjusted to map them to the new
subdivided network.
Value
Object of the same kind as X.
Author(s)
Greg McSwiggan, Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
linnet, lpp.
Examples
A <- lixellate(simplenet, nsplit=4)
plot(A, main="lixellate(simplenet, nsplit=4)")
points(vertices(A), pch=16)
spiders
lixellate(spiders, nsplit=3)

726

localK

localK

Neighbourhood density function

Description
Computes the neighbourhood density function, a local version of the K-function or L-function,
defined by Getis and Franklin (1987).
Usage
localK(X, ..., correction = "Ripley", verbose = TRUE, rvalue=NULL)
localL(X, ..., correction = "Ripley", verbose = TRUE, rvalue=NULL)
Arguments
X

A point pattern (object of class "ppp").

...

Ignored.

correction

String specifying the edge correction to be applied. Options are "none", "translate",
"translation", "Ripley", "isotropic" or "best". Only one correction may
be specified.

verbose

Logical flag indicating whether to print progress reports during the calculation.

rvalue

Optional. A single value of the distance argument r at which the function L or
K should be computed.

Details
The command localL computes the neighbourhood density function, a local version of the Lfunction (Besag’s transformation of Ripley’s K-function) that was proposed by Getis and Franklin
(1987). The command localK computes the corresponding local analogue of the K-function.
Given a spatial point pattern X, the neighbourhood density function Li (r) associated with the ith
point in X is computed by
s
X
a
Li (r) =
eij
(n − 1)π j
where the sum is over all points j 6= i that lie within a distance r of the ith point, a is the area of the
observation window, n is the number of points in X, and eij is an edge correction term (as described
in Kest). The value of Li (r) can also be interpreted as one of the summands that contributes to the
global estimate of the L function.
By default, the function Li (r) or Ki (r) is computed for a range of r values for each point i. The results are stored as a function value table (object of class "fv") with a column of the table containing
the function estimates for each point of the pattern X.
Alternatively, if the argument rvalue is given, and it is a single number, then the function will only
be computed for this value of r, and the results will be returned as a numeric vector, with one entry
of the vector for each point of the pattern X.
Inhomogeneous counterparts of localK and localL are computed by localKinhom and localLinhom.

localK

727

Value
If rvalue is given, the result is a numeric vector of length equal to the number of points in the point
pattern.
If rvalue is absent, the result is an object of class "fv", see fv.object, which can be plotted
directly using plot.fv. Essentially a data frame containing columns
r

the vector of values of the argument r at which the function K has been estimated

theo

the theoretical value K(r) = πr2 or L(r) = r for a stationary Poisson process

together with columns containing the values of the neighbourhood density function for each point
in the pattern. Column i corresponds to the ith point. The last two columns contain the r and theo
values.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Getis, A. and Franklin, J. (1987) Second-order neighbourhood analysis of mapped point patterns.
Ecology 68, 473–477.
See Also
Kest, Lest, localKinhom, localLinhom.
Examples
data(ponderosa)
X <- ponderosa
# compute all the local L functions
L <- localL(X)
# plot all the local L functions against r
plot(L, main="local L functions for ponderosa", legend=FALSE)
# plot only the local L function for point number 7
plot(L, iso007 ~ r)
# compute the values of L(r) for r = 12 metres
L12 <- localL(X, rvalue=12)
# Spatially interpolate the values of L12
# Compare Figure 5(b) of Getis and Franklin (1987)
X12 <- X %mark% L12
Z <- Smooth(X12, sigma=5, dimyx=128)
plot(Z, col=topo.colors(128), main="smoothed neighbourhood density")
contour(Z, add=TRUE)
points(X, pch=16, cex=0.5)

728

localKinhom

localKinhom

Inhomogeneous Neighbourhood Density Function

Description
Computes spatially-weighted versions of the the local K-function or L-function.
Usage
localKinhom(X, lambda, ...,
correction = "Ripley",
sigma = NULL, varcov =
localLinhom(X, lambda, ...,
correction = "Ripley",
sigma = NULL, varcov =

verbose = TRUE, rvalue=NULL,
NULL)
verbose = TRUE, rvalue=NULL,
NULL)

Arguments
X

A point pattern (object of class "ppp").

lambda

Optional. Values of the estimated intensity function. Either a vector giving the
intensity values at the points of the pattern X, a pixel image (object of class "im")
giving the intensity values at all locations, a fitted point process model (object of
class "ppm") or a function(x,y) which can be evaluated to give the intensity
value at any location.

...

Extra arguments. Ignored if lambda is present. Passed to density.ppp if
lambda is omitted.

correction

String specifying the edge correction to be applied. Options are "none", "translate",
"Ripley", "translation", "isotropic" or "best". Only one correction may
be specified.

verbose

Logical flag indicating whether to print progress reports during the calculation.

rvalue

Optional. A single value of the distance argument r at which the function L or
K should be computed.

sigma, varcov

Optional arguments passed to density.ppp to control the kernel smoothing procedure for estimating lambda, if lambda is missing.

Details
The functions localKinhom and localLinhom are inhomogeneous or weighted versions of the
neighbourhood density function implemented in localK and localL.
Given a spatial point pattern X, the inhomogeneous neighbourhood density function Li (r) associated with the ith point in X is computed by
s
1 X eij
Li (r) =
π j λj
where the sum is over all points j 6= i that lie within a distance r of the ith point, λj is the estimated
intensity of the point pattern at the point j, and eij is an edge correction term (as described in Kest).
The value of Li (r) can also be interpreted as one of the summands that contributes to the global
estimate of the inhomogeneous L function (see Linhom).

localKinhom

729

By default, the function Li (r) or Ki (r) is computed for a range of r values for each point i. The results are stored as a function value table (object of class "fv") with a column of the table containing
the function estimates for each point of the pattern X.
Alternatively, if the argument rvalue is given, and it is a single number, then the function will only
be computed for this value of r, and the results will be returned as a numeric vector, with one entry
of the vector for each point of the pattern X.
Value
If rvalue is given, the result is a numeric vector of length equal to the number of points in the point
pattern.
If rvalue is absent, the result is an object of class "fv", see fv.object, which can be plotted
directly using plot.fv. Essentially a data frame containing columns
r

the vector of values of the argument r at which the function K has been estimated

theo

the theoretical value K(r) = πr2 or L(r) = r for a stationary Poisson process

together with columns containing the values of the neighbourhood density function for each point
in the pattern. Column i corresponds to the ith point. The last two columns contain the r and theo
values.
Author(s)
Mike Kuhn, Adrian Baddeley 
and Rolf Turner 
See Also
Kinhom, Linhom, localK, localL.
Examples
data(ponderosa)
X <- ponderosa
# compute all the local L functions
L <- localLinhom(X)
# plot all the local L functions against r
plot(L, main="local L functions for ponderosa", legend=FALSE)
# plot only the local L function for point number 7
plot(L, iso007 ~ r)
# compute the values of L(r) for r = 12 metres
L12 <- localL(X, rvalue=12)

730

localpcf

localpcf

Local pair correlation function

Description
Computes individual contributions to the pair correlation function from each data point.
Usage
localpcf(X, ..., delta=NULL, rmax=NULL, nr=512, stoyan=0.15)
localpcfinhom(X, ..., delta=NULL, rmax=NULL, nr=512, stoyan=0.15,
lambda=NULL, sigma=NULL, varcov=NULL)
Arguments
X

A point pattern (object of class "ppp").

delta

Smoothing bandwidth for pair correlation. The halfwidth of the Epanechnikov
kernel.

rmax

Optional. Maximum value of distance r for which pair correlation values g(r)
should be computed.

nr

Optional. Number of values of distance r for which pair correlation g(r) should
be computed.

stoyan

Optional. The value of the constant c in Stoyan’s rule of thumb for selecting the
smoothing bandwidth delta.

lambda

Optional. Values of the estimated intensity function, for the inhomogeneous pair
correlation. Either a vector giving the intensity values at the points of the pattern
X, a pixel image (object of class "im") giving the intensity values at all locations,
a fitted point process model (object of class "ppm") or a function(x,y) which
can be evaluated to give the intensity value at any location.
sigma,varcov,...
These arguments are ignored by localpcf but are passed by localpcfinhom
(when lambda=NULL) to the function density.ppp to control the kernel smoothing estimation of lambda.

Details
localpcf computes the contribution, from each individual data point in a point pattern X, to the
empirical pair correlation function of X. These contributions are sometimes known as LISA (local
indicator of spatial association) functions based on pair correlation.
localpcfinhom computes the corresponding contribution to the inhomogeneous empirical pair correlation function of X.
Given a spatial point pattern X, the local pcf gi (r) associated with the ith point in X is computed by
a X
k(di,j − r)
gi (r) =
2πn j
where the sum is over all points j 6= i, a is the area of the observation window, n is the number of
points in X, and dij is the distance between points i and j. Here k is the Epanechnikov kernel,
k(t) =

3
t2
max(0, 1 − 2 ).
4δ
δ

localpcf

731

Edge correction is performed using the border method (for the sake of computational efficiency):
the estimate gi (r) is set to NA if r > bi , where bi is the distance from point i to the boundary of the
observation window.
The smoothing bandwidth δ may be specified. If not, it is chosen by Stoyan’s rule of thumb δ = c/λ̂
where λ̂ = n/a is the estimated intensity and c is a constant, usually taken to be 0.15. The value of
c is controlled by the argument stoyan.
For localpcfinhom, the optional argument lambda specifies the values of the estimated intensity
function. If lambda is given, it should be either a numeric vector giving the intensity values at
the points of the pattern X, a pixel image (object of class "im") giving the intensity values at all
locations, a fitted point process model (object of class "ppm") or a function(x,y) which can be
evaluated to give the intensity value at any location. If lambda is not given, then it will be estimated
using a leave-one-out kernel density smoother as described in pcfinhom.
Value
An object of class "fv", see fv.object, which can be plotted directly using plot.fv. Essentially
a data frame containing columns
r

the vector of values of the argument r at which the function K has been estimated

theo

the theoretical value K(r) = πr2 or L(r) = r for a stationary Poisson process

together with columns containing the values of the local pair correlation function for each point in
the pattern. Column i corresponds to the ith point. The last two columns contain the r and theo
values.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
localK, localKinhom, pcf, pcfinhom
Examples
data(ponderosa)
X <- ponderosa
g <- localpcf(X, stoyan=0.5)
colo <- c(rep("grey", npoints(X)), "blue")
a <- plot(g, main=c("local pair correlation functions", "Ponderosa pines"),
legend=FALSE, col=colo, lty=1)
# plot only the local pair correlation function for point number 7
plot(g, est007 ~ r)
gi <- localpcfinhom(X, stoyan=0.5)
a <- plot(gi, main=c("inhomogeneous local pair correlation functions",
"Ponderosa pines"),
legend=FALSE, col=colo, lty=1)

732

logLik.dppm

logLik.dppm

Log Likelihood and AIC for Fitted Determinantal Point Process Model

Description
Extracts the log Palm likelihood, deviance, and AIC of a fitted determinantal point process model.
Usage
## S3 method for class 'dppm'
logLik(object, ...)
## S3 method for class 'dppm'
AIC(object, ..., k=2)
## S3 method for class 'dppm'
extractAIC(fit, scale=0, k=2, ...)
## S3 method for class 'dppm'
nobs(object, ...)
Arguments
object,fit

Fitted point process model. An object of class "dppm".

...

Ignored.

scale

Ignored.

k

Numeric value specifying the weight of the equivalent degrees of freedom in the
AIC. See Details.

Details
These functions are methods for the generic commands logLik, extractAIC and nobs for the class
"dppm".
An object of class "dppm" represents a fitted Cox or cluster point process model. It is obtained from
the model-fitting function dppm.
These methods apply only when the model was fitted by maximising the Palm likelihood (Tanaka
et al, 2008) by calling dppm with the argument method="palm".
The method logLik.dppm computes the maximised value of the log Palm likelihood for the fitted
model object.
The methods AIC.dppm and extractAIC.dppm compute the Akaike Information Criterion AIC for
the fitted model based on the Palm likelihood (Tanaka et al, 2008)
AIC = −2 log(P L) + k × edf
where P L is the maximised Palm likelihood of the fitted model, and edf is the effective degrees of
freedom of the model.
The method nobs.dppm returns the number of points in the original data point pattern to which the
model was fitted.
The R function step uses these methods, but it does not work for determinantal models yet due to
a missing implementation of update.dppm.

logLik.kppm

733

Value
logLik returns a numerical value, belonging to the class "logLik", with an attribute "df" giving
the degrees of freedom.
AIC returns a numerical value.
extractAIC returns a numeric vector of length 2 containing the degrees of freedom and the AIC
value.
nobs returns an integer value.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Tanaka, U. and Ogata, Y. and Stoyan, D. (2008) Parameter estimation and model selection for
Neyman-Scott point processes. Biometrical Journal 50, 43–57.
See Also
dppm, logLik.ppm
Examples
fit <- dppm(swedishpines ~ x, dppGauss(), method="palm")
nobs(fit)
logLik(fit)
extractAIC(fit)
AIC(fit)

logLik.kppm

Log Likelihood and AIC for Fitted Cox or Cluster Point Process Model

Description
Extracts the log Palm likelihood, deviance, and AIC of a fitted Cox or cluster point process model.
Usage
## S3 method for class 'kppm'
logLik(object, ...)
## S3 method for class 'kppm'
AIC(object, ..., k=2)
## S3 method for class 'kppm'
extractAIC(fit, scale=0, k=2, ...)
## S3 method for class 'kppm'
nobs(object, ...)

734

logLik.kppm

Arguments
object,fit

Fitted point process model. An object of class "kppm".

...

Ignored.

scale

Ignored.

k

Numeric value specifying the weight of the equivalent degrees of freedom in the
AIC. See Details.

Details
These functions are methods for the generic commands logLik, extractAIC and nobs for the class
"kppm".
An object of class "kppm" represents a fitted Cox or cluster point process model. It is obtained from
the model-fitting function kppm.
These methods apply only when the model was fitted by maximising the Palm likelihood (Tanaka
et al, 2008) by calling kppm with the argument method="palm".
The method logLik.kppm computes the maximised value of the log Palm likelihood for the fitted
model object.
The methods AIC.kppm and extractAIC.kppm compute the Akaike Information Criterion AIC for
the fitted model based on the Palm likelihood (Tanaka et al, 2008)
AIC = −2 log(P L) + k × edf
where P L is the maximised Palm likelihood of the fitted model, and edf is the effective degrees of
freedom of the model.
The method nobs.kppm returns the number of points in the original data point pattern to which the
model was fitted.
The R function step uses these methods.
Value
logLik returns a numerical value, belonging to the class "logLik", with an attribute "df" giving
the degrees of freedom.
AIC returns a numerical value.
extractAIC returns a numeric vector of length 2 containing the degrees of freedom and the AIC
value.
nobs returns an integer value.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Tanaka, U. and Ogata, Y. and Stoyan, D. (2008) Parameter estimation and model selection for
Neyman-Scott point processes. Biometrical Journal 50, 43–57.

logLik.mppm

735

See Also
kppm, logLik.ppm
Examples
fit <- kppm(redwood ~ x, "Thomas", method="palm")
nobs(fit)
logLik(fit)
extractAIC(fit)
AIC(fit)
step(fit)

logLik.mppm

Log Likelihood and AIC for Multiple Point Process Model

Description
For a point process model that has been fitted to multiple point patterns, these functions extract the
log likelihood and AIC, or analogous quantities based on the pseudolikelihood.
Usage
## S3 method for class 'mppm'
logLik(object, ..., warn=TRUE)
## S3 method for class 'mppm'
AIC(object, ..., k=2, takeuchi=TRUE)
## S3 method for class 'mppm'
extractAIC(fit, scale = 0, k = 2, ..., takeuchi = TRUE)
## S3 method for class 'mppm'
nobs(object, ...)
## S3 method for class 'mppm'
getCall(x, ...)
## S3 method for class 'mppm'
terms(x, ...)
Arguments
object,fit,x
...
warn
scale
k
takeuchi

Fitted point process model (fitted to multiple point patterns). An object of class
"mppm".
Ignored.
If TRUE, a warning is given when the pseudolikelihood is returned instead of the
likelihood.
Ignored.
Numeric value specifying the weight of the equivalent degrees of freedom in the
AIC. See Details.
Logical value specifying whether to use the Takeuchi penalty (takeuchi=TRUE)
or the number of fitted parameters (takeuchi=FALSE) in calculating AIC.

736

logLik.mppm

Details
These functions are methods for the generic commands logLik, AIC, extractAIC, terms and
getCall for the class "mppm".
An object of class "mppm" represents a fitted Poisson or Gibbs point process model fitted to several
point patterns. It is obtained from the model-fitting function mppm.
The method logLik.mppm extracts the maximised value of the log likelihood for the fitted model (as
approximated by quadrature using the Berman-Turner approximation). If object is not a Poisson
process, the maximised log pseudolikelihood is returned, with a warning.
The Akaike Information Criterion AIC for a fitted model is defined as
AIC = −2 log(L) + k × penalty
where L is the maximised likelihood of the fitted model, and penalty is a penalty for model complexity, usually equal to the effective degrees of freedom of the model. The method extractAIC.mppm
returns the analogous quantity AIC∗ in which L is replaced by L∗, the quadrature approximation
to the likelihood (if fit is a Poisson model) or the pseudolikelihood (if fit is a Gibbs model).
The penalty term is calculated as follows. If takeuchi=FALSE then penalty is the number of fitted
parameters. If takeuchi=TRUE then penalty = trace(JH −1 ) where J and H are the estimated
variance and hessian, respectively, of the composite score. These two choices are equivalent for a
Poisson process.
The method nobs.mppm returns the total number of points in the original data point patterns to
which the model was fitted.
The method getCall.mppm extracts the original call to mppm which caused the model to be fitted.
The method terms.mppm extracts the covariate terms in the model formula as a terms object. Note
that these terms do not include the interaction component of the model.
The R function step uses these methods.
Value
See the help files for the corresponding generic functions.
Author(s)

Adrian Baddeley, Ida-Maria Sintorn and Leanne Bischoff. Implemented by Adrian Baddeley  and Ege Rubak .
References
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. London: Chapman and Hall/CRC Press.
See Also
mppm
Examples
fit <- mppm(Bugs ~ x, hyperframe(Bugs=waterstriders))
logLik(fit)
AIC(fit)
nobs(fit)
getCall(fit)

logLik.ppm

logLik.ppm

737

Log Likelihood and AIC for Point Process Model

Description
Extracts the log likelihood, deviance, and AIC of a fitted Poisson point process model, or analogous quantities based on the pseudolikelihood or logistic likelihood for a fitted Gibbs point process
model.
Usage
## S3 method for class 'ppm'
logLik(object, ..., new.coef=NULL, warn=TRUE, absolute=FALSE)
## S3 method for class 'ppm'
deviance(object, ...)
## S3 method for class 'ppm'
AIC(object, ..., k=2, takeuchi=TRUE)
## S3 method for class 'ppm'
extractAIC(fit, scale=0, k=2, ..., takeuchi=TRUE)
## S3 method for class 'ppm'
nobs(object, ...)
Arguments
object,fit

Fitted point process model. An object of class "ppm".

...

Ignored.

warn

If TRUE, a warning is given when the pseudolikelihood or logistic likelihood is
returned instead of the likelihood.

absolute

Logical value indicating whether to include constant terms in the loglikelihood.

scale

Ignored.

k

Numeric value specifying the weight of the equivalent degrees of freedom in the
AIC. See Details.

new.coef

New values for the canonical parameters of the model. A numeric vector of the
same length as coef(object).

takeuchi

Logical value specifying whether to use the Takeuchi penalty (takeuchi=TRUE)
or the number of fitted parameters (takeuchi=FALSE) in calculating AIC.

Details
These functions are methods for the generic commands logLik, deviance, extractAIC and nobs
for the class "ppm".
An object of class "ppm" represents a fitted Poisson or Gibbs point process model. It is obtained
from the model-fitting function ppm.
The method logLik.ppm computes the maximised value of the log likelihood for the fitted model
object (as approximated by quadrature using the Berman-Turner approximation) is extracted. If

738

logLik.ppm
object is not a Poisson process, the maximised log pseudolikelihood is returned, with a warning
(if warn=TRUE).
The Akaike Information Criterion AIC for a fitted model is defined as
AIC = −2 log(L) + k × penalty
where L is the maximised likelihood of the fitted model, and penalty is a penalty for model complexity, usually equal to the effective degrees of freedom of the model. The method extractAIC.ppm
returns the analogous quantity AIC∗ in which L is replaced by L∗, the quadrature approximation
to the likelihood (if fit is a Poisson model) or the pseudolikelihood or logistic likelihood (if fit is
a Gibbs model).
The penalty term is calculated as follows. If takeuchi=FALSE then penalty is the number of fitted
parameters. If takeuchi=TRUE then penalty = trace(JH −1 ) where J and H are the estimated
variance and hessian, respectively, of the composite score. These two choices are equivalent for a
Poisson process.
The method nobs.ppm returns the number of points in the original data point pattern to which the
model was fitted.
The R function step uses these methods.

Value
logLik returns a numerical value, belonging to the class "logLik", with an attribute "df" giving
the degrees of freedom.
AIC returns a numerical value.
extractAIC returns a numeric vector of length 2 containing the degrees of freedom and the AIC
value.
nobs returns an integer value.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Varin, C. and Vidoni, P. (2005) A note on composite likelihood inference and model selection.
Biometrika 92, 519–528.
See Also
ppm, as.owin, coef.ppm, fitted.ppm, formula.ppm, model.frame.ppm, model.matrix.ppm,
plot.ppm, predict.ppm, residuals.ppm, simulate.ppm, summary.ppm, terms.ppm, update.ppm,
vcov.ppm.
Examples
data(cells)
fit <- ppm(cells, ~x)
nobs(fit)
logLik(fit)
deviance(fit)

logLik.slrm

739

extractAIC(fit)
AIC(fit)
step(fit)

logLik.slrm

Loglikelihood of Spatial Logistic Regression

Description
Computes the (maximised) loglikelihood of a fitted Spatial Logistic Regression model.
Usage
## S3 method for class 'slrm'
logLik(object, ..., adjust = TRUE)
Arguments
object
...
adjust

a fitted spatial logistic regression model. An object of class "slrm".
Ignored.
Logical value indicating whether to adjust the loglikelihood of the model to
make it comparable with a point process likelihood. See Details.

Details
This is a method for logLik for fitted spatial logistic regression models (objects of class "slrm",
usually obtained from the function slrm). It computes the log-likelihood of a fitted spatial logistic
regression model.
If adjust=FALSE, the loglikelihood is computed using the standard formula for the loglikelihood of
a logistic regression model for a finite set of (pixel) observations.
If adjust=TRUE then the loglikelihood is adjusted so that it is approximately comparable with the
likelihood of a point process in continuous space, by subtracting the value n log(a) where n is the
number of points in the original point pattern dataset, and a is the area of one pixel.
Value
A numerical value.
Author(s)
Adrian Baddeley  
and Rolf Turner 
See Also
slrm
Examples
X <- rpoispp(42)
fit <- slrm(X ~ x+y)
logLik(fit)
logLik(fit, adjust=FALSE)

740

lohboot

lohboot

Bootstrap Confidence Bands for Summary Function

Description
Computes a bootstrap confidence band for a summary function of a point process.
Usage
lohboot(X,
fun=c("pcf", "Kest", "Lest", "pcfinhom", "Kinhom", "Linhom"),
...,
block=FALSE, global=FALSE, basicboot=FALSE, Vcorrection=FALSE,
confidence=0.95, nx = 4, ny = nx, nsim=200, type=7)
Arguments
X

A point pattern (object of class "ppp").

fun

Name of the summary function for which confidence intervals are desired: one
of the strings "pcf", "Kest", "Lest", "pcfinhom", "Kinhom" or "Linhom".
Alternatively, the function itself; it must be one of the functions listed here.

...

Arguments passed to the corresponding local version of the summary function
(see Details).

block

Logical value indicating whether to use Loh’s block bootstrap as originally proposed. Default is FALSE for consistency with older code. See Details.

global

Logical. If FALSE (the default), pointwise confidence intervals are constructed.
If TRUE, a global (simultaneous) confidence band is constructed.

basicboot

Logical value indicating whether to use the so-called basic bootstrap confidence
interval. See Details.

Vcorrection

Logical value indicating whether to use a variance correction when fun="Kest"
or fun="Kinhom". See Details.

confidence

Confidence level, as a fraction between 0 and 1.

nx,ny

Integers. If block=TRUE, divide the window into nx*ny rectangles.

nsim

Number of bootstrap simulations.

type

Integer. Type of quantiles. Argument passed to quantile.default controlling
the way the quantiles are calculated.

Details
This algorithm computes confidence bands for the true value of the summary function fun using the
bootstrap method of Loh (2008) and a modification described in Baddeley, Rubak, Turner (2015).
If fun="pcf", for example, the algorithm computes a pointwise (100 * confidence)% confidence
interval for the true value of the pair correlation function for the point process, normally estimated
by pcf. It starts by computing the array of local pair correlation functions, localpcf, of the data
pattern X. This array consists of the contributions to the estimate of the pair correlation function
from each data point.

lohboot

741

If block=FALSE, these contributions are resampled nsim times with replacement as described in
Baddeley, Rubak, Turner (2015); from each resampled dataset the total contribution is computed,
yielding nsim random pair correlation functions.
If block=TRUE, the (bounding box of the) window is divided into nx ∗ ny rectangles (blocks). The
average contribution of a block is obtained by averaging the contribution of each point included in
the block. Then, the average contributions on each block are resampled nsim times with replacement as described in Loh (2008) and Loh (2010); from each resampled dataset the total contribution
is computed, yielding nsim random pair correlation functions. Notice that for non-rectangular windows any blocks not fully contained in the window are discarded before doing the resampling, so
the effective number of blocks may be substantially smaller than nx ∗ ny in this case.
The pointwise alpha/2 and 1 - alpha/2 quantiles of these functions are computed, where
alpha = 1 - confidence. The average of the local functions is also computed as an estimate of
the pair correlation function.
There are several ways to define a bootstrap confidence interval. If basicbootstrap=TRUE, the
so-called basic confidence bootstrap interval is used as described in Loh (2008).
It has been noticed in Loh (2010) that when the intensity of the point process is unknown, the
bootstrap error estimate is larger than it should be. When the K function is used, an adjustment
procedure has been proposed in Loh (2010) that is used if Vcorrection=TRUE. In this case, the
basic confidence bootstrap interval is implicitly used.
To control the estimation algorithm, use the arguments ..., which are passed to the local version
of the summary function, as shown below:
fun
pcf
Kest
Lest
pcfinhom
Kinhom
Linhom

local version
localpcf
localK
localK
localpcfinhom
localKinhom
localKinhom

For fun="Lest", the calculations are first performed as if fun="Kest", and then the square-root
transformation is applied to obtain the L-function.
Note that the confidence bands computed by lohboot(fun="pcf") may not contain the estimate
of the pair correlation function computed by pcf, because of differences between the algorithm parameters (such as the choice of edge correction) in localpcf and pcf. If you are using lohboot, the
appropriate point estimate of the pair correlation itself is the pointwise mean of the local estimates,
which is provided in the result of lohboot and is shown in the default plot.
If the confidence bands seem unbelievably narrow, this may occur because the point pattern has a
hard core (the true pair correlation function is zero for certain values of distance) or because of an
optical illusion when the function is steeply sloping (remember the width of the confidence bands
should be measured vertically).
An alternative to lohboot is varblock.
Value
A function value table (object of class "fv") containing columns giving the estimate of the summary
function, the upper and lower limits of the bootstrap confidence interval, and the theoretical value
of the summary function for a Poisson process.

742

lpp

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak  and Christophe Biscio.
References
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
Loh, J.M. (2008) A valid and fast spatial bootstrap for correlation functions. The Astrophysical
Journal, 681, 726–734.
Loh, J.M. (2010) Bootstrapping an inhomogeneous point process. Journal of Statistical Planning
and Inference, 140, 734–749.
See Also
Summary functions Kest, pcf, Kinhom, pcfinhom, localK, localpcf, localKinhom, localpcfinhom.
See varblock for an alternative bootstrap technique.
Examples
p <- lohboot(simdat, stoyan=0.5)
plot(p)

lpp

Create Point Pattern on Linear Network

Description
Creates an object of class "lpp" that represents a point pattern on a linear network.
Usage
lpp(X, L, ...)
Arguments
X

Locations of the points. A matrix or data frame of coordinates, or a point pattern
object (of class "ppp") or other data acceptable to as.ppp.

L

Linear network (object of class "linnet").

...

Ignored.

Details
This command creates an object of class "lpp" that represents a point pattern on a linear network.
Normally X is a point pattern. The points of X should lie on the lines of L.
Alternatively X may be a matrix or data frame containing at least two columns.
• Usually the first two columns of X will be interpreted as spatial coordinates, and any remaining
columns as marks.

lpp

743
• An exception occurs if X is a data frame with columns named x, y, seg and tp. Then x and
y will be interpreted as spatial coordinates, and seg and tp as local coordinates, with seg
indicating which line segment of L the point lies on, and tp indicating how far along the
segment the point lies (normalised to 1). Any remaining columns will be interpreted as marks.
• Another exception occurs if X is a data frame with columns named seg and tp. Then seg and
tp will be interpreted as local coordinates, as above, and the spatial coordinates x,y will be
computed from them. Any remaining columns will be interpreted as marks.
If X is missing or NULL, the result is an empty point pattern (i.e. containing no points).

Value
An object of class "lpp". Also inherits the class "ppx".
Note on changed format
The internal format of "lpp" objects was changed in spatstat version 1.28-0. Objects in the old
format are still handled correctly, but computations are faster in the new format. To convert an
object X from the old format to the new format, use X <- lpp(as.ppp(X), as.linnet(X)).
Author(s)
Ang Qi Wei  and Adrian Baddeley 
See Also
Installed datasets which are "lpp" objects: chicago, dendrite, spiders.
See as.lpp for converting data to an lpp object.
See methods.lpp and methods.ppx for other methods applicable to lpp objects.
Calculations on an lpp object: intensity.lpp, distfun.lpp, nndist.lpp, nnwhich.lpp, nncross.lpp,
nnfun.lpp.
Summary functions: linearK, linearKinhom, linearpcf, linearKdot, linearKcross, linearmarkconnect,
etc.
Random point patterns on a linear network can be generated by rpoislpp or runiflpp.
See linnet for linear networks.
Examples
# letter 'A'
v <- ppp(x=(-2):2, y=3*c(0,1,2,1,0), c(-3,3), c(-1,7))
edg <- cbind(1:4, 2:5)
edg <- rbind(edg, c(2,4))
letterA <- linnet(v, edges=edg)
# points on letter A
xx <- list(x=c(-1.5,0,0.5,1.5), y=c(1.5,3,4.5,1.5))
X <- lpp(xx, letterA)
plot(X)
X
summary(X)
# empty pattern
lpp(L=letterA)

744

lppm

lppm

Fit Point Process Model to Point Pattern on Linear Network

Description
Fit a point process model to a point pattern dataset on a linear network
Usage
lppm(X, ...)
## S3 method for class 'formula'
lppm(X, interaction=NULL, ..., data=NULL)
## S3 method for class 'lpp'
lppm(X, ..., eps=NULL, nd=1000, random=FALSE)
Arguments
X

Either an object of class "lpp" specifying a point pattern on a linear network, or
a formula specifying the point process model.

...

Arguments passed to ppm.

interaction

An object of class "interact" describing the point process interaction structure, or NULL indicating that a Poisson process (stationary or nonstationary)
should be fitted.

data

Optional. The values of spatial covariates (other than the Cartesian coordinates)
required by the model. A list whose entries are images, functions, windows,
tessellations or single numbers.

eps

Optional. Spacing between dummy points along each segment of the network.

nd

Optional. Total number of dummy points placed on the network. Ignored if eps
is given.

random

Logical value indicating whether the grid of dummy points should be placed at
a randomised starting position.

Details
This function fits a point process model to data that specify a point pattern on a linear network. It is
a counterpart of the model-fitting function ppm designed to work with objects of class "lpp" instead
of "ppp".
The function lppm is generic, with methods for the classes formula and lppp.
In lppm.lpp the first argument X should be an object of class "lpp" (created by the command lpp)
specifying a point pattern on a linear network.
In lppm.formula, the first argument is a formula in the R language describing the spatial trend
model to be fitted. It has the general form pattern ~ trend where the left hand side pattern is
usually the name of a point pattern on a linear network (object of class "lpp") to which the model
should be fitted, or an expression which evaluates to such a point pattern; and the right hand side
trend is an expression specifying the spatial trend of the model.
Other arguments ... are passed from lppm.formula to lppm.lpp and from lppm.lpp to ppm.

lurking

745

Value
An object of class "lppm" representing the fitted model. There are methods for print, predict,
coef and similar functions.
Author(s)
Adrian Baddeley  and Greg McSwiggan.
References
Ang, Q.W. (2010) Statistical methodology for events on a network. Master’s thesis, School of
Mathematics and Statistics, University of Western Australia.
Ang, Q.W., Baddeley, A. and Nair, G. (2012) Geometrically corrected second-order analysis of
events on a linear network, with applications to ecology and criminology. Scandinavian Journal of
Statistics 39, 591–617.
McSwiggan, G., Nair, M.G. and Baddeley, A. (2012) Fitting Poisson point process models to events
on a linear network. Manuscript in preparation.
See Also
methods.lppm, predict.lppm, ppm, lpp.
Examples
X <- runiflpp(15, simplenet)
lppm(X ~1)
lppm(X ~x)
marks(X) <- factor(rep(letters[1:3], 5))
lppm(X ~ marks)
lppm(X ~ marks * x)

lurking

Lurking Variable Plot

Description
Plot spatial point process residuals against a covariate
Usage
lurking(object, ...)
## S3 method for class 'ppm'
lurking(object, covariate,
type="eem",
cumulative=TRUE,
...,
plot.it = TRUE,
plot.sd = is.poisson(object),
clipwindow=default.clipwindow(object),
rv = NULL,

746

lurking
envelope=FALSE, nsim=39, nrank=1,
typename,
covname,
oldstyle=FALSE,
check=TRUE,
verbose=TRUE,
nx=128,
splineargs=list(spar=0.5),
internal=NULL)
## S3 method for class 'ppp'
lurking(object, covariate,
type="eem",
cumulative=TRUE,
...,
plot.it = TRUE,
plot.sd = is.poisson(object),
clipwindow=default.clipwindow(object),
rv = NULL,
envelope=FALSE, nsim=39, nrank=1,
typename,
covname,
oldstyle=FALSE,
check=TRUE,
verbose=TRUE,
nx=128,
splineargs=list(spar=0.5),
internal=NULL)

Arguments
object

The fitted point process model (an object of class "ppm") for which diagnostics
should be produced. This object is usually obtained from ppm. Alternatively,
object may be a point pattern (object of class "ppp").

covariate

The covariate against which residuals should be plotted. Either a numeric vector,
a pixel image, or an expression. See Details below.

type

String indicating the type of residuals or weights to be computed. Choices include "eem", "raw", "inverse" and "pearson". See diagnose.ppm for all
possible choices.

cumulative

Logical flag indicating whether to plot a cumulative sum of marks (cumulative=TRUE)
or the derivative of this sum, a marginal density of the smoothed residual field
(cumulative=FALSE).

...

Arguments passed to plot.default and lines to control the plot behaviour.

plot.it

Logical value indicating whether plots should be shown. If plot.it=FALSE,
only the computed coordinates for the plots are returned. See Value.

plot.sd

Logical value indicating whether error bounds should be added to plot. The
default is TRUE for Poisson models and FALSE for non-Poisson models. See
Details.

clipwindow

If not NULL this argument indicates that residuals shall only be computed inside a subregion of the window containing the original point pattern data. Then

lurking

rv

envelope

nsim
nrank
typename
covname
oldstyle

check
verbose
nx
splineargs
internal

747
clipwindow should be a window object of class "owin".
Usually absent. If this argument is present, the point process residuals will not
be calculated from the fitted model object, but will instead be taken directly
from rv.
Logical value indicating whether to compute simulation envelopes for the plot.
Alternatively envelope may be a list of point patterns to use for computing the
simulation envelopes, or an object of class "envelope" containing simulated
point patterns.
Number of simulated point patterns to be generated to produce the simulation
envelope, if envelope=TRUE.
Integer. Rank of the envelope value amongst the nsim simulated values. A rank
of 1 means that the minimum and maximum simulated values will be used.
Usually absent. If this argument is present, it should be a string, and will be used
(in the axis labels of plots) to describe the type of residuals.
A string name for the covariate, to be used in axis labels of plots.
Logical flag indicating whether error bounds should be plotted using the approximation given in the original paper (oldstyle=TRUE), or using the correct
asymptotic formula (oldstyle=FALSE).
Logical flag indicating whether the integrity of the data structure in object
should be checked.
Logical value indicating whether to print progress reports during Monte Carlo
simulation.
Integer. Number of covariate values to be used in the plot.
A list of arguments passed to smooth.spline for the estimation of the derivatives in the case cumulative=FALSE.
Internal use only.

Details
This function generates a ‘lurking variable’ plot for a fitted point process model. Residuals from the
model represented by object are plotted against the covariate specified by covariate. This plot
can be used to reveal departures from the fitted model, in particular, to reveal that the point pattern
depends on the covariate.
The function lurking is generic, with methods for ppm and ppp documented here, and possibly
other methods.
The argument object would usually be a fitted point process model (object of class "ppm") produced by the model-fitting algorithm ppm). If object is a point pattern (object of class "ppp") then
the model is taken to be the uniform Poisson process (Complete Spatial Randomness) fitted to this
point pattern.
First the residuals from the fitted model (Baddeley et al, 2004) are computed at each quadrature
point, or alternatively the ‘exponential energy marks’ (Stoyan and Grabarnik, 1991) are computed
at each data point. The argument type selects the type of residual or weight. See diagnose.ppm
for options and explanation.
A lurking variable plot for point processes (Baddeley et al, 2004) displays either the cumulative sum
of residuals/weights (if cumulative = TRUE) or a kernel-weighted average of the residuals/weights
(if cumulative = FALSE) plotted against the covariate. The empirical plot (solid lines) is shown
together with its expected value assuming the model is true (dashed lines) and optionally also the
pointwise two-standard-deviation limits (grey shading).
To be more precise, let Z(u) denote the value of the covariate at a spatial location u.

748

lurking
• If cumulative=TRUE then we plot H(z) against z, where H(z) is the sum of the residuals
over all quadrature points where the covariate takes a value less than or equal to z, or the sum
of the exponential energy weights over all data points where the covariate takes a value less
than or equal to z.
• If cumulative=FALSE then we plot h(z) against z, where h(z) is the derivative of H(z),
computed approximately by spline smoothing.
For the point process residuals E(H(z)) = 0, while for the exponential energy weights E(H(z)) =
area of the subset of the window satisfying Z(u) <= z.
If the empirical and theoretical curves deviate substantially from one another, the interpretation is
that the fitted model does not correctly account for dependence on the covariate. The correct form
(of the spatial trend part of the model) may be suggested by the shape of the plot.
If plot.sd = TRUE, then superimposed on the lurking variable plot are the pointwise two-standarddeviation error limits for H(x) calculated for the inhomogeneous Poisson process. The default is
plot.sd = TRUE for Poisson models and plot.sd = FALSE for non-Poisson models.
By default, the two-standard-deviation limits are calculated from the exact formula for the asymptotic variance of the residuals under the asymptotic normal approximation, equation (37) of Baddeley et al (2006). However, for compatibility with the original paper of Baddeley et al (2005), if
oldstyle=TRUE, the two-standard-deviation limits are calculated using the innovation variance, an
over-estimate of the true variance of the residuals.
The argument covariate is either a numeric vector, a pixel image, or an R language expression. If
it is a numeric vector, it is assumed to contain the values of the covariate for each of the quadrature
points in the fitted model. The quadrature points can be extracted by quad.ppm(object).
If covariate is a pixel image, it is assumed to contain the values of the covariate at each location
in the window. The values of this image at the quadrature points will be extracted.
Alternatively, if covariate is an expression, it will be evaluated in the same environment as
the model formula used in fitting the model object. It must yield a vector of the same length as
the number of quadrature points. The expression may contain the terms x and y representing the
cartesian coordinates, and may also contain other variables that were available when the model was
fitted. Certain variable names are reserved words; see ppm.
Note that lurking variable plots for the x and y coordinates are also generated by diagnose.ppm,
amongst other types of diagnostic plots. This function is more general in that it enables the user to
plot the residuals against any chosen covariate that may have been present.
For advanced use, even the values of the residuals/weights can be altered. If the argument rv is
present, the residuals will not be calculated from the fitted model object but will instead be taken
directly from the object rv. If type = "eem" then rv should be similar to the return value of
eem, namely, a numeric vector with length equal to the number of data points in the original point
pattern. Otherwise, rv should be similar to the return value of residuals.ppm, that is, rv should
be an object of class "msr" (see msr) representing a signed measure.

Value
The (invisible) return value is an object belonging to the class "lurk", for which there are methods
for plot and print.
This object is a list containing two dataframes empirical and theoretical. The first dataframe
empirical contains columns covariate and value giving the coordinates of the lurking variable
plot. The second dataframe theoretical contains columns covariate, mean and sd giving the
coordinates of the plot of the theoretical mean and standard deviation.

lurking.mppm

749

Author(s)
Adrian Baddeley  and Rolf Turner .
References
Baddeley, A., Turner, R., Møller, J. and Hazelton, M. (2005) Residual analysis for spatial point
processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
Baddeley, A., Møller, J. and Pakes, A.G. (2006) Properties of residuals for spatial point processes.
Annals of the Institute of Statistical Mathematics 60, 627–649.
Stoyan, D. and Grabarnik, P. (1991) Second-order characteristics for stochastic structures connected
with Gibbs point processes. Mathematische Nachrichten, 151:95–100.
See Also
residuals.ppm, diagnose.ppm, residuals.ppm, qqplot.ppm, eem, ppm
Examples
lurking(nztrees, expression(x))
fit <- ppm(nztrees, ~x, Poisson())
lurking(fit, expression(x))
lurking(fit, expression(x), cumulative=FALSE)

lurking.mppm

Lurking Variable Plot for Multiple Point Patterns

Description
Generate a lurking variable plot of spatial point process residuals against a covariate, for a model
fitted to several point patterns.
Usage
## S3 method for class 'mppm'
lurking(object, covariate, type="eem",
...,
separate = FALSE,
plot.it = TRUE,
covname, oldstyle = FALSE, nx = 512, main="")
Arguments
object

The fitted model. An object of class "mppm" representing a point process model
fitted to several point patterns.

covariate

The covariate to be used on the horizontal axis. Either an expression which
can be evaluated in the original data, or a list of pixel images, one image for
each point pattern in the original data.

type

String indicating the type of residuals or weights to be computed. Choices include "eem", "raw", "inverse" and "pearson". See diagnose.ppm for all
possible choices.

750

lurking.mppm
...

Additional arguments passed to lurking.ppm, including arguments controlling
the plot.

separate

Logical value indicating whether to compute a separate lurking variable plot
for each of the original point patterns. If FALSE (the default), a single lurkingvariable plot is produced by combining residuals from all patterns.

plot.it

Logical value indicating whether plots should be shown. If plot.it=FALSE,
only the computed coordinates for the plots are returned. See Value.

covname

A string name for the covariate, to be used in axis labels of plots.

oldstyle

Logical flag indicating whether error bounds should be plotted using the approximation given in the original paper (oldstyle=TRUE), or using the correct
asymptotic formula (oldstyle=FALSE).

nx

Integer. Number of covariate values to be used in the plot.

main

Character string giving a main title for the plot.

Details
This function generates a ‘lurking variable’ plot for a point process model fitted to several point
patterns. Residuals from the model represented by object are plotted against the covariate specified
by covariate. This plot can be used to reveal departures from the fitted model.
The function lurking is generic. This is the method for the class mppm. The argument object must
be a fitted point process model object of class "mppm") produced by the model-fitting algorithm
mppm.
Value
If separate=FALSE (the default), the return value is an object belonging to the class "lurk", for
which there are methods for plot and print. See lurking for details of the format.
If separate=TRUE, the result is a list of such objects, and also belongs to the class anylist so that
it can be printed and plotted.
Author(s)
Adrian Baddeley , with thanks to Nicholas Read.
See Also
lurking.ppm
Examples
fit <- mppm(Points ~ Image + Group, demohyper)
lurking(fit, expression(Image), type="P")
lurking(fit, expression(Image), type="P", separate=TRUE)

lut

751

lut

Lookup Tables

Description
Create a lookup table.
Usage
lut(outputs, ..., range=NULL, breaks=NULL, inputs=NULL, gamma=1)
Arguments
outputs
...
range
inputs
breaks
gamma

Vector of output values
Ignored.
Interval of numbers to be mapped. A numeric vector of length 2, specifying the
ends of the range of values to be mapped. Incompatible with breaks or inputs.
Input values to which the output values are associated. A factor or vector of the
same length as outputs. Incompatible with breaks or range.
Breakpoints for the lookup table. A numeric vector of length equal to length(outputs)+1.
Incompatible with range or inputs.
Exponent for gamma correction, when range is given. A single positive number.
See Details.

Details
A lookup table is a function, mapping input values to output values.
The command lut creates an object representing a lookup table, which can then be used to control
various behaviour in the spatstat package. It can also be used to compute the output value assigned
to any input value.
The argument outputs specifies the output values to which input data values will be mapped. It
should be a vector of any atomic type (e.g. numeric, logical, character, complex) or factor values.
Exactly one of the arguments range, inputs or breaks must be specified by name.
If inputs is given, then it should be a vector or factor, of the same length as outputs. The entries
of inputs can be any atomic type (e.g. numeric, logical, character, complex) or factor values. The
resulting lookup table associates the value inputs[i] with the value outputs[i].
If range is given, then it determines the interval of the real number line that will be mapped. It
should be a numeric vector of length 2. The interval will be divided evenly into bands, each of
which is mapped to an entry of outputs. (If gamma is given, then the bands are equally spaced on a
scale where the original values are raised to the power gamma.)
If breaks is given, then it determines intervals of the real number line which are mapped to each
output value. It should be a numeric vector, of length at least 2, with entries that are in increasing
order. Infinite values are allowed. Any number in the range between breaks[i] and breaks[i+1]
will be mapped to the value outputs[i].
The result is an object of class "lut". There is a print method for this class. Some plot commands
in the spatstat package accept an object of this class as a specification of a lookup table.
The result is also a function f which can be used to compute the output value assigned to any input
data value. That is, f(x) returns the output value assigned to x. This also works for vectors of input
data values.

752

markconnect

Value
A function, which is also an object of class "lut".
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
colourmap.
Examples
# lookup table for real numbers, using breakpoints
cr <- lut(factor(c("low", "medium", "high")), breaks=c(0,5,10,15))
cr
cr(3.2)
cr(c(3,5,7))
# lookup table for discrete set of values
ct <- lut(c(0,1), inputs=c(FALSE, TRUE))
ct(TRUE)

markconnect

Mark Connection Function

Description
Estimate the marked connection function of a multitype point pattern.
Usage
markconnect(X, i, j, r=NULL,
correction=c("isotropic", "Ripley", "translate"),
method="density", ..., normalise=FALSE)
Arguments
X

The observed point pattern. An object of class "ppp" or something acceptable
to as.ppp.

i

Number or character string identifying the type (mark value) of the points in X
from which distances are measured.

j

Number or character string identifying the type (mark value) of the points in X
to which distances are measured.

r

numeric vector. The values of the argument r at which the mark connection
function pij (r) should be evaluated. There is a sensible default.

correction

A character vector containing any selection of the options "isotropic", "Ripley"
or "translate". It specifies the edge correction(s) to be applied.

method

A character vector indicating the user’s choice of density estimation technique
to be used. Options are "density", "loess", "sm" and "smrep".

markconnect

753

...

Arguments passed to the density estimation routine (density, loess or sm.density)
selected by method.

normalise

If TRUE, normalise the pair connection function by dividing it by pi pj , the estimated probability that randomly-selected points will have marks i and j.

Details
The mark connection function pij (r) of a multitype point process X is a measure of the dependence
between the types of two points of the process a distance r apart.
Informally pij (r) is defined as the conditional probability, given that there is a point of the process
at a location u and another point of the process at a location v separated by a distance ||u − v|| = r,
that the first point is of type i and the second point is of type j. See Stoyan and Stoyan (1994).
If the marks attached to the points of X are independent and identically distributed, then pij (r) ≡
pi pj where pi denotes the probability that a point is of type i. Values larger than this, pij (r) >
pi pj , indicate positive association between the two types, while smaller values indicate negative
association.
The argument X must be a point pattern (object of class "ppp") or any data that are acceptable to
as.ppp. It must be a multitype point pattern (a marked point pattern with factor-valued marks).
The argument r is the vector of values for the distance r at which pij (r) is estimated. There is a
sensible default.
This algorithm assumes that X can be treated as a realisation of a stationary (spatially homogeneous)
random spatial point process in the plane, observed through a bounded window. The window (which
is specified in X as Window(X)) may have arbitrary shape.
Biases due to edge effects are treated in the same manner as in Kest. The edge corrections implemented here are
isotropic/Ripley Ripley’s isotropic correction (see Ripley, 1988; Ohser, 1983). This is implemented only for rectangular and polygonal windows (not for binary masks).
translate Translation correction (Ohser, 1983). Implemented for all window geometries, but slow
for complex windows.
Note that the estimator assumes the process is stationary (spatially homogeneous).
The mark connection function is estimated using density estimation techniques. The user can
choose between
"density" which uses the standard kernel density estimation routine density, and works only for
evenly-spaced r values;
"loess" which uses the function loess in the package modreg;
"sm" which uses the function sm.density in the package sm and is extremely slow;
"smrep" which uses the function sm.density in the package sm and is relatively fast, but may
require manual control of the smoothing parameter hmult.
Value
An object of class "fv" (see fv.object).
Essentially a data frame containing numeric columns
r

the values of the argument r at which the mark connection function pij (r) has
been estimated

754

markcorr
theo

the theoretical value of pij (r) when the marks attached to different points are
independent

together with a column or columns named "iso" and/or "trans", according to the selected edge
corrections. These columns contain estimates of the function pij (r) obtained by the edge corrections named.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical
statistics. John Wiley and Sons.
See Also
Multitype pair correlation pcfcross and multitype K-functions Kcross, Kdot.
Use alltypes to compute the mark connection functions between all pairs of types.
Mark correlation markcorr and mark variogram markvario for numeric-valued marks.
Examples
# Hughes' amacrine data
# Cells marked as 'on'/'off'
data(amacrine)
M <- markconnect(amacrine, "on", "off")
plot(M)
# Compute for all pairs of types at once
plot(alltypes(amacrine, markconnect))

markcorr

Mark Correlation Function

Description
Estimate the marked correlation function of a marked point pattern.
Usage
markcorr(X, f = function(m1, m2) { m1 * m2}, r=NULL,
correction=c("isotropic", "Ripley", "translate"),
method="density", ..., weights=NULL,
f1=NULL, normalise=TRUE, fargs=NULL)

markcorr

755

Arguments
X

The observed point pattern. An object of class "ppp" or something acceptable
to as.ppp.

f

Optional. Test function f used in the definition of the mark correlation function.
An R function with at least two arguments. There is a sensible default.

r

Optional. Numeric vector. The values of the argument r at which the mark
correlation function kf (r) should be evaluated. There is a sensible default.

correction

A character vector containing any selection of the options "isotropic", "Ripley",
"translate", "translation", "none" or "best". It specifies the edge correction(s) to be applied. Alternatively correction="all" selects all options.

method

A character vector indicating the user’s choice of density estimation technique
to be used. Options are "density", "loess", "sm" and "smrep".

...

Arguments passed to the density estimation routine (density, loess or sm.density)
selected by method.

weights

Optional numeric vector of weights for each data point in X.

f1

An alternative to f. If this argument is given, then f is assumed to take the form
f (u, v) = f1 (u)f1 (v).

normalise

If normalise=FALSE, compute only the numerator of the expression for the
mark correlation.

fargs

Optional. A list of extra arguments to be passed to the function f or f1.

Details
By default, this command calculates an estimate of Stoyan’s mark correlation kmm (r) for the point
pattern.
Alternatively if the argument f or f1 is given, then it calculates Stoyan’s generalised mark correlation kf (r) with test function f .
Theoretical definitions are as follows (see Stoyan and Stoyan (1994, p. 262)):
• For a point process X with numeric marks, Stoyan’s mark correlation function kmm (r), is
kmm (r) =

E0u [M (0)M (u)]
E[M, M 0 ]

where E0u denotes the conditional expectation given that there are points of the process at the
locations 0 and u separated by a distance r, and where M (0), M (u) denote the marks attached
to these two points. On the denominator, M, M 0 are random marks drawn independently from
the marginal distribution of marks, and E is the usual expectation.
• For a multitype point process X, the mark correlation is
kmm (r) =

P0u [M (0)M (u)]
P [M = M 0 ]

where P and P0u denote the probability and conditional probability.
• The generalised mark correlation function kf (r) of a marked point process X, with test function f , is
E0u [f (M (0), M (u))]
kf (r) =
E[f (M, M 0 )]

756

markcorr
The test function f is any function f (m1 , m2 ) with two arguments which are possible marks of
the pattern, and which returns a nonnegative real value. Common choices of f are: for continuous
nonnegative real-valued marks,
f (m1 , m2 ) = m1 m2
for discrete marks (multitype point patterns),
f (m1 , m2 ) = 1(m1 = m2 )
and for marks taking values in [0, 2π),
f (m1 , m2 ) = sin(m1 − m2 )
.
Note that kf (r) is not a “correlation” in the usual statistical sense. It can take any nonnegative
real value. The value 1 suggests “lack of correlation”: if the marks attached to the points of X
are independent and identically distributed, then kf (r) ≡ 1. The interpretation of values larger or
smaller than 1 depends on the choice of function f .
The argument X must be a point pattern (object of class "ppp") or any data that are acceptable to
as.ppp. It must be a marked point pattern.
The argument f determines the function to be applied to pairs of marks. It has a sensible default,
which depends on the kind of marks in X. If the marks are numeric values, then f <- function(m1, m2) { m1 * m2}
computes the product of two marks. If the marks are a factor (i.e. if X is a multitype point pattern)
then f <- function(m1, m2) { m1 == m2} yields the value 1 when the two marks are equal, and
0 when they are unequal. These are the conventional definitions for numerical marks and multitype
points respectively.
The argument f may be specified by the user. It must be an R function, accepting two arguments
m1 and m2 which are vectors of equal length containing mark values (of the same type as the marks
of X). (It may also take additional arguments, passed through fargs). It must return a vector of
numeric values of the same length as m1 and m2. The values must be non-negative, and NA values
are not permitted.
Alternatively the user may specify the argument f1 instead of f. This indicates that the test function
f should take the form f (u, v) = f1 (u)f1 (v) where f1 (u) is given by the argument f1. The
argument f1 should be an R function with at least one argument. (It may also take additional
arguments, passed through fargs).
The argument r is the vector of values for the distance r at which kf (r) is estimated.
This algorithm assumes that X can be treated as a realisation of a stationary (spatially homogeneous)
random spatial point process in the plane, observed through a bounded window. The window (which
is specified in X as Window(X)) may have arbitrary shape.
Biases due to edge effects are treated in the same manner as in Kest. The edge corrections implemented here are
isotropic/Ripley Ripley’s isotropic correction (see Ripley, 1988; Ohser, 1983). This is implemented only for rectangular and polygonal windows (not for binary masks).
translate Translation correction (Ohser, 1983). Implemented for all window geometries, but slow
for complex windows.
Note that the estimator assumes the process is stationary (spatially homogeneous).
The numerator and denominator of the mark correlation function (in the expression above) are
estimated using density estimation techniques. The user can choose between
"density" which uses the standard kernel density estimation routine density, and works only for
evenly-spaced r values;

markcorr

757

"loess" which uses the function loess in the package modreg;
"sm" which uses the function sm.density in the package sm and is extremely slow;
"smrep" which uses the function sm.density in the package sm and is relatively fast, but may
require manual control of the smoothing parameter hmult.
If normalise=FALSE then the algorithm will compute only the numerator
cf (r) = E0u f (M (0), M (u))
of the expression for the mark correlation function.
Value
A function value table (object of class "fv") or a list of function value tables, one for each column
of marks.
An object of class "fv" (see fv.object) is essentially a data frame containing numeric columns
r

the values of the argument r at which the mark correlation function kf (r) has
been estimated

theo

the theoretical value of kf (r) when the marks attached to different points are
independent, namely 1

together with a column or columns named "iso" and/or "trans", according to the selected edge
corrections. These columns contain estimates of the mark correlation function kf (r) obtained by
the edge corrections named.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
References
Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical
statistics. John Wiley and Sons.
See Also
Mark variogram markvario for numeric marks.
Mark connection function markconnect and multitype K-functions Kcross, Kdot for factor-valued
marks.
Mark cross-correlation function markcrosscorr for point patterns with several columns of marks.
Kmark to estimate a cumulative function related to the mark correlation function.
Examples
# CONTINUOUS-VALUED MARKS:
# (1) Spruces
# marks represent tree diameter
# mark correlation function
ms <- markcorr(spruces)
plot(ms)
# (2) simulated data with independent marks

758

markcrosscorr
X <- rpoispp(100)
X <- X %mark% runif(npoints(X))
## Not run:
Xc <- markcorr(X)
plot(Xc)
## End(Not run)
#
#
#
#
#
M

MULTITYPE DATA:
Hughes' amacrine data
Cells marked as 'on'/'off'
(3) Kernel density estimate with Epanecnikov kernel
(as proposed by Stoyan & Stoyan)
<- markcorr(amacrine, function(m1,m2) {m1==m2},
correction="translate", method="density",
kernel="epanechnikov")
plot(M)
# Note: kernel="epanechnikov" comes from help(density)
# (4) Same again with explicit control over bandwidth
## Not run:
M <- markcorr(amacrine,
correction="translate", method="density",
kernel="epanechnikov", bw=0.02)
# see help(density) for correct interpretation of 'bw'
## End(Not run)

#
Y
a
v

weighted mark correlation
<- subset(betacells, select=type)
<- marks(betacells)$area
<- markcorr(Y, weights=a)

markcrosscorr

Mark Cross-Correlation Function

Description
Given a spatial point pattern with several columns of marks, this function computes the mark correlation function between each pair of columns of marks.
Usage
markcrosscorr(X, r = NULL,
correction = c("isotropic", "Ripley", "translate"),
method = "density", ..., normalise = TRUE, Xname = NULL)
Arguments
X

The observed point pattern. An object of class "ppp" or something acceptable
to as.ppp.

markcrosscorr

759

r

Optional. Numeric vector. The values of the argument r at which the mark
correlation function kf (r) should be evaluated. There is a sensible default.

correction

A character vector containing any selection of the options "isotropic", "Ripley",
"translate", "translation", "none" or "best". It specifies the edge correction(s) to be applied. Alternatively correction="all" selects all options.

method

A character vector indicating the user’s choice of density estimation technique
to be used. Options are "density", "loess", "sm" and "smrep".

...

Arguments passed to the density estimation routine (density, loess or sm.density)
selected by method.

normalise

If normalise=FALSE, compute only the numerator of the expression for the
mark correlation.

Xname

Optional character string name for the dataset X.

Details
First, all columns of marks are converted to numerical values. A factor with m possible levels is
converted to m columns of dummy (indicator) values.
Next, each pair of columns is considered, and the mark cross-correlation is defined as
kmm (r) =

E0u [Mi (0)Mj (u)]
E[Mi , Mj ]

where E0u denotes the conditional expectation given that there are points of the process at the
locations 0 and u separated by a distance r. On the numerator, Mi (0) and Mj (u) are the marks
attached to locations 0 and u respectively in the ith and jth columns of marks respectively. On the
denominator, Mi and Mj are independent random values drawn from the ith and jth columns of
marks, respectively, and E is the usual expectation.
Note that kmm (r) is not a “correlation” in the usual statistical sense. It can take any nonnegative
real value. The value 1 suggests “lack of correlation”: if the marks attached to the points of X are
independent and identically distributed, then kmm (r) ≡ 1.
The argument X must be a point pattern (object of class "ppp") or any data that are acceptable to
as.ppp. It must be a marked point pattern.
The cross-correlations are estimated in the same manner as for markcorr.
Value
A function array (object of class "fasp") containing the mark cross-correlation functions for each
possible pair of columns of marks.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
markcorr

760

marks

Examples
# The dataset 'betacells' has two columns of marks:
#
'type' (factor)
#
'area' (numeric)
if(interactive()) plot(betacells)
plot(markcrosscorr(betacells))

marks

Marks of a Point Pattern

Description
Extract or change the marks attached to a point pattern dataset.
Usage
marks(x, ...)
## S3 method for class 'ppp'
marks(x, ..., dfok=TRUE, drop=TRUE)
## S3 method for class 'ppx'
marks(x, ..., drop=TRUE)
marks(x, ...) <- value
## S3 replacement method for class 'ppp'
marks(x, ..., dfok=TRUE, drop=TRUE) <- value
## S3 replacement method for class 'ppx'
marks(x, ...) <- value
setmarks(x, value)
x %mark% value
Arguments
x

Point pattern dataset (object of class "ppp" or "ppx").

...

Ignored.

dfok

Logical. If FALSE, data frames of marks are not permitted and will generate an
error.

drop

Logical. If TRUE, a data frame consisting of a single column of marks will be
converted to a vector or factor.

value

Replacement value. A vector, data frame or hyperframe of mark values, or NULL.

marks

761

Details
These functions extract or change the marks attached to the points of the point pattern x.
The expression marks(x) extracts the marks of x. The assignment marks(x) <- value assigns
new marks to the dataset x, and updates the dataset x in the current environment. The expression
setmarks(x,value) or equivalently x %mark% value returns a point pattern obtained by replacing
the marks of x by value, but does not change the dataset x itself.
For point patterns in two-dimensional space (objects of class "ppp") the marks can be a vector, a
factor, or a data frame.
For general point patterns (objects of class "ppx") the marks can be a vector, a factor, a data frame
or a hyperframe.
For the assignment marks(x) <- value, the value should be a vector or factor of length equal to
the number of points in x, or a data frame or hyperframe with as many rows as there are points in x.
If value is a single value, or a data frame or hyperframe with one row, then it will be replicated so
that the same marks will be attached to each point.
To remove marks, use marks(x) <- NULL or unmark(x).
Use ppp or ppx to create point patterns in more general situations.
Value
For marks(x), the result is a vector, factor, data frame or hyperframe, containing the mark values
attached to the points of x.
For marks(x) <- value, the result is the updated point pattern x (with the side-effect that the
dataset x is updated in the current environment).
For setmarks(x,value) and x %mark% value, the return value is the point pattern obtained by
replacing the marks of x by value.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
ppp.object, ppx, unmark, hyperframe
Examples
X <- amacrine
# extract marks
m <- marks(X)
# recode the mark values "off", "on" as 0, 1
marks(X) <- as.integer(m == "on")

762

marks.psp

marks.psp

Marks of a Line Segment Pattern

Description
Extract or change the marks attached to a line segment pattern.
Usage
## S3 method for class 'psp'
marks(x, ..., dfok=TRUE)
## S3 replacement method for class 'psp'
marks(x, ...) <- value
Arguments
x

Line segment pattern dataset (object of class "psp").

...

Ignored.

dfok

Logical. If FALSE, data frames of marks are not permitted and will generate an
error.

value

Vector or data frame of mark values, or NULL.

Details
These functions extract or change the marks attached to each of the line segments in the pattern x.
They are methods for the generic functions marks and marks<- for the class "psp" of line segment
patterns.
The expression marks(x) extracts the marks of x. The assignment marks(x) <- value assigns
new marks to the dataset x, and updates the dataset x in the current environment.
The marks can be a vector, a factor, or a data frame.
For the assignment marks(x) <- value, the value should be a vector or factor of length equal to
the number of segments in x, or a data frame with as many rows as there are segments in x. If value
is a single value, or a data frame with one row, then it will be replicated so that the same marks will
be attached to each segment.
To remove marks, use marks(x) <- NULL or unmark(x).
Value
For marks(x), the result is a vector, factor or data frame, containing the mark values attached to the
line segments of x. If there are no marks, the result is NULL.
For marks(x) <- value, the result is the updated line segment pattern x (with the side-effect that
the dataset x is updated in the current environment).
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .

marks.tess

763

See Also
psp.object, marks, marks
Rolf Turner 
and Ege Rubak 
See Also
marks, marks, Rolf Turner 
and Ege Rubak .
See Also
applynbd, marktable, ppp.object, apply
Examples
trees <- longleaf
# average diameter of 5 closest neighbours of each tree
md <- markstat(trees, mean, N=5)
# range of diameters of trees within 10 metre radius
rd <- markstat(trees, range, R=10)

766

marktable

marktable

Tabulate Marks in Neighbourhood of Every Point in a Point Pattern

Description
Visit each point in a point pattern, find the neighbouring points, and compile a frequency table of
the marks of these neighbour points.
Usage
marktable(X, R, N, exclude=TRUE, collapse=FALSE)
Arguments
X

A marked point pattern. An object of class "ppp".

R

Neighbourhood radius. Incompatible with N.

N

Number of neighbours of each point. Incompatible with R.

exclude

Logical. If exclude=TRUE, the neighbours of a point do not include the point
itself. If exclude=FALSE, a point belongs to its own neighbourhood.

collapse

Logical. If collapse=FALSE (the default) the results for each point are returned
as separate rows of a table. If collapse=TRUE, the results are aggregated according to the type of point.

Details
This algorithm visits each point in the point pattern X, inspects all the neighbouring points within
a radius R of the current point (or the N nearest neighbours of the current point), and compiles a
frequency table of the marks attached to the neighbours.
The dataset X must be a multitype point pattern, that is, marks(X) must be a factor.
If collapse=FALSE (the default), the result is a two-dimensional contingency table with one row
for each point in the pattern, and one column for each possible mark value. The [i,j] entry in the
table gives the number of neighbours of point i that have mark j.
If collapse=TRUE, this contingency table is aggregated according to the type of point, so that the
result is a contingency table with one row and one column for each possible mark value. The [i,j]
entry in the table gives the number of neighbours of a point with mark i that have mark j.
To perform more complicated calculations on the neighbours of every point, use markstat or
applynbd.
Value
A contingency table (object of class "table"). If collapse=FALSE, the table has one row for each
point in X, and one column for each possible mark value. If collapse=TRUE, the table has one row
and one column for each possible mark value.
Author(s)
Adrian Baddeley 
and Rolf Turner 

markvario

767

See Also
markstat, applynbd, Kcross, ppp.object, table
Examples
head(marktable(amacrine, 0.1))
head(marktable(amacrine, 0.1, exclude=FALSE))
marktable(amacrine, N=1, collapse=TRUE)

markvario

Mark Variogram

Description
Estimate the mark variogram of a marked point pattern.
Usage
markvario(X, correction = c("isotropic", "Ripley", "translate"),
r = NULL, method = "density", ..., normalise=FALSE)
Arguments
X

The observed point pattern. An object of class "ppp" or something acceptable
to as.ppp. It must have marks which are numeric.

correction

A character vector containing any selection of the options "isotropic", "Ripley"
or "translate". It specifies the edge correction(s) to be applied.

r

numeric vector. The values of the argument r at which the mark variogram γ(r)
should be evaluated. There is a sensible default.

method

A character vector indicating the user’s choice of density estimation technique
to be used. Options are "density", "loess", "sm" and "smrep".

...

Arguments passed to the density estimation routine (density, loess or sm.density)
selected by method.

normalise

If TRUE, normalise the variogram by dividing it by the estimated mark variance.

Details
The mark variogram γ(r) of a marked point process X is a measure of the dependence between the
marks of two points of the process a distance r apart. It is informally defined as
1
γ(r) = E[ (M1 − M2 )2 ]
2
where E[] denotes expectation and M1 , M2 are the marks attached to two points of the process a
distance r apart.
The mark variogram of a marked point process is analogous, but not equivalent, to the variogram
of a random field in geostatistics. See Waelder and Stoyan (1996).

768

markvario

Value
An object of class "fv" (see fv.object).
Essentially a data frame containing numeric columns
r

the values of the argument r at which the mark variogram γ(r) has been estimated

theo

the theoretical value of γ(r) when the marks attached to different points are
independent; equal to the sample variance of the marks

together with a column or columns named "iso" and/or "trans", according to the selected edge
corrections. These columns contain estimates of the function γ(r) obtained by the edge corrections
named.

Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Cressie, N.A.C. (1991) Statistics for spatial data. John Wiley and Sons, 1991.
Mase, S. (1996) The threshold method for estimating annual rainfall. Annals of the Institute of
Statistical Mathematics 48 (1996) 201-213.
Waelder, O. and Stoyan, D. (1996) On variograms in point process statistics. Biometrical Journal
38 (1996) 895-905.

See Also
Mark correlation function markcorr for numeric marks.
Mark connection function markconnect and multitype K-functions Kcross, Kdot for factor-valued
marks.

Examples
# Longleaf Pine data
# marks represent tree diameter
data(longleaf)
# Subset of this large pattern
swcorner <- owin(c(0,100),c(0,100))
sub <- longleaf[ , swcorner]
# mark correlation function
mv <- markvario(sub)
plot(mv)

matchingdist

matchingdist

769

Distance for a Point Pattern Matching

Description
Computes the distance associated with a matching between two point patterns.
Usage
matchingdist(matching, type = NULL, cutoff = NULL, q = NULL)
Arguments
matching

A point pattern matching (an object of class "pppmatching").

type

A character string giving the type of distance to be computed. One of "spa",
"ace" or "mat". See details below.

cutoff

The value > 0 at which interpoint distances are cut off.

q

The order of the average that is applied to the interpoint distances. May be Inf,
in which case the maximum of the interpoint distances is taken.

Details
Computes the distance specified by type, cutoff, and order for a point matching. If any of
these arguments are not provided, the function uses the corresponding elements of matching (if
available).
For the type "spa" (subpattern assignment) it is assumed that the points of the point pattern with
the smaller cardinality m are matched to a m-point subpattern of the point pattern with the larger
cardinality n in a 1-1 way. The distance is then given as the q-th order average of the m distances between matched points (minimum of Euclidean distance and cutoff) and n − m "penalty distances"
of value cutoff.
For the type "ace" (assignment only if cardinalities equal) the matching is assumed to be 1-1 if the
cardinalities of the point patterns are the same, in which case the q-th order average of the matching
distances (minimum of Euclidean distance and cutoff) is taken. If the cardinalities are different,
the matching may be arbitrary and the distance returned is always equal to cutoff.
For the type mat (mass transfer) it is assumed that each point of the point pattern with the smaller
cardinality m has mass 1, each point of the point pattern with the larger cardinality n has mass
m/n, and fractions of these masses are matched in such a way that each point contributes exactly
its mass. The distance is then given as the q-th order weighted average of all distances (minimum of
Euclidean distance and cutoff) of (partially) matched points with weights equal to the fractional
masses divided by m.
If the cardinalities of the two point patterns are equal, matchingdist(m, type, cutoff, q)
yields the same result no matter if type is "spa", "ace" or "mat".
Value
Numeric value of the distance associated with the matching.

770

matclust.estK

Author(s)
Dominic Schuhmacher  http://www.dominic.schuhmacher.
name
See Also
pppdist pppmatching.object
Examples
# an optimal matching
X <- runifpoint(20)
Y <- runifpoint(20)
m.opt <- pppdist(X, Y)
summary(m.opt)
matchingdist(m.opt)
# is the same as the distance given by summary(m.opt)
# sequential nearest neighbour matching
# (go through all points of point pattern X in sequence
# and match each point with the closest point of Y that is
# still unmatched)
am <- matrix(0, 20, 20)
h <- matrix(c(1:20, rep(0,20)), 20, 2)
h[1,2] = nncross(X[1],Y)[1,2]
for (i in 2:20) {
nn <- nncross(X[i],Y[-h[1:(i-1),2]])[1,2]
h[i,2] <- ((1:20)[-h[1:(i-1),2]])[nn]
}
am[h] <- 1
m.nn <- pppmatching(X, Y, am)
matchingdist(m.nn, type="spa", cutoff=1, q=1)
# is >= the distance obtained for m.opt
# in most cases strictly >
## Not run:
par(mfrow=c(1,2))
plot(m.opt)
plot(m.nn)
text(X$x, X$y, 1:20, pos=1, offset=0.3, cex=0.8)
## End(Not run)

matclust.estK

Fit the Matern Cluster Point Process by Minimum Contrast

Description
Fits the Matern Cluster point process to a point pattern dataset by the Method of Minimum Contrast.
Usage
matclust.estK(X, startpar=c(kappa=1,scale=1), lambda=NULL,
q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...)

matclust.estK

771

Arguments
X

Data to which the Matern Cluster model will be fitted. Either a point pattern or
a summary statistic. See Details.

startpar

Vector of starting values for the parameters of the Matern Cluster process.

lambda

Optional. An estimate of the intensity of the point process.

q,p

Optional. Exponents for the contrast criterion.

rmin, rmax

Optional. The interval of r values for the contrast criterion.

...

Optional arguments passed to optim to control the optimisation algorithm. See
Details.

Details
This algorithm fits the Matern Cluster point process model to a point pattern dataset by the Method
of Minimum Contrast, using the K function.
The argument X can be either
a point pattern: An object of class "ppp" representing a point pattern dataset. The K function of
the point pattern will be computed using Kest, and the method of minimum contrast will be
applied to this.
a summary statistic: An object of class "fv" containing the values of a summary statistic, computed for a point pattern dataset. The summary statistic should be the K function, and this
object should have been obtained by a call to Kest or one of its relatives.
The algorithm fits the Matern Cluster point process to X, by finding the parameters of the Matern
Cluster model which give the closest match between the theoretical K function of the Matern Cluster process and the observed K function. For a more detailed explanation of the Method of Minimum Contrast, see mincontrast.
The Matern Cluster point process is described in Møller and Waagepetersen (2003, p. 62). It is
a cluster process formed by taking a pattern of parent points, generated according to a Poisson
process with intensity κ, and around each parent point, generating a random number of offspring
points, such that the number of offspring of each parent is a Poisson random variable with mean µ,
and the locations of the offspring points of one parent are independent and uniformly distributed
inside a circle of radius R centred on the parent point, where R is equal to the parameter scale.
The named vector of stating values can use either R or scale as the name of the second component,
but the latter is recommended for consistency with other cluster models.
The theoretical K-function of the Matern Cluster process is
K(r) = πr2 +

1
r
h(
)
κ 2R

where the radius R is the parameter scale and
h(z) = 2 +

p
p
1
[(8z 2 − 4)arccos(z) − 2arcsin(z) + 4z (1 − z 2 )3 − 6z 1 − z 2 ]
π

for z <= 1, and h(z) = 1 for z > 1. The theoretical intensity of the Matern Cluster process is
λ = κµ.
In this algorithm, the Method of Minimum Contrast is first used to find optimal values of the parameters κ and R. Then the remaining parameter µ is inferred from the estimated intensity λ.
If the argument lambda is provided, then this is used as the value of λ. Otherwise, if X is a point
pattern, then λ will be estimated from X. If X is a summary statistic and lambda is missing, then the
intensity λ cannot be estimated, and the parameter µ will be returned as NA.

772

matclust.estpcf
The remaining arguments rmin,rmax,q,p control the method of minimum contrast; see mincontrast.
The Matern Cluster process can be simulated, using rMatClust.
Homogeneous or inhomogeneous Matern Cluster models can also be fitted using the function kppm.
The optimisation algorithm can be controlled through the additional arguments "..." which are
passed to the optimisation function optim. For example, to constrain the parameter values to a
certain range, use the argument method="L-BFGS-B" to select an optimisation algorithm that respects box constraints, and use the arguments lower and upper to specify (vectors of) minimum
and maximum values for each parameter.

Value
An object of class "minconfit". There are methods for printing and plotting this object. It contains
the following main components:
par

Vector of fitted parameter values.

fit

Function value table (object of class "fv") containing the observed values of the
summary statistic (observed) and the theoretical values of the summary statistic
computed from the fitted model parameters.

Author(s)

Rasmus Waagepetersen  Adapted for spatstat by Adrian Baddeley  1. The theoretical intensity of the Matern Cluster process is
λ = κµ.
In this algorithm, the Method of Minimum Contrast is first used to find optimal values of the parameters κ and R. Then the remaining parameter µ is inferred from the estimated intensity λ.
If the argument lambda is provided, then this is used as the value of λ. Otherwise, if X is a point
pattern, then λ will be estimated from X. If X is a summary statistic and lambda is missing, then the
intensity λ cannot be estimated, and the parameter µ will be returned as NA.
The remaining arguments rmin,rmax,q,p control the method of minimum contrast; see mincontrast.
The Matern Cluster process can be simulated, using rMatClust.
Homogeneous or inhomogeneous Matern Cluster models can also be fitted using the function kppm.
The optimisation algorithm can be controlled through the additional arguments "..." which are
passed to the optimisation function optim. For example, to constrain the parameter values to a
certain range, use the argument method="L-BFGS-B" to select an optimisation algorithm that respects box constraints, and use the arguments lower and upper to specify (vectors of) minimum
and maximum values for each parameter.

Value
An object of class "minconfit". There are methods for printing and plotting this object. It contains
the following main components:
par

Vector of fitted parameter values.

fit

Function value table (object of class "fv") containing the observed values of the
summary statistic (observed) and the theoretical values of the summary statistic
computed from the fitted model parameters.

Author(s)
Adrian Baddeley 
References
Møller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point
Processes. Chapman and Hall/CRC, Boca Raton.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous NeymanScott processes. Biometrics 63, 252–258.
See Also
kppm, matclust.estK, thomas.estpcf, thomas.estK, lgcp.estK, mincontrast, pcf, rMatClust
to simulate the fitted model.
Examples
data(redwood)
u <- matclust.estpcf(redwood, c(kappa=10, R=0.1))
u
plot(u, legendpos="topright")

Math.im

775

Math.im

S3 Group Generic methods for images

Description
These are group generic methods for images of class "im", which allows for usual mathematical
functions and operators to be applied directly to images. See Details for a list of implemented
functions.
Usage
## S3 methods for group generics have prototypes:
Math(x, ...)
Ops(e1, e2)
Complex(z)
Summary(..., na.rm=FALSE, drop=TRUE)

Arguments
x, z, e1, e2

objects of class "im".

...

further arguments passed to methods.

na.rm,drop

Logical values specifying whether missing values should be removed. This will
happen if either na.rm=TRUE or drop=TRUE. See Details.

Details
Below is a list of mathematical functions and operators which are defined for images. Not all
functions will make sense for all types of images. For example, none of the functions in the "Math"
group make sense for character-valued images. Note that the "Ops" group methods are implemented
using eval.im, which tries to harmonise images via harmonise.im if they aren’t compatible to
begin with.
1. Group "Math":
• abs, sign, sqrt,
floor, ceiling, trunc,
round, signif
• exp, log, expm1, log1p,
cos, sin, tan,
cospi, sinpi, tanpi,
acos, asin, atan
cosh, sinh, tanh,
acosh, asinh, atanh
• lgamma, gamma, digamma, trigamma
• cumsum, cumprod, cummax, cummin
2. Group "Ops":

776

Math.imlist
• "+", "-", "*", "/", "^", "%%", "%/%"
• "&", "|", "!"
• "==", "!=", "<", "<=", ">=", ">"
3. Group "Summary":
•
•
•
•

all, any
sum, prod
min, max
range

4. Group "Complex":
• Arg, Conj, Im, Mod, Re
For the Summary group, the generic has an argument na.rm=FALSE, but for pixel images it makes
sense to set na.rm=TRUE so that pixels outside the domain of the image are ignored. To enable this,
we added the argument drop. Pixel values that are NA are removed if drop=TRUE or if na.rm=TRUE.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak  and Kassel Hingee.
See Also
eval.im for evaluating expressions involving images.
Examples
## Convert gradient values to angle of inclination:
V <- atan(bei.extra$grad) * 180/pi
## Make logical image which is TRUE when heat equals 'Moderate':
A <- (gorillas.extra$heat == "Moderate")
## Summary:
any(A)
## Complex:
Z <- exp(1 + V * 1i)
Z
Re(Z)

Math.imlist

S3 Group Generic methods for List of Images

Description
These are group generic methods for the class "imlist" of lists of images. These methods allows
the usual mathematical functions and operators to be applied directly to lists of images. See Details
for a list of implemented functions.

Math.imlist

777

Usage
## S3 methods for group generics have prototypes:
Math(x, ...)
Ops(e1, e2)
Complex(z)
Summary(..., na.rm = TRUE)

Arguments
x, z, e1, e2

objects of class "imlist".

...

further arguments passed to methods.

na.rm

logical: should missing values be removed?

Details
Below is a list of mathematical functions and operators which are defined for lists of images. Not
all functions will make sense for all types of images. For example, none of the functions in the
"Math" group make sense for character-valued images. Note that the "Ops" group methods are
implemented using eval.im, which tries to harmonise images via harmonise.im if they aren’t
compatible to begin with.
1. Group "Math":
• abs, sign, sqrt,
floor, ceiling, trunc,
round, signif
• exp, log, expm1, log1p,
cos, sin, tan,
cospi, sinpi, tanpi,
acos, asin, atan
cosh, sinh, tanh,
acosh, asinh, atanh
• lgamma, gamma, digamma, trigamma
• cumsum, cumprod, cummax, cummin
2. Group "Ops":
• "+", "-", "*", "/", "^", "%%", "%/%"
• "&", "|", "!"
• "==", "!=", "<", "<=", ">=", ">"
3. Group "Summary":
•
•
•
•

all, any
sum, prod
min, max
range

4. Group "Complex":
• Arg, Conj, Im, Mod, Re

778

Math.linim

Value
The result of "Math", "Ops" and "Complex" group operations is another list of images. The result
of "Summary" group operations is a numeric vector of length 1 or 2.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
Math.im or eval.im for evaluating expressions involving images.
Examples
a <- Smooth(finpines, 2)
log(a)/2 - sqrt(a)
range(a)

Math.linim

S3 Group Generic Methods for Images on a Linear Network

Description
These are group generic methods for images of class "linim", which allows for usual mathematical
functions and operators to be applied directly to pixel images on a linear network. See Details for a
list of implemented functions.
Usage
## S3 methods for group generics have prototypes:
Math(x, ...)
Ops(e1, e2)
Complex(z)
Summary(..., na.rm = FALSE)

Arguments
x, z, e1, e2

objects of class "linim".

...

further arguments passed to methods.

na.rm

logical: should missing values be removed?

Math.linim

779

Details
An object of class "linim" represents a pixel image on a linear network. See linim.
Below is a list of mathematical functions and operators which are defined for these images. Not
all functions will make sense for all types of images. For example, none of the functions in the
"Math" group make sense for character-valued images. Note that the "Ops" group methods are
implemented using eval.linim.
1. Group "Math":
• abs, sign, sqrt,
floor, ceiling, trunc,
round, signif
• exp, log, expm1, log1p,
cos, sin, tan,
cospi, sinpi, tanpi,
acos, asin, atan
cosh, sinh, tanh,
acosh, asinh, atanh
• lgamma, gamma, digamma, trigamma
• cumsum, cumprod, cummax, cummin
2. Group "Ops":
• "+", "-", "*", "/", "^", "%%", "%/%"
• "&", "|", "!"
• "==", "!=", "<", "<=", ">=", ">"
3. Group "Summary":
• all, any
• sum, prod
• min, max
• range
4. Group "Complex":
• Arg, Conj, Im, Mod, Re
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
eval.linim for evaluating expressions involving images.
Examples
fx <- function(x,y,seg,tp) { (x - y)^2 }
fL <- linfun(fx, simplenet)
Z <- as.linim(fL)
A <- Z+2
A <- -Z
A <- sqrt(Z)
A <- !(Z > 0.1)

780

matrixpower

matrixpower

Power of a Matrix

Description
Evaluate a specified power of a matrix.
Usage
matrixpower(x, power, complexOK = TRUE)
matrixsqrt(x, complexOK = TRUE)
matrixinvsqrt(x, complexOK = TRUE)
Arguments
x

A square matrix containing numeric or complex values.

power

A numeric value giving the power (exponent) to which x should be raised.

complexOK

Logical value indicating whether the result is allowed to be complex.

Details
These functions raise the matrix x to the desired power: matrixsqrt takes the square root, matrixinvsqrt
takes the inverse square root, and matrixpower takes the specified power of x.
Up to numerical error, matrixpower(x, 2) should be equivalent to x %*% x, and matrixpower(x, -1)
should be equivalent to solve(x), the inverse of x.
The square root y <- matrixsqrt(x) should satisfy y %*% y = x. The inverse square root
z <- matrixinvsqrt(x) should satisfy z %*% z = solve(x).
Computations are performed using the eigen decomposition (eigen).
Value
A matrix of the same size as x containing numeric or complex values.
Author(s)
Adrian Baddeley .
See Also
eigen, svd
Examples
x <- matrix(c(10,2,2,1), 2, 2)
y <- matrixsqrt(x)
y
y %*% y
z <- matrixinvsqrt(x)
z %*% y
matrixpower(x, 0.1)

maxnndist

maxnndist

781

Compute Minimum or Maximum Nearest-Neighbour Distance

Description
A faster way to compute the minimum or maximum nearest-neighbour distance in a point pattern.
Usage
minnndist(X, positive=FALSE)
maxnndist(X, positive=FALSE)
Arguments
X

A point pattern (object of class "ppp").

positive

Logical. If FALSE (the default), compute the usual nearest-neighbour distance.
If TRUE, ignore coincident points, so that the nearest neighbour distance for each
point is greater than zero.

Details
These functions find the minimum and maximum values of nearest-neighbour distances in the point
pattern X. minnndist(X) and maxnndist(X) are equivalent to, but faster than, min(nndist(X))
and max(nndist(X)) respectively.
The value is NA if npoints(X) < 2.
Value
A single numeric value (possibly NA).
Author(s)
Adrian Baddeley 
, Rolf Turner  and Ege Rubak .
See Also
nndist
Examples
min(nndist(swedishpines))
minnndist(swedishpines)
max(nndist(swedishpines))
maxnndist(swedishpines)
minnndist(lansing, positive=TRUE)
if(interactive()) {
X <- rpoispp(1e6)
system.time(min(nndist(X)))

782

mean.im

}

system.time(minnndist(X))

mean.im

Mean and Median of Pixel Values in an Image

Description
Calculates the mean or median of the pixel values in a pixel image.
Usage
## S3 method for class 'im'
## mean(x, trim=0, na.rm=TRUE, ...)
## S3 method for class 'im'
## median(x, na.rm=TRUE)
## median(x, na.rm=TRUE, ...)

[R < 3.4.0]
[R >= 3.4.0]

Arguments
x

A pixel image (object of class "im").

na.rm

Logical value indicating whether NA values should be stripped before the computation proceeds.

trim

The fraction (0 to 0.5) of pixel values to be trimmed from each end of their
range, before the mean is computed.

...

Ignored.

Details
These functions calculate the mean and median of the pixel values in the image x.
An object of class "im" describes a pixel image. See im.object) for details of this class.
The function mean.im is a method for the generic function mean for the class "im". Similarly
median.im is a method for the generic median.
If the image x is logical-valued, the mean value of x is the fraction of pixels that have the value
TRUE. The median is not defined.
If the image x is factor-valued, then the mean of x is the mean of the integer codes of the pixel
values. The median is are not defined.
Other mathematical operations on images are supported by Math.im, Summary.im and Complex.im.
Other information about an image can be obtained using summary.im or quantile.im.
Value
A single number.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak  and Kassel Hingee.

mean.linim

783

See Also
Math.im for other operations.
Generics and default methods: mean, median.
quantile.im, anyNA.im, im.object, summary.im.
Examples
X <- as.im(function(x,y) {x^2}, unit.square())
mean(X)
median(X)
mean(X, trim=0.05)

mean.linim

Mean, Median, Quantiles of Pixel Values on a Linear Network

Description
Calculates the mean, median, or quantiles of the pixel values in a pixel image on a linear network.
Usage
## S3 method for class 'linim'
mean(x, ...)
## S3 method for class 'linim'
median(x, ...)
## S3 method for class 'linim'
quantile(x, probs=seq(0,1,0.25), ...)
Arguments
x

A pixel image on a linear network (object of class "linim").

probs

Vector of probabilities for which quantiles should be calculated.

...

Arguments passed to other methods.

Details
These functions calculate the mean, median and quantiles of the pixel values in the image x on a
linear network.
An object of class "linim" describes a pixel image on a linear network. See linim.
The functions described here are methods for the generic mean, median and quantile for the class
"linim".
Value
For mean and median, a single number. For quantile, a numeric vector of the same length as
probs.

784

measureVariation

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
mean, median, quantile,
mean.im.
Examples
M <- as.mask.psp(as.psp(simplenet))
Z <- as.im(function(x,y) {x-y}, W=M)
X <- linim(simplenet, Z)
X
mean(X)
median(X)
quantile(X)

measureVariation

Positive and Negative Parts, and Variation, of a Measure

Description
Given a measure A (object of class "msr") these functions find the positive part, negative part and
variation of A.
Usage
measurePositive(x)
measureNegative(x)
measureVariation(x)
totalVariation(x)
Arguments
x

A measure (object of class "msr").

Details
The functions measurePositive and measureNegative return the positive and negative parts of
the measure, and measureVariation returns the variation (sum of positive and negative parts). The
function totalVariation returns the total variation norm.
If µ is a signed measure, it can be represented as
µ = µ+ − µ−
where µ+ and µ− are nonnegative measures called the positive and negative parts of µ. In a nutshell,
the positive part of µ consists of all positive contributions or increments, and the negative part
consists of all negative contributions multiplied by -1.
The variation |µ| is defined by
µ = µ+ + µ−

mergeLevels

785

and is also a nonnegative measure.
The total variation norm is the integral of the variation.
Value
The result of measurePositive, measureNegative and measureVariation is another measure
(object of class "msr") on the same spatial domain. The result of totalVariation is a non-negative
number.
Author(s)
Adrian Baddeley .
References
Halmos, P.R. (1950) Measure Theory. Van Nostrand.
See Also
msr, with.msr, split.msr
Examples
X <- rpoispp(function(x,y) { exp(3+3*x) })
fit <- ppm(X, ~x+y)
rp <- residuals(fit, type="pearson")
measurePositive(rp)
measureNegative(rp)
measureVariation(rp)
# total variation norm
totalVariation(rp)

mergeLevels

Merge Levels of a Factor

Description
Specified levels of the factor will be merged into a single level.
Usage
mergeLevels(.f, ...)
Arguments
.f

A factor (or a factor-valued pixel image or a point pattern with factor-valued
marks).

...

List of name=value pairs, where name is the new merged level, and value is the
vector of old levels that will be merged.

786

methods.box3

Details
This utility function takes a factor .f and merges specified levels of the factor.
The grouping is specified by the arguments ... which must each be given in the form new=old,
where new is the name for the new merged level, and old is a character vector containing the old
levels that are to be merged.
The result is a new factor (or factor-valued object), in which the levels listed in old have been
replaced by a single level new.
An argument of the form name=character(0) or name=NULL is interpreted to mean that all other
levels of the old factor should be mapped to name.
Value
Another factor of the same length as .f (or object of the same kind as .f).
Tips for manipulating factor levels
To remove unused levels from a factor f, just type f <- factor(f).
To change the ordering of levels in a factor, use factor(f, levels=l) or relevel(f, ref).
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
factor, relevel
Examples
likert <- c("Strongly Agree", "Agree", "Neutral",
"Disagree", "Strongly Disagree")
answers <- factor(sample(likert, 15, replace=TRUE), levels=likert)
answers
mergeLevels(answers, Positive=c("Strongly Agree", "Agree"),
Negative=c("Strongly Disagree", "Disagree"))

methods.box3

Methods for Three-Dimensional Box

Description
Methods for class "box3".

methods.box3

787

Usage
## S3 method for class 'box3'
print(x, ...)
## S3 method for class 'box3'
unitname(x)
## S3 replacement method for class 'box3'
unitname(x) <- value

Arguments
x

Object of class "box3" representing a three-dimensional box.

...

Other arguments passed to print.default.

value

Name of the unit of length. See unitname.

Details
These are methods for the generic functions print and unitname for the class "box3" of threedimensional boxes.
The print method prints a description of the box, while the unitname method extracts the name of
the unit of length in which the box coordinates are expressed.

Value
For print.box3 the value is NULL. For unitname.box3 an object of class "units".

Author(s)
Adrian Baddeley 
and Rolf Turner 

See Also
box3, print, unitname

Examples
X <- box3(c(0,10),c(0,10),c(0,5), unitname=c("metre", "metres"))
X
unitname(X)
# Northern European usage
unitname(X) <- "meter"

788

methods.boxx

methods.boxx

Methods for Multi-Dimensional Box

Description
Methods for class "boxx".
Usage
## S3 method for class 'boxx'
print(x, ...)
## S3 method for class 'boxx'
unitname(x)
## S3 replacement method for class 'boxx'
unitname(x) <- value
Arguments
x

Object of class "boxx" representing a multi-dimensional box.

...

Other arguments passed to print.default.

value

Name of the unit of length. See unitname.

Details
These are methods for the generic functions print and unitname for the class "boxx" of multidimensional boxes.
The print method prints a description of the box, while the unitname method extracts the name of
the unit of length in which the box coordinates are expressed.
Value
For print.boxx the value is NULL. For unitname.boxx an object of class "units".
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
boxx, print, unitname
Examples
X <- boxx(c(0,10),c(0,10),c(0,5),c(0,1), unitname=c("metre", "metres"))
X
unitname(X)
# Northern European usage
unitname(X) <- "meter"

methods.dppm

methods.dppm

789

Methods for Determinantal Point Process Models

Description
These are methods for the class "dppm".
Usage
## S3 method for class
coef(object, ...)
## S3 method for class
formula(x, ...)
## S3 method for class
print(x, ...)
## S3 method for class
terms(x, ...)
## S3 method for class
labels(object, ...)

'dppm'
'dppm'
'dppm'
'dppm'
'dppm'

Arguments
x,object

An object of class "dppm", representing a fitted determinantal point process
model.

...

Arguments passed to other methods.

Details
These functions are methods for the generic commands coef, formula, print, terms and labels
for the class "dppm".
An object of class "dppm" represents a fitted determinantal point process model. It is obtained from
dppm.
The method coef.dppm returns the vector of regression coefficients of the fitted model. It does not
return the interaction parameters.
Value
See the help files for the corresponding generic functions.
Author(s)
Adrian Baddeley 
See Also
dppm, plot.dppm, predict.dppm, simulate.dppm, as.ppm.dppm.

790

methods.fii

Examples
fit <- dppm(swedishpines ~ x + y, dppGauss())
coef(fit)
formula(fit)
tf <- terms(fit)
labels(fit)

methods.fii

Methods for Fitted Interactions

Description
These are methods specifically for the class "fii" of fitted interpoint interactions.
Usage
## S3 method for class 'fii'
print(x, ...)
## S3 method for class 'fii'
coef(object, ...)
## S3 method for class 'fii'
plot(x, ...)
## S3 method for class 'fii'
summary(object,...)
## S3 method for class 'summary.fii'
print(x, ...)
## S3 method for class 'summary.fii'
coef(object, ...)
Arguments
x,object

An object of class "fii" representing a fitted interpoint interaction.

...

Arguments passed to other methods.

Details
These are methods for the class "fii". An object of class "fii" represents a fitted interpoint
interaction. It is usually obtained by using the command fitin to extract the fitted interaction part
of a fitted point process model. See fitin for further explanation of this class.
The commands listed here are methods for the generic functions print, summary, plot and coef
for objects of the class "fii".
Following the usual convention, summary.fii returns an object of class summary.fii, for which
there is a print method. The effect is that, when the user types summary(x), the summary is printed,
but when the user types y <- summary(x), the summary information is saved.

methods.funxy

791

The method coef.fii extracts the canonical coefficients of the fitted interaction, and returns them
as a numeric vector. The method coef.summary.fii transforms these values into quantities that
are more easily interpretable, in a format that depends on the particular model.
There are also methods for the generic commands reach and as.interact, described elsewhere.
Value
The print and plot methods return NULL.
The summary method returns an object of class summary.fii.
coef.fii returns a numeric vector. coef.summary.fii returns data whose structure depends on
the model.
Author(s)
Adrian Baddeley 
See Also
fitin, reach.fii, as.interact.fii
Examples
mod <- ppm(cells, ~1, Strauss(0.1))
f <- fitin(mod)
f
summary(f)
plot(f)
coef(f)
coef(summary(f))

methods.funxy

Methods for Spatial Functions

Description
Methods for objects of the class "funxy".
Usage
## S3 method for class 'funxy'
contour(x, ...)
## S3 method for class 'funxy'
persp(x, ...)
## S3 method for class 'funxy'
plot(x, ...)
Arguments
x

Object of class "funxy" representing a function of x, y coordinates.

...

Named arguments controlling the plot. See Details.

792

methods.kppm

Details
These are methods for the generic functions plot, contour and persp for the class "funxy" of
spatial functions.
Objects of class "funxy" are created, for example, by the commands distfun and funxy.
The plot, contour and persp methods first convert x to a pixel image object using as.im, then
display it using plot.im, contour.im or persp.im.
Additional arguments ... are either passed to as.im.function to control the spatial resolution of
the pixel image, or passed to contour.im, persp.im or plot.im to control the appearance of the
plot.
Value
NULL.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
funxy, distfun, as.im, plot.im, persp.im, contour.im, spatstat.options
Examples
data(letterR)
f <- distfun(letterR)
contour(f)
contour(f, W=owin(c(1,5),c(-1,4)), eps=0.1)

methods.kppm

Methods for Cluster Point Process Models

Description
These are methods for the class "kppm".
Usage
## S3 method for class
coef(object, ...)
## S3 method for class
formula(x, ...)
## S3 method for class
print(x, ...)
## S3 method for class
terms(x, ...)
## S3 method for class
labels(object, ...)

'kppm'
'kppm'
'kppm'
'kppm'
'kppm'

methods.layered

793

Arguments
x,object

An object of class "kppm", representing a fitted cluster point process model.

...

Arguments passed to other methods.

Details
These functions are methods for the generic commands coef, formula, print, terms and labels
for the class "kppm".
An object of class "kppm" represents a fitted cluster point process model. It is obtained from kppm.
The method coef.kppm returns the vector of regression coefficients of the fitted model. It does not
return the clustering parameters.
Value
See the help files for the corresponding generic functions.
Author(s)
Adrian Baddeley 
See Also
kppm, plot.kppm, predict.kppm, simulate.kppm, update.kppm, vcov.kppm, as.ppm.kppm.
Examples
data(redwood)
fit <- kppm(redwood ~ x, "MatClust")
coef(fit)
formula(fit)
tf <- terms(fit)
labels(fit)

methods.layered

Methods for Layered Objects

Description
Methods for geometrical transformations of layered objects (class "layered").
Usage
## S3 method for class 'layered'
shift(X, vec=c(0,0), ...)
## S3 method for class 'layered'
rotate(X, ..., centre=NULL)
## S3 method for class 'layered'
affine(X, ...)

794

methods.layered
## S3 method for class 'layered'
reflect(X)
## S3 method for class 'layered'
flipxy(X)
## S3 method for class 'layered'
rescale(X, s, unitname)
## S3 method for class 'layered'
scalardilate(X, ...)

Arguments
X

Object of class "layered".

...

Arguments passed to the relevant methods when applying the operation to each
layer of X.

s

Rescaling factor passed to the relevant method for rescale. May be missing.

vec

Shift vector (numeric vector of length 2).

centre

Centre of rotation. Either a vector of length 2, or a character string (partially
matched to "centroid", "midpoint" or "bottomleft"). The default is the
coordinate origin c(0,0).

unitname

Optional. New name for the unit of length. A value acceptable to the function
unitname<-

Details
These are methods for the generic functions shift, rotate, reflect, affine, rescale, scalardilate
and flipxy for the class of layered objects.
A layered object represents data that should be plotted in successive layers, for example, a background and a foreground. See layered.
Value
Another object of class "layered".
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
layered
Examples
L <- layered(letterR, runifpoint(20, letterR))
plot(L)
plot(rotate(L, pi/4))

methods.linfun

methods.linfun

795

Methods for Functions on Linear Network

Description
Methods for the class "linfun" of functions on a linear network.
Usage
## S3 method for class 'linfun'
print(x, ...)
## S3 method for class 'linfun'
summary(object, ...)
## S3 method for class 'linfun'
plot(x, ..., L=NULL, main)
## S3 method for class 'linfun'
as.data.frame(x, ...)
## S3 method for class 'linfun'
as.owin(W, ...)
## S3 method for class 'linfun'
as.function(x, ...)
Arguments
x,object,W

A function on a linear network (object of class "linfun").

L

A linear network

...

Extra arguments passed to as.linim, plot.linim, plot.im or print.default,
or arguments passed to x if it is a function.

main

Main title for plot.

Details
These are methods for the generic functions plot, print, summary as.data.frame and as.function,
and for the spatstat generic function as.owin.
An object of class "linfun" represents a mathematical function that could be evaluated at any
location on a linear network. It is essentially an R function with some extra attributes.
The method as.owin.linfun extracts the two-dimensional spatial window containing the linear
network.
The method plot.linfun first converts the function to a pixel image using as.linim.linfun, then
plots the image using plot.linim.
Note that a linfun function may have additional arguments, other than those which specify the
location on the network (see linfun). These additional arguments may be passed to plot.linfun.

796

methods.linim

Value
For print.linfun and summary.linfun the result is NULL.
For plot.linfun the result is the same as for plot.linim.
For the conversion methods, the result is an object of the required type: as.owin.linfun returns
an object of class "owin", and so on.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
Examples
X <- runiflpp(3, simplenet)
f <- nnfun(X)
f
plot(f)
as.function(f)
as.owin(f)
head(as.data.frame(f))

methods.linim

Methods for Images on a Linear Network

Description
Methods for the class "linim" of functions on a linear network.
Usage
## S3 method for class 'linim'
print(x, ...)
## S3 method for class 'linim'
summary(object, ...)
## S3 method for class 'linim'
as.im(X, ...)
## S3 method for class 'linim'
as.data.frame(x, ...)
## S3 method for class 'linim'
shift(X, ...)
## S3 method for class 'linim'
scalardilate(X, f, ..., origin=NULL)
## S3 method for class 'linim'
affine(X, mat=diag(c(1,1)), vec=c(0,0), ...)

methods.linim

797

Arguments
X,x,object

A pixel image on a linear network (object of class "linim").

...

Extra arguments passed to other methods.

f

Numeric. Scalar dilation factor.

mat

Numeric matrix representing the linear transformation.

vec

Numeric vector of length 2 specifying the shift vector.

origin

Character string determining a location that will be shifted to the origin. Options
are "centroid", "midpoint" and "bottomleft". Partially matched.

Details
These are methods for the generic functions print, summary and as.data.frame, and the spatstat
generic functions as.im, shift, scalardilate and affine.
An object of class "linfun" represents a pixel image defined on a linear network.
The method as.im.linim extracts the pixel values and returns a pixel image of class "im".
The method as.data.frame.linim returns a data frame giving spatial locations (in cartesian and
network coordinates) and corresponding function values.
The methods shift.linim, scalardilate.linim and affine.linim apply geometric transformations to the pixels and the underlying linear network, without changing the pixel values.
Value
For print.linim the result is NULL.
The function summary.linim returns an object of class "summary.linim". In normal usage this
summary is automatically printed by print.summary.linim.
For as.im.linim the result is an object of class "im".
For the geometric transformations shift.linim, scalardilate.linim and affine.linim, the
result is another object of class "linim".
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
Examples
M <- as.mask.psp(as.psp(simplenet))
Z <- as.im(function(x,y) {x-y}, W=M)
X <- linim(simplenet, Z)
X
shift(X, c(1,1))
scalardilate(X, 2)
head(as.data.frame(X))

798

methods.linnet

methods.linnet

Methods for Linear Networks

Description
These are methods for the class "linnet" of linear networks.
Usage
as.linnet(X, ...)
## S3 method for class 'linnet'
as.linnet(X, ..., sparse)
## S3 method for class 'linnet'
as.owin(W, ...)
## S3 method for class 'linnet'
as.psp(x, ..., fatal=TRUE)
## S3 method for class 'linnet'
nsegments(x)
## S3 method for class 'linnet'
nvertices(x, ...)
## S3 method for class 'linnet'
pixellate(x, ...)
## S3 method for class 'linnet'
print(x, ...)
## S3 method for class 'linnet'
summary(object, ...)
## S3 method for class 'linnet'
unitname(x)
## S3 replacement method for class 'linnet'
unitname(x) <- value
vertexdegree(x)
## S3 method for class 'linnet'
vertices(w)
## S3 method for class 'linnet'
volume(x)
## S3 method for class 'linnet'
Window(X, ...)

methods.linnet

799

Arguments
x,X,object,w,W An object of class "linnet" representing a linear network.
...

Arguments passed to other methods.

value

A valid name for the unit of length for x. See unitname.

fatal

Logical value indicating whether data in the wrong format should lead to an
error (fatal=TRUE) or a warning (fatal=FALSE).

sparse

Logical value indicating whether to use a sparse matrix representation, as explained in linnet. Default is to keep the same representation as in X.

Details
The function as.linnet is generic. It converts data from some other format into an object of class
"linnet". The method as.linnet.lpp extracts the linear network information from an lpp object.
The other functions are methods for the generic commands as.owin, as.psp, nsegments, nvertices,
pixellate, print, summary, unitname, unitname<-, vertices, volume and Window for the class
"linnet".
The methods as.owin.linnet and Window.linnet extract the window containing the linear network, and return it as an object of class "owin".
The method as.psp.linnet extracts the lines of the linear network as a line segment pattern (object
of class "psp") while nsegments.linnet simply counts the number of line segments.
The method vertices.linnet extracts the vertices (nodes) of the linear network and nvertices.linnet
simply counts the vertices. The function vertexdegree calculates the topological degree of each
vertex (the number of lines emanating from that vertex) and returns these values as an integer vector.
The method pixellate.linnet applies as.psp.linnet to convert the network to a collection of
line segments, then invokes pixellate.psp.
Value
For as.linnet the value is an object of class "linnet". For other functions, see the help file for
the corresponding generic function.
Author(s)
Adrian Baddeley 
See Also
linnet.
Generic functions: as.owin, as.psp, nsegments, nvertices, pixellate, print, summary, unitname,
unitname<-, vertices, volume and Window.
Special tools: thinNetwork, insertVertices, connected.linnet.
lixellate for dividing segments into shorter segments.
Examples
simplenet
summary(simplenet)
nsegments(simplenet)
nvertices(simplenet)
pixellate(simplenet)

800

methods.lpp
volume(simplenet)
unitname(simplenet) <- c("cubit", "cubits")
Window(simplenet)

methods.lpp

Methods for Point Patterns on a Linear Network

Description
These are methods specifically for the class "lpp" of point patterns on linear networks.
Usage
## S3 method for class 'lpp'
as.ppp(X, ..., fatal=TRUE)
## S3 method for class 'lpp'
as.psp(x, ..., fatal=TRUE)
## S3 replacement method for class 'lpp'
marks(x, ...) <- value
## S3 method for class 'lpp'
nsegments(x)
## S3 method for class 'lpp'
print(x, ...)
## S3 method for class 'summary.lpp'
print(x, ...)
## S3 method for class 'lpp'
summary(object, ...)
## S3 method for class 'lpp'
unitname(x)
## S3 replacement method for class 'lpp'
unitname(x) <- value
## S3 method for class 'lpp'
unmark(X)
Arguments
x,X,object

An object of class "lpp" representing a point pattern on a linear network.

...

Arguments passed to other methods.

value

Replacement value for the marks or unitname of x. See Details.

fatal

Logical value indicating whether data in the wrong format should lead to an
error (fatal=TRUE) or a warning (fatal=FALSE).

methods.lppm

801

Details
These are methods for the generic functions as.ppp, as.psp, marks<-, nsegments, print, summary,
unitname, unitname<- and unmark for objects of the class "lpp".
For "marks<-.lpp" the replacement value should be either NULL, or a vector of length equal to the
number of points in x, or a data frame with one row for each point in x.
For "unitname<-.lpp" the replacement value should be a valid name for the unit of length, as
described in unitname.
Value
See the documentation on the corresponding generic function.
Other methods
An object of class "lpp" also inherits the class "ppx" for which many other methods are available.
See methods.ppx.
Author(s)
Adrian Baddeley 
See Also
lpp, intensity.lpp, methods.ppx
Examples
X <- runiflpp(10, simplenet)
X
as.ppp(X)
summary(X)
unitname(X) <- c("furlong", "furlongs")

methods.lppm

Methods for Fitted Point Process Models on a Linear Network

Description
These are methods for the class "lppm" of fitted point process models on a linear network.
Usage
## S3 method for class 'lppm'
coef(object, ...)
## S3 method for class 'lppm'
emend(object, ...)
## S3 method for class 'lppm'
extractAIC(fit, ...)

802

methods.lppm
## S3 method for class 'lppm'
formula(x, ...)
## S3 method for class 'lppm'
logLik(object, ...)
## S3 method for class 'lppm'
deviance(object, ...)
## S3 method for class 'lppm'
nobs(object, ...)
## S3 method for class 'lppm'
print(x, ...)
## S3 method for class 'lppm'
summary(object, ...)
## S3 method for class 'lppm'
terms(x, ...)
## S3 method for class 'lppm'
update(object, ...)
## S3 method for class 'lppm'
valid(object, ...)
## S3 method for class 'lppm'
vcov(object, ...)
## S3 method for class 'lppm'
as.linnet(X, ...)

Arguments
object,fit,x,X An object of class "lppm" representing a fitted point process model on a linear
network.
...

Arguments passed to other methods, usually the method for the class "ppm".

Details
These are methods for the generic commands coef, emend, extractAIC, formula, logLik, deviance,
nobs, print, summary, terms, update, valid and vcov for the class "lppm".
Value
See the default methods.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 

methods.objsurf

803

See Also
lppm, plot.lppm.
Examples
X <- runiflpp(15, simplenet)
fit <- lppm(X ~ x)
print(fit)
coef(fit)
formula(fit)
terms(fit)
logLik(fit)
deviance(fit)
nobs(fit)
extractAIC(fit)
update(fit, ~1)
valid(fit)
vcov(fit)

methods.objsurf

Methods for Objective Function Surfaces

Description
Methods for printing and plotting an objective function surface.
Usage
## S3 method for
print(x, ...)
## S3 method for
plot(x, ...)
## S3 method for
image(x, ...)
## S3 method for
contour(x, ...)
## S3 method for
persp(x, ...)

class 'objsurf'
class 'objsurf'
class 'objsurf'
class 'objsurf'
class 'objsurf'

Arguments
x

Object of class "objsurf" representing an objective function surface.

...

Additional arguments passed to plot methods.

Details
These are methods for the generic functions print, plot, image, contour and persp for the class
"objsurf".

804

methods.pp3

Value
For print.objsurf, plot.objsurf and image.objsurf the value is NULL.
For contour.objsurf and persp.objsurf the value is described in the help for contour.default
and persp.default respectively.
Author(s)
Adrian Baddeley 
and Ege Rubak .
See Also
objsurf
Examples
fit <- kppm(redwood ~ 1, "Thomas")
os <- objsurf(fit)
os
plot(os)
contour(os, add=TRUE)
persp(os)

methods.pp3

Methods for three-dimensional point patterns

Description
Methods for class "pp3".
Usage
## S3 method for class 'pp3'
print(x, ...)
## S3 method for class 'summary.pp3'
print(x, ...)
## S3 method for class 'pp3'
summary(object, ...)
## S3 method for class 'pp3'
unitname(x)
## S3 replacement method for class 'pp3'
unitname(x) <- value
Arguments
x,object

Object of class "pp3".

...

Ignored.

value

Name of the unit of length. See unitname.

methods.ppx

805

Details
These are methods for the generic functions print, summary, unitname and unitname<- for the
class "pp3" of three-dimensional point patterns.
The print and summary methods print a description of the point pattern.
The unitname method extracts the name of the unit of length in which the point coordinates are
expressed. The unitname<- method assigns the name of the unit of length.
Value
For print.pp3 the value is NULL. For unitname.pp3 an object of class "units".
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
pp3, print, unitname unitname
and Rolf Turner 
See Also
ppx, unitname

methods.rho2hat

Methods for Intensity Functions of Two Spatial Covariates

Description
These are methods for the class "rho2hat".
Usage
## S3 method for class 'rho2hat'
plot(x, ..., do.points=FALSE)
## S3 method for class 'rho2hat'
print(x, ...)
## S3 method for class 'rho2hat'
predict(object, ..., relative=FALSE)
Arguments
x,object

An object of class "rho2hat".

...

Arguments passed to other methods.

do.points

Logical value indicating whether to plot the observed values of the covariates at
the data points.

relative

Logical value indicating whether to compute the estimated point process intensity (relative=FALSE) or the relative risk (relative=TRUE) in the case of a
relative risk estimate.

methods.rhohat

807

Details
These functions are methods for the generic commands print, predict and plot for the class
"rho2hat".
An object of class "rho2hat" is an estimate of the intensity of a point process, as a function of two
given spatial covariates. See rho2hat.
The method plot.rho2hat displays the estimated function ρ using plot.fv, and optionally adds a
rug plot of the observed values of the covariate. In this plot the two axes represent possible values
of the two covariates.
The method predict.rho2hat computes a pixel image of the intensity ρ(Z1 (u), Z2 (u)) at each
spatial location u, where Z1 (u) and Z2 (u) are the two spatial covariates.
Value
For predict.rho2hat the value is a pixel image (object of class "im"). For other functions, the
value is NULL.
Author(s)
Adrian Baddeley 
See Also
rho2hat
Examples
r2 <- with(bei.extra, rho2hat(bei, elev, grad))
r2
plot(r2)
plot(predict(r2))

methods.rhohat

Methods for Intensity Functions of Spatial Covariate

Description
These are methods for the class "rhohat".
Usage
## S3 method for class 'rhohat'
print(x, ...)
## S3 method for class 'rhohat'
plot(x, ..., do.rug=TRUE)
## S3 method for class 'rhohat'
predict(object, ..., relative=FALSE,
what=c("rho", "lo", "hi", "se"))
## S3 method for class 'rhohat'
simulate(object, nsim=1, ..., drop=TRUE)

808

methods.rhohat

Arguments
x,object

An object of class "rhohat" representing a smoothed estimate of the intensity
function of a point process.

...

Arguments passed to other methods.

do.rug

Logical value indicating whether to plot the observed values of the covariate as
a rug plot along the horizontal axis.

relative

Logical value indicating whether to compute the estimated point process intensity (relative=FALSE) or the relative risk (relative=TRUE) in the case of a
relative risk estimate.

nsim

Number of simulations to be generated.

drop

Logical value indicating what to do when nsim=1. If drop=TRUE (the default),
a point pattern is returned. If drop=FALSE, a list of length 1 containing a point
pattern is returned.

what

Optional character string (partially matched) specifying which value should be
calculated: either the function estimate (what="rho", the default), the lower or
upper end of the confidence interval (what="lo" or what="hi") or the standard
error (what="se").

Details
These functions are methods for the generic commands print, plot, predict and simulate for
the class "rhohat".
An object of class "rhohat" is an estimate of the intensity of a point process, as a function of a
given spatial covariate. See rhohat.
The method plot.rhohat displays the estimated function ρ using plot.fv, and optionally adds a
rug plot of the observed values of the covariate.
The method predict.rhohat computes a pixel image of the intensity ρ(Z(u)) at each spatial
location u, where Z is the spatial covariate.
The method simulate.rhohat invokes predict.rhohat to determine the predicted intensity, and
then simulates a Poisson point process with this intensity.
Value
For predict.rhohat the value is a pixel image (object of class "im" or "linim"). For simulate.rhohat
the value is a point pattern (object of class "ppp" or "lpp"). For other functions, the value is NULL.
Author(s)
Adrian Baddeley 
See Also
rhohat
Examples
X <- rpoispp(function(x,y){exp(3+3*x)})
rho <- rhohat(X, function(x,y){x})
rho
plot(rho)

methods.slrm

809

Y <- predict(rho)
plot(Y)
plot(simulate(rho), add=TRUE)
#
fit <- ppm(X, ~x)
rho <- rhohat(fit, "y")
opa <- par(mfrow=c(1,2))
plot(predict(rho))
plot(predict(rho, relative=TRUE))
par(opa)
plot(predict(rho, what="se"))

methods.slrm

Methods for Spatial Logistic Regression Models

Description
These are methods for the class "slrm".
Usage
## S3 method for class 'slrm'
formula(x, ...)
## S3 method for class 'slrm'
print(x, ...)
## S3 method for class 'slrm'
terms(x, ...)
## S3 method for class 'slrm'
labels(object, ...)
## S3 method for class 'slrm'
update(object, ..., evaluate = TRUE, env = parent.frame())
Arguments
x,object

An object of class "slrm", representing a fitted spatial logistic regression model.

...

Arguments passed to other methods.

evaluate

Logical value. If TRUE, evaluate the updated call to slrm, so that the model is
refitted; if FALSE, simply return the updated call.

env

Optional environment in which the model should be updated.

Details
These functions are methods for the generic commands formula, update, print, terms and labels
for the class "slrm".
An object of class "slrm" represents a fitted spatial logistic regression model. It is obtained from
slrm.
Value
See the help files for the corresponding generic functions.

810

methods.ssf

Author(s)
Adrian Baddeley 
See Also
slrm, plot.slrm, predict.slrm, simulate.slrm, vcov.slrm, coef.slrm.
Examples
data(redwood)
fit <- slrm(redwood ~ x)
coef(fit)
formula(fit)
tf <- terms(fit)
labels(fit)

methods.ssf

Methods for Spatially Sampled Functions

Description
Methods for various generic commands, for the class "ssf" of spatially sampled functions.
Usage
## S3 method for class 'ssf'
marks(x, ...)
## S3 replacement method for class 'ssf'
marks(x, ...) <- value
## S3 method for class 'ssf'
unmark(X)
## S3 method for class 'ssf'
as.im(X, ...)
## S3 method for class 'ssf'
as.function(x, ...)
## S3 method for class 'ssf'
as.ppp(X, ...)
## S3 method for class 'ssf'
print(x, ..., brief=FALSE)
## S3 method for class 'ssf'
range(x, ...)
## S3 method for class 'ssf'
min(x, ...)

methods.ssf

811

## S3 method for class 'ssf'
max(x, ...)
## S3 method for class 'ssf'
integral(f, domain=NULL, ..., weights=attr(f, "weights"))
Arguments
x,X,f
...
brief
value
domain
weights

A spatially sampled function (object of class "ssf").
Arguments passed to the default method.
Logical value controlling the amount of detail printed.
Matrix of replacement values for the function.
Optional. Domain of integration. An object of class"owin".
Optional. Numeric vector of weights associated with the sample points.

Details
An object of class "ssf" represents a function (real- or vector-valued) that has been sampled at a
finite set of points.
The commands documented here are methods for this class, for the generic commands marks,
marks<-, unmark, as.im, as.function, as.ppp, print, range, min, max and integral.
Value
marks returns a matrix.
marks(x) <- value returns an object of class "ssf".
as.owin returns a window (object of class "owin").
as.ppp and unmark return a point pattern (object of class "ppp").
as.function returns a function(x,y) of class "funxy".
print returns NULL.
range returns a numeric vector of length 2. min and max return a single numeric value.
integral returns a numeric value (if x had numeric values) or a numeric vector (if x had vector
values).
Author(s)
Adrian Baddeley
See Also
ssf
Examples
X <- cells[1:4]
f <- ssf(X, nndist(X, k=1:3))
f
marks(f)
as.ppp(f)
as.im(f)

812

methods.unitname

methods.unitname

Methods for Units

Description
Methods for class "unitname".
Usage
## S3 method for class 'unitname'
print(x, ...)
## S3 method for class 'unitname'
summary(object, ...)
## S3 method for class 'unitname'
rescale(X, s, unitname)
## S3 method for class 'unitname'
compatible(A,B, ..., coerce=TRUE)
## S3 method for class 'unitname'
harmonise(..., coerce=TRUE, single=FALSE)
## S3 method for class 'unitname'
harmonize(..., coerce=TRUE, single=FALSE)
Arguments
x,X,A,B,object Objects of class "unitname" representing units of length.
...

Other arguments. For print.unitname these arguments are passed to print.default.
For summary.unitname they are ignored. For compatible.unitname and harmonise.unitname
these arguments are other objects of class "unitname".

s

Conversion factor: the new units are s times the old units.

unitname

Optional new name for the unit. If present, this overrides the rescaling operation
and simply substitutes the new name for the old one.

coerce

Logical. If TRUE, a null unit of length is compatible with any non-null unit.

single

Logical value indicating whether to return a single unitname, or a list of unitnames.

Details
These are methods for the generic functions print, summary, rescale and compatible for the
class "unitname".
An object of class "unitname" represents a unit of length.
The print method prints a description of the unit of length, and the summary method gives a more
detailed description.
The rescale method changes the unit of length by rescaling it.
The compatible method tests whether two or more units of length are compatible.
The harmonise method returns the common unit of length if there is one. For consistency with other
methods for harmonise, the result is a list of unitname objects, with one entry for each argument
in .... All of these entries are identical. This can be overridden by setting single=TRUE when the
result will be a single unitname object.

methods.zclustermodel

813

Value
For print.unitname the value is NULL. For summary.unitname the value is an object of class
summary.unitname (with its own print method). For rescale.unitname the value is another object
of class "unitname". For compatible.unitname the result is logical. For harmonise.unitname
the result is a list of identical unitnames if single=FALSE (the default), or a single unitname if
single=TRUE.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
box3, print, unitname

methods.zclustermodel Methods for Cluster Models

Description
Methods for the experimental class of cluster models.
Usage
## S3 method for class 'zclustermodel'
pcfmodel(model, ...)
## S3 method for class 'zclustermodel'
predict(object, ...,
locations, type = "intensity", ngrid = NULL)
## S3 method for class 'zclustermodel'
print(x, ...)
Arguments
model,object,x Object of class "zclustermodel".
...

Arguments passed to other methods.

locations

Locations where prediction should be performed. A window or a point pattern.

type

Currently must equal "intensity".

ngrid

Pixel grid dimensions for prediction, if locations is a rectangle or polygon.

Details
Experimental.
Value
Same as for other methods.

814

midpoints.psp

Author(s)
Adrian Baddeley 
See Also
zclustermodel
Examples
m <- zclustermodel("Thomas", kappa=10, mu=5, scale=0.1)
m2 <- zclustermodel("VarGamma", kappa=10, mu=10, scale=0.1, nu=0.7)
m
m2
g <- pcfmodel(m)
g(0.2)
g2 <- pcfmodel(m2)
g2(1)
Z <- predict(m, locations=square(2))
Z2 <- predict(m2, locations=square(1))
varcount(m, square(1))
varcount(m2, square(1))

midpoints.psp

Midpoints of Line Segment Pattern

Description
Computes the midpoints of each line segment in a line segment pattern.
Usage
midpoints.psp(x)
Arguments
x

A line segment pattern (object of class "psp").

Details
The midpoint of each line segment is computed.
Value
Point pattern (object of class "ppp").
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
summary.psp, lengths.psp, angles.psp

mincontrast

815

Examples
a <- psp(runif(10), runif(10), runif(10), runif(10), window=owin())
b <- midpoints.psp(a)

mincontrast

Method of Minimum Contrast

Description
A general low-level algorithm for fitting theoretical point process models to point pattern data by
the Method of Minimum Contrast.
Usage
mincontrast(observed, theoretical, startpar, ...,
ctrl=list(q = 1/4, p = 2, rmin=NULL, rmax=NULL),
fvlab=list(label=NULL, desc="minimum contrast fit"),
explain=list(dataname=NULL, modelname=NULL, fname=NULL),
adjustment=NULL)
Arguments
observed

Summary statistic, computed for the data. An object of class "fv".

theoretical

An R language function that calculates the theoretical expected value of the
summary statistic, given the model parameters. See Details.

startpar

Vector of initial values of the parameters of the point process model (passed to
theoretical).

...

Additional arguments passed to the function theoretical and to the optimisation algorithm optim.

ctrl

Optional. List of arguments controlling the optimisation. See Details.

fvlab

Optional. List containing some labels for the return value. See Details.

explain

Optional. List containing strings that give a human-readable description of the
model, the data and the summary statistic.

adjustment

Internal use only.

Details
This function is a general algorithm for fitting point process models by the Method of Minimum
Contrast. If you want to fit the Thomas process, see thomas.estK. If you want to fit a log-Gaussian
Cox process, see lgcp.estK. If you want to fit the Matern cluster process, see matclust.estK.
The Method of Minimum Contrast (Diggle and Gratton, 1984) is a general technique for fitting
a point process model to point pattern data. First a summary function (typically the K function)
is computed from the data point pattern. Second, the theoretical expected value of this summary
statistic under the point process model is derived (if possible, as an algebraic expression involving
the parameters of the model) or estimated from simulations of the model. Then the model is fitted by
finding the optimal parameter values for the model to give the closest match between the theoretical
and empirical curves.

816

mincontrast
The argument observed should be an object of class "fv" (see fv.object) containing the values
of a summary statistic computed from the data point pattern. Usually this is the function K(r)
computed by Kest or one of its relatives.
The argument theoretical should be a user-supplied function that computes the theoretical expected value of the summary statistic. It must have an argument named par that will be the vector
of parameter values for the model (the length and format of this vector are determined by the starting values in startpar). The function theoretical should also expect a second argument (the
first argument other than par) containing values of the distance r for which the theoretical value of
the summary statistic K(r) should be computed. The value returned by theoretical should be a
vector of the same length as the given vector of r values.
The argument ctrl determines the contrast criterion (the objective function that will be minimised).
The algorithm minimises the criterion
Z rmax
D(θ) =
|F̂ (r)q − Fθ (r)q |p dr
rmin

where θ is the vector of parameters of the model, F̂ (r) is the observed value of the summary statistic
computed from the data, Fθ (r) is the theoretical expected value of the summary statistic, and p, q
are two exponents. The default is q = 1/4, p=2 so that the contrast criterion is the integrated
squared difference between the fourth roots of the two functions (Waagepetersen, 2006).
The other arguments just make things print nicely. The argument fvlab contains labels for the component fit of the return value. The argument explain contains human-readable strings describing
the data, the model and the summary statistic.
The "..." argument of mincontrast can be used to pass extra arguments to the function theoretical
and/or to the optimisation function optim. In this case, the function theoretical should also have
a "..." argument and should ignore it (so that it ignores arguments intended for optim).
Value
An object of class "minconfit". There are methods for printing and plotting this object. It contains
the following components:
par
fit

opt
crtl
info

Vector of fitted parameter values.
Function value table (object of class "fv") containing the observed values of the
summary statistic (observed) and the theoretical values of the summary statistic
computed from the fitted model parameters.
The return value from the optimizer optim.
The control parameters of the algorithm.
List of explanatory strings.

Author(s)

Rasmus Waagepetersen , adapted for spatstat by Adrian Baddeley 
See Also
dilation, erosionAny
Examples
B <- square(0.2)
RplusB <- letterR %(+)% B
opa <- par(mfrow=c(1,2))
FR <- grow.rectangle(Frame(letterR), 0.3)
plot(FR, main="")
plot(letterR, add=TRUE, lwd=2, hatch=TRUE, hatchargs=list(texture=5))
plot(shift(B, vec=c(3.675, 3)),
add=TRUE, border="red", lwd=2)
plot(FR, main="")
plot(letterR, add=TRUE, lwd=2, hatch=TRUE, hatchargs=list(texture=5))
plot(RplusB, add=TRUE, border="blue", lwd=2,
hatch=TRUE, hatchargs=list(col="blue"))
par(opa)
plot(cells %(+)% square(0.1))

miplot

Morisita Index Plot

Description
Displays the Morisita Index Plot of a spatial point pattern.
Usage
miplot(X, ...)
Arguments
X

A point pattern (object of class "ppp") or something acceptable to as.ppp.

...

Optional arguments to control the appearance of the plot.

miplot

819

Details
Morisita (1959) defined an index of spatial aggregation for a spatial point pattern based on quadrat
counts. The spatial domain of the point pattern is first divided into Q subsets (quadrats) of equal
size and shape. The numbers of points falling in each quadrat are counted. Then the Morisita Index
is computed as
PQ
ni (ni − 1)
MI = Q i=1
N (N − 1)
where ni is the number of points falling in the i-th quadrat, and N is the total number of points. If
the pattern is completely random, MI should be approximately equal to 1. Values of MI greater than
1 suggest clustering.
The Morisita Index plot is a plot of the Morisita Index MI against the linear dimension of the
quadrats. The point pattern dataset is divided into 2 × 2 quadrats, then 3 × 3 quadrats, etc, and
the Morisita Index is computed each time. This plot is an attempt to discern different scales of
dependence in the point pattern data.
Value
None.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
M. Morisita (1959) Measuring of the dispersion of individuals and analysis of the distributional
patterns. Memoir of the Faculty of Science, Kyushu University, Series E: Biology. 2: 215–235.
See Also
quadratcount
Examples
data(longleaf)
miplot(longleaf)
opa <- par(mfrow=c(2,3))
data(cells)
data(japanesepines)
data(redwood)
plot(cells)
plot(japanesepines)
plot(redwood)
miplot(cells)
miplot(japanesepines)
miplot(redwood)
par(opa)

820

model.depends

model.depends

Identify Covariates Involved in each Model Term

Description
Given a fitted model (of any kind), identify which of the covariates is involved in each term of the
model.
Usage
model.depends(object)
model.is.additive(object)
model.covariates(object, fitted=TRUE, offset=TRUE)
has.offset.term(object)
has.offset(object)
Arguments
object

A fitted model of any kind.

fitted,offset

Logical values determining which type of covariates to include.

Details
The object can be a fitted model of any kind, including models of the classes lm, glm and ppm.
To be precise, object must belong to a class for which there are methods for formula, terms and
model.matrix.
The command model.depends determines the relationship between the original covariates (the data
supplied when object was fitted) and the canonical covariates (the columns of the design matrix).
It returns a logical matrix, with one row for each canonical covariate, and one column for each of
the original covariates, with the i,j entry equal to TRUE if the ith canonical covariate depends on
the jth original covariate.
If the model formula of object includes offset terms (see offset), then the return value of model.depends
also has an attribute "offset". This is a logical value or matrix with one row for each offset term
and one column for each of the original covariates, with the i,j entry equal to TRUE if the ith offset
term depends on the jth original covariate.
The command model.covariates returns a character vector containing the names of all (original)
covariates that were actually used to fit the model. By default, this includes all covariates that appear
in the model formula, including offset terms as well as canonical covariate terms. To omit the offset
terms, set offset=FALSE. To omit the canonical covariate terms, set fitted=FALSE.
The command model.is.additive determines whether the model is additive, in the sense that
there is no canonical covariate that depends on two or more original covariates. It returns a logical
value.
The command has.offset.term is a faster way to determine whether the model formula includes
an offset term.
The functions model.depends and has.offset.term only detect offset terms which are present
in the model formula. They do not detect numerical offsets in the model object, that were inserted
using the offset argument in lm, glm etc. To detect the presence of offsets of both kinds, use
has.offset.

model.frame.ppm

821

Value
A logical value or matrix.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
ppm, model.matrix
Examples
x <- 1:10
y <- 3*x + 2
z <- rep(c(-1,1), 5)
fit <- lm(y ~ poly(x,2) + sin(z))
model.depends(fit)
model.covariates(fit)
model.is.additive(fit)
fitoff1 <- lm(y ~ x + offset(z))
fitoff2 <- lm(y ~ x, offset=z)
has.offset.term(fitoff1)
has.offset(fitoff1)
has.offset.term(fitoff2)
has.offset(fitoff2)

model.frame.ppm

Extract the Variables in a Point Process Model

Description
Given a fitted point process model, this function returns a data frame containing all the variables
needed to fit the model using the Berman-Turner device.
Usage
## S3 method for class 'ppm'
model.frame(formula, ...)
## S3 method for class 'kppm'
model.frame(formula, ...)
## S3 method for class 'dppm'
model.frame(formula, ...)
## S3 method for class 'lppm'
model.frame(formula, ...)

822

model.frame.ppm

Arguments
formula

A fitted point process model. An object of class "ppm" or "kppm" or "dppm" or
"lppm".

...

Additional arguments passed to model.frame.glm.

Details
The function model.frame is generic. These functions are method for model.frame for fitted point
process models (objects of class "ppm" or "kppm" or "dppm" or "lppm").
The first argument should be a fitted point process model; it has to be named formula for consistency with the generic function.
The result is a data frame containing all the variables used in fitting the model. The data frame has
one row for each quadrature point used in fitting the model. The quadrature scheme can be extracted
using quad.ppm.
Value
A data.frame containing all the variables used in the fitted model, plus additional variables specified in .... It has an additional attribute "terms" containing information about the model formula.
For details see model.frame.glm.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns.
Australian and New Zealand Journal of Statistics 42, 283–322.

See Also
ppm, kppm, dppm, lppm, model.frame, model.matrix.ppm
Examples
fit <- ppm(cells ~ x)
mf <- model.frame(fit)
kfit <- kppm(redwood ~ x, "Thomas")
kmf <- model.frame(kfit)

model.images

model.images

823

Compute Images of Constructed Covariates

Description
For a point process model fitted to spatial point pattern data, this function computes pixel images of
the covariates in the design matrix.
Usage
model.images(object, ...)
## S3 method for class 'ppm'
model.images(object, W = as.owin(object), ...)
## S3 method for class 'kppm'
model.images(object, W = as.owin(object), ...)
## S3 method for class 'dppm'
model.images(object, W = as.owin(object), ...)
## S3 method for class 'lppm'
model.images(object, L = as.linnet(object), ...)
## S3 method for class 'slrm'
model.images(object, ...)
Arguments
object

The fitted point process model. An object of class "ppm" or "kppm" or "lppm"
or "slrm" or "dppm".

W

A window (object of class "owin") in which the images should be computed.
Defaults to the window in which the model was fitted.

L

A linear network (object of class "linnet") in which the images should be
computed. Defaults to the network in which the model was fitted.

...

Other arguments (such as na.action) passed to model.matrix.lm.

Details
This command is similar to model.matrix.ppm except that it computes pixel images of the covariates, instead of computing the covariate values at certain points only.
The object must be a fitted spatial point process model object of class "ppm" (produced by the
model-fitting function ppm) or class "kppm" (produced by the fitting function kppm) or class "dppm"
(produced by the fitting function dppm) or class "lppm" (produced by lppm) or class "slrm" (produced by slrm).
The spatial covariates required by the model-fitting procedure are computed at every pixel location
in the window W. For lppm objects, the covariates are computed at every location on the network L.
For slrm objects, the covariates are computed on the pixels that were used to fit the model.

824

model.images
Note that the spatial covariates computed here are not the original covariates that were supplied
when fitting the model. Rather, they are the covariates that actually appear in the loglinear representation of the (conditional) intensity and in the columns of the design matrix. For example,
they might include dummy or indicator variables for different levels of a factor, depending on the
contrasts that are in force.
The pixel resolution is determined by W if W is a mask (that is W$type = "mask"). Otherwise, the
pixel resolution is determined by spatstat.options.
The format of the result depends on whether the original point pattern data were marked or unmarked.
• If the original dataset was unmarked, the result is a named list of pixel images (objects of
class "im") containing the values of the spatial covariates. The names of the list elements
are the names of the covariates determined by model.matrix.lm. The result is also of class
"solist" so that it can be plotted immediately.
• If the original dataset was a multitype point pattern, the result is a hyperframe with one
column for each possible type of points. Each column is a named list of pixel images (objects
of class "im") containing the values of the spatial covariates. The row names of the hyperframe
are the names of the covariates determined by model.matrix.lm.

Value
A list (of class "solist") or array (of class "hyperframe") containing pixel images (objects of
class "im"). For model.images.lppm, the images are also of class "linim".
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
model.matrix.ppm, model.matrix, ppm, ppm.object, lppm, dppm, kppm, slrm, im, im.object,
plot.solist, spatstat.options
Examples
fit <- ppm(cells ~ x)
model.images(fit)
B <- owin(c(0.2, 0.4), c(0.3, 0.8))
model.images(fit, B)
fit2 <- ppm(cells ~ cut(x,3))
model.images(fit2)
fit3 <- slrm(japanesepines ~ x)
model.images(fit3)
fit4 <- ppm(amacrine ~ marks + x)
model.images(fit4)

model.matrix.mppm

825

model.matrix.mppm

Extract Design Matrix of Point Process Model for Several Point Patterns

Description
Given a point process model fitted to a list of point patterns, this function extracts the design matrix.
Usage
## S3 method for class 'mppm'
model.matrix(object, ..., keepNA=TRUE, separate=FALSE)
Arguments
object
...
keepNA
separate

A point process model fitted to several point patterns. An object of class "mppm".
Other arguments (such as na.action) passed to model.matrix.lm.
Logical. Determines whether rows containing NA values will be deleted or retained.
Logical value indicating whether to split the model matrix into sub-matrices
corresponding to each of the original point patterns.

Details
This command is a method for the generic function model.matrix. It extracts the design matrix of
a point process model fitted to several point patterns.
The argument object must be a fitted point process model (object of class "mppm") produced by
the fitting algorithm mppm). This represents a point process model that has been fitted to a list of
several point pattern datasets. See mppm for information.
The result is a matrix with one column for every constructed covariate in the model, and one row
for every quadrature point.
If separate=TRUE this matrix will be split into sub-matrices corresponding to the original point
patterns, and the result will be a list containing these matrices.
Value
A matrix (or list of matrices). Columns of the matrix are canonical covariates in the model.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
model.matrix, mppm.
Examples
fit <- mppm(Points ~ Image + x, demohyper)
head(model.matrix(fit))
# matrix with three columns: '(Intercept)', 'x' and 'Image'

826

model.matrix.ppm

model.matrix.ppm

Extract Design Matrix from Point Process Model

Description
Given a point process model that has been fitted to spatial point pattern data, this function extracts
the design matrix of the model.
Usage
## S3 method for class 'ppm'
model.matrix(object,
data=model.frame(object, na.action=NULL),
...,
Q=NULL, keepNA=TRUE)
## S3 method for class 'kppm'
model.matrix(object,
data=model.frame(object, na.action=NULL),
...,
Q=NULL, keepNA=TRUE)
## S3 method for class 'dppm'
model.matrix(object,
data=model.frame(object, na.action=NULL),
...,
Q=NULL, keepNA=TRUE)
## S3 method for class 'lppm'
model.matrix(object,
data=model.frame(object, na.action=NULL),
...,
keepNA=TRUE)
## S3 method for class 'ippm'
model.matrix(object,
data=model.frame(object, na.action=NULL),
...,
Q=NULL, keepNA=TRUE,
irregular=FALSE)
Arguments
object

The fitted point process model. An object of class "ppm" or "kppm" or "dppm"
or "ippm" or "lppm".

data

A model frame, containing the data required for the Berman-Turner device.

Q

A point pattern (class "ppp") or quadrature scheme (class "quad") specifying
new locations where the covariates should be computed.

keepNA

Logical. Determines whether rows containing NA values will be deleted or
retained.

model.matrix.ppm

827

...

Other arguments (such as na.action) passed to model.matrix.lm.

irregular

Logical value indicating whether to include the irregular score components.

Details
These commands are methods for the generic function model.matrix. They extract the design
matrix of a spatial point process model (class "ppm" or "kppm" or "dppm" or "lppm").
More precisely, this command extracts the design matrix of the generalised linear model associated
with a spatial point process model.
The object must be a fitted point process model (object of class "ppm" or "kppm" or "dppm" or
"lppm") fitted to spatial point pattern data. Such objects are produced by the model-fitting functions
ppm, kppm, dppm and lppm.
The methods model.matrix.ppm, model.matrix.kppm, model.matrix.dppm and model.matrix.lppm
extract the model matrix for the GLM.
The result is a matrix, with one row for every quadrature point in the fitting procedure, and one
column for every constructed covariate in the design matrix.
If there are NA values in the covariates, the argument keepNA determines whether to retain or delete
the corresponding rows of the model matrix. The default keepNA=TRUE is to retain them. Note that
this differs from the default behaviour of many other methods for model.matrix, which typically
delete rows containing NA.
The quadrature points themselves can be extracted using quad.ppm.
Value
A matrix. Columns of the matrix are canonical covariates in the model. Rows of the matrix correspond to quadrature points in the fitting procedure (provided keepNA=TRUE).
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
model.matrix, model.images, ppm, kppm, dppm, lppm, ippm, ppm.object, quad.ppm, residuals.ppm
Examples
fit <- ppm(cells ~ x)
head(model.matrix(fit))
model.matrix(fit, Q=runifpoint(5))
kfit <- kppm(redwood ~ x, "Thomas")
m <- model.matrix(kfit)

828

model.matrix.slrm

model.matrix.slrm

Extract Design Matrix from Spatial Logistic Regression Model

Description
This function extracts the design matrix of a spatial logistic regression model.
Usage
## S3 method for class 'slrm'
model.matrix(object, ..., keepNA=TRUE)
Arguments
object

A fitted spatial logistic regression model. An object of class "slrm".

...

Other arguments (such as na.action) passed to model.matrix.lm.

keepNA

Logical. Determines whether rows containing NA values will be deleted or retained.

Details
This command is a method for the generic function model.matrix. It extracts the design matrix of
a spatial logistic regression.
The object must be a fitted spatial logistic regression (object of class "slrm"). Such objects are
produced by the model-fitting function slrm.
Usually the result is a matrix with one column for every constructed covariate in the model, and one
row for every pixel in the grid used to fit the model.
If object was fitted using split pixels (by calling slrm using the argument splitby) then the matrix
has one row for every pixel or half-pixel.
Value
A matrix. Columns of the matrix are canonical covariates in the model.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
model.matrix, model.images, slrm.
Examples
fit <- slrm(japanesepines ~x)
head(model.matrix(fit))
# matrix with two columns: '(Intercept)' and 'x'

mppm

829

mppm

Fit Point Process Model to Several Point Patterns

Description
Fits a Gibbs point process model to several point patterns simultaneously.
Usage
mppm(formula, data, interaction=Poisson(), ...,
iformula=NULL,
random=NULL,
use.gam = FALSE,
reltol.pql=1e-3,
gcontrol=list())
Arguments
formula

A formula describing the systematic part of the model. Variables in the formula
are names of columns in data.

data

A hyperframe (object of class "hyperframe", see hyperframe) containing the
point pattern responses and the explanatory variables.

interaction

Interpoint interaction(s) appearing in the model. Either an object of class "interact"
describing the point process interaction structure, or a hyperframe (with the
same number of rows as data) whose entries are objects of class "interact".

...

Arguments passed to ppm controlling the fitting procedure.

iformula

Optional. A formula (with no left hand side) describing the interaction to be
applied to each case. Each variable name in the formula should either be the
name of a column in the hyperframe interaction, or the name of a column in
the hyperframe data that is a vector or factor.

random

Optional. A formula (with no left hand side) describing a random effect. Variable names in the formula may be any of the column names of data and interaction.
The formula must be recognisable to lme.

use.gam

Logical flag indicating whether to fit the model using gam or glm.

reltol.pql

Relative tolerance for successive steps in the penalised quasi-likelihood algorithm, used when the model includes random effects. The algorithm terminates
when the root mean square of the relative change in coefficients is less than
reltol.pql.

gcontrol

List of arguments to control the fitting algorithm. Arguments are passed to
glm.control or gam.control or lmeControl depending on the kind of model
being fitted. If the model has random effects, the arguments are passed to
lmeControl. Otherwise, if use.gam=TRUE the arguments are passed to gam.control,
and if use.gam=FALSE (the default) they are passed to glm.control.

Details
This function fits a common point process model to a dataset containing several different point
patterns.
It extends the capabilities of the function ppm to deal with data such as

830

mppm
• replicated observations of spatial point patterns
• two groups of spatial point patterns
• a designed experiment in which the response from each unit is a point pattern.
The syntax of this function is similar to that of standard R model-fitting functions like lm and glm.
The first argument formula is an R formula describing the systematic part of the model. The second
argument data contains the responses and the explanatory variables. Other arguments determine
the stochastic structure of the model.
Schematically, the data are regarded as the results of a designed experiment involving n experimental units. Each unit has a ‘response’, and optionally some ‘explanatory variables’ (covariates)
describing the experimental conditions for that unit. In this context, the response from each unit is
a point pattern. The value of a particular covariate for each unit can be either a single value (numerical, logical or factor), or a spatial covariate. A ‘spatial’ covariate is a quantity that depends on
spatial location, for example, the soil acidity or altitude at each location. For the purposes of mppm,
a spatial covariate must be stored as a pixel image (object of class "im") which gives the values of
the covariate at a fine grid of locations.
The argument data is a hyperframe (a generalisation of a data frame, see hyperframe). This is like
a data frame except that the entries can be objects of any class. The hyperframe has one row for
each experimental unit, and one column for each variable (response or explanatory variable).
The formula should be an R formula. The left hand side of formula determines the ‘response’
variable. This should be a single name, which should correspond to a column in data.
The right hand side of formula determines the spatial trend of the model. It specifies the linear
predictor, and effectively represents the logarithm of the spatial trend. Variables in the formula
must be the names of columns of data, or one of the reserved names
x,y Cartesian coordinates of location
marks Mark attached to point
id which is a factor representing the serial number (1 to n) of the point pattern, i.e. the row number
in the data hyperframe.
The column of responses in data must consist of point patterns (objects of class "ppp"). The
individual point pattern responses can be defined in different spatial windows. If some of the point
patterns are marked, then they must all be marked, and must have the same type of marks.
The scope of models that can be fitted to each pattern is the same as the scope of ppm, that is,
Gibbs point processes with interaction terms that belong to a specified list, including for example
the Poisson process, Strauss process, Geyer’s saturation model, and piecewise constant pairwise
interaction models. Additionally, it is possible to include random effects as explained in the section
on Random Effects below.
The stochastic part of the model is determined by the arguments interaction and (optionally)
iformula.
• In the simplest case, interaction is an object of class "interact", determining the interpoint interaction structure of the point process model, for all experimental units.
• Alternatively, interaction may be a hyperframe, whose entries are objects of class "interact".
It should have the same number of rows as data.
– If interaction consists of only one column, then the entry in row i is taken to be the
interpoint interaction for the ith experimental unit (corresponding to the ith row of data).
– If interaction has more than one column, then the argument iformula is also required.
Each row of interaction determines several interpoint interaction structures that might
be applied to the corresponding row of data. The choice of interaction is determined

mppm

831
by iformula; this should be an R formula, without a left hand side. For example if
interaction has two columns called A and B then iformula = ~B indicates that the
interpoint interactions are taken from the second column.

Variables in iformula typically refer to column names of interaction. They can also be names
of columns in data, but only for columns of numeric, logical or factor values. For example
iformula = ~B * group (where group is a column of data that contains a factor) causes the
model with interpoint interaction B to be fitted with different interaction parameters for each level
of group.
Value
An object of class "mppm" representing the fitted model.
There are methods for print, summary, coef, AIC, anova, fitted, fixef, logLik, plot, predict,
ranef, residuals, summary, terms and vcov for this class.
The default methods for update and formula also work on this class.
Random Effects
It is also possible to include random effects in the trend term. The argument random is a formula,
with no left-hand side, that specifies the structure of the random effects. The formula should be
recognisable to lme (see the description of the argument random for lme).
The names in the formula random may be any of the covariates supplied by data. Additionally the
formula may involve the name id, which is a factor representing the serial number (1 to n) of the
point pattern in the list X.
Author(s)
Adrian Baddeley, Ida-Maria Sintorn and Leanne Bischoff. Implemented in spatstat by Adrian
Baddeley , Rolf Turner  and
Ege Rubak .
References
Baddeley, A. and Turner, R. Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42 (2000) 283–322.
Baddeley, A., Bischof, L., Sintorn, I.-M., Haggarty, S., Bell, M. and Turner, R. Analysis of a
designed experiment where the response is a spatial point pattern. In preparation.
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. London: Chapman and Hall/CRC Press.
Bell, M. and Grunwald, G. (2004) Mixed models for the analysis of replicated spatial point patterns.
Biostatistics 5, 633–648.

See Also
ppm, print.mppm, summary.mppm, coef.mppm,

832

msr

Examples
# Waterstriders data
H <- hyperframe(Y = waterstriders)
mppm(Y ~ 1, data=H)
mppm(Y ~ 1, data=H, Strauss(7))
mppm(Y ~ id, data=H)
mppm(Y ~ x, data=H)
# Synthetic data from known model
n <- 10
H <- hyperframe(V=1:n,
U=runif(n, min=-1, max=1),
M=factor(letters[1 + (1:n) %% 3]))
H$Z <- setcov(square(1))
H$U <- with(H, as.im(U, as.rectangle(Z)))
H$Y <- with(H, rpoispp(eval.im(exp(2+3*Z))))
fit <- mppm(Y ~Z + U + V, data=H)

msr

Signed or Vector-Valued Measure

Description
Defines an object representing a signed measure or vector-valued measure on a spatial domain.
Usage
msr(qscheme, discrete, density, check=TRUE)
Arguments
qscheme

A quadrature scheme (object of class "quad" usually extracted from a fitted
point process model).

discrete

Vector or matrix containing the values (masses) of the discrete component of the
measure, for each of the data points in qscheme.

density

Vector or matrix containing values of the density of the diffuse component of
the measure, for each of the quadrature points in qscheme.

check

Logical. Whether to check validity of the arguments.

Details
This function creates an object that represents a signed or vector valued measure on the twodimensional plane. It is not normally called directly by the user.
A signed measure is a classical mathematical object (Diestel and Uhl, 1977) which can be visualised
as a collection of electric charges, positive and/or negative, spread over the plane. Electric charges
may be concentrated at specific points (atoms), or spread diffusely over a region.
An object of class "msr" represents a signed (i.e. real-valued) or vector-valued measure in the
spatstat package.

msr

833
Spatial residuals for point process models (Baddeley et al, 2005, 2008) take the form of a realvalued or vector-valued measure. The function residuals.ppm returns an object of class "msr"
representing the residual measure.
The function msr would not normally be called directly by the user. It is the low-level creator
function that makes an object of class "msr" from raw data.
The first argument qscheme is a quadrature scheme (object of class "quad"). It is typically created by quadscheme or extracted from a fitted point process model using quad.ppm. A quadrature
scheme contains both data points and dummy points. The data points of qscheme are used as the
locations of the atoms of the measure. All quadrature points (i.e. both data points and dummy
points) of qscheme are used as sampling points for the density of the continuous component of the
measure.
The argument discrete gives the values of the atomic component of the measure for each data
point in qscheme. It should be either a numeric vector with one entry for each data point, or a
numeric matrix with one row for each data point.
The argument density gives the values of the density of the diffuse component of the measure, at
each quadrature point in qscheme. It should be either a numeric vector with one entry for each
quadrature point, or a numeric matrix with one row for each quadrature point.
If both discrete and density are vectors (or one-column matrices) then the result is a signed (realvalued) measure. Otherwise, the result is a vector-valued measure, with the dimension of the vector
space being determined by the number of columns in the matrices discrete and/or density. (If
one of these is a k-column matrix and the other is a 1-column matrix, then the latter is replicated to
k columns).
The class "msr" has methods for print, plot and [. There is also a function Smooth.msr for
smoothing a measure.

Value
An object of class "msr" that can be plotted by plot.msr.
Author(s)
Adrian Baddeley 
References
Baddeley, A., Turner, R., Møller, J. and Hazelton, M. (2005) Residual analysis for spatial point
processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
Baddeley, A., Møller, J. and Pakes, A.G. (2008) Properties of residuals for spatial point processes.
Annals of the Institute of Statistical Mathematics 60, 627–649.
Diestel, J. and Uhl, J.J. Jr (1977) Vector measures. Providence, RI, USA: American Mathematical
Society.
Halmos, P.R. (1950) Measure Theory. Van Nostrand.
See Also
plot.msr, Smooth.msr, [.msr, with.msr, split.msr, Ops.msr, measureVariation.

834

MultiHard

Examples
X <- rpoispp(function(x,y) { exp(3+3*x) })
fit <- ppm(X, ~x+y)
rp <- residuals(fit, type="pearson")
rp
rs <- residuals(fit, type="score")
rs
colnames(rs)
# An equivalent way to construct the Pearson residual measure by hand
Q <- quad.ppm(fit)
lambda <- fitted(fit)
slam <- sqrt(lambda)
Z <- is.data(Q)
m <- msr(Q, discrete=1/slam[Z], density = -slam)
m

MultiHard

The Multitype Hard Core Point Process Model

Description
Creates an instance of the multitype hard core point process model which can then be fitted to point
pattern data.
Usage
MultiHard(hradii, types=NULL)
Arguments
hradii

Matrix of hard core radii

types

Optional; vector of all possible types (i.e. the possible levels of the marks variable in the data)

Details
This is a multitype version of the hard core process. A pair of points of types i and j must not lie
closer than hij units apart.
The argument types need not be specified in normal use. It will be determined automatically from
the point pattern data set to which the MultiStrauss interaction is applied, when the user calls ppm.
However, the user should be confident that the ordering of types in the dataset corresponds to the
ordering of rows and columns in the matrix hradii.
The matrix hradii must be symmetric, with entries which are either positive numbers or NA. A
value of NA indicates that no distance constraint should be applied for this combination of types.
Note that only the hardcore radii are specified in MultiHard. The canonical parameters log(βj ) are
estimated by ppm(), not fixed in MultiHard().

multiplicity.ppp

835

Value
An object of class "interact" describing the interpoint interaction structure of the multitype hard
core process with hard core radii hradii[i, j].

Warnings
In order that ppm can fit the multitype hard core model correctly to a point pattern X, this pattern
must be marked, with markformat equal to vector and the mark vector marks(X) must be a factor.
If the argument types is specified it is interpreted as a set of factor levels and this set must equal
levels(marks(X)).
Changed Syntax
Before spatstat version 1.37-0, the syntax of this function was different: MultiHard(types=NULL, hradii).
The new code attempts to handle the old syntax as well.

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
ppm, pairwise.family, ppm.object, MultiStrauss, MultiStraussHard, Strauss.
See ragsMultiHard and rmh for simulation.
Examples
h <- matrix(c(1,2,2,1), nrow=2,ncol=2)
# prints a sensible description of itself
MultiHard(h)
# Fit the stationary multitype hardcore process to `amacrine'
# with hard core operating only between cells of the same type.
h <- 0.02 * matrix(c(1, NA, NA, 1), nrow=2,ncol=2)
ppm(amacrine ~1, MultiHard(h))

multiplicity.ppp

Count Multiplicity of Duplicate Points

Description
Counts the number of duplicates for each point in a spatial point pattern.

836

multiplicity.ppp

Usage
multiplicity(x)
## S3 method for class 'ppp'
multiplicity(x)
## S3 method for class 'ppx'
multiplicity(x)
## S3 method for class 'data.frame'
multiplicity(x)
## Default S3 method:
multiplicity(x)
Arguments
x

A spatial point pattern (object of class "ppp" or "ppx") or a vector, matrix or
data frame.

Details
Two points in a point pattern are deemed to be identical if their x, y coordinates are the same, and
their marks are also the same (if they carry marks). The Examples section illustrates how it is
possible for a point pattern to contain a pair of identical points.
For each point in x, the function multiplicity counts how many points are identical to it, and
returns the vector of counts.
The argument x can also be a vector, a matrix or a data frame. When x is a vector, m <- multiplicity(x)
is a vector of the same length as x, and m[i] is the number of elements of x that are identical to
x[i]. When x is a matrix or data frame, m <- multiplicity(x) is a vector of length equal to the
number of rows of x, and m[i] is the number of rows of x that are identical to the ith row.
Value
A vector of integers (multiplicities) of length equal to the number of points in x.
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Sebastian Meyer.
See Also
ppp.object, duplicated.ppp, unique.ppp
Examples
X <- ppp(c(1,1,0.5,1), c(2,2,1,2), window=square(3), check=FALSE)
m <- multiplicity(X)
# unique points in X, marked by their multiplicity

MultiStrauss

837

first <- !duplicated(X)
Y <- X[first] %mark% m[first]

MultiStrauss

The Multitype Strauss Point Process Model

Description
Creates an instance of the multitype Strauss point process model which can then be fitted to point
pattern data.
Usage
MultiStrauss(radii, types=NULL)
Arguments
radii

Matrix of interaction radii

types

Optional; vector of all possible types (i.e. the possible levels of the marks variable in the data)

Details
The (stationary) multitype Strauss process with m types, with interaction radii rij and parameters
βj and γij is the pairwise interaction point process in which each point of type j contributes a factor
βj to the probability density of the point pattern, and a pair of points of types i and j closer than rij
units apart contributes a factor γij to the density.
The nonstationary multitype Strauss process is similar except that the contribution of each individual
point xi is a function β(xi ) of location and type, rather than a constant beta.
The function ppm(), which fits point process models to point pattern data, requires an argument of
class "interact" describing the interpoint interaction structure of the model to be fitted. The appropriate description of the multitype Strauss process pairwise interaction is yielded by the function
MultiStrauss(). See the examples below.
The argument types need not be specified in normal use. It will be determined automatically from
the point pattern data set to which the MultiStrauss interaction is applied, when the user calls ppm.
However, the user should be confident that the ordering of types in the dataset corresponds to the
ordering of rows and columns in the matrix radii.
The matrix radii must be symmetric, with entries which are either positive numbers or NA. A value
of NA indicates that no interaction term should be included for this combination of types.
Note that only the interaction radii are specified in MultiStrauss. The canonical parameters
log(βj ) and log(γij ) are estimated by ppm(), not fixed in MultiStrauss().
Value
An object of class "interact" describing the interpoint interaction structure of the multitype
Strauss process with interaction radii radii[i, j].

838

MultiStraussHard

Warnings
In order that ppm can fit the multitype Strauss model correctly to a point pattern X, this pattern must
be marked, with markformat equal to vector and the mark vector marks(X) must be a factor. If
the argument types is specified it is interpreted as a set of factor levels and this set must equal
levels(marks(X)).
Changed Syntax
Before spatstat version 1.37-0, the syntax of this function was different: MultiStrauss(types=NULL, radii).
The new code attempts to handle the old syntax as well.
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak 
See Also
ppm, pairwise.family, ppm.object, Strauss, MultiHard
Examples
r <- matrix(c(1,2,2,1), nrow=2,ncol=2)
MultiStrauss(r)
# prints a sensible description of itself
r <- 0.03 * matrix(c(1,2,2,1), nrow=2,ncol=2)
X <- amacrine
ppm(X ~1, MultiStrauss(r))
# fit the stationary multitype Strauss process to `amacrine'
## Not run:
ppm(X ~polynom(x,y,3), MultiStrauss(r, c("off","on")))
# fit a nonstationary multitype Strauss process with log-cubic trend
## End(Not run)

MultiStraussHard

The Multitype/Hard Core Strauss Point Process Model

Description
Creates an instance of the multitype/hard core Strauss point process model which can then be fitted
to point pattern data.
Usage
MultiStraussHard(iradii, hradii, types=NULL)

MultiStraussHard

839

Arguments
iradii

Matrix of interaction radii

hradii

Matrix of hard core radii

types

Optional; vector of all possible types (i.e. the possible levels of the marks variable in the data)

Details
This is a hybrid of the multitype Strauss process (see MultiStrauss) and the hard core process
(case γ = 0 of the Strauss process). A pair of points of types i and j must not lie closer than hij
units apart; if the pair lies more than hij and less than rij units apart, it contributes a factor γij to
the probability density.
The argument types need not be specified in normal use. It will be determined automatically from
the point pattern data set to which the MultiStraussHard interaction is applied, when the user calls
ppm. However, the user should be confident that the ordering of types in the dataset corresponds to
the ordering of rows and columns in the matrices iradii and hradii.
The matrices iradii and hradii must be symmetric, with entries which are either positive numbers
or NA. A value of NA indicates that no interaction term should be included for this combination of
types.
Note that only the interaction radii and hardcore radii are specified in MultiStraussHard. The
canonical parameters log(βj ) and log(γij ) are estimated by ppm(), not fixed in MultiStraussHard().
Value
An object of class "interact" describing the interpoint interaction structure of the multitype/hard
core Strauss process with interaction radii iradii[i, j] and hard core radii hradii[i, j].
Warnings
In order that ppm can fit the multitype/hard core Strauss model correctly to a point pattern X, this
pattern must be marked, with markformat equal to vector and the mark vector marks(X) must be
a factor. If the argument types is specified it is interpreted as a set of factor levels and this set must
equal levels(marks(X)).
Changed Syntax

Before spatstat version 1.37-0, the syntax of this function was different: MultiStraussHard(types=NULL, iradii, h
The new code attempts to handle the old syntax as well.
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak 
See Also
ppm, pairwise.family, ppm.object, MultiStrauss, MultiHard, Strauss

840

nearest.raster.point

Examples
r <- matrix(3, nrow=2,ncol=2)
h <- matrix(c(1,2,2,1), nrow=2,ncol=2)
MultiStraussHard(r,h)
# prints a sensible description of itself
r <- 0.04 * matrix(c(1,2,2,1), nrow=2,ncol=2)
h <- 0.02 * matrix(c(1,NA,NA,1), nrow=2,ncol=2)
X <- amacrine
fit <- ppm(X ~1, MultiStraussHard(r,h))
# fit stationary multitype hardcore Strauss process to `amacrine'

nearest.raster.point

Find Pixel Nearest to a Given Point

Description
Given cartesian coordinates, find the nearest pixel.
Usage
nearest.raster.point(x,y,w, indices=TRUE)
Arguments
x

Numeric vector of x coordinates of any points

y

Numeric vector of y coordinates of any points

w

An image (object of class "im") or a binary mask window (an object of class
"owin" of type "mask").

indices

Logical flag indicating whether to return the row and column indices, or the
actual x, y coordinates.

Details
The argument w should be either a pixel image (object of class "im") or a window (an object of
class "owin", see owin.object for details) of type "mask".
The arguments x and y should be numeric vectors of equal length. They are interpreted as the
coordinates of points in space. For each point (x[i], y[i]), the function finds the nearest pixel in
the grid of pixels for w.
If indices=TRUE, this function returns a list containing two vectors rr and cc giving row and
column positions (in the image matrix). For the location (x[i],y[i]) the nearest pixel is at row
rr[i] and column cc[i] of the image.
If indices=FALSE, the function returns a list containing two vectors x and y giving the actual
coordinates of the pixels.
Value
If indices=TRUE, a list containing two vectors rr and cc giving row and column positions (in the
image matrix). If indices=FALSE, a list containing vectors x and y giving actual coordinates of the
pixels.

nearestsegment

841

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
owin.object, as.mask
Examples
w <- owin(c(0,1), c(0,1), mask=matrix(TRUE, 100,100))
nearest.raster.point(0.5, 0.3, w)
nearest.raster.point(0.5, 0.3, w, indices=FALSE)

nearestsegment

# 100 x 100 grid

Find Line Segment Nearest to Each Point

Description
Given a point pattern and a line segment pattern, this function finds the nearest line segment for
each point.
Usage
nearestsegment(X, Y)
Arguments
X

A point pattern (object of class "ppp").

Y

A line segment pattern (object of class "psp").

Details
The distance between a point x and a straight line segment y is defined to be the shortest Euclidean
distance between x and any location on y. This algorithm first calculates the distance from each
point of X to each segment of Y. Then it determines, for each point x in X, which segment of Y is
closest. The index of this segment is returned.
Value
Integer vector v (of length equal to the number of points in X) identifying the nearest segment to
each point. If v[i] = j, then Y[j] is the line segment lying closest to X[i].
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
project2segment to project each point of X to a point lying on one of the line segments.
Use distmap.psp to identify the nearest line segment for each pixel in a grid.

842

nestsplit

Examples
X <- runifpoint(3)
Y <- as.psp(matrix(runif(20), 5, 4), window=owin())
v <- nearestsegment(X,Y)
plot(Y)
plot(X, add=TRUE)
plot(X[1], add=TRUE, col="red")
plot(Y[v[1]], add=TRUE, lwd=2, col="red")

nestsplit

Nested Split

Description
Applies two splitting operations to a point pattern, producing a list of lists of patterns.
Usage
nestsplit(X, ...)
Arguments
X

Point pattern to be split. Object of class "ppp".

...

Data determining the splitting factors or splitting regions. See Details.

Details
This function splits the point pattern X into several sub-patterns using split.ppp, then splits each of
the sub-patterns into sub-sub-patterns using split.ppp again. The result is a hyperframe containing
the sub-sub-patterns and two factors indicating the grouping.
The arguments ... determine the two splitting factors or splitting regions. Each argument may be:
• a factor (of length equal to the number of points in X)
• the name of a column of marks of X (provided this column contains factor values)
• a tessellation (class "tess")
• a pixel image (class "im") with factor values
• a window (class "owin")
• identified by name (in the form name=value) as one of the formal arguments of quadrats or
tess
The arguments will be processed to yield a list of two splitting factors/tessellations. The splits will
be applied to X consecutively to produce the sub-sub-patterns.
Value
A hyperframe with three columns. The first column contains the sub-sub-patterns. The second and
third columns are factors which identify the grouping according to the two splitting factors.

nnclean

843

Author(s)
Original idea by Ute Hahn. Code by Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
split.ppp, quantess
Examples
# factor and tessellation
Nft <- nestsplit(amacrine, marks(amacrine), quadrats(amacrine, 3, 1))
Ntf <- nestsplit(amacrine, quadrats(amacrine, 3, 1), marks(amacrine))
Ntf
# two factors
big <- with(marks(betacells), area > 300)
Nff <- nestsplit(betacells, "type", factor(big))
# two tessellations
Tx <- quantess(redwood, "x", 4)
Td <- dirichlet(runifpoint(5, Window(redwood)))
Ntt <- nestsplit(redwood, Td, Tx)
Ntt2 <- nestsplit(redwood, Td, ny=3)

nnclean

Nearest Neighbour Clutter Removal

Description
Detect features in a 2D or 3D spatial point pattern using nearest neighbour clutter removal.
Usage
nnclean(X, k, ...)
## S3 method for class 'ppp'
nnclean(X, k, ...,
edge.correct = FALSE, wrap = 0.1,
convergence = 0.001, plothist = FALSE,
verbose = TRUE, maxit = 50)
## S3 method for class 'pp3'
nnclean(X, k, ...,
convergence = 0.001, plothist = FALSE,
verbose = TRUE, maxit = 50)

844

nnclean

Arguments
X
k
...
edge.correct
wrap

convergence
maxit
plothist
verbose

A two-dimensional spatial point pattern (object of class "ppp") or a three-dimensional
point pattern (object of class "pp3").
Degree of neighbour: k=1 means nearest neighbour, k=2 means second nearest,
etc.
Arguments passed to hist.default to control the appearance of the histogram,
if plothist=TRUE.
Logical flag specifying whether periodic edge correction should be performed
(only implemented in 2 dimensions).
Numeric value specifying the relative size of the margin in which data will be
replicated for the periodic edge correction (if edge.correct=TRUE). A fraction
of window width and window height.
Relative tolerance threshold for testing convergence of EM algorithm.
Maximum number of iterations for EM algorithm.
Logical flag specifying whether to plot a diagnostic histogram of the nearest
neighbour distances and the fitted distribution.
Logical flag specifying whether to print progress reports.

Details
Byers and Raftery (1998) developed a technique for recognising features in a spatial point pattern
in the presence of random clutter.
For each point in the pattern, the distance to the kth nearest neighbour is computed. Then the E-M
algorithm is used to fit a mixture distribution to the kth nearest neighbour distances. The mixture
components represent the feature and the clutter. The mixture model can be used to classify each
point as belong to one or other component.
The function nnclean is generic, with methods for two-dimensional point patterns (class "ppp")
and three-dimensional point patterns (class "pp3") currently implemented.
The result is a point pattern (2D or 3D) with two additional columns of marks:
class A factor, with levels "noise" and "feature", indicating the maximum likelihood classification of each point.
prob Numeric vector giving the estimated probabilities that each point belongs to a feature.
The object also has extra information stored in attributes: "theta" contains the fitted parameters of
the mixture model, "info" contains information about the fitting procedure, and "hist" contains
the histogram structure returned from hist.default if plothist = TRUE.
Value
An object of the same kind as X, obtained by attaching marks to the points of X.
The object also has attributes, as described under Details.
Author(s)

Original by Simon Byers and Adrian Raftery. Adapted for spatstat by Adrian Baddeley 
and Rolf Turner 
References
Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical
statistics. John Wiley and Sons.
Examples
data(finpines)
nncorr(finpines)
# heights of neighbouring trees are slightly negatively correlated
data(amacrine)
nncorr(amacrine)
# neighbouring cells are usually of different type

848

nncross

nncross

Nearest Neighbours Between Two Patterns

Description
Given two point patterns X and Y, finds the nearest neighbour in Y of each point of X. Alternatively
Y may be a line segment pattern.
Usage
nncross(X, Y, ...)
## S3 method for class 'ppp'
nncross(X, Y,
iX=NULL, iY=NULL,
what = c("dist", "which"),
...,
k = 1,
sortby=c("range", "var", "x", "y"),
is.sorted.X = FALSE,
is.sorted.Y = FALSE)
## Default S3 method:
nncross(X, Y, ...)
Arguments
X

Point pattern (object of class "ppp").

Y

Either a point pattern (object of class "ppp") or a line segment pattern (object of
class "psp").

iX, iY

Optional identifiers, applicable only in the case where Y is a point pattern, used
to determine whether a point in X is identical to a point in Y. See Details.

what

Character string specifying what information should be returned. Either the
nearest neighbour distance ("dist"), the identifier of the nearest neighbour
("which"), or both.

k

Integer, or integer vector. The algorithm will compute the distance to the kth
nearest neighbour.

sortby
Determines which coordinate to use to sort the point patterns. See Details.
is.sorted.X, is.sorted.Y
Logical values attesting whether the point patterns X and Y have been sorted. See
Details.
...

Ignored.

Details
Given two point patterns X and Y this function finds, for each point of X, the nearest point of Y. The
distance between these points is also computed. If the argument k is specified, then the k-th nearest
neighbours will be found.

nncross

849

Alternatively if X is a point pattern and Y is a line segment pattern, the function finds the nearest line
segment to each point of X, and computes the distance.
The return value is a data frame, with rows corresponding to the points of X. The first column gives
the nearest neighbour distances (i.e. the ith entry is the distance from the ith point of X to the
nearest element of Y). The second column gives the indices of the nearest neighbours (i.e.\ the ith
entry is the index of the nearest element in Y.) If what="dist" then only the vector of distances is
returned. If what="which" then only the vector of indices is returned.
The argument k may be an integer or an integer vector. If it is a single integer, then the k-th nearest
neighbours are computed. If it is a vector, then the k[i]-th nearest neighbours are computed for
each entry k[i]. For example, setting k=1:3 will compute the nearest, second-nearest and thirdnearest neighbours. The result is a data frame.
Note that this function is not symmetric in X and Y. To find the nearest neighbour in X of each point
in Y, where Y is a point pattern, use nncross(Y,X).
The arguments iX and iY are used when the two point patterns X and Y have some points in common.
In this situation nncross(X, Y) would return some zero distances. To avoid this, attach a unique
integer identifier to each point, such that two points are identical if their identifying numbers are
equal. Let iX be the vector of identifier values for the points in X, and iY the vector of identifiers
for points in Y. Then the code will only compare two points if they have different values of the
identifier. See the Examples.
Value
A data frame, or a vector if the data frame would contain only one column.
By default (if what=c("dist", "which") and k=1) a data frame with two columns:
dist

Nearest neighbour distance

which

Nearest neighbour index in Y

If what="dist" and k=1, a vector of nearest neighbour distances.
If what="which" and k=1, a vector of nearest neighbour indices.
If k is specified, the result is a data frame with columns containing the k-th nearest neighbour
distances and/or nearest neighbour indices.
Sorting data and pre-sorted data
Read this section if you care about the speed of computation.
For efficiency, the algorithm sorts the point patterns X and Y into increasing order of the x coordinate
or increasing order of the the y coordinate. Sorting is only an intermediate step; it does not affect
the output, which is always given in the same order as the original data.
By default (if sortby="range"), the sorting will occur on the coordinate that has the larger range
of values (according to the frame of the enclosing window of Y). If sortby = "var"), sorting
will occur on the coordinate that has the greater variance (in the pattern Y). Setting sortby="x" or
sortby = "y" will specify that sorting should occur on the x or y coordinate, respectively.
If the point pattern X is already sorted, then the corresponding argument is.sorted.X should be set
to TRUE, and sortby should be set equal to "x" or "y" to indicate which coordinate is sorted.
Similarly if Y is already sorted, then is.sorted.Y should be set to TRUE, and sortby should be set
equal to "x" or "y" to indicate which coordinate is sorted.
If both X and Y are sorted on the same coordinate axis then both is.sorted.X and is.sorted.Y
should be set to TRUE, and sortby should be set equal to "x" or "y" to indicate which coordinate is
sorted.

850

nncross.lpp

Author(s)
Adrian Baddeley 
, Rolf Turner  , and Jens Oehlschlaegel
See Also
nndist for nearest neighbour distances in a single point pattern.
Examples
# two different point patterns
X <- runifpoint(15)
Y <- runifpoint(20)
N <- nncross(X,Y)$which
# note that length(N) = 15
plot(superimpose(X=X,Y=Y), main="nncross", cols=c("red","blue"))
arrows(X$x, X$y, Y[N]$x, Y[N]$y, length=0.15)
# third-nearest neighbour
NXY <- nncross(X, Y, k=3)
NXY[1:3,]
# second and third nearest neighbours
NXY <- nncross(X, Y, k=2:3)
NXY[1:3,]
# two patterns with some points in common
Z <- runifpoint(50)
X <- Z[1:30]
Y <- Z[20:50]
iX <- 1:30
iY <- 20:50
N <- nncross(X,Y, iX, iY)$which
N <- nncross(X,Y, iX, iY, what="which") #faster
plot(superimpose(X=X, Y=Y), main="nncross", cols=c("red","blue"))
arrows(X$x, X$y, Y[N]$x, Y[N]$y, length=0.15)
#
X
Y
N

point pattern and line segment pattern
<- runifpoint(15)
<- rpoisline(10)
<- nncross(X,Y)

nncross.lpp

Nearest Neighbours on a Linear Network

Description
Given two point patterns X and Y on a linear network, finds the nearest neighbour in Y of each point
of X using the shortest path in the network.

nncross.lpp

851

Usage
## S3 method for class 'lpp'
nncross(X, Y,
iX=NULL, iY=NULL,
what = c("dist", "which"),
...,
k = 1,
method="C")
Arguments
X,Y

Point patterns on a linear network (objects of class "lpp"). They must lie on the
same linear network.

iX, iY

Optional identifiers, used to determine whether a point in X is identical to a point
in Y. See Details.

what

Character string specifying what information should be returned. Either the
nearest neighbour distance ("dist"), the identifier of the nearest neighbour
("which"), or both.

...

Ignored.

k

Integer, or integer vector. The algorithm will compute the distance to the kth
nearest neighbour, for each value of k.

method

Internal use only.

Details
Given two point patterns X and Y on the same linear network, this function finds, for each point of X,
the nearest point of Y, measuring distance by the shortest path in the network. The distance between
these points is also computed.
The return value is a data frame, with rows corresponding to the points of X. The first column gives
the nearest neighbour distances (i.e. the ith entry is the distance from the ith point of X to the
nearest element of Y). The second column gives the indices of the nearest neighbours (i.e.\ the ith
entry is the index of the nearest element in Y.) If what="dist" then only the vector of distances is
returned. If what="which" then only the vector of indices is returned.
Note that this function is not symmetric in X and Y. To find the nearest neighbour in X of each point
in Y, use nncross(Y,X).
The arguments iX and iY are used when the two point patterns X and Y have some points in common.
In this situation nncross(X, Y) would return some zero distances. To avoid this, attach a unique
integer identifier to each point, such that two points are identical if their identifying numbers are
equal. Let iX be the vector of identifier values for the points in X, and iY the vector of identifiers
for points in Y. Then the code will only compare two points if they have different values of the
identifier. See the Examples.
The kth nearest neighbour may be undefined, for example if there are fewer than k+1 points in the
dataset, or if the linear network is not connected. In this case, the kth nearest neighbour distance is
infinite.
Value
By default (if what=c("dist", "which") and k=1) a data frame with two columns:
dist

Nearest neighbour distance

852

nncross.pp3
which

Nearest neighbour index in Y

If what="dist", a vector of nearest neighbour distances.
If what="which", a vector of nearest neighbour indices.
If k is a vector of integers, the result is a matrix with one row for each point in X, giving the distances
and/or indices of the kth nearest neighbours in Y.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
nndist.lpp for nearest neighbour distances in a single point pattern.
nnwhich.lpp to identify which points are nearest neighbours in a single point pattern.
Examples
# two different point patterns
X <- runiflpp(3, simplenet)
Y <- runiflpp(5, simplenet)
nn <- nncross(X,Y)
nn
plot(simplenet, main="nncross")
plot(X, add=TRUE, cols="red")
plot(Y, add=TRUE, cols="blue", pch=16)
XX <- as.ppp(X)
YY <- as.ppp(Y)
i <- nn$which
arrows(XX$x, XX$y, YY[i]$x, YY[i]$y, length=0.15)
# nearest and second-nearest neighbours
nncross(X, Y, k=1:2)
# two patterns with some points in common
X <- Y[1:2]
iX <- 1:2
iY <- 1:5
nncross(X,Y, iX, iY)

nncross.pp3

Nearest Neighbours Between Two Patterns in 3D

Description
Given two point patterns X and Y in three dimensions, finds the nearest neighbour in Y of each point
of X.

nncross.pp3

853

Usage
## S3 method for class 'pp3'
nncross(X, Y,
iX=NULL, iY=NULL,
what = c("dist", "which"),
...,
k = 1,
sortby=c("range", "var", "x", "y", "z"),
is.sorted.X = FALSE,
is.sorted.Y = FALSE)
Arguments
X,Y

Point patterns in three dimensions (objects of class "pp3").

iX, iY

Optional identifiers, used to determine whether a point in X is identical to a point
in Y. See Details.

what

Character string specifying what information should be returned. Either the
nearest neighbour distance ("dist"), the identifier of the nearest neighbour
("which"), or both.

k

Integer, or integer vector. The algorithm will compute the distance to the kth
nearest neighbour.

sortby
Determines which coordinate to use to sort the point patterns. See Details.
is.sorted.X, is.sorted.Y
Logical values attesting whether the point patterns X and Y have been sorted. See
Details.
...

Ignored.

Details
Given two point patterns X and Y in three dimensions, this function finds, for each point of X,
the nearest point of Y. The distance between these points is also computed. If the argument k is
specified, then the k-th nearest neighbours will be found.
The return value is a data frame, with rows corresponding to the points of X. The first column gives
the nearest neighbour distances (i.e. the ith entry is the distance from the ith point of X to the
nearest element of Y). The second column gives the indices of the nearest neighbours (i.e.\ the ith
entry is the index of the nearest element in Y.) If what="dist" then only the vector of distances is
returned. If what="which" then only the vector of indices is returned.
The argument k may be an integer or an integer vector. If it is a single integer, then the k-th nearest
neighbours are computed. If it is a vector, then the k[i]-th nearest neighbours are computed for
each entry k[i]. For example, setting k=1:3 will compute the nearest, second-nearest and thirdnearest neighbours. The result is a data frame.
Note that this function is not symmetric in X and Y. To find the nearest neighbour in X of each point
in Y, use nncross(Y,X).
The arguments iX and iY are used when the two point patterns X and Y have some points in common.
In this situation nncross(X, Y) would return some zero distances. To avoid this, attach a unique
integer identifier to each point, such that two points are identical if their identifying numbers are
equal. Let iX be the vector of identifier values for the points in X, and iY the vector of identifiers
for points in Y. Then the code will only compare two points if they have different values of the
identifier. See the Examples.

854

nncross.pp3

Value
A data frame, or a vector if the data frame would contain only one column.
By default (if what=c("dist", "which") and k=1) a data frame with two columns:
dist

Nearest neighbour distance

which

Nearest neighbour index in Y

If what="dist" and k=1, a vector of nearest neighbour distances.
If what="which" and k=1, a vector of nearest neighbour indices.
If k is specified, the result is a data frame with columns containing the k-th nearest neighbour
distances and/or nearest neighbour indices.
Sorting data and pre-sorted data
Read this section if you care about the speed of computation.
For efficiency, the algorithm sorts both the point patterns X and Y into increasing order of the x
coordinate, or both into increasing order of the y coordinate, or both into increasing order of the z
coordinate. Sorting is only an intermediate step; it does not affect the output, which is always given
in the same order as the original data.
By default (if sortby="range"), the sorting will occur on the coordinate that has the largest range
of values (according to the frame of the enclosing window of Y). If sortby = "var"), sorting
will occur on the coordinate that has the greater variance (in the pattern Y). Setting sortby="x" or
sortby = "y" or sortby = "z" will specify that sorting should occur on the x, y or z coordinate,
respectively.
If the point pattern X is already sorted, then the corresponding argument is.sorted.X should be set
to TRUE, and sortby should be set equal to "x", "y" or "z" to indicate which coordinate is sorted.
Similarly if Y is already sorted, then is.sorted.Y should be set to TRUE, and sortby should be set
equal to "x", "y" or "z" to indicate which coordinate is sorted.
If both X and Y are sorted on the same coordinate axis then both is.sorted.X and is.sorted.Y
should be set to TRUE, and sortby should be set equal to "x", "y" or "z" to indicate which coordinate is sorted.
Author(s)
Adrian Baddeley 
, Rolf Turner  , and Jens Oehlschlaegel
See Also
nndist for nearest neighbour distances in a single point pattern.
Examples
#
X
Y
N
N
#

two different point patterns
<- pp3(runif(10), runif(10), runif(10), box3(c(0,1)))
<- pp3(runif(20), runif(20), runif(20), box3(c(0,1)))
<- nncross(X,Y)$which
<- nncross(X,Y, what="which") #faster
note that length(N) = 10

# k-nearest neighbours

nndensity.ppp

855

N3 <- nncross(X, Y, k=1:3)
# two patterns with some points in common
Z <- pp3(runif(20), runif(20), runif(20), box3(c(0,1)))
X <- Z[1:15]
Y <- Z[10:20]
iX <- 1:15
iY <- 10:20
N <- nncross(X,Y, iX, iY, what="which")

nndensity.ppp

Estimate Intensity of Point Pattern Using Nearest Neighbour Distances

Description
Estimates the intensity of a point pattern using the distance from each spatial location to the kth
nearest data point.
Usage
nndensity(x, ...)
## S3 method for class 'ppp'
nndensity(x, k, ..., verbose = TRUE)
Arguments
x
k
...
verbose

A point pattern (object of class "ppp") or some other spatial object.
Integer. The distance to the kth nearest data point will be computed. There is a
sensible default.
Arguments passed to nnmap and as.mask controlling the pixel resolution.
Logical. If TRUE, print the value of k when it is automatically selected. If FALSE,
remain silent.

Details
This function computes a quick estimate of the intensity of the point process that generated the
point pattern x.
For each spatial location s, let d(s) be the distance from s to the k-th nearest point in the dataset x.
If the data came from a homogeneous Poisson process with intensity λ, then πd(s)2 would follow a
negative exponential distribution with mean 1/λ, and the maximum likelihood estimate of λ would
be 1/(πd(s)2 ). This is the estimate computed by nndensity, apart from an edge effect correction.
This estimator of intensity is relatively fast to compute, and is spatially adaptive (so that it can handle
wide variation in the intensity function). However, it implicitly assumes the points are independent,
so it does not perform well if the pattern is strongly clustered or strongly inhibited.
The value of k should be greater than 1 in order to avoid infinite peaks in the intensity estimate
around each data point. The default value of k is the square root of the number of points in x, which
seems to work well in many cases.
The window of x is digitised using as.mask and the values d(s) are computed using nnmap. To
control the pixel resolution, see as.mask.

856

nndist

Value
A pixel image (object of class "im") giving the estimated intensity of the point process at each
spatial location. Pixel values are intensities (number of points per unit area).
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
NEED REFERENCES. TRY CRESSIE
See Also
density.ppp, intensity for alternative estimates of point process intensity.
Examples
plot(nndensity(swedishpines))

nndist

Nearest neighbour distances

Description
Computes the distance from each point to its nearest neighbour in a point pattern. Alternatively
computes the distance to the second nearest neighbour, or third nearest, etc.
Usage
nndist(X, ...)
## S3 method for class 'ppp'
nndist(X, ..., k=1, by=NULL, method="C")
## Default S3 method:
nndist(X, Y=NULL, ..., k=1, by=NULL, method="C")
Arguments
X,Y

Arguments specifying the locations of a set of points. For nndist.ppp, the argument X should be a point pattern (object of class "ppp"). For nndist.default,
typically X and Y would be numeric vectors of equal length. Alternatively Y may
be omitted and X may be a list with two components x and y, or a matrix with
two columns.

...

Ignored by nndist.ppp and nndist.default.

k

Integer, or integer vector. The algorithm will compute the distance to the kth
nearest neighbour.

by

Optional. A factor, which separates X into groups. The algorithm will compute
the distance to the nearest point in each group.

method

String specifying which method of calculation to use. Values are "C" and "interpreted".

nndist

857

Details
This function computes the Euclidean distance from each point in a point pattern to its nearest
neighbour (the nearest other point of the pattern). If k is specified, it computes the distance to the
kth nearest neighbour.
The function nndist is generic, with a method for point patterns (objects of class "ppp"), and a default method for coordinate vectors. There is also a method for line segment patterns, nndist.psp.
The method for point patterns expects a single point pattern argument X and returns the vector of its
nearest neighbour distances.
The default method expects that X and Y will determine the coordinates of a set of points. Typically
X and Y would be numeric vectors of equal length. Alternatively Y may be omitted and X may be a
list with two components named x and y, or a matrix or data frame with two columns.
The argument k may be a single integer, or an integer vector. If it is a vector, then the kth nearest
neighbour distances are computed for each value of k specified in the vector.
If the argument by is given, it should be a factor, of length equal to the number of points in X.
This factor effectively partitions X into subsets, each subset associated with one of the levels of X.
The algorithm will then compute, for each point of X, the distance to the nearest neighbour in each
subset.
The argument method is not normally used. It is retained only for checking the validity of the
software. If method = "interpreted" then the distances are computed using interpreted R code
only. If method="C" (the default) then C code is used. The C code is faster by two to three orders
of magnitude and uses much less memory.
If there is only one point (if x has length 1), then a nearest neighbour distance of Inf is returned. If
there are no points (if x has length zero) a numeric vector of length zero is returned.
To identify which point is the nearest neighbour of a given point, use nnwhich.
To use the nearest neighbour distances for statistical inference, it is often advisable to use the edgecorrected empirical distribution, computed by Gest.
To find the nearest neighbour distances from one point pattern to another point pattern, use nncross.
Value
Numeric vector or matrix containing the nearest neighbour distances for each point.
If k = 1 (the default), the return value is a numeric vector v such that v[i] is the nearest neighbour
distance for the ith data point.
If k is a single integer, then the return value is a numeric vector v such that v[i] is the kth nearest
neighbour distance for the ith data point.
If k is a vector, then the return value is a matrix m such that m[i,j] is the k[j]th nearest neighbour
distance for the ith data point.
If the argument by is given, then the result is a data frame containing the distances described above,
from each point of X, to the nearest point in each subset of X defined by the factor by.
Nearest neighbours of each type
If X is a multitype point pattern and by=marks(X), then the algorithm will compute, for each point
of X, the distance to the nearest neighbour of each type. See the Examples.
To find the minimum distance from any point of type i to the nearest point of type j, for all combinations of i and j, use the R function aggregate as suggested in the Examples.

858

nndist.lpp

Warnings
An infinite or NA value is returned if the distance is not defined (e.g. if there is only one point in the
point pattern).
Author(s)
Pavel Grabarnik  and Adrian Baddeley .
See Also
nndist.psp, pairdist, Gest, nnwhich, nncross.
Examples
data(cells)
# nearest neighbours
d <- nndist(cells)
# second nearest neighbours
d2 <- nndist(cells, k=2)
# first, second and third nearest
d1to3 <- nndist(cells, k=1:3)
x <- runif(100)
y <- runif(100)
d <- nndist(x, y)
# Stienen diagram
plot(cells %mark% (nndist(cells)/2), markscale=1)
# distance to nearest neighbour of each type
nnda <- nndist(ants, by=marks(ants))
head(nnda)
# For nest number 1, the nearest Cataglyphis nest is 87.32125 units away
# Use of 'aggregate':
# minimum distance between each pair of types
aggregate(nnda, by=list(from=marks(ants)), min)
# Always a symmetric matrix
# mean nearest neighbour distances
aggregate(nnda, by=list(from=marks(ants)), mean)
# The mean distance from a Messor nest to
# the nearest Cataglyphis nest is 59.02549 units

nndist.lpp

Nearest neighbour distances on a linear network

Description
Given a pattern of points on a linear network, compute the nearest-neighbour distances, measured
by the shortest path in the network.

nndist.lpp

859

Usage
## S3 method for class 'lpp'
nndist(X, ..., k=1, method="C")
Arguments
X

Point pattern on linear network (object of class "lpp").

method

Optional string determining the method of calculation. Either "interpreted"
or "C".

k

Integer, or integer vector. The algorithm will compute the distance to the kth
nearest neighbour.

...

Ignored.

Details
Given a pattern of points on a linear network, this function computes the nearest neighbour distance
for each point (i.e. the distance from each point to the nearest other point), measuring distance by
the shortest path in the network.
If method="C" the distances are computed using code in the C language. If method="interpreted"
then the computation is performed using interpreted R code. The R code is much slower, but is provided for checking purposes.
The kth nearest neighbour distance is infinite if the kth nearest neighbour does not exist. This can
occur if there are fewer than k+1 points in the dataset, or if the linear network is not connected.
Value
A numeric vector, of length equal to the number of points in X, or a matrix, with one row for each
point in X and one column for each entry of k. Entries are nonnegative numbers or infinity (Inf).
Author(s)
Adrian Baddeley 
See Also
lpp
Examples
X <- runiflpp(12, simplenet)
nndist(X)
nndist(X, k=2)

860

nndist.pp3

nndist.pp3

Nearest neighbour distances in three dimensions

Description
Computes the distance from each point to its nearest neighbour in a three-dimensional point pattern.
Alternatively computes the distance to the second nearest neighbour, or third nearest, etc.
Usage
## S3 method for class 'pp3'
nndist(X, ..., k=1)
Arguments
X

Three-dimensional point pattern (object of class "pp3").

...

Ignored.

k

Integer, or integer vector. The algorithm will compute the distance to the kth
nearest neighbour.

Details
This function computes the Euclidean distance from each point in a three-dimensional point pattern
to its nearest neighbour (the nearest other point of the pattern). If k is specified, it computes the
distance to the kth nearest neighbour.
The function nndist is generic; this function nndist.pp3 is the method for the class "pp3".
The argument k may be a single integer, or an integer vector. If it is a vector, then the kth nearest
neighbour distances are computed for each value of k specified in the vector.
If there is only one point (if x has length 1), then a nearest neighbour distance of Inf is returned. If
there are no points (if x has length zero) a numeric vector of length zero is returned.
To identify which point is the nearest neighbour of a given point, use nnwhich.
To use the nearest neighbour distances for statistical inference, it is often advisable to use the edgecorrected empirical distribution, computed by G3est.
To find the nearest neighbour distances from one point pattern to another point pattern, use nncross.
Value
Numeric vector or matrix containing the nearest neighbour distances for each point.
If k = 1 (the default), the return value is a numeric vector v such that v[i] is the nearest neighbour
distance for the ith data point.
If k is a single integer, then the return value is a numeric vector v such that v[i] is the kth nearest
neighbour distance for the ith data point.
If k is a vector, then the return value is a matrix m such that m[i,j] is the k[j]th nearest neighbour
distance for the ith data point.
Warnings
An infinite or NA value is returned if the distance is not defined (e.g. if there is only one point in the
point pattern).

nndist.ppx

861

Author(s)
Adrian Baddeley 
based on code for two dimensions by Pavel Grabarnik
See Also
nndist, pairdist, G3est, nnwhich
Examples
X <- runifpoint3(40)
# nearest neighbours
d <- nndist(X)
# second nearest neighbours
d2 <- nndist(X, k=2)
# first, second and third nearest
d1to3 <- nndist(X, k=1:3)

nndist.ppx

Nearest Neighbour Distances in Any Dimensions

Description
Computes the distance from each point to its nearest neighbour in a multi-dimensional point pattern.
Alternatively computes the distance to the second nearest neighbour, or third nearest, etc.
Usage
## S3 method for class 'ppx'
nndist(X, ..., k=1)
Arguments
X

Multi-dimensional point pattern (object of class "ppx").

...

Arguments passed to coords.ppx to determine which coordinates should be
used.

k

Integer, or integer vector. The algorithm will compute the distance to the kth
nearest neighbour.

Details
This function computes the Euclidean distance from each point in a multi-dimensional point pattern
to its nearest neighbour (the nearest other point of the pattern). If k is specified, it computes the
distance to the kth nearest neighbour.
The function nndist is generic; this function nndist.ppx is the method for the class "ppx".
The argument k may be a single integer, or an integer vector. If it is a vector, then the kth nearest
neighbour distances are computed for each value of k specified in the vector.

862

nndist.ppx
If there is only one point (if x has length 1), then a nearest neighbour distance of Inf is returned. If
there are no points (if x has length zero) a numeric vector of length zero is returned.
To identify which point is the nearest neighbour of a given point, use nnwhich.
To find the nearest neighbour distances from one point pattern to another point pattern, use nncross.
By default, both spatial and temporal coordinates are extracted. To obtain the spatial distance
between points in a space-time point pattern, set temporal=FALSE.

Value
Numeric vector or matrix containing the nearest neighbour distances for each point.
If k = 1 (the default), the return value is a numeric vector v such that v[i] is the nearest neighbour
distance for the ith data point.
If k is a single integer, then the return value is a numeric vector v such that v[i] is the kth nearest
neighbour distance for the ith data point.
If k is a vector, then the return value is a matrix m such that m[i,j] is the k[j]th nearest neighbour
distance for the ith data point.

Warnings
An infinite or NA value is returned if the distance is not defined (e.g. if there is only one point in the
point pattern).

Author(s)
Adrian Baddeley 

See Also
nndist, pairdist, nnwhich

Examples
df <- data.frame(x=runif(5),y=runif(5),z=runif(5),w=runif(5))
X <- ppx(data=df)
# nearest neighbours
d <- nndist(X)
# second nearest neighbours
d2 <- nndist(X, k=2)
# first, second and third nearest
d1to3 <- nndist(X, k=1:3)

nndist.psp

nndist.psp

863

Nearest neighbour distances between line segments

Description
Computes the distance from each line segment to its nearest neighbour in a line segment pattern.
Alternatively finds the distance to the second nearest, third nearest etc.
Usage
## S3 method for class 'psp'
nndist(X, ..., k=1, method="C")
Arguments
X

A line segment pattern (object of class "psp").

...

Ignored.

k

Integer, or integer vector. The algorithm will compute the distance to the kth
nearest neighbour.

method

String specifying which method of calculation to use. Values are "C" and "interpreted".
Usually not specified.

Details
This is a method for the generic function nndist for the class "psp".
If k=1, this function computes the distance from each line segment to the nearest other line segment
in X. In general it computes the distance from each line segment to the kth nearest other line segment.
The argument k can also be a vector, and this computation will be performed for each value of k.
Distances are calculated using the Hausdorff metric. The Hausdorff distance between two line
segments is the maximum distance from any point on one of the segments to the nearest point on
the other segment.
If there are fewer than max(k)+1 line segments in the pattern, some of the nearest neighbour distances will be infinite (Inf).
The argument method is not normally used. It is retained only for checking the validity of the
software. If method = "interpreted" then the distances are computed using interpreted R code
only. If method="C" (the default) then compiled C code is used. The C code is somewhat faster.
Value
Numeric vector or matrix containing the nearest neighbour distances for each line segment.
If k = 1 (the default), the return value is a numeric vector v such that v[i] is the nearest neighbour
distance for the ith segment.
If k is a single integer, then the return value is a numeric vector v such that v[i] is the kth nearest
neighbour distance for the ith segment.
If k is a vector, then the return value is a matrix m such that m[i,j] is the k[j]th nearest neighbour
distance for the ith segment.

864

nnfromvertex

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
nndist, nndist.ppp
Examples
L <- psp(runif(10), runif(10), runif(10), runif(10), owin())
D <- nndist(L)
D <- nndist(L, k=1:3)

nnfromvertex

Nearest Data Point From Each Vertex in a Network

Description
Given a point pattern on a linear network, for each vertex of the network find the nearest data point.
Usage
nnfromvertex(X, what = c("dist", "which"), k = 1)
Arguments
X

Point pattern on a linear network (object of class "lpp").

what

Character string specifying whether to return the nearest-neighbour distances,
nearest-neighbour identifiers, or both.

k

Integer, or integer vector, specifying that the kth nearest neighbour should be
returned.

Details
For each vertex (node) of the linear network, this algorithm finds the nearest data point to the vertex,
and returns either the distance from the vertex to its nearest neighbour in X, or the serial number of
the nearest neighbour in X, or both.
If k is an integer, then the k-th nearest neighbour is found instead.
If k is an integer vector, this is repeated for each integer in k.
Value
A numeric vector, matrix, or data frame.
Author(s)
Adrian Baddeley .

nnfun

865

See Also
nndist.lpp
Examples
X <- runiflpp(5, simplenet)
nnfromvertex(X)
nnfromvertex(X, k=1:3)

nnfun

Nearest Neighbour Index Map as a Function

Description
Compute the nearest neighbour index map of an object, and return it as a function.
Usage
nnfun(X, ...)
## S3 method for class 'ppp'
nnfun(X, ..., k=1)
## S3 method for class 'psp'
nnfun(X, ...)
Arguments
X

Any suitable dataset representing a two-dimensional collection of objects, such
as a point pattern (object of class "ppp") or a line segment pattern (object of
class "psp").

k

A single integer. The kth nearest neighbour will be found.

...

Extra arguments are ignored.

Details
For a collection X of two dimensional objects (such as a point pattern or a line segment pattern),
the “nearest neighbour index function” of X is the mathematical function f such that, for any twodimensional spatial location (x, y), the function value f(x,y) is the index i identifying the closest
member of X. That is, if i = f (x, y) then X[i] is the closest member of the collection X to the
location (x, y).
The command f <- nnfun(X) returns a function in the R language, with arguments x,y, that represents the nearest neighbour index function of X. Evaluating the function f in the form v <- f(x,y),
where x and y are any numeric vectors of equal length containing coordinates of spatial locations,
yields the indices of the nearest neighbours to these locations.
If the argument k is specified then the k-th nearest neighbour will be found.
The result of f <- nnfun(X) also belongs to the class "funxy" and to the special class "nnfun".
It can be printed and plotted immediately as shown in the Examples.
A nnfun object can be converted to a pixel image using as.im.

866

nnfun.lpp

Value
A function with arguments x,y. The function also belongs to the class "nnfun" which has a
method for print. It also belongs to the class "funxy" which has methods for plot, contour and
persp.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
distfun, plot.funxy
Examples
f <- nnfun(cells)
f
plot(f)
f(0.2, 0.3)
g <- nnfun(cells, k=2)
g(0.2, 0.3)
L <- psp(runif(10), runif(10), runif(10), runif(10), window=owin())
h <- nnfun(L)
h(0.2, 0.3)

nnfun.lpp

Nearest Neighbour Map on Linear Network

Description
Compute the nearest neighbour function of a point pattern on a linear network.
Usage
## S3 method for class 'lpp'
nnfun(X, ..., k=1)
Arguments
X

A point pattern on a linear network (object of class "lpp").

k

Integer. The algorithm finds the kth nearest neighbour in X from any spatial
location.

...

Other arguments are ignored.

nnmap

867

Details
The (geodesic) nearest neighbour function of a point pattern X on a linear network L tells us which
point of X is closest to any given location.
If X is a point pattern on a linear network L, the nearest neighbour function of X is the mathematical
function f defined for any location s on the network by f(s) = i, where X[i] is the closest point
of X to the location s measured by the shortest path. In other words the value of f(s) is the identifier
or serial number of the closest point of X.
The command nnfun.lpp is a method for the generic command nnfun for the class "lpp" of point
patterns on a linear network.
If X is a point pattern on a linear network, f <- nnfun(X) returns a function in the R language, with
arguments x,y, ..., that represents the nearest neighbour function of X. Evaluating the function
f in the form v <- f(x,y), where x and y are any numeric vectors of equal length containing
coordinates of spatial locations, yields a vector of identifiers or serial numbers of the data points
closest to these spatial locations. More efficiently f can take the arguments x, y, seg, tp where
seg and tp are the local coordinates on the network.
The result of f <- nnfun(X) also belongs to the class "linfun". It can be printed and plotted
immediately as shown in the Examples. It can be converted to a pixel image using as.linim.
Value
A function in the R language, with arguments x,y and optional arguments seg,tp. It also belongs
to the class "linfun" which has methods for plot, print etc.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
linfun, methods.linfun.
To compute the distance to the nearest neighbour, see distfun.lpp.
Examples
data(letterR)
X <- runiflpp(3, simplenet)
f <- nnfun(X)
f
plot(f)

nnmap

K-th Nearest Point Map

Description
Given a point pattern, this function constructs pixel images giving the distance from each pixel to
its k-th nearest neighbour in the point pattern, and the index of the k-th nearest neighbour.

868

nnmap

Usage
nnmap(X, k = 1, what = c("dist", "which"),
..., W = as.owin(X),
is.sorted.X = FALSE, sortby = c("range", "var", "x", "y"))
Arguments
X

Point pattern (object of class "ppp").

k

Integer, or integer vector. The algorithm will find the kth nearest neighbour.

what

Character string specifying what information should be returned. Either the
nearest neighbour distance ("dist"), the index of the nearest neighbour ("which"),
or both.

...

Arguments passed to as.mask to determine the pixel resolution of the result.

W

Window (object of class "owin") specifying the spatial domain in which the
distances will be computed. Defaults to the window of X.

is.sorted.X

Logical value attesting whether the point pattern X has been sorted. See Details.

sortby

Determines which coordinate to use to sort the point pattern. See Details.

Details
Given a point pattern X, this function constructs two pixel images:
• a distance map giving, for each pixel, the distance to the nearest point of X;
• a nearest neighbour map giving, for each pixel, the identifier of the nearest point of X.
If the argument k is specified, then the k-th nearest neighbours will be found.
If what="dist" then only the distance map is returned. If what="which" then only the nearest
neighbour map is returned.
The argument k may be an integer or an integer vector. If it is a single integer, then the k-th nearest
neighbours are computed. If it is a vector, then the k[i]-th nearest neighbours are computed for
each entry k[i]. For example, setting k=1:3 will compute the nearest, second-nearest and thirdnearest neighbours.
Value
A pixel image, or a list of pixel images.
By default (if what=c("dist", "which")), the result is a list with two components dist and which
containing the distance map and the nearest neighbour map.
If what="dist" then the result is a real-valued pixel image containing the distance map.
If what="which" then the result is an integer-valued pixel image containing the nearest neighbour
map.
If k is a vector of several integers, then the result is similar except that each pixel image is replaced
by a list of pixel images, one for each entry of k.

nnmark

869

Sorting data and pre-sorted data
Read this section if you care about the speed of computation.
For efficiency, the algorithm sorts the point pattern X into increasing order of the x coordinate or
increasing order of the the y coordinate. Sorting is only an intermediate step; it does not affect the
output, which is always given in the same order as the original data.
By default (if sortby="range"), the sorting will occur on the coordinate that has the larger range
of values (according to the frame of the enclosing window of X). If sortby = "var"), sorting
will occur on the coordinate that has the greater variance (in the pattern X). Setting sortby="x" or
sortby = "y" will specify that sorting should occur on the x or y coordinate, respectively.
If the point pattern X is already sorted, then the argument is.sorted.X should be set to TRUE, and
sortby should be set equal to "x" or "y" to indicate which coordinate is sorted.
Warning About Ties
Ties are possible: there may be two data points which lie exactly the same distance away from a particular pixel. This affects the results from nnmap(what="which"). The handling of ties is not welldefined: it is not consistent between different computers and different installations of R. If there are
ties, then different calls to nnmap(what="which") may give inconsistent results. For example, you
may get a different answer from nnmap(what="which",k=1) and nnmap(what="which", k=1:2)[[1]].
Author(s)
Adrian Baddeley 
, Rolf Turner  , and Jens Oehlschlaegel
See Also
distmap
Examples
plot(nnmap(cells, 2, what="which"))

nnmark

Mark of Nearest Neighbour

Description
Given a marked point pattern dataset X this function computes, for each desired location y, the mark
attached to the nearest neighbour of y in X. The desired locations y can be either a pixel grid or the
point pattern X itself.
Usage
nnmark(X, ..., k = 1, at=c("pixels", "points"))

870

nnmark

Arguments
X

A marked point pattern (object of class "ppp").

...

Arguments passed to as.mask to determine the pixel resolution.

k

Single integer. The kth nearest data point will be used.

at

String specifying whether to compute the values at a grid of pixel locations
(at="pixels") or only at the points of X (at="points").

Details
Given a marked point pattern dataset X this function computes, for each desired location y, the mark
attached to the point of X that is nearest to y. The desired locations y can be either a pixel grid or
the point pattern X itself.
The argument X must be a marked point pattern (object of class "ppp", see ppp.object). The marks
are allowed to be a vector or a data frame.
• If at="points", then for each point in X, the algorithm finds the nearest other point in X, and
extracts the mark attached to it. The result is a vector or data frame containing the marks of
the neighbours of each point.
• If at="pixels" (the default), then for each pixel in a rectangular grid, the algorithm finds the
nearest point in X, and extracts the mark attached to it. The result is an image or a list of images
containing the marks of the neighbours of each pixel. The pixel resolution is controlled by the
arguments ... passed to as.mask.
If the argument k is given, then the k-th nearest neighbour will be used.
Value
If X has a single column of marks:
• If at="pixels" (the default), the result is a pixel image (object of class "im"). The value at
each pixel is the mark attached to the nearest point of X.
• If at="points", the result is a vector or factor of length equal to the number of points in X.
Entries are the mark values of the nearest neighbours of each point of X.
If X has a data frame of marks:
• If at="pixels" (the default), the result is a named list of pixel images (object of class "im").
There is one image for each column of marks. This list also belongs to the class "solist",
for which there is a plot method.
• If at="points", the result is a data frame with one row for each point of X, Entries are the
mark values of the nearest neighbours of each point of X.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
Smooth.ppp, marktable, nnwhich

nnorient

871

Examples
plot(nnmark(ants))
v <- nnmark(ants, at="points")
v[1:10]
plot(nnmark(finpines))
vf <- nnmark(finpines, at="points")
vf[1:5,]

nnorient

Nearest Neighbour Orientation Distribution

Description
Computes the distribution of the orientation of the vectors from each point to its nearest neighbour.
Usage
nnorient(X, ..., cumulative = FALSE, correction, k = 1,
unit = c("degree", "radian"),
domain = NULL, ratio = FALSE)
Arguments
X

Point pattern (object of class "ppp").

...

Arguments passed to circdensity to control the kernel smoothing, if cumulative=FALSE.

cumulative

Logical value specifying whether to estimate the probability density (cumulative=FALSE,
the default) or the cumulative distribution function (cumulative=TRUE).

correction

Character vector specifying edge correction or corrections. Options are "none",
"bord.modif", "good" and "best". Alternatively correction="all" selects
all options.

k

Integer. The kth nearest neighbour will be used.

ratio

Logical. If TRUE, the numerator and denominator of each edge-corrected estimate will also be saved, for use in analysing replicated point patterns.

unit

Unit in which the angles should be expressed. Either "degree" or "radian".

domain

Optional window. The first point xi of each pair of points will be constrained to
lie in domain.

Details
This algorithm considers each point in the pattern X and finds its nearest neighbour (or kth nearest
neighour). The direction of the arrow joining the data point to its neighbour is measured, as an
angle in degrees or radians, anticlockwise from the x axis.
If cumulative=FALSE (the default), a kernel estimate of the probability density of the angles is
calculated using circdensity. This is the function ϑ(φ) defined in Illian et al (2008), equation
(4.5.3), page 253.
If cumulative=TRUE, then the cumulative distribution function of these angles is calculated.
In either case the result can be plotted as a rose diagram by rose, or as a function plot by plot.fv.

872

nnwhich
The algorithm gives each observed direction a weight, determined by an edge correction, to adjust
for the fact that some interpoint distances are more likely to be observed than others. The choice of
edge correction or corrections is determined by the argument correction.
It is also possible to calculate an estimate of the probability density from the cumulative distribution
function, by numerical differentiation. Use deriv.fv with the argument Dperiodic=TRUE.

Value
A function value table (object of class "fv") containing the estimates of the probability density
or the cumulative distribution function of angles, in degrees (if unit="degree") or radians (if
unit="radian").
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Illian, J., Penttinen, A., Stoyan, H. and Stoyan, D. (2008) Statistical Analysis and Modelling of
Spatial Point Patterns. Wiley.
See Also
pairorient
Examples
rose(nnorient(redwood, adjust=0.6), col="grey")
plot(CDF <- nnorient(redwood, cumulative=TRUE))

nnwhich

Nearest neighbour

Description
Finds the nearest neighbour of each point in a point pattern.
Usage
nnwhich(X, ...)
## S3 method for class 'ppp'
nnwhich(X, ..., k=1, by=NULL, method="C")
## Default S3 method:
nnwhich(X, Y=NULL, ..., k=1, by=NULL, method="C")

nnwhich

873

Arguments
X,Y

Arguments specifying the locations of a set of points. For nnwhich.ppp, the argument X should be a point pattern (object of class "ppp"). For nnwhich.default,
typically X and Y would be numeric vectors of equal length. Alternatively Y may
be omitted and X may be a list with two components x and y, or a matrix with
two columns.

...

Ignored by nnwhich.ppp and nnwhich.default.

k

Integer, or integer vector. The algorithm will compute the distance to the kth
nearest neighbour.

by

Optional. A factor, which separates X into groups. The algorithm will find the
nearest neighbour in each group.

method

String specifying which method of calculation to use. Values are "C" and "interpreted".

Details
For each point in the given point pattern, this function finds its nearest neighbour (the nearest other
point of the pattern). By default it returns a vector giving, for each point, the index of the point’s
nearest neighbour. If k is specified, the algorithm finds each point’s kth nearest neighbour.
The function nnwhich is generic, with method for point patterns (objects of class "ppp") and a
default method which are described here, as well as a method for three-dimensional point patterns
(objects of class "pp3", described in nnwhich.pp3.
The method nnwhich.ppp expects a single point pattern argument X. The default method expects
that X and Y will determine the coordinates of a set of points. Typically X and Y would be numeric
vectors of equal length. Alternatively Y may be omitted and X may be a list with two components
named x and y, or a matrix or data frame with two columns.
The argument k may be a single integer, or an integer vector. If it is a vector, then the kth nearest
neighbour distances are computed for each value of k specified in the vector.
If the argument by is given, it should be a factor, of length equal to the number of points in X. This
factor effectively partitions X into subsets, each subset associated with one of the levels of X. The
algorithm will then find, for each point of X, the nearest neighbour in each subset.
If there are no points (if x has length zero) a numeric vector of length zero is returned. If there
is only one point (if x has length 1), then the nearest neighbour is undefined, and a value of NA
is returned. In general if the number of points is less than or equal to k, then a vector of NA’s is
returned.
The argument method is not normally used. It is retained only for checking the validity of the
software. If method = "interpreted" then the distances are computed using interpreted R code
only. If method="C" (the default) then C code is used. The C code is faster by two to three orders
of magnitude and uses much less memory.
To evaluate the distance between a point and its nearest neighbour, use nndist.
To find the nearest neighbours from one point pattern to another point pattern, use nncross.
Value
Numeric vector or matrix giving, for each point, the index of its nearest neighbour (or kth nearest
neighbour).
If k = 1 (the default), the return value is a numeric vector v giving the indices of the nearest
neighbours (the nearest neighbout of the ith point is the jth point where j = v[i]).

874

nnwhich
If k is a single integer, then the return value is a numeric vector giving the indices of the kth nearest
neighbours.
If k is a vector, then the return value is a matrix m such that m[i,j] is the index of the k[j]th nearest
neighbour for the ith data point.
If the argument by is given, then the result is a data frame containing the indices described above,
from each point of X, to the nearest point in each subset of X defined by the factor by.

Nearest neighbours of each type
If X is a multitype point pattern and by=marks(X), then the algorithm will find, for each point of X,
the nearest neighbour of each type. See the Examples.

Warnings
A value of NA is returned if there is only one point in the point pattern.

Author(s)
Pavel Grabarnik  and Adrian Baddeley 

See Also
nndist, nncross

Examples
data(cells)
plot(cells)
m <- nnwhich(cells)
m2 <- nnwhich(cells, k=2)
# plot nearest neighbour links
b <- cells[m]
arrows(cells$x, cells$y, b$x, b$y, angle=15, length=0.15, col="red")
# find points which are the neighbour of their neighbour
self <- (m[m] == seq(m))
# plot them
A <- cells[self]
B <- cells[m[self]]
plot(cells)
segments(A$x, A$y, B$x, B$y)
# nearest neighbours of each type
head(nnwhich(ants, by=marks(ants)))

nnwhich.lpp

nnwhich.lpp

875

Identify Nearest Neighbours on a Linear Network

Description
Given a pattern of points on a linear network, identify the nearest neighbour for each point, measured by the shortest path in the network.
Usage
## S3 method for class 'lpp'
nnwhich(X, ..., k=1, method="C")
Arguments
X
method
k
...

Point pattern on linear network (object of class "lpp").
Optional string determining the method of calculation. Either "interpreted"
or "C".
Integer, or integer vector. The algorithm will find the kth nearest neighbour.
Ignored.

Details
Given a pattern of points on a linear network, this function finds the nearest neighbour of each point
(i.e. for each point it identifies the nearest other point) measuring distance by the shortest path in
the network.
If method="C" the task is performed using code in the C language. If method="interpreted" then
the computation is performed using interpreted R code. The R code is much slower, but is provided
for checking purposes.
The result is NA if the kth nearest neighbour does not exist. This can occur if there are fewer than
k+1 points in the dataset, or if the linear network is not connected.
Value
An integer vector, of length equal to the number of points in X, identifying the nearest neighbour of
each point. If nnwhich(X)[2] = 4 then the nearest neighbour of point 2 is point 4.
Alternatively a matrix with one row for each point in X and one column for each entry of k.
Author(s)
Adrian Baddeley 
See Also
lpp
Examples
X <- runiflpp(10, simplenet)
nnwhich(X)
nnwhich(X, k=2)

876

nnwhich.pp3

nnwhich.pp3

Nearest neighbours in three dimensions

Description
Finds the nearest neighbour of each point in a three-dimensional point pattern.
Usage
## S3 method for class 'pp3'
nnwhich(X, ..., k=1)
Arguments
X

Three-dimensional point pattern (object of class "pp3").

...

Ignored.

k

Integer, or integer vector. The algorithm will compute the distance to the kth
nearest neighbour.

Details
For each point in the given three-dimensional point pattern, this function finds its nearest neighbour
(the nearest other point of the pattern). By default it returns a vector giving, for each point, the
index of the point’s nearest neighbour. If k is specified, the algorithm finds each point’s kth nearest
neighbour.
The function nnwhich is generic. This is the method for the class "pp3".
If there are no points in the pattern, a numeric vector of length zero is returned. If there is only
one point, then the nearest neighbour is undefined, and a value of NA is returned. In general if the
number of points is less than or equal to k, then a vector of NA’s is returned.
To evaluate the distance between a point and its nearest neighbour, use nndist.
To find the nearest neighbours from one point pattern to another point pattern, use nncross.
Value
Numeric vector or matrix giving, for each point, the index of its nearest neighbour (or kth nearest
neighbour).
If k = 1 (the default), the return value is a numeric vector v giving the indices of the nearest
neighbours (the nearest neighbout of the ith point is the jth point where j = v[i]).
If k is a single integer, then the return value is a numeric vector giving the indices of the kth nearest
neighbours.
If k is a vector, then the return value is a matrix m such that m[i,j] is the index of the k[j]th nearest
neighbour for the ith data point.
Warnings
A value of NA is returned if there is only one point in the point pattern.

nnwhich.ppx

877

Author(s)
Adrian Baddeley 
based on two-dimensional code by Pavel Grabarnik
See Also
nnwhich, nndist, nncross
Examples
X <- runifpoint3(30)
m <- nnwhich(X)
m2 <- nnwhich(X, k=2)

nnwhich.ppx

Nearest Neighbours in Any Dimensions

Description
Finds the nearest neighbour of each point in a multi-dimensional point pattern.
Usage
## S3 method for class 'ppx'
nnwhich(X, ..., k=1)
Arguments
X

Multi-dimensional point pattern (object of class "ppx").

...

Arguments passed to coords.ppx to determine which coordinates should be
used.

k

Integer, or integer vector. The algorithm will compute the distance to the kth
nearest neighbour.

Details
For each point in the given multi-dimensional point pattern, this function finds its nearest neighbour
(the nearest other point of the pattern). By default it returns a vector giving, for each point, the
index of the point’s nearest neighbour. If k is specified, the algorithm finds each point’s kth nearest
neighbour.
The function nnwhich is generic. This is the method for the class "ppx".
If there are no points in the pattern, a numeric vector of length zero is returned. If there is only
one point, then the nearest neighbour is undefined, and a value of NA is returned. In general if the
number of points is less than or equal to k, then a vector of NA’s is returned.
To evaluate the distance between a point and its nearest neighbour, use nndist.
To find the nearest neighbours from one point pattern to another point pattern, use nncross.
By default, both spatial and temporal coordinates are extracted. To obtain the spatial distance
between points in a space-time point pattern, set temporal=FALSE.

878

nobjects

Value
Numeric vector or matrix giving, for each point, the index of its nearest neighbour (or kth nearest
neighbour).
If k = 1 (the default), the return value is a numeric vector v giving the indices of the nearest
neighbours (the nearest neighbout of the ith point is the jth point where j = v[i]).
If k is a single integer, then the return value is a numeric vector giving the indices of the kth nearest
neighbours.
If k is a vector, then the return value is a matrix m such that m[i,j] is the index of the k[j]th nearest
neighbour for the ith data point.
Warnings
A value of NA is returned if there is only one point in the point pattern.
Author(s)
Adrian Baddeley 
See Also
nnwhich, nndist, nncross
Examples
df <- data.frame(x=runif(5),y=runif(5),z=runif(5),w=runif(5))
X <- ppx(data=df)
m <- nnwhich(X)
m2 <- nnwhich(X, k=2)

nobjects

Count Number of Geometrical Objects in a Spatial Dataset

Description
A generic function to count the number of geometrical objects in a spatial dataset.
Usage
nobjects(x)
## S3 method for class 'ppp'
nobjects(x)
## S3 method for class 'ppx'
nobjects(x)
## S3 method for class 'psp'
nobjects(x)
## S3 method for class 'tess'
nobjects(x)

npfun

879

Arguments
x

A dataset.

Details
The generic function nobjects counts the number of geometrical objects in the spatial dataset x.
The methods for point patterns (classes "ppp" and "ppx", embracing "pp3" and "lpp") count the
number of points in the pattern.
The method for line segment patterns (class "psp") counts the number of line segments in the
pattern.
The method for tessellations (class "tess") counts the number of tiles of the tessellation.
Value
A single integer.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
npoints
Examples
nobjects(redwood)
nobjects(edges(letterR))
nobjects(dirichlet(cells))

npfun

Dummy Function Returns Number of Points

Description
Returns a summary function which is constant with value equal to the number of points in the point
pattern.
Usage
npfun(X, ..., r)
Arguments
X

Point pattern.

...

Ignored.

r

Vector of values of the distance argument r.

880

npoints

Details
This function is normally not called by the user. Instead it is passed as an argument to the function
psst.
Value
Object of class "fv" representing a constant function.
Author(s)
Adrian Baddeley 
Ege Rubak  and Jesper Møller.
See Also
psst
Examples
fit0 <- ppm(cells, ~1, nd=10)
v <- psst(fit0, npfun)

npoints

Number of Points in a Point Pattern

Description
Returns the number of points in a point pattern of any kind.
Usage
npoints(x)
## S3 method for class 'ppp'
npoints(x)
## S3 method for class 'pp3'
npoints(x)
## S3 method for class 'ppx'
npoints(x)
Arguments
x

A point pattern (object of class "ppp", "pp3", "ppx" or some other suitable
class).

Details
This function returns the number of points in a point pattern. The function npoints is generic with
methods for the classes "ppp", "pp3", "ppx" and possibly other classes.
Value
Integer.

nsegments

881

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
ppp.object, print.pp3, print.ppx.
Examples
data(cells)
npoints(cells)

nsegments

Number of Line Segments in a Line Segment Pattern

Description
Returns the number of line segments in a line segment pattern.
Usage
nsegments(x)
## S3 method for class 'psp'
nsegments(x)
Arguments
x

A line segment pattern, i.e. an object of class psp, or an object containing a
linear network.

Details
This function is generic, with methods for classes psp, linnet and lpp.
Value
Integer.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
npoints(), psp.object()
Examples
nsegments(copper$Lines)
nsegments(copper$SouthLines)

882

nvertices

nvertices

Count Number of Vertices

Description
Count the number of vertices in an object for which vertices are well-defined.
Usage
nvertices(x, ...)
## S3 method for class 'owin'
nvertices(x, ...)
## Default S3 method:
nvertices(x, ...)
Arguments
x

A window (object of class "owin"), or some other object which has vertices.

...

Currently ignored.

Details
This function counts the number of vertices of x as they would be returned by vertices(x). It is
more efficient than executing npoints(vertices(x)).
Value
A single integer.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak  and Suman Rakshit.
See Also
vertices
Examples
nvertices(square(2))
nvertices(letterR)

objsurf

883

objsurf

Objective Function Surface

Description
For a model that was fitted by optimisation, compute the values of the objective function in a
neighbourhood of the optimal value.
Usage
objsurf(x, ...)
## S3 method for class 'dppm'
objsurf(x, ..., ngrid = 32, ratio = 1.5, verbose = TRUE)
## S3 method for class 'kppm'
objsurf(x, ..., ngrid = 32, ratio = 1.5, verbose = TRUE)
## S3 method for class 'minconfit'
objsurf(x, ..., ngrid = 32, ratio = 1.5, verbose = TRUE)
Arguments
x

Some kind of model that was fitted by finding the optimal value of an objective
function. An object of class "dppm", "kppm" or "minconfit".

...

Extra arguments are usually ignored.

ngrid

Number of grid points to evaluate along each axis. Either a single integer, or a
pair of integers. For example ngrid=32 would mean a 32 * 32 grid.

ratio

Number greater than 1 determining the range of parameter values to be considered. If the optimal parameter value is opt then the objective function will be
evaluated for values between opt/ratio and opt * ratio.

verbose

Logical value indicating whether to print progress reports.

Details
The object x should be some kind of model that was fitted by maximising or minimising the value
of an objective function. The objective function will be evaluated on a grid of values of the model
parameters.
Currently the following types of objects are accepted:
• an object of class "dppm" representing a determinantal point process. See dppm.
• an object of class "kppm" representing a cluster point process or Cox point process. See kppm.
• an object of class "minconfit" representing a minimum-contrast fit between a summary function and its theoretical counterpart. See mincontrast.
The result is an object of class "objsurf" which can be printed and plotted: see methods.objsurf.
Value
An object of class "objsurf" which can be printed and plotted. Essentially a list containing entries
x, y, z giving the parameter values and objective function values.

884

opening

Author(s)
Adrian Baddeley  and Ege Rubak .
See Also
methods.objsurf, kppm, mincontrast
Examples
fit <- kppm(redwood ~ 1, "Thomas")
os <- objsurf(fit)
if(interactive()) {
plot(os)
contour(os, add=TRUE)
persp(os)
}

opening

Morphological Opening

Description
Perform morphological opening of a window, a line segment pattern or a point pattern.
Usage
opening(w, r, ...)
## S3 method for class 'owin'
opening(w, r, ..., polygonal=NULL)
## S3 method for class 'ppp'
opening(w, r, ...)
## S3 method for class 'psp'
opening(w, r, ...)
Arguments
w

A window (object of class "owin" or a line segment pattern (object of class
"psp") or a point pattern (object of class "ppp").

r

positive number: the radius of the opening.

...

extra arguments passed to as.mask controlling the pixel resolution, if a pixel
approximation is used

polygonal

Logical flag indicating whether to compute a polygonal approximation to the
erosion (polygonal=TRUE) or a pixel grid approximation (polygonal=FALSE).

Ops.msr

885

Details
The morphological opening (Serra, 1982) of a set W by a distance r > 0 is the subset of points in
W that can be separated from the boundary of W by a circle of radius r. That is, a point x belongs
to the opening if it is possible to draw a circle of radius r (not necessarily centred on x) that has x
on the inside and the boundary of W on the outside. The opened set is a subset of W.
For a small radius r, the opening operation has the effect of smoothing out irregularities in the
boundary of W . For larger radii, the opening operation removes promontories in the boundary. For
very large radii, the opened set is empty.
The algorithm applies erosion followed by dilation.
Value
If r > 0, an object of class "owin" representing the opened region. If r=0, the result is identical to
w.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Serra, J. (1982) Image analysis and mathematical morphology. Academic Press.
See Also
closing for the opposite operation.
dilation, erosion for the basic operations.
owin, as.owin for information about windows.
Examples
v <- opening(letterR, 0.3)
plot(letterR, type="n", main="opening")
plot(v, add=TRUE, col="grey")
plot(letterR, add=TRUE)

Ops.msr

Arithmetic Operations on Measures

Description
These group generic methods for the class "msr" allow the arithmetic operators +, -, * and / to be
applied directly to measures.
Usage
## S3 methods for group generics have prototypes:
Ops(e1, e2)

886

Ord

Arguments
e1, e2

objects of class "msr".

Details
Arithmetic operators on a measure A are only defined in some cases. The arithmetic operator is
effectively applied to the value of A(W) for every spatial domain W. If the result is a measure, then
this operation is valid.
If A is a measure (object of class "msr") then the operations -A and +A are defined.
If A and B are measures with the same dimension (i.e. both are scalar-valued, or both are kdimensional vector-valued) then A + B and A - B are defined.
If A is a measure and z is a numeric value, then A * z and A / z are defined, and z * A is defined.
Value
Another measure (object of class "msr").
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
with.msr
Examples
X <- rpoispp(function(x,y) { exp(3+3*x) })
fit <- ppm(X, ~x+y)
rp <- residuals(fit, type="pearson")
rp
-rp
2 * rp
rp /2
rp - rp
rr <- residuals(fit, type="raw")
rp - rr

Ord

Generic Ord Interaction model

Description
Creates an instance of an Ord-type interaction point process model which can then be fitted to point
pattern data.

Ord

887

Usage
Ord(pot, name)

Arguments
pot

An S language function giving the user-supplied interaction potential.

name

Character string.

Details
Ord’s point process model (Ord, 1977) is a Gibbs point process of infinite order. Each point xi in
the point pattern x contributes a factor g(ai ) where ai = a(xi , x) is the area of the tile associated
with xi in the Dirichlet tessellation of x.
Ord (1977) proposed fitting this model to forestry data when g(a) has a simple “threshold” form.
That model is implemented in our function OrdThresh. The present function Ord implements the
case of a completely general Ord potential g(a) specified as an S language function pot.
This is experimental.

Value
An object of class "interact" describing the interpoint interaction structure of a point process.

Author(s)
Adrian Baddeley 
and Rolf Turner 

References
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns.
Australian and New Zealand Journal of Statistics 42, 283–322.
Ord, J.K. (1977) Contribution to the discussion of Ripley (1977).
Ord, J.K. (1978) How many trees in a forest? Mathematical Scientist 3, 23–33.
Ripley, B.D. (1977) Modelling spatial patterns (with discussion). Journal of the Royal Statistical
Society, Series B, 39, 172 – 212.

See Also
ppm, ppm.object, OrdThresh

888

OrdThresh

ord.family

Ord Interaction Process Family

Description
An object describing the family of all Ord interaction point processes
Details
Advanced Use Only!
This structure would not normally be touched by the user. It describes the family of point process
models introduced by Ord (1977).
If you need to create a specific Ord-type model for use in analysis, use the function OrdThresh or
Ord.
Anyway, ord.family is an object of class "isf" containing a function ord.family$eval for evaluating the sufficient statistics of any Ord type point process model taking an exponential family
form.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns.
Australian and New Zealand Journal of Statistics 42, 283–322.
Ord, J.K. (1977) Contribution to the discussion of Ripley (1977).
Ord, J.K. (1978) How many trees in a forest? Mathematical Scientist 3, 23–33.
Ripley, B.D. (1977) Modelling spatial patterns (with discussion). Journal of the Royal Statistical
Society, Series B, 39, 172 – 212.
See Also
pairwise.family, pairsat.family, Poisson, Pairwise, PairPiece, Strauss, StraussHard,
Softcore, Geyer, SatPiece, Saturated, Ord, OrdThresh

OrdThresh

Ord’s Interaction model

Description
Creates an instance of Ord’s point process model which can then be fitted to point pattern data.
Usage
OrdThresh(r)

overlap.owin

889

Arguments
r

Positive number giving the threshold value for Ord’s model.

Details
Ord’s point process model (Ord, 1977) is a Gibbs point process of infinite order. Each point xi in the
point pattern x contributes a factor g(ai ) where ai = a(xi , x) is the area of the tile associated with
xi in the Dirichlet tessellation of x. The function g is simply g(a) = 1 if a ≥ r and g(a) = γ < 1
if a < r, where r is called the threshold value.
This function creates an instance of Ord’s model with a given value of r. It can then be fitted to
point process data using ppm.
Value
An object of class "interact" describing the interpoint interaction structure of a point process.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns.
Australian and New Zealand Journal of Statistics 42, 283–322.
Ord, J.K. (1977) Contribution to the discussion of Ripley (1977).
Ord, J.K. (1978) How many trees in a forest? Mathematical Scientist 3, 23–33.
Ripley, B.D. (1977) Modelling spatial patterns (with discussion). Journal of the Royal Statistical
Society, Series B, 39, 172 – 212.
See Also
ppm, ppm.object

overlap.owin

Compute Area of Overlap

Description
Computes the area of the overlap (intersection) of two windows.
Usage
overlap.owin(A, B)
Arguments
A,B

Windows (objects of class "owin").

890

owin

Details
This function computes the area of the overlap between the two windows A and B.
If one of the windows is a binary mask, then both windows are converted to masks on the same grid,
and the area is computed by counting pixels. Otherwise, the area is computed analytically (using
the discrete Stokes theorem).
Value
A single numeric value.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
intersect.owin, area.owin, setcov.
Examples
A <- square(1)
B <- shift(A, c(0.3, 0.2))
overlap.owin(A, B)

owin

Create a Window

Description
Creates an object of class "owin" representing an observation window in the two-dimensional plane
Usage
owin(xrange=c(0,1), yrange=c(0,1), ..., poly=NULL, mask=NULL,
unitname=NULL, xy=NULL)
Arguments
xrange

x coordinate limits of enclosing box

yrange

y coordinate limits of enclosing box

...

Ignored.

poly

Optional. Polygonal boundary of window. Incompatible with mask.

mask

Optional. Logical matrix giving binary image of window. Incompatible with
poly.

unitname

Optional. Name of unit of length. Either a single character string, or a vector of
two character strings giving the singular and plural forms, respectively.

xy

Optional. List with components x and y specifying the pixel coordinates for
mask.

owin

891

Details
In the spatstat library, a point pattern dataset must include information about the window of observation. This is represented by an object of class "owin". See owin.object for an overview.
To create a window in its own right, users would normally invoke owin, although sometimes
as.owin may be convenient.
A window may be rectangular, polygonal, or a mask (a binary image).
• rectangular windows: If only xrange and yrange are given, then the window will be rectangular, with its x and y coordinate dimensions given by these two arguments (which must
be vectors of length 2). If no arguments are given at all, the default is the unit square with
dimensions xrange=c(0,1) and yrange=c(0,1).
• polygonal windows: If poly is given, then the window will be polygonal.
– single polygon: If poly is a matrix or data frame with two columns, or a structure with
two component vectors x and y of equal length, then these values are interpreted as the
cartesian coordinates of the vertices of a polygon circumscribing the window. The vertices must be listed anticlockwise. No vertex should be repeated (i.e. do not repeat the
first vertex).
– multiple polygons or holes: If poly is a list, each entry poly[[i]] of which is a matrix or
data frame with two columns or a structure with two component vectors x and y of equal
length, then the successive list members poly[[i]] are interpreted as separate polygons
which together make up the boundary of the window. The vertices of each polygon must
be listed anticlockwise if the polygon is part of the external boundary, but clockwise if the
polygon is the boundary of a hole in the window. Again, do not repeat any vertex.
• binary masks: If mask is given, then the window will be a binary image.
– Specified by logical matrix: Normally the argument mask should be a logical matrix such
that mask[i,j] is TRUE if the point (x[j],y[i]) belongs to the window, and FALSE if it
does not. Note carefully that rows of mask correspond to the y coordinate, and columns
to the x coordinate. Here x and y are vectors of x and y coordinates equally spaced over
xrange and yrange respectively. The pixel coordinate vectors x and y may be specified
explicitly using the argument xy, which should be a list containing components x and y.
Alternatively there is a sensible default.
– Specified by list of pixel coordinates: Alternatively the argument mask can be a data frame
with 2 or 3 columns. If it has 2 columns, it is expected to contain the spatial coordinates
of all the pixels which are inside the window. If it has 3 columns, it should contain the
spatial coordinates (x, y) of every pixel in the grid, and the logical value associated with
each pixel. The pixels may be listed in any order.
To create a window which is mathematically defined by inequalities in the Cartesian coordinates,
use raster.x() and raster.y() as in the examples below.
Functions square and disc will create square and circular windows, respectively.
Value
An object of class "owin" describing a window in the two-dimensional plane.
Validity of polygon data
Polygon data may contain geometrical inconsistencies such as self-intersections and overlaps. These
inconsistencies must be removed to prevent problems in other spatstat functions. By default, polygon data will be repaired automatically using polygon-clipping code. The repair process may
change the number of vertices in a polygon and the number of polygon components. To disable
the repair process, set spatstat.options(fixpolygons=FALSE).

892

owin

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
owin.object, as.owin, complement.owin, ppp.object, ppp
square, hexagon, regularpolygon, disc, ellipse.
Examples
w <- owin()
w <- owin(c(0,1), c(0,1))
# the unit square
w <- owin(c(10,20), c(10,30), unitname=c("foot","feet"))
# a rectangle of dimensions 10 x 20 feet
# with lower left corner at (10,10)
# polygon (diamond shape)
w <- owin(poly=list(x=c(0.5,1,0.5,0),y=c(0,1,2,1)))
w <- owin(c(0,1), c(0,2), poly=list(x=c(0.5,1,0.5,0),y=c(0,1,2,1)))
# polygon with hole
ho <- owin(poly=list(list(x=c(0,1,1,0), y=c(0,0,1,1)),
list(x=c(0.6,0.4,0.4,0.6), y=c(0.2,0.2,0.4,0.4))))
w <- owin(c(-1,1), c(-1,1), mask=matrix(TRUE, 100,100))
# 100 x 100 image, all TRUE
X <- raster.x(w)
Y <- raster.y(w)
wm <- owin(w$xrange, w$yrange, mask=(X^2 + Y^2 <= 1))
# discrete approximation to the unit disc
## Not run:
if(FALSE) {
plot(c(0,1),c(0,1),type="n")
bdry <- locator()
# click the vertices of a polygon (anticlockwise)
}
## End(Not run)
w <- owin(poly=bdry)
## Not run: plot(w)
## Not run:
im <- as.logical(matrix(scan("myfile"), nrow=128, ncol=128))
# read in an arbitrary 128 x 128 digital image from text file
rim <- im[, 128:1]
# Assuming it was given in row-major order in the file
# i.e. scanning left-to-right in rows from top-to-bottom,
# the use of matrix() has effectively transposed rows & columns,
# so to convert it to our format just reverse the column order.
w <- owin(mask=rim)

owin.object

893

plot(w)
# display it to check!
## End(Not run)

owin.object

Class owin

Description
A class owin to define the “observation window” of a point pattern
Details
In the spatstat library, a point pattern dataset must include information about the window or region
in which the pattern was observed. A window is described by an object of class "owin". Windows
of arbitrary shape are supported.
An object of class "owin" has one of three types:
"rectangle":
"polygonal":
"mask":

a rectangle in the two-dimensional plane with edges parallel to the axes
a region whose boundary is a polygon or several polygons. The region may have holes and may consist o
a binary image (a logical matrix) set to TRUE for pixels inside the window and FALSE outside the window.

Objects of class "owin" may be created by the function owin and converted from other types of
data by the function as.owin.
They may be manipulated by the functions as.rectangle, as.mask, complement.owin, rotate,
shift, affine, erosion, dilation, opening and closing.
Geometrical calculations available for windows include area.owin, perimeter, diameter.owin,
boundingbox, eroded.areas, bdist.points, bdist.pixels, and even.breaks.owin. The mapping between continuous coordinates and pixel raster indices is facilitated by the functions raster.x,
raster.y and nearest.raster.point.
There is a plot method for window objects, plot.owin. This may be useful if you wish to plot a
point pattern’s window without the points for graphical purposes.
There are also methods for summary and print.
Warnings
In a window of type "mask", the row index corresponds to increasing y coordinate, and the column
index corresponds to increasing x coordinate.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
owin, as.owin, as.rectangle, as.mask, summary.owin, print.owin, complement.owin, erosion,
dilation, opening, closing, affine.owin, shift.owin, rotate.owin, raster.x, raster.y,

894

padimage
nearest.raster.point, plot.owin, area.owin, boundingbox, diameter, eroded.areas, bdist.points,
bdist.pixels

Examples
w <- owin()
w <- owin(c(0,1), c(0,1))
# the unit square
w <- owin(c(0,1), c(0,2))
## Not run:
if(FALSE) {
plot(w)
# plots edges of a box 1 unit x 2 units
v <- locator()
# click on points in the plot window
# to be the vertices of a polygon
# traversed in anticlockwise order
u <- owin(c(0,1), c(0,2), poly=v)
plot(u)
# plots polygonal boundary using polygon()
plot(as.mask(u, eps=0.02))
# plots discrete pixel approximation to polygon
}
## End(Not run)

padimage

Pad the Border of a Pixel Image

Description
Fills the border of a pixel image with a given value or values, or extends a pixel image to fill a larger
window.
Usage
padimage(X, value=NA, n=1, W=NULL)
Arguments
X

Pixel image (object of class "im").

value

Single value to be placed around the border of X.

n

Width of border, in pixels. See Details.

W

Window for the resulting image. Incompatible with n.

Details
The image X will be expanded by a margin of n pixels, or extended to fill the window W, with new
pixel values set to value.
The argument value should be a single value (a vector of length 1), normally a value of the same
type as the pixel values of X. It may be NA. Alternatively if X is a factor-valued image, value can be
one of the levels of X.

pairdist

895

If n is given, it may be a single number, specifying the width of the border in pixels. Alternatively
it may be a vector of length 2 or 4. It will be replicated to length 4, and these numbers will be
interpreted as the border widths for the (left, right, top, bottom) margins respectively.
Alternatively if W is given, the image will be extended to the window W.
Value
Another object of class "im", of the same type as X.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
im
Examples
Z <- setcov(owin())
plot(padimage(Z, 1, 10))

pairdist

Pairwise distances

Description
Computes the matrix of distances between all pairs of ‘things’ in a dataset
Usage
pairdist(X, ...)
Arguments
X

Object specifying the locations of a set of ‘things’ (such as a set of points or a
set of line segments).

...

Further arguments depending on the method.

Details
Given a dataset X and Y (representing either a point pattern or a line segment pattern) pairdist
computes the distance between each pair of ‘things’ in the dataset, and returns a matrix containing
these distances.
The function pairdist is generic, with methods for point patterns (objects of class "ppp"), line segment patterns (objects of class "psp") and a default method. See the documentation for pairdist.ppp,
pairdist.psp or pairdist.default for details.

896

pairdist.default

Value
A square matrix whose [i,j] entry is the distance between the ‘things’ numbered i and j.
Author(s)
Pavel Grabarnik  and Adrian Baddeley 
See Also
pairdist.ppp, pairdist.psp, pairdist.default, crossdist, nndist, Kest
pairdist.default

Pairwise distances

Description
Computes the matrix of distances between all pairs of points in a set of points
Usage
## Default S3 method:
pairdist(X, Y=NULL, ..., period=NULL, method="C", squared=FALSE)
Arguments
X,Y

...
period
method
squared

Arguments specifying the coordinates of a set of points. Typically X and Y would
be numeric vectors of equal length. Alternatively Y may be omitted and X may
be a list with two components x and y, or a matrix with two columns.
Ignored.
Optional. Dimensions for periodic edge correction.
String specifying which method of calculation to use. Values are "C" and "interpreted".
Usually not specified.
Logical. If squared=TRUE, the squared distances are returned instead (this computation is faster).

Details
Given the coordinates of a set of points, this function computes the Euclidean distances between all
pairs of points, and returns the matrix of distances. It is a method for the generic function pairdist.
The arguments X and Y must determine the coordinates of a set of points. Typically X and Y would
be numeric vectors of equal length. Alternatively Y may be omitted and X may be a list with two
components named x and y, or a matrix or data frame with two columns.
Alternatively if period is given, then the distances will be computed in the ‘periodic’ sense (also
known as ‘torus’ distance). The points will be treated as if they are in a rectangle of width
period[1] and height period[2]. Opposite edges of the rectangle are regarded as equivalent.
If squared=TRUE then the squared Euclidean distances d2 are returned, instead of the Euclidean
distances d. The squared distances are faster to calculate, and are sufficient for many purposes
(such as finding the nearest neighbour of a point).
The argument method is not normally used. It is retained only for checking the validity of the
software. If method = "interpreted" then the distances are computed using interpreted R code
only. If method="C" (the default) then C code is used. The C code is somewhat faster.

pairdist.lpp

897

Value
A square matrix whose [i,j] entry is the distance between the points numbered i and j.
Author(s)
Pavel Grabarnik  and Adrian Baddeley 
See Also
crossdist, nndist, Kest
Examples
x
y
d
d
d
d

<<<<<<-

runif(100)
runif(100)
pairdist(x, y)
pairdist(cbind(x,y))
pairdist(x, y, period=c(1,1))
pairdist(x, y, squared=TRUE)

pairdist.lpp

Pairwise shortest-path distances between points on a linear network

Description
Given a pattern of points on a linear network, compute the matrix of distances between all pairs of
points, measuring distance by the shortest path in the network.
Usage
## S3 method for class 'lpp'
pairdist(X, ..., method="C")
Arguments
X

Point pattern on linear network (object of class "lpp").

method

Optional string determining the method of calculation. Either "interpreted"
or "C".

...

Ignored.

Details
Given a pattern of points on a linear network, this function computes the matrix of distances between
all pairs of points, measuring distance by the shortest path in the network.
If method="C" the distances are computed using code in the C language. If method="interpreted"
then the computation is performed using interpreted R code. The R code is much slower, but is provided for checking purposes.
If two points cannot be joined by a path, the distance between them is infinite (Inf).
Value
A symmetric matrix, whose values are nonnegative numbers or infinity (Inf).

898

pairdist.pp3

Author(s)
Ang Qi Wei  and Adrian Baddeley .
See Also
lpp
Examples
X <- runiflpp(12, simplenet)
pairdist(X)

pairdist.pp3

Pairwise distances in Three Dimensions

Description
Computes the matrix of distances between all pairs of points in a three-dimensional point pattern.
Usage
## S3 method for class 'pp3'
pairdist(X, ..., periodic=FALSE, squared=FALSE)
Arguments
X

A point pattern (object of class "pp3").

...

Ignored.

periodic

Logical. Specifies whether to apply a periodic edge correction.

squared

Logical. If squared=TRUE, the squared distances are returned instead (this computation is faster).

Details
This is a method for the generic function pairdist.
Given a three-dimensional point pattern X (an object of class "pp3"), this function computes the
Euclidean distances between all pairs of points in X, and returns the matrix of distances.
Alternatively if periodic=TRUE and the window containing X is a box, then the distances will be
computed in the ‘periodic’ sense (also known as ‘torus’ distance): opposite faces of the box are
regarded as equivalent. This is meaningless if the window is not a box.
If squared=TRUE then the squared Euclidean distances d2 are returned, instead of the Euclidean
distances d. The squared distances are faster to calculate, and are sufficient for many purposes
(such as finding the nearest neighbour of a point).
Value
A square matrix whose [i,j] entry is the distance between the points numbered i and j.

pairdist.ppp

899

Author(s)
Adrian Baddeley 
based on two-dimensional code by Pavel Grabarnik.
See Also
pairdist, crossdist, nndist, K3est
Examples
X
d
d
d

<<<<-

runifpoint3(20)
pairdist(X)
pairdist(X, periodic=TRUE)
pairdist(X, squared=TRUE)

pairdist.ppp

Pairwise distances

Description
Computes the matrix of distances between all pairs of points in a point pattern.
Usage
## S3 method for class 'ppp'
pairdist(X, ..., periodic=FALSE, method="C", squared=FALSE)
Arguments
X
...
periodic
method
squared

A point pattern (object of class "ppp").
Ignored.
Logical. Specifies whether to apply a periodic edge correction.
String specifying which method of calculation to use. Values are "C" and "interpreted".
Usually not specified.
Logical. If squared=TRUE, the squared distances are returned instead (this computation is faster).

Details
This is a method for the generic function pairdist.
Given a point pattern X (an object of class "ppp"), this function computes the Euclidean distances
between all pairs of points in X, and returns the matrix of distances.
Alternatively if periodic=TRUE and the window containing X is a rectangle, then the distances will
be computed in the ‘periodic’ sense (also known as ‘torus’ distance): opposite edges of the rectangle
are regarded as equivalent. This is meaningless if the window is not a rectangle.
If squared=TRUE then the squared Euclidean distances d2 are returned, instead of the Euclidean
distances d. The squared distances are faster to calculate, and are sufficient for many purposes
(such as finding the nearest neighbour of a point).
The argument method is not normally used. It is retained only for checking the validity of the
software. If method = "interpreted" then the distances are computed using interpreted R code
only. If method="C" (the default) then C code is used. The C code is somewhat faster.

900

pairdist.ppx

Value
A square matrix whose [i,j] entry is the distance between the points numbered i and j.
Author(s)
Pavel Grabarnik  and Adrian Baddeley 
See Also
pairdist, pairdist.default, pairdist.psp, crossdist, nndist, Kest
Examples
data(cells)
d <- pairdist(cells)
d <- pairdist(cells, periodic=TRUE)
d <- pairdist(cells, squared=TRUE)

pairdist.ppx

Pairwise Distances in Any Dimensions

Description
Computes the matrix of distances between all pairs of points in a multi-dimensional point pattern.
Usage
## S3 method for class 'ppx'
pairdist(X, ...)
Arguments
X

A point pattern (object of class "ppx").

...

Arguments passed to coords.ppx to determine which coordinates should be
used.

Details
This is a method for the generic function pairdist.
Given a multi-dimensional point pattern X (an object of class "ppx"), this function computes the
Euclidean distances between all pairs of points in X, and returns the matrix of distances.
By default, both spatial and temporal coordinates are extracted. To obtain the spatial distance
between points in a space-time point pattern, set temporal=FALSE.
Value
A square matrix whose [i,j] entry is the distance between the points numbered i and j.
Author(s)
Adrian Baddeley 

pairdist.psp

901

See Also
pairdist, crossdist, nndist
Examples
df <- data.frame(x=runif(4),y=runif(4),z=runif(4),w=runif(4))
X <- ppx(data=df)
pairdist(X)

pairdist.psp

Pairwise distances between line segments

Description
Computes the matrix of distances between all pairs of line segments in a line segment pattern.
Usage
## S3 method for class 'psp'
pairdist(X, ..., method="C", type="Hausdorff")
Arguments
X

A line segment pattern (object of class "psp").

...

Ignored.

method

String specifying which method of calculation to use. Values are "C" and "interpreted".
Usually not specified.

type

Type of distance to be computed. Options are "Hausdorff" and "separation".
Partial matching is used.

Details
This function computes the distance between each pair of line segments in X, and returns the matrix
of distances.
This is a method for the generic function pairdist for the class "psp".
The distances between line segments are measured in one of two ways:
• if type="Hausdorff", distances are computed in the Hausdorff metric. The Hausdorff distance between two line segments is the maximum distance from any point on one of the segments to the nearest point on the other segment.
• if type="separation", distances are computed as the minimum distance from a point on one
line segment to a point on the other line segment. For example, line segments which cross
over each other have separation zero.
The argument method is not normally used. It is retained only for checking the validity of the
software. If method = "interpreted" then the distances are computed using interpreted R code
only. If method="C" (the default) then compiled C code is used, which is somewhat faster.
Value
A square matrix whose [i,j] entry is the distance between the line segments numbered i and j.

902

pairorient

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
crossdist, nndist, pairdist.ppp
Examples
L <- psp(runif(10), runif(10), runif(10), runif(10), owin())
D <- pairdist(L)
S <- pairdist(L, type="sep")

pairorient

Point Pair Orientation Distribution

Description
Computes the distribution of the orientation of vectors joining pairs of points at a particular range
of distances.
Usage
pairorient(X, r1, r2, ..., cumulative=FALSE,
correction, ratio = FALSE,
unit=c("degree", "radian"), domain=NULL)
Arguments
X

Point pattern (object of class "ppp").

r1,r2

Minimum and maximum values of distance to be considered.

...

Arguments passed to circdensity to control the kernel smoothing, if cumulative=FALSE.

cumulative

Logical value specifying whether to estimate the probability density (cumulative=FALSE,
the default) or the cumulative distribution function (cumulative=TRUE).

correction

Character vector specifying edge correction or corrections. Options are "none",
"isotropic", "translate", "good" and "best". Alternatively correction="all"
selects all options.

ratio

Logical. If TRUE, the numerator and denominator of each edge-corrected estimate will also be saved, for use in analysing replicated point patterns.

unit

Unit in which the angles should be expressed. Either "degree" or "radian".

domain

Optional window. The first point xi of each pair of points will be constrained to
lie in domain.

pairorient

903

Details
This algorithm considers all pairs of points in the pattern X that lie more than r1 and less than r2
units apart. The direction of the arrow joining the points is measured, as an angle in degrees or
radians, anticlockwise from the x axis.
If cumulative=FALSE (the default), a kernel estimate of the probability density of the orientations
is calculated using circdensity.
If cumulative=TRUE, then the cumulative distribution function of these directions is calculated.
This is the function Or1,r2 (φ) defined in Stoyan and Stoyan (1994), equation (14.53), page 271.
In either case the result can be plotted as a rose diagram by rose, or as a function plot by plot.fv.
The algorithm gives each observed direction a weight, determined by an edge correction, to adjust
for the fact that some interpoint distances are more likely to be observed than others. The choice of
edge correction or corrections is determined by the argument correction.
It is also possible to calculate an estimate of the probability density from the cumulative distribution
function, by numerical differentiation. Use deriv.fv with the argument Dperiodic=TRUE.

Value
A function value table (object of class "fv") containing the estimates of the probability density
or the cumulative distribution function of angles, in degrees (if unit="degree") or radians (if
unit="radian").

Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 

References
Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical
statistics. John Wiley and Sons.

See Also
Kest, Ksector, nnorient

Examples
rose(pairorient(redwood, 0.05, 0.15, sigma=8), col="grey")
plot(CDF <- pairorient(redwood, 0.05, 0.15, cumulative=TRUE))
plot(f <- deriv(CDF, spar=0.6, Dperiodic=TRUE))

904

PairPiece

PairPiece

The Piecewise Constant Pairwise Interaction Point Process Model

Description
Creates an instance of a pairwise interaction point process model with piecewise constant potential
function. The model can then be fitted to point pattern data.
Usage
PairPiece(r)
Arguments
r

vector of jump points for the potential function

Details
A pairwise interaction point process in a bounded region is a stochastic point process with probability density of the form
Y
Y
f (x1 , . . . , xn ) = α
b(xi )
h(xi , xj )
i

i rmax where rmax is the maximum
value in r. Thus the model has as many regular parameters (see ppm) as there are entries in r. The
i-th regular parameter θi is the logarithm of the value of the interaction function H on the interval
[ri−1 , ri ).
If r is a single number, this model is similar to the Strauss process, see Strauss. The difference
is that in PairPiece the interaction function is continuous on the right, while in Strauss it is
continuous on the left.
The analogue of this model for multitype point processes has not yet been implemented.
Value
An object of class "interact" describing the interpoint interaction structure of a point process.
The process is a pairwise interaction process, whose interaction potential is piecewise constant,
with jumps at the distances given in the vector r.

pairs.im

905

Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Takacs, R. (1986) Estimator for the pair potential of a Gibbsian point process. Statistics 17, 429–
433.
See Also
ppm, pairwise.family, ppm.object, Strauss rmh.ppm
Examples
PairPiece(c(0.1,0.2))
# prints a sensible description of itself
data(cells)
## Not run:
ppm(cells, ~1, PairPiece(r = c(0.05, 0.1, 0.2)))
# fit a stationary piecewise constant pairwise interaction process
## End(Not run)
ppm(cells, ~polynom(x,y,3), PairPiece(c(0.05, 0.1)))
# nonstationary process with log-cubic polynomial trend

pairs.im

Scatterplot Matrix for Pixel Images

Description
Produces a scatterplot matrix of the pixel values in two or more pixel images.
Usage
## S3 method for class 'im'
pairs(..., plot=TRUE)
Arguments
...

Any number of arguments, each of which is either a pixel image (object of class
"im") or a named argument to be passed to pairs.default.

plot

Logical. If TRUE, the scatterplot matrix is plotted.

906

pairs.im

Details
This is a method for the generic function pairs for the class of pixel images.
It produces a square array of plot panels, in which each panel shows a scatterplot of the pixel values
of one image against the corresponding pixel values of another image.
At least two of the arguments ... should be pixel images (objects of class "im"). Their spatial
domains must overlap, but need not have the same pixel dimensions.
First the pixel image domains are intersected, and converted to a common pixel resolution. Then
the corresponding pixel values of each image are extracted. Then pairs.default is called to plot
the scatterplot matrix.
Any arguments in ... which are not pixel images will be passed to pairs.default to control the
plot.
Value
Invisible. A data.frame containing the corresponding pixel values for each image. The return
value also belongs to the class plotpairsim which has a plot method, so that it can be re-plotted.
Image or Contour Plots
Since the scatterplots may show very dense concentrations of points, it may be useful to set panel=panel.image
or panel=panel.contour to draw a colour image or contour plot of the kernel-smoothed density
of the scatterplot in each panel. The argument panel is passed to pairs.default. See the help for
panel.image and panel.contour.
Low Level Control of Graphics
To control the appearance of the individual scatterplot panels, see pairs.default, points or par.
To control the plotting symbol for the points in the scatterplot, use the arguments pch, col, bg as
described under points (because the default panel plotter is the function points). To suppress the
tick marks on the plot axes, type par(xaxt="n", yaxt="n") before calling pairs.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
pairs, pairs.default, panel.contour, panel.image, plot.im, im, par
Examples
X <- density(rpoispp(30))
Y <- density(rpoispp(40))
Z <- density(rpoispp(30))
pairs(X,Y,Z)

pairs.linim

pairs.linim

907

Scatterplot Matrix for Pixel Images on a Linear Network

Description
Produces a scatterplot matrix of the pixel values in two or more pixel images on a linear network.
Usage
## S3 method for class 'linim'
pairs(..., plot=TRUE, eps=NULL)
Arguments
...

Any number of arguments, each of which is either a pixel image on a linear
network (object of class "linim"), a pixel image (object of class "im"), or a
named argument to be passed to pairs.default.

plot

Logical. If TRUE, the scatterplot matrix is plotted.

eps

Optional. Spacing between sample points on the network. A positive number.

Details
This is a method for the generic function pairs for the class of pixel images on a linear network.
It produces a square array of plot panels, in which each panel shows a scatterplot of the pixel values
of one image against the corresponding pixel values of another image.
At least two of the arguments ... should be a pixel image on a linear network (object of class
"linim"). They should be defined on the same linear network, but may have different pixel resolutions.
First the pixel values of each image are extracted at a set of sample points equally-spaced across the
network. Then pairs.default is called to plot the scatterplot matrix.
Any arguments in ... which are not pixel images will be passed to pairs.default to control the
plot.
Value
Invisible. A data.frame containing the corresponding pixel values for each image. The return
value also belongs to the class plotpairsim which has a plot method, so that it can be re-plotted.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
pairs.default, pairs.im

908

pairsat.family

Examples
fit <- lppm(chicago ~ marks * (x+y))
lam <- predict(fit)
do.call(pairs, lam)

pairsat.family

Saturated Pairwise Interaction Point Process Family

Description
An object describing the Saturated Pairwise Interaction family of point process models
Details
Advanced Use Only!
This structure would not normally be touched by the user. It describes the “saturated pairwise
interaction” family of point process models.
If you need to create a specific interaction model for use in spatial pattern analysis, use the function Saturated() or the two existing implementations of models in this family, Geyer() and
SatPiece().
Geyer (1999) introduced the “saturation process”, a modification of the Strauss process in which
the total contribution to the potential from each point (from its pairwise interaction with all other
points) is trimmed to a maximum value c. This model is implemented in the function Geyer().
The present class pairsat.family is the extension of this saturation idea to all pairwise interactions. Note that the resulting models are no longer pairwise interaction processes - they have
interactions of infinite order.
pairsat.family is an object of class "isf" containing a function pairwise$eval for evaluating the sufficient statistics of any saturated pairwise interaction point process model in which the
original pair potentials take an exponential family form.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Geyer, C.J. (1999) Likelihood Inference for Spatial Point Processes. Chapter 3 in O.E. BarndorffNielsen, W.S. Kendall and M.N.M. Van Lieshout (eds) Stochastic Geometry: Likelihood and Computation, Chapman and Hall / CRC, Monographs on Statistics and Applied Probability, number 80.
Pages 79–140.
See Also
Geyer to create the Geyer saturation process.
SatPiece to create a saturated process with piecewise constant pair potential.
Saturated to create a more general saturation model.
Other families: inforder.family, ord.family, pairwise.family.

Pairwise

Pairwise

909

Generic Pairwise Interaction model

Description
Creates an instance of a pairwise interaction point process model which can then be fitted to point
pattern data.
Usage
Pairwise(pot, name, par, parnames, printfun)
Arguments
pot

An R language function giving the user-supplied pairwise interaction potential.

name

Character string.

par

List of numerical values for irregular parameters

parnames

Vector of names of irregular parameters

printfun

Do not specify this argument: for internal use only.

Details
This code constructs a member of the pairwise interaction family pairwise.family with arbitrary
pairwise interaction potential given by the user.
Each pair of points in the point pattern contributes a factor h(d) to the probability density, where d
is the distance between the two points. The factor term h(d) is
h(d) = exp(−θpot(d))
provided pot(d) is finite, where θ is the coefficient vector in the model.
The function pot must take as its first argument a matrix of interpoint distances, and evaluate the
potential for each of these distances. The result must be either a matrix with the same dimensions as
its input, or an array with its first two dimensions the same as its input (the latter case corresponds
to a vector-valued potential).
If irregular parameters are present, then the second argument to pot should be a vector of the same
type as par giving those parameter values.
The values returned by pot may be finite numeric values, or -Inf indicating a hard core (that is,
the corresponding interpoint distance is forbidden). We define h(d) = 0 if pot(d) = −∞. Thus, a
potential value of minus infinity is always interpreted as corresponding to h(d) = 0, regardless of
the sign and magnitude of θ.
Value
An object of class "interact" describing the interpoint interaction structure of a point process.
Author(s)
Adrian Baddeley 
and Rolf Turner 

910

pairwise.family

See Also
ppm, pairwise.family, ppm.object
Examples
#This is the same as StraussHard(r=0.7,h=0.05)
strpot <- function(d,par) {
r <- par$r
h <- par$h
value <- (d <= r)
value[d < h] <- -Inf
value
}
mySH <- Pairwise(strpot, "StraussHard process", list(r=0.7,h=0.05),
c("interaction distance r", "hard core distance h"))
data(cells)
ppm(cells, ~ 1, mySH, correction="isotropic")
# Fiksel (1984) double exponential interaction
# see Stoyan, Kendall, Mecke 1987 p 161
fikspot <- function(d, par) {
r <- par$r
h <- par$h
zeta <- par$zeta
value <- exp(-zeta * d)
value[d < h] <- -Inf
value[d > r] <- 0
value
}
Fiksel <- Pairwise(fikspot, "Fiksel double exponential process",
list(r=3.5, h=1, zeta=1),
c("interaction distance r",
"hard core distance h",
"exponential coefficient zeta"))
data(spruces)
fit <- ppm(unmark(spruces), ~1, Fiksel, rbord=3.5)
fit
plot(fitin(fit), xlim=c(0,4))
coef(fit)
# corresponding values obtained by Fiksel (1984) were -1.9 and -6.0

pairwise.family

Pairwise Interaction Process Family

Description
An object describing the family of all pairwise interaction Gibbs point processes.
Details
Advanced Use Only!
This structure would not normally be touched by the user. It describes the pairwise interaction
family of point process models.

panel.contour

911

If you need to create a specific pairwise interaction model for use in modelling, use the function
Pairwise or one of the existing functions listed below.
Anyway, pairwise.family is an object of class "isf" containing a function pairwise.family$eval
for evaluating the sufficient statistics of any pairwise interaction point process model taking an exponential family form.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
Other families: pairsat.family, ord.family, inforder.family.
Pairwise interactions: Poisson, Pairwise, PairPiece, Fiksel, Hardcore, LennardJones, MultiHard,
MultiStrauss, MultiStraussHard, Strauss, StraussHard, Softcore.
Other interactions: AreaInter, Geyer, Saturated, Ord, OrdThresh.
panel.contour

Panel Plots using Colour Image or Contour Lines

Description
These functions can be passed to pairs or coplot to determine what kind of plotting is done in
each panel of a multi-panel graphical display.
Usage
panel.contour(x, y, ..., sigma = NULL)
panel.image(x, y, ..., sigma = NULL)
panel.histogram(x, ...)
Arguments
x,y
...
sigma

Coordinates of points in a scatterplot.
Extra graphics arguments, passed to contour.im, plot.im or rect, respectively, to control the appearance of the panel.
Bandwidth of kernel smoother, on a scale where x and y range between 0 and 1.

Details
These functions can serve as one of the arguments panel, lower.panel, upper.panel, diag.panel
passed to graphics commands like pairs or coplot, to determine what kind of plotting is done in
each panel of a multi-panel graphical display. In particular they work with pairs.im.
The functions panel.contour and panel.contour are suitable for the off-diagonal plots which
involve two datasets x and y. They first rescale x and y to the unit square, then apply kernel
smoothing with bandwidth sigma using density.ppp. Then panel.contour draws a contour plot
while panel.image draws a colour image.
The function panel.histogram is suitable for the diagonal plots which involve a single dataset x.
It displays a histogram of the data.

912

parameters

Value
Null.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
pairs.im, pairs.default, panel.smooth
Examples
with(bei.extra,
pairs(grad, elev,
panel
= panel.contour,
diag.panel = panel.histogram))

parameters

Extract Model Parameters in Understandable Form

Description
Given a fitted model of some kind, this function extracts all the parameters needed to specify the
model, and returns them as a list.
Usage
parameters(model, ...)
## S3 method for class 'dppm'
parameters(model, ...)
## S3 method for class 'kppm'
parameters(model, ...)
## S3 method for class 'ppm'
parameters(model, ...)
## S3 method for class 'profilepl'
parameters(model, ...)
## S3 method for class 'fii'
parameters(model, ...)
## S3 method for class 'interact'
parameters(model, ...)

parres

913

Arguments
model

A fitted model of some kind.

...

Arguments passed to methods.

Details
The argument model should be a fitted model of some kind. This function extracts all the parameters
that would be needed to specify the model, and returns them as a list.
The function parameters is generic, with methods for class "ppm", "kppm", "dppm" and "profilepl"
and other classes.
Value
A named list, whose format depends on the fitted model.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
coef
Examples
fit1 <- ppm(cells ~ x, Strauss(0.1))
parameters(fit1)
fit2 <- kppm(redwood ~ x, "Thomas")
parameters(fit2)

parres

Partial Residuals for Point Process Model

Description
Computes the smoothed partial residuals, a diagnostic for transformation of a covariate in a Poisson
point process model.
Usage
parres(model, covariate, ...,
smooth.effect=FALSE, subregion=NULL,
bw = "nrd0", adjust=1, from = NULL, to = NULL, n = 512,
bw.input = c("points", "quad"), bw.restrict=FALSE, covname)

914

parres

Arguments
model

Fitted point process model (object of class "ppm").

covariate

The covariate of interest. Either a character string matching the name of one of
the canonical covariates in the model, or one of the names "x" or "y" referring
to the Cartesian coordinates, or one of the names of the covariates given when
model was fitted, or a pixel image (object of class "im") or function(x,y)
supplying the values of a covariate at any location.

smooth.effect

Logical. Determines the choice of algorithm. See Details.

subregion

Optional. A window (object of class "owin") specifying a subset of the spatial
domain of the data. The calculation will be confined to the data in this subregion.

bw

Smoothing bandwidth or bandwidth rule (passed to density.default).

adjust

Smoothing bandwidth adjustment factor (passed to density.default).

n, from, to

Arguments passed to density.default to control the number and range of values at which the function will be estimated.

...

Additional arguments passed to density.default.

bw.input

Character string specifying the input data used for automatic bandwidth selection.

bw.restrict

Logical value, specifying whether bandwidth selection is performed using data
from the entire spatial domain or from the subregion.

covname

Optional. Character string to use as the name of the covariate.

Details
This command computes the smoothed partial residual diagnostic (Baddeley, Chang, Song and
Turner, 2012) for the transformation of a covariate in a Poisson point process model.
The argument model must be a fitted Poisson point process model.
The diagnostic works in two different ways:
Canonical covariate: The argument covariate may be a character string which is the name of
one of the canonical covariates in the model. The canonical covariates are the functions Zj
that appear in the expression for the Poisson point process intensity
λ(u) = exp(β1 Z1 (u) + . . . + βp Zp (u))
at spatial location u. Type names(coef(model)) to see the names of the canonical covariates
in model. If the selected covariate is Zj , then the diagnostic plot concerns the model term
βj Zj (u). The plot shows a smooth estimate of a function h(z) that should replace this linear
term, that is, βj Zj (u) should be replaced by h(Zj (u)). The linear function is also plotted as
a dotted line.
New covariate: If the argument covariate is a pixel image (object of class "im") or a function(x,y),
it is assumed to provide the values of a covariate that is not present in the model. Alternatively
covariate can be the name of a covariate that was supplied when the model was fitted (i.e.
in the call to ppm) but which does not feature in the model formula. In either case we speak
of a new covariate Z(u). If the fitted model intensity is λ(u) then we consider modifying this
to λ(u) exp(h(Z(u))) where h(z) is some function. The diagnostic plot shows an estimate
of h(z). Warning: in this case the diagnostic is not theoretically justified. This option is
provided for research purposes.

parres

915

Alternatively covariate can be one of the character strings "x" or "y" signifying the Cartesian
coordinates. The behaviour here depends on whether the coordinate was one of the canonical covariates in the model.
If there is more than one canonical covariate in the model that depends on the specified covariate,
then the covariate effect is computed using all these canonical covariates. For example in a logquadratic model which includes the terms x and I(x^2), the quadratic effect involving both these
terms will be computed.
There are two choices for the algorithm. If smooth.effect=TRUE, the fitted covariate effect (according to model) is added to the point process residuals, then smoothing is applied to these values.
If smooth.effect=FALSE, the point process residuals are smoothed first, and then the fitted covariate effect is added to the result.
The smoothing bandwidth is controlled by the arguments bw, adjust, bw.input and bw.restrict.
If bw is a numeric value, then the bandwidth is taken to be adjust * bw. If bw is a string representing
a bandwidth selection rule (recognised by density.default) then the bandwidth is selected by this
rule.
The data used for automatic bandwidth selection are specified by bw.input and bw.restrict. If
bw.input="points" (the default) then bandwidth selection is based on the covariate values at the
points of the original point pattern dataset to which the model was fitted. If bw.input="quad" then
bandwidth selection is based on the covariate values at every quadrature point used to fit the model.
If bw.restrict=TRUE then the bandwidth selection is performed using only data from inside the
subregion.
Value
A function value table (object of class "fv") containing the values of the smoothed partial residual,
the estimated variance, and the fitted effect of the covariate. Also belongs to the class "parres"
which has methods for print and plot.
Slow computation
In a large dataset, computation can be very slow if the default settings are used, because the smoothing bandwidth is selected automatically. To avoid this, specify a numerical value for the bandwidth
bw. One strategy is to use a coarser subset of the data to select bw automatically. The selected
bandwidth can be read off the print output for parres.
Author(s)
Adrian Baddeley 
, Rolf Turner  , Ya-Mei Chang and Yong Song.
References
Baddeley, A., Chang, Y.-M., Song, Y. and Turner, R. (2013) Residual diagnostics for covariate
effects in spatial point process models. Journal of Computational and Graphical Statistics, 22,
886–905.
See Also
addvar, rhohat, rho2hat

916

pcf

Examples
X <- rpoispp(function(x,y){exp(3+x+2*x^2)})
model <- ppm(X, ~x+y)
tra <- parres(model, "x")
plot(tra)
plot(parres(model, "x", subregion=square(0.5)))
model2 <- ppm(X, ~x+I(x^2)+y)
plot(parres(model2, "x"))
Z <- setcov(owin())
plot(parres(model2, Z))

pcf

Pair Correlation Function

Description
Estimate the pair correlation function.
Usage
pcf(X, ...)
Arguments
X

Either the observed data point pattern, or an estimate of its K function, or an
array of multitype K functions (see Details).

...

Other arguments passed to the appropriate method.

Details
The pair correlation function of a stationary point process is
g(r) =

K 0 (r)
2πr

where K 0 (r) is the derivative of K(r), the reduced second moment function (aka “Ripley’s K
function”) of the point process. See Kest for information about K(r). For a stationary Poisson
process, the pair correlation function is identically equal to 1. Values g(r) < 1 suggest inhibition
between points; values greater than 1 suggest clustering.
We also apply the same definition to other variants of the classical K function, such as the multitype
K functions (see Kcross, Kdot) and the inhomogeneous K function (see Kinhom). For all these
variants, the benchmark value of K(r) = πr2 corresponds to g(r) = 1.
This routine computes an estimate of g(r) either directly from a point pattern, or indirectly from an
estimate of K(r) or one of its variants.
This function is generic, with methods for the classes "ppp", "fv" and "fasp".
If X is a point pattern (object of class "ppp") then the pair correlation function is estimated using a
traditional kernel smoothing method (Stoyan and Stoyan, 1994). See pcf.ppp for details.
If X is a function value table (object of class "fv"), then it is assumed to contain estimates of the K
function or one of its variants (typically obtained from Kest or Kinhom). This routine computes an
estimate of g(r) using smoothing splines to approximate the derivative. See pcf.fv for details.

pcf.fasp

917

If X is a function value array (object of class "fasp"), then it is assumed to contain estimates of
several K functions (typically obtained from Kmulti or alltypes). This routine computes an
estimate of g(r) for each cell in the array, using smoothing splines to approximate the derivatives.
See pcf.fasp for details.
Value
Either a function value table (object of class "fv", see fv.object) representing a pair correlation
function, or a function array (object of class "fasp", see fasp.object) representing an array of
pair correlation functions.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
References
Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical
statistics. John Wiley and Sons.
See Also
pcf.ppp, pcf.fv, pcf.fasp, Kest, Kinhom, Kcross, Kdot, Kmulti, alltypes
Examples
# ppp object
X <- simdat
p <- pcf(X)
plot(p)
# fv object
K <- Kest(X)
p2 <- pcf(K, spar=0.8, method="b")
plot(p2)
# multitype pattern; fasp object
amaK <- alltypes(amacrine, "K")
amap <- pcf(amaK, spar=1, method="b")
plot(amap)

pcf.fasp

Pair Correlation Function obtained from array of K functions

Description
Estimates the (bivariate) pair correlation functions of a point pattern, given an array of (bivariate) K
functions.

918

pcf.fasp

Usage
## S3 method for class 'fasp'
pcf(X, ..., method="c")
Arguments
X
...
method

An array of multitype K functions (object of class "fasp").
Arguments controlling the smoothing spline function smooth.spline.
Letter "a", "b", "c" or "d" indicating the method for deriving the pair correlation function from the K function.

Details
The pair correlation function of a stationary point process is
K 0 (r)
2πr
0
where K (r) is the derivative of K(r), the reduced second moment function (aka “Ripley’s K
function”) of the point process. See Kest for information about K(r). For a stationary Poisson
process, the pair correlation function is identically equal to 1. Values g(r) < 1 suggest inhibition
between points; values greater than 1 suggest clustering.
g(r) =

We also apply the same definition to other variants of the classical K function, such as the multitype
K functions (see Kcross, Kdot) and the inhomogeneous K function (see Kinhom). For all these
variants, the benchmark value of K(r) = πr2 corresponds to g(r) = 1.
This routine computes an estimate of g(r) from an array of estimates of K(r) or its variants, using
smoothing splines to approximate the derivatives. It is a method for the generic function pcf.
The argument X should be a function array (object of class "fasp", see fasp.object) containing
several estimates of K functions. This should have been obtained from alltypes with the argument
fun="K".
The smoothing spline operations are performed by smooth.spline and predict.smooth.spline
from the modreg library. Four numerical methods are available:
• "a" apply smoothing to K(r), estimate its derivative, and plug in to the formula above;
• "b" apply smoothing to Y (r) = K(r)
2πr constraining Y (0) = 0, estimate the derivative of Y ,
and solve;
• "c" apply smoothing to Z(r) = K(r)
πr 2 constraining Z(0) = 1, estimate its derivative, and
solve.
p
• "d" apply smoothing to V (r) = K(r), estimate its derivative, and solve.
Method "c" seems to be the best at suppressing variability for small values of r. However it effectively constrains g(0) = 1. If the point pattern seems to have inhibition at small distances, you may
wish to experiment with method "b" which effectively constrains g(0) = 0. Method "a" seems
comparatively unreliable.
Useful arguments to control the splines include the smoothing tradeoff parameter spar and the
degrees of freedom df. See smooth.spline for details.
Value
A function array (object of class "fasp", see fasp.object) representing an array of pair correlation
functions. This can be thought of as a matrix Y each of whose entries Y[i,j] is a function value
table (class "fv") representing the pair correlation function between points of type i and points of
type j.

pcf.fv

919

Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Stoyan, D, Kendall, W.S. and Mecke, J. (1995) Stochastic geometry and its applications. 2nd
edition. Springer Verlag.
Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical
statistics. John Wiley and Sons.
See Also
Kest, Kinhom, Kcross, Kdot, Kmulti, alltypes, smooth.spline, predict.smooth.spline
Examples
# multitype point pattern
KK <- alltypes(amacrine, "K")
p <- pcf.fasp(KK, spar=0.5, method="b")
plot(p)
# strong inhibition between points of the same type

pcf.fv

Pair Correlation Function obtained from K Function

Description
Estimates the pair correlation function of a point pattern, given an estimate of the K function.
Usage
## S3 method for class 'fv'
pcf(X, ..., method="c")
Arguments
X

An estimate of the K function or one of its variants. An object of class "fv".

...

Arguments controlling the smoothing spline function smooth.spline.

method

Letter "a", "b", "c" or "d" indicating the method for deriving the pair correlation function from the K function.

Details
The pair correlation function of a stationary point process is
g(r) =

K 0 (r)
2πr

where K 0 (r) is the derivative of K(r), the reduced second moment function (aka “Ripley’s K
function”) of the point process. See Kest for information about K(r). For a stationary Poisson

920

pcf.fv
process, the pair correlation function is identically equal to 1. Values g(r) < 1 suggest inhibition
between points; values greater than 1 suggest clustering.
We also apply the same definition to other variants of the classical K function, such as the multitype
K functions (see Kcross, Kdot) and the inhomogeneous K function (see Kinhom). For all these
variants, the benchmark value of K(r) = πr2 corresponds to g(r) = 1.
This routine computes an estimate of g(r) from an estimate of K(r) or its variants, using smoothing
splines to approximate the derivative. It is a method for the generic function pcf for the class "fv".
The argument X should be an estimated K function, given as a function value table (object of
class "fv", see fv.object). This object should be the value returned by Kest, Kcross, Kmulti or
Kinhom.
The smoothing spline operations are performed by smooth.spline and predict.smooth.spline
from the modreg library. Four numerical methods are available:
• "a" apply smoothing to K(r), estimate its derivative, and plug in to the formula above;
• "b" apply smoothing to Y (r) = K(r)
2πr constraining Y (0) = 0, estimate the derivative of Y ,
and solve;
• "c" apply smoothing to Z(r) = K(r)
πr 2 constraining Z(0) = 1, estimate its derivative, and
solve.
p
• "d" apply smoothing to V (r) = K(r), estimate its derivative, and solve.
Method "c" seems to be the best at suppressing variability for small values of r. However it effectively constrains g(0) = 1. If the point pattern seems to have inhibition at small distances, you may
wish to experiment with method "b" which effectively constrains g(0) = 0. Method "a" seems
comparatively unreliable.
Useful arguments to control the splines include the smoothing tradeoff parameter spar and the
degrees of freedom df. See smooth.spline for details.

Value
A function value table (object of class "fv", see fv.object) representing a pair correlation function.
Essentially a data frame containing (at least) the variables
r
pcf

the vector of values of the argument r at which the pair correlation function g(r)
has been estimated
vector of values of g(r)

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
References
Stoyan, D, Kendall, W.S. and Mecke, J. (1995) Stochastic geometry and its applications. 2nd
edition. Springer Verlag.
Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical
statistics. John Wiley and Sons.
See Also
pcf, pcf.ppp, Kest, Kinhom, Kcross, Kdot, Kmulti, alltypes, smooth.spline, predict.smooth.spline

pcf.ppp

921

Examples
# univariate point pattern
X <- simdat
K <- Kest(X)
p <- pcf.fv(K, spar=0.5, method="b")
plot(p, main="pair correlation function for simdat")
# indicates inhibition at distances r < 0.3

pcf.ppp

Pair Correlation Function of Point Pattern

Description
Estimates the pair correlation function of a point pattern using kernel methods.
Usage
## S3 method for class 'ppp'
pcf(X, ..., r = NULL, kernel="epanechnikov", bw=NULL,
stoyan=0.15,
correction=c("translate", "Ripley"),
divisor = c("r", "d"),
var.approx = FALSE,
domain=NULL,
ratio=FALSE, close=NULL)
Arguments
X

A point pattern (object of class "ppp").

r

Vector of values for the argument r at which g(r) should be evaluated. There is
a sensible default.

kernel

Choice of smoothing kernel, passed to density.default.

bw

Bandwidth for smoothing kernel, passed to density.default. Either a single numeric value, or a character string specifying a bandwidth selection rule
recognised by density.default. If bw is missing or NULL, the default value is
computed using Stoyan’s rule of thumb: see Details.

...

Other arguments passed to the kernel density estimation function density.default.

stoyan

Coefficient for Stoyan’s bandwidth selection rule; see Details.

correction

Choice of edge correction.

divisor

Choice of divisor in the estimation formula: either "r" (the default) or "d". See
Details.

var.approx

Logical value indicating whether to compute an analytic approximation to the
variance of the estimated pair correlation.

domain

Optional. Calculations will be restricted to this subset of the window. See Details.

ratio

Logical. If TRUE, the numerator and denominator of each edge-corrected estimate will also be saved, for use in analysing replicated point patterns.

close

Advanced use only. Precomputed data. See section on Advanced Use.

922

pcf.ppp

Details
The pair correlation function g(r) is a summary of the dependence between points in a spatial point
process. The best intuitive interpretation is the following: the probability p(r) of finding two points
at locations x and y separated by a distance r is equal to
p(r) = λ2 g(r) dx dy
where λ is the intensity of the point process. For a completely random (uniform Poisson) process,
p(r) = λ2 dx dy so g(r) = 1. Formally, the pair correlation function of a stationary point process
is defined by
K 0 (r)
g(r) =
2πr
0
where K (r) is the derivative of K(r), the reduced second moment function (aka “Ripley’s K
function”) of the point process. See Kest for information about K(r).
For a stationary Poisson process, the pair correlation function is identically equal to 1. Values
g(r) < 1 suggest inhibition between points; values greater than 1 suggest clustering.
This routine computes an estimate of g(r) by kernel smoothing.
• If divisor="r" (the default), then the standard kernel estimator (Stoyan and Stoyan, 1994,
pages 284–285) is used. By default, the recommendations of Stoyan and Stoyan (1994) are
followed exactly.
• If divisor="d" then a modified estimator is used: the contribution from an interpoint distance
dij to the estimate of g(r) is divided by dij instead of dividing by r. This usually improves
the bias of the estimator when r is close to zero.
There is also a choice of spatial edge corrections (which are needed to avoid bias due to edge effects
associated with the boundary of the spatial window):
• If correction="translate" or correction="translation" then the translation correction
is used. For divisor="r" the translation-corrected estimate is given in equation (15.15), page
284 of Stoyan and Stoyan (1994).
• If correction="Ripley" then Ripley’s isotropic edge correction is used. For divisor="r"
the isotropic-corrected estimate is given in equation (15.18), page 285 of Stoyan and Stoyan
(1994).
• If correction=c("translate", "Ripley") then both estimates will be computed.
Alternatively correction="all" selects all options.
The choice of smoothing kernel is controlled by the argument kernel which is passed to density.default.
The default is the Epanechnikov kernel, recommended by Stoyan and Stoyan (1994, page 285).
The bandwidth of the smoothing kernel can be controlled by the argument bw. Its precise interpretation is explained in the documentation
for density.default. For the Epanechnikov kernel, the
√
argument bw is equivalent to h/ 5.
Stoyan and Stoyan (1994, page 285) recommend
using the Epanechnikov kernel with support
√
[−h, h] chosen by the rule of thumn h = c/ λ, where λ is the (estimated) intensity of the point
process, and c is a constant in the range from 0.1 to 0.2. See equation (15.16). If bw is missing or
NULL, then this rule of thumb will be applied. The argument stoyan determines the value of c. The
smoothing bandwidth that was used in the calculation is returned as an attribute of the final result.
The argument r is the vector of values for the distance r at which g(r) should be evaluated. There
is a sensible default. If it is specified, r must be a vector of increasing numbers starting from
r[1] = 0, and max(r) must not exceed half the diameter of the window.

pcf.ppp

923

If the argument domain is given, estimation will be restricted to this region. That is, the estimate of
g(r) will be based on pairs of points in which the first point lies inside domain and the second point
is unrestricted. The argument domain should be a window (object of class "owin") or something
acceptable to as.owin. It must be a subset of the window of the point pattern X.
To compute a confidence band for the true value of the pair correlation function, use lohboot.
If var.approx = TRUE, the variance of the estimate of the pair correlation will also be calculated
using an analytic approximation (Illian et al, 2008, page 234) which is valid for stationary point
processes which are not too clustered. This calculation is not yet implemented when the argument
domain is given.
Value
A function value table (object of class "fv"). Essentially a data frame containing the variables
r

the vector of values of the argument r at which the pair correlation function g(r)
has been estimated

theo

vector of values equal to 1, the theoretical value of g(r) for the Poisson process

trans

vector of values of g(r) estimated by translation correction

iso

vector of values of g(r) estimated by Ripley isotropic correction

v

vector of approximate values of the variance of the estimate of g(r)

as required.
If ratio=TRUE then the return value also has two attributes called "numerator" and "denominator"
which are "fv" objects containing the numerators and denominators of each estimate of g(r).
The return value also has an attribute "bw" giving the smoothing bandwidth that was used.
Advanced Use
To perform the same computation using several different bandwidths bw, it is efficient to use the
argument close. This should be the result of closepairs(X, rmax) for a suitably large value of
rmax, namely rmax >= max(r) + 3 * bw.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak  and Martin Hazelton.
References
Illian, J., Penttinen, A., Stoyan, H. and Stoyan, D. (2008) Statistical Analysis and Modelling of
Spatial Point Patterns. Wiley.
Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical
statistics. John Wiley and Sons.
See Also
Kest, pcf, density.default, bw.stoyan, bw.pcf, lohboot.

924

pcf3est

Examples
X <- simdat
p <- pcf(X)
plot(p, main="pair correlation function for X")
# indicates inhibition at distances r < 0.3
pd <- pcf(X, divisor="d")
# compare estimates
plot(p, cbind(iso, theo) ~ r, col=c("blue", "red"),
ylim.covers=0, main="", lwd=c(2,1), lty=c(1,3), legend=FALSE)
plot(pd, iso ~ r, col="green", lwd=2, add=TRUE)
legend("center", col=c("blue", "green"), lty=1, lwd=2,
legend=c("divisor=r","divisor=d"))
# calculate approximate variance and show POINTWISE confidence bands
pv <- pcf(X, var.approx=TRUE)
plot(pv, cbind(iso, iso+2*sqrt(v), iso-2*sqrt(v)) ~ r)

pcf3est

Pair Correlation Function of a Three-Dimensional Point Pattern

Description
Estimates the pair correlation function from a three-dimensional point pattern.
Usage
pcf3est(X, ..., rmax = NULL, nrval = 128, correction = c("translation",
"isotropic"), delta=NULL, adjust=1, biascorrect=TRUE)
Arguments
X

Three-dimensional point pattern (object of class "pp3").

...

Ignored.

rmax

Optional. Maximum value of argument r for which g3 (r) will be estimated.

nrval

Optional. Number of values of r for which g3 (r) will be estimated.

correction

Optional. Character vector specifying the edge correction(s) to be applied. See
Details.

delta

Optional. Half-width of the Epanechnikov smoothing kernel.

adjust

Optional. Adjustment factor for the default value of delta.

biascorrect

Logical value. Whether to correct for underestimation due to truncation of the
kernel near r = 0.

pcf3est

925

Details
For a stationary point process Φ in three-dimensional space, the pair correlation function is
g3 (r) =

K30 (r)
4πr2

where K30 is the derivative of the three-dimensional K-function (see K3est).
The three-dimensional point pattern X is assumed to be a partial realisation of a stationary point process Φ. The distance between each pair of distinct points is computed. Kernel smoothing is applied
to these distance values (weighted by an edge correction factor) and the result is renormalised to
give the estimate of g3 (r).
The available edge corrections are:
"translation": the Ohser translation correction estimator (Ohser, 1983; Baddeley et al, 1993)
"isotropic": the three-dimensional counterpart of Ripley’s isotropic edge correction (Ripley,
1977; Baddeley et al, 1993).
Kernel smoothing is performed using the Epanechnikov kernel with half-width delta. If delta is
missing, the default is to use the rule-of-thumb δ = 0.26/λ1/3 where λ = n/v is the estimated
intensity, computed from the number n of data points and the volume v of the enclosing box. This
default value of delta is multiplied by the factor adjust.
The smoothing estimate of the pair correlation g3 (r) is typically an underestimate when r is small,
due to truncation of the kernel at r = 0. If biascorrect=TRUE, the smoothed estimate is approximately adjusted for this bias. This is advisable whenever the dataset contains a sufficiently large
number of points.
Value
A function value table (object of class "fv") that can be plotted, printed or coerced to a data frame
containing the function values.
Additionally the value of delta is returned as an attribute of this object.
Author(s)
Adrian Baddeley 
and Rana Moyeed.
References
Baddeley, A.J, Moyeed, R.A., Howard, C.V. and Boyde, A. (1993) Analysis of a three-dimensional
point pattern with replication. Applied Statistics 42, 641–668.
Ohser, J. (1983) On estimators for the reduced second moment measure of point processes. Mathematische Operationsforschung und Statistik, series Statistics, 14, 63 – 71.
Ripley, B.D. (1977) Modelling spatial patterns (with discussion). Journal of the Royal Statistical
Society, Series B, 39, 172 – 212.
See Also
K3est, pcf

926

pcfcross

Examples
X <- rpoispp3(250)
Z <- pcf3est(X)
Zbias <- pcf3est(X, biascorrect=FALSE)
if(interactive()) {
opa <- par(mfrow=c(1,2))
plot(Z,
ylim.covers=c(0, 1.2))
plot(Zbias, ylim.covers=c(0, 1.2))
par(opa)
}
attr(Z, "delta")

pcfcross

Multitype pair correlation function (cross-type)

Description
Calculates an estimate of the cross-type pair correlation function for a multitype point pattern.
Usage
pcfcross(X, i, j, ...,
r = NULL,
kernel = "epanechnikov", bw = NULL, stoyan = 0.15,
correction = c("isotropic", "Ripley", "translate"),
divisor = c("r", "d"))
Arguments
X

The observed point pattern, from which an estimate of the cross-type pair correlation function gij (r) will be computed. It must be a multitype point pattern (a
marked point pattern whose marks are a factor).

i

The type (mark value) of the points in X from which distances are measured.
A character string (or something that will be converted to a character string).
Defaults to the first level of marks(X).

j

The type (mark value) of the points in X to which distances are measured. A
character string (or something that will be converted to a character string). Defaults to the second level of marks(X).

...

Ignored.

r

Vector of values for the argument r at which g(r) should be evaluated. There is
a sensible default.

kernel

Choice of smoothing kernel, passed to density.default.

bw

Bandwidth for smoothing kernel, passed to density.default.

stoyan

Coefficient for default bandwidth rule; see Details.

correction

Choice of edge correction.

divisor

Choice of divisor in the estimation formula: either "r" (the default) or "d". See
Details.

pcfcross

927

Details
The cross-type pair correlation function is a generalisation of the pair correlation function pcf to
multitype point patterns.
For two locations x and y separated by a distance r, the probability p(r) of finding a point of type i
at location x and a point of type j at location y is
p(r) = λi λj gi,j (r) dx dy
where λi is the intensity of the points of type i. For a completely random Poisson marked point
process, p(r) = λi λj so gi,j (r) = 1. Indeed for any marked point pattern in which the points of
type i are independent of the points of type j, the theoretical value of the cross-type pair correlation
is gi,j (r) = 1.
For a stationary multitype point process, the cross-type pair correlation function between marks i
and j is formally defined as
0
Ki,j
(r)
gi,j (r) =
2πr
0
where Ki,j
is the derivative of the cross-type K function Ki,j (r). of the point process. See Kest
for information about K(r).
The command pcfcross computes a kernel estimate of the cross-type pair correlation function
between marks i and j.
• If divisor="r" (the default), then the multitype counterpart of the standard kernel estimator (Stoyan and Stoyan, 1994, pages 284–285) is used. By default, the recommendations of
Stoyan and Stoyan (1994) are followed exactly.
• If divisor="d" then a modified estimator is used: the contribution from an interpoint distance
dij to the estimate of g(r) is divided by dij instead of dividing by r. This usually improves
the bias of the estimator when r is close to zero.
There is also a choice of spatial edge corrections (which are needed to avoid bias due to edge effects
associated with the boundary of the spatial window): correction="translate" is the OhserStoyan translation correction, and correction="isotropic" or "Ripley" is Ripley’s isotropic
correction.
The choice of smoothing kernel is controlled by the argument kernel which is passed to density.
The default is the Epanechnikov kernel.
The bandwidth of the smoothing kernel can be controlled by the argument bw. Its precise interpretation is explained in the documentation for density.default.
For the Epanechnikov kernel with
√
support [−h, h], the argument bw is equivalent to h/ 5.
If bw is not specified, the default bandwidth is determined by Stoyan’s rule of√thumb (Stoyan and
Stoyan, 1994, page 285) applied to the points of type j. That is, h = c/ λ, where λ is the
(estimated) intensity of the point process of type j, and c is a constant in the range from 0.1 to 0.2.
The argument stoyan determines the value of c.
The companion function pcfdot computes the corresponding analogue of Kdot.
Value
An object of class "fv", see fv.object, which can be plotted directly using plot.fv.
Essentially a data frame containing columns
r

the vector of values of the argument r at which the function gi,j has been estimated

928

pcfcross.inhom
theo

the theoretical value gi,j (r) = 1 for independent marks.

together with columns named "border", "bord.modif", "iso" and/or "trans", according to the
selected edge corrections. These columns contain estimates of the function gi,j obtained by the
edge corrections named.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
Mark connection function markconnect.
Multitype pair correlation pcfdot, pcfmulti.
Pair correlation pcf,pcf.ppp.
Kcross
Examples
data(amacrine)
p <- pcfcross(amacrine, "off", "on")
p <- pcfcross(amacrine, "off", "on", stoyan=0.1)
plot(p)

pcfcross.inhom

Inhomogeneous Multitype Pair Correlation Function (Cross-Type)

Description
Estimates the inhomogeneous cross-type pair correlation function for a multitype point pattern.
Usage
pcfcross.inhom(X, i, j, lambdaI = NULL, lambdaJ = NULL, ...,
r = NULL, breaks = NULL,
kernel="epanechnikov", bw=NULL, stoyan=0.15,
correction = c("isotropic", "Ripley", "translate"),
sigma = NULL, varcov = NULL)
Arguments
X

The observed point pattern, from which an estimate of the inhomogeneous crosstype pair correlation function gij (r) will be computed. It must be a multitype
point pattern (a marked point pattern whose marks are a factor).

i

The type (mark value) of the points in X from which distances are measured.
A character string (or something that will be converted to a character string).
Defaults to the first level of marks(X).

j

The type (mark value) of the points in X to which distances are measured. A
character string (or something that will be converted to a character string). Defaults to the second level of marks(X).

pcfcross.inhom

929

lambdaI

Optional. Values of the estimated intensity function of the points of type i. Either a vector giving the intensity values at the points of type i, a pixel image (object of class "im") giving the intensity values at all locations, or a function(x,y)
which can be evaluated to give the intensity value at any location.

lambdaJ

Optional. Values of the estimated intensity function of the points of type j. A
numeric vector, pixel image or function(x,y).

r

Vector of values for the argument r at which gij (r) should be evaluated. There
is a sensible default.

breaks

This argument is for internal use only.

kernel

Choice of smoothing kernel, passed to density.default.

bw

Bandwidth for smoothing kernel, passed to density.default.

...

Other arguments passed to the kernel density estimation function density.default.

stoyan

Bandwidth coefficient; see Details.

correction

Choice of edge correction.

sigma,varcov

Optional arguments passed to density.ppp to control the smoothing bandwidth, when lambdaI or lambdaJ is estimated by kernel smoothing.

Details
The inhomogeneous cross-type pair correlation function gij (r) is a summary of the dependence
between two types of points in a multitype spatial point process that does not have a uniform density
of points.
The best intuitive interpretation is the following: the probability p(r) of finding two points, of types
i and j respectively, at locations x and y separated by a distance r is equal to
p(r) = λi (x)lambdaj (y)g(r) dx dy
where λi is the intensity function of the process of points of type i. For a multitype Poisson point
process, this probability is p(r) = λi (x)λj (y) so gij (r) = 1.
The command pcfcross.inhom estimates the inhomogeneous pair correlation using a modified
version of the algorithm in pcf.ppp.
If the arguments lambdaI and lambdaJ are missing or null, they are estimated from X by kernel
smoothing using a leave-one-out estimator.
Value
A function value table (object of class "fv"). Essentially a data frame containing the variables
r

the vector of values of the argument r at which the inhomogeneous cross-type
pair correlation function gij (r) has been estimated

theo

vector of values equal to 1, the theoretical value of gij (r) for the Poisson process

trans

vector of values of gij (r) estimated by translation correction

iso

vector of values of gij (r) estimated by Ripley isotropic correction

as required.
Author(s)
Adrian Baddeley 
and Rolf Turner 

930

pcfdot

See Also
pcf.ppp, pcfinhom, pcfcross, pcfdot.inhom
Examples
data(amacrine)
plot(pcfcross.inhom(amacrine, "on", "off", stoyan=0.1),
legendpos="bottom")

pcfdot

Multitype pair correlation function (i-to-any)

Description
Calculates an estimate of the multitype pair correlation function (from points of type i to points of
any type) for a multitype point pattern.
Usage
pcfdot(X, i, ..., r = NULL,
kernel = "epanechnikov", bw = NULL, stoyan = 0.15,
correction = c("isotropic", "Ripley", "translate"),
divisor = c("r", "d"))
Arguments
X

The observed point pattern, from which an estimate of the dot-type pair correlation function gi• (r) will be computed. It must be a multitype point pattern (a
marked point pattern whose marks are a factor).

i

The type (mark value) of the points in X from which distances are measured.
A character string (or something that will be converted to a character string).
Defaults to the first level of marks(X).

...

Ignored.

r

Vector of values for the argument r at which g(r) should be evaluated. There is
a sensible default.

kernel

Choice of smoothing kernel, passed to density.default.

bw

Bandwidth for smoothing kernel, passed to density.default.

stoyan

Coefficient for default bandwidth rule; see Details.

correction

Choice of edge correction.

divisor

Choice of divisor in the estimation formula: either "r" (the default) or "d". See
Details.

pcfdot

931

Details
This is a generalisation of the pair correlation function pcf to multitype point patterns.
For two locations x and y separated by a nonzero distance r, the probability p(r) of finding a point
of type i at location x and a point of any type at location y is
p(r) = λi λgi• (r) dx dy
where λ is the intensity of all points, and λi is the intensity of the points of type i. For a completely
random Poisson marked point process, p(r) = λi λ so gi• (r) = 1.
For a stationary multitype point process, the type-i-to-any-type pair correlation function between
marks i and j is formally defined as
0
Ki•
(r)
2πr
0
where Ki•
is the derivative of the type-i-to-any-type K function Ki• (r). of the point process. See
Kdot for information about Ki• (r).

gi• (r) =

The command pcfdot computes a kernel estimate of the multitype pair correlation function from
points of type i to points of any type.
• If divisor="r" (the default), then the multitype counterpart of the standard kernel estimator (Stoyan and Stoyan, 1994, pages 284–285) is used. By default, the recommendations of
Stoyan and Stoyan (1994) are followed exactly.
• If divisor="d" then a modified estimator is used: the contribution from an interpoint distance
dij to the estimate of g(r) is divided by dij instead of dividing by r. This usually improves
the bias of the estimator when r is close to zero.
There is also a choice of spatial edge corrections (which are needed to avoid bias due to edge effects
associated with the boundary of the spatial window): correction="translate" is the OhserStoyan translation correction, and correction="isotropic" or "Ripley" is Ripley’s isotropic
correction.
The choice of smoothing kernel is controlled by the argument kernel which is passed to density.
The default is the Epanechnikov kernel.
The bandwidth of the smoothing kernel can be controlled by the argument bw. Its precise interpretation is explained in the documentation for density.default.
For the Epanechnikov kernel with
√
support [−h, h], the argument bw is equivalent to h/ 5.
If bw is not specified, the default bandwidth
√ is determined by Stoyan’s rule of thumb (Stoyan and
Stoyan, 1994, page 285). That is, h = c/ λ, where λ is the (estimated) intensity of the unmarked
point process, and c is a constant in the range from 0.1 to 0.2. The argument stoyan determines the
value of c.
The companion function pcfcross computes the corresponding analogue of Kcross.
Value
An object of class "fv", see fv.object, which can be plotted directly using plot.fv.
Essentially a data frame containing columns
r
theo

the vector of values of the argument r at which the function gi• has been estimated
the theoretical value gi• (r) = 1 for independent marks.

together with columns named "border", "bord.modif", "iso" and/or "trans", according to the
selected edge corrections. These columns contain estimates of the function gi,j obtained by the
edge corrections named.

932

pcfdot.inhom

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
Mark connection function markconnect.
Multitype pair correlation pcfcross, pcfmulti.
Pair correlation pcf,pcf.ppp.
Kdot
Examples
data(amacrine)
p <- pcfdot(amacrine, "on")
p <- pcfdot(amacrine, "on", stoyan=0.1)
plot(p)

pcfdot.inhom

Inhomogeneous Multitype Pair Correlation Function (Type-i-To-AnyType)

Description
Estimates the inhomogeneous multitype pair correlation function (from type i to any type) for a
multitype point pattern.
Usage
pcfdot.inhom(X, i, lambdaI = NULL, lambdadot = NULL, ...,
r = NULL, breaks = NULL,
kernel="epanechnikov", bw=NULL, stoyan=0.15,
correction = c("isotropic", "Ripley", "translate"),
sigma = NULL, varcov = NULL)
Arguments
X

The observed point pattern, from which an estimate of the inhomogeneous multitype pair correlation function gi• (r) will be computed. It must be a multitype
point pattern (a marked point pattern whose marks are a factor).

i

The type (mark value) of the points in X from which distances are measured.
A character string (or something that will be converted to a character string).
Defaults to the first level of marks(X).

lambdaI

Optional. Values of the estimated intensity function of the points of type i. Either a vector giving the intensity values at the points of type i, a pixel image (object of class "im") giving the intensity values at all locations, or a function(x,y)
which can be evaluated to give the intensity value at any location.

lambdadot

Optional. Values of the estimated intensity function of the point pattern X. A
numeric vector, pixel image or function(x,y).

pcfdot.inhom

933

r

Vector of values for the argument r at which gi• (r) should be evaluated. There
is a sensible default.

breaks

This argument is for internal use only.

kernel

Choice of smoothing kernel, passed to density.default.

bw

Bandwidth for smoothing kernel, passed to density.default.

...

Other arguments passed to the kernel density estimation function density.default.

stoyan

Bandwidth coefficient; see Details.

correction

Choice of edge correction.

sigma,varcov

Optional arguments passed to density.ppp to control the smoothing bandwidth, when lambdaI or lambdadot is estimated by kernel smoothing.

Details
The inhomogeneous multitype (type i to any type) pair correlation function gi• (r) is a summary of
the dependence between different types of points in a multitype spatial point process that does not
have a uniform density of points.
The best intuitive interpretation is the following: the probability p(r) of finding a point of type i at
location x and another point of any type at location y, where x and y are separated by a distance r,
is equal to
p(r) = λi (x)lambda(y)g(r) dx dy
where λi is the intensity function of the process of points of type i, and where λ is the intensity
function of the points of all types. For a multitype Poisson point process, this probability is p(r) =
λi (x)λ(y) so gi• (r) = 1.
The command pcfdot.inhom estimates the inhomogeneous multitype pair correlation using a modified version of the algorithm in pcf.ppp.
If the arguments lambdaI and lambdadot are missing or null, they are estimated from X by kernel
smoothing using a leave-one-out estimator.
Value
A function value table (object of class "fv"). Essentially a data frame containing the variables
r

the vector of values of the argument r at which the inhomogeneous multitype
pair correlation function gi• (r) has been estimated

theo

vector of values equal to 1, the theoretical value of gi• (r) for the Poisson process

trans

vector of values of gi• (r) estimated by translation correction

iso

vector of values of gi• (r) estimated by Ripley isotropic correction

as required.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
pcf.ppp, pcfinhom, pcfdot, pcfcross.inhom

934

pcfinhom

Examples
data(amacrine)
plot(pcfdot.inhom(amacrine, "on", stoyan=0.1), legendpos="bottom")

pcfinhom

Inhomogeneous Pair Correlation Function

Description
Estimates the inhomogeneous pair correlation function of a point pattern using kernel methods.
Usage
pcfinhom(X, lambda = NULL, ..., r = NULL,
kernel = "epanechnikov", bw = NULL, stoyan = 0.15,
correction = c("translate", "Ripley"),
divisor = c("r", "d"),
renormalise = TRUE, normpower=1,
update = TRUE, leaveoneout = TRUE,
reciplambda = NULL,
sigma = NULL, varcov = NULL, close=NULL)
Arguments
X
lambda

r
kernel
bw

...
stoyan
correction
divisor
renormalise
normpower
update

A point pattern (object of class "ppp").
Optional. Values of the estimated intensity function. Either a vector giving the
intensity values at the points of the pattern X, a pixel image (object of class "im")
giving the intensity values at all locations, a fitted point process model (object of
class "ppm") or a function(x,y) which can be evaluated to give the intensity
value at any location.
Vector of values for the argument r at which g(r) should be evaluated. There is
a sensible default.
Choice of smoothing kernel, passed to density.default.
Bandwidth for smoothing kernel, passed to density.default. Either a single numeric value, or a character string specifying a bandwidth selection rule
recognised by density.default. If bw is missing or NULL, the default value is
computed using Stoyan’s rule of thumb: see bw.stoyan.
Other arguments passed to the kernel density estimation function density.default.
Coefficient for Stoyan’s bandwidth selection rule; see bw.stoyan.
Choice of edge correction.
Choice of divisor in the estimation formula: either "r" (the default) or "d". See
pcf.ppp.
Logical. Whether to renormalise the estimate. See Details.
Integer (usually either 1 or 2). Normalisation power. See Details.
Logical. If lambda is a fitted model (class "ppm", "kppm" or "dppm") and
update=TRUE (the default), the model will first be refitted to the data X (using update.ppm or update.kppm) before the fitted intensity is computed. If
update=FALSE, the fitted intensity of the model will be computed without refitting it to X.

pcfinhom

935

leaveoneout

Logical value (passed to density.ppp or fitted.ppm) specifying whether to
use a leave-one-out rule when calculating the intensity.

reciplambda

Alternative to lambda. Values of the estimated reciprocal 1/λ of the intensity
function. Either a vector giving the reciprocal intensity values at the points of
the pattern X, a pixel image (object of class "im") giving the reciprocal intensity
values at all locations, or a function(x,y) which can be evaluated to give the
reciprocal intensity value at any location.

sigma,varcov

Optional arguments passed to density.ppp to control the smoothing bandwidth, when lambda is estimated by kernel smoothing.

close

Advanced use only. Precomputed data. See section on Advanced Use.

Details
The inhomogeneous pair correlation function ginhom (r) is a summary of the dependence between
points in a spatial point process that does not have a uniform density of points.
The best intuitive interpretation is the following: the probability p(r) of finding two points at locations x and y separated by a distance r is equal to
p(r) = λ(x)lambda(y)g(r) dx dy
where λ is the intensity function of the point process. For a Poisson point process with intensity
function λ, this probability is p(r) = λ(x)λ(y) so ginhom (r) = 1.
The inhomogeneous pair correlation function is related to the inhomogeneous K function through
ginhom (r) =

0
Kinhom
(r)
2πr

0
where Kinhom
(r) is the derivative of Kinhom (r), the inhomogeneous K function. See Kinhom for
information about Kinhom (r).

The command pcfinhom estimates the inhomogeneous pair correlation using a modified version of
the algorithm in pcf.ppp.
If renormalise=TRUE
(the default), then the estimates are multiplied by cnormpower where c =
P
area(W )/ (1/λ(xi )). This rescaling reduces the variability and bias of the estimate in small samples and in cases of very strong inhomogeneity. The default value of normpower isP
1 but the most
sensible value is 2, which would correspond to rescaling the lambda values so that (1/λ(xi )) =
area(W ).
Value
A function value table (object of class "fv"). Essentially a data frame containing the variables
r

the vector of values of the argument r at which the inhomogeneous pair correlation function ginhom (r) has been estimated

theo

vector of values equal to 1, the theoretical value of ginhom (r) for the Poisson
process

trans

vector of values of ginhom (r) estimated by translation correction

iso

vector of values of ginhom (r) estimated by Ripley isotropic correction

as required.

936

pcfmulti

Advanced Use
To perform the same computation using several different bandwidths bw, it is efficient to use the
argument close. This should be the result of closepairs(X, rmax) for a suitably large value of
rmax, namely rmax >= max(r) + 3 * bw.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
pcf, pcf.ppp, bw.stoyan, bw.pcf, Kinhom
Examples
data(residualspaper)
X <- residualspaper$Fig4b
plot(pcfinhom(X, stoyan=0.2, sigma=0.1))
fit <- ppm(X, ~polynom(x,y,2))
plot(pcfinhom(X, lambda=fit, normpower=2))

pcfmulti

Marked pair correlation function

Description
For a marked point pattern, estimate the multitype pair correlation function using kernel methods.
Usage
pcfmulti(X, I, J, ..., r = NULL,
kernel = "epanechnikov", bw = NULL, stoyan = 0.15,
correction = c("translate", "Ripley"),
divisor = c("r", "d"),
Iname = "points satisfying condition I",
Jname = "points satisfying condition J")
Arguments
X

The observed point pattern, from which an estimate of the cross-type pair correlation function gij (r) will be computed. It must be a multitype point pattern (a
marked point pattern whose marks are a factor).

I

Subset index specifying the points of X from which distances are measured.

J

Subset index specifying the points in X to which distances are measured.

...

Ignored.

r

Vector of values for the argument r at which g(r) should be evaluated. There is
a sensible default.

kernel

Choice of smoothing kernel, passed to density.default.

bw

Bandwidth for smoothing kernel, passed to density.default.

pcfmulti

937

stoyan

Coefficient for default bandwidth rule.

correction

Choice of edge correction.

divisor

Choice of divisor in the estimation formula: either "r" (the default) or "d".

Iname,Jname

Optional. Character strings describing the members of the subsets I and J.

Details
This is a generalisation of pcfcross to arbitrary collections of points.
The algorithm measures the distance from each data point in subset I to each data point in subset J,
excluding identical pairs of points. The distances are kernel-smoothed and renormalised to form a
pair correlation function.
• If divisor="r" (the default), then the multitype counterpart of the standard kernel estimator (Stoyan and Stoyan, 1994, pages 284–285) is used. By default, the recommendations of
Stoyan and Stoyan (1994) are followed exactly.
• If divisor="d" then a modified estimator is used: the contribution from an interpoint distance
dij to the estimate of g(r) is divided by dij instead of dividing by r. This usually improves
the bias of the estimator when r is close to zero.
There is also a choice of spatial edge corrections (which are needed to avoid bias due to edge effects
associated with the boundary of the spatial window): correction="translate" is the OhserStoyan translation correction, and correction="isotropic" or "Ripley" is Ripley’s isotropic
correction.
The arguments I and J specify two subsets of the point pattern X. They may be any type of subset
indices, for example, logical vectors of length equal to npoints(X), or integer vectors with entries
in the range 1 to npoints(X), or negative integer vectors.
Alternatively, I and J may be functions that will be applied to the point pattern X to obtain index
vectors. If I is a function, then evaluating I(X) should yield a valid subset index. This option is
useful when generating simulation envelopes using envelope.
The choice of smoothing kernel is controlled by the argument kernel which is passed to density.
The default is the Epanechnikov kernel.
The bandwidth of the smoothing kernel can be controlled by the argument bw. Its precise interpretation is explained in the documentation for density.default.
For the Epanechnikov kernel with
√
support [−h, h], the argument bw is equivalent to h/ 5.
If bw is not specified, the default bandwidth is determined by Stoyan’s rule of√thumb (Stoyan and
Stoyan, 1994, page 285) applied to the points of type j. That is, h = c/ λ, where λ is the
(estimated) intensity of the point process of type j, and c is a constant in the range from 0.1 to 0.2.
The argument stoyan determines the value of c.
Value
An object of class "fv".
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
pcfcross, pcfdot, pcf.ppp.

938

Penttinen

Examples
adult <- (marks(longleaf) >= 30)
juvenile <- !adult
p <- pcfmulti(longleaf, adult, juvenile)

Penttinen

Penttinen Interaction

Description
Creates an instance of the Penttinen pairwise interaction point process model, which can then be
fitted to point pattern data.
Usage
Penttinen(r)
Arguments
r

circle radius

Details
Penttinen (1984, Example 2.1, page 18), citing Cormack (1979), described the pairwise interaction
point process with interaction factor
h(d) = eθA(d) = γ A(d)
between each pair of points separated by a distance $d$. Here A(d) is the area of intersection
between two discs of radius r separated by a distance d, normalised so that A(0) = 1.
The scale of interaction is controlled by the disc radius r: two points interact if they are closer than
2r apart. The strength of interaction is controlled by the canonical parameter θ, which must be less
than or equal to zero, or equivalently by the parameter γ = eθ , which must lie between 0 and 1.
The potential is inhibitory, i.e.\ this model is only appropriate for regular point patterns. For γ = 0
the model is a hard core process with hard core diameter 2r. For γ = 1 the model is a Poisson
process.
The irregular parameter r must be given in the call to Penttinen, while the regular parameter θ
will be estimated.
This model can be considered as a pairwise approximation to the area-interaction model AreaInter.
Value
An object of class "interact" describing the interpoint interaction structure of a point process.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 

perimeter

939

References
Cormack, R.M. (1979) Spatial aspects of competition between individuals. Pages 151–212 in Spatial and Temporal Analysis in Ecology, eds. R.M. Cormack and J.K. Ord, International Co-operative
Publishing House, Fairland, MD, USA.
Penttinen, A. (1984) Modelling Interaction in Spatial Point Patterns: Parameter Estimation by the
Maximum Likelihood Method. Jyväskylä Studies in Computer Science, Economics and Statistics 7,
University of Jyväskylä, Finland.
See Also
ppm, ppm.object, Pairwise, AreaInter.
Examples
fit <- ppm(cells ~ 1, Penttinen(0.07))
fit
reach(fit) # interaction range is circle DIAMETER

perimeter

Perimeter Length of Window

Description
Computes the perimeter length of a window
Usage
perimeter(w)
Arguments
w

A window (object of class "owin") or data that can be converted to a window
by as.owin.

Details
This function computes the perimeter (length of the boundary) of the window w. If w is a rectangle
or a polygonal window, the perimeter is the sum of the lengths of the edges of w. If w is a mask, it is
first converted to a polygonal window using as.polygonal, then staircase edges are removed using
simplify.owin, and the perimeter of the resulting polygon is computed.
Value
A numeric value giving the perimeter length of the window.
Author(s)
Adrian Baddeley 
and Rolf Turner 

940

periodify

See Also
area.owin diameter.owin, owin.object, as.owin
Examples
perimeter(square(3))
data(letterR)
perimeter(letterR)
if(interactive()) print(perimeter(as.mask(letterR)))

periodify

Make Periodic Copies of a Spatial Pattern

Description
Given a spatial pattern (point pattern, line segment pattern, window, etc) make shifted copies of the
pattern and optionally combine them to make a periodic pattern.
Usage
periodify(X, ...)
## S3 method for class 'ppp'
periodify(X, nx = 1, ny = 1, ...,
combine=TRUE, warn=TRUE, check=TRUE,
ix=(-nx):nx, iy=(-ny):ny,
ixy=expand.grid(ix=ix,iy=iy))
## S3 method for class 'psp'
periodify(X, nx = 1, ny = 1, ...,
combine=TRUE, warn=TRUE, check=TRUE,
ix=(-nx):nx, iy=(-ny):ny,
ixy=expand.grid(ix=ix,iy=iy))
## S3 method for class 'owin'
periodify(X, nx = 1, ny = 1, ...,
combine=TRUE, warn=TRUE,
ix=(-nx):nx, iy=(-ny):ny,
ixy=expand.grid(ix=ix,iy=iy))
Arguments
X

An object representing a spatial pattern (point pattern, line segment pattern or
window).

nx,ny

Integers. Numbers of additional copies of X in each direction. The result will be
a grid of 2 * nx + 1 by 2 * ny + 1 copies of the original object. (Overruled
by ix, iy, ixy).

...

Ignored.

combine

Logical flag determining whether the copies should be superimposed to make an
object like X (if combine=TRUE) or simply returned as a list of objects (combine=FALSE).

warn

Logical flag determining whether to issue warnings.

check

Logical flag determining whether to check the validity of the combined pattern.

periodify

941

ix, iy

Integer vectors determining the grid positions of the copies of X. (Overruled by
ixy).

ixy

Matrix or data frame with two columns, giving the grid positions of the copies
of X.

Details
Given a spatial pattern (point pattern, line segment pattern, etc) this function makes a number of
shifted copies of the pattern and optionally combines them. The function periodify is generic,
with methods for various kinds of spatial objects.
The default is to make a 3 by 3 array of copies of X and combine them into a single pattern of the
same kind as X. This can be used (for example) to compute toroidal or periodic edge corrections for
various operations on X.
If the arguments nx, ny are given and other arguments are missing, the original object will be copied
nx times to the right and nx times to the left, then ny times upward and ny times downward, making
(2 * nx + 1) * (2 * ny + 1) copies altogether, arranged in a grid, centred on the original
object.
If the arguments ix, iy or ixy are specified, then these determine the grid positions of the copies
of X that will be made. For example (ix,iy) = (1, 2) means a copy of X shifted by the vector
(ix * w, iy * h) where w,h are the width and height of the bounding rectangle of X.
If combine=TRUE (the default) the copies of X are superimposed to create an object of the same kind
as X. If combine=FALSE the copies of X are returned as a list.
Value
If combine=TRUE, an object of the same class as X. If combine=FALSE, a list of objects of the same
class as X.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
shift
Examples
data(cells)
plot(periodify(cells))
a <- lapply(periodify(Window(cells), combine=FALSE),
plot, add=TRUE,lty=2)

942

persp.im

persp.im

Perspective Plot of Pixel Image

Description
Displays a perspective plot of a pixel image.
Usage
## S3 method for class 'im'
persp(x, ...,
colmap=NULL, colin=x, apron=FALSE, visible=FALSE)
Arguments
x

The pixel image to be plotted as a surface. An object of class "im" (see im.object).

...

Extra arguments passed to persp.default to control the display.

colmap

Optional data controlling the colour map. See Details.

colin

Optional. Colour input. Another pixel image (of the same dimensions as x)
containing the values that will be mapped to colours.

apron

Logical. If TRUE, a grey apron is placed around the sides of the perspective plot.

visible

Logical value indicating whether to compute which pixels of x are visible in the
perspective view. See Details.

Details
This is the persp method for the class "im".
The pixel image x must have real or integer values. These values are treated as heights of a surface,
and the surface is displayed as a perspective plot on the current plot device, using equal scales on
the x and y axes.
The optional argument colmap gives an easy way to display different altitudes in different colours
(if this is what you want).
• If colmap is a colour map (object of class "colourmap", created by the function colourmap)
then this colour map will be used to associate altitudes with colours.
• If colmap is a character vector, then the range of altitudes in the perspective plot will be
divided into length(colmap) intervals, and those parts of the surface which lie in a particular
altitude range will be assigned the corresponding colour from colmap.
• If colmap is a function in the R language of the form function(n, ...), this function will
be called with an appropriate value of n to generate a character vector of n colours. Examples
of such functions are heat.colors, terrain.colors, topo.colors and cm.colors.
• If colmap is a function in the R language of the form function(range, ...) then it will
be called with range equal to the range of altitudes, to determine the colour values or colour
map. Examples of such functions are beachcolours and beachcolourmap.
• If colmap is a list with entries breaks and col, then colmap$breaks determines the breakpoints of the altitude intervals, and colmap$col provides the corresponding colours.

persp.im

943

Alternatively, if the argument colin (colour input) is present, then the colour map colmap will be
applied to the pixel values of colin instead of the pixel values of x. The result is a perspective view
of a surface with heights determined by x and colours determined by colin (mapped by colmap).
If apron=TRUE, vertical surface is drawn around the boundary of the perspective plot, so that the
terrain appears to have been cut out of a solid material. If colour data were supplied, then the apron
is coloured light grey.
Graphical parameters controlling the perspective plot are passed through the ... arguments directly
to the function persp.default. See the examples in persp.default or in demo(persp).
The vertical scale is controlled by the argument expand: setting expand=1 will interpret the pixel
values as being in the same units as the spatial coordinates x and y and represent them at the same
scale.
If visible=TRUE, the algorithm also computes whether each pixel in x is visible in the perspective
view. In order to be visible, a pixel must not be obscured by another pixel which lies in front of it (as
seen from the viewing direction), and the three-dimensional vector normal to the surface must be
pointing toward the viewer. The return value of persp.im then has an attribute "visible" which
is a pixel image, compatible with x, with pixel value equal to TRUE if the corresponding pixel in x
is visible, and FALSE if it is not visible.
Value
(invisibly) the 3D transformation matrix returned by persp.default, together with an attribute
"expand" which gives the relative scale of the z coordinate.
If argument visible=TRUE was given, the return value also has an attribute "visible" which is a
pixel image, compatible with x, with logical values which are TRUE when the corresponding pixel
is visible in the perspective view, and FALSE when it is obscured.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
perspPoints, perspLines for drawing additional points or lines on the surface.
im.object, plot.im, contour.im
Examples
# an image
Z <- setcov(owin())
persp(Z, colmap=terrain.colors(128))
co <- colourmap(range=c(0,1), col=rainbow(128))
persp(Z, colmap=co, axes=FALSE, shade=0.3)
## Terrain elevation
persp(bei.extra$elev, colmap=terrain.colors(128),
apron=TRUE, theta=-30, phi=20,
zlab="Elevation", main="", ticktype="detailed",
expand=6)

944

perspPoints

perspPoints

Draw Points or Lines on a Surface Viewed in Perspective

Description
After a surface has been plotted in a perspective view using persp.im, these functions can be used
to draw points or lines on the surface.
Usage
perspPoints(x, y=NULL, ..., Z, M)
perspLines(x, y = NULL, ..., Z, M)
perspSegments(x0, y0 = NULL, x1 = NULL, y1 = NULL, ..., Z, M)
perspContour(Z, M, ...,
nlevels=10, levels=pretty(range(Z), nlevels))
Arguments
x,y

Spatial coordinates, acceptable to xy.coords, for the points or lines on the horizontal plane.

Z

Pixel image (object of class "im") specifying the surface heights.

M

Projection matrix returned from persp.im when Z was plotted.

...

Graphical arguments passed to points, lines or segments to control the drawing.

x0,y0,x1,y1

Spatial coordinates of the line segments, on the horizontal plane. Alternatively
x0 can be a line segment pattern (object of class "psp") and y0,x1,y1 can be
NULL.

nlevels

Number of contour levels

levels

Vector of heights of contours.

Details
After a surface has been plotted in a perspective view, these functions can be used to draw points or
lines on the surface.
The user should already have called persp.im in the form M <- persp(Z, visible=TRUE, ...)
to display the perspective view of the surface Z.
Only points and lines which are visible from the viewer’s standpoint will be drawn.
Value
Same as the return value from points or segments.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 

pixelcentres

945

See Also
persp.im
Examples
M <- persp(bei.extra$elev, colmap=terrain.colors(128),
apron=TRUE, theta=-30, phi=20,
zlab="Elevation", main="",
expand=6, visible=TRUE, shade=0.3)
perspContour(bei.extra$elev, M=M, col="pink", nlevels=12)
perspPoints(bei, Z=bei.extra$elev, M=M, pch=16, cex=0.3, col="chartreuse")

pixelcentres

Extract Pixel Centres as Point Pattern

Description
Given a pixel image or binary mask window, extract the centres of all pixels and return them as a
point pattern.
Usage
pixelcentres(X, W = NULL, ...)
Arguments
X

Pixel image (object of class "im") or window (object of class "owin").

W

Optional window to contain the resulting point pattern.

...

Optional arguments defining the pixel resolution.

Details
If the argument X is a pixel image, the result is a point pattern, consisting of the centre of every pixel
whose pixel value is not NA.
If X is a window which is a binary mask, the result is a point pattern consisting of the centre of every
pixel inside the window (i.e. every pixel for which the mask value is TRUE).
Otherwise, X is first converted to a window, then converted to a mask using as.mask, then handled
as above.
Value
A point pattern (object of class "ppp").
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak 

946

pixellate

See Also
raster.xy
Examples
pixelcentres(letterR, dimyx=5)

pixellate

Convert Spatial Object to Pixel Image

Description
Convert a spatial object to a pixel image by measuring the amount of stuff in each pixel.
Usage
pixellate(x, ...)
Arguments
x

Spatial object to be converted. A point pattern (object of class "ppp"), a window
(object of class "owin"), a line segment pattern (object of class "psp"), or some
other suitable data.

...

Arguments passed to methods.

Details
The function pixellate converts a geometrical object x into a pixel image, by measuring the
amount of x that is inside each pixel.
If x is a point pattern, pixellate(x) counts the number of points of x falling in each pixel. If x is
a window, pixellate(x) measures the area of intersection of each pixel with the window.
The function pixellate is generic, with methods for point patterns (pixellate.ppp), windows
(pixellate.owin), and line segment patterns (pixellate.psp), See the separate documentation
for these methods.
The related function as.im also converts x into a pixel image, but typically measures only the
presence or absence of x inside each pixel.
Value
A pixel image (object of class "im").
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
pixellate.ppp, pixellate.owin, pixellate.psp, pixellate.linnet, as.im

pixellate.owin

pixellate.owin

947

Convert Window to Pixel Image

Description
Convert a window to a pixel image by measuring the area of intersection between the window and
each pixel in a raster.
Usage
## S3 method for class 'owin'
pixellate(x, W = NULL, ..., DivideByPixelArea=FALSE)
Arguments
x

Window (object of class "owin") to be converted.

W

Optional. Window determining the pixel raster on which the conversion should
occur.

...
Optional. Extra arguments passed to as.mask to determine the pixel raster.
DivideByPixelArea
Logical value, indicating whether the resulting pixel values should be divided
by the pixel area.
Details
This is a method for the generic function pixellate.
It converts a window x into a pixel image, by measuring the amount of x that is inside each pixel.
(The related function as.im also converts x into a pixel image, but records only the presence or
absence of x in each pixel.)
The pixel raster for the conversion is determined by the argument W and the extra arguments ....
• If W is given, and it is a binary mask (a window of type "mask") then it determines the pixel
raster.
• If W is given, but it is not a binary mask (it is a window of another type) then it will be converted
to a binary mask using as.mask(W, ...).
• If W is not given, it defaults to as.mask(as.rectangle(x), ...)
In the second and third cases it would be common to use the argument dimyx to control the number
of pixels. See the Examples.
The algorithm then computes the area of intersection of each pixel with the window.
The result is a pixel image with pixel entries equal to these intersection areas.
Value
A pixel image (object of class "im").
Author(s)
Adrian Baddeley  and Rolf Turner 

948

pixellate.ppp

See Also
pixellate.ppp, pixellate, as.im
Examples
data(letterR)
plot(pixellate(letterR, dimyx=15))
W <- grow.rectangle(as.rectangle(letterR), 0.2)
plot(pixellate(letterR, W, dimyx=15))

pixellate.ppp

Convert Point Pattern to Pixel Image

Description
Converts a point pattern to a pixel image. The value in each pixel is the number of points falling in
that pixel, and is typically either 0 or 1.
Usage
## S3 method for class 'ppp'
pixellate(x, W=NULL, ..., weights = NULL,
padzero=FALSE, fractional=FALSE, preserve=FALSE,
DivideByPixelArea=FALSE)
## S3 method for class 'ppp'
as.im(X, ...)
Arguments
x,X

Point pattern (object of class "ppp").

...

Arguments passed to as.mask to determine the pixel resolution

W

Optional window mask (object of class "owin") determining the pixel raster.

weights

Optional vector of weights associated with the points.

padzero
Logical value indicating whether to set pixel values to zero outside the window.
fractional,preserve
Logical values determining the type of discretisation. See Details.
DivideByPixelArea
Logical value, indicating whether the resulting pixel values should be divided
by the pixel area.
Details
The functions pixellate.ppp and as.im.ppp convert a spatial point pattern x into a pixel image,
by counting the number of points (or the total weight of points) falling in each pixel.
Calling as.im.ppp is equivalent to calling pixellate.ppp with its default arguments. Note that
pixellate.ppp is more general than as.im.ppp (it has additional arguments for greater flexibility).
The functions as.im.ppp and pixellate.ppp are methods for the generic functions as.im and
pixellate respectively, for the class of point patterns.

pixellate.ppp

949

The pixel raster (in which points are counted) is determined by the argument W if it is present (for
pixellate.ppp only). In this case W should be a binary mask (a window object of class "owin"
with type "mask"). Otherwise the pixel raster is determined by extracting the window containing x
and converting it to a binary pixel mask using as.mask. The arguments ... are passed to as.mask
to control the pixel resolution.
If weights is NULL, then for each pixel in the mask, the algorithm counts how many points in x fall
in the pixel. This count is usually either 0 (for a pixel with no data points in it) or 1 (for a pixel
containing one data point) but may be greater than 1. The result is an image with these counts as its
pixel values.
If weights is given, it should be a numeric vector of the same length as the number of points in x.
For each pixel, the algorithm finds the total weight associated with points in x that fall in the given
pixel. The result is an image with these total weights as its pixel values.
By default (if zeropad=FALSE) the resulting pixel image has the same spatial domain as the window
of the point pattern x. If zeropad=TRUE then the resulting pixel image has a rectangular domain;
pixels outside the original window are assigned the value zero.
The discretisation procedure is controlled by the arguments fractional and preserve.
• The argument fractional specifies how data points are mapped to pixels. If fractional=FALSE
(the default), each data point is allocated to the nearest pixel centre. If fractional=TRUE, each
data point is allocated with fractional weight to four pixel centres (the corners of a rectangle
containing the data point).
• The argument preserve specifies what to do with pixels lying near the boundary of the window, if the window is not a rectangle. If preserve=FALSE (the default), any contributions that
are attributed to pixel centres lying outside the window are reset to zero. If preserve=TRUE,
any such contributions are shifted to the nearest pixel lying inside the window, so that the total
mass is preserved.
Value
A pixel image (object of class "im").
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
pixellate, im, as.im, density.ppp, Smooth.ppp.
Examples
data(humberside)
plot(pixellate(humberside))
plot(pixellate(humberside, fractional=TRUE))

950

pixellate.psp

pixellate.psp

Convert Line Segment Pattern to Pixel Image

Description
Converts a line segment pattern to a pixel image by measuring the length or number of lines intersecting each pixel.
Usage
## S3 method for class 'psp'
pixellate(x, W=NULL, ..., weights = NULL,
what=c("length", "number"),
DivideByPixelArea=FALSE)
Arguments
x

Line segment pattern (object of class "psp").

W

Optional window (object of class "owin") determining the pixel resolution.

...

Optional arguments passed to as.mask to determine the pixel resolution.

weights

Optional vector of weights associated with each line segment.

what

String (partially matched) indicating whether to compute the total length of intersection (what="length", the default) or the total number of segments intersecting each pixel (what="number").
DivideByPixelArea
Logical value, indicating whether the resulting pixel values should be divided
by the pixel area.

Details
This function converts a line segment pattern to a pixel image by computing, for each pixel, the
total length of intersection between the pixel and the line segments. Alternatively it can count the
number of line segments intersecting each pixel.
This is a method for the generic function pixellate for the class of line segment patterns.
The pixel raster is determined by W and the optional arguments .... If W is missing or NULL, it
defaults to the window containing x. Then W is converted to a binary pixel mask using as.mask.
The arguments ... are passed to as.mask to control the pixel resolution.
If weights are given, then the length of the intersection between line segment i and pixel j is
multiplied by weights[i] before the lengths are summed for each pixel.
Value
A pixel image (object of class "im") with numeric values.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .

pixelquad

951

See Also
pixellate, as.mask, as.mask.psp.
Use as.mask.psp if you only want to know which pixels are intersected by lines.
Examples
X <- psp(runif(10),runif(10), runif(10), runif(10), window=owin())
plot(pixellate(X))
plot(X, add=TRUE)
sum(lengths.psp(X))
sum(pixellate(X))
plot(pixellate(X, what="n"))

pixelquad

Quadrature Scheme Based on Pixel Grid

Description
Makes a quadrature scheme with a dummy point at every pixel of a pixel image.
Usage
pixelquad(X, W = as.owin(X))
Arguments
X

Point pattern (object of class "ppp") containing the data points for the quadrature
scheme.

W

Specifies the pixel grid. A pixel image (object of class "im"), a window (object
of class "owin"), or anything that can be converted to a window by as.owin.

Details
This is a method for producing a quadrature scheme for use by ppm. It is an alternative to quadscheme.
The function ppm fits a point process model to an observed point pattern using the Berman-Turner
quadrature approximation (Berman and Turner, 1992; Baddeley and Turner, 2000) to the pseudolikelihood of the model. It requires a quadrature scheme consisting of the original data point pattern,
an additional pattern of dummy points, and a vector of quadrature weights for all these points. Such
quadrature schemes are represented by objects of class "quad". See quad.object for a description
of this class.
Given a grid of pixels, this function creates a quadrature scheme in which there is one dummy point
at the centre of each pixel. The counting weights are used (the weight attached to each quadrature
point is 1 divided by the number of quadrature points falling in the same pixel).
The argument X specifies the locations of the data points for the quadrature scheme. Typically this
would be a point pattern dataset.
The argument W specifies the grid of pixels for the dummy points of the quadrature scheme. It
should be a pixel image (object of class "im"), a window (object of class "owin"), or anything that
can be converted to a window by as.owin. If W is a pixel image or a binary mask (a window of type
"mask") then the pixel grid of W will be used. If W is a rectangular or polygonal window, then it will
first be converted to a binary mask using as.mask at the default pixel resolution.

952

plot.anylist

Value
An object of class "quad" describing the quadrature scheme (data points, dummy points, and
quadrature weights) suitable as the argument Q of the function ppm() for fitting a point process
model.
The quadrature scheme can be inspected using the print and plot methods for objects of class
"quad".
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
quadscheme, quad.object, ppm
Examples
W <- owin(c(0,1),c(0,1))
X <- runifpoint(42, W)
W <- as.mask(W,dimyx=128)
pixelquad(X,W)

plot.anylist

Plot a List of Things

Description
Plots a list of things
Usage
## S3 method for class 'anylist'
plot(x, ..., main, arrange=TRUE,
nrows=NULL, ncols=NULL, main.panel=NULL,
mar.panel=c(2,1,1,2), hsep=0, vsep=0,
panel.begin=NULL, panel.end=NULL, panel.args=NULL,
panel.begin.args=NULL, panel.end.args=NULL, panel.vpad=0.2,
plotcommand="plot",
adorn.left=NULL, adorn.right=NULL, adorn.top=NULL, adorn.bottom=NULL,
adorn.size=0.2, equal.scales=FALSE, halign=FALSE, valign=FALSE)
Arguments
x

An object of the class "anylist". Essentially a list of objects.

...

Arguments passed to plot when generating each plot panel.

main

Overall heading for the plot.

arrange

Logical flag indicating whether to plot the objects side-by-side on a single page
(arrange=TRUE) or plot them individually in a succession of frames (arrange=FALSE).

plot.anylist

953

nrows,ncols

Optional. The number of rows/columns in the plot layout (assuming arrange=TRUE).
You can specify either or both of these numbers.

main.panel

Optional. A character string, or a vector of character strings, giving the headings
for each of the objects.

mar.panel

Size of the margins outside each plot panel. A numeric vector of length 4 giving
the bottom, left, top, and right margins in that order. (Alternatively the vector
may have length 1 or 2 and will be replicated to length 4). See the section on
Spacing between plots.

hsep,vsep

Additional horizontal and vertical separation between plot panels, expressed in
the same units as mar.panel.
panel.begin,panel.end
Optional. Functions that will be executed before and after each panel is plotted.
See Details.
panel.args

Optional. Function that determines different plot arguments for different panels.
See Details.
panel.begin.args
Optional. List of additional arguments for panel.begin when it is a function.
panel.end.args Optional. List of additional arguments for panel.end when it is a function.
panel.vpad

Amount of extra vertical space that should be allowed for the title of each panel,
if a title will be displayed. Expressed as a fraction of the height of the panel. Applies only when equal.scales=FALSE (the default) and requires that the height
of each panel can be determined.

plotcommand

Optional. Character string containing the name of the command that should be
executed to plot each panel.
adorn.left,adorn.right,adorn.top,adorn.bottom
Optional. Functions (with no arguments) that will be executed to generate additional plots at the margins (left, right, top and/or bottom, respectively) of the
array of plots.
adorn.size

Relative width (as a fraction of the other panels’ widths) of the margin plots.

equal.scales

Logical value indicating whether the components should be plotted at (approximately) the same physical scale.

halign,valign

Logical values indicating whether panels in a column should be aligned to the
same x coordinate system (halign=TRUE) and whether panels in a row should be
aligned to the same y coordinate system (valign=TRUE). These are applicable
only if equal.scales=TRUE.

Details
This is the plot method for the class "anylist".
An object of class "anylist" represents a list of objects intended to be treated in the same way.
This is the method for plot.
In the spatstat package, various functions produce an object of class "anylist", essentially a list of
objects of the same kind. These objects can be plotted in a nice arrangement using plot.anylist.
See the Examples.
The argument panel.args determines extra graphics parameters for each panel. It should be a
function that will be called as panel.args(i) where i is the panel number. Its return value should
be a list of graphics parameters that can be passed to the relevant plot method. These parameters
override any parameters specified in the ... arguments.

954

plot.anylist
The arguments panel.begin and panel.end determine graphics that will be plotted before and after each panel is plotted. They may be objects of some class that can be plotted with the generic plot
command. Alternatively they may be functions that will be called as panel.begin(i, y, main=main.panel[i])
and panel.end(i, y, add=TRUE) where i is the panel number and y = x[[i]].
If all entries of x are pixel images, the function image.listof is called to control the plotting. The
arguments equal.ribbon and col can be used to determine the colour map or maps applied.
If equal.scales=FALSE (the default), then the plot panels will have equal height on the plot device
(unless there is only one column of panels, in which case they will have equal width on the plot
device). This means that the objects are plotted at different physical scales, by default.
If equal.scales=TRUE, then the dimensions of the plot panels on the plot device will be proportional to the spatial dimensions of the corresponding components of x. This means that the objects
will be plotted at approximately equal physical scales. If these objects have very different spatial
sizes, the plot command could fail (when it tries to plot the smaller objects at a tiny scale), with an
error message that the figure margins are too large.
The objects will be plotted at exactly equal physical scales, and exactly aligned on the device, under
the following conditions:
• every component of x is a spatial object whose position can be shifted by shift;
• panel.begin and panel.end are either NULL or they are spatial objects whose position can
be shifted by shift;
• adorn.left, adorn.right, adorn.top and adorn.bottom are all NULL.
Another special case is when every component of x is an object of class "fv" representing a function. If equal.scales=TRUE then all these functions will be plotted with the same axis scales (i.e.
with the same xlim and the same ylim).

Value
Null.
Spacing between plots
The spacing between individual plots is controlled by the parameters mar.panel, hsep and vsep.
If equal.scales=FALSE, the plot panels are logically separate plots. The margins for each panel
are determined by the argument mar.panel which becomes the graphics parameter mar described
in the help file for par. One unit of mar corresponds to one line of text in the margin. If hsep or
vsep are present, mar.panel is augmented by c(vsep, hsep, vsep, hsep)/2.
If equal.scales=TRUE, all the plot panels are drawn in the same coordinate system which represents a physical scale. The unit of measurement for mar.panel[1,3] is one-sixth of the greatest
height of any object plotted in the same row of panels, and the unit for mar.panel[2,4] is onesixth of the greatest width of any object plotted in the same column of panels. If hsep or vsep are
present, they are interpreted in the same units as mar.panel[2] and mar.panel[1] respectively.
Error messages
If the error message ‘Figure margins too large’ occurs, this generally means that one of the objects
had a much smaller physical scale than the others. Ensure that equal.scales=FALSE and increase
the values of mar.panel.

plot.bermantest

955

Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
contour.listof, image.listof, density.splitppp
Examples
trichotomy <- list(regular=cells,
random=japanesepines,
clustered=redwood)
K <- lapply(trichotomy, Kest)
K <- as.anylist(K)
plot(K, main="")
# list of 3D point patterns
ape1 <- osteo[osteo$shortid==4, "pts", drop=TRUE]
class(ape1)
plot(ape1, main.panel="", mar.panel=0.1, hsep=0.7, vsep=1,
cex=1.5, pch=21, bg='white')

plot.bermantest

Plot Result of Berman Test

Description
Plot the result of Berman’s test of goodness-of-fit
Usage
## S3 method for class 'bermantest'
plot(x, ...,
lwd=par("lwd"), col=par("col"), lty=par("lty"),
lwd0=lwd, col0=2, lty0=2)
Arguments
x

Object to be plotted. An object of class "bermantest" produced by berman.test.

...

extra arguments that will be passed to the plotting function plot.ecdf.

col,lwd,lty

The width, colour and type of lines used to plot the empirical distribution curve.

col0,lwd0,lty0 The width, colour and type of lines used to plot the predicted (null) distribution
curve.

956

plot.bermantest

Details
This is the plot method for the class "bermantest". An object of this class represents the outcome of Berman’s test of goodness-of-fit of a spatial Poisson point process model, computed by
berman.test.
For the Z1 test (i.e. if x was computed using berman.test( ,which="Z1")), the plot displays
the two cumulative distribution functions that are compared by the test: namely the empirical cumulative distribution function of the covariate at the data points, F̂ , and the predicted cumulative
distribution function of the covariate under the model, F0 , both plotted against the value of the covariate. Two vertical lines show the mean values of these two distributions. If the model is correct,
the two curves should be close; the test is based on comparing the two vertical lines.
For the Z2 test (i.e. if x was computed using berman.test( ,which="Z2")), the plot displays the
empirical cumulative distribution function of the values Ui = F0 (Yi ) where Yi is the value of the
covariate at the i-th data point. The diagonal line with equation y = x is also shown. Two vertical
lines show the mean of the values Ui and the value 1/2. If the model is correct, the two curves
should be close. The test is based on comparing the two vertical lines.
Value
NULL.
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak 
See Also
berman.test
Examples
# synthetic data: nonuniform Poisson process
X <- rpoispp(function(x,y) { 100 * exp(-x) }, win=square(1))
# fit uniform Poisson process
fit0 <- ppm(X, ~1)
# test covariate = x coordinate
xcoord <- function(x,y) { x }
# test wrong model
k <- berman.test(fit0, xcoord, "Z1")
# plot result of test
plot(k, col="red", col0="green")
# Z2 test
k2 <- berman.test(fit0, xcoord, "Z2")
plot(k2, col="red", col0="green")

plot.cdftest

plot.cdftest

957

Plot a Spatial Distribution Test

Description
Plot the result of a spatial distribution test computed by cdf.test.
Usage
## S3 method for class 'cdftest'
plot(x, ...,
style=c("cdf", "PP", "QQ"),
lwd=par("lwd"), col=par("col"), lty=par("lty"),
lwd0=lwd, col0=2, lty0=2,
do.legend)
Arguments
x

Object to be plotted. An object of class "cdftest" produced by a method for
cdf.test.

...

extra arguments that will be passed to the plotting function plot.default.

style

Style of plot. See Details.

col,lwd,lty

The width, colour and type of lines used to plot the empirical curve (the empirical distribution, or PP plot or QQ plot).

col0,lwd0,lty0 The width, colour and type of lines used to plot the reference curve (the predicted
distribution, or the diagonal).
do.legend

Logical value indicating whether to add an explanatory legend. Applies only
when style="cdf".

Details
This is the plot method for the class "cdftest". An object of this class represents the outcome of
a spatial distribution test, computed by cdf.test, and based on either the Kolmogorov-Smirnov,
Cramér-von Mises or Anderson-Darling test.
If style="cdf" (the default), the plot displays the two cumulative distribution functions that are
compared by the test: namely the empirical cumulative distribution function of the covariate at
the data points, and the predicted cumulative distribution function of the covariate under the model,
both plotted against the value of the covariate. The Kolmogorov-Smirnov test statistic (for example)
is the maximum vertical separation between the two curves.
If style="PP" then the P-P plot is drawn. The x coordinates of the plot are cumulative probabilities
for the covariate under the model. The y coordinates are cumulative probabilities for the covariate
at the data points. The diagonal line y = x is also drawn for reference. The Kolmogorov-Smirnov
test statistic is the maximum vertical separation between the P-P plot and the diagonal reference
line.
If style="QQ" then the Q-Q plot is drawn. The x coordinates of the plot are quantiles of the
covariate under the model. The y coordinates are quantiles of the covariate at the data points. The
diagonal line y = x is also drawn for reference. The Kolmogorov-Smirnov test statistic cannot be
read off the Q-Q plot.

958

plot.colourmap

Value
NULL.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
cdf.test
Examples
op <- options(useFancyQuotes=FALSE)
# synthetic data: nonuniform Poisson process
X <- rpoispp(function(x,y) { 100 * exp(x) }, win=square(1))
# fit uniform Poisson process
fit0 <- ppm(X, ~1)
# test covariate = x coordinate
xcoord <- function(x,y) { x }
# test wrong model
k <- cdf.test(fit0, xcoord)
# plot result of test
plot(k, lwd0=3)
plot(k, style="PP")
plot(k, style="QQ")
options(op)

plot.colourmap

Plot a Colour Map

Description
Displays a colour map as a colour ribbon
Usage
## S3 method for class 'colourmap'
plot(x, ...,
main, xlim = NULL, ylim = NULL, vertical = FALSE, axis = TRUE,
labelmap=NULL, gap=0.25, add=FALSE)

plot.colourmap

959

Arguments
x

Colour map to be plotted. An object of class "colourmap".

...

Graphical arguments passed to image.default or axis.

main

Main title for plot. A character string.

xlim

Optional range of x values for the location of the colour ribbon.

ylim

Optional range of y values for the location of the colour ribbon.

vertical

Logical flag determining whether the colour ribbon is plotted as a horizontal
strip (FALSE) or a vertical strip (TRUE).

axis

Logical flag determining whether an axis should be plotted showing the numerical values that are mapped to the colours.

labelmap

Function. If this is present, then the labels on the plot, which indicate the input
values corresponding to particular colours, will be transformed by labelmap
before being displayed on the plot. Typically used to simplify or shorten the
labels on the plot.

gap

Distance between separate blocks of colour, as a fraction of the width of one
block, if the colourmap is discrete.

add

Logical value indicating whether to add the colourmap to the existing plot (add=TRUE),
or to start a new plot (add=FALSE, the default).

Details
This is the plot method for the class "colourmap". An object of this class (created by the function
colourmap) represents a colour map or colour lookup table associating colours with each data value.
The command plot.colourmap displays the colour map as a colour ribbon or as a colour legend (a
sequence of blocks of colour). This plot can be useful on its own to inspect the colour map.
If the domain of the colourmap is an interval of real numbers, the colourmap is displayed as a
continuous ribbon of colour. If the domain of the colourmap is a finite set of inputs, the colours
are displayed as separate blocks of colour. The separation between blocks is equal to gap times the
width of one block.
To annotate an existing plot with an explanatory colour ribbon or colour legend, specify add=TRUE
and use the arguments xlim and/or ylim to control the physical position of the ribbon on the plot.
Labels explaining the colour map are drawn by axis and can be modified by specifying arguments
that will be passed to this function.
Value
None.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
colourmap

960

plot.dppm

Examples
co <- colourmap(rainbow(100), breaks=seq(-1,1,length=101))
plot(co)
plot(co, col.ticks="pink")
ca <- colourmap(rainbow(8), inputs=letters[1:8])
plot(ca, vertical=TRUE)

plot.dppm

Plot a fitted determinantal point process

Description
Plots a fitted determinantal point process model, displaying the fitted intensity and the fitted summary function.
Usage
## S3 method for class 'dppm'
plot(x, ..., what=c("intensity", "statistic"))
Arguments
x

Fitted determinantal point process model. An object of class "dppm".

...

Arguments passed to plot.ppm and plot.fv to control the plot.

what

Character vector determining what will be plotted.

Details
This is a method for the generic function plot for the class "dppm" of fitted determinantal point
process models.
The argument x should be a determinantal point process model (object of class "dppm") obtained
using the function dppm.
The choice of plots (and the order in which they are displayed) is controlled by the argument what.
The options (partially matched) are "intensity" and "statistic".
This command is capable of producing two different plots:
what="intensity" specifies the fitted intensity of the model, which is plotted using plot.ppm. By
default this plot is not produced for stationary models.
what="statistic" specifies the empirical and fitted summary statistics, which are plotted using
plot.fv. This is only meaningful if the model has been fitted using the Method of Minimum
Contrast, and it is turned off otherwise.
Value
Null.
Author(s)
Adrian Baddeley 
and Rolf Turner 

plot.envelope

961

See Also
dppm, plot.ppm,
Examples
fit <- dppm(swedishpines ~ x + y, dppGauss())
plot(fit)

plot.envelope

Plot a Simulation Envelope

Description
Plot method for the class "envelope".
Usage
## S3 method for class 'envelope'
plot(x, ..., main)
Arguments
x

An object of class "envelope", containing the variables to be plotted or variables from which the plotting coordinates can be computed.

main

Main title for plot.

...

Extra arguments passed to plot.fv.

Details
This is the plot method for the class "envelope" of simulation envelopes. Objects of this class are
created by the command envelope.
This plot method is currently identical to plot.fv.
Its default behaviour is to shade the region between the upper and lower envelopes in a light grey
colour. To suppress the shading and plot the upper and lower envelopes as curves, set shade=NULL.
To change the colour of the shading, use the argument shadecol which is passed to plot.fv.
See plot.fv for further information on how to control the plot.
Value
Either NULL, or a data frame giving the meaning of the different line types and colours.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
envelope, plot.fv

962

plot.fasp

Examples
data(cells)
E <- envelope(cells, Kest, nsim=19)
plot(E)
plot(E, sqrt(./pi) ~ r)

plot.fasp

Plot a Function Array

Description
Plots an array of summary functions, usually associated with a point pattern, stored in an object of
class "fasp". A method for plot.
Usage
## S3 method for class 'fasp'
plot(x,formule=NULL, ...,
subset=NULL, title=NULL, banner=TRUE,
transpose=FALSE,
samex=FALSE, samey=FALSE,
mar.panel=NULL,
outerlabels=TRUE, cex.outerlabels=1.25,
legend=FALSE)
Arguments
x

An object of class "fasp" representing a function array.

formule

A formula or list of formulae indicating what variables are to be plotted against
what variable. Each formula is either an R language formula object, or a string
that can be parsed as a formula. If formule is a list, its k th component should
be applicable to the (i, j)th plot where x$which[i,j]=k. If the formula is left
as NULL, then plot.fasp attempts to use the component default.formula of
x. If that component is NULL as well, it gives up.

...

Arguments passed to plot.fv to control the individual plot panels.

subset

A logical vector, or a vector of indices, or an expression or a character string,
or a list of such, indicating a subset of the data to be included in each plot. If
subset is a list, its k th component should be applicable to the (i, j)th plot where
x$which[i,j]=k.

title

Overall title for the plot.

banner

Logical. If TRUE, the overall title is plotted. If FALSE, the overall title is not
plotted and no space is allocated for it.

transpose

Logical. If TRUE, rows and columns will be exchanged.

samex,samey

Logical values indicating whether all individual plot panels should have the
same x axis limits and the same y axis limits, respectively. This makes it easier
to compare the plots.

mar.panel

Vector of length 4 giving the value of the graphics parameter mar controlling the
size of plot margins for each individual plot panel. See par.

plot.fasp
outerlabels

963
Logical. If TRUE, the row and column names of the array of functions are plotted
in the margins of the array of plot panels. If FALSE, each individual plot panel is
labelled by its row and column name.

cex.outerlabels
Character expansion factor for row and column labels of array.
legend

Logical flag determining whether to plot a legend in each panel.

Details
An object of class "fasp" represents an array of summary functions, usually associated with a point
pattern. See fasp.object for details. Such an object is created, for example, by alltypes.
The function plot.fasp is a method for plot. It calls plot.fv to plot the individual panels.
For information about the interpretation of the arguments formule and subset, see plot.fv.
Arguments that are often passed through ... include col to control the colours of the different
lines in a panel, and lty and lwd to control the line type and line width of the different lines in a
panel. The argument shade can also be used to display confidence intervals or significance bands
as filled grey shading. See plot.fv.
The argument title, if present, will determine the overall title of the plot. If it is absent, it defaults
to x$title. Titles for the individual plot panels will be taken from x$titles.
Value
None.
Warnings
(Each component of) the subset argument may be a logical vector (of the same length as the
vectors of data which are extracted from x), or a vector of indices, or an expression such as
expression(r<=0.2), or a text string, such as "r<=0.2".
Attempting a syntax such as subset = r<=0.2 (without wrapping r<=0.2 either in quote marks or
in expression()) will cause this function to fall over.
Variables referred to in any formula must exist in the data frames stored in x. What the names of
these variables are will of course depend upon the nature of x.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
alltypes, plot.fv, fasp.object
Examples
## Not run:
# Bramble Canes data.
data(bramblecanes)
X.G <- alltypes(bramblecanes,"G",dataname="Bramblecanes",verb=TRUE)
plot(X.G)
plot(X.G,subset="r<=0.2")

964

plot.fv
plot(X.G,formule=asin(sqrt(cbind(km,theo))) ~ asin(sqrt(theo)))
plot(X.G,fo=cbind(km,theo) - theo~r,subset="r<=0.2")
# Simulated data.
pp <- runifpoint(350, owin(c(0,1),c(0,1)))
pp <- pp %mark% factor(c(rep(1,50),rep(2,100),rep(3,200)))
X.K <- alltypes(pp,"K",verb=TRUE,dataname="Fake Data")
plot(X.K,fo=cbind(border,theo)~theo,subset="theo<=0.75")
## End(Not run)

plot.fv

Plot Function Values

Description
Plot method for the class "fv".
Usage
## S3 method for class 'fv'
plot(x, fmla, ..., subset=NULL, lty=NULL, col=NULL, lwd=NULL,
xlim=NULL, ylim=NULL, xlab=NULL, ylab=NULL, ylim.covers=NULL,
legend=!add, legendpos="topleft", legendavoid=missing(legendpos),
legendmath=TRUE, legendargs=list(),
shade=fvnames(x, ".s"), shadecol="grey",
add=FALSE, log="",
mathfont=c("italic", "plain", "bold", "bolditalic"),
limitsonly=FALSE)
Arguments
x

An object of class "fv", containing the variables to be plotted or variables from
which the plotting coordinates can be computed.

fmla

an R language formula determining which variables or expressions are plotted.
Either a formula object, or a string that can be parsed as a formula. See Details.

subset

(optional) subset of rows of the data frame that will be plotted.

lty

(optional) numeric vector of values of the graphical parameter lty controlling
the line style of each plot.

col

(optional) numeric vector of values of the graphical parameter col controlling
the colour of each plot.

lwd

(optional) numeric vector of values of the graphical parameter lwd controlling
the line width of each plot.

xlim

(optional) range of x axis

ylim

(optional) range of y axis

xlab

(optional) label for x axis

ylab

(optional) label for y axis

...

Extra arguments passed to plot.default.

plot.fv

965

ylim.covers

Optional vector of y values that must be included in the y axis. For example
ylim.covers=0 will ensure that the y axis includes the origin.

legend

Logical flag or NULL. If legend=TRUE, the algorithm plots a legend in the top left
corner of the plot, explaining the meaning of the different line types and colours.

legendpos

The position of the legend. Either a character string keyword (see legend for
keyword options) or a pair of coordinates in the format list(x,y). Alternatively if legendpos="float", a location will be selected inside the plot region,
avoiding the graphics.

legendavoid

Whether to avoid collisions between the legend and the graphics. Logical value.
If TRUE, the code will check for collisions between the legend box and the graphics, and will override legendpos if a collision occurs. If FALSE, the value of
legendpos is always respected.

legendmath

Logical. If TRUE, the legend will display the mathematical notation for each
curve. If FALSE, the legend text is the identifier (column name) for each curve.

legendargs

Named list containing additional arguments to be passed to legend controlling
the appearance of the legend.

shade

A character vector giving the names of two columns of x, or another type of
index that identifies two columns. When the corresponding curves are plotted,
the region between the curves will be shaded in light grey. The object x may or
may not contain two columns which are designated as boundaries for shading;
they are identified by fvnames(x, ".s"). The default is to shade between these
two curves if they exist. To suppress this behaviour, set shade=NULL.

shadecol

The colour to be used in the shade plot. A character string or an integer specifying a colour.

add

Logical. Whether the plot should be added to an existing plot

log

A character string which contains "x" if the x axis is to be logarithmic, "y" if the
y axis is to be logarithmic and "xy" or "yx" if both axes are to be logarithmic.

mathfont

Character string. The font to be used for mathematical expressions in the axis
labels and the legend.

limitsonly

Logical. If FALSE, plotting is performed normally. If TRUE, no plotting is performed at all; just the x and y limits of the plot are computed and returned.

Details
This is the plot method for the class "fv".
The use of the argument fmla is like plot.formula, but offers some extra functionality.
The left and right hand sides of fmla are evaluated, and the results are plotted against each other
(the left side on the y axis against the right side on the x axis).
The left and right hand sides of fmla may be the names of columns of the data frame x, or expressions involving these names. If a variable in fmla is not the name of a column of x, the algorithm
will search for an object of this name in the environment where plot.fv was called, and then in the
enclosing environment, and so on.
Multiple curves may be specified by a single formula of the form cbind(y1,y2,...,yn) ~ x,
where x,y1,y2,...,yn are expressions involving the variables in the data frame. Each of the
variables y1,y2,...,yn in turn will be plotted against x. See the examples.
Convenient abbreviations which can be used in the formula are
• the symbol . which represents all the columns in the data frame that will be plotted by default;

966

plot.fv
• the symbol .x which represents the function argument;
• the symbol .y which represents the recommended value of the function.
For further information, see fvnames.
The value returned by this plot function indicates the meaning of the line types and colours in the
plot. It can be used to make a suitable legend for the plot if you want to do this by hand. See the
examples.
The argument shade can be used to display critical bands or confidence intervals. If it is not NULL,
then it should be a subset index for the columns of x, that identifies exactly 2 columns. When the
corresponding curves are plotted, the region between the curves will be shaded in light grey. See
the Examples.
The default values of lty, col and lwd can be changed using spatstat.options("plot.fv").
Use type = "n" to create the plot region and draw the axes without plotting any data.
Use limitsonly=TRUE to suppress all plotting and just compute the x and y limits. This can be
used to calculate common x and y scales for several plots.
To change the kind of parenthesis enclosing the explanatory text about the unit of length, use
spatstat.options('units.paren')

Value
Invisible: either NULL, or a data frame giving the meaning of the different line types and colours.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
fv.object, Kest
Examples
K <- Kest(cells)
# K is an object of class "fv"
plot(K, iso ~ r)

# plots iso against r

plot(K, sqrt(iso/pi) ~ r)

# plots sqrt(iso/r)

plot(K, cbind(iso,theo) ~ r)
plot(K, .

~ r)

plot(K, sqrt(./pi) ~ r)

against r

# plots iso against r

AND theo against r

# plots all available estimates of K against r
# plots all estimates of L-function
# L(r) = sqrt(K(r)/pi)

plot(K, cbind(iso,theo) ~ r, col=c(2,3))
# plots iso against r in colour 2
# and theo against r in colour 3
plot(K, iso ~ r, subset=quote(r < 0.2))
# plots iso against r for r < 10

plot.hyperframe

967

# Can't remember the names of the columns? No problem..
plot(K, sqrt(./pi) ~ .x)
# making a legend by hand
v <- plot(K, . ~ r, legend=FALSE)
legend("topleft", legend=v$meaning, lty=v$lty, col=v$col)
# significance bands
KE <- envelope(cells, Kest, nsim=19)
plot(KE, shade=c("hi", "lo"))
# how to display two functions on a common scale
Kr <- Kest(redwood)
a <- plot(K, limitsonly=TRUE)
b <- plot(Kr, limitsonly=TRUE)
xlim <- range(a$xlim, b$xlim)
ylim <- range(a$ylim, b$ylim)
opa <- par(mfrow=c(1,2))
plot(K, xlim=xlim, ylim=ylim)
plot(Kr, xlim=xlim, ylim=ylim)
par(opa)

plot.hyperframe

Plot Entries in a Hyperframe

Description
Plots the entries in a hyperframe, in a series of panels, one panel for each row of the hyperframe.
Usage
## S3 method for class 'hyperframe'
plot(x, e, ..., main, arrange=TRUE,
nrows=NULL, ncols=NULL,
parargs=list(mar=mar * marsize),
marsize=1, mar=c(1,1,3,1))
Arguments
x

Data to be plotted. A hyperframe (object of class "hyperframe", see hyperframe).

e

How to plot each row. Optional. An R language call or expression (typically
enclosed in quote() that will be evaluated in each row of the hyperframe to
generate the plots.

...

Extra arguments controlling the plot (when e is missing).

main

Overall title for the array of plots.

arrange

Logical flag indicating whether to plot the objects side-by-side on a single page
(arrange=TRUE) or plot them individually in a succession of frames (arrange=FALSE).

nrows,ncols

Optional. The number of rows/columns in the plot layout (assuming arrange=TRUE).
You can specify either or both of these numbers.

968

plot.hyperframe
parargs

Optional list of arguments passed to par before plotting each panel. Can be used
to control margin sizes, etc.

marsize

Optional scale parameter controlling the sizes of margins around the panels.
Incompatible with parargs.

mar

Optional numeric vector of length 1, 2 or 4 controlling the relative sizes of margins between the panels. Incompatible with parargs.

Details
This is the plot method for the class "hyperframe".
The argument x must be a hyperframe (like a data frame, except that the entries can be objects of
any class; see hyperframe).
This function generates a series of plots, one plot for each row of the hyperframe. If arrange=TRUE
(the default), then these plots are arranged in a neat array of panels within a single plot frame. If
arrange=FALSE, the plots are simply executed one after another.
Exactly what is plotted, and how it is plotted, depends on the argument e. The default (if e is
missing) is to plot only the first column of x. Each entry in the first column is plotted using the
generic plot command, together with any extra arguments given in ....
If e is present, it should be an R language expression involving the column names of x. (It is
typically created using quote or expression.) The expression will be evaluated once for each row
of x. It will be evaluated in an environment where each column name of x is interpreted as meaning
the object in that column in the current row. See the Examples.
Value
NULL.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
hyperframe, with.hyperframe
Examples
H <- hyperframe(id=1:10)
H$X <- with(H, rpoispp(100))
H$D <- with(H, distmap(X))
# points only
plot(H[,"X"])
plot(H, quote(plot(X, main=id)))
# points superimposed on images
plot(H, quote({plot(D, main=id); plot(X, add=TRUE)}))

plot.im

plot.im

969

Plot a Pixel Image

Description
Plot a pixel image.
Usage
## S3 method for class 'im'
plot(x, ...,
main,
add=FALSE, clipwin=NULL,
col=NULL, valuesAreColours=NULL, log=FALSE,
ncolours=256, gamma=1,
ribbon=show.all, show.all=!add,
ribside=c("right", "left", "bottom", "top"),
ribsep=0.15, ribwid=0.05, ribn=1024,
ribscale=1, ribargs=list(), colargs=list(),
useRaster=NULL, workaround=FALSE,
do.plot=TRUE)
## S3 method for class 'im'
image(x, ...,
main,
add=FALSE, clipwin=NULL,
col=NULL, valuesAreColours=NULL, log=FALSE,
ncolours=256, gamma=1,
ribbon=show.all, show.all=!add,
ribside=c("right", "left", "bottom", "top"),
ribsep=0.15, ribwid=0.05, ribn=1024,
ribscale=1, ribargs=list(), colargs=list(),
useRaster=NULL, workaround=FALSE,
do.plot=TRUE)
Arguments
x

The pixel image to be plotted. An object of class "im" (see im.object).

...

Extra arguments passed to image.default to control the plot. See Details.

main

Main title for the plot.

add

Logical value indicating whether to superimpose the image on the existing plot
(add=TRUE) or to initialise a new plot (add=FALSE, the default).

clipwin

Optional. A window (object of class "owin"). Only this subset of the image will
be displayed.

col

Colours for displaying the pixel values. Either a character vector of colour values, an object of class colourmap, or a function as described under Details.
valuesAreColours
Logical value. If TRUE, the pixel values of x are to be interpreted as colour
values.

970

plot.im
log

Logical value. If TRUE, the colour map will be evenly-spaced on a logarithmic
scale.

ncolours

Integer. The default number of colours in the colour map for a real-valued image.

gamma

Exponent for the gamma correction of the colours. A single positive number.

ribbon

Logical flag indicating whether to display a ribbon showing the colour map.
Default is TRUE for new plots and FALSE for added plots.

show.all

Logical value indicating whether to display all plot elements including the main
title and colour ribbon. Default is TRUE for new plots and FALSE for added plots.

ribside

Character string indicating where to display the ribbon relative to the main image.

ribsep

Factor controlling the space between the ribbon and the image.

ribwid

Factor controlling the width of the ribbon.

ribn

Number of different values to display in the ribbon.

ribscale

Rescaling factor for tick marks. The values on the numerical scale printed beside
the ribbon will be multiplied by this rescaling factor.

ribargs

List of additional arguments passed to image.default, axis and axisTicks to
control the display of the ribbon and its scale axis. These may override the ...
arguments.

colargs

List of additional arguments passed to col if it is a function.

useRaster

Logical value, passed to image.default. Images are plotted using a bitmap
raster if useRaster=TRUE or by drawing polygons if useRaster=FALSE. Bitmap
raster display tends to produce better results, but is not supported on all graphics
devices. The default is to use bitmap raster display if it is supported.

workaround

Logical value, specifying whether to use a workaround to avoid a bug which
occurs with some device drivers in R, in which the image has the wrong spatial
orientation. See the section on Image is Displayed in Wrong Spatial Orientation below.

do.plot

Logical value indicating whether to actually plot the image and colour ribbon.
Setting do.plot=FALSE will simply return the colour map and the bounding box
that were chosen for the plot.

Details
This is the plot method for the class "im". [It is also the image method for "im".]
The pixel image x is displayed on the current plot device, using equal scales on the x and y axes.
If ribbon=TRUE, a legend will be plotted. The legend consists of a colour ribbon and an axis with
tick-marks, showing the correspondence between the pixel values and the colour map.
Arguments ribside, ribsep, ribwid control the placement of the colour ribbon. By default, the
ribbon is placed at the right of the main image. This can be changed using the argument ribside.
The width of the ribbon is ribwid times the size of the pixel image, where ‘size’ means the larger
of the width and the height. The distance separating the ribbon and the image is ribsep times the
size of the pixel image.
The ribbon contains the colours representing ribn different numerical values, evenly spaced between the minimum and maximum pixel values in the image x, rendered according to the chosen
colour map.
The argument ribargs controls the annotation of the colour ribbon. It is a list of arguments to be
passed to image.default, axis and axisTicks. To plot the colour ribbon without the axis and

plot.im

971

tick-marks, use ribargs=list(axes=FALSE). To ensure that the numerals or symbols printed next
to the colour map are oriented horizontally, use ribargs=list(las=1). To control the number of
tick-marks, use ribargs=list(nint=N) where N is the desired number of intervals (so there will
be N+1 tickmarks, subject to the vagaries of R internal code).
The argument ribscale is used to rescale the numerals printed next to the colour map.
Normally the pixel values are displayed using the colours given in the argument col. This may be
either
• an explicit colour map (an object of class "colourmap", created by the command colourmap).
This is the best way to ensure that when we plot different images, the colour maps are consistent.
• a character vector or integer vector that specifies a set of colours. The colour mapping will be
stretched to match the range of pixel values in the image x. The mapping of pixel values to
colours is determined as follows.
logical-valued images: the values FALSE and TRUE are mapped to the colours col[1] and
col[2] respectively. The vector col should have length 2.
factor-valued images: the factor levels levels(x) are mapped to the entries of col in order.
The vector col should have the same length as levels(x).
numeric-valued images: By default, the range of pixel values in x is divided into n = length(col)
equal subintervals, which are mapped to the colours in col. (If col was not specified, it
defaults to a vector of 255 colours.)
Alternatively if the argument zlim is given, it should be a vector of length 2 specifying
an interval of real numbers. This interval will be used instead of the range of pixel values.
The interval from zlim[1] to zlim[2] will be mapped to the colours in col. This facility
enables the user to plot several images using a consistent colour map.
Alternatively if the argument breaks is given, then this specifies the endpoints of the
subintervals that are mapped to each colour. This is incompatible with zlim.
The arguments col and zlim or breaks are then passed to the function image.default.
For examples of the use of these arguments, see image.default.
• a function in the R language with an argument named range or inputs.
If col is a function with an argument named range, and if the pixel values of x are numeric values, then the colour values will be determined by evaluating col(range=range(x)). The result of this evaluation should be a character vector containing colour values, or a "colourmap"
object. Examples of such functions are beachcolours and beachcolourmap.
If col is a function with an argument named inputs, and if the pixel values of x are discrete
values (integer, logical, factor or character), then the colour values will be determined by
evaluating col(inputs=p) where p is the set of possible pixel values. The result should be a
character vector containing colour values, or a "colourmap" object.
• a function in the R language with first argument named n. The colour values will be determined by evaluating col(n) where n is the number of distinct pixel values, up to a maximum
of 128. The result of this evaluation should be a character vector containing color values. Examples of such functions are heat.colors, terrain.colors, topo.colors and cm.colors.
If spatstat.options("monochrome") has been set to TRUE then all colours will be converted to
grey scale values.
Other graphical parameters controlling the display of both the pixel image and the ribbon can be
passed through the ... arguments to the function image.default. A parameter is handled only if
it is one of the following:
• a formal argument of image.default that is operative when add=TRUE.

972

plot.im
• one of the parameters "main", "asp", "sub", "axes", "ann",
described in par.

"cex", "font", "cex.axis", "cex.lab

• the argument box, a logical value specifying whether a box should be drawn.
Images are plotted using a bitmap raster if useRaster=TRUE or by drawing polygons if useRaster=FALSE.
Bitmap raster display (performed by rasterImage) tends to produce better results, but is not supported on all graphics devices. The default is to use bitmap raster display if it is supported according
to dev.capabilities.
Alternatively, the pixel values could be directly interpretable as colour values in R. That is, the
pixel values could be character strings that represent colours, or values of a factor whose levels are
character strings representing colours.
• If valuesAreColours=TRUE, then the pixel values will be interpreted as colour values and
displayed using these colours.
• If valuesAreColours=FALSE, then the pixel values will not be interpreted as colour values,
even if they could be.
• If valuesAreColours=NULL, the algorithm will guess what it should do. If the argument col
is given, the pixel values will not be interpreted as colour values. Otherwise, if all the pixel
values are strings that represent colours, then they will be interpreted and displayed as colours.
If pixel values are interpreted as colours, the arguments col and ribbon will be ignored, and a
ribbon will not be plotted.
Value
The colour map used. An object of class "colourmap".
Also has an attribute "bbox" giving a bounding box for the colour image (including the ribbon if
present).
Complex-valued images
If the pixel values in x are complex numbers, they will be converted into four images containing the real and imaginary parts and the modulus and argument, and plotted side-by-side using
plot.imlist.
Monochrome colours
If spatstat.options("monochrome") has been set to TRUE, then the image will be plotted in
greyscale. The colours are converted to grey scale values using to.grey. The choice of colour map
still has an effect, since it determines the final grey scale values.
Monochrome display can also be achieved by setting the graphics device parameter colormodel="grey"
when starting a new graphics device, or in a call to ps.options or pdf.options.
Image Rendering Errors and Problems
The help for image.default and rasterImage explains that errors may occur, or images may be
rendered incorrectly, on some devices, depending on the availability of colours and other devicespecific constraints.
If the image is not displayed at all, try setting useRaster=FALSE in the call to plot.im. If the
ribbon colours are not displayed, set ribargs=list(useRaster=FALSE).
Errors may occur on some graphics devices if the image is very large. If this happens, try setting
useRaster=FALSE in the call to plot.im.

plot.im

973

The error message useRaster=TRUE can only be used with a regular grid means that the
x and y coordinates of the pixels in the image are not perfectly equally spaced, due to numerical
rounding. This occurs with some images created by earlier versions of spatstat. To repair the
coordinates in an image X, type X <- as.im(X).
Image is Displayed in Wrong Spatial Orientation
If the image is displayed in the wrong spatial orientation, and you created the image data directly,
please check that you understand the spatstat convention for the spatial orientation of pixel images.
The row index of the matrix of pixel values corresponds to the increasing y coordinate; the column
index of the matrix corresponds to the increasing x coordinate (Baddeley, Rubak and Turner, 2015,
section 3.6.3, pages 66–67).
Images can be displayed in the wrong spatial orientation on some devices, due to a bug in the device
driver. This occurs only when the plot coordinates are reversed, that is, when the plot was initialised
with coordinate limits xlim, ylim such that xlim[1] > xlim[2] or ylim[1] > ylim[2] or both.
This bug is reported to occur only when useRaster=TRUE. To fix this, try setting workaround=TRUE,
or if that is unsuccessful, useRaster=FALSE.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
References
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
See Also
im.object, colourmap, contour.im, persp.im, hist.im, image.default, spatstat.options
Examples
# an image
Z <- setcov(owin())
plot(Z)
plot(Z, ribside="bottom")
# stretchable colour map
plot(Z, col=terrain.colors(128), axes=FALSE)
# fixed colour map
tc <- colourmap(rainbow(128), breaks=seq(-1,2,length=129))
plot(Z, col=tc)
# colour map function, with argument 'range'
plot(Z, col=beachcolours, colargs=list(sealevel=0.5))
# tweaking the plot
plot(Z, main="La vie en bleu", col.main="blue", cex.main=1.5,
box=FALSE,
ribargs=list(col.axis="blue", col.ticks="blue", cex.axis=0.75))
# log scale
V <- eval.im(exp(exp(Z+2))/1e4)
plot(V, log=TRUE, main="Log scale")
# it's complex
Y <- exp(Z + V * 1i)
plot(Y)

974

plot.imlist

plot.imlist

Plot a List of Images

Description
Plots an array of pixel images.
Usage
## S3 method for class 'imlist'
plot(x, ..., plotcommand="image",
equal.ribbon=FALSE, ribmar=NULL)
## S3 method for class 'imlist'
image(x, ..., equal.ribbon=FALSE, ribmar=NULL)
## S3 method for class 'listof'
image(x, ..., equal.ribbon=FALSE, ribmar=NULL)
Arguments
x

An object of the class "imlist" representing a list of pixel images. Alternatively
x may belong to the outdated class "listof".

...

Arguments passed to plot.solist to control the spatial arrangement of panels,
and arguments passed to plot.im to control the display of each panel.

equal.ribbon

Logical. If TRUE, the colour maps of all the images will be the same. If FALSE,
the colour map of each image is adjusted to the range of values of that image.

ribmar

Numeric vector of length 4 specifying the margins around the colour ribbon, if
equal.ribbon=TRUE. Entries in the vector give the margin at the bottom, left,
top, and right respectively, as a multiple of the height of a line of text.

plotcommand

Character string giving the name of a function to be used to display each image.
Recognised by plot.imlist only.

Details
These are methods for the generic plot commands plot and image for the class "imlist". They
are currently identical.
An object of class "imlist" represents a list of pixel images. (The outdated class "listof" is also
handled.)
Each entry in the list x will be displayed as a pixel image, in an array of panels laid out on the same
graphics display, using plot.solist. Individual panels are plotted by plot.im.
If equal.ribbon=FALSE (the default), the images are rendered using different colour maps, which
are displayed as colour ribbons beside each image. If equal.ribbon=TRUE, the images are rendered
using the same colour map, and a single colour ribbon will be displayed at the right side of the array.
The colour maps and the placement of the colour ribbons are controlled by arguments ... passed
to plot.im.
Value
Null.

plot.influence.ppm

975

Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
plot.solist, plot.im
Examples
D <- density(split(amacrine))
image(D, equal.ribbon=TRUE, main="", col.ticks="red", col.axis="red")

plot.influence.ppm

Plot Influence Measure

Description
Plots an influence measure that has been computed by influence.ppm.
Usage
## S3 method for class 'influence.ppm'
plot(x, ..., multiplot=TRUE)
Arguments
x

Influence measure (object of class "influence.ppm") computed by influence.ppm.

...

Arguments passed to plot.ppp to control the plotting.

multiplot

Logical value indicating whether it is permissible to plot more than one panel.
This happens if the original point process model is multitype.

Details
This is the plot method for objects of class "influence.ppm". These objects are computed by the
command influence.ppm.
For a point process model fitted by maximum likelihood or maximum pseudolikelihood (the default), influence values are associated with the data points. The display shows circles centred at the
data points with radii proportional to the influence values. If the original data were a multitype point
pattern, then if multiplot=TRUE (the default), there is one such display for each possible type of
point, while if multiplot=FALSE there is a single plot combining all data points regardless of type.
For a model fitted by logistic composite likelihood (method="logi" in ppm) influence values are
associated with the data points and also with the dummy points used to fit the model. The display
consist of two panels, for the data points and dummy points respectively, showing circles with radii
proportional to the influence values. If the original data were a multitype point pattern, then if
multiplot=TRUE (the default), there is one pair of panels for each possible type of point, while if
multiplot=FALSE there is a single plot combining all data and dummy points regardless of type.
Use the argument clipwin to restrict the plot to a subset of the full data.

976

plot.kppm

Value
None.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
References
Baddeley, A. and Chang, Y.M. and Song, Y. (2013) Leverage and influence diagnostics for spatial
point process models. Scandinavian Journal of Statistics 40, 86–104.
See Also
influence.ppm
Examples
X <- rpoispp(function(x,y) { exp(3+3*x) })
fit <- ppm(X, ~x+y)
plot(influence(fit))

plot.kppm

Plot a fitted cluster point process

Description
Plots a fitted cluster point process model, displaying the fitted intensity and the fitted K-function.
Usage
## S3 method for class 'kppm'
plot(x, ...,
what=c("intensity", "statistic", "cluster"),
pause=interactive(),
xname)
Arguments
x

Fitted cluster point process model. An object of class "kppm".

...

Arguments passed to plot.ppm and plot.fv to control the plot.

what

Character vector determining what will be plotted.

pause

Logical value specifying whether to pause between plots.

xname

Optional. Character string. The name of the object x for use in the title of the
plot.

plot.kppm

977

Details
This is a method for the generic function plot for the class "kppm" of fitted cluster point process
models.
The argument x should be a cluster point process model (object of class "kppm") obtained using the
function kppm.
The choice of plots (and the order in which they are displayed) is controlled by the argument what.
The options (partially matched) are "intensity", "statistic" and "cluster".
This command is capable of producing three different plots:
what="intensity" specifies the fitted intensity of the model, which is plotted using plot.ppm. By
default this plot is not produced for stationary models.
what="statistic" specifies the empirical and fitted summary statistics, which are plotted using
plot.fv. This is only meaningful if the model has been fitted using the Method of Minimum
Contrast, and it is turned off otherwise.
what="cluster" specifies a fitted cluster, which is computed by clusterfield and plotted by
plot.im. It is only meaningful for Poisson cluster (incl. Neyman-Scott) processes, and it is
turned off for log-Gaussian Cox processes (LGCP). If the model is stationary (and non-LGCP)
this option is turned on by default and shows a fitted cluster positioned at the centroid of the
observation window. For non-stationary (and non-LGCP) models this option is only invoked
if explicitly told so, and in that case an additional argument locations (see clusterfield)
must be given to specify where to position the parent point(s) .
Alternatively what="all" selects all available options.
Value
Null.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
kppm, plot.ppm,
Examples
data(redwood)
fit <- kppm(redwood~1, "Thomas")
plot(fit)

978

plot.laslett

plot.laslett

Plot Laslett Transform

Description
Plot the result of Laslett’s Transform.
Usage
## S3 method for class 'laslett'
plot(x, ...,
Xpars = list(box = TRUE, col = "grey"),
pointpars = list(pch = 3, cols = "blue"),
rectpars = list(lty = 3, border = "green"))
Arguments
x

Object of class "laslett" produced by laslett representing the result of Laslett’s
transform.

...

Additional plot arguments passed to plot.solist.

Xpars

A list of plot arguments passed to plot.owin or plot.im to display the original
region X before transformation.

pointpars

A list of plot arguments passed to plot.ppp to display the tangent points.

rectpars

A list of plot arguments passed to plot.owin to display the maximal rectangle.

Details
This is the plot method for the class "laslett".
The function laslett applies Laslett’s Transform to a spatial region X and returns an object of class
"laslett" representing the result of the transformation. The result is plotted by this method.
The plot function plot.solist is used to align the before-and-after pictures. See plot.solist for
further options to control the plot.
Value
None.
Author(s)
Kassel Hingee and Adrian Baddeley .
See Also
laslett
Examples
b <- laslett(heather$coarse, plotit=FALSE)
plot(b, main="Heather Data")

plot.layered

plot.layered

979

Layered Plot

Description
Generates a layered plot. The plot method for objects of class "layered".
Usage
## S3 method for class 'layered'
plot(x, ..., which = NULL, plotargs = NULL,
add=FALSE, show.all=!add, main=NULL,
do.plot=TRUE)
Arguments
x

An object of class "layered" created by the function layered.

...

Arguments to be passed to the plot method for every layer.

which

Subset index specifying which layers should be plotted.

plotargs

Arguments to be passed to the plot methods for individual layers. A list of lists
of arguments of the form name=value.

add

Logical value indicating whether to add the graphics to an existing plot.

show.all

Logical value indicating whether the first layer should be displayed in full (including the main title, bounding window, coordinate axes, colour ribbon, and so
on).

main

Main title for the plot

do.plot

Logical value indicating whether to actually do the plotting.

Details
Layering is a simple mechanism for controlling a high-level plot that is composed of several successive plots, for example, a background and a foreground plot. The layering mechanism makes
it easier to plot, to switch on or off the plotting of each individual layer, to control the plotting
arguments that are passed to each layer, and to zoom in on a subregion.
The layers of data to be plotted should first be converted into a single object of class "layered"
using the function layered. Then the layers can be plotted using the method plot.layered.
To zoom in on a subregion, apply the subset operator [.layered to x before plotting.
Graphics parameters for each layer are determined by (in order of precedence) ..., plotargs, and
layerplotargs(x).
The graphics parameters may also include the special argument .plot specifying (the name of) a
function which will be used to perform the plotting instead of the generic plot.
The argument show.all is recognised by many plot methods in spatstat. It determines whether a
plot is drawn with all its additional components such as the main title, bounding window, coordinate
axes, colour ribbons and legends. The default is TRUE for new plots and FALSE for added plots.
In plot.layered, the argument show.all applies only to the first layer. The subsequent layers are
plotted with show.all=FALSE.
To override this, that is, if you really want to draw all the components of all layers of x, insert the
argument show.all=TRUE in each entry of plotargs or layerplotargs(x).

980

plot.leverage.ppm

Value
(Invisibly) a list containing the return values from the plot commands for each layer. This list has
an attribute "bbox" giving a bounding box for the entire plot.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
layered, layerplotargs, [.layered, plot.
Examples
data(cells)
D <- distmap(cells)
L <- layered(D, cells)
plot(L)
plot(L, which = 2)
plot(L, plotargs=list(list(ribbon=FALSE), list(pch=3, cols="white")))
# plot a subregion
plot(L[, square(0.5)])

plot.leverage.ppm

Plot Leverage Function

Description
Generate a pixel image plot, or a contour plot, or a perspective plot, of a leverage function that has
been computed by leverage.ppm.
Usage
## S3 method for class 'leverage.ppm'
plot(x, ...,
what=c("smooth", "nearest", "exact"),
showcut=TRUE,
args.cut=list(drawlabels=FALSE),
multiplot=TRUE)
## S3 method for class 'leverage.ppm'
contour(x, ...,
what=c("smooth", "nearest"),
showcut=TRUE,
args.cut=list(col=3, lwd=3, drawlabels=FALSE),
multiplot=TRUE)
## S3 method for class 'leverage.ppm'
persp(x, ...,
what=c("smooth", "nearest"),
main, zlab="leverage")

plot.leverage.ppm

981

Arguments
x

Leverage function (object of class "leverage.ppm") computed by leverage.ppm.

...

Arguments passed to plot.im or contour.im or persp.im controlling the plot.

what

Character string (partially matched) specifying the values to be plotted. See
Details.

showcut

Logical. If TRUE, a contour line is plotted at the level equal to the theoretical
mean of the leverage.

args.cut

Optional list of arguments passed to contour.default to control the plotting
of the contour line for the mean leverage.

multiplot

Logical value indicating whether it is permissible to display several plot panels.

main

Optional main title. A character string or character vector.

zlab

Label for the z axis. A character string.

Details
These functions are the plot, contour and persp methods for objects of class "leverage.ppm".
Such objects are computed by the command leverage.ppm.
The plot method displays the leverage function as a colour pixel image using plot.im, and draws a
single contour line at the mean leverage value using contour.default. Use the argument clipwin
to restrict the plot to a subset of the full data.
The contour method displays the leverage function as a contour plot, and also draws a single
contour line at the mean leverage value, using contour.im.
The persp method displays the leverage function as a surface in perspective view, using persp.im.
Since the exact values of leverage are computed only at a finite set of quadrature locations, there
are several options for these plots:
what="smooth": (the default) an image plot showing a smooth function, obtained by applying
kernel smoothing to the exact leverage values;
what="nearest": an image plot showing a piecewise-constant function, obtained by taking the
exact leverage value at the nearest quadrature point;
what="exact": a symbol plot showing the exact values of leverage as circles, centred at the quadrature points, with diameters proportional to leverage.
The pixel images are already contained in the object x and were computed by leverage.ppm; the
resolution of these images is controlled by arguments to leverage.ppm.
Value
Same as for plot.im, contour.im and persp.im respectively.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
References
Baddeley, A., Chang, Y.M. and Song, Y. (2013) Leverage and influence diagnostics for spatial point
process models. Scandinavian Journal of Statistics 40, 86–104.

982

plot.linim

See Also
leverage.ppm.
Examples
X <- rpoispp(function(x,y) { exp(3+3*x) })
fit <- ppm(X ~x+y)
lef <- leverage(fit)
plot(lef)
contour(lef)
persp(lef)

plot.linim

Plot Pixel Image on Linear Network

Description
Given a pixel image on a linear network, the pixel values are displayed either as colours or as line
widths.
Usage
## S3 method for class 'linim'
plot(x, ..., style = c("colour", "width"),
scale, adjust = 1,
negative.args = list(col=2),
legend=TRUE,
leg.side=c("right", "left", "bottom", "top"),
leg.sep=0.1,
leg.wid=0.1,
leg.args=list(),
leg.scale=1,
zlim,
do.plot=TRUE)
Arguments
x

The pixel image to be plotted. An object of class "linim".

...

Extra graphical parameters, passed to plot.im if style="colour", or to polygon
if style="width".

style

Character string specifying the type of plot. See Details.

scale

Physical scale factor for representing the pixel values as line widths.

adjust

Adjustment factor for the default scale.

negative.args

A list of arguments to be passed to polygon specifying how to plot negative
values of x when style="width".

legend

Logical value indicating whether to plot a legend (colour ribbon or scale bar).

leg.side

Character string indicating where to display the legend relative to the main image.

leg.sep

Factor controlling the space between the legend and the image.

plot.linim
leg.wid
leg.scale
leg.args
zlim
do.plot

983
Factor controlling the width of the legend.
Rescaling factor for annotations on the legend. The values on the numerical
scale printed beside the legend will be multiplied by this rescaling factor.
List of additional arguments passed to image.default, axis or text.default
to control the display of the legend. These may override the ... arguments.
The range of numerical values that should be mapped. A numeric vector of
length 2. Defaults to the range of values of x.
Logical value indicating whether to actually perform the plot.

Details
This is the plot method for objects of class "linim". Such an object represents a pixel image
defined on a linear network.
If style="colour" (the default) then the pixel values of x are plotted as colours, using plot.im.
If style="width" then the pixel values of x are used to determine the widths of thick lines centred
on the line segments of the linear network.
Value
If style="colour", the result is an object of class "colourmap" specifying the colour map used.
If style="width", the result is a numeric value v giving the physical scale: one unit of pixel value
is represented as v physical units on the plot.
The result also has an attribute "bbox" giving a bounding box for the plot. The bounding box
includes the ribbon or scale bar, if present, but not the main title.
Author(s)
Adrian Baddeley 
References
Ang, Q.W., Baddeley, A. and Nair, G. (2012) Geometrically corrected second-order analysis of
events on a linear network, with applications to ecology and criminology. Scandinavian Journal of
Statistics 39, 591–617.
See Also
linim, plot.im, polygon
Examples
X <- linfun(function(x,y,seg,tp){y^2+x}, simplenet)
X <- as.linim(X)
plot(X)
plot(X, style="width", main="Width proportional to function value")
# signed values
f <- linfun(function(x,y,seg,tp){y-x}, simplenet)
plot(f, style="w", main="Negative values in red")
plot(f, style="w", negative.args=list(density=10),
main="Negative values are hatched")

984

plot.linnet

plot.linnet

Plot a linear network

Description
Plots a linear network
Usage
## S3 method for class 'linnet'
plot(x, ..., main=NULL, add=FALSE,
vertices=FALSE, window=FALSE,
do.plot=TRUE)
Arguments
x

Linear network (object of class "linnet").

...

Arguments passed to plot.psp controlling the plot.

main

Main title for plot. Use main="" to suppress it.

add

Logical. If codeTRUE, superimpose the graphics over the current plot. If FALSE,
generate a new plot.

vertices

Logical. Whether to plot the vertices as well.

window

Logical. Whether to plot the window containing the linear network.

do.plot

Logical. Whether to actually perform the plot.

Details
This is the plot method for class "linnet".
Value
An (invisible) object of class "owin" giving the bounding box of the network.
Author(s)
Ang Qi Wei  and Adrian Baddeley 
See Also
linnet
Examples
plot(simplenet)

plot.lintess

985

plot.lintess

Plot a Tessellation on a Linear Network

Description
Plot a tessellation or division of a linear network into tiles.
Usage
## S3 method for class 'lintess'
plot(x, ...,
main, add = FALSE, style = c("segments", "image"), col = NULL)
Arguments
x

Tessellation on a linear network (object of class "lintess").

...

Arguments passed to segments (if style="segments") or to plot.im (if style="image")
to control the plot.

main

Optional main title for the plot.

add

Logical value indicating whether the plot is to be added to an existing plot.

style

Character string (partially matched) indicating whether to plot the tiles of the
tessellation using segments or to convert the tessellation to a pixel image and
use plot.im.

col

Vector of colours, or colour map, determining the colours used to plot the different tiles of the tessellation.

Details
A tessellation on a linear network L is a partition of the network into non-overlapping pieces (tiles).
Each tile consists of one or more line segments which are subsets of the line segments making up
the network. A tile can consist of several disjoint pieces.
This function plots the tessellation on the current device. It is a method for the generic plot.
If style="segments", each tile is plotted using segments. Colours distinguish the different tiles.
If style="image", the tessellation is converted to a pixel image, and plotted using plot.im.
Value
(Invisible) colour map.
Author(s)
Adrian Baddeley 
See Also
lintess

986

plot.listof

Examples
X <- runiflpp(7, simplenet)
Z <- divide.linnet(X)
plot(Z, main="tessellation on network")
points(as.ppp(X))

plot.listof

Plot a List of Things

Description
Plots a list of things
Usage
## S3 method for class 'listof'
plot(x, ..., main, arrange=TRUE,
nrows=NULL, ncols=NULL, main.panel=NULL,
mar.panel=c(2,1,1,2), hsep=0, vsep=0,
panel.begin=NULL, panel.end=NULL, panel.args=NULL,
panel.begin.args=NULL, panel.end.args=NULL, panel.vpad=0.2,
plotcommand="plot",
adorn.left=NULL, adorn.right=NULL, adorn.top=NULL, adorn.bottom=NULL,
adorn.size=0.2, equal.scales=FALSE, halign=FALSE, valign=FALSE)
Arguments
x

An object of the class "listof". Essentially a list of objects.

...

Arguments passed to plot when generating each plot panel.

main

Overall heading for the plot.

arrange

Logical flag indicating whether to plot the objects side-by-side on a single page
(arrange=TRUE) or plot them individually in a succession of frames (arrange=FALSE).

nrows,ncols

Optional. The number of rows/columns in the plot layout (assuming arrange=TRUE).
You can specify either or both of these numbers.

main.panel

Optional. A character string, or a vector of character strings, giving the headings
for each of the objects.

mar.panel

Size of the margins outside each plot panel. A numeric vector of length 4 giving
the bottom, left, top, and right margins in that order. (Alternatively the vector
may have length 1 or 2 and will be replicated to length 4). See the section on
Spacing between plots.

hsep,vsep

Additional horizontal and vertical separation between plot panels, expressed in
the same units as mar.panel.
panel.begin,panel.end
Optional. Functions that will be executed before and after each panel is plotted.
See Details.
panel.args

Optional. Function that determines different plot arguments for different panels.
See Details.

plot.listof

987

panel.begin.args
Optional. List of additional arguments for panel.begin when it is a function.
panel.end.args Optional. List of additional arguments for panel.end when it is a function.
panel.vpad

Amount of extra vertical space that should be allowed for the title of each panel,
if a title will be displayed. Expressed as a fraction of the height of the panel. Applies only when equal.scales=FALSE (the default) and requires that the height
of each panel can be determined.

plotcommand

Optional. Character string containing the name of the command that should be
executed to plot each panel.
adorn.left,adorn.right,adorn.top,adorn.bottom
Optional. Functions (with no arguments) that will be executed to generate additional plots at the margins (left, right, top and/or bottom, respectively) of the
array of plots.
adorn.size

Relative width (as a fraction of the other panels’ widths) of the margin plots.

equal.scales

Logical value indicating whether the components should be plotted at (approximately) the same physical scale.

halign,valign

Logical values indicating whether panels in a column should be aligned to the
same x coordinate system (halign=TRUE) and whether panels in a row should be
aligned to the same y coordinate system (valign=TRUE). These are applicable
only if equal.scales=TRUE.

Details
This is the plot method for the class "listof".
An object of class "listof" (defined in the base R package) represents a list of objects, all belonging to a common class. The base R package defines a method for printing these objects,
print.listof, but does not define a method for plot. So here we have provided a method for
plot.
In the spatstat package, various functions produce an object of class "listof", essentially a list
of spatial objects of the same kind. These objects can be plotted in a nice arrangement using
plot.listof. See the Examples.
The argument panel.args determines extra graphics parameters for each panel. It should be a
function that will be called as panel.args(i) where i is the panel number. Its return value should
be a list of graphics parameters that can be passed to the relevant plot method. These parameters
override any parameters specified in the ... arguments.
The arguments panel.begin and panel.end determine graphics that will be plotted before and after each panel is plotted. They may be objects of some class that can be plotted with the generic plot
command. Alternatively they may be functions that will be called as panel.begin(i, y, main=main.panel[i])
and panel.end(i, y, add=TRUE) where i is the panel number and y = x[[i]].
If all entries of x are pixel images, the function image.listof is called to control the plotting. The
arguments equal.ribbon and col can be used to determine the colour map or maps applied.
If equal.scales=FALSE (the default), then the plot panels will have equal height on the plot device
(unless there is only one column of panels, in which case they will have equal width on the plot
device). This means that the objects are plotted at different physical scales, by default.
If equal.scales=TRUE, then the dimensions of the plot panels on the plot device will be proportional to the spatial dimensions of the corresponding components of x. This means that the objects
will be plotted at approximately equal physical scales. If these objects have very different spatial
sizes, the plot command could fail (when it tries to plot the smaller objects at a tiny scale), with an
error message that the figure margins are too large.

988

plot.listof
The objects will be plotted at exactly equal physical scales, and exactly aligned on the device, under
the following conditions:
• every component of x is a spatial object whose position can be shifted by shift;
• panel.begin and panel.end are either NULL or they are spatial objects whose position can
be shifted by shift;
• adorn.left, adorn.right, adorn.top and adorn.bottom are all NULL.
Another special case is when every component of x is an object of class "fv" representing a function. If equal.scales=TRUE then all these functions will be plotted with the same axis scales (i.e.
with the same xlim and the same ylim).

Value
Null.
Spacing between plots
The spacing between individual plots is controlled by the parameters mar.panel, hsep and vsep.
If equal.scales=FALSE, the plot panels are logically separate plots. The margins for each panel
are determined by the argument mar.panel which becomes the graphics parameter mar described
in the help file for par. One unit of mar corresponds to one line of text in the margin. If hsep or
vsep are present, mar.panel is augmented by c(vsep, hsep, vsep, hsep)/2.
If equal.scales=TRUE, all the plot panels are drawn in the same coordinate system which represents a physical scale. The unit of measurement for mar.panel[1,3] is one-sixth of the greatest
height of any object plotted in the same row of panels, and the unit for mar.panel[2,4] is onesixth of the greatest width of any object plotted in the same column of panels. If hsep or vsep are
present, they are interpreted in the same units as mar.panel[2] and mar.panel[1] respectively.
Error messages
If the error message ‘Figure margins too large’ occurs, this generally means that one of the objects
had a much smaller physical scale than the others. Ensure that equal.scales=FALSE and increase
the values of mar.panel.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
print.listof, contour.listof, image.listof, density.splitppp
Examples
# Intensity estimate of multitype point pattern
plot(D <- density(split(amacrine)))
plot(D, main="", equal.ribbon=TRUE,
panel.end=function(i,y,...){contour(y, ...)})
# list of 3D point patterns
ape1 <- osteo[osteo$shortid==4, "pts", drop=TRUE]
class(ape1)

plot.lpp

989

plot(ape1, main.panel="", mar.panel=0.1, hsep=0.7, vsep=1,
cex=1.5, pch=21, bg='white')

plot.lpp

Plot Point Pattern on Linear Network

Description
Plots a point pattern on a linear network. Plot method for the class "lpp" of point patterns on a
linear network.
Usage
## S3 method for class 'lpp'
plot(x, ..., main, add = FALSE,
use.marks=TRUE, which.marks=NULL,
show.all = !add, show.window=FALSE, show.network=TRUE,
do.plot = TRUE, multiplot=TRUE)
Arguments
x

Point pattern on a linear network (object of class "lpp").

...

Additional arguments passed to plot.linnet or plot.ppp.

main

Main title for plot.

add

Logical value indicating whether the plot is to be added to the existing plot
(add=TRUE) or whether a new plot should be initialised (add=FALSE, the default).

use.marks

logical flag; if TRUE, plot points using a different plotting symbol for each mark;
if FALSE, only the locations of the points will be plotted, using points().

which.marks

Index determining which column of marks to use, if the marks of x are a data
frame. A character or integer vector identifying one or more columns of marks.
If add=FALSE then the default is to plot all columns of marks, in a series of
separate plots. If add=TRUE then only one column of marks can be plotted, and
the default is which.marks=1 indicating the first column of marks.

show.all

Logical value indicating whether to plot everything including the main title and
the window containing the network.

show.window

Logical value indicating whether to plot the window containing the network.
Overrides show.all.

show.network

Logical value indicating whether to plot the network.

do.plot

Logical value determining whether to actually perform the plotting.

multiplot

Logical value giving permission to display multiple plots.

Details
The linear network is plotted by plot.linnet, then the points are plotted by plot.ppp.
Commonly-used arguments include:
• col and lwd for the colour and width of lines in the linear network
• cols for the colour or colours of the points

990

plot.lppm
• chars for the plot characters representing different types of points
• legend and leg.side to control the graphics legend
Note that the linear network will be plotted even when add=TRUE, unless show.network=FALSE.

Value
(Invisible) object of class "symbolmap" giving the correspondence between mark values and plotting characters.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
lpp.
See plot.ppp for options for representing the points.
See also points.lpp, text.lpp.
Examples
plot(chicago, cols=1:6)

plot.lppm

Plot a Fitted Point Process Model on a Linear Network

Description
Plots the fitted intensity of a point process model on a linear network.
Usage
## S3 method for class 'lppm'
plot(x, ..., type="trend")
Arguments
x

An object of class "lppm" representing a fitted point process model on a linear
network.

...

Arguments passed to plot.linim to control the plot.

type

Character string (either "trend" or "cif") determining whether to plot the fitted
first order trend or the conditional intensity.

Details
This function is the plot method for the class "lppm". It computes the fitted intensity of the point
process model, and displays it using plot.linim.
The default is to display intensity values as colours. Alternatively if the argument style="width"
is given, intensity values are displayed as the widths of thick lines drawn over the network.

plot.mppm

991

Value
Null.
Author(s)
Adrian Baddeley 
See Also
lppm, plot.linim, methods.lppm, predict.lppm.
Examples
X <- runiflpp(10, simplenet)
fit <- lppm(X ~x)
plot(fit)
plot(fit, style="width")

plot.mppm

plot a Fitted Multiple Point Process Model

Description
Given a point process model fitted to multiple point patterns by mppm, compute spatial trend or
conditional intensity surface of the model, in a form suitable for plotting, and (optionally) plot this
surface.
Usage
## S3 method for class 'mppm'
plot(x, ...,
trend=TRUE, cif=FALSE, se=FALSE,
how=c("image", "contour", "persp"))
Arguments
x

A point process model fitted to multiple point patterns, typically obtained from
the model-fitting algorithm mppm. An object of class "mppm".

...

Arguments passed to plot.ppm or plot.anylist controlling the plot.

trend

Logical value indicating whether to plot the fitted trend.

cif

Logical value indicating whether to plot the fitted conditional intensity.

se

Logical value indicating whether to plot the standard error of the fitted trend.

how

Single character string indicating the style of plot to be performed.

Details
This is the plot method for the class "mppm" of point process models fitted to multiple point patterns
(see mppm).
It invokes subfits to compute the fitted model for each individual point pattern dataset, then calls
plot.ppm to plot these individual models. These individual plots are displayed using plot.anylist,
which generates either a series of separate plot frames or an array of plot panels on a single page.

992

plot.msr

Value
NULL.
Author(s)
Adrian Baddeley , Ida-Maria Sintorn and Leanne Bischoff.
Implemented by Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. London: Chapman and Hall/CRC Press.
See Also
plot.ppm, mppm, plot.listof
Examples
# Synthetic data from known model
n <- 9
H <- hyperframe(V=1:n,
U=runif(n, min=-1, max=1))
H$Z <- setcov(square(1))
H$U <- with(H, as.im(U, as.rectangle(Z)))
H$Y <- with(H, rpoispp(eval.im(exp(2+3*Z))))
fit <- mppm(Y ~Z + U + V, data=H)
plot(fit)

plot.msr

Plot a Signed or Vector-Valued Measure

Description
Plot a signed measure or vector-valued measure.
Usage
## S3 method for class 'msr'
plot(x, ...,
add = FALSE,
how = c("image", "contour", "imagecontour"),
main = NULL,
do.plot = TRUE,
multiplot = TRUE,
massthresh = 0,
equal.markscale = FALSE,
equal.ribbon = FALSE)

plot.msr

993

Arguments
x

The signed or vector measure to be plotted. An object of class "msr" (see msr).

...

Extra arguments passed to Smooth.ppp to control the interpolation of the continuous density component of x, or passed to plot.im or plot.ppp to control
the appearance of the plot.

add

Logical flag; if TRUE, the graphics are added to the existing plot. If FALSE (the
default) a new plot is initialised.

how

String indicating how to display the continuous density component.

main

String. Main title for the plot.

do.plot

Logical value determining whether to actually perform the plotting.

multiplot

Logical value indicating whether it is permissible to display a plot with multiple panels (representing different components of a vector-valued measure, or
different types of points in a multitype measure.)

massthresh

Threshold for plotting atoms. A single numeric value or NULL. If massthresh=0
(the default) then only atoms with nonzero mass will be plotted. If massthresh > 0
then only atoms whose absolute mass exceeds massthresh will be plotted. If
massthresh=NULL, then all atoms of the measure will be plotted.

equal.markscale
Logical value indicating whether different panels should use the same symbol
map (to represent the masses of atoms of the measure).
equal.ribbon

Logical value indicating whether different panels should use the same colour
map (to represent the density values in the diffuse component of the measure).

Details
This is the plot method for the class "msr".
The continuous density component of x is interpolated from the existing data by Smooth.ppp, and
then displayed as a colour image by plot.im.
The discrete atomic component of x is then superimposed on this image by plotting the atoms as
circles (for positive mass) or squares (for negative mass) by plot.ppp. By default, atoms with zero
mass are not plotted at all.
To smooth both the discrete and continuous components, use Smooth.msr.
Use the argument clipwin to restrict the plot to a subset of the full data.
To remove atoms with tiny masses, use the argument massthresh.
Value
(Invisible) colour map (object of class "colourmap") for the colour image.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
msr, Smooth.ppp, Smooth.msr, plot.im, plot.ppp

994

plot.onearrow

Examples
X <- rpoispp(function(x,y) { exp(3+3*x) })
fit <- ppm(X, ~x+y)
rp <- residuals(fit, type="pearson")
rs <- residuals(fit, type="score")
plot(rp)
plot(rs)
plot(rs, how="contour")

plot.onearrow

Plot an Arrow

Description
Plots an object of class "onearrow".
Usage
## S3 method for class 'onearrow'
plot(x, ...,
add = FALSE, main = "",
retract = 0.05, headfraction = 0.25, headangle = 12, headnick = 0.1,
col.head = NA, lwd.head = lwd, lwd = 1, col = 1,
zap = FALSE, zapfraction = 0.07,
pch = 1, cex = 1, do.plot = TRUE, do.points = FALSE, show.all = !add)
Arguments
x

Object of class "onearrow" to be plotted. This object is created by the command
onearrow.

...

Additional graphics arguments passed to segments to control the appearance of
the line.

add

Logical value indicating whether to add graphics to the existing plot (add=TRUE)
or to start a new plot (add=FALSE).

main

Main title for the plot.

retract

Fraction of length of arrow to remove at each end.

headfraction

Length of arrow head as a fraction of overall length of arrow.

headangle

Angle (in degrees) between the outer edge of the arrow head and the shaft of the
arrow.

headnick

Size of the nick in the trailing edge of the arrow head as a fraction of length of
arrow head.
col.head,lwd.head
Colour and line style of the filled arrow head.
col,lwd

Colour and line style of the arrow shaft.

zap

Logical value indicating whether the arrow should include a Z-shaped (lightningbolt) feature in the middle of the shaft.

zapfraction

Size of Z-shaped deviation as a fraction of total arrow length.

plot.owin

995

pch,cex

Plot character and character size for the two end points of the arrow, if do.points=TRUE.

do.plot

Logical. Whether to actually perform the plot.

do.points

Logical. Whether to display the two end points of the arrow as well.

show.all

Internal use only.

Details
The argument x should be an object of class "onearrow" created by the command onearrow.
Value
A window (class "owin") enclosing the plotted graphics.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
onearrow, yardstick
Examples
oa <- onearrow(cells[c(1, 42)])
plot(oa)
plot(oa, zap=TRUE, do.points=TRUE, col.head="pink", col="red")

plot.owin

Plot a Spatial Window

Description
Plot a two-dimensional window of observation for a spatial point pattern
Usage
## S3 method for class 'owin'
plot(x, main, add=FALSE, ..., box, edge=0.04,
type=c("w","n"), show.all=!add,
hatch=FALSE,
hatchargs=list(),
invert=FALSE, do.plot=TRUE,
claim.title.space=FALSE)

996

plot.owin

Arguments
x

The window to be plotted. An object of class owin, or data which can be converted into this format by as.owin().
main
text to be displayed as a title above the plot.
add
logical flag: if TRUE, draw the window in the current plot; if FALSE, generate a
new plot.
...
extra arguments controlling the appearance of the plot. These arguments are
passed to polygon if x is a polygonal or rectangular window, or passed to
image.default if x is a binary mask. See Details.
box
logical flag; if TRUE, plot the enclosing rectangular box
edge
nonnegative number; the plotting region will have coordinate limits that are
1 + edge times as large as the limits of the rectangular box that encloses the
pattern.
type
Type of plot: either "w" or "n". If type="w" (the default), the window is plotted.
If type="n" and add=TRUE, a new plot is initialised and the coordinate system
is established, but nothing is drawn.
show.all
Logical value indicating whether to plot everything including the main title.
hatch
logical flag; if TRUE, the interior of the window will be shaded by texture, such
as a grid of parallel lines.
hatchargs
List of arguments passed to add.texture to control the texture shading when
hatch=TRUE.
invert
logical flag; when the window is a binary pixel mask, the mask colours will be
inverted if invert=TRUE.
do.plot
Logical value indicating whether to actually perform the plot.
claim.title.space
Logical value indicating whether extra space for the main title should be allocated when declaring the plot dimensions. Should be set to FALSE under normal
conditions.
Details
This is the plot method for the class owin. The action is to plot the boundary of the window on the
current plot device, using equal scales on the x and y axes.
If the window x is of type "rectangle" or "polygonal", the boundary of the window is plotted
as a polygon or series of polygons. If x is of type "mask" the discrete raster approximation of the
window is displayed as a binary image (white inside the window, black outside).
Graphical parameters controlling the display (e.g. setting the colours) may be passed directly via
the ... arguments, or indirectly reset using spatstat.options.
When x is of type "rectangle" or "polygonal", it is plotted by the R function polygon. To control
the appearance (colour, fill density, line density etc) of the polygon plot, determine the required
argument of polygon and pass it through ... For example, to paint the interior of the polygon in
red, use the argument col="red". To draw the polygon edges in green, use border="green". To
suppress the drawing of polygon edges, use border=NA.
When x is of type "mask", it is plotted by image.default. The appearance of the image plot can
be controlled by passing arguments to image.default through .... The default appearance can
also be changed by setting the parameter par.binary of spatstat.options.
To zoom in (to view only a subset of the window at higher magnification), use the graphical arguments xlim and ylim to specify the desired rectangular field of view. (The actual field of view may
be larger, depending on the graphics device).

plot.owin

997

Value
none.
Notes on Filled Polygons with Holes
The function polygon can only handle polygons without holes. To plot polygons with holes in a
solid colour, we have implemented two workarounds.
polypath function: The first workaround uses the relatively new function polypath which does
have the capability to handle polygons with holes. However, not all graphics devices support
polypath. The older devices xfig and pictex do not support polypath. On a Windows system, the default graphics device windows supports polypath. On a Linux system, the default
graphics device X11(type="Xlib") does not support polypath but X11(type="cairo")
does support it. See X11 and the section on Cairo below.
polygon decomposition: The other workaround involves decomposing the polygonal window into
pieces which do not have holes. This code is experimental but works in all our test cases. If
this code fails, a warning will be issued, and the filled colours will not be plotted.
Cairo graphics on a Linux system
Linux systems support the graphics device X11(type="cairo") (see X11) provided the external
library cairo is installed on the computer. See www.cairographics.org for instructions on obtaining and installing cairo. After having installed cairo one needs to re-install R from source so that
it has cairo capabilites. To check whether your current installation of R has cairo capabilities, type
(in R) capabilities()["cairo"]. The default type for X11 is controlled by X11.options. You
may find it convenient to make cairo the default, e.g. via your .Rprofile. The magic incantation
to put into .Rprofile is
setHook(packageEvent("graphics", "onLoad"),
function(...) grDevices::X11.options(type="cairo"))

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
owin.object, plot.ppp, polygon, image.default, spatstat.options
Examples
# rectangular window
plot(Window(nztrees))
abline(v=148, lty=2)
# polygonal window
w <- Window(demopat)
plot(w)
plot(w, col="red", border="green", lwd=2)
plot(w, hatch=TRUE, lwd=2)
# binary mask

998

plot.plotppm
we <- as.mask(w)
plot(we)
op <- spatstat.options(par.binary=list(col=grey(c(0.5,1))))
plot(we)
spatstat.options(op)

plot.plotppm

Plot a plotppm Object Created by plot.ppm

Description
The function plot.ppm produces objects which specify plots of fitted point process models. The
function plot.plotppm carries out the actual plotting of these objects.
Usage
## S3 method for class 'plotppm'
plot(x, data = NULL, trend = TRUE, cif = TRUE,
se = TRUE, pause = interactive(),
how = c("persp", "image", "contour"),
..., pppargs)
Arguments
x

An object of class plotppm produced by plot.ppm() .

data

The point pattern (an object of class ppp) to which the point process model was
fitted (by ppm).

trend

Logical scalar; should the trend component of the fitted model be plotted?

cif

Logical scalar; should the complete conditional intensity of the fitted model be
plotted?

se

Logical scalar; should the estimated standard error of the fitted intensity be plotted?

pause

Logical scalar indicating whether to pause with a prompt after each plot. Set
pause=FALSE if plotting to a file.

how

Character string or character vector indicating the style or styles of plots to be
performed.

...

Extra arguments to the plotting functions persp, image and contour.

pppargs

List of extra arguments passed to plot.ppp when displaying the original point
pattern data.

Details
If argument data is supplied then the point pattern will be superimposed on the image and contour
plots.
Sometimes a fitted model does not have a trend component, or the trend component may constitute
all of the conditional intensity (if the model is Poisson). In such cases the object x will not contain
a trend component, or will contain only a trend component. This will also be the case if one of
the arguments trend and cif was set equal to FALSE in the call to plot.ppm() which produced x.
If this is so then only the item which is present will be plotted. Explicitly setting trend=TRUE, or
cif=TRUE, respectively, will then give an error.

plot.pp3

999

Value
None.
Warning
Arguments which are passed to persp, image, and contour via the . . . argument get passed to any
of the other functions listed in the how argument, and won’t be recognized by them. This leads
to a lot of annoying but harmless warning messages. Arguments to persp may be supplied via
spatstat.options() which alleviates the warning messages in this instance.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
plot.ppm()
Examples
## Not run:
m <- ppm(cells ~ 1, Strauss(0.05))
mpic <- plot(m)
# Perspective plot only, with altered parameters:
plot(mpic,how="persp", theta=-30,phi=40,d=4)
# All plots, with altered parameters for perspective plot:
op <- spatstat.options(par.persp=list(theta=-30,phi=40,d=4))
plot(mpic)
# Revert
spatstat.options(op)
## End(Not run)

plot.pp3

Plot a Three-Dimensional Point Pattern

Description
Plots a three-dimensional point pattern.
Usage
## S3 method for class 'pp3'
plot(x, ..., eye=NULL, org=NULL, theta=25, phi=15,
type=c("p", "n", "h"),
box.back=list(col="pink"),
box.front=list(col="blue", lwd=2))

1000

plot.pp3

Arguments
x

Three-dimensional point pattern (object of class "pp3").

...

Arguments passed to points controlling the appearance of the points.

eye

Optional. Eye position. A numeric vector of length 3 giving the location from
which the scene is viewed.

org

Optional. Origin (centre) of the view. A numeric vector of length 3 which will
be at the centre of the view.

theta,phi

Optional angular coordinates (in degrees) specifying the direction from which
the scene is viewed: theta is the azimuth and phi is the colatitude. Ignored if
eye is given.

type

Type of plot: type="p" for points, type="h" for points on vertical lines, type="n"
for box only.
box.front,box.back
How to plot the three-dimensional box that contains the points. A list of graphical arguments passed to segments, or a logical value indicating whether or not
to plot the relevant part of the box. See Details.

Details
This is the plot method for objects of class "pp3". It generates a two-dimensional plot of the point
pattern x and its containing box as if they had been viewed from the location specified by eye (or
from the direction specified by theta and phi).
The edges of the box at the ‘back’ of the scene (as viewed from the eye position) are plotted first.
Then the points are added. Finally the remaining ‘front’ edges are plotted. The arguments box.back
and box.front specify graphical parameters for drawing the back and front edges, respectively.
Alternatively box.back=FALSE specifies that the back edges shall not be drawn.
Note that default values of arguments to plot.pp3 can be set by spatstat.options("par.pp3").
Value
Null.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
pp3, spatstat.options.
Examples
X <- osteo$pts[[1]]
plot(X, main="Osteocyte lacunae, animal 1, brick 1",
cex=1.5, pch=16)
plot(X, main="", box.back=list(lty=3))

plot.ppm

plot.ppm

1001

plot a Fitted Point Process Model

Description
Given a fitted point process model obtained by ppm, create spatial trend and conditional intensity
surfaces of the model, in a form suitable for plotting, and (optionally) plot these surfaces.
Usage
## S3 method for class 'ppm'
plot(x, ngrid = c(40,40), superimpose = TRUE,
trend = TRUE, cif = TRUE, se = TRUE, pause = interactive(),
how=c("persp","image", "contour"), plot.it = TRUE,
locations = NULL, covariates=NULL, ...)
Arguments
x

A fitted point process model, typically obtained from the model-fitting algorithm
ppm. An object of class "ppm".

ngrid

The dimensions for a grid on which to evaluate, for plotting, the spatial trend
and conditional intensity. A vector of 1 or 2 integers. If it is of length 1, ngrid
is replaced by c(ngrid,ngrid).

superimpose

logical flag; if TRUE (and if plot=TRUE) the original data point pattern will be
superimposed on the plots.

trend

logical flag; if TRUE, the spatial trend surface will be produced.

cif

logical flag; if TRUE, the conditional intensity surface will be produced.

se

logical flag; if TRUE, the estimated standard error of the spatial trend surface will
be produced.

pause

logical flag indicating whether to pause with a prompt after each plot. Set
pause=FALSE if plotting to a file. (This flag is ignored if plot=FALSE).

how

character string or character vector indicating the style or styles of plots to be
performed. Ignored if plot=FALSE.

plot.it

logical scalar; should a plot be produced immediately?

locations

If present, this determines the locations of the pixels at which predictions are
computed. It must be a binary pixel image (an object of class "owin" with type
"mask"). (Incompatible with ngrid).

covariates

Values of external covariates required by the fitted model. Passed to predict.ppm.

...

extra arguments to the plotting functions persp, image and contour.

Details
This is the plot method for the class "ppm" (see ppm.object for details of this class).
It invokes predict.ppm to compute the spatial trend and conditional intensity of the fitted point
process model. See predict.ppm for more explanation about spatial trend and conditional intensity.
The default action is to create a rectangular grid of points in (the bounding box of) the observation
window of the data point pattern, and evaluate the spatial trend and conditional intensity of the fitted

1002

plot.ppm

spatial point process model x at these locations. If the argument locations= is supplied, then the
spatial trend and conditional intensity are calculated at the grid of points specified by this argument.
The argument locations, if present, should be a binary image mask (an object of class "owin" and
type "mask"). This determines a rectangular grid of locations, or a subset of such a grid, at which
predictions will be computed. Binary image masks are conveniently created using as.mask.
The argument covariates gives the values of any spatial covariates at the prediction locations. If
the trend formula in the fitted model involves spatial covariates (other than the Cartesian coordinates
x, y) then covariates is required.
The argument covariates has the same format and interpretation as in predict.ppm. It may be
either a data frame (the number of whose rows must match the number of pixels in locations
multiplied by the number of possible marks in the point pattern), or a list of images. If argument
locations is not supplied, and covariates is supplied, then it must be a list of images.
If the fitted model was a marked (multitype) point process, then predictions are made for each
possible mark value in turn.
If the fitted model had no spatial trend, then the default is to omit calculating this (flat) surface,
unless trend=TRUE is set explicitly.
If the fitted model was Poisson, so that there were no spatial interactions, then the conditional
intensity and spatial trend are identical, and the default is to omit the conditional intensity, unless
cif=TRUE is set explicitly.
If plot.it=TRUE then plot.plotppm() is called upon to plot the class plotppm object which is
produced. (That object is also returned, silently.)
Plots are produced successively using persp, image and contour (or only a selection of these three,
if how is given). Extra graphical parameters controlling the display may be passed directly via the
arguments ... or indirectly reset using spatstat.options.
Value
An object of class plotppm. Such objects may be plotted by plot.plotppm().
This is a list with components named trend and cif, either of which may be missing. They will be
missing if the corresponding component does not make sense for the model, or if the corresponding
argument was set equal to FALSE.
Both trend and cif are lists of images. If the model is an unmarked point process, then they are
lists of length 1, so that trend[[1]] is an image of the spatial trend and cif[[1]] is an image of
the conditional intensity.
If the model is a marked point process, then trend[[i]] is an image of the spatial trend for the
mark m[i], and cif[[1]] is an image of the conditional intensity for the mark m[i], where m is the
vector of levels of the marks.
Warnings
See warnings in predict.ppm.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
plot.plotppm, ppm, ppm.object, predict.ppm, print.ppm, persp, image, contour, plot, spatstat.options

plot.ppp

1003

Examples
m <- ppm(cells ~1, Strauss(0.05))
pm <- plot(m) # The object ``pm'' will be plotted as well as saved
# for future plotting.

plot.ppp

plot a Spatial Point Pattern

Description
Plot a two-dimensional spatial point pattern
Usage
## S3 method for class 'ppp'
plot(x, main, ..., clipwin=NULL,
chars=NULL, cols=NULL,
use.marks=TRUE, which.marks=NULL,
add=FALSE, type=c("p","n"),
legend=TRUE,
leg.side=c("left", "bottom", "top", "right"),
leg.args=list(),
symap=NULL, maxsize=NULL, meansize=NULL, markscale=NULL,
zap=0.01,
show.window=show.all, show.all=!add, do.plot=TRUE,
multiplot=TRUE)
Arguments
x

The spatial point pattern to be plotted. An object of class "ppp", or data which
can be converted into this format by as.ppp().

main

text to be displayed as a title above the plot.

...

extra arguments that will be passed to the plotting functions plot.default,
points and/or symbols.

clipwin

Optional. A window (object of class "owin"). Only this subset of the image will
be displayed.

chars

plotting character(s) used to plot points.

cols

the colour(s) used to plot points.

use.marks

logical flag; if TRUE, plot points using a different plotting symbol for each mark;
if FALSE, only the locations of the points will be plotted, using points().

which.marks

Index determining which column of marks to use, if the marks of x are a data
frame. A character or integer vector identifying one or more columns of marks.
If add=FALSE then the default is to plot all columns of marks, in a series of
separate plots. If add=TRUE then only one column of marks can be plotted, and
the default is which.marks=1 indicating the first column of marks.

add

logical flag; if TRUE, just the points are plotted, over the existing plot. A new
plot is not created, and the window is not plotted.

1004

plot.ppp

type

Type of plot: either "p" or "n". If type="p" (the default), both the points and
the observation window are plotted. If type="n", only the window is plotted.

legend

Logical value indicating whether to add a legend showing the mapping between
mark values and graphical symbols (for a marked point pattern).

leg.side

Position of legend relative to main plot.

leg.args

List of additional arguments passed to plot.symbolmap or symbolmap to control the legend. In addition to arguments documented under plot.symbolmap,
and graphical arguments recognised by symbolmap, the list may also include the
argument sep giving the separation between the main plot and the legend, or
sep.frac giving the separation as a fraction of the relevant dimension (width or
height) of the main plot.

symap

Optional. The graphical symbol map to be applied to the marks. An object of
class "symbolmap"; see symbolmap.

maxsize

Maximum physical size of the circles/squares plotted when x is a marked point
pattern with numerical marks. Incompatible with meansize and markscale.
Ignored if symap is given.

meansize

Average physical size of the circles/squares plotted when x is a marked point
pattern with numerical marks. Incompatible with maxsize and markscale. Ignored if symap is given.

markscale

physical scale factor determining the sizes of the circles/squares plotted when
x is a marked point pattern with numerical marks. Mark value will be multiplied by markscale to determine physical size. Incompatible with maxsize and
meansize. Ignored if symap is given.

zap

Fraction between 0 and 1. When x is a marked point pattern with numerical
marks, zap is the smallest mark value (expressed as a fraction of the maximum
possible mark) that will be plotted. Any points which have marks smaller in
absolute value than zap * max(abs(marks(x))) will not be plotted.

show.window

Logical value indicating whether to plot the observation window of x.

show.all

Logical value indicating whether to plot everything including the main title and
the observation window of x.

do.plot

Logical value determining whether to actually perform the plotting.

multiplot

Logical value giving permission to display multiple plots.

Details
This is the plot method for point pattern datasets (of class "ppp", see ppp.object).
First the observation window Window(x) is plotted (if show.window=TRUE). Then the points themselves are plotted, in a fashion that depends on their marks, as follows.
unmarked point pattern: If the point pattern does not have marks, or if use.marks = FALSE,
then the locations of all points will be plotted using a single plot character
multitype point pattern: If x$marks is a factor, then each level of the factor is represented by a
different plot character.
continuous marks: If x$marks is a numeric vector, the marks are rescaled to the unit interval and
each point is represented by a circle with diameter proportional to the rescaled mark (if the
value is positive) or a square with side length proportional to the absolute value of the rescaled
mark (if the value is negative).

plot.ppp

1005

other kinds of marks: If x$marks is neither numeric nor a factor, then each possible mark will be
represented by a different plotting character. The default is to represent the ith smallest mark
value by points(..., pch=i).
If there are several columns of marks, and if which.marks is missing or NULL, then
• if add=FALSE and multiplot=TRUE the default is to plot all columns of marks, in a series of
separate plots, placed side-by-side. The plotting is coordinated by plot.listof, which calls
plot.ppp to make each of the individual plots.
• Otherwise, only one column of marks can be plotted, and the default is which.marks=1 indicating the first column of marks.
Plotting of the window Window(x) is performed by plot.owin. This plot may be modified through
the ... arguments. In particular the extra argument border determines the colour of the window,
if the window is not a binary mask.
Plotting of the points themselves is performed by the function points, except for the case of continuous marks, where it is performed by symbols. Their plotting behaviour may be modified through
the ... arguments.
The argument chars determines the plotting character or characters used to display the points (in
all cases except for the case of continuous marks). For an unmarked point pattern, this should be a
single integer or character determining a plotting character (see par("pch")). For a multitype point
pattern, chars should be a vector of integers or characters, of the same length as levels(x$marks),
and then the ith level or type will be plotted using character chars[i].
If chars is absent, but there is an extra argument pch, then this will determine the plotting character
for all points.
The argument cols determines the colour or colours used to display the points. For an unmarked
point pattern, cols should be a character string determining a colour. For a multitype point pattern,
cols should be a character vector, of the same length as levels(marks(x)): that is, there is one
colour for each possible mark value. The ith level or type will be plotted using colour cols[i]. For
a point pattern with continuous marks, cols can be either a character string or a character vector
specifying colour values: the range of mark values will be mapped to the specified colours.
If cols is absent, the colours used to plot the points may be determined by the extra argument fg
(for multitype point patterns) or the extra argument col (for all other cases). Note that specifying
col will also apply this colour to the window itself.
The default colour for the points is a semi-transparent grey, if this is supported by the plot device. This behaviour can be suppressed (so that the default colour is non-transparent) by setting
spatstat.options(transparent=FALSE).
The arguments maxsize, meansize and markscale incompatible. They control the physical size
of the circles and squares which represent the marks in a point pattern with continuous marks. The
size of a circle is defined as its diameter; the size of a square is its side length. If markscale is
given, then a mark value of m is plotted as a circle of diameter m * markscale (if m is positive)
or a square of side abs(m) * markscale (if m is negative). If maxsize is given, then the largest
mark in absolute value, mmax=max(abs(marks(x))), will be scaled to have physical size maxsize.
If meansize is given, then the average absolute mark value, mmean=mean(abs(marks(x))), will be
scaled to have physical size meansize.
The user can set the default values of these plotting parameters using spatstat.options("par.points").
To zoom in (to view only a subset of the point pattern at higher magnification), use the graphical
arguments xlim and ylim to specify the rectangular field of view.
The value returned by this plot function is an object of class "symbolmap" representing the mapping
from mark values to graphical symbols. See symbolmap. It can be used to make a suitable legend,
or to ensure that two plots use the same graphics map.

1006

plot.ppp

Value
(Invisible) object of class "symbolmap" giving the correspondence between mark values and plotting characters.
Removing White Space Around The Plot
A frequently-asked question is: How do I remove the white space around the plot? Currently
plot.ppp uses the base graphics system of R, so the space around the plot is controlled by parameters to par. To reduce the white space, change the parameter mar. Typically, par(mar=rep(0.5, 4))
is adequate, if there are no annotations or titles outside the window.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
iplot, ppp.object, plot, par, points, text.ppp, plot.owin, symbols
Examples
plot(cells)
plot(cells, pch=16)
# make the plotting symbols larger (for publication at reduced scale)
plot(cells, cex=2)
# set it in spatstat.options
oldopt <- spatstat.options(par.points=list(cex=2))
plot(cells)
spatstat.options(oldopt)
# multitype
plot(lansing)
# marked by a real number
plot(longleaf)
# just plot the points
plot(longleaf, use.marks=FALSE)
plot(unmark(longleaf)) # equivalent
# point pattern with multiple marks
plot(finpines)
plot(finpines, which.marks="height")
# controlling COLOURS of points
plot(cells, cols="blue")
plot(lansing, cols=c("black", "yellow", "green",
"blue","red","pink"))
plot(longleaf, fg="blue")
# make window purple
plot(lansing, border="purple")

plot.psp

1007

# make everything purple
plot(lansing, border="purple", cols="purple", col.main="purple",
leg.args=list(col.axis="purple"))
# controlling PLOT CHARACTERS for multitype pattern
plot(lansing, chars = 11:16)
plot(lansing, chars = c("o","h","m",".","o","o"))
## multitype pattern mapped to symbols
plot(amacrine, shape=c("circles", "squares"), size=0.04)
plot(amacrine, shape="arrows", direction=c(0,90), size=0.07)
## plot trees as trees!
plot(lansing, shape="arrows", direction=90, cols=1:6)
# controlling MARK SCALE for pattern with numeric marks
plot(longleaf, markscale=0.1)
plot(longleaf, maxsize=5)
plot(longleaf, meansize=2)
# draw circles of diameter equal to nearest neighbour distance
plot(cells %mark% nndist(cells), markscale=1, legend=FALSE)
# inspecting the symbol map
v <- plot(amacrine)
v
## variable colours ('cols' not 'col')
plot(longleaf, cols=function(x) ifelse(x < 30, "red", "black"))
## re-using the same mark scale
a <- plot(longleaf)
juveniles <- longleaf[marks(longleaf) < 30]
plot(juveniles, symap=a)
## numerical marks mapped to symbols of fixed size with variable colour
ra <- range(marks(longleaf))
colmap <- colourmap(terrain.colors(20), range=ra)
## filled plot characters are the codes 21-25
## fill colour is indicated by 'bg'
sy <- symbolmap(pch=21, bg=colmap, range=ra)
plot(longleaf, symap=sy)
## or more compactly..
plot(longleaf, bg=terrain.colors(20), pch=21, cex=1)
## clipping
plot(humberside)
B <- owin(c(4810, 5190), c(4180, 4430))
plot(B, add=TRUE, border="red")
plot(humberside, clipwin=B, main="Humberside (clipped)")

plot.psp

plot a Spatial Line Segment Pattern

1008

plot.psp

Description
Plot a two-dimensional line segment pattern
Usage
## S3 method for class 'psp'
plot(x, ..., main, add=FALSE,
show.all=!add, show.window=show.all,
which.marks=1, ribbon=show.all,
ribsep=0.15, ribwid=0.05, ribn=1024,
do.plot=TRUE)
Arguments
x

The line segment pattern to be plotted. An object of class "psp", or data which
can be converted into this format by as.psp().

...

extra arguments that will be passed to the plotting functions segments (to plot
the segments) and plot.owin (to plot the observation window).

main

Character string giving a title for the plot.

add

Logical. If TRUE, the current plot is not erased; the segments are plotted on top
of the current plot, and the window is not plotted (by default).

show.all

Logical value specifying whether to plot everything including the window, main
title, and colour ribbon.

show.window

Logical value specifying whether to plot the window.

which.marks

Index determining which column of marks to use, if the marks of x are a data
frame. A character string or an integer. Defaults to 1 indicating the first column
of marks.

ribbon

Logical flag indicating whether to display a ribbon showing the colour map (in
which mark values are associated with colours).

ribsep

Factor controlling the space between the ribbon and the image.

ribwid

Factor controlling the width of the ribbon.

ribn

Number of different values to display in the ribbon.

do.plot

Logical value indicating whether to actually perform the plot.

Details
This is the plot method for line segment pattern datasets (of class "psp", see psp.object). It plots
both the observation window Window(x) and the line segments themselves.
Plotting of the window Window(x) is performed by plot.owin. This plot may be modified through
the ... arguments.
Plotting of the segments themselves is performed by the standard R function segments. Its plotting
behaviour may also be modified through the ... arguments.
For a marked line segment pattern (i.e. if marks(x) is not NULL) the line segments are plotted
in colours determined by the mark values. If marks(x) is a data frame, the default is to use the
first column of marks(x) to determine the colours. To specify another column, use the argument
which.marks. The colour map (associating mark values with colours) will be displayed as a vertical
colour ribbon to the right of the plot, if ribbon=TRUE.

plot.quad

1009

Value
(Invisibly) a colour map object specifying the association between marks and colours, if any. The
return value also has an attribute "bbox" giving a bounding box for the plot.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
psp.object, plot, par, plot.owin, text.psp, symbols
Examples
X <- psp(runif(20), runif(20), runif(20), runif(20), window=owin())
plot(X)
plot(X, lwd=3)
lettuce <- sample(letters[1:4], 20, replace=TRUE)
marks(X) <- data.frame(A=1:20, B=factor(lettuce))
plot(X)
plot(X, which.marks="B")

plot.quad

Plot a Spatial Quadrature Scheme

Description
Plot a two-dimensional spatial quadrature scheme.
Usage
## S3 method for class 'quad'
plot(x, ..., main, add=FALSE, dum=list(), tiles=FALSE)
Arguments
x

The spatial quadrature scheme to be plotted. An object of class "quad".

...

extra arguments controlling the plotting of the data points of the quadrature
scheme.

main

text to be displayed as a title above the plot.

add

Logical value indicating whether the graphics should be added to the current
plot if there is one (add=TRUE) or whether a new plot should be initialised
(add=FALSE, the default).

dum

list of extra arguments controlling the plotting of the dummy points of the quadrature scheme. See below.

tiles

Logical value indicating whether to display the tiles used to compute the quadrature weights.

1010

plot.quadratcount

Details
This is the plot method for quadrature schemes (objects of class "quad", see quad.object).
First the data points of the quadrature scheme are plotted (in their observation window) using
plot.ppp with any arguments specified in ...
Then the dummy points of the quadrature scheme are plotted using plot.ppp with any arguments
specified in dum.
By default the dummy points are superimposed onto the plot of data points. This can be overridden
by including the argument add=FALSE in the list dum as shown in the examples. In this case the data
and dummy point patterns are plotted separately.
See par and plot.ppp for other possible arguments controlling the plots.
Value
NULL.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
quad.object, plot.ppp, par
Examples
data(nztrees)
Q <- quadscheme(nztrees)
plot(Q, main="NZ trees: quadrature scheme")
oldpar <- par(mfrow=c(2,1))
plot(Q, main="NZ trees", dum=list(add=FALSE))
par(oldpar)

plot.quadratcount

Plot Quadrat Counts

Description
Given a table of quadrat counts for a spatial point pattern, plot the quadrats which were used, and
display the quadrat count as text in the centre of each quadrat.
Usage
## S3 method for class 'quadratcount'
plot(x, ..., add = FALSE,
entries = as.vector(t(as.table(x))),
dx = 0, dy = 0, show.tiles = TRUE,
textargs = list())

plot.quadratcount

1011

Arguments
x

Object of class "quadratcount" produced by the function quadratcount.

...

Additional arguments passed to plot.tess to plot the quadrats.

add

Logical. Whether to add the graphics to an existing plot.

entries

Vector of numbers to be plotted in each quadrat. The default is to plot the quadrat
counts.

dx,dy

Horizontal and vertical displacement of text relative to centroid of quadrat.

show.tiles

Logical value indicating whether to plot the quadrats.

textargs

List containing extra arguments passed to text.default to control the annotation.

Details
This is the plot method for the objects of class "quadratcount" that are produced by the function
quadratcount. Given a spatial point pattern, quadratcount divides the observation window into
disjoint tiles or quadrats, counts the number of points in each quadrat, and stores the result as a
contingency table which also belongs to the class "quadratcount".
First the quadrats are plotted (provided show.tiles=TRUE, the default). This display can be controlled by passing additional arguments ... to plot.tess.
Then the quadrat counts are printed using text.default. This display can be controlled using the
arguments dx,dy and textargs.

Value
Null.

Author(s)
Adrian Baddeley 
and Rolf Turner 

See Also
quadratcount, plot.tess, text.default, plot.quadrattest.

Examples
plot(quadratcount(swedishpines, 5))

1012

plot.quadrattest

plot.quadrattest

Display the result of a quadrat counting test.

Description
Given the result of a quadrat counting test, graphically display the quadrats that were used, the
observed and expected counts, and the residual in each quadrat.
Usage
## S3 method for class 'quadrattest'
plot(x, ..., textargs=list())
Arguments
x

Object of class "quadrattest" containing the result of quadrat.test.

...

Additional arguments passed to plot.tess to control the display of the quadrats.

textargs

List of additional arguments passed to text.default to control the appearance
of the text.

Details
This is the plot method for objects of class "quadrattest". Such an object is produced by
quadrat.test and represents the result of a χ2 test for a spatial point pattern.
The quadrats are first plotted using plot.tess. Then in each quadrat, the observed and expected
counts and the Pearson residual are displayed as text using text.default. Observed count is
displayed at top left; expected count at top right; and Pearson residual at bottom.
Value
Null.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
quadrat.test, plot.tess, text.default, plot.quadratcount
Examples
plot(quadrat.test(swedishpines, 3))

plot.rppm

plot.rppm

1013

Plot a Recursively Partitioned Point Process Model

Description
Given a model which has been fitted to point pattern data by recursive partitioning, plot the partition
tree or the fitted intensity.
Usage
## S3 method for class 'rppm'
plot(x, ..., what = c("tree", "spatial"), treeplot=NULL)
Arguments
x

Fitted point process model of class "rppm" produced by the function rppm.

what

Character string (partially matched) specifying whether to plot the partition tree
or the fitted intensity.

...

Arguments passed to plot.rpart and text.rpart (if what="tree") or passed
to plot.im (if what="spatial") controlling the appearance of the plot.

treeplot

Optional. A function to be used to plot and label the partition tree, replacing the
two functions plot.rpart and text.rpart.

Details
If what="tree" (the default), the partition tree will be plotted using plot.rpart, and labelled using
text.rpart.
If the argument treeplot is given, then plotting and labelling will be performed by treeplot
instead. A good choice is the function prp in package rpart.plot.
If what="spatial", the predicted intensity will be computed using predict.rppm, and this intensity will be plotted as an image using plot.im.
Value
If what="tree", a list containing x and y coordinates of the plotted nodes of the tree. If what="spatial",
the return value of plot.im.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
rppm

1014

plot.scan.test

Examples
# Murchison gold data
mur <- solapply(murchison, rescale, s=1000, unitname="km")
mur$dfault <- distfun(mur$faults)
#
fit <- rppm(gold ~ dfault + greenstone, data=mur)
#
opa <- par(mfrow=c(1,2))
plot(fit)
plot(fit, what="spatial")
par(opa)

plot.scan.test

Plot Result of Scan Test

Description
Computes or plots an image showing the likelihood ratio test statistic for the scan test, or the optimal
circle radius.
Usage
## S3 method for class 'scan.test'
plot(x, ..., what=c("statistic", "radius"),
do.window = TRUE)
## S3 method for class 'scan.test'
as.im(X, ..., what=c("statistic", "radius"))
Arguments
x,X

Result of a scan test. An object of class "scan.test" produced by scan.test.

...

Arguments passed to plot.im to control the appearance of the plot.

what

Character string indicating whether to produce an image of the (profile) likelihood ratio test statistic (what="statistic", the default) or an image of the
optimal value of circle radius (what="radius").

do.window

Logical value indicating whether to plot the original window of the data as well.

Details
These functions extract, and plot, the spatially-varying value of the likelihood ratio test statistic
which forms the basis of the scan test.
If the test result X was based on circles of the same radius r, then as.im(X) is a pixel image of the
likelihood ratio test statistic as a function of the position of the centre of the circle.
If the test result X was based on circles of several different radii r, then as.im(X) is a pixel image
of the profile (maximum value over all radii r) likelihood ratio test statistic as a function of the
position of the centre of the circle, and as.im(X, what="radius") is a pixel image giving for each
location u the value of r which maximised the likelihood ratio test statistic at that location.
The plot method plots the corresponding image.

plot.slrm

1015

Value
The value of as.im.scan.test is a pixel image (object of class "im"). The value of plot.scan.test
is NULL.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
scan.test, scanLRTS
Examples
if(interactive()) {
a <- scan.test(redwood, seq(0.04, 0.1, by=0.01),
method="poisson", nsim=19)
} else {
a <- scan.test(redwood, c(0.05, 0.1), method="poisson", nsim=2)
}
plot(a)
as.im(a)
plot(a, what="radius")

plot.slrm

Plot a Fitted Spatial Logistic Regression

Description
Plots a fitted Spatial Logistic Regression model.
Usage
## S3 method for class 'slrm'
plot(x, ..., type = "intensity")
Arguments
x

a fitted spatial logistic regression model. An object of class "slrm".

...

Extra arguments passed to plot.im to control the appearance of the plot.

type

Character string (partially) matching one of "probabilities", "intensity"
or "link".

Details
This is a method for plot for fitted spatial logistic regression models (objects of class "slrm",
usually obtained from the function slrm).
This function plots the result of predict.slrm.

1016

plot.solist

Value
None.
Author(s)
Adrian Baddeley  
and Rolf Turner 
See Also
slrm, predict.slrm, plot.im
Examples
data(copper)
X <- copper$SouthPoints
Y <- copper$SouthLines
Z <- distmap(Y)
fit <- slrm(X ~ Z)
plot(fit)
plot(fit, type="link")

plot.solist

Plot a List of Spatial Objects

Description
Plots a list of two-dimensional spatial objects.
Usage
## S3 method for class 'solist'
plot(x, ..., main, arrange=TRUE,
nrows=NULL, ncols=NULL, main.panel=NULL,
mar.panel=c(2,1,1,2), hsep=0, vsep=0,
panel.begin=NULL, panel.end=NULL, panel.args=NULL,
panel.begin.args=NULL, panel.end.args=NULL, panel.vpad = 0.2,
plotcommand="plot",
adorn.left=NULL, adorn.right=NULL, adorn.top=NULL, adorn.bottom=NULL,
adorn.size=0.2, equal.scales=FALSE, halign=FALSE, valign=FALSE)
Arguments
x

An object of the class "solist", essentially a list of two-dimensional spatial
datasets.

...

Arguments passed to plot when generating each plot panel.

main

Overall heading for the plot.

arrange

Logical flag indicating whether to plot the objects side-by-side on a single page
(arrange=TRUE) or plot them individually in a succession of frames (arrange=FALSE).

nrows,ncols

Optional. The number of rows/columns in the plot layout (assuming arrange=TRUE).
You can specify either or both of these numbers.

plot.solist

1017

main.panel

Optional. A character string, or a vector of character strings, giving the headings
for each of the objects.

mar.panel

Size of the margins outside each plot panel. A numeric vector of length 4 giving
the bottom, left, top, and right margins in that order. (Alternatively the vector
may have length 1 or 2 and will be replicated to length 4). See the section on
Spacing between plots.

hsep,vsep

Additional horizontal and vertical separation between plot panels, expressed in
the same units as mar.panel.
panel.begin,panel.end
Optional. Functions that will be executed before and after each panel is plotted.
See Details.
panel.args

Optional. Function that determines different plot arguments for different panels.
See Details.
panel.begin.args
Optional. List of additional arguments for panel.begin when it is a function.
panel.end.args Optional. List of additional arguments for panel.end when it is a function.
panel.vpad

Amount of extra vertical space that should be allowed for the title of each panel,
if a title will be displayed. Expressed as a fraction of the height of the panel.
Applies only when equal.scales=FALSE (the default).

plotcommand

Optional. Character string containing the name of the command that should be
executed to plot each panel.
adorn.left,adorn.right,adorn.top,adorn.bottom
Optional. Functions (with no arguments) that will be executed to generate additional plots at the margins (left, right, top and/or bottom, respectively) of the
array of plots.
adorn.size

Relative width (as a fraction of the other panels’ widths) of the margin plots.

equal.scales

Logical value indicating whether the components should be plotted at (approximately) the same physical scale.

halign,valign

Logical values indicating whether panels in a column should be aligned to the
same x coordinate system (halign=TRUE) and whether panels in a row should be
aligned to the same y coordinate system (valign=TRUE). These are applicable
only if equal.scales=TRUE.

Details
This is the plot method for the class "solist".
An object of class "solist" represents a list of two-dimensional spatial datasets. This is the plot
method for such objects.
In the spatstat package, various functions produce an object of class "solist". These objects can
be plotted in a nice arrangement using plot.solist. See the Examples.
The argument panel.args determines extra graphics parameters for each panel. It should be a
function that will be called as panel.args(i) where i is the panel number. Its return value should
be a list of graphics parameters that can be passed to the relevant plot method. These parameters
override any parameters specified in the ... arguments.
The arguments panel.begin and panel.end determine graphics that will be plotted before and after each panel is plotted. They may be objects of some class that can be plotted with the generic plot
command. Alternatively they may be functions that will be called as panel.begin(i, y, main=main.panel[i])
and panel.end(i, y, add=TRUE) where i is the panel number and y = x[[i]].

1018

plot.solist

If all entries of x are pixel images, the function image.listof is called to control the plotting. The
arguments equal.ribbon and col can be used to determine the colour map or maps applied.
If equal.scales=FALSE (the default), then the plot panels will have equal height on the plot device
(unless there is only one column of panels, in which case they will have equal width on the plot
device). This means that the objects are plotted at different physical scales, by default.
If equal.scales=TRUE, then the dimensions of the plot panels on the plot device will be proportional to the spatial dimensions of the corresponding components of x. This means that the objects
will be plotted at approximately equal physical scales. If these objects have very different spatial
sizes, the plot command could fail (when it tries to plot the smaller objects at a tiny scale), with an
error message that the figure margins are too large.
The objects will be plotted at exactly equal physical scales, and exactly aligned on the device, under
the following conditions:
• every component of x is a spatial object whose position can be shifted by shift;
• panel.begin and panel.end are either NULL or they are spatial objects whose position can
be shifted by shift;
• adorn.left, adorn.right, adorn.top and adorn.bottom are all NULL.
Another special case is when every component of x is an object of class "fv" representing a function. If equal.scales=TRUE then all these functions will be plotted with the same axis scales (i.e.
with the same xlim and the same ylim).
Value
Null.
Spacing between plots
The spacing between individual plots is controlled by the parameters mar.panel, hsep and vsep.
If equal.scales=FALSE, the plot panels are logically separate plots. The margins for each panel
are determined by the argument mar.panel which becomes the graphics parameter mar described
in the help file for par. One unit of mar corresponds to one line of text in the margin. If hsep or
vsep are present, mar.panel is augmented by c(vsep, hsep, vsep, hsep)/2.
If equal.scales=TRUE, all the plot panels are drawn in the same coordinate system which represents a physical scale. The unit of measurement for mar.panel[1,3] is one-sixth of the greatest
height of any object plotted in the same row of panels, and the unit for mar.panel[2,4] is onesixth of the greatest width of any object plotted in the same column of panels. If hsep or vsep are
present, they are interpreted in the same units as mar.panel[2] and mar.panel[1] respectively.
Error messages
If the error message ‘Figure margins too large’ occurs, this generally means that one of the objects
had a much smaller physical scale than the others. Ensure that equal.scales=FALSE and increase
the values of mar.panel.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 

plot.splitppp

1019

See Also
plot.anylist, contour.listof, image.listof, density.splitppp
Examples
# Intensity estimate of multitype point pattern
plot(D <- density(split(amacrine)))
plot(D, main="", equal.ribbon=TRUE,
panel.end=function(i,y,...){contour(y, ...)})

plot.splitppp

Plot a List of Point Patterns

Description
Plots a list of point patterns.
Usage
## S3 method for class 'splitppp'
plot(x, ..., main)
Arguments
x

A named list of point patterns, typically obtained from split.ppp.

...

Arguments passed to plot.listof which control the layout of the plot panels,
their appearance, and the plot behaviour in individual plot panels.

main

Optional main title for the plot.

Details
This is the plot method for the class "splitppp". It is typically used to plot the result of the
function split.ppp.
The argument x should be a named list of point patterns (objects of class "ppp", see ppp.object).
Each of these point patterns will be plotted in turn using plot.ppp.
Plotting is performed by plot.listof.
Value
Null.
Error messages
If the error message ‘Figure margins too large’ occurs, ensure that equal.scales=FALSE and increase the values of mar.panel.
Author(s)
Adrian Baddeley 
and Rolf Turner 

1020

plot.ssf

See Also
plot.listof for arguments controlling the plot.
split.ppp, plot.ppp, ppp.object.
Examples
# Multitype point pattern
plot(split(amacrine))
plot(split(amacrine), main="",
panel.begin=function(i, y, ...) { plot(density(y), ribbon=FALSE, ...) })

plot.ssf

Plot a Spatially Sampled Function

Description
Plot a spatially sampled function object.
Usage
## S3 method for class 'ssf'
plot(x, ...,
how = c("smoothed", "nearest", "points"),
style = c("image", "contour", "imagecontour"),
sigma = NULL, contourargs=list())
## S3 method for class 'ssf'
image(x, ...)
## S3 method for class 'ssf'
contour(x, ..., main, sigma = NULL)
Arguments
x

Spatially sampled function (object of class "ssf").

...

Arguments passed to image.default or plot.ppp to control the plot.

how

Character string determining whether to display the function values at the data
points (how="points"), a smoothed interpolation of the function (how="smoothed"),
or the function value at the nearest data point (how="nearest").

style

Character string indicating whether to plot the smoothed function as a colour
image, a contour map, or both.

contourargs

Arguments passed to contour.default to control the contours, if style="contour"
or style="imagecontour".

sigma

Smoothing bandwidth for smooth interpolation.

main

Optional main title for the plot.

plot.symbolmap

1021

Details
These are methods for the generic plot, image and contour for the class "ssf".
An object of class "ssf" represents a function (real- or vector-valued) that has been sampled at a
finite set of points.
For plot.ssf there are three types of display. If how="points" the exact function values will be
displayed as circles centred at the locations where they were computed. If how="smoothed" (the
default) these values will be kernel-smoothed using Smooth.ppp and displayed as a pixel image.
If how="nearest" the values will be interpolated by nearest neighbour interpolation using nnmark
and displayed as a pixel image.
For image.ssf and contour.ssf the values are kernel-smoothed before being displayed.
Value
NULL.
Author(s)
Adrian Baddeley .
References
Baddeley, A. (2016) Local composite likelihood for spatial point processes. Spatial Statistics, in
press.
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.
See Also
ssf
Examples
a <- ssf(cells, nndist(cells, k=1:3))
plot(a, how="points")
plot(a, how="smoothed")
plot(a, how="nearest")

plot.symbolmap

Plot a Graphics Symbol Map

Description
Plot a representation of a graphics symbol map, similar to a plot legend.
Usage
## S3 method for class 'symbolmap'
plot(x, ..., main, xlim = NULL, ylim = NULL,
vertical = FALSE,
side = c("bottom", "left", "top", "right"),
annotate = TRUE, labelmap = NULL, add = FALSE,
nsymbols = NULL)

1022

plot.symbolmap

Arguments
x

Graphics symbol map (object of class "symbolmap").

...

Additional graphics arguments passed to points, symbols or axis.

main

Main title for the plot. A character string.

xlim,ylim

Coordinate limits for the plot. Numeric vectors of length 2.

vertical

Logical. Whether to plot the symbol map in a vertical orientation.

side

Character string specifying the position of the text that annotates the symbols.

annotate

Logical. Whether to annotate the symbols with labels.

labelmap

Transformation of the labels. A function or a scale factor which will be applied
to the data values corresponding to the plotted symbols.

add

Logical value indicating whether to add the plot to the current plot (add=TRUE)
or to initialise a new plot.

nsymbols

Optional. The number of symbols that should be displayed. (This may not be
exactly obeyed.)

Details
A graphics symbol map is an association between data values and graphical symbols.
This command plots the graphics symbol map itself, in the style of a plot legend.
Value
None.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
symbolmap to create a symbol map.
invoke.symbolmap to apply the symbol map to some data and plot the resulting symbols.
Examples
g <- symbolmap(inputs=letters[1:10], pch=11:20)
plot(g)
g2 <- symbolmap(range=c(-1,1),
shape=function(x) ifelse(x > 0, "circles", "squares"),
size=function(x) sqrt(ifelse(x > 0, x/pi, -x)),
bg = function(x) ifelse(abs(x) < 1, "red", "black"))
plot(g2, vertical=TRUE, side="left", col.axis="blue", cex.axis=2)

plot.tess

plot.tess

1023

Plot a tessellation

Description
Plots a tessellation.
Usage
## S3 method for class 'tess'
plot(x, ..., main, add=FALSE,
show.all=!add, col=NULL, do.plot=TRUE,
do.labels=FALSE,
labels=tilenames(x), labelargs=list())
Arguments
x

Tessellation (object of class "tess") to be plotted.

...

Arguments controlling the appearance of the plot.

main

Heading for the plot. A character string.

add

Logical. Determines whether the tessellation plot is added to the existing plot.

show.all

Logical value indicating whether to plot everything including the main title and
the observation window of x.

col

Colour of the tile boundaries. A character string. Ignored for pixel tessellations.

do.plot

Logical value indicating whether to actually perform the plot.

do.labels

Logical value indicating whether to show a text label for each tile of the tessellation.

labels

Character vector of labels for the tiles.

labelargs

List of arguments passed to text.default to control display of the text labels.

Details
This is a method for the generic plot function for the class "tess" of tessellations (see tess).
The arguments ... control the appearance of the plot. They are passed to segments, plot.owin or
plot.im, depending on the type of tessellation.
Value
(Invisible) window of class "owin" specifying a bounding box for the plot (including a colour
ribbon if plotted).
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
tess

1024

plot.textstring

Examples
A <- tess(xgrid=0:4,ygrid=0:4)
plot(A, col="blue", lwd=2, lty=2)
B <- A[c(1, 2, 5, 7, 9)]
plot(B, hatch=TRUE)
v <- as.im(function(x,y){factor(round(5 * (x^2 + y^2)))}, W=owin())
levels(v) <- letters[seq(length(levels(v)))]
E <- tess(image=v)
plot(E)

plot.textstring

Plot a Text String

Description
Plots an object of class "textstring".
Usage
## S3 method for class 'textstring'
plot(x, ..., do.plot = TRUE)
Arguments
x
...
do.plot

Object of class "textstring" to be plotted. This object is created by the command textstring.
Additional graphics arguments passed to text to control the plotting of text.
Logical value indicating whether to actually plot the text.

Details
The argument x should be an object of class "textstring" created by the command textstring.
This function displays the text using text.
Value
A window (class "owin") enclosing the plotted graphics.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
onearrow, yardstick
Examples
W <- Window(humberside)
te <- textstring(centroid.owin(W), txt="Humberside", cex=2.5)
plot(layered(W, te), main="")

plot.texturemap

plot.texturemap

1025

Plot a Texture Map

Description
Plot a representation of a texture map, similar to a plot legend.
Usage
## S3 method for class 'texturemap'
plot(x, ..., main, xlim = NULL, ylim = NULL,
vertical = FALSE, axis = TRUE,
labelmap = NULL, gap = 0.25,
spacing = NULL, add = FALSE)
Arguments
x

Texture map object (class "texturemap").

...

Additional graphics arguments passed to add.texture or axis.default.

main

Main title for plot.

xlim,ylim

Optional vectors of length 2 giving the x and y limits of the plot.

vertical

Logical value indicating whether to arrange the texture boxes in a vertical column (vertical=TRUE or a horizontal row (vertical=FALSE, the default).

axis

Logical value indicating whether to plot an axis line joining the texture boxes.

labelmap

Optional. A function which will be applied to the data values (the inputs of the
texture map) before they are displayed on the plot.

gap

Separation between texture boxes, as a fraction of the width or height of a box.

spacing

Argument passed to add.texture controlling the density of lines in a texture.
Expressed in spatial coordinate units.

add

Logical value indicating whether to add the graphics to an existing plot (add=TRUE)
or to initialise a new plot (add=FALSE, the default).

Details
A texture map is an association between data values and graphical textures. An object of class
"texturemap" represents a texture map. Such objects are returned from the plotting function
textureplot, and can be created directly by the function texturemap.
This function plot.texturemap is a method for the generic plot for the class "texturemap". It
displays a sample of each of the textures in the texture map, in a separate box, annotated by the data
value which is mapped to that texture.
The arrangement and position of the boxes is controlled by the arguments vertical, xlim, ylim
and gap.
Value
Null.

1026

plot.yardstick

Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
texturemap, textureplot, add.texture.
Examples
tm <- texturemap(c("First", "Second", "Third"), 2:4, col=2:4)
plot(tm, vertical=FALSE)
## abbreviate the labels
plot(tm, labelmap=function(x) substr(x, 1, 2))

plot.yardstick

Plot a Yardstick or Scale Bar

Description
Plots an object of class "yardstick".
Usage
## S3 method for class 'yardstick'
plot(x, ...,
angle = 20, frac = 1/8,
split = FALSE, shrink = 1/4,
pos = NULL,
txt.args=list(),
txt.shift=c(0,0),
do.plot = TRUE)
Arguments
x

Object of class "yardstick" to be plotted. This object is created by the command yardstick.

...

Additional graphics arguments passed to segments to control the appearance of
the line.

angle

Angle between the arrows and the line segment, in degrees.

frac

Length of arrow as a fraction of total length of the line segment.

split

Logical. If TRUE, then the line will be broken in the middle, and the text will
be placed in this gap. If FALSE, the line will be unbroken, and the text will be
placed beside the line.

shrink

Fraction of total length to be removed from the middle of the line segment, if
split=TRUE.

pos

Integer (passed to text) determining the position of the annotation text relative
to the line segment, if split=FALSE. Values of 1, 2, 3 and 4 indicate positions
below, to the left of, above and to the right of the line, respectively.

points.lpp

1027

txt.args

Optional list of additional arguments passed to text controlling the appearance
of the text. Examples include adj, srt, col, cex, font.

txt.shift

Optional numeric vector of length 2 specifying displacement of the text position
relative to the centre of the yardstick.

do.plot

Logical. Whether to actually perform the plot (do.plot=TRUE).

Details
A yardstick or scale bar is a line segment, drawn on any spatial graphics display, indicating the scale
of the plot.
The argument x should be an object of class "yardstick" created by the command yardstick.
Value
A window (class "owin") enclosing the plotted graphics.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
yardstick
Examples
plot(owin(), main="Yardsticks")
ys <- yardstick(as.psp(list(xmid=0.5, ymid=0.1, length=0.4, angle=0),
window=owin(c(0.2, 0.8), c(0, 0.2))),
txt="1 km")
plot(ys)
ys <- shift(ys, c(0, 0.3))
plot(ys, angle=90, frac=0.08)
ys <- shift(ys, c(0, 0.3))
plot(ys, split=TRUE)

points.lpp

Draw Points on Existing Plot

Description
For a point pattern on a linear network, this function draws the coordinates of the points only, on
the existing plot display.
Usage
## S3 method for class 'lpp'
points(x, ...)

1028

pointsOnLines

Arguments
x

A point pattern on a linear network (object of class "lpp").

...

Additional arguments passed to points.default.

Details
This is a method for the generic function points for the class "lpp" of point patterns on a linear
network.
If x is a point pattern on a linear network, then points(x) plots the spatial coordinates of the points
only, on the existing plot display, without plotting the underlying network. It is an error to call this
function if a plot has not yet been initialised.
The spatial coordinates are extracted and passed to points.default along with any extra arguments. Arguments controlling the colours and the plot symbols are interpreted by points.default.
For example, if the argument col is a vector, then the ith point is drawn in the colour col[i].
Value
Null.
Difference from plot method
The more usual way to plot the points is using plot.lpp. For example plot(x) would plot both
the points and the underlying network, while plot(x, add=TRUE) would plot only the points.
The interpretation of arguments controlling the colours and plot symbols is different here: they
determine a symbol map, as explained in the help for plot.ppp.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
plot.lpp, points.default
Examples
plot(Frame(spiders), main="Spiders on a Brick Wall")
points(spiders)

pointsOnLines

Place Points Evenly Along Specified Lines

Description
Given a line segment pattern, place a series of points at equal distances along each line segment.
Usage
pointsOnLines(X, eps = NULL, np = 1000, shortok=TRUE)

Poisson

1029

Arguments
X

A line segment pattern (object of class "psp").

eps

Spacing between successive points.

np

Approximate total number of points (incompatible with eps).

shortok

Logical. If FALSE, very short segments (of length shorter than eps) will not
generate any points. If TRUE, a very short segment will be represented by its
midpoint.

Details
For each line segment in the pattern X, a succession of points is placed along the line segment. These
points are equally spaced at a distance eps, except for the first and last points in the sequence.
The spacing eps is measured in coordinate units of X.
If eps is not given, then it is determined by eps = len/np where len is the total length of the
segments in X. The actual number of points will then be slightly larger than np.
Value
A point pattern (object of class "ppp") in the same window as X.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
psp, ppp, runifpointOnLines
Examples
X <- psp(runif(20), runif(20), runif(20), runif(20),
Y <- pointsOnLines(X, eps=0.05)
plot(X, main="")
plot(Y, add=TRUE, pch="+")

Poisson

window=owin())

Poisson Point Process Model

Description
Creates an instance of the Poisson point process model which can then be fitted to point pattern
data.
Usage
Poisson()

1030

Poisson

Details
The function ppm, which fits point process models to point pattern data, requires an argument
interaction of class "interact" describing the interpoint interaction structure of the model to be
fitted. The appropriate description of the Poisson process is provided by the value of the function
Poisson.
This works for all types of Poisson processes including multitype and nonstationary Poisson processes.
Value
An object of class "interact" describing the interpoint interaction structure of the Poisson point
process (namely, there are no interactions).
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
ppm, Strauss
Examples
ppm(nztrees ~1, Poisson())
# fit the stationary Poisson process to 'nztrees'
# no edge correction needed
lon <- longleaf
longadult <- unmark(subset(lon, marks >= 30))
ppm(longadult ~ x, Poisson())
# fit the nonstationary Poisson process
# with intensity lambda(x,y) = exp( a + bx)
# trees marked by species
lans <- lansing
ppm(lans ~ marks, Poisson())
# fit stationary marked Poisson process
# with different intensity for each species
## Not run:
ppm(lansing ~ marks * polynom(x,y,3), Poisson())
## End(Not run)
# fit nonstationary marked Poisson process
# with different log-cubic trend for each species

polynom

1031

polynom

Polynomial in One or Two Variables

Description
This function is used to represent a polynomial term in a model formula. It computes the homogeneous terms in the polynomial of degree n in one variable x or two variables x,y.
Usage
polynom(x, ...)
Arguments
x

A numerical vector.

...

Either a single integer n specifying the degree of the polynomial, or two arguments y,n giving another vector of data y and the degree of the polynomial.

Details
This function is typically used inside a model formula in order to specify the most general possible
polynomial of order n involving one numerical variable x or two numerical variables x,y.
It is equivalent to poly(, raw=TRUE).
If only one numerical vector argument x is given, the function computes the vectors x^k for k = 1, 2, ..., n.
These vectors are combined into a matrix with n columns.
If two numerical vector arguments x,y are given, the function computes the vectors x^k * y^m for
k >= 0 and m >= 0 satisfying 0 < k + m <= n. These vectors are combined into a matrix with
one column for each homogeneous term.
Value
A numeric matrix, with rows corresponding to the entries of x, and columns corresponding to the
terms in the polynomial.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
poly, harmonic
Examples
x <- 1:4
y <- 10 * (0:3)
polynom(x, 3)
polynom(x, y, 3)

1032

pool

pool.anylist

Pool Data

Description
Pool the data from several objects of the same class.
Usage
pool(...)
Arguments
...

Objects of the same type.

Details
The function pool is generic. There are methods for several classes, listed below.
pool is used to combine the data from several objects of the same type, and to compute statistics
based on the combined dataset. It may be used to pool the estimates obtained from replicated
datasets. It may also be used in high-performance computing applications, when the objects ...
have been computed on different processors or in different batch runs, and we wish to combine
them.
Value
An object of the same class as the arguments ....
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
pool.envelope, pool.fasp, pool.rat, pool.fv

pool.anylist

Pool Data from a List of Objects

Description
Pool the data from the objects in a list.
Usage
## S3 method for class 'anylist'
pool(x, ...)

pool.envelope

1033

Arguments
x

A list, belonging to the class "anylist", containing objects that can be pooled.

...

Optional additional objects which can be pooled with the elements of x.

Details
The function pool is generic. Its purpose is to combine data from several objects of the same type
(typically computed from different datasets) into a common, pooled estimate.
The function pool.anyist is the method for the class "anylist". It is used when the objects to be
pooled are given in a list x.
Each of the elements of the list x, and each of the subsequent arguments ... if provided, must be
an object of the same class.
Value
An object of the same class as each of the entries in x.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
anylist, pool.
Examples
Keach <- anylapply(waterstriders, Kest, ratio=TRUE, correction="iso")
K <- pool(Keach)

pool.envelope

Pool Data from Several Envelopes

Description
Pool the simulation data from several simulation envelopes (objects of class "envelope") and compute a new envelope.
Usage
## S3 method for class 'envelope'
pool(..., savefuns=FALSE, savepatterns=FALSE)
Arguments
...

Objects of class "envelope".

savefuns

Logical flag indicating whether to save all the simulated function values.

savepatterns

Logical flag indicating whether to save all the simulated point patterns.

1034

pool.envelope

Details
The function pool is generic. This is the method for the class "envelope" of simulation envelopes.
It is used to combine the simulation data from several simulation envelopes and to compute an
envelope based on the combined data.
Each of the arguments ... must be an object of class "envelope". These envelopes must be
compatible, in that they are envelopes for the same function, and were computed using the same
options.
• In normal use, each envelope object will have been created by running the command envelope
with the argument savefuns=TRUE. This ensures that each object contains the simulated data
(summary function values for the simulated point patterns) that were used to construct the
envelope.
The simulated data are extracted from each object and combined. A new envelope is computed
from the combined set of simulations.
• Alternatively, if each envelope object was created by running envelope with VARIANCE=TRUE,
then the saved functions are not required.
The sample means and sample variances from each envelope will be pooled. A new envelope
is computed from the pooled mean and variance.
Warnings or errors will be issued if the envelope objects ... appear to be incompatible. Apart from
these basic checks, the code is not smart enough to decide whether it is sensible to pool the data.
To modify the envelope parameters or the type of envelope that is computed, first pool the envelope
data using pool.envelope, then use envelope.envelope to modify the envelope parameters.
Value
An object of class "envelope".
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
envelope, envelope.envelope, pool, pool.fasp
Examples
E1 <- envelope(cells, Kest, nsim=10, savefuns=TRUE)
E2 <- envelope(cells, Kest, nsim=20, savefuns=TRUE)
pool(E1, E2)
V1 <- envelope(E1, VARIANCE=TRUE)
V2 <- envelope(E2, VARIANCE=TRUE)
pool(V1, V2)

pool.fasp

pool.fasp

1035

Pool Data from Several Function Arrays

Description
Pool the simulation data from several function arrays (objects of class "fasp") and compute a new
function array.
Usage
## S3 method for class 'fasp'
pool(...)
Arguments
...

Objects of class "fasp".

Details
The function pool is generic. This is the method for the class "fasp" of function arrays. It is used
to combine the simulation data from several arrays of simulation envelopes and to compute a new
array of envelopes based on the combined data.
Each of the arguments ... must be a function array (object of class "fasp") containing simulation envelopes. This is typically created by running the command alltypes with the arguments
envelope=TRUE and savefuns=TRUE. This ensures that each object is an array of simulation envelopes, and that each envelope contains the simulated data (summary function values) that were
used to construct the envelope.
The simulated data are extracted from each object and combined. A new array of envelopes is
computed from the combined set of simulations.
Warnings or errors will be issued if the objects ... appear to be incompatible. However, the code
is not smart enough to decide whether it is sensible to pool the data.
Value
An object of class "fasp".
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
fasp, alltypes, pool.envelope, pool
Examples
data(amacrine)
A1 <- alltypes(amacrine,"K",nsim=9,envelope=TRUE,savefuns=TRUE)
A2 <- alltypes(amacrine,"K",nsim=10,envelope=TRUE,savefuns=TRUE)
pool(A1, A2)

1036

pool.fv

pool.fv

Pool Several Functions

Description
Combine several summary functions into a single function.
Usage
## S3 method for class 'fv'
pool(..., weights=NULL, relabel=TRUE, variance=TRUE)
Arguments
...

Objects of class "fv".

weights

Optional numeric vector of weights for the functions.

relabel

Logical value indicating whether the columns of the resulting function should
be labelled to show that they were obtained by pooling.

variance

Logical value indicating whether to compute the sample variance and related
terms.

Details
The function pool is generic. This is the method for the class "fv" of summary functions. It is used
to combine several estimates of the same function into a single function.
Each of the arguments ... must be an object of class "fv". They must be compatible, in that they
are estimates of the same function, and were computed using the same options.
The sample mean and sample variance of the corresponding estimates will be computed.
Value
An object of class "fv".
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
pool, pool.anylist, pool.rat
Examples
K <- lapply(waterstriders, Kest, correction="iso")
Kall <- pool(K[[1]], K[[2]], K[[3]])
Kall <- pool(as.anylist(K))
plot(Kall, cbind(pooliso, pooltheo) ~ r,
shade=c("loiso", "hiiso"),
main="Pooled K function of waterstriders")

pool.quadrattest

pool.quadrattest

1037

Pool Several Quadrat Tests

Description
Pool several quadrat tests into a single quadrat test.
Usage
## S3 method for class 'quadrattest'
pool(..., df=NULL, df.est=NULL, nsim=1999,
Xname=NULL, CR=NULL)
Arguments
...

Any number of objects, each of which is a quadrat test (object of class "quadrattest").

df

Optional. Number of degrees of freedom of the test statistic. Relevant only for
χ2 tests. Incompatible with df.est.

df.est

Optional. The number of fitted parameters, or the number of degrees of freedom
lost by estimation of parameters. Relevant only for χ2 tests. Incompatible with
df.

nsim

Number of simulations, for Monte Carlo test.

Xname

Optional. Name of the original data.

CR

Optional. Numeric value of the Cressie-Read exponent CR overriding the value
used in the tests.

Details
The function pool is generic. This is the method for the class "quadrattest".
An object of class "quadrattest" represents a χ2 test or Monte Carlo test of goodness-of-fit
for a point process model, based on quadrat counts. Such objects are created by the command
quadrat.test.
Each of the arguments ... must be an object of class "quadrattest". They must all be the same
type of test (chi-squared test or Monte Carlo test, conditional or unconditional) and must all have
the same type of alternative hypothesis.
The test statistic of the pooled test is the Pearson X 2 statistic taken over all cells (quadrats) of all
tests. The p value of the pooled test is then computed using either a Monte Carlo test or a χ2 test.
For a pooled χ2 test, the number of degrees of freedom of the combined test is computed by adding
the degrees of freedom of all the tests (equivalent to assuming the tests are independent) unless it is
determined by the arguments df or df.est. The resulting p value is computed to obtain the pooled
test.
For a pooled Monte Carlo test, new simulations are performed to determine the pooled Monte Carlo
p value.
Value
Another object of class "quadrattest".

1038

pool.rat

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
pool, quadrat.test
Examples
Y <- split(humberside)
test1 <- quadrat.test(Y[[1]])
test2 <- quadrat.test(Y[[2]])
pool(test1, test2, Xname="Humberside")

pool.rat

Pool Data from Several Ratio Objects

Description
Pool the data from several ratio objects (objects of class "rat") and compute a pooled estimate.
Usage
## S3 method for class 'rat'
pool(..., weights=NULL, relabel=TRUE, variance=TRUE)
Arguments
...

Objects of class "rat".

weights

Numeric vector of weights.

relabel

Logical value indicating whether the result should be relabelled to show that it
was obtained by pooling.

variance

Logical value indicating whether to compute the sample variance and related
terms.

Details
The function pool is generic. This is the method for the class "rat" of ratio objects. It is used to
combine several estimates of the same quantity when each estimate is a ratio.
Each of the arguments ... must be an object of class "rat" representing a ratio object (basically a
numerator and a denominator; see rat). We assume that these ratios are all estimates of the same
quantity.
If the objects are called R1 , . . . , Rn and if Ri has numerator Yi and denominator Xi , so that notionally Ri = Yi /Xi , then the pooled estimate is the ratio-of-sums estimator
P
Yi
R = Pi .
X
i i
The standard error of R is computed using the delta method as described in Baddeley et al. (1993)
or Cochran (1977, pp 154, 161).

pp3

1039
If the argument weights is given, it should be a numeric vector of length equal to the number of
objects to be pooled. The pooled estimator is the ratio-of-sums estimator
P
i wi Yi
P
R=
i wi Xi
where w_iw[i] is the ith weight.
This calculation is implemented only for certain classes of objects where the arithmetic can be
performed.
This calculation is currently implemented only for objects which also belong to the class "fv"
(function value tables). For example, if Kest is called with argument ratio=TRUE, the result is a
suitable object (belonging to the classes "rat" and "fv").
Warnings or errors will be issued if the ratio objects ... appear to be incompatible. However, the
code is not smart enough to decide whether it is sensible to pool the data.

Value
An object of the same class as the input.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
References
Baddeley, A.J, Moyeed, R.A., Howard, C.V. and Boyde, A. (1993) Analysis of a three-dimensional
point pattern with replication. Applied Statistics 42, 641–668.
Cochran, W.G. (1977) Sampling techniques, 3rd edition. New York: John Wiley and Sons.
See Also
rat, pool, pool.fv, Kest
Examples
K1 <- Kest(runifpoint(42), ratio=TRUE, correction="iso")
K2 <- Kest(runifpoint(42), ratio=TRUE, correction="iso")
K3 <- Kest(runifpoint(42), ratio=TRUE, correction="iso")
K <- pool(K1, K2, K3)
plot(K, pooliso ~ r, shade=c("hiiso", "loiso"))

pp3

Three Dimensional Point Pattern

Description
Create a three-dimensional point pattern
Usage
pp3(x, y, z, ...)

1040

ppm

Arguments
x,y,z

Numeric vectors of equal length, containing Cartesian coordinates of points in
three-dimensional space.

...

Arguments passed to as.box3 to determine the three-dimensional box in which
the points have been observed.

Details
An object of class "pp3" represents a pattern of points in three-dimensional space. The points are
assumed to have been observed by exhaustively inspecting a three-dimensional rectangular box.
The boundaries of the box are included as part of the dataset.
Value
Object of class "pp3" representing a three dimensional point pattern. Also belongs to class "ppx".
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
box3, print.pp3, ppx
Examples
X <- pp3(runif(10), runif(10), runif(10), box3(c(0,1)))

ppm

Fit Point Process Model to Data

Description
Fits a point process model to an observed point pattern.
Usage
ppm(Q, ...)
## S3 method for class 'formula'
ppm(Q, interaction=NULL, ..., data=NULL, subset)
Arguments
Q

A formula in the R language describing the model to be fitted.

interaction

An object of class "interact" describing the point process interaction structure, or a function that makes such an object, or NULL indicating that a Poisson
process (stationary or nonstationary) should be fitted.

...

Arguments passed to ppm.ppp or ppm.quad to control the model-fitting process.

ppm

1041
data

Optional. The values of spatial covariates (other than the Cartesian coordinates)
required by the model. Either a data frame, or a list whose entries are images,
functions, windows, tessellations or single numbers. See Details.

subset

Optional. An expression (which may involve the names of the Cartesian coordinates x and y and the names of entries in data) defining a subset of the spatial
domain, to which the model-fitting should be restricted. The result of evaluating
the expression should be either a logical vector, or a window (object of class
"owin") or a logical-valued pixel image (object of class "im").

Details
This function fits a point process model to an observed point pattern. The model may include spatial
trend, interpoint interaction, and dependence on covariates.
The model fitted by ppm is either a Poisson point process (in which different points do not interact
with each other) or a Gibbs point process (in which different points typically inhibit each other).
For clustered point process models, use kppm.
The function ppm is generic, with methods for the classes formula, ppp and quad. This page
describes the method for a formula.
The first argument is a formula in the R language describing the spatial trend model to be fitted. It
has the general form pattern ~ trend where the left hand side pattern is usually the name of a
spatial point pattern (object of class "ppp") to which the model should be fitted, or an expression
which evaluates to a point pattern; and the right hand side trend is an expression specifying the
spatial trend of the model.
Systematic effects (spatial trend and/or dependence on spatial covariates) are specified by the trend
expression on the right hand side of the formula. The trend may involve the Cartesian coordinates
x, y, the marks marks, the names of entries in the argument data (if supplied), or the names of
objects that exist in the R session. The trend formula specifies the logarithm of the intensity of a
Poisson process, or in general, the logarithm of the first order potential of the Gibbs process. The
formula should not use any names beginning with .mpl as these are reserved for internal use. If the
formula is pattern~1, then the model to be fitted is stationary (or at least, its first order potential is
constant).
The symbol . in the trend expression stands for all the covariates supplied in the argument data.
For example the formula pattern ~ . indicates an additive model with a main effect for each
covariate in data.
Stochastic interactions between random points of the point process are defined by the argument
interaction. This is an object of class "interact" which is initialised in a very similar way to the
usage of family objects in glm and gam. The interaction models currently available are: AreaInter,
BadGey, Concom, DiggleGatesStibbard, DiggleGratton, Fiksel, Geyer, Hardcore, Hybrid,
LennardJones, MultiStrauss, MultiStraussHard, OrdThresh, Ord, Pairwise, PairPiece, Penttinen,
Poisson, Saturated, SatPiece, Softcore, Strauss, StraussHard and Triplets. See the examples below. Note that it is possible to combine several interactions using Hybrid.
If interaction is missing or NULL, then the model to be fitted has no interpoint interactions, that is,
it is a Poisson process (stationary or nonstationary according to trend). In this case the methods of
maximum pseudolikelihood and maximum logistic likelihood coincide with maximum likelihood.
The fitted point process model returned by this function can be printed (by the print method print.ppm)
to inspect the fitted parameter values. If a nonparametric spatial trend was fitted, this can be extracted using the predict method predict.ppm.
To fit a model involving spatial covariates other than the Cartesian coordinates x and y, the values
of the covariates should either be supplied in the argument data, or should be stored in objects
that exist in the R session. Note that it is not sufficient to have observed the covariate only at the

1042

ppm

points of the data point pattern; the covariate must also have been observed at other locations in the
window.
If it is given, the argument data is typically a list, with names corresponding to variables in the
trend formula. Each entry in the list is either
a pixel image, giving the values of a spatial covariate at a fine grid of locations. It should be an
object of class "im", see im.object.
a function, which can be evaluated at any location (x,y) to obtain the value of the spatial covariate.
It should be a function(x, y) or function(x, y, ...) in the R language. The first two
arguments of the function should be the Cartesian coordinates x and y. The function may
have additional arguments; if the function does not have default values for these additional
arguments, then the user must supply values for them, in covfunargs. See the Examples.
a window, interpreted as a logical variable which is TRUE inside the window and FALSE outside it.
This should be an object of class "owin".
a tessellation, interpreted as a factor covariate. For each spatial location, the factor value indicates
which tile of the tessellation it belongs to. This should be an object of class "tess".
a single number, indicating a covariate that is constant in this dataset.
The software will look up the values of each covariate at the required locations (quadrature points).
Note that, for covariate functions, only the name of the function appears in the trend formula. A
covariate function is treated as if it were a single variable. The function arguments do not appear in
the trend formula. See the Examples.
If data is a list, the list entries should have names corresponding to (some of) the names of covariates in the model formula trend. The variable names x, y and marks are reserved for the Cartesian
coordinates and the mark values, and these should not be used for variables in data.
Alternatively, data may be a data frame giving the values of the covariates at specified locations.
Then pattern should be a quadrature scheme (object of class "quad") giving the corresponding
locations. See ppm.quad for details.
Value
An object of class "ppm" describing a fitted point process model.
See ppm.object for details of the format of this object and methods available for manipulating it.
Interaction parameters
Apart from the Poisson model, every point process model fitted by ppm has parameters that determine the strength and range of ‘interaction’ or dependence between points. These parameters are
of two types:
regular parameters: A parameter φ is called regular if the log likelihood is a linear function of θ
where θ = θ(ψ) is some transformation of ψ. [Then θ is called the canonical parameter.]
irregular parameters Other parameters are called irregular.
Typically, regular parameters determine the ‘strength’ of the interaction, while irregular parameters
determine the ‘range’ of the interaction. For example, the Strauss process has a regular parameter γ
controlling the strength of interpoint inhibition, and an irregular parameter r determining the range
of interaction.
The ppm command is only designed to estimate regular parameters of the interaction. It requires
the values of any irregular parameters of the interaction to be fixed. For example, to fit a Strauss
process model to the cells dataset, you could type ppm(cells ~ 1, Strauss(r=0.07)). Note

ppm

1043
that the value of the irregular parameter r must be given. The result of this command will be a fitted
model in which the regular parameter γ has been estimated.
To determine the irregular parameters, there are several practical techniques, but no general statistical theory available. Useful techniques include maximum profile pseudolikelihood, which is
implemented in the command profilepl, and Newton-Raphson maximisation, implemented in the
experimental command ippm.
Some irregular parameters can be estimated directly from data: the hard-core radius in the model
Hardcore and the matrix of hard-core radii in MultiHard can be estimated easily from data. In
these cases, ppm allows the user to specify the interaction without giving the value of the irregular parameter. The user can give the hard core interaction as interaction=Hardcore() or even
interaction=Hardcore, and the hard core radius will then be estimated from the data.

Technical Warnings and Error Messages
See ppm.ppp for some technical warnings about the weaknesses of the algorithm, and explanation
of some common error messages.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
References
Baddeley, A., Coeurjolly, J.-F., Rubak, E. and Waagepetersen, R. (2014) Logistic regression for
spatial Gibbs point processes. Biometrika 101 (2) 377–392.
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns.
Australian and New Zealand Journal of Statistics 42 283–322.
Berman, M. and Turner, T.R. (1992) Approximating point process likelihoods with GLIM. Applied
Statistics 41, 31–38.
Besag, J. (1975) Statistical analysis of non-lattice data. The Statistician 24, 179-195.
Diggle, P.J., Fiksel, T., Grabarnik, P., Ogata, Y., Stoyan, D. and Tanemura, M. (1994) On parameter
estimation for pairwise interaction processes. International Statistical Review 62, 99-117.
Huang, F. and Ogata, Y. (1999) Improvements of the maximum pseudo-likelihood estimators in
various spatial statistical models. Journal of Computational and Graphical Statistics 8, 510–530.
Jensen, J.L. and Moeller, M. (1991) Pseudolikelihood for exponential family models of spatial point
processes. Annals of Applied Probability 1, 445–461.
Jensen, J.L. and Kuensch, H.R. (1994) On asymptotic normality of pseudo likelihood estimates for
pairwise interaction processes, Annals of the Institute of Statistical Mathematics 46, 475–486.
See Also
ppm.ppp and ppm.quad for more details on the fitting technique and edge correction.
ppm.object for details of how to print, plot and manipulate a fitted model.
ppp and quadscheme for constructing data.
Interactions: AreaInter, BadGey, Concom, DiggleGatesStibbard, DiggleGratton, Fiksel, Geyer,
Hardcore, Hybrid, LennardJones, MultiStrauss, MultiStraussHard, OrdThresh, Ord, Pairwise,
PairPiece, Penttinen, Poisson, Saturated, SatPiece, Softcore, Strauss, StraussHard and
Triplets.

1044

ppm

See profilepl for advice on fitting nuisance parameters in the interaction, and ippm for irregular
parameters in the trend.
See valid.ppm and project.ppm for ensuring the fitted model is a valid point process.
See kppm for fitting Cox point process models and cluster point process models, and dppm for fitting
determinantal point process models.
Examples
# fit the stationary Poisson process
# to point pattern 'nztrees'
ppm(nztrees ~ 1)
## Not run:
Q <- quadscheme(nztrees)
ppm(Q ~ 1)
# equivalent.
## End(Not run)
fit1 <- ppm(nztrees ~ x)
# fit the nonstationary Poisson process
# with intensity function lambda(x,y) = exp(a + bx)
# where x,y are the Cartesian coordinates
# and a,b are parameters to be estimated
fit1
coef(fit1)
coef(summary(fit1))
## Not run:
ppm(nztrees ~ polynom(x,2))
## End(Not run)
# fit the nonstationary Poisson process
# with intensity function lambda(x,y) = exp(a + bx + cx^2)
## Not run:
library(splines)
ppm(nztrees ~ bs(x,df=3))
##
#
#
#
#

End(Not run)
WARNING: do not use predict.ppm() on this result
Fits the nonstationary Poisson process
with intensity function lambda(x,y) = exp(B(x))
where B is a B-spline with df = 3

## Not run:
ppm(nztrees ~ 1, Strauss(r=10), rbord=10)
## End(Not run)
# Fit the stationary Strauss process with interaction range r=10
# using the border method with margin rbord=10

ppm

1045

## Not run:
ppm(nztrees ~ x, Strauss(13), correction="periodic")
## End(Not run)
# Fit the nonstationary Strauss process with interaction range r=13
# and exp(first order potential) = activity = beta(x,y) = exp(a+bx)
# using the periodic correction.
# Compare Maximum Pseudolikelihood, Huang-Ogata and Variational Bayes fits:
## Not run: ppm(swedishpines ~ 1, Strauss(9))
## Not run: ppm(swedishpines ~ 1, Strauss(9), method="ho")
ppm(swedishpines ~ 1, Strauss(9), method="VBlogi")
# COVARIATES
#
X <- rpoispp(42)
weirdfunction <- function(x,y){ 10 * x^2 + 5 * sin(10 * y) }
#
# (a) covariate values as function
ppm(X ~ y + weirdfunction)
#
# (b) covariate values in pixel image
Zimage <- as.im(weirdfunction, unit.square())
ppm(X ~ y + Z, covariates=list(Z=Zimage))
#
# (c) covariate values in data frame
Q <- quadscheme(X)
xQ <- x.quad(Q)
yQ <- y.quad(Q)
Zvalues <- weirdfunction(xQ,yQ)
ppm(Q ~ y + Z, data=data.frame(Z=Zvalues))
# Note Q not X
# COVARIATE FUNCTION WITH EXTRA ARGUMENTS
#
f <- function(x,y,a){ y - a }
ppm(X ~ x + f, covfunargs=list(a=1/2))
# COVARIATE: inside/outside window
b <- owin(c(0.1, 0.6), c(0.1, 0.9))
ppm(X ~ b)
## MULTITYPE POINT PROCESSES ###
# fit stationary marked Poisson process
# with different intensity for each species
## Not run: ppm(lansing ~ marks, Poisson())
# fit nonstationary marked Poisson process
# with different log-cubic trend for each species
## Not run: ppm(lansing ~ marks * polynom(x,y,3), Poisson())

1046

ppm.object

ppm.object

Class of Fitted Point Process Models

Description
A class ppm to represent a fitted stochastic model for a point process. The output of ppm.
Details
An object of class ppm represents a stochastic point process model that has been fitted to a point
pattern dataset. Typically it is the output of the model fitter, ppm.
The class ppm has methods for the following standard generic functions:
generic
print
plot
predict
fitted
coef
anova
formula
terms
labels
residuals
simulate
update
vcov
model.frame
model.matrix
logLik
extractAIC
nobs

method
print.ppm
plot.ppm
predict.ppm
fitted.ppm
coef.ppm
anova.ppm
formula.ppm
terms.ppm
labels.ppm
residuals.ppm
simulate.ppm
update.ppm
vcov.ppm
model.frame.ppm
model.matrix.ppm
logLik.ppm
extractAIC.ppm
nobs.ppm

description
print details
plot fitted model
fitted intensity and conditional intensity
fitted intensity
fitted coefficients of model
Analysis of Deviance
Extract model formula
Terms in the model formula
Names of estimable terms in the model formula
Point process residuals
Simulate the fitted model
Change or refit the model
Variance/covariance matrix of parameter estimates
Model frame
Design matrix
log pseudo likelihood
pseudolikelihood counterpart of AIC
number of observations

Objects of class ppm can also be handled by the following standard functions, without requiring a
special method:
name
confint
step
drop1
add1

description
Confidence intervals for parameters
Stepwise model selection
One-step model improvement
One-step model improvement

The class ppm also has methods for the following generic functions defined in the spatstat package:
generic
as.interact
as.owin

method
as.interact.ppm
as.owin.ppm

description
Interpoint interaction structure
Observation window of data

ppm.object

1047

berman.test
envelope
fitin
is.marked
is.multitype
is.poisson
is.stationary
cdf.test
quadrat.test
reach
rmhmodel
rmh
unitname

berman.test.ppm
envelope.ppm
fitin.ppm
is.marked.ppm
is.multitype.ppm
is.poisson.ppm
is.stationary.ppm
cdf.test.ppm
quadrat.test.ppm
reach.ppm
rmhmodel.ppm
rmh.ppm
unitname.ppm

Berman’s test
Simulation envelopes
Fitted interaction
Determine whether the model is marked
Determine whether the model is multitype
Determine whether the model is Poisson
Determine whether the model is stationary
Spatial distribution test
Quadrat counting test
Interaction range of model
Model in a form that can be simulated
Perform simulation
Name of unit of length

Information about the data (to which the model was fitted) can be extracted using data.ppm,
dummy.ppm and quad.ppm.
Internal format
If you really need to get at the internals, a ppm object contains at least the following entries:
coef
trend
interaction
Q
maxlogpl
correction

the fitted regular parameters (as returned by glm)
the trend formula or NULL
the point process interaction family (an object of class "interact") or NULL
the quadrature scheme used
the maximised value of log pseudolikelihood
name of edge correction method used

See ppm for explanation of these concepts. The irregular parameters (e.g. the interaction radius of
the Strauss process) are encoded in the interaction entry. However see the Warnings.
Warnings
The internal representation of ppm objects may change slightly between releases of the spatstat
package.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
ppm, coef.ppm, fitted.ppm, print.ppm, predict.ppm, plot.ppm.
Examples
data(cells)
fit <- ppm(cells, ~ x, Strauss(0.1), correction="periodic")
fit
coef(fit)
## Not run:

1048

ppm.ppp
pred <- predict(fit)

## End(Not run)
pred <- predict(fit, ngrid=20, type="trend")
## Not run:
plot(fit)
## End(Not run)

ppm.ppp

Fit Point Process Model to Point Pattern Data

Description
Fits a point process model to an observed point pattern.
Usage
## S3 method for class 'ppp'
ppm(Q, trend=~1, interaction=Poisson(),
...,
covariates=data,
data=NULL,
covfunargs = list(),
subset,
clipwin,
correction="border",
rbord=reach(interaction),
use.gam=FALSE,
method="mpl",
forcefit=FALSE,
emend=project,
project=FALSE,
prior.mean = NULL,
prior.var = NULL,
nd = NULL,
eps = NULL,
gcontrol=list(),
nsim=100, nrmh=1e5, start=NULL, control=list(nrep=nrmh),
verb=TRUE,
callstring=NULL)
## S3 method for class 'quad'
ppm(Q, trend=~1, interaction=Poisson(),
...,
covariates=data,
data=NULL,
covfunargs = list(),
subset,
clipwin,
correction="border",
rbord=reach(interaction),

ppm.ppp

1049
use.gam=FALSE,
method="mpl",
forcefit=FALSE,
emend=project,
project=FALSE,
prior.mean = NULL,
prior.var = NULL,
nd = NULL,
eps = NULL,
gcontrol=list(),
nsim=100, nrmh=1e5, start=NULL, control=list(nrep=nrmh),
verb=TRUE,
callstring=NULL)

Arguments
Q

A data point pattern (of class "ppp") to which the model will be fitted, or a
quadrature scheme (of class "quad") containing this pattern.

trend

An R formula object specifying the spatial trend to be fitted. The default formula, ~1, indicates the model is stationary and no trend is to be fitted.

interaction

An object of class "interact" describing the point process interaction structure, or a function that makes such an object, or NULL indicating that a Poisson
process (stationary or nonstationary) should be fitted.

...
Ignored.
data,covariates
The values of any spatial covariates (other than the Cartesian coordinates) required by the model. Either a data frame, or a list whose entries are images,
functions, windows, tessellations or single numbers. See Details.
subset

Optional. An expression (which may involve the names of the Cartesian coordinates x and y and the names of entries in data) defining a subset of the spatial
domain, to which the likelihood or pseudolikelihood should be restricted. See
Details. The result of evaluating the expression should be either a logical vector,
or a window (object of class "owin") or a logical-valued pixel image (object of
class "im").

clipwin

Optional. A spatial window (object of class "owin") to which data will be restricted, before model-fitting is performed. See Details.

covfunargs

A named list containing the values of any additional arguments required by covariate functions.

correction

The name of the edge correction to be used. The default is "border" indicating
the border correction. Other possibilities may include "Ripley", "isotropic",
"periodic", "translate" and "none", depending on the interaction.

rbord

If correction = "border" this argument specifies the distance by which the
window should be eroded for the border correction.

use.gam

Logical flag; if TRUE then computations are performed using gam instead of glm.

method

The method used to fit the model. Options are "mpl" for the method of Maximum PseudoLikelihood, "logi" for the Logistic Likelihood method, "VBlogi"
for the Variational Bayes Logistic Likelihood method, and "ho" for the HuangOgata approximate maximum likelihood method.

1050

ppm.ppp

forcefit

Logical flag for internal use. If forcefit=FALSE, some trivial models will be
fitted by a shortcut. If forcefit=TRUE, the generic fitting method will always
be used.

emend,project

(These are equivalent: project is an older name for emend.) Logical value.
Setting emend=TRUE will ensure that the fitted model is always a valid point
process by applying emend.ppm.

prior.mean

Optional vector of prior means for canonical parameters (for method="VBlogi").
See Details.

prior.var

Optional prior variance covariance matrix for canonical parameters (for method="VBlogi").
See Details.

nd

Optional. Integer or pair of integers. The dimension of the grid of dummy points
(nd * nd or nd[1] * nd[2]) used to evaluate the integral in the pseudolikelihood. Incompatible with eps.

eps

Optional. A positive number, or a vector of two positive numbers, giving the
horizontal and vertical spacing, respectively, of the grid of dummy points. Incompatible with nd.

gcontrol

Optional. List of parameters passed to glm.control (or passed to gam.control
if use.gam=TRUE) controlling the model-fitting algorithm.

nsim

Number of simulated realisations to generate (for method="ho")

nrmh

Number of Metropolis-Hastings iterations for each simulated realisation (for
method="ho")

start,control

Arguments passed to rmh controlling the behaviour of the Metropolis-Hastings
algorithm (for method="ho")

verb

Logical flag indicating whether to print progress reports (for method="ho")

callstring

Internal use only.

Details
NOTE: This help page describes the old syntax of the function ppm, described in many older
documents. This old syntax is still supported. However, if you are learning about ppm for the first
time, we recommend you use the new syntax described in the help file for ppm.
This function fits a point process model to an observed point pattern. The model may include spatial
trend, interpoint interaction, and dependence on covariates.
basic use: In basic use, Q is a point pattern dataset (an object of class "ppp") to which we wish to
fit a model.
The syntax of ppm() is closely analogous to the R functions glm and gam. The analogy is:
glm
formula
family

ppm
trend
interaction

The point process model to be fitted is specified by the arguments trend and interaction
which are respectively analogous to the formula and family arguments of glm().
Systematic effects (spatial trend and/or dependence on spatial covariates) are specified by
the argument trend. This is an R formula object, which may be expressed in terms of the
Cartesian coordinates x, y, the marks marks, or the variables in covariates (if supplied), or
both. It specifies the logarithm of the first order potential of the process. The formula should
not use any names beginning with .mpl as these are reserved for internal use. If trend is

ppm.ppp

1051
absent or equal to the default, ~1, then the model to be fitted is stationary (or at least, its first
order potential is constant).
The symbol . in the trend expression stands for all the covariates supplied in the argument
data. For example the formula ~ . indicates an additive model with a main effect for each
covariate in data.
Stochastic interactions between random points of the point process are defined by the argument interaction. This is an object of class "interact" which is initialised in a very
similar way to the usage of family objects in glm and gam. The models currently available
are: AreaInter, BadGey, Concom, DiggleGatesStibbard, DiggleGratton, Fiksel, Geyer,
Hardcore, Hybrid, LennardJones, MultiStrauss, MultiStraussHard, OrdThresh, Ord,
Pairwise, PairPiece, Penttinen, Poisson, Saturated, SatPiece, Softcore, Strauss,
StraussHard and Triplets. See the examples below. It is also possible to combine several
interactions using Hybrid.
If interaction is missing or NULL, then the model to be fitted has no interpoint interactions,
that is, it is a Poisson process (stationary or nonstationary according to trend). In this case
the methods of maximum pseudolikelihood and maximum logistic likelihood coincide with
maximum likelihood.
The fitted point process model returned by this function can be printed (by the print method
print.ppm) to inspect the fitted parameter values. If a nonparametric spatial trend was fitted,
this can be extracted using the predict method predict.ppm.

Models with covariates: To fit a model involving spatial covariates other than the Cartesian coordinates x and y, the values of the covariates should be supplied in the argument covariates.
Note that it is not sufficient to have observed the covariate only at the points of the data point
pattern; the covariate must also have been observed at other locations in the window.
Typically the argument covariates is a list, with names corresponding to variables in the
trend formula. Each entry in the list is either
a pixel image, giving the values of a spatial covariate at a fine grid of locations. It should be
an object of class "im", see im.object.
a function, which can be evaluated at any location (x,y) to obtain the value of the spatial
covariate. It should be a function(x, y) or function(x, y, ...) in the R language.
The first two arguments of the function should be the Cartesian coordinates x and y. The
function may have additional arguments; if the function does not have default values for
these additional arguments, then the user must supply values for them, in covfunargs.
See the Examples.
a window, interpreted as a logical variable which is TRUE inside the window and FALSE outside it. This should be an object of class "owin".
a tessellation, interpreted as a factor covariate. For each spatial location, the factor value
indicates which tile of the tessellation it belongs to. This should be an object of class
"tess".
a single number, indicating a covariate that is constant in this dataset.
The software will look up the values of each covariate at the required locations (quadrature
points).
Note that, for covariate functions, only the name of the function appears in the trend formula.
A covariate function is treated as if it were a single variable. The function arguments do not
appear in the trend formula. See the Examples.
If covariates is a list, the list entries should have names corresponding to the names of
covariates in the model formula trend. The variable names x, y and marks are reserved for
the Cartesian coordinates and the mark values, and these should not be used for variables in
covariates.

1052

ppm.ppp
If covariates is a data frame, Q must be a quadrature scheme (see under Quadrature Schemes
below). Then covariates must have as many rows as there are points in Q. The ith row of
covariates should contain the values of spatial variables which have been observed at the ith
point of Q.

Quadrature schemes: In advanced use, Q may be a ‘quadrature scheme’. This was originally just
a technicality but it has turned out to have practical uses, as we explain below.
Quadrature schemes are required for our implementation of the method of maximum pseudolikelihood. The definition of the pseudolikelihood involves an integral over the spatial window
containing the data. In practice this integral must be approximated by a finite sum over a set
of quadrature points. We use the technique of Baddeley and Turner (2000), a generalisation
of the Berman-Turner (1992) device. In this technique the quadrature points for the numerical approximation include all the data points (points of the observed point pattern) as well as
additional ‘dummy’ points.
Quadrature schemes are also required for the method of maximum logistic likelihood, which
combines the data points with additional ‘dummy’ points.
A quadrature scheme is an object of class "quad" (see quad.object) which specifies both
the data point pattern and the dummy points for the quadrature scheme, as well as the quadrature weights associated with these points. If Q is simply a point pattern (of class "ppp", see
ppp.object) then it is interpreted as specifying the data points only; a set of dummy points
specified by default.dummy() is added, and the default weighting rule is invoked to compute
the quadrature weights.
Finer quadrature schemes (i.e. those with more dummy points) generally yield a better approximation, at the expense of higher computational load.
An easy way to fit models using a finer quadrature scheme is to let Q be the original point pattern data, and use the argument nd to determine the number of dummy points in the quadrature
scheme.
Complete control over the quadrature scheme is possible. See quadscheme for an overview.
Use quadscheme(X, D, method="dirichlet") to compute quadrature weights based on
the Dirichlet tessellation, or quadscheme(X, D, method="grid") to compute quadrature
weights by counting points in grid squares, where X and D are the patterns of data points and
dummy points respectively. Alternatively use pixelquad to make a quadrature scheme with a
dummy point at every pixel in a pixel image.
A practical advantage of quadrature schemes arises when we want to fit a model involving
covariates (e.g. soil pH). Suppose we have only been able to observe the covariates at a small
number of locations. Suppose cov.dat is a data frame containing the values of the covariates
at the data points (i.e.\ cov.dat[i,] contains the observations for the ith data point) and
cov.dum is another data frame (with the same columns as cov.dat) containing the covariate
values at another set of points whose locations are given by the point pattern Y. Then setting
Q = quadscheme(X,Y) combines the data points and dummy points into a quadrature scheme,
and covariates = rbind(cov.dat, cov.dum) combines the covariate data frames. We can
then fit the model by calling ppm(Q, ..., covariates).
Model-fitting technique: There are several choices for the technique used to fit the model.
method="mpl" (the default): the model will be fitted by maximising the pseudolikelihood
(Besag, 1975) using the Berman-Turner computational approximation (Berman and Turner,
1992; Baddeley and Turner, 2000). Maximum pseudolikelihood is equivalent to maximum likelihood if the model is a Poisson process. Maximum pseudolikelihood is biased
if the interpoint interaction is very strong, unless there is a large number of dummy points.
The default settings for method='mpl' specify a moderately large number of dummy
points, striking a compromise between speed and accuracy.
method="logi": the model will be fitted by maximising the logistic likelihood (Baddeley
et al, 2014). This technique is roughly equivalent in speed to maximum pseudolikeli-

ppm.ppp

1053
hood, but is believed to be less biased. Because it is less biased, the default settings for
method='logi' specify a relatively small number of dummy points, so that this method
is the fastest, in practice.
method="VBlogi": the model will be fitted in a Bayesian setup by maximising the posterior
probability density for the canonical model parameters. This uses the variational Bayes
approximation to the posterior derived from the logistic likelihood as described in Rajala
(2014). The prior is assumed to be multivariate Gaussian with mean vector prior.mean
and variance-covariance matrix prior.var.
method="ho": the model will be fitted by applying the approximate maximum likelihood
method of Huang and Ogata (1999). See below. The Huang-Ogata method is slower than
the other options, but has better statistical properties.
Note that method='logi', method='VBlogi' and method='ho' involve randomisation, so
that the results are subject to random variation.

Huang-Ogata method: If method="ho" then the model will be fitted using the Huang-Ogata (1999)
approximate maximum likelihood method. First the model is fitted by maximum pseudolikelihood as described above, yielding an initial estimate of the parameter vector θ0 . From this
initial model, nsim simulated realisations are generated. The score and Fisher information
of the model at θ = θ0 are estimated from the simulated realisations. Then one step of the
Fisher scoring algorithm is taken, yielding an updated estimate θ1 . The corresponding model
is returned.
Simulated realisations are generated using rmh. The iterative behaviour of the MetropolisHastings algorithm is controlled by the arguments start and control which are passed to
rmh.
As a shortcut, the argument nrmh determines the number of Metropolis-Hastings iterations run
to produce one simulated realisation (if control is absent). Also if start is absent or equal
to NULL, it defaults to list(n.start=N) where N is the number of points in the data point
pattern.
Edge correction Edge correction should be applied to the sufficient statistics of the model, to reduce bias. The argument correction is the name of an edge correction method. The default correction="border" specifies the border correction, in which the quadrature window
(the domain of integration of the pseudolikelihood) is obtained by trimming off a margin of
width rbord from the observation window of the data pattern. Not all edge corrections are
implemented (or implementable) for arbitrary windows. Other options depend on the argument interaction, but these generally include correction="periodic" (the periodic or
toroidal edge correction in which opposite edges of a rectangular window are identified) and
correction="translate" (the translation correction, see Baddeley 1998 and Baddeley and
Turner 2000). For pairwise interaction models there is also Ripley’s isotropic correction, identified by correction="isotropic" or "Ripley".
Subsetting The arguments subset and clipwin specify that the model should be fitted to a restricted subset of the available data. These arguments are equivalent for Poisson point process
models, but different for Gibbs models. If clipwin is specified, then all the available data will
be restricted to this spatial region, and data outside this region will be discarded, before the
model is fitted. If subset is specified, then no data are deleted, but the domain of integration
of the likelihood or pseudolikelihood is restricted to the subset. For Poisson models, these
two arguments have the same effect; but for a Gibbs model, interactions between points inside
and outside the subset are taken into account, while interactions between points inside and
outside the clipwin are ignored.
Value
An object of class "ppm" describing a fitted point process model.

1054

ppm.ppp

See ppm.object for details of the format of this object and methods available for manipulating it.
Interaction parameters
Apart from the Poisson model, every point process model fitted by ppm has parameters that determine the strength and range of ‘interaction’ or dependence between points. These parameters are
of two types:
regular parameters: A parameter φ is called regular if the log likelihood is a linear function of θ
where θ = θ(ψ) is some transformation of ψ. [Then θ is called the canonical parameter.]
irregular parameters Other parameters are called irregular.
Typically, regular parameters determine the ‘strength’ of the interaction, while irregular parameters
determine the ‘range’ of the interaction. For example, the Strauss process has a regular parameter γ
controlling the strength of interpoint inhibition, and an irregular parameter r determining the range
of interaction.
The ppm command is only designed to estimate regular parameters of the interaction. It requires
the values of any irregular parameters of the interaction to be fixed. For example, to fit a Strauss
process model to the cells dataset, you could type ppm(cells, ~1, Strauss(r=0.07)). Note
that the value of the irregular parameter r must be given. The result of this command will be a fitted
model in which the regular parameter γ has been estimated.
To determine the irregular parameters, there are several practical techniques, but no general statistical theory available. Useful techniques include maximum profile pseudolikelihood, which is
implemented in the command profilepl, and Newton-Raphson maximisation, implemented in the
experimental command ippm.
Some irregular parameters can be estimated directly from data: the hard-core radius in the model
Hardcore and the matrix of hard-core radii in MultiHard can be estimated easily from data. In
these cases, ppm allows the user to specify the interaction without giving the value of the irregular parameter. The user can give the hard core interaction as interaction=Hardcore() or even
interaction=Hardcore, and the hard core radius will then be estimated from the data.
Error and Warning Messages
Some common error messages and warning messages are listed below, with explanations.
“System is computationally singular” The Fisher information matrix of the fitted model has a
determinant close to zero, so that the matrix cannot be inverted, and the software cannot calculate standard errors or confidence intervals. This error is usually reported when the model
is printed, because the print method calculates standard errors for the fitted parameters. Singularity usually occurs because the spatial coordinates in the original data were very large
numbers (e.g. expressed in metres) so that the fitted coefficients were very small numbers.
The simple remedy is to rescale the data, for example, to convert from metres to kilometres
by X <- rescale(X, 1000), then re-fit the model. Singularity can also occur if the covariate
values are very large numbers, or if the covariates are approximately collinear.
“Covariate values were NA or undefined at X% (M out of N) of the quadrature points” The covariate data (typically a pixel image) did not provide values of the covariate at some of the
spatial locations in the observation window of the point pattern. This means that the spatial
domain of the pixel image does not completely cover the observation window of the point
pattern. If the percentage is small, this warning can be ignored - typically it happens because
of rounding effects which cause the pixel image to be one-pixel-width narrower than the observation window. However if more than a few percent of covariate values are undefined, it
would be prudent to check that the pixel images are correct, and are correctly registered in
their spatial relation to the observation window.

ppm.ppp

1055

“Model is unidentifiable” It is not possible to estimate all the model parameters from this dataset.
The error message gives a further explanation, such as “data pattern is empty”. Choose a
simpler model, or check the data.
“N data points are illegal (zero conditional intensity)” In a Gibbs model (i.e. with interaction
between points), the conditional intensity may be zero at some spatial locations, indicating
that the model forbids the presence of a point at these locations. However if the conditional
intensity is zero at a data point, this means that the model is inconsistent with the data. Modify
the interaction parameters so that the data point is not illegal (e.g. reduce the value of the hard
core radius) or choose a different interaction.
Warnings
The implementation of the Huang-Ogata method is experimental; several bugs were fixed in spatstat 1.19-0.
See the comments above about the possible inefficiency and bias of the maximum pseudolikelihood
estimator.
The accuracy of the Berman-Turner approximation to the pseudolikelihood depends on the number
of dummy points used in the quadrature scheme. The number of dummy points should at least equal
the number of data points.
The parameter values of the fitted model do not necessarily determine a valid point process. Some
of the point process models are only defined when the parameter values lie in a certain subset. For
example the Strauss process only exists when the interaction parameter γ is less than or equal to 1,
corresponding to a value of ppm()$theta[2] less than or equal to 0.
By default (if emend=FALSE) the algorithm maximises the pseudolikelihood without constraining
the parameters, and does not apply any checks for sanity after fitting the model. This is because the
fitted parameter value could be useful information for data analysis. To constrain the parameters to
ensure that the model is a valid point process, set emend=TRUE. See also the functions valid.ppm
and emend.ppm.
The trend formula should not use any variable names beginning with the prefixes .mpl or Interaction
as these names are reserved for internal use. The data frame covariates should have as many rows
as there are points in Q. It should not contain variables called x, y or marks as these names are
reserved for the Cartesian coordinates and the marks.
If the model formula involves one of the functions poly(), bs() or ns() (e.g. applied to spatial
coordinates x and y), the fitted coefficients can be misleading. The resulting fit is not to the raw
spatial variates (x, x^2, x*y, etc.) but to a transformation of these variates. The transformation
is implemented by poly() in order to achieve better numerical stability. However the resulting
coefficients are appropriate for use with the transformed variates, not with the raw variates. This
affects the interpretation of the constant term in the fitted model, logbeta. Conventionally, β is the
background intensity, i.e. the value taken by the conditional intensity function when all predictors
(including spatial or “trend” predictors) are set equal to 0. However the coefficient actually produced
is the value that the log conditional intensity takes when all the predictors, including the transformed
spatial predictors, are set equal to 0, which is not the same thing.
Worse still, the result of predict.ppm can be completely wrong if the trend formula contains one of
the functions poly(), bs() or ns(). This is a weakness of the underlying function predict.glm.
If you wish to fit a polynomial trend, we offer an alternative to poly(), namely polynom(), which
avoids the difficulty induced by transformations. It is completely analogous to poly except that it
does not orthonormalise. The resulting coefficient estimates then have their natural interpretation
and can be predicted correctly. Numerical stability may be compromised.
Values of the maximised pseudolikelihood are not comparable if they have been obtained with
different values of rbord.

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ppm.ppp

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
References
Baddeley, A., Coeurjolly, J.-F., Rubak, E. and Waagepetersen, R. (2014) Logistic regression for
spatial Gibbs point processes. Biometrika 101 (2) 377–392.
Baddeley, A. and Turner, R. Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42 (2000) 283–322.
Berman, M. and Turner, T.R. Approximating point process likelihoods with GLIM. Applied Statistics 41 (1992) 31–38.
Besag, J. Statistical analysis of non-lattice data. The Statistician 24 (1975) 179-195.
Diggle, P.J., Fiksel, T., Grabarnik, P., Ogata, Y., Stoyan, D. and Tanemura, M. On parameter estimation for pairwise interaction processes. International Statistical Review 62 (1994) 99-117.
Huang, F. and Ogata, Y. Improvements of the maximum pseudo-likelihood estimators in various
spatial statistical models. Journal of Computational and Graphical Statistics 8 (1999) 510-530.
Jensen, J.L. and Moeller, M. Pseudolikelihood for exponential family models of spatial point processes. Annals of Applied Probability 1 (1991) 445–461.
Jensen, J.L. and Kuensch, H.R. On asymptotic normality of pseudo likelihood estimates for pairwise
interaction processes, Annals of the Institute of Statistical Mathematics 46 (1994) 475-486.
Rajala T. (2014) A note on Bayesian logistic regression for spatial exponential family Gibbs point
processes, Preprint on ArXiv.org. http://arxiv.org/abs/1411.0539
See Also
ppm.object for details of how to print, plot and manipulate a fitted model.
ppp and quadscheme for constructing data.
Interactions: AreaInter, BadGey, Concom, DiggleGatesStibbard, DiggleGratton, Fiksel, Geyer,
Hardcore, Hybrid, LennardJones, MultiStrauss, MultiStraussHard, OrdThresh, Ord, Pairwise,
PairPiece, Penttinen, Poisson, Saturated, SatPiece, Softcore, Strauss, StraussHard and
Triplets.
See profilepl for advice on fitting nuisance parameters in the interaction, and ippm for irregular
parameters in the trend.
See valid.ppm and emend.ppm for ensuring the fitted model is a valid point process.
Examples
# fit the stationary Poisson process
# to point pattern 'nztrees'
ppm(nztrees)
ppm(nztrees ~ 1)
## Not run:
Q <- quadscheme(nztrees)
ppm(Q)
# equivalent.
## End(Not run)

ppm.ppp

## Not run:
ppm(nztrees, nd=128)
## End(Not run)
fit1 <- ppm(nztrees, ~ x)
# fit the nonstationary Poisson process
# with intensity function lambda(x,y) = exp(a + bx)
# where x,y are the Cartesian coordinates
# and a,b are parameters to be estimated
fit1
coef(fit1)
coef(summary(fit1))
## Not run:
ppm(nztrees, ~ polynom(x,2))
## End(Not run)
# fit the nonstationary Poisson process
# with intensity function lambda(x,y) = exp(a + bx + cx^2)
## Not run:
library(splines)
ppm(nztrees, ~ bs(x,df=3))
##
#
#
#
#

End(Not run)
WARNING: do not use predict.ppm() on this result
Fits the nonstationary Poisson process
with intensity function lambda(x,y) = exp(B(x))
where B is a B-spline with df = 3

## Not run:
ppm(nztrees, ~1, Strauss(r=10), rbord=10)
## End(Not run)
# Fit the stationary Strauss process with interaction range r=10
# using the border method with margin rbord=10
## Not run:
ppm(nztrees, ~ x, Strauss(13), correction="periodic")
## End(Not run)
# Fit the nonstationary Strauss process with interaction range r=13
# and exp(first order potential) = activity = beta(x,y) = exp(a+bx)
# using the periodic correction.
# Compare Maximum Pseudolikelihood, Huang-Ogata and VB fits:
## Not run: ppm(swedishpines, ~1, Strauss(9))

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ppmInfluence

## Not run: ppm(swedishpines, ~1, Strauss(9), method="ho")
ppm(swedishpines, ~1, Strauss(9), method="VBlogi")
# COVARIATES
#
X <- rpoispp(42)
weirdfunction <- function(x,y){ 10 * x^2 + 5 * sin(10 * y) }
#
# (a) covariate values as function
ppm(X, ~ y + Z, covariates=list(Z=weirdfunction))
#
# (b) covariate values in pixel image
Zimage <- as.im(weirdfunction, unit.square())
ppm(X, ~ y + Z, covariates=list(Z=Zimage))
#
# (c) covariate values in data frame
Q <- quadscheme(X)
xQ <- x.quad(Q)
yQ <- y.quad(Q)
Zvalues <- weirdfunction(xQ,yQ)
ppm(Q, ~ y + Z, covariates=data.frame(Z=Zvalues))
# Note Q not X
# COVARIATE FUNCTION WITH EXTRA ARGUMENTS
#
f <- function(x,y,a){ y - a }
ppm(X, ~x + f, covariates=list(f=f), covfunargs=list(a=1/2))
# COVARIATE: inside/outside window
b <- owin(c(0.1, 0.6), c(0.1, 0.9))
ppm(X, ~w, covariates=list(w=b))
## MULTITYPE POINT PROCESSES ###
# fit stationary marked Poisson process
# with different intensity for each species
## Not run: ppm(lansing, ~ marks, Poisson())
# fit nonstationary marked Poisson process
# with different log-cubic trend for each species
## Not run: ppm(lansing, ~ marks * polynom(x,y,3), Poisson())

ppmInfluence

Leverage and Influence Measures for Spatial Point Process Model

Description
Calculates all the leverage and influence measures described in influence.ppm, leverage.ppm
and dfbetas.ppm.

ppmInfluence

1059

Usage
ppmInfluence(fit,
what = c("leverage", "influence", "dfbetas"),
...,
iScore = NULL, iHessian = NULL, iArgs = NULL,
drop = FALSE,
fitname = NULL)
Arguments
fit

A fitted point process model of class "ppm".

what

Character vector specifying which quantities are to be calculated. Default is to
calculate all quantities.

...
Ignored.
iScore,iHessian
Components of the score vector and Hessian matrix for the irregular parameters,
if required. See Details.
iArgs

List of extra arguments for the functions iScore, iHessian if required.

drop

Logical. Whether to include (drop=FALSE) or exclude (drop=TRUE) contributions from quadrature points that were not used to fit the model.

fitname

Optional character string name for the fitted model fit.

Details
This function calculates all the leverage and influence measures described in influence.ppm,
leverage.ppm and dfbetas.ppm.
When analysing large datasets, the user can call ppmInfluence to perform the calculations efficiently, then extract the leverage and influence values as desired. For example the leverage can be
extracted either as result$leverage or leverage(result).
If the point process model trend has irregular parameters that were fitted (using ippm) then the
influence calculation requires the first and second derivatives of the log trend with respect to the
irregular parameters. The argument iScore should be a list, with one entry for each irregular
parameter, of R functions that compute the partial derivatives of the log trend (i.e. log intensity or
log conditional intensity) with respect to each irregular parameter. The argument iHessian should
be a list, with p2 entries where p is the number of irregular parameters, of R functions that compute
the second order partial derivatives of the log trend with respect to each pair of irregular parameters.
Value
A list containing the leverage and influence measures specified by what. The result also belongs to
the class "ppmInfluence".
Author(s)
Adrian Baddeley 
See Also
leverage.ppm, influence.ppm, dfbetas.ppm

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ppp

Examples
X <- rpoispp(function(x,y) { exp(3+3*x) })
fit <- ppm(X ~ x+y)
fI <- ppmInfluence(fit)
fI$influence
influence(fI)
fI$leverage
fI$dfbetas

ppp

Create a Point Pattern

Description
Creates an object of class "ppp" representing a point pattern dataset in the two-dimensional plane.
Usage
ppp(x,y, ..., window, marks,
check=TRUE, checkdup=check, drop=TRUE)
Arguments
x

Vector of x coordinates of data points

y

Vector of y coordinates of data points

window

window of observation, an object of class "owin"

...

arguments passed to owin to create the window, if window is missing

marks

(optional) mark values for the points. A vector or data frame.

check

Logical value indicating whether to check that all the (x, y) points lie inside the
specified window. Do not set this to FALSE unless you are absolutely sure that
this check is unnecessary. See Warnings below.

checkdup

Logical value indicating whether to check for duplicated coordinates. See Warnings below.

drop

Logical flag indicating whether to simplify data frames of marks. See Details.

Details
In the spatstat library, a point pattern dataset is described by an object of class "ppp". This function
creates such objects.
The vectors x and y must be numeric vectors of equal length. They are interpreted as the cartesian coordinates of the points in the pattern. Note that x and y are permitted to have length zero,
corresponding to an empty point pattern; this is the default if these arguments are missing.
A point pattern dataset is assumed to have been observed within a specific region of the plane called
the observation window. An object of class "ppp" representing a point pattern contains information
specifying the observation window. This window must always be specified when creating a point
pattern dataset; there is intentionally no default action of “guessing” the window dimensions from
the data points alone.
You can specify the observation window in several (mutually exclusive) ways:

ppp

1061
• xrange, yrange specify a rectangle with these dimensions;
• poly specifies a polygonal boundary. If the boundary is a single polygon then poly must be
a list with components x,y giving the coordinates of the vertices. If the boundary consists of
several disjoint polygons then poly must be a list of such lists so that poly[[i]]$x gives the
x coordinates of the vertices of the ith boundary polygon.
• mask specifies a binary pixel image with entries that are TRUE if the corresponding pixel is
inside the window.
• window is an object of class "owin" specifying the window. A window object can be created
by owin from raw coordinate data. Special shapes of windows can be created by the functions
square, hexagon, regularpolygon, disc and ellipse. See the Examples.
The arguments xrange, yrange or poly or mask are passed to the window creator function owin
for interpretation. See owin for further details.
The argument window, if given, must be an object of class "owin". It is a full description of the
window geometry, and could have been obtained from owin or as.owin, or by just extracting the
observation window of another point pattern, or by manipulating such windows. See owin or the
Examples below.
The points with coordinates x and y must lie inside the specified window, in order to define a valid
object of this class. Any points which do not lie inside the window will be removed from the point
pattern, and a warning will be issued. See the section on Rejected Points.
The name of the unit of length for the x and y coordinates can be specified in the dataset, using the
argument unitname, which is passed to owin. See the examples below, or the help file for owin.
The optional argument marks is given if the point pattern is marked, i.e. if each data point carries
additional information. For example, points which are classified into two or more different types, or
colours, may be regarded as having a mark which identifies which colour they are. Data recording
the locations and heights of trees in a forest can be regarded as a marked point pattern where the
mark is the tree height.
The argument marks can be either
• a vector, of the same length as x and y, which is interpreted so that marks[i] is the mark
attached to the point (x[i],y[i]). If the mark is a real number then marks should be a
numeric vector, while if the mark takes only a finite number of possible values (e.g. colours
or types) then marks should be a factor.
• a data frame, with the number of rows equal to the number of points in the point pattern. The
ith row of the data frame is interpreted as containing the mark values for the ith point in the
point pattern. The columns of the data frame correspond to different mark variables (e.g. tree
species and tree diameter).
If drop=TRUE (the default), then a data frame with only one column will be converted to a vector,
and a data frame with no columns will be converted to NULL.
See ppp.object for a description of the class "ppp".
Users would normally invoke ppp to create a point pattern, but the functions as.ppp and scanpp
may sometimes be convenient.

Value
An object of class "ppp" describing a point pattern in the two-dimensional plane (see ppp.object).
Invalid coordinate values
The coordinate vectors x and y must contain only finite numerical values. If the coordinates include
any of the values NA, NaN, Inf or -Inf, these will be removed.

1062

ppp

Rejected points
The points with coordinates x and y must lie inside the specified window, in order to define a valid
object of class "ppp". Any points which do not lie inside the window will be removed from the
point pattern, and a warning will be issued.
The rejected points are still accessible: they are stored as an attribute of the point pattern called
"rejects" (which is an object of class "ppp" containing the rejected points in a large window).
However, rejected points in a point pattern will be ignored by all other functions except plot.ppp.
To remove the rejected points altogether, use as.ppp. To include the rejected points, you will need
to find a larger window that contains them, and use this larger window in a call to ppp.
Warnings
The code will check for problems with the data, and issue a warning if any problems are found. The
checks and warnings can be switched off, for efficiency’s sake, but this should only be done if you
are confident that the data do not have these problems.
Setting check=FALSE will disable all the checking procedures: the check for points outside the
window, and the check for duplicated points. This is extremely dangerous, because points lying
outside the window will break many of the procedures in spatstat, causing crashes and strange
errors. Set check=FALSE only if you are absolutely sure that there are no points outside the window.
If duplicated points are found, a warning is issued, but no action is taken. Duplicated points are not
illegal, but may cause unexpected problems later. Setting checkdup=FALSE will disable the check
for duplicated points. Do this only if you already know the answer.
Methodology and software for spatial point patterns often assume that all points are distinct so
that there are no duplicated points. If duplicated points are present, the consequence could be
an incorrect result or a software crash. To the best of our knowledge, all spatstat code handles
duplicated points correctly. However, if duplicated points are present, we advise using unique.ppp
or multiplicity.ppp to eliminate duplicated points and re-analyse the data.
Author(s)
Adrian Baddeley  and Rolf Turner 
See Also
ppp.object, as.ppp, owin.object, owin, as.owin
Examples
# some arbitrary coordinates in [0,1]
x <- runif(20)
y <- runif(20)
#
X
X
X

the following are equivalent
<- ppp(x, y, c(0,1), c(0,1))
<- ppp(x, y)
<- ppp(x, y, window=owin(c(0,1),c(0,1)))

# specify that the coordinates are given in metres
X <- ppp(x, y, c(0,1), c(0,1), unitname=c("metre","metres"))
## Not run: plot(X)

ppp.object

1063

# marks
m <- sample(1:2, 20, replace=TRUE)
m <- factor(m, levels=1:2)
X <- ppp(x, y, c(0,1), c(0,1), marks=m)
## Not run: plot(X)
# polygonal window
X <- ppp(x, y, poly=list(x=c(0,10,0), y=c(0,0,10)))
## Not run: plot(X)
# circular window of radius 2
X <- ppp(x, y, window=disc(2))
# copy the window from another pattern
data(cells)
X <- ppp(x, y, window=Window(cells))

ppp.object

Class of Point Patterns

Description
A class "ppp" to represent a two-dimensional point pattern. Includes information about the window
in which the pattern was observed. Optionally includes marks.
Details
This class represents a two-dimensional point pattern dataset. It specifies
• the locations of the points
• the window in which the pattern was observed
• optionally, “marks” attached to each point (extra information such as a type label).
If X is an object of type ppp, it contains the following elements:
x
y
n
window
marks

vector of x coordinates of data points
vector of y coordinates of data points
number of points
window of observation
(an object of class owin)
optional vector or data frame of marks

Users are strongly advised not to manipulate these entries directly.
Objects of class "ppp" may be created by the function ppp and converted from other types of data
by the function as.ppp. Note that you must always specify the window of observation; there is
intentionally no default action of “guessing” the window dimensions from the data points alone.
Standard point pattern datasets provided with the package include amacrine, betacells, bramblecanes,
cells, demopat, ganglia, lansing, longleaf, nztrees, redwood, simdat and swedishpines.
Point patterns may be scanned from your own data files by scanpp or by using read.table and
as.ppp.
They may be manipulated by the functions [.ppp and superimpose.

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ppp.object

Point pattern objects can be plotted just by typing plot(X) which invokes the plot method for
point pattern objects, plot.ppp. See plot.ppp for further information.
There are also methods for summary and print for point patterns. Use summary(X) to see a useful
description of the data.
Patterns may be generated at random by runifpoint, rpoispp, rMaternI, rMaternII, rSSI,
rNeymanScott, rMatClust, and rThomas.
Most functions which are intended to operate on a window (of class owin) will, if presented with a
ppp object instead, automatically extract the window information from the point pattern.
Warnings
The internal representation of marks is likely to change in the next release of this package.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
owin, ppp, as.ppp, [.ppp
Examples
x <- runif(100)
y <- runif(100)
X <- ppp(x, y, c(0,1),c(0,1))
X
## Not run: plot(X)
mar <- sample(1:3, 100, replace=TRUE)
mm <- ppp(x, y, c(0,1), c(0,1), marks=mar)
## Not run: plot(mm)
# points with mark equal to 2
ss <- mm[ mm$marks == 2 , ]
## Not run: plot(ss)
# left half of pattern 'mm'
lu <- owin(c(0,0.5),c(0,1))
mmleft <- mm[ , lu]
## Not run: plot(mmleft)
## Not run:
if(FALSE) {
# input data from file
qq <- scanpp("my.table", unit.square())
# interactively build a point pattern
plot(unit.square())
X <- as.ppp(locator(10), unit.square())
plot(X)
}
## End(Not run)

pppdist

pppdist

1065

Distance Between Two Point Patterns

Description
Given two point patterns, find the distance between them based on optimal point matching.
Usage
pppdist(X, Y, type = "spa", cutoff = 1, q = 1, matching = TRUE,
ccode = TRUE, auction = TRUE, precision = NULL, approximation = 10,
show.rprimal = FALSE, timelag = 0)
Arguments
X,Y

Two point patterns (objects of class "ppp").

type

A character string giving the type of distance to be computed. One of "spa" (default), "ace" or "mat", indicating whether the algorithm should find the optimal
matching based on “subpattern assignment”, “assignment only if cardinalities
are equal” or “mass transfer”. See Details.

cutoff

The value > 0 at which interpoint distances are cut off.

q

The order of the average that is applied to the interpoint distances. May be Inf,
in which case the maximum of the interpoint distances is taken.

matching

Logical. Whether to return the optimal matching or only the associated distance.

ccode

Logical. If FALSE, R code is used which allows for higher precision, but is much
slower.

auction

Logical. By default a version of Bertsekas’ auction algorithm is used to compute
an optimal point matching if type is either "spa" or "ace". If auction is FALSE
(or type is "mat") a specialized primal-dual algorithm is used instead. This was
the standard in earlier versions of spatstat, but is several orders of magnitudes
slower.

precision

Index controlling accuracy of algorithm. The q-th powers of interpoint distances
will be rounded to the nearest multiple of 10^(-precision). There is a sensible
default which depends on ccode.

approximation

If q = Inf, compute distance based on the optimal matching for the corresponding distance of order approximation. Can be Inf, but this makes computations
extremely slow.

show.rprimal

Logical. Whether to plot the progress of the primal-dual algorithm. If TRUE,
slow primal-dual R code is used, regardless of the arguments ccode and auction.

timelag

Time lag, in seconds, between successive displays of the iterative solution of the
restricted primal problem.

Details
Computes the distance between point patterns X and Y based on finding the matching between
them which minimizes the average of the distances between matched points (if q=1), the maximum
distance between matched points (if q=Inf), and in general the q-th order average (i.e. the 1/qth

1066

pppdist

power of the sum of the qth powers) of the distances between matched points. Distances between
matched points are Euclidean distances cut off at the value of cutoff.
The parameter type controls the behaviour of the algorithm if the cardinalities of the point patterns
are different. For the type "spa" (subpattern assignment) the subpattern of the point pattern with the
larger cardinality n that is closest to the point pattern with the smaller cardinality m is determined;
then the q-th order average is taken over n values: the m distances of matched points and n − m
"penalty distances" of value cutoff for the unmatched points. For the type "ace" (assignment only
if cardinalities equal) the matching is empty and the distance returned is equal to cutoff if the
cardinalities differ. For the type "mat" (mass transfer) each point pattern is assumed to have total
mass m (= the smaller cardinality) distributed evenly among its points; the algorithm finds then
the "mass transfer plan" that minimizes the q-th order weighted average of the distances, where the
weights are given by the transferred mass divided by m. The result is a fractional matching (each
match of two points has a weight in (0, 1]) with the minimized quantity as the associated distance.
The central problem to be solved is the assignment problem (for types "spa" and "ace") or the more
general transport problem (for type "mat"). Both are well-known problems in discrete optimization,
see e.g. Luenberger (2003).
For the assignment problem pppdist uses by default the forward/backward version of Bertsekas’
auction algorithm with automated epsilon scaling; see Bertsekas (1992). The implemented version
gives good overall performance and can handle point patterns with several thousand points.
For the transport problem a specialized primal-dual algorithm is employed; see Luenberger (2003),
Section 5.9. The C implementation used by default can handle patterns with a few hundreds of
points, but should not be used with thousands of points. By setting show.rprimal = TRUE, some
insight in the working of the algorithm can be gained.
For a broader selection of optimal transport algorithms that are not restricted to spatial point patterns
and allow for additional fine tuning, we recommend the R package transport.
For moderate and large values of q there can be numerical issues based on the fact that the q-th
powers of distances are taken and some positive values enter the optimization algorithm as zeroes
because they are too small in comparison with the larger values. In this case the number of zeroes
introduced is given in a warning message, and it is possible then that the matching obtained is not
optimal and the associated distance is only a strict upper bound of the true distance. As a general
guideline (which can be very wrong in special situations) a small number of zeroes (up to about
50% of the smaller point pattern cardinality m) usually still results in the right matching, and the
number can even be quite a bit higher and usually still provides a highly accurate upper bound for
the distance. These numerical problems can be reduced by enforcing (much slower) R code via the
argument ccode = FALSE.
For q = Inf there is no fast algorithm available, which is why approximation is normally used: for
finding the optimal matching, q is set to the value of approximation. The resulting distance is still
given as the maximum rather than the q-th order average in the corresponding distance computation.
If approximation = Inf, approximation is suppressed and a very inefficient exhaustive search for
the best matching is performed.
The value of precision should normally not be supplied by the user. If ccode = TRUE, this value is
preset to the highest exponent of 10 that the C code still can handle (usually 9). If ccode = FALSE,
the value is preset according to q (usually 15 if q is small), which can sometimes be changed to
obtain less severe warning messages.
Value
Normally an object of class pppmatching that contains detailed information about the parameters
used and the resulting distance. See pppmatching.object for details. If matching = FALSE, only
the numerical value of the distance is returned.

pppdist

1067

Author(s)
Dominic Schuhmacher 
http://www.dominic.schuhmacher.name
References
Bertsekas, D.P. (1992). Auction algorithms for network flow problems: a tutorial introduction.
Computational Optimization and Applications 1, 7-66.
Luenberger, D.G. (2003). Linear and nonlinear programming. Second edition. Kluwer.
Schuhmacher, D. (2014). transport: optimal transport in various forms. R package version 0.6-2
(or later)
Schuhmacher, D. and Xia, A. (2008). A new metric between distributions of point processes. Advances in Applied Probability 40, 651–672
Schuhmacher, D., Vo, B.-T. and Vo, B.-N. (2008). A consistent metric for performance evaluation
of multi-object filters. IEEE Transactions on Signal Processing 56, 3447–3457.
See Also
pppmatching.object, matchingdist
Examples
# equal cardinalities
set.seed(140627)
X <- runifpoint(500)
Y <- runifpoint(500)
m <- pppdist(X, Y)
m
## Not run:
plot(m)
## End(Not run)
# differing cardinalities
X <- runifpoint(14)
Y <- runifpoint(10)
m1 <- pppdist(X, Y, type="spa")
m2 <- pppdist(X, Y, type="ace")
m3 <- pppdist(X, Y, type="mat", auction=FALSE)
summary(m1)
summary(m2)
summary(m3)
## Not run:
m1$matrix
m2$matrix
m3$matrix
## End(Not run)
# q = Inf
X <- runifpoint(10)
Y <- runifpoint(10)
mx1 <- pppdist(X, Y, q=Inf, matching=FALSE)
mx2 <- pppdist(X, Y, q=Inf, matching=FALSE, ccode=FALSE, approximation=50)
mx3 <- pppdist(X, Y, q=Inf, matching=FALSE, approximation=Inf)
all.equal(mx1,mx2,mx3)

1068

pppmatching

# sometimes TRUE
all.equal(mx2,mx3)
# very often TRUE

pppmatching

Create a Point Matching

Description
Creates an object of class "pppmatching" representing a matching of two planar point patterns
(objects of class "ppp").
Usage
pppmatching(X, Y, am, type = NULL, cutoff = NULL, q = NULL,
mdist = NULL)
Arguments
X,Y

Two point patterns (objects of class "ppp").

am

An npoints(X) by npoints(Y) matrix with entries ≥ 0 that specifies which
points are matched and with what weight; alternatively, an object that can be
coerced to this form by as.matrix.

type

A character string giving the type of the matching. One of "spa", "ace" or
"mat", or NULL for a generic or unknown matching.

cutoff, q

Numerical values specifying the cutoff value > 0 for interpoint distances and the
order q ∈ [1, ∞] of the average that is applied to them. NULL if not applicable or
unknown.

mdist

Numerical value for the distance to be associated with the matching.

Details
The argument am is interpreted as a "generalized adjacency matrix": if the [i,j]-th entry is positive,
then the i-th point of X and the j-th point of Y are matched and the value of the entry gives the
corresponding weight of the match. For an unweighted matching all the weights should be set to 1.
The remaining arguments are optional and allow to save additional information about the matching.
See the help files for pppdist and matchingdist for details on the meaning of these parameters.
Author(s)
Dominic Schuhmacher  http://www.dominic.schuhmacher.
name
See Also
pppmatching.object matchingdist

pppmatching.object

1069

Examples
# a random unweighted complete matching
X <- runifpoint(10)
Y <- runifpoint(10)
am <- r2dtable(1, rep(1,10), rep(1,10))[[1]]
# generates a random permutation matrix
m <- pppmatching(X, Y, am)
summary(m)
m$matrix
## Not run:
plot(m)
## End(Not run)
# a random weighted complete matching
X <- runifpoint(7)
Y <- runifpoint(7)
am <- r2dtable(1, rep(10,7), rep(10,7))[[1]]/10
# generates a random doubly stochastic matrix
m2 <- pppmatching(X, Y, am)
summary(m2)
m2$matrix
## Not run:
# Note: plotting does currently not distinguish
# between different weights
plot(m2)
## End(Not run)

pppmatching.object

Class of Point Matchings

Description
A class "pppmatching" to represent a matching of two planar point patterns. Optionally includes
information about the construction of the matching and its associated distance between the point
patterns.
Details
This class represents a (possibly weighted and incomplete) matching between two planar point
patterns (objects of class "ppp").
A matching can be thought of as a bipartite weighted graph where the vertices are given by the two
point patterns and edges of positive weights are drawn each time a point of the first point pattern is
"matched" with a point of the second point pattern.
If m is an object of type pppmatching, it contains the following elements
pp1, pp2
matrix
type

the two point patterns to be matched (vertices)
a matrix specifying which points are matched
and with what weights (edges)
(optional) a character string for the type of
the matching (one of "spa", "ace" or "mat")

1070

pppmatching.object
cutoff
q
distance

(optional) cutoff value for interpoint distances
(optional) the order for taking averages of
interpoint distances
(optional) the distance associated with the matching

The element matrix is a "generalized adjacency matrix". The numbers of rows and columns match
the cardinalities of the first and second point patterns, respectively. The [i,j]-th entry is positive
if the i-th point of X and the j-th point of Y are matched (zero otherwise) and its value then gives
the corresponding weight of the match. For an unweighted matching all the weights are set to 1.
The optional elements are for saving details about matchings in the context of optimal point matching techniques. type can be one of "spa" (for "subpattern assignment"), "ace" (for "assignment
only if cardinalities differ") or "mat" (for "mass transfer"). cutoff is a positive numerical value
that specifies the maximal interpoint distance and q is a value in [1, ∞] that gives the order of the
average applied to the interpoint distances. See the help files for pppdist and matchingdist for
detailed information about these elements.
Objects of class "pppmatching" may be created by the function pppmatching, and are most commonly obtained as output of the function pppdist. There are methods plot, print and summary
for this class.
Author(s)
Dominic Schuhmacher  http://www.dominic.schuhmacher.
name
See Also
matchingdist pppmatching
Examples
# a random complete unweighted matching
X <- runifpoint(10)
Y <- runifpoint(10)
am <- r2dtable(1, rep(1,10), rep(1,10))[[1]]
# generates a random permutation matrix
m <- pppmatching(X, Y, am)
summary(m)
m$matrix
## Not run:
plot(m)
## End(Not run)
# an optimal complete unweighted matching
m2 <- pppdist(X,Y)
summary(m2)
m2$matrix
## Not run:
plot(m2)
## End(Not run)

PPversion

PPversion

1071

Transform a Function into its P-P or Q-Q Version

Description
Given a function object f containing both the estimated and theoretical versions of a summary
function, these operations combine the estimated and theoretical functions into a new function.
When plotted, the new function gives either the P-P plot or Q-Q plot of the original f.
Usage
PPversion(f, theo = "theo", columns = ".")
QQversion(f, theo = "theo", columns = ".")
Arguments
f

The function to be transformed. An object of class "fv".

theo

The name of the column of f that should be treated as the theoretical value of
the function.

columns

Character vector, specifying the columns of f to which the transformation will
be applied. Either a vector of names of columns of f, or one of the abbreviations
recognised by fvnames.

Details
The argument f should be an object of class "fv", containing both empirical estimates fb(r) and a
theoretical value f0 (r) for a summary function.
The P–P version of f is the function g(x) = fb(f0−1 (x)) where f0−1 is the inverse function of f0 .
A plot of g(x) against x is equivalent to a plot of fb(r) against f0 (r) for all r. If f is a cumulative
distribution function (such as the result of Fest or Gest) then this is a P–P plot, a plot of the
observed versus theoretical probabilities for the distribution. The diagonal line y = x corresponds
to perfect agreement between observed and theoretical distribution.
The Q–Q version of f is the function h(x) = f0−1 (fb(x)). If f is a cumulative distribution function, a plot of h(x) against x is a Q–Q plot, a plot of the observed versus theoretical quantiles of
the distribution. The diagonal line y = x corresponds to perfect agreement between observed and
theoretical distribution. Another straight line corresponds to the situation where the observed variable is a linear transformation of the theoretical variable. For a point pattern X, the Q–Q version of
Kest(X) is essentially equivalent to Lest(X).
Value
Another object of class "fv".
Author(s)
Tom Lawrence and Adrian Baddeley.
Implemented by Adrian Baddeley , Rolf Turner 
and Ege Rubak .

1072

ppx

See Also
plot.fv
Examples
opa <- par(mar=0.1+c(5,5,4,2))
G <- Gest(redwoodfull)
plot(PPversion(G))
plot(QQversion(G))
par(opa)

ppx

Multidimensional Space-Time Point Pattern

Description
Creates a multidimensional space-time point pattern with any kind of coordinates and marks.
Usage
ppx(data, domain=NULL, coord.type=NULL, simplify=FALSE)
Arguments
data
domain
coord.type

simplify

The coordinates and marks of the points. A data.frame or hyperframe.
Optional. The space-time domain containing the points. An object in some
appropriate format, or NULL.
Character vector specifying how each column of data should be interpreted:
as a spatial coordinate, a temporal coordinate, a local coordinate or a mark.
Entries are partially matched to the values "spatial", "temporal", "local"
and "mark".
Logical value indicating whether to simplify the result in special cases. If
simplify=TRUE, a two-dimensional point pattern will be returned as an object
of class "ppp", and a three-dimensional point pattern will be returned as an object of class "pp3". If simplify=FALSE (the default) then the result is always
an object of class "ppx".

Details
An object of class "ppx" represents a marked point pattern in multidimensional space and/or time.
There may be any number of spatial coordinates, any number of temporal coordinates, any number
of local coordinates, and any number of mark variables. The individual marks may be atomic
(numeric values, factor values, etc) or objects of any kind.
The argument data should contain the coordinates and marks of the points. It should be a data.frame
or more generally a hyperframe (see hyperframe) with one row of data for each point.
Each column of data is either a spatial coordinate, a temporal coordinate, a local coordinate, or a
mark variable. The argument coord.type determines how each column is interpreted. It should be
a character vector, of length equal to the number of columns of data. It should contain strings that
partially match the values "spatial", "temporal", "local" and "mark". (The first letters will be
sufficient.)
By default (if coord.type is missing or NULL), columns of numerical data are assumed to represent
spatial coordinates, while other columns are assumed to be marks.

predict.dppm

1073

Value
Usually an object of class "ppx". If simplify=TRUE the result may be an object of class "ppp" or
"pp3".
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
pp3, print.ppx
Examples
df <- data.frame(x=runif(4),y=runif(4),t=runif(4),
age=rep(c("old", "new"), 2),
size=runif(4))
X <- ppx(data=df, coord.type=c("s","s","t","m","m"))
X
val <- 20 * runif(4)
E <- lapply(val, function(s) { rpoispp(s) })
hf <- hyperframe(t=val, e=as.listof(E))
Z <- ppx(data=hf, domain=c(0,1))
Z

predict.dppm

Prediction from a Fitted Determinantal Point Process Model

Description
Given a fitted determinantal point process model, these functions compute the fitted intensity.
Usage
## S3 method for class 'dppm'
fitted(object, ...)
## S3 method for class 'dppm'
predict(object, ...)
Arguments
object
...

Fitted determinantal point process model. An object of class "dppm".
Arguments passed to fitted.ppm or predict.ppm respectively.

Details
These functions are methods for the generic functions fitted and predict. The argument object
should be a determinantal point process model (object of class "dppm") obtained using the function
dppm.
The intensity of the fitted model is computed, using fitted.ppm or predict.ppm respectively.

1074

predict.kppm

Value
The value of fitted.dppm is a numeric vector giving the fitted values at the quadrature points.
The value of predict.dppm is usually a pixel image (object of class "im"), but see predict.ppm
for details.
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak 
See Also
dppm, plot.dppm, fitted.ppm, predict.ppm
Examples
fit <- dppm(swedishpines ~ x + y, dppGauss())
predict(fit)

predict.kppm

Prediction from a Fitted Cluster Point Process Model

Description
Given a fitted cluster point process model, these functions compute the fitted intensity.
Usage
## S3 method for class 'kppm'
fitted(object, ...)
## S3 method for class 'kppm'
predict(object, ...)
Arguments
object

Fitted cluster point process model. An object of class "kppm".

...

Arguments passed to fitted.ppm or predict.ppm respectively.

Details
These functions are methods for the generic functions fitted and predict. The argument object
should be a cluster point process model (object of class "kppm") obtained using the function kppm.
The intensity of the fitted model is computed, using fitted.ppm or predict.ppm respectively.
Value
The value of fitted.kppm is a numeric vector giving the fitted values at the quadrature points.
The value of predict.kppm is usually a pixel image (object of class "im"), but see predict.ppm
for details.

predict.lppm

1075

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
kppm, plot.kppm, vcov.kppm, fitted.ppm, predict.ppm
Examples
data(redwood)
fit <- kppm(redwood ~ x, "Thomas")
predict(fit)

predict.lppm

Predict Point Process Model on Linear Network

Description
Given a fitted point process model on a linear network, compute the fitted intensity or conditional
intensity of the model.
Usage
## S3 method for class 'lppm'
predict(object, ...,
type = "trend", locations = NULL, new.coef=NULL)
Arguments
object

The fitted model. An object of class "lppm", see lppm.

type

Type of values to be computed. Either "trend", "cif" or "se".

locations

Optional. Locations at which predictions should be computed. Either a data
frame with two columns of coordinates, or a binary image mask.

new.coef

Optional. Numeric vector of model coefficients, to be used instead of the fitted
coefficients coef(object) when calculating the prediction.

...

Optional arguments passed to as.mask to determine the pixel resolution (if
locations is missing).

Details
This function computes the fitted poin process intensity, fitted conditional intensity, or standard
error of the fitted intensity, for a point process model on a linear network. It is a method for the
generic predict for the class "lppm".
The argument object should be an object of class "lppm" (produced by lppm) representing a point
process model on a linear network.
Predicted values are computed at the locations given by the argument locations. If this argument
is missing, then predicted values are computed at a fine grid of points on the linear network.

1076

predict.mppm
• If locations is missing or NULL (the default), the return value is a pixel image (object of class
"linim" which inherits class "im") corresponding to a discretisation of the linear network,
with numeric pixel values giving the predicted values at each location on the linear network.
• If locations is a data frame, the result is a numeric vector of predicted values at the locations
specified by the data frame.
• If locations is a binary mask, the result is a pixel image with predicted values computed at
the pixels of the mask.

Value
A pixel image (object of class "linim" which inherits class "im") or a numeric vector, depending
on the argument locations. See Details.
Author(s)
Adrian Baddeley 
References
Ang, Q.W. (2010) Statistical methodology for events on a network. Master’s thesis, School of
Mathematics and Statistics, University of Western Australia.
Ang, Q.W., Baddeley, A. and Nair, G. (2012) Geometrically corrected second-order analysis of
events on a linear network, with applications to ecology and criminology. Scandinavian Journal of
Statistics 39, 591–617.
McSwiggan, G., Nair, M.G. and Baddeley, A. (2012) Fitting Poisson point process models to events
on a linear network. Manuscript in preparation.
See Also
lpp, linim
Examples
X <- runiflpp(12, simplenet)
fit <- lppm(X ~ x)
v <- predict(fit, type="trend")
plot(v)

predict.mppm

Prediction for Fitted Multiple Point Process Model

Description
Given a fitted multiple point process model obtained by mppm, evaluate the spatial trend and/or the
conditional intensity of the model. By default, predictions are evaluated over a grid of locations,
yielding pixel images of the trend and conditional intensity. Alternatively predictions may be evaluated at specified locations with specified values of the covariates.

predict.mppm

1077

Usage
## S3 method for class 'mppm'
predict(object, ..., newdata = NULL, type = c("trend", "cif"),
ngrid = 40, locations=NULL, verbose=FALSE)
Arguments
object

The fitted model. An object of class "mppm" obtained from mppm.

...

Ignored.

newdata

New values of the covariates, for which the predictions should be computed. If
newdata=NULL, predictions are computed for the original values of the covariates, to which the model was fitted. Otherwise newdata should be a hyperframe
(see hyperframe) containing columns of covariates as required by the model. If
type includes "cif", then newdata must also include a column of spatial point
pattern responses, in order to compute the conditional intensity.

type

Type of predicted values required. A character string or vector of character
strings. Options are "trend" for the spatial trend (first-order term) and "cif"
or "lambda" for the conditional intensity. Alternatively type="all" selects all
options.

ngrid

Dimensions of the grid of spatial locations at which prediction will be performed
(if locations=NULL). An integer or a pair of integers.

locations

Optional. The locations at which predictions should be performed. A list of
point patterns, with one entry for each row of newdata.

verbose

Logical flag indicating whether to print progress reports.

Details
This function computes the spatial trend and the conditional intensity of a fitted multiple spatial
point process model. See Baddeley and Turner (2000) and Baddeley et al (2007) for explanation
and examples.
Note that by “spatial trend” we mean the (exponentiated) first order potential and not the intensity
of the process. [For example if we fit the stationary Strauss process with parameters β and γ, then
the spatial trend is constant and equal to β.] The conditional intensity λ(u, X) of the fitted model
is evaluated at each required spatial location u, with respect to the response point pattern X.
If locations=NULL, then predictions are performed at an ngrid by ngrid grid of locations in the
window for each response point pattern. The result will be a hyperframe containing a column of
images of the trend (if selected) and a column of images of the conditional intensity (if selected).
The result can be plotted.
If locations is given, then it should be a list of point patterns (objects of class "ppp"). Predictions
are performed at these points. The result is a hyperframe containing a column of marked point
patterns where the locations each point.
Value
A hyperframe with columns named trend and cif.
If locations=NULL, the entries of the hyperframe are pixel images.
If locations is not null, the entries are marked point patterns constructed by attaching the predicted
values to the locations point patterns.

1078

predict.ppm

Author(s)
Adrian Baddeley , Ida-Maria Sintorn and Leanne Bischoff.
Implemented by Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Baddeley, A. and Turner, R. Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42 (2000) 283–322.
Baddeley, A., Bischof, L., Sintorn, I.-M., Haggarty, S., Bell, M. and Turner, R. Analysis of a
designed experiment where the response is a spatial point pattern. In preparation.
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. London: Chapman and Hall/CRC Press.
See Also
mppm, fitted.mppm, hyperframe
Examples
h <- hyperframe(Bugs=waterstriders)
fit <- mppm(Bugs ~ x, data=h, interaction=Strauss(7))
# prediction on a grid
p <- predict(fit)
plot(p$trend)
# prediction at specified locations
loc <- with(h, runifpoint(20, Window(Bugs)))
p2 <- predict(fit, locations=loc)
plot(p2$trend)

predict.ppm

Prediction from a Fitted Point Process Model

Description
Given a fitted point process model obtained by ppm, evaluate the spatial trend or the conditional
intensity of the model at new locations.
Usage
## S3 method for class 'ppm'
predict(object, window=NULL, ngrid=NULL, locations=NULL,
covariates=NULL,
type=c("trend", "cif", "intensity", "count"),
se=FALSE,
interval=c("none", "confidence", "prediction"),
level = 0.95,
X=data.ppm(object), correction,
...,
dimyx=NULL, eps=NULL,
new.coef=NULL, check=TRUE, repair=TRUE)

predict.ppm

1079

Arguments
object

A fitted point process model, typically obtained from the model-fitting algorithm
ppm. An object of class "ppm" (see ppm.object).

window

Optional. A window (object of class "owin") delimiting the locations where
predictions should be computed. Defaults to the window of the original data
used to fit the model object.

ngrid

Optional. Dimensions of a rectangular grid of locations inside window where
the predictions should be computed. An integer, or an integer vector of length
2, specifying the number of grid points in the y and x directions. (Incompatible
with locations. Equivalent to dimyx.)

locations

Optional. Data giving the exact x, y coordinates (and marks, if required) of
locations at which predictions should be computed. Either a point pattern, or
a data frame with columns named x and y, or a binary image mask, or a pixel
image. (Incompatible with ngrid, dimyx and eps).

covariates

Values of external covariates required by the model. Either a data frame or a list
of images. See Details.

type

Character string. Indicates which property of the fitted model should be predicted. Options are "trend" for the spatial trend, "cif" or "lambda" for the
conditional intensity, "intensity" for the intensity, and "count" for the total
number of points in window.

se

Logical value indicating whether to calculate standard errors as well.

interval

String (partially matched) indicating whether to produce estimates (interval="none",
the default) or a confidence interval (interval="confidence") or a prediction
interval (interval="prediction").

level

Coverage probability for the confidence or prediction interval.

X

Optional. A point pattern (object of class "ppp") to be taken as the data point
pattern when calculating the conditional intensity. The default is to use the original data to which the model was fitted.

correction

Name of the edge correction to be used in calculating the conditional intensity.
Options include "border" and "none". Other options may include "periodic",
"isotropic" and "translate" depending on the model. The default correction
is the one that was used to fit object.

...

Ignored.

dimyx

Equivalent to ngrid.

eps

Width and height of pixels in the prediction grid. A numerical value, or numeric
vector of length 2.

new.coef

Numeric vector of parameter values to replace the fitted model parameters coef(object).

check

Logical value indicating whether to check the internal format of object. If
there is any possibility that this object has been restored from a dump file, or
has otherwise lost track of the environment where it was originally computed,
set check=TRUE.

repair

Logical value indicating whether to repair the internal format of object, if it is
found to be damaged.

1080

predict.ppm

Details
This function computes properties of a fitted spatial point process model (object of class "ppm").
For a Poisson point process it can compute the fitted intensity function, or the expected number of
points in a region. For a Gibbs point process it can compute the spatial trend (first order potential),
conditional intensity, and approximate intensity of the process. Point estimates, standard errors,
confidence intervals and prediction intervals are available.
Given a point pattern dataset, we may fit a point process model to the data using the model-fitting
algorithm ppm. This returns an object of class "ppm" representing the fitted point process model (see
ppm.object). The parameter estimates in this fitted model can be read off simply by printing the
ppm object. The spatial trend, conditional intensity and intensity of the fitted model are evaluated
using this function predict.ppm.
The default action is to create a rectangular grid of points in the observation window of the data
point pattern, and evaluate the spatial trend at these locations.
The argument type specifies the values that are desired:
If type="trend": the “spatial trend” of the fitted model is evaluated at each required spatial location u. See below.
If type="cif": the conditional intensity λ(u, X) of the fitted model is evaluated at each required
spatial location u, with respect to the data point pattern X.
If type="intensity": the intensity λ(u) of the fitted model is evaluated at each required spatial
location u.
If type="count": the expected total number of points (or the expected number of points falling in
window) is evaluated. If window is a tessellation, the expected number of points in each tile of
the tessellation is evaluated.
The spatial trend, conditional intensity, and intensity are all equivalent if the fitted model is a Poisson
point process. However, if the model is not a Poisson process, then they are all different. The
“spatial trend” is the (exponentiated) first order potential, and not the intensity of the process. [For
example if we fit the stationary Strauss process with parameters β and γ, then the spatial trend is
constant and equal to β, while the intensity is a smaller value.]
The default is to compute an estimate of the desired quantity. If interval="confidence" or
interval="prediction", the estimate is replaced by a confidence interval or prediction interval.
If se=TRUE, then a standard error is also calculated, and is returned together with the (point or
interval) estimate.
The spatial locations where predictions are required, are determined by the (incompatible) arguments ngrid and locations.
• If the argument ngrid is present, then predictions are performed at a rectangular grid of locations in the window window. The result of prediction will be a pixel image or images.
• If locations is present, then predictions will be performed at the spatial locations given by
this dataset. These may be an arbitrary list of spatial locations, or they may be a rectangular
grid. The result of prediction will be either a numeric vector or a pixel image or images.
• If neither ngrid nor locations is given, then ngrid is assumed. The value of ngrid defaults
to spatstat.options("npixel"), which is initialised to 128 when spatstat is loaded.
The argument locations may be a point pattern, a data frame or a list specifying arbitrary locations;
or it may be a binary image mask (an object of class "owin" with type "mask") or a pixel image
(object of class "im") specifying (a subset of) a rectangular grid of locations.

predict.ppm

1081

• If locations is a point pattern (object of class "ppp"), then prediction will be performed at
the points of the point pattern. The result of prediction will be a vector of predicted values,
one value for each point. If the model is a marked point process, then locations should be a
marked point pattern, with marks of the same kind as the model; prediction will be performed
at these marked points. The result of prediction will be a vector of predicted values, one value
for each (marked) point.
• If locations is a data frame or list, then it must contain vectors locations$x and locations$y
specifying the x, y coordinates of the prediction locations. Additionally, if the model is a
marked point process, then locations must also contain a factor locations$marks specifying the marks of the prediction locations. These vectors must have equal length. The result of
prediction will be a vector of predicted values, of the same length.
• If locations is a binary image mask, then prediction will be performed at each pixel in this
binary image where the pixel value is TRUE (in other words, at each pixel that is inside the
window). If the fitted model is an unmarked point process, then the result of prediction will
be an image. If the fitted model is a marked point process, then prediction will be performed
for each possible value of the mark at each such location, and the result of prediction will be
a list of images, one for each mark value.
• If locations is a pixel image (object of class "im"), then prediction will be performed at each
pixel in this image where the pixel value is defined (i.e.\ where the pixel value is not NA).
The argument covariates gives the values of any spatial covariates at the prediction locations. If
the trend formula in the fitted model involves spatial covariates (other than the Cartesian coordinates
x, y) then covariates is required. The format and use of covariates are analogous to those of
the argument of the same name in ppm. It is either a data frame or a list of images.
• If covariates is a list of images, then the names of the entries should correspond to the names
of covariates in the model formula trend. Each entry in the list must be an image object (of
class "im", see im.object). The software will look up the pixel values of each image at the
quadrature points.
• If covariates is a data frame, then the ith row of covariates is assumed to contain covariate
data for the ith location. When locations is a data frame, this just means that each row of
covariates contains the covariate data for the location specified in the corresponding row
of locations. When locations is a binary image mask, the row covariates[i,] must
correspond to the location x[i],y[i] where x = as.vector(raster.x(locations)) and
y = as.vector(raster.y(locations)).
Note that if you only want to use prediction in order to generate a plot of the predicted values, it
may be easier to use plot.ppm which calls this function and plots the results.
Value
If total is given: a numeric vector or matrix.
If locations is given and is a data frame: a vector of predicted values for the spatial locations
(and marks, if required) given in locations.
If ngrid is given, or if locations is given and is a binary image mask or a pixel image: If object
is an unmarked point process, the result is a pixel image object (of class "im", see im.object)
containing the predictions. If object is a multitype point process, the result is a list of pixel images,
containing the predictions for each type at the same grid of locations.
The “predicted values” are either values of the spatial trend (if type="trend"), values of the conditional intensity (if type="cif" or type="lambda"), values of the intensity (if type="intensity")
or numbers of points (if type="count").

1082

predict.ppm

If se=TRUE, then the result is a list with two entries, the first being the predicted values in the format
described above, and the second being the standard errors in the same format.
Warnings
The current implementation invokes predict.glm so that prediction is wrong if the trend formula
in object involves terms in ns(), bs() or poly(). This is a weakness of predict.glm itself!
Error messages may be very opaque, as they tend to come from deep in the workings of predict.glm.
If you are passing the covariates argument and the function crashes, it is advisable to start by
checking that all the conditions listed above are satisfied.
Author(s)
Adrian Baddeley  and Rolf Turner 
References
Baddeley, A. and Turner, R. Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42 (2000) 283–322.
Berman, M. and Turner, T.R. Approximating point process likelihoods with GLIM. Applied Statistics 41 (1992) 31–38.
See Also
ppm, ppm.object, plot.ppm, print.ppm, fitted.ppm, spatstat.options
Examples
m <- ppm(cells ~ polynom(x,y,2), Strauss(0.05))
trend <- predict(m, type="trend")
## Not run:
image(trend)
points(cells)
## End(Not run)
cif <- predict(m, type="cif")
## Not run:
persp(cif)
## End(Not run)
data(japanesepines)
mj <- ppm(japanesepines ~ harmonic(x,y,2))
se <- predict(mj, se=TRUE)
# prediction interval for total number of points
predict(mj, type="count", interval="p")
# prediction at arbitrary locations
predict(mj, locations=data.frame(x=0.3, y=0.4))
X <- runifpoint(5, Window(japanesepines))
predict(mj, locations=X, se=TRUE)
# multitype

predict.rppm

1083

rr <- matrix(0.06, 2, 2)
ma <- ppm(amacrine ~ marks, MultiStrauss(rr))
Z <- predict(ma)
Z <- predict(ma, type="cif")
predict(ma, locations=data.frame(x=0.8, y=0.5,marks="on"), type="cif")

predict.rppm

Make Predictions From a Recursively Partitioned Point Process Model

Description
Given a model which has been fitted to point pattern data by recursive partitioning, compute the
predicted intensity of the model.
Usage
## S3 method for class 'rppm'
predict(object, ...)
## S3 method for class 'rppm'
fitted(object, ...)
Arguments
object

Fitted point process model of class "rppm" produced by the function rppm.

...

Optional arguments passed to predict.ppm to specify the locations where prediction is required. (Ignored by fitted.rppm)

Details
These functions are methods for the generic functions fitted and predict. They compute the
fitted intensity of a point process model. The argument object should be a fitted point process
model of class "rppm" produced by the function rppm.
The fitted method computes the fitted intensity at the original data points, yielding a numeric
vector with one entry for each data point.
The predict method computes the fitted intensity at any locations. By default, predictions are
calculated at a regular grid of spatial locations, and the result is a pixel image giving the predicted
intensity values at these locations.
Alternatively, predictions can be performed at other locations, or a finer grid of locations, or
only at certain specified locations, using additional arguments ... which will be interpreted by
predict.ppm. Common arguments are ngrid to increase the grid resolution, window to specify the
prediction region, and locations to specify the exact locations of predictions. See predict.ppm
for details of these arguments.
Predictions are computed by evaluating the explanatory covariates at each desired location, and
applying the recursive partitioning rule to each set of covariate values.

1084

predict.slrm

Value
The result of fitted.rppm is a numeric vector.
The result of predict.rppm is a pixel image, a list of pixel images, or a numeric vector.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
rppm, plot.rppm
Examples
fit <- rppm(unmark(gorillas) ~ vegetation, data=gorillas.extra)
plot(predict(fit))
lambdaX <- fitted(fit)
lambdaX[1:5]
# Mondriaan pictures
plot(predict(rppm(redwoodfull ~ x + y)))
points(redwoodfull)

predict.slrm

Predicted or Fitted Values from Spatial Logistic Regression

Description
Given a fitted Spatial Logistic Regression model, this function computes the fitted probabilities for
each pixel, or the fitted point process intensity, or the values of the linear predictor in each pixel.
Usage
## S3 method for class 'slrm'
predict(object, ..., type = "intensity",
newdata=NULL, window=NULL)
Arguments
object

a fitted spatial logistic regression model. An object of class "slrm".

...

Optional arguments passed to pixellate determining the pixel resolution for
the discretisation of the point pattern.

type

Character string (partially) matching one of "probabilities", "intensity"
or "link".

newdata

Optional. List containing new covariate values for the prediction. See Details.

window

Optional. New window in which to predict. An object of class "owin".

print.im

1085

Details
This is a method for predict for spatial logistic regression models (objects of class "slrm", usually
obtained from the function slrm).
The argument type determines which quantity is computed. If type="intensity"), the value of
the point process intensity is computed at each pixel. If type="probabilities") the probability of
the presence of a random point in each pixel is computed. If type="link", the value of the linear
predictor is computed at each pixel.
If newdata = NULL (the default), the algorithm computes fitted values of the model (based on the
data that was originally used to fit the model object).
If newdata is given, the algorithm computes predicted values of the model, using the new values
of the covariates provided by newdata. The argument newdata should be a list; names of entries
in the list should correspond to variables appearing in the model formula of the object. Each list
entry may be a pixel image or a single numeric value.
Value
A pixel image (object of class "im") containing the predicted values for each pixel.
Author(s)
Adrian Baddeley  
and Rolf Turner 
See Also
slrm
Examples
X <- rpoispp(42)
fit <- slrm(X ~ x+y)
plot(predict(fit))
data(copper)
X <- copper$SouthPoints
Y <- copper$SouthLines
Z <- distmap(Y)
fitc <- slrm(X ~ Z)
pc <- predict(fitc)
Znew <- distmap(copper$Lines)[copper$SouthWindow]
pcnew <- predict(fitc, newdata=list(Z=Znew))

print.im

Print Brief Details of an Image

Description
Prints a very brief description of a pixel image object.

1086

print.owin

Usage
## S3 method for class 'im'
print(x, ...)
Arguments
x

Pixel image (object of class "im").

...

Ignored.

Details
A very brief description of the pixel image x is printed.
This is a method for the generic function print.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
print, im.object, summary.im
Examples
data(letterR)
U <- as.im(letterR)
U

print.owin

Print Brief Details of a Spatial Window

Description
Prints a very brief description of a window object.
Usage
## S3 method for class 'owin'
print(x, ..., prefix="window: ")
Arguments
x

Window (object of class "owin").

...

Ignored.

prefix

Character string to be printed at the start of the output.

Details
A very brief description of the window x is printed.
This is a method for the generic function print.

print.ppm

1087

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
print, print.ppp, summary.owin
Examples
owin()

# the unit square

data(demopat)
W <- Window(demopat)
W
# just says it is polygonal
as.mask(W)
# just says it is a binary image

print.ppm

Print a Fitted Point Process Model

Description
Default print method for a fitted point process model.
Usage
## S3 method for class 'ppm'
print(x,...,
what=c("all", "model", "trend", "interaction", "se", "errors"))
Arguments
x

A fitted point process model, typically obtained from the model-fittingg algorithm ppm. An object of class "ppm".

what

Character vector (partially-matched) indicating what information should be printed.

...

Ignored.

Details
This is the print method for the class "ppm". It prints information about the fitted model in a
sensible format.
The argument what makes it possible to print only some of the information.
If what is missing, then by default, standard errors for the estimated coefficients of the model will be
printed only if the model is a Poisson point process. To print the standard errors for a non-Poisson
model, call print.ppm with the argument what given explicitly, or reset the default rule by typing
spatstat.options(print.ppm.SE="always").
Value
none.

1088

print.ppp

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
ppm.object for details of the class "ppm".
ppm for generating these objects.
plot.ppm, predict.ppm
Examples
## Not run:
m <- ppm(cells, ~1, Strauss(0.05))
m
## End(Not run)

print.ppp

Print Brief Details of a Point Pattern Dataset

Description
Prints a very brief description of a point pattern dataset.
Usage
## S3 method for class 'ppp'
print(x, ...)
Arguments
x

Point pattern (object of class "ppp").

...

Ignored.

Details
A very brief description of the point pattern x is printed.
This is a method for the generic function print.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
print, print.owin, summary.ppp

print.psp

1089

Examples
data(cells)
cells

# plain vanilla point pattern

data(lansing)
lansing

# multitype point pattern

data(longleaf)
longleaf

# numeric marks

data(demopat)
demopat

# weird polygonal window

print.psp

Print Brief Details of a Line Segment Pattern Dataset

Description
Prints a very brief description of a line segment pattern dataset.
Usage
## S3 method for class 'psp'
print(x, ...)
Arguments
x

Line segment pattern (object of class "psp").

...

Ignored.

Details
A very brief description of the line segment pattern x is printed.
This is a method for the generic function print.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
print, print.owin, summary.psp
Examples
a <- psp(runif(10), runif(10), runif(10), runif(10), window=owin())
a

1090

print.quad

print.quad

Print a Quadrature Scheme

Description
print method for a quadrature scheme.
Usage
## S3 method for class 'quad'
print(x,...)
Arguments
x

A quadrature scheme object, typically obtained from quadscheme. An object of
class "quad".

...

Ignored.

Details
This is the print method for the class "quad". It prints simple information about the quadrature
scheme.
See quad.object for details of the class "quad".
Value
none.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
quadscheme, quad.object, plot.quad, summary.quad
Examples
data(cells)
Q <- quadscheme(cells)
Q

profilepl

profilepl

1091

Fit Models by Profile Maximum Pseudolikelihood or AIC

Description
Fits point process models by maximising the profile likelihood, profile pseudolikelihood, profile
composite likelihood or AIC.
Usage
profilepl(s, f, ..., aic=FALSE, rbord=NULL, verbose = TRUE, fast=TRUE)
Arguments
s

Data frame containing values of the irregular parameters over which the criterion
will be computed.

f

Function (such as Strauss) that generates an interpoint interaction object, given
values of the irregular parameters.

...

Data passed to ppm to fit the model.

aic

Logical value indicating whether to find the parameter values which minimise
the AIC (aic=TRUE) or maximise the profile likelihood (aic=FALSE, the default).

rbord

Radius for border correction (same for all models). If omitted, this will be computed from the interactions.

verbose

Logical value indicating whether to print progress reports.

fast

Logical value indicating whether to use a faster, less accurate model-fitting technique when computing the profile pseudolikelihood. See Section on Speed and
Accuracy.

Details
The model-fitting function ppm fits point process models to point pattern data. However, only the
‘regular’ parameters of the model can be fitted by ppm. The model may also depend on ‘irregular’
parameters that must be fixed in any call to ppm.
This function profilepl is a wrapper which finds the values of the irregular parameters that give
the best fit. If aic=FALSE (the default), the best fit is the model which maximises the likelihood
(if the models are Poisson processes) or maximises the pseudolikelihood or logistic likelihood. If
aic=TRUE then the best fit is the model which minimises the Akaike Information Criterion AIC.ppm.
The argument s must be a data frame whose columns contain values of the irregular parameters
over which the maximisation is to be performed.
An irregular parameter may affect either the interpoint interaction or the spatial trend.
interaction parameters: in a call to ppm, the argument interaction determines the interaction
between points. It is usually a call to a function such as Strauss. The arguments of this
call are irregular parameters. For example, the interaction radius parameter r of the Strauss
process, determined by the argument r to the function Strauss, is an irregular parameter.
trend parameters: in a call to ppm, the spatial trend may depend on covariates, which are supplied
by the argument covariates. These covariates may be functions written by the user, of the
form function(x,y,...), and the extra arguments ... are irregular parameters.

1092

profilepl

The argument f determines the interaction for each model to be fitted. It would typically be one
of the functions Poisson, AreaInter, BadGey, DiggleGatesStibbard, DiggleGratton, Fiksel,
Geyer, Hardcore, LennardJones, OrdThresh, Softcore, Strauss or StraussHard. Alternatively
it could be a function written by the user.
Columns of s which match the names of arguments of f will be interpreted as interaction parameters. Other columns will be interpreted as trend parameters.
The data frame s must provide values for each argument of f, except for the optional arguments,
which are those arguments of f that have the default value NA.
To find the best fit, each row of s will be taken in turn. Interaction parameters in this row will be
passed to f, resulting in an interaction object. Then ppm will be applied to the data ... using this
interaction. Any trend parameters will be passed to ppm through the argument covfunargs. This
results in a fitted point process model. The value of the log pseudolikelihood or AIC from this
model is stored. After all rows of s have been processed in this way, the row giving the maximum
value of log pseudolikelihood will be found.
The object returned by profilepl contains the profile pseudolikelihood (or profile AIC) function,
the best fitting model, and other data. It can be plotted (yielding a plot of the log pseudolikelihood or
AIC values against the irregular parameters) or printed (yielding information about the best fitting
values of the irregular parameters).
In general, f may be any function that will return an interaction object (object of class "interact")
that can be used in a call to ppm. Each argument of f must be a single value.
Value
An object of class "profilepl". There are methods for plot, print, summary, simulate, as.ppm,
fitin and parameters for objects of this class.
The components of the object include
fit

Best-fitting model

param

The data frame s

iopt

Row index of the best-fitting parameters in s

To extract the best fitting model you can also use as.ppm.
Speed and Accuracy
Computation of the profile pseudolikelihood can be time-consuming. We recommend starting with
a small experiment in which s contains only a few rows of values. This will indicate roughly
the optimal values of the parameters. Then a full calculation using more finely spaced values can
identify the exact optimal values.
It is normal that the procedure appears to slow down at the end. During the computation of the
profile pseudolikelihood, the model-fitting procedure is accelerated by omitting some calculations
that are not needed for computing the pseudolikelihood. When the optimal parameter values have
been identified, they are used to fit the final model in its entirety. Fitting the final model can take
longer than computing the profile pseudolikelihood.
If fast=TRUE (the default), then additional shortcuts are taken in order to accelerate the computation
of the profile log pseudolikelihood. These shortcuts mean that the values of the profile log pseudolikelihood in the result ($prof) may not be equal to the values that would be obtained if the model
was fitted normally. Currently this happens only for the area interaction AreaInter. It may be wise
to do a small experiment with fast=TRUE and then a definitive calculation with fast=FALSE.

progressreport

1093

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
Examples
# one irregular parameter
rr <- data.frame(r=seq(0.05,0.15, by=0.01))
ps <- profilepl(rr, Strauss, cells)
ps
if(interactive()) plot(ps)
# two irregular parameters
rs <- expand.grid(r=seq(0.05,0.15, by=0.01),sat=1:3)
pg <- profilepl(rs, Geyer, cells)
pg
if(interactive()) {
plot(pg)
as.ppm(pg)
}
# multitype pattern with a common interaction radius
## Not run:
RR <- data.frame(R=seq(0.03,0.05,by=0.01))
MS <- function(R) { MultiStrauss(radii=diag(c(R,R))) }
pm <- profilepl(RR, MS, amacrine ~marks)
## End(Not run)
## more information
summary(pg)

progressreport

Print Progress Reports

Description
Prints Progress Reports during a loop or iterative calculation.
Usage
progressreport(i, n,
every = min(100,max(1, ceiling(n/100))),
tick = 1,
nperline = NULL,
charsperline = getOption("width"),
style = spatstat.options("progress"),
showtime = NULL,
state=NULL)

1094

progressreport

Arguments
i

Integer. The current iteration number (from 1 to n).

n

Integer. The (maximum) number of iterations to be computed.

every

Optional integer. Iteration number will be printed when i is a multiple of every.

tick

Optional integer. A tick mark or dot will be printed when i is a multiple of tick.

nperline

Optional integer. Number of iterations per line of output.

charsperline

Optional integer. The number of characters in a line of output.

style

Character string determining the style of display. Options are "tty" (the default), "tk" and "txtbar". See Details.

showtime

Optional. Logical value indicating whether to print the estimated time remaining. Applies only when style="tty".

state

Optional. A list containing the internal data.

Details
This is a convenient function for reporting progress during an iterative sequence of calculations or
a suite of simulations.
• If style="tk" then tcltk::tkProgressBar is used to pop-up a new graphics window showing a progress bar. This requires the package tcltk. As i increases from 1 to n, the bar will
lengthen. The arguments every, tick, nperline, showtime are ignored.
• If style="txtbar" then txtProgressBar is used to represent progress as a bar made of text
characters in the R interpreter window. As i increases from 1 to n, the bar will lengthen. The
arguments every, tick, nperline, showtime are ignored.
• If style="tty" (the default), then progress reports are printed to the console. This only seems
to work well under Linux. As i increases from 1 to n, the output will be a sequence of dots
(one dot for every tick iterations), iteration numbers (printed when iteration number is a
multiple of every or is less than 4), and optionally the estimated time remaining. For example
[etd 1:20:05] means an estimated time of 1 hour, 20 minutes and 5 seconds until finished.
The estimated time remaining will be printed only if style="tty", and the argument state
is given, and either showtime=TRUE, or showtime=NULL and the iterations are slow (defined
as: the estimated time remaining is longer than 3 minutes, or the average time per iteration is
longer than 20 seconds).
It is optional, but strongly advisable, to use the argument state to store and update the internal data
for the progress reports (such as the cumulative time taken for computation) as shown in the last
example below. This avoids conflicts with other programs that might be calling progressreport
at the same time.
Value
If state was NULL, the result is NULL. Otherwise the result is the updated value of state.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .

project2segment

1095

Examples
for(i in 1:40) {
#
# code that does something...
#
progressreport(i, 40)
}
# saving internal state: *recommended*
sta <- list()
for(i in 1:20) {
# some code ...
sta <- progressreport(i, 20, state=sta)
}

project2segment

Move Point To Nearest Line

Description
Given a point pattern and a line segment pattern, this function moves each point to the closest
location on a line segment.
Usage
project2segment(X, Y)
Arguments
X
Y

A point pattern (object of class "ppp").
A line segment pattern (object of class "psp").

Details
For each point x in the point pattern X, this function finds the closest line segment y in the line
segment pattern Y. It then ‘projects’ the point x onto the line segment y by finding the position z
along y which is closest to x. This position z is returned, along with supplementary information.
Value
A list with the following components. Each component has length equal to the number of points in
X, and its entries correspond to the points of X.
Xproj
mapXY
d
tp

Point pattern (object of class "ppp" containing the projected points.
Integer vector identifying the nearest segment to each point.
Numeric vector of distances from each point of X to the corresponding projected
point.
Numeric vector giving the scaled parametric coordinate 0 ≤ tp ≤ 1 of the
position of the projected point along the segment.

For example suppose mapXY[2] = 5 and tp[2] = 0.33. Then Y[5] is the line segment lying
closest to X[2]. The projection of the point X[2] onto the segment Y[5] is the point Xproj[2],
which lies one-third of the way between the first and second endpoints of the line segment Y[5].

1096

project2set

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
nearestsegment for a faster way to determine which segment is closest to each point.
Examples
X <- rstrat(square(1), 5)
Y <- as.psp(matrix(runif(20), 5, 4), window=owin())
plot(Y, lwd=3, col="green")
plot(X, add=TRUE, col="red", pch=16)
v <- project2segment(X,Y)
Xproj <- v$Xproj
plot(Xproj, add=TRUE, pch=16)
arrows(X$x, X$y, Xproj$x, Xproj$y, angle=10, length=0.15, col="red")

project2set

Find Nearest Point in a Region

Description
For each data point in a point pattern X, find the nearest location in a given spatial region W.
Usage
project2set(X, W, ...)
Arguments
X
W
...

Point pattern (object of class "ppp").
Window (object of class "owin") or something acceptable to as.owin.
Arguments passed to as.mask controlling the pixel resolution.

Details
The window W is first discretised as a binary mask using as.mask.
For each data point X[i] in the point pattern X, the algorithm finds the nearest pixel in W.
The result is a point pattern Y containing these nearest points, that is, Y[i] is the nearest point in W
to the point X[i].
Value
A point pattern (object of class "ppp") with the same number of points as X in the window W.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 

prune.rppm

1097

See Also
project2segment, nncross
Examples
He <- heather$fine[owin(c(2.8, 7.4), c(4.0, 7.8))]
plot(He, main="project2set")
X <- runifpoint(4, erosion(complement.owin(He), 0.2))
points(X, col="red")
Y <- project2set(X, He)
points(Y, col="green")
arrows(X$x, X$y, Y$x, Y$y, angle=15, length=0.2)

prune.rppm

Prune a Recursively Partitioned Point Process Model

Description
Given a model which has been fitted to point pattern data by recursive partitioning, apply pruning
to reduce the complexity of the partition tree.
Usage
## S3 method for class 'rppm'
prune(tree, ...)
Arguments
tree

Fitted point process model of class "rppm" produced by the function rppm.

...

Arguments passed to prune.rpart to control the pruning procedure.

Details
This is a method for the generic function prune for the class "rppm". An object of this class is a
point process model, fitted to point pattern data by recursive partitioning, by the function rppm.
The recursive partition tree will be pruned using prune.rpart. The result is another object of class
"rppm".
Value
Object of class "rppm".
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
rppm, plot.rppm, predict.rppm.

1098

pseudoR2

Examples
# Murchison gold data
mur <- solapply(murchison, rescale, s=1000, unitname="km")
mur$dfault <- distfun(mur$faults)
fit <- rppm(gold ~ dfault + greenstone, data=mur)
fit
prune(fit, cp=0.1)

pseudoR2

Calculate Pseudo-R-Squared for Point Process Model

Description
Given a fitted point process model, calculate the pseudo-R-squared value, which measures the fraction of variation in the data that is explained by the model.
Usage
pseudoR2(object, ...)
## S3 method for class 'ppm'
pseudoR2(object, ...)
## S3 method for class 'lppm'
pseudoR2(object, ...)
Arguments
object
...

Fitted point process model. An object of class "ppm" or "lppm".
Additional arguments passed to deviance.ppm or deviance.lppm.

Details
The function pseudoR2 is generic, with methods for fitted point process models of class "ppm" and
"lppm".
This function computes McFadden’s pseudo-Rsquared
R2 = 1 −

D
D0

where D is the deviance of the fitted model object, and D0 is the deviance of the null model
(obtained by refitting object using the trend formula ~1). Deviance is defined as twice the negative
log-likelihood or log-pseudolikelihood.
Value
A single numeric value.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 

psib

1099

See Also
deviance.ppm, deviance.lppm.
Examples
fit <- ppm(swedishpines ~ x+y)
pseudoR2(fit)

psib

Sibling Probability of Cluster Point Process

Description
Computes the sibling probability of a cluster point process model.
Usage
psib(object)
## S3 method for class 'kppm'
psib(object)
Arguments
object

Fitted cluster point process model (object of class "kppm").

Details
In a Poisson cluster process, two points are called siblings if they belong to the same cluster, that
is, if they had the same parent point. If two points of the process are separated by a distance r, the
probability that they are siblings is p(r) = 1 − 1/g(r) where g is the pair correlation function of
the process.
The value p(0) = 1 − 1/g(0) is the probability that, if two points of the process are situated very
close to each other, they came from the same cluster. This probability is an index of the strength of
clustering, with high values suggesting strong clustering.
This concept was proposed in Baddeley, Rubak and Turner (2015, page 479) and Baddeley (2016).
Value
A single number.
Author(s)
Adrian Baddeley .
References
Baddeley, A. (2016) Local composite likelihood for spatial point processes. Spatial Statistics, in
press.
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. Chapman and Hall/CRC Press.

1100

psp

See Also
kppm
Examples
fit <- kppm(redwood ~1, "Thomas")
psib(fit)

psp

Create a Line Segment Pattern

Description
Creates an object of class "psp" representing a line segment pattern in the two-dimensional plane.
Usage
psp(x0,y0, x1, y1, window, marks=NULL,
check=spatstat.options("checksegments"))
Arguments
x0

Vector of x coordinates of first endpoint of each segment

y0

Vector of y coordinates of first endpoint of each segment

x1

Vector of x coordinates of second endpoint of each segment

y1

Vector of y coordinates of second endpoint of each segment

window

window of observation, an object of class "owin"

marks

(optional) vector or data frame of mark values

check

Logical value indicating whether to check that the line segments lie inside the
window.

Details
In the spatstat library, a spatial pattern of line segments is described by an object of class "psp".
This function creates such objects.
The vectors x0, y0, x1 and y1 must be numeric vectors of equal length. They are interpreted as the
cartesian coordinates of the endpoints of the line segments.
A line segment pattern is assumed to have been observed within a specific region of the plane called
the observation window. An object of class "psp" representing a point pattern contains information
specifying the observation window. This window must always be specified when creating a point
pattern dataset; there is intentionally no default action of “guessing” the window dimensions from
the data points alone.
The argument window must be an object of class "owin". It is a full description of the window
geometry, and could have been obtained from owin or as.owin, or by just extracting the observation
window of another dataset, or by manipulating such windows. See owin or the Examples below.
The optional argument marks is given if the line segment pattern is marked, i.e. if each line segment
carries additional information. For example, line segments which are classified into two or more

psp.object

1101

different types, or colours, may be regarded as having a mark which identifies which colour they
are.
The object marks must be a vector of the same length as x0, or a data frame with number of rows
equal to the length of x0. The interpretation is that marks[i] or marks[i,] is the mark attached to
the ith line segment. If the marks are real numbers then marks should be a numeric vector, while
if the marks takes only a finite number of possible values (e.g. colours or types) then marks should
be a factor.
See psp.object for a description of the class "psp".
Users would normally invoke psp to create a line segment pattern, and the function as.psp to
convert data in another format into a line segment pattern.
Value
An object of class "psp" describing a line segment pattern in the two-dimensional plane (see
psp.object).
Author(s)
Adrian Baddeley  and Rolf Turner .
See Also
psp.object, as.psp, owin.object, owin, as.owin, marks.psp
Examples
X <- psp(runif(10), runif(10), runif(10), runif(10), window=owin())
m <- data.frame(A=1:10, B=letters[1:10])
X <- psp(runif(10), runif(10), runif(10), runif(10), window=owin(), marks=m)

psp.object

Class of Line Segment Patterns

Description
A class "psp" to represent a spatial pattern of line segments in the plane. Includes information
about the window in which the pattern was observed. Optionally includes marks.
Details
An object of this class represents a two-dimensional pattern of line segments. It specifies
• the locations of the line segments (both endpoints)
• the window in which the pattern was observed
• optionally, a “mark” attached to each line segment (extra information such as a type label).
If X is an object of type psp, it contains the following elements:
ends
window

data frame with entries x0, y0, x1, y1
giving coordinates of segment endpoints
window of observation

1102

psst

n
marks
markformat

(an object of class owin)
number of line segments
optional vector or data frame of marks
character string specifying the format of the
marks; “none”, “vector”, or “dataframe”

Users are strongly advised not to manipulate these entries directly.
Objects of class "psp" may be created by the function psp and converted from other types of data
by the function as.psp. Note that you must always specify the window of observation; there is
intentionally no default action of “guessing” the window dimensions from the line segments alone.
Subsets of a line segment pattern may be obtained by the functions [.psp and clip.psp.
Line segment pattern objects can be plotted just by typing plot(X) which invokes the plot method
for line segment pattern objects, plot.psp. See plot.psp for further information.
There are also methods for summary and print for line segment patterns. Use summary(X) to see a
useful description of the data.
Utilities for line segment patterns include midpoints.psp (to compute the midpoints of each segment), lengths.psp, (to compute the length of each segment), angles.psp, (to compute the angle
of orientation of each segment), and distmap.psp to compute the distance map of a line segment
pattern.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
psp, as.psp, [.psp
Examples
# creating
a <- psp(runif(20),runif(20),runif(20),runif(20), window=owin())
# converting from other formats
a <- as.psp(matrix(runif(80), ncol=4), window=owin())
a <- as.psp(data.frame(x0=runif(20), y0=runif(20),
x1=runif(20), y1=runif(20)), window=owin())
# clipping
w <- owin(c(0.1,0.7), c(0.2, 0.8))
b <- clip.psp(a, w)
b <- a[w]
# the last two lines are equivalent.

psst

Pseudoscore Diagnostic For Fitted Model against General Alternative

Description
Given a point process model fitted to a point pattern dataset, and any choice of functional summary
statistic, this function computes the pseudoscore test statistic of goodness-of-fit for the model.

psst

1103

Usage
psst(object, fun, r = NULL, breaks = NULL, ...,
model=NULL,
trend = ~1, interaction = Poisson(), rbord = reach(interaction),
truecoef=NULL, hi.res=NULL, funargs = list(correction="best"),
verbose=TRUE)
Arguments
object

Object to be analysed. Either a fitted point process model (object of class "ppm")
or a point pattern (object of class "ppp") or quadrature scheme (object of class
"quad").

fun

Summary function to be applied to each point pattern.

r

Optional. Vector of values of the argument r at which the function S(r) should
be computed. This argument is usually not specified. There is a sensible default.

breaks

Optional alternative to r for advanced use.

...

Ignored.

model

Optional. A fitted point process model (object of class "ppm") to be re-fitted to
the data using update.ppm, if object is a point pattern. Overrides the arguments trend,interaction,rbord.
trend,interaction,rbord
Optional. Arguments passed to ppm to fit a point process model to the data, if
object is a point pattern. See ppm for details.
truecoef

Optional. Numeric vector. If present, this will be treated as if it were the true
coefficient vector of the point process model, in calculating the diagnostic. Incompatible with hi.res.

hi.res

Optional. List of parameters passed to quadscheme. If this argument is present,
the model will be re-fitted at high resolution as specified by these parameters.
The coefficients of the resulting fitted model will be taken as the true coefficients. Then the diagnostic will be computed for the default quadrature scheme,
but using the high resolution coefficients.

funargs

List of additional arguments to be passed to fun.

verbose

Logical value determining whether to print progress reports during the computation.

Details
Let x be a point pattern dataset consisting of points x1 , . . . , xn in a window W . Consider a point
process model fitted to x, with conditional intensity λ(u, x) at location u. For the purpose of testing
goodness-of-fit, we regard the fitted model as the null hypothesis. Given a functional summary
statistic S, consider a family of alternative models obtained by exponential tilting of the null model
by S. The pseudoscore for the null model is
Z
X
V (r) =
∆S(xi , x, r) −
∆S(u, x, r)λ(u, x)du
i

W

where the ∆ operator is
∆S(u, x, r) = S(x ∪ {u}, r) − S(x \ u, r)
the difference between the values of S for the point pattern with and without the point u.

1104

psstA

According to the Georgii-Nguyen-Zessin formula, V (r) should have mean zero if the model is
correct (ignoring the fact that the parameters of the model have been estimated). Hence V (r) can
be used as a diagnostic for goodness-of-fit.
This algorithm computes V (r) by direct evaluation of the sum and integral. It is computationally
intensive, but it is available for any summary statistic S(r).
The diagnostic V (r) is also called the pseudoresidual of S. On the right hand side of the equation
for V (r) given above, the sum over points of x is called the pseudosum and the integral is called
the pseudocompensator.
Value
A function value table (object of class "fv"), essentially a data frame of function values.
Columns in this data frame include dat for the pseudosum, com for the compensator and res for
the pseudoresidual.
There is a plot method for this class. See fv.object.
Author(s)
Adrian Baddeley 
Ege Rubak  and Jesper Møller.
References
Baddeley, A., Rubak, E. and Møller, J. (2011) Score, pseudo-score and residual diagnostics for
spatial point process models. Statistical Science 26, 613–646.
See Also
Special cases: psstA, psstG.
Alternative functions: Kres, Gres.
Examples
data(cells)
fit0 <- ppm(cells, ~1) # uniform Poisson
G0 <- psst(fit0, Gest)
G0
if(interactive()) plot(G0)

psstA

Pseudoscore Diagnostic For Fitted Model against Area-Interaction Alternative

Description
Given a point process model fitted to a point pattern dataset, this function computes the pseudoscore
diagnostic of goodness-of-fit for the model, against moderately clustered or moderately inhibited
alternatives of area-interaction type.

psstA

1105

Usage
psstA(object, r = NULL, breaks = NULL, ...,
model = NULL,
trend = ~1, interaction = Poisson(),
rbord = reach(interaction), ppmcorrection = "border",
correction = "all",
truecoef = NULL, hi.res = NULL,
nr=spatstat.options("psstA.nr"),
ngrid=spatstat.options("psstA.ngrid"))
Arguments
object

Object to be analysed. Either a fitted point process model (object of class "ppm")
or a point pattern (object of class "ppp") or quadrature scheme (object of class
"quad").

r

Optional. Vector of values of the argument r at which the diagnostic should be
computed. This argument is usually not specified. There is a sensible default.

breaks

This argument is for internal use only.

...

Extra arguments passed to quadscheme to determine the quadrature scheme, if
object is a point pattern.

model

Optional. A fitted point process model (object of class "ppm") to be re-fitted to
the data using update.ppm, if object is a point pattern. Overrides the arguments trend,interaction,rbord,ppmcorrection.
trend,interaction,rbord
Optional. Arguments passed to ppm to fit a point process model to the data, if
object is a point pattern. See ppm for details.
ppmcorrection

Optional. Character string specifying the edge correction for the pseudolikelihood to be used in fitting the point process model. Passed to ppm.

correction

Optional. Character string specifying which diagnostic quantities will be computed. Options are "all" and "best". The default is to compute all diagnostic
quantities.

truecoef

Optional. Numeric vector. If present, this will be treated as if it were the true
coefficient vector of the point process model, in calculating the diagnostic. Incompatible with hi.res.

hi.res

Optional. List of parameters passed to quadscheme. If this argument is present,
the model will be re-fitted at high resolution as specified by these parameters.
The coefficients of the resulting fitted model will be taken as the true coefficients. Then the diagnostic will be computed for the default quadrature scheme,
but using the high resolution coefficients.

nr

Optional. Number of r values to be used if r is not specified.

ngrid

Integer. Number of points in the square grid used to compute the approximate
area.

Details
This function computes the pseudoscore test statistic which can be used as a diagnostic for goodnessof-fit of a fitted point process model.
Let x be a point pattern dataset consisting of points x1 , . . . , xn in a window W . Consider a point
process model fitted to x, with conditional intensity λ(u, x) at location u. For the purpose of testing

1106

psstA

goodness-of-fit, we regard the fitted model as the null hypothesis. The alternative hypothesis is a
family of hybrid models obtained by combining the fitted model with the area-interaction process
(see AreaInter). The family of alternatives includes models that are slightly more regular than the
fitted model, and others that are slightly more clustered than the fitted model.
The pseudoscore, evaluated at the null model, is
Z
X
V (r) =
A(xi , x, r) −
i

A(u, x, r)λ(u, x)du

W

where
A(u, x, r) = B(x ∪ {u}, r) − B(x \ u, r)
where B(x, r) is the area of the union of the discs of radius r centred at the points of x (i.e. B(x, r)
is the area of the dilation of x by a distance r). Thus A(u, x, r) is the unclaimed area associated
with u, that is, the area of that part of the disc of radius r centred at the point u that is not covered
by any of the discs of radius r centred at points of x.
According to the Georgii-Nguyen-Zessin formula, V (r) should have mean zero if the model is
correct (ignoring the fact that the parameters of the model have been estimated). Hence V (r) can
be used as a diagnostic for goodness-of-fit.
The diagnostic V (r) is also called the pseudoresidual of S. On the right hand side of the equation
for V (r) given above, the sum over points of x is called the pseudosum and the integral is called
the pseudocompensator.
Value
A function value table (object of class "fv"), essentially a data frame of function values.
Columns in this data frame include dat for the pseudosum, com for the compensator and res for
the pseudoresidual.
There is a plot method for this class. See fv.object.
Warning
This computation can take a very long time.
To shorten the computation time, choose smaller values of the arguments nr and ngrid, or reduce
the values of their defaults spatstat.options("psstA.nr") and spatstat.options("psstA.ngrid").
Computation time is roughly proportional to nr * npoints * ngrid^2 where npoints is the
number of points in the point pattern.
Author(s)
Adrian Baddeley 
Ege Rubak  and Jesper Møller.
References
Baddeley, A., Rubak, E. and Møller, J. (2011) Score, pseudo-score and residual diagnostics for
spatial point process models. Statistical Science 26, 613–646.
See Also
Alternative functions: psstG, psst, Gres, Kres.
Point process models: ppm.
Options: spatstat.options

psstG

1107

Examples
pso <- spatstat.options(psstA.ngrid=16,psstA.nr=10)
X <- rStrauss(200,0.1,0.05)
plot(psstA(X))
plot(psstA(X, interaction=Strauss(0.05)))
spatstat.options(pso)

psstG

Pseudoscore Diagnostic For Fitted Model against Saturation Alternative

Description
Given a point process model fitted to a point pattern dataset, this function computes the pseudoscore
diagnostic of goodness-of-fit for the model, against moderately clustered or moderately inhibited
alternatives of saturation type.
Usage
psstG(object, r = NULL, breaks = NULL, ...,
model=NULL,
trend = ~1, interaction = Poisson(), rbord = reach(interaction),
truecoef = NULL, hi.res = NULL)
Arguments
object

Object to be analysed. Either a fitted point process model (object of class "ppm")
or a point pattern (object of class "ppp") or quadrature scheme (object of class
"quad").

r

Optional. Vector of values of the argument r at which the diagnostic should be
computed. This argument is usually not specified. There is a sensible default.

breaks

Optional alternative to r for advanced use.

...

Ignored.

model

Optional. A fitted point process model (object of class "ppm") to be re-fitted to
the data using update.ppm, if object is a point pattern. Overrides the arguments trend,interaction,rbord,ppmcorrection.
trend,interaction,rbord
Optional. Arguments passed to ppm to fit a point process model to the data, if
object is a point pattern. See ppm for details.
truecoef

Optional. Numeric vector. If present, this will be treated as if it were the true
coefficient vector of the point process model, in calculating the diagnostic. Incompatible with hi.res.

hi.res

Optional. List of parameters passed to quadscheme. If this argument is present,
the model will be re-fitted at high resolution as specified by these parameters.
The coefficients of the resulting fitted model will be taken as the true coefficients. Then the diagnostic will be computed for the default quadrature scheme,
but using the high resolution coefficients.

1108

psstG

Details
This function computes the pseudoscore test statistic which can be used as a diagnostic for goodnessof-fit of a fitted point process model.
Consider a point process model fitted to x, with conditional intensity λ(u, x) at location u. For the
purpose of testing goodness-of-fit, we regard the fitted model as the null hypothesis. The alternative
hypothesis is a family of hybrid models obtained by combining the fitted model with the Geyer
saturation process (see Geyer) with saturation parameter 1. The family of alternatives includes
models that are more regular than the fitted model, and others that are more clustered than the fitted
model.
For any point pattern x, and any r > 0, let S(x, r) be the number of points in x whose nearest
neighbour (the nearest other point in x) is closer than r units. Then the pseudoscore for the null
model is
Z
X
V (r) =
∆S(xi , x, r) −
∆S(u, x, r)λ(u, x)du
i

W

where the ∆ operator is
∆S(u, x, r) = S(x ∪ {u}, r) − S(x \ u, r)
the difference between the values of S for the point pattern with and without the point u.
According to the Georgii-Nguyen-Zessin formula, V (r) should have mean zero if the model is
correct (ignoring the fact that the parameters of the model have been estimated). Hence V (r) can
be used as a diagnostic for goodness-of-fit.
The diagnostic V (r) is also called the pseudoresidual of S. On the right hand side of the equation
for V (r) given above, the sum over points of x is called the pseudosum and the integral is called
the pseudocompensator.
Value
A function value table (object of class "fv"), essentially a data frame of function values.
Columns in this data frame include dat for the pseudosum, com for the compensator and res for
the pseudoresidual.
There is a plot method for this class. See fv.object.
Author(s)
Adrian Baddeley 
Ege Rubak  and Jesper Møller.
References
Baddeley, A., Rubak, E. and Møller, J. (2011) Score, pseudo-score and residual diagnostics for
spatial point process models. Statistical Science 26, 613–646.
See Also
Alternative functions: psstA, psst, Kres, Gres.
Examples
X <- rStrauss(200,0.1,0.05)
plot(psstG(X))
plot(psstG(X, interaction=Strauss(0.05)))

qqplot.ppm

qqplot.ppm

1109

Q-Q Plot of Residuals from Fitted Point Process Model

Description
Given a point process model fitted to a point pattern, produce a Q-Q plot based on residuals from
the model.
Usage
qqplot.ppm(fit, nsim=100, expr=NULL, ..., type="raw",
style="mean", fast=TRUE, verbose=TRUE, plot.it=TRUE,
dimyx=NULL, nrep=if(fast) 5e4 else 1e5,
control=update(default.rmhcontrol(fit), nrep=nrep),
saveall=FALSE,
monochrome=FALSE,
limcol=if(monochrome) "black" else "red",
maxerr=max(100, ceiling(nsim/10)),
check=TRUE, repair=TRUE, envir.expr)
Arguments
fit

The fitted point process model, which is to be assessed using the Q-Q plot. An
object of class "ppm". Smoothed residuals obtained from this fitted model will
provide the “data” quantiles for the Q-Q plot.

nsim

The number of simulations from the “reference” point process model.

expr

Determines the simulation mechanism which provides the “theoretical” quantiles for the Q-Q plot. See Details.

...

Arguments passed to diagnose.ppm influencing the computation of residuals.

type

String indicating the type of residuals or weights to be used. Current options are
"eem" for the Stoyan-Grabarnik exponential energy weights, "raw" for the raw
residuals, "inverse" for the inverse-lambda residuals, and "pearson" for the
Pearson residuals. A partial match is adequate.

style

Character string controlling the type of Q-Q plot. Options are "classical" and
"mean". See Details.

fast

Logical flag controlling the speed and accuracy of computation. Use fast=TRUE
for interactive use and fast=FALSE for publication standard plots. See Details.

verbose

Logical flag controlling whether the algorithm prints progress reports during
long computations.

plot.it

Logical flag controlling whether the function produces a plot or simply returns
a value (silently).

dimyx

Dimensions of the pixel grid on which the smoothed residual field will be calculated. A vector of two integers.

nrep

If control is absent, then nrep gives the number of iterations of the MetropolisHastings algorithm that should be used to generate one simulation of the fitted
point process.

1110

qqplot.ppm

control

List of parameters controlling the Metropolis-Hastings algorithm rmh which
generates each simulated realisation from the model (unless the model is Poisson). This list becomes the argument control of rmh.default. It overrides
nrep.

saveall

Logical flag indicating whether to save all the intermediate calculations.

monochrome

Logical flag indicating whether the plot should be in black and white (monochrome=TRUE),
or in colour (monochrome=FALSE).

limcol

String. The colour to be used when plotting the 95% limit curves.

maxerr

Maximum number of failures tolerated while generating simulated realisations.
See Details.

check

Logical value indicating whether to check the internal format of fit. If there
is any possibility that this object has been restored from a dump file, or has
otherwise lost track of the environment where it was originally computed, set
check=TRUE.

repair

Logical value indicating whether to repair the internal format of fit, if it is
found to be damaged.

envir.expr

Optional. An environment in which the expression expr should be evaluated.

Details
This function generates a Q-Q plot of the residuals from a fitted point process model. It is an
addendum to the suite of diagnostic plots produced by the function diagnose.ppm, kept separate
because it is computationally intensive. The quantiles of the theoretical distribution are estimated
by simulation.
In classical statistics, a Q-Q plot of residuals is a useful diagnostic for checking the distributional
assumptions. Analogously, in spatial statistics, a Q-Q plot of the (smoothed) residuals from a fitted
point process model is a useful way to check the interpoint interaction part of the model (Baddeley
et al, 2005). The systematic part of the model (spatial trend, covariate effects, etc) is assessed using
other plots made by diagnose.ppm.
The argument fit represents the fitted point process model. It must be an object of class "ppm"
(typically produced by the maximum pseudolikelihood fitting algorithm ppm). Residuals will be
computed for this fitted model using residuals.ppm, and the residuals will be kernel-smoothed to
produce a “residual field”. The values of this residual field will provide the “data” quantiles for the
Q-Q plot.
The argument expr is not usually specified. It provides a way to modify the “theoretical” or “reference” quantiles for the Q-Q plot.
In normal usage we set expr=NULL. The default is to generate nsim simulated realisations of the
fitted model fit, re-fit this model to each of the simulated patterns, evaluate the residuals from
these fitted models, and use the kernel-smoothed residual field from these fitted models as a sample
from the reference distribution for the Q-Q plot.
In advanced use, expr may be an expression. It will be re-evaluated nsim times, and should
include random computations so that the results are not identical each time. The result of evaluating
expr should be either a point pattern (object of class "ppp") or a fitted point process model (object
of class "ppm"). If the value is a point pattern, then the original fitted model fit will be fitted to this
new point pattern using update.ppm, to yield another fitted model. Smoothed residuals obtained
from these nsim fitted models will yield the “theoretical” quantiles for the Q-Q plot.
Alternatively expr can be a list of point patterns, or an envelope object that contains a list of point
patterns (typically generated by calling envelope with savepatterns=TRUE). These point patterns
will be used as the simulated patterns.

qqplot.ppm

1111

Simulation is performed (if expr=NULL) using the Metropolis-Hastings algorithm rmh. Each simulated realisation is the result of running the Metropolis-Hastings algorithm from an independent
random starting state each time. The iterative and termination behaviour of the Metropolis-Hastings
algorithm are governed by the argument control. See rmhcontrol for information about this argument. As a shortcut, the argument nrep determines the number of Metropolis-Hastings iterations
used to generate each simulated realisation, if control is absent.
By default, simulations are generated in an expanded window. Use the argument control to change
this, as explained in the section on Warning messages.
The argument type selects the type of residual or weight that will be computed. For options, see
diagnose.ppm.
The argument style determines the type of Q-Q plot. It is highly recommended to use the default,
style="mean".
style="classical" The quantiles of the residual field for the data (on the y axis) are plotted
against the quantiles of the pooled simulations (on the x axis). This plot is biased, and therefore difficult to interpret, because of strong autocorrelations in the residual field and the large
differences in sample size.
style="mean" The order statistics of the residual field for the data are plotted against the sample
means, over the nsim simulations, of the corresponding order statistics of the residual field
for the simulated datasets. Dotted lines show the 2.5 and 97.5 percentiles, over the nsim
simulations, of each order statistic.
The argument fast is a simple way to control the accuracy and speed of computation. If fast=FALSE,
the residual field is computed on a fine grid of pixels (by default 100 by 100 pixels, see below)
and the Q-Q plot is based on the complete set of order statistics (usually 10,000 quantiles). If
fast=TRUE, the residual field is computed on a coarse grid (at most 40 by 40 pixels) and the QQ plot is based on the percentiles only. This is about 7 times faster. It is recommended to use
fast=TRUE for interactive data analysis and fast=FALSE for definitive plots for publication.
The argument dimyx gives full control over the resolution of the pixel grid used to calculate the
smoothed residuals. Its interpretation is the same as the argument dimyx to the function as.mask.
Note that dimyx[1] is the number of pixels in the y direction, and dimyx[2] is the number in
the x direction. If dimyx is not present, then the default pixel grid dimensions are controlled by
spatstat.options("npixel").
Since the computation is so time-consuming, qqplot.ppm returns a list containing all the data
necessary to re-display the Q-Q plot. It is advisable to assign the result of qqplot.ppm to something
(or use .Last.value if you forgot to.) The return value is an object of class "qqppm". There are
methods for plot.qqppm and print.qqppm. See the Examples.
The argument saveall is usually set to FALSE. If saveall=TRUE, then the intermediate results
of calculation for each simulated realisation are saved and returned. The return value includes a
3-dimensional array sim containing the smoothed residual field images for each of the nsim realisations. When saveall=TRUE, the return value is an object of very large size, and should not be
saved on disk.
Errors may occur during the simulation process, because random data are generated. For example:
• one of the simulated patterns may be empty.
• one of the simulated patterns may cause an error in the code that fits the point process model.
• the user-supplied argument expr may have a bug.
Empty point patterns do not cause a problem for the code, but they are reported. Other problems that
would lead to a crash are trapped; the offending simulated data are discarded, and the simulation is
retried. The argument maxerr determines the maximum number of times that such errors will be
tolerated (mainly as a safeguard against an infinite loop).

1112

qqplot.ppm

Value
An object of class "qqppm" containing the information needed to reproduce the Q-Q plot. Entries x
and y are numeric vectors containing quantiles of the simulations and of the data, respectively.
Side Effects
Produces a Q-Q plot if plot.it is TRUE.
Warning messages
A warning message will be issued if any of the simulations trapped an error (a potential crash).
A warning message will be issued if all, or many, of the simulated point patterns are empty. This
usually indicates a problem with the simulation procedure.
The default behaviour of qqplot.ppm is to simulate patterns on an expanded window (specified
through the argument control) in order to avoid edge effects. The model’s trend is extrapolated
over this expanded window. If the trend is strongly inhomogeneous, the extrapolated trend may
have very large (or even infinite) values. This can cause the simulation algorithm to produce empty
patterns.
The only way to suppress this problem entirely is to prohibit the expansion of the window, by
setting the control argument to something like control=list(nrep=1e6, expand=1). Here
expand=1 means there will be no expansion. See rmhcontrol for more information about the
argument control.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Baddeley, A., Turner, R., Møller, J. and Hazelton, M. (2005) Residual analysis for spatial point
processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
Stoyan, D. and Grabarnik, P. (1991) Second-order characteristics for stochastic structures connected
with Gibbs point processes. Mathematische Nachrichten, 151:95–100.
See Also
diagnose.ppm, lurking, residuals.ppm, eem, ppm.object, ppm, rmh, rmhcontrol
Examples
data(cells)
fit <- ppm(cells, ~1, Poisson())
diagnose.ppm(fit) # no suggestion of departure from stationarity
## Not run: qqplot.ppm(fit, 80) # strong evidence of non-Poisson interaction
## Not run:
diagnose.ppm(fit, type="pearson")
qqplot.ppm(fit, type="pearson")
## End(Not run)

quad.object

1113

###########################################
## oops, I need the plot coordinates
mypreciousdata <- .Last.value
## Not run: mypreciousdata <- qqplot.ppm(fit, type="pearson")
plot(mypreciousdata)
######################################################
# Q-Q plots based on fixed n
# The above QQ plots used simulations from the (fitted) Poisson process.
# But I want to simulate conditional on n, instead of Poisson
# Do this by setting rmhcontrol(p=1)
fixit <- list(p=1)
## Not run: qqplot.ppm(fit, 100, control=fixit)
######################################################
# Inhomogeneous Poisson data
X <- rpoispp(function(x,y){1000 * exp(-3*x)}, 1000)
plot(X)
# Inhomogeneous Poisson model
fit <- ppm(X, ~x, Poisson())
## Not run: qqplot.ppm(fit, 100)
# conclusion: fitted inhomogeneous Poisson model looks OK
######################################################
# Advanced use of 'expr' argument
#
# set the initial conditions in Metropolis-Hastings algorithm
#
expr <- expression(rmh(fit, start=list(n.start=42), verbose=FALSE))
## Not run: qqplot.ppm(fit, 100, expr)

quad.object

Class of Quadrature Schemes

Description
A class "quad" to represent a quadrature scheme.
Details
A (finite) quadrature scheme is a list of quadrature points uj and associated weights wj which is
used to approximate an integral by a finite sum:
Z
X
f (x)dx ≈
f (uj )wj
j

Given a point pattern dataset, a Berman-Turner quadrature scheme is one which includes all these
data points, as well as a nonzero number of other (“dummy”) points.

1114

quad.ppm

These quadrature schemes are used to approximate the pseudolikelihood of a point process, in the
method of Baddeley and Turner (2000) (see Berman and Turner (1992)). Accuracy and computation
time both increase with the number of points in the quadrature scheme.
An object of class "quad" represents a Berman-Turner quadrature scheme. It can be passed as an
argument to the model-fitting function ppm, which requires a quadrature scheme.
An object of this class contains at least the following elements:
data:
dummy:
w:

an object of class "ppp"
giving the locations (and marks) of the data points.
an object of class "ppp"
giving the locations (and marks) of the dummy points.
vector of nonnegative weights for the quadrature points

Users are strongly advised not to manipulate these entries directly.
The domain of quadrature is specified by Window(dummy) while the observation window (if this
needs to be specified separately) is taken to be Window(data).
The weights vector w may also have an attribute attr(w, "zeroes") equivalent to the logical
vector (w == 0). If this is absent then all points are known to have positive weights.
To create an object of class "quad", users would typically call the high level function quadscheme.
(They are actually created by the low level function quad.)
Entries are extracted from a "quad" object by the functions x.quad, y.quad, w.quad and marks.quad,
which extract the x coordinates, y coordinates, weights, and marks, respectively. The function
n.quad returns the total number of quadrature points (dummy plus data).
An object of class "quad" can be converted into an ordinary point pattern by the function union.quad
which simply takes the union of the data and dummy points.
Quadrature schemes can be plotted using plot.quad (a method for the generic plot).
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
quadscheme, ppm

quad.ppm

Extract Quadrature Scheme Used to Fit a Point Process Model

Description
Given a fitted point process model, this function extracts the quadrature scheme used to fit the
model.
Usage
quad.ppm(object, drop=FALSE, clip=FALSE)

quad.ppm

1115

Arguments
object

fitted point process model (an object of class "ppm" or "kppm" or "lppm").

drop

Logical value determining whether to delete quadrature points that were not
used to fit the model.

clip

Logical value determining whether to erode the window, if object was fitted
using the border correction. See Details.

Details
An object of class "ppm" represents a point process model that has been fitted to data. It is typically
produced by the model-fitting algorithm ppm.
The maximum pseudolikelihood algorithm in ppm approximates the pseudolikelihood integral by
a sum over a finite set of quadrature points, which is constructed by augmenting the original data
point pattern by a set of “dummy” points. The fitted model object returned by ppm contains complete
information about this quadrature scheme. See ppm or ppm.object for further information.
This function quad.ppm extracts the quadrature scheme. A typical use of this function would be to
inspect the quadrature scheme (points and weights) to gauge the accuracy of the approximation to
the exact pseudolikelihood.
Some quadrature points may not have been used in fitting the model. This happens if the border
correction is used, and in other cases (e.g. when the value of a covariate is NA at these points). The
argument drop specifies whether these unused quadrature points shall be deleted (drop=TRUE) or
retained (drop=FALSE) in the return value.
The quadrature scheme has a window, which by default is set to equal the window of the original
data. However this window may be larger than the actual domain of integration of the pseudolikelihood or composite likelihood that was used to fit the model. If clip=TRUE then the window of the
quadrature scheme is set to the actual domain of integration. This option only has an effect when the
model was fitted using the border correction; then the window is obtained by eroding the original
data window by the border correction distance.
See ppm.object for a list of all operations that can be performed on objects of class "ppm". See
quad.object for a list of all operations that can be performed on objects of class "quad".
This function can also be applied to objects of class "kppm" and "lppm".
Value
A quadrature scheme (object of class "quad").
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
ppm.object, quad.object, ppm
Examples
fit <- ppm(cells ~1, Strauss(r=0.1))
Q <- quad.ppm(fit)
## Not run: plot(Q)

1116

quadrat.test
npoints(Q$data)
npoints(Q$dummy)

quadrat.test

Dispersion Test for Spatial Point Pattern Based on Quadrat Counts

Description
Performs a test of Complete Spatial Randomness for a given point pattern, based on quadrat counts.
Alternatively performs a goodness-of-fit test of a fitted inhomogeneous Poisson model. By default
performs chi-squared tests; can also perform Monte Carlo based tests.
Usage
quadrat.test(X, ...)
## S3 method for class 'ppp'
quadrat.test(X, nx=5, ny=nx,
alternative=c("two.sided", "regular", "clustered"),
method=c("Chisq", "MonteCarlo"),
conditional=TRUE, CR=1,
lambda=NULL,
...,
xbreaks=NULL, ybreaks=NULL, tess=NULL,
nsim=1999)
## S3 method for class 'ppm'
quadrat.test(X, nx=5, ny=nx,
alternative=c("two.sided", "regular", "clustered"),
method=c("Chisq", "MonteCarlo"),
conditional=TRUE, CR=1,
...,
xbreaks=NULL, ybreaks=NULL, tess=NULL,
nsim=1999)
## S3 method for class 'quadratcount'
quadrat.test(X,
alternative=c("two.sided", "regular", "clustered"),
method=c("Chisq", "MonteCarlo"),
conditional=TRUE, CR=1,
lambda=NULL,
...,
nsim=1999)
Arguments
X

A point pattern (object of class "ppp") to be subjected to the goodness-of-fit test.
Alternatively a fitted point process model (object of class "ppm") to be tested.
Alternatively X can be the result of applying quadratcount to a point pattern.

nx,ny

Numbers of quadrats in the x and y directions. Incompatible with xbreaks and
ybreaks.

quadrat.test

1117

alternative

Character string (partially matched) specifying the alternative hypothesis.

method

Character string (partially matched) specifying the test to use: either method="Chisq"
for the chi-squared test (the default), or method="MonteCarlo" for a Monte
Carlo test.

conditional

Logical. Should the Monte Carlo test be conducted conditionally upon the observed number of points of the pattern? Ignored if method="Chisq".

CR

Optional. Numerical value of the index λ for the Cressie-Read test statistic.

lambda

Optional. Pixel image (object of class "im") or function (class "funxy") giving
the predicted intensity of the point process.

...

Ignored.

xbreaks

Optional. Numeric vector giving the x coordinates of the boundaries of the
quadrats. Incompatible with nx.

ybreaks

Optional. Numeric vector giving the y coordinates of the boundaries of the
quadrats. Incompatible with ny.

tess

Tessellation (object of class "tess" or something acceptable to as.tess) determining the quadrats. Incompatible with nx, ny, xbreaks, ybreaks.

nsim

The number of simulated samples to generate when method="MonteCarlo".

Details
These functions perform χ2 tests or Monte Carlo tests of goodness-of-fit for a point process model,
based on quadrat counts.
The function quadrat.test is generic, with methods for point patterns (class "ppp"), split point
patterns (class "splitppp"), point process models (class "ppm") and quadrat count tables (class
"quadratcount").
• if X is a point pattern, we test the null hypothesis that the data pattern is a realisation of
Complete Spatial Randomness (the uniform Poisson point process). Marks in the point pattern
are ignored. (If lambda is given then the null hypothesis is the Poisson process with intensity
lambda.)
• if X is a split point pattern, then for each of the component point patterns (taken separately)
we test the null hypotheses of Complete Spatial Randomness. See quadrat.test.splitppp
for documentation.
• If X is a fitted point process model, then it should be a Poisson point process model. The data
to which this model was fitted are extracted from the model object, and are treated as the data
point pattern for the test. We test the null hypothesis that the data pattern is a realisation of the
(inhomogeneous) Poisson point process specified by X.
In all cases, the window of observation is divided into tiles, and the number of data points in each
tile is counted, as described in quadratcount. The quadrats are rectangular by default, or may be
regions of arbitrary shape specified by the argument tess. The expected number of points in each
quadrat is also calculated, as determined by CSR (in the first case) or by the fitted model (in the
second case). Then the Pearson X 2 statistic
X 2 = sum((observed − expected)2 /expected)
is computed.
If method="Chisq" then a χ2 test of goodness-of-fit is performed by comparing the test statistic to
the χ2 distribution with m − k degrees of freedom, where m is the number of quadrats and k is the

1118

quadrat.test

number of fitted parameters (equal to 1 for quadrat.test.ppp). The default is to compute the twosided p-value, so that the test will be declared significant if X 2 is either very large or very small.
One-sided p-values can be obtained by specifying the alternative. An important requirement of
the χ2 test is that the expected counts in each quadrat be greater than 5.
If method="MonteCarlo" then a Monte Carlo test is performed, obviating the need for all expected
counts to be at least 5. In the Monte Carlo test, nsim random point patterns are generated from
the null hypothesis (either CSR or the fitted point process model). The Pearson X 2 statistic is
computed as above. The p-value is determined by comparing the X 2 statistic for the observed
point pattern, with the values obtained from the simulations. Again the default is to compute the
two-sided p-value.
If conditional is TRUE then the simulated samples are generated from the multinomial distribution
with the number of “trials” equal to the number of observed points and the vector of probabilities
equal to the expected counts divided by the sum of the expected counts. Otherwise the simulated
samples are independent Poisson counts, with means equal to the expected counts.
If the argument CR is given, then instead of the Pearson X 2 statistic, the Cressie-Read (1984) power
divergence test statistic
"
#
X  Xi λ
2
−1
2nI =
λ(λ + 1) i
Ei
is computed, where Xi is the ith observed count and Ei is the corresponding expected count, and the
exponent λ is equal to CR. The value CR=1 gives the Pearson X 2 statistic; CR=0 gives the likelihood
ratio test statistic G2 ; CR=-1/2 gives the Freeman-Tukey statistic T 2 ; CR=-1 gives the modified
likelihood ratio test statistic GM 2 ; and CR=-2 gives Neyman’s modified statistic N M 2 . In all cases
the asymptotic distribution of this test statistic is the same χ2 distribution as above.
The return value is an object of class "htest". Printing the object gives comprehensible output
about the outcome of the test.
The return value also belongs to the special class "quadrat.test". Plotting the object will display
the quadrats, annotated by their observed and expected counts and the Pearson residuals. See the
examples.
Value
An object of class "htest". See chisq.test for explanation.
The return value is also an object of the special class "quadrattest", and there is a plot method
for this class. See the examples.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Cressie, N. and Read, T.R.C. (1984) Multinomial goodness-of-fit tests. Journal of the Royal Statistical Society, Series B 46, 440–464.
See Also
quadrat.test.splitppp, quadratcount, quadrats, quadratresample, chisq.test, cdf.test.
To test a Poisson point process model against a specific alternative, use anova.ppm.

quadrat.test.mppm

1119

Examples
data(simdat)
quadrat.test(simdat)
quadrat.test(simdat, 4, 3)
quadrat.test(simdat, alternative="regular")
quadrat.test(simdat, alternative="clustered")
# Using Monte Carlo p-values
quadrat.test(swedishpines) # Get warning, small expected values.
## Not run:
quadrat.test(swedishpines, method="M", nsim=4999)
quadrat.test(swedishpines, method="M", nsim=4999, conditional=FALSE)
## End(Not run)
# quadrat counts
qS <- quadratcount(simdat, 4, 3)
quadrat.test(qS)
# fitted model: inhomogeneous Poisson
fitx <- ppm(simdat, ~x, Poisson())
quadrat.test(fitx)
te <- quadrat.test(simdat, 4)
residuals(te) # Pearson residuals
plot(te)
plot(simdat, pch="+", cols="green", lwd=2)
plot(te, add=TRUE, col="red", cex=1.4, lty=2, lwd=3)
sublab <- eval(substitute(expression(p[chi^2]==z),
list(z=signif(te$p.value,3))))
title(sub=sublab, cex.sub=3)
# quadrats of irregular shape
B <- dirichlet(runifpoint(6, Window(simdat)))
qB <- quadrat.test(simdat, tess=B)
plot(simdat, main="quadrat.test(simdat, tess=B)", pch="+")
plot(qB, add=TRUE, col="red", lwd=2, cex=1.2)

quadrat.test.mppm

Chi-Squared Test for Multiple Point Process Model Based on Quadrat
Counts

Description
Performs a chi-squared goodness-of-fit test of a Poisson point process model fitted to multiple point
patterns.

1120

quadrat.test.mppm

Usage
## S3 method for class 'mppm'
quadrat.test(X, ...)
Arguments
X

An object of class "mppm" representing a point process model fitted to multiple
point patterns. It should be a Poisson model.

...

Arguments passed to quadrat.test.ppm which determine the size of the quadrats.

Details
This function performs a χ2 test of goodness-of-fit for a Poisson point process model, based on
quadrat counts. It can also be used to perform a test of Complete Spatial Randomness for a list of
point patterns.
The function quadrat.test is generic, with methods for point patterns (class "ppp"), point process
models (class "ppm") and multiple point process models (class "mppm").
For this function, the argument X should be a multiple point process model (object of class "mppm")
obtained by fitting a point process model to a list of point patterns using the function mppm.
To perform the test, the data point patterns are extracted from X. For each point pattern
• the window of observation is divided into rectangular tiles, and the number of data points in
each tile is counted, as described in quadratcount.
• The expected number of points in each quadrat is calculated, as determined by the fitted model.
Then we perform a single χ2 test of goodness-of-fit based on these observed and expected counts.
Value
An object of class "htest". Printing the object gives comprehensible output about the outcome of
the test. The p-value of the test is stored in the component p.value.
The return value also belongs to the special class "quadrat.test". Plotting the object will display,
for each window, the position of the quadrats, annotated by their observed and expected counts and
the Pearson residuals. See the examples.
The return value also has an attribute "components" which is a list containing the results of χ2 tests
of goodness-of-fit for each individual point pattern.
Testing Complete Spatial Randomness
If the intention is to test Complete Spatial Randomness (CSR) there are two options:
• CSR with the same intensity of points in each point pattern;
• CSR with a different, unrelated intensity of points in each point pattern.
In the first case, suppose P is a list of point patterns we want to test. Then fit the multiple model
fit1 <- mppm(P, ~1) which signifies a Poisson point process model with a constant intensity.
Then apply quadrat.test(fit1).
In the second case, fit the model codefit2 <- mppm(P, ~id) which signifies a Poisson point process
with a different constant intensity for each point pattern. Then apply quadrat.test(fit2).

quadrat.test.splitppp

1121

Author(s)
Adrian Baddeley , Ida-Maria Sintorn and Leanne Bischoff.
Implemented by Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. London: Chapman and Hall/CRC Press.
See Also
mppm, quadrat.test
Examples
H <- hyperframe(X=waterstriders)
# Poisson with constant intensity for all patterns
fit1 <- mppm(X~1, H)
quadrat.test(fit1, nx=2)
# uniform Poisson with different intensity for each pattern
fit2 <- mppm(X ~ id, H)
quadrat.test(fit2, nx=2)

quadrat.test.splitppp Dispersion Test of CSR for Split Point Pattern Based on Quadrat
Counts

Description
Performs a test of Complete Spatial Randomness for each of the component patterns in a split point
pattern, based on quadrat counts. By default performs chi-squared tests; can also perform Monte
Carlo based tests.
Usage
## S3 method for class 'splitppp'
quadrat.test(X, ..., df=NULL, df.est=NULL, Xname=NULL)
Arguments
X

A split point pattern (object of class "splitppp"), each component of which
will be subjected to the goodness-of-fit test.

...
Arguments passed to quadrat.test.ppp.
df,df.est,Xname
Arguments passed to pool.quadrattest.

1122

quadratcount

Details
The function quadrat.test is generic, with methods for point patterns (class "ppp"), split point
patterns (class "splitppp") and point process models (class "ppm").
If X is a split point pattern, then for each of the component point patterns (taken separately) we test
the null hypotheses of Complete Spatial Randomness, then combine the result into a single test.
The method quadrat.test.ppp is applied to each component point pattern. Then the results are
pooled using pool.quadrattest to obtain a single test.
Value
An object of class "quadrattest" which can be printed and plotted.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
quadrat.test, quadratcount, quadrats, quadratresample, chisq.test, cdf.test.
To test a Poisson point process model against a specific Poisson alternative, use anova.ppm.
Examples
data(humberside)
qH <- quadrat.test(split(humberside), 2, 3)
plot(qH)
qH

quadratcount

Quadrat counting for a point pattern

Description
Divides window into quadrats and counts the numbers of points in each quadrat.
Usage
quadratcount(X, ...)
## S3 method for class 'ppp'
quadratcount(X, nx=5, ny=nx, ...,
xbreaks=NULL, ybreaks=NULL, tess=NULL)
## S3 method for class 'splitppp'
quadratcount(X, ...)

quadratcount

1123

Arguments
X

A point pattern (object of class "ppp") or a split point pattern (object of class
"splitppp").

nx,ny

Numbers of rectangular quadrats in the x and y directions. Incompatible with
xbreaks and ybreaks.

...

Additional arguments passed to quadratcount.ppp.

xbreaks

Numeric vector giving the x coordinates of the boundaries of the rectangular
quadrats. Incompatible with nx.

ybreaks

Numeric vector giving the y coordinates of the boundaries of the rectangular
quadrats. Incompatible with ny.

tess

Tessellation (object of class "tess" or something acceptable to as.tess) determining the quadrats. Incompatible with nx,ny,xbreaks,ybreaks.

Details
Quadrat counting is an elementary technique for analysing spatial point patterns. See Diggle (2003).
If X is a point pattern, then by default, the window containing the point pattern X is divided into an
nx * ny grid of rectangular tiles or ‘quadrats’. (If the window is not a rectangle, then these tiles are
intersected with the window.) The number of points of X falling in each quadrat is counted. These
numbers are returned as a contingency table.
If xbreaks is given, it should be a numeric vector giving the x coordinates of the quadrat boundaries. If it is not given, it defaults to a sequence of nx+1 values equally spaced over the range of x
coordinates in the window Window(X).
Similarly if ybreaks is given, it should be a numeric vector giving the y coordinates of the quadrat
boundaries. It defaults to a vector of ny+1 values equally spaced over the range of y coordinates in
the window. The lengths of xbreaks and ybreaks may be different.
Alternatively, quadrats of any shape may be used. The argument tess can be a tessellation (object
of class "tess") whose tiles will serve as the quadrats.
The algorithm counts the number of points of X falling in each quadrat, and returns these counts as
a contingency table.
The return value is a table which can be printed neatly. The return value is also a member of
the special class "quadratcount". Plotting the object will display the quadrats, annotated by their
counts. See the examples.
To perform a chi-squared test based on the quadrat counts, use quadrat.test.
To calculate an estimate of intensity based on the quadrat counts, use intensity.quadratcount.
To extract the quadrats used in a quadratcount object, use as.tess.
If X is a split point pattern (object of class "splitppp" then quadrat counting will be performed
on each of the components point patterns, and the resulting contingency tables will be returned in a
list. This list can be printed or plotted.
Marks attached to the points are ignored by quadratcount.ppp. To obtain a separate contingency
table for each type of point in a multitype point pattern, first separate the different points using
split.ppp, then apply quadratcount.splitppp. See the Examples.
Value
The value of quadratcount.ppp is a contingency table containing the number of points in each
quadrat. The table is also an object of the special class "quadratcount" and there is a plot method
for this class.

1124

quadratcount

The value of quadratcount.splitppp is a list of such contingency tables, each containing the
quadrat counts for one of the component point patterns in X. This list also has the class "solist"
which has print and plot methods.
Warning
If Q is the result of quadratcount using rectangular tiles, then as.numeric(Q) extracts the counts
in the wrong order. To obtain the quadrat counts in the same order as the tiles of the corresponding
tessellation would be listed, use as.vector(t(Q)), which works in all cases.
Note
To perform a chi-squared test based on the quadrat counts, use quadrat.test.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Diggle, P.J. Statistical analysis of spatial point patterns. Academic Press, 2003.
Stoyan, D. and Stoyan, H. (1994) Fractals, random shapes and point fields: methods of geometrical
statistics. John Wiley and Sons.
See Also
plot.quadratcount, intensity.quadratcount, quadrats, quadrat.test, tess, hextess, quadratresample,
miplot
Examples
X <- runifpoint(50)
quadratcount(X)
quadratcount(X, 4, 5)
quadratcount(X, xbreaks=c(0, 0.3, 1), ybreaks=c(0, 0.4, 0.8, 1))
qX <- quadratcount(X, 4, 5)
# plotting:
plot(X, pch="+")
plot(qX, add=TRUE, col="red", cex=1.5, lty=2)
# irregular window
data(humberside)
plot(humberside)
qH <- quadratcount(humberside, 2, 3)
plot(qH, add=TRUE, col="blue", cex=1.5, lwd=2)
# multitype - split
plot(quadratcount(split(humberside), 2, 3))
# quadrats determined by tessellation:
B <- dirichlet(runifpoint(6))
qX <- quadratcount(X, tess=B)
plot(X, pch="+")

quadratresample

1125

plot(qX, add=TRUE, col="red", cex=1.5, lty=2)

quadratresample

Resample a Point Pattern by Resampling Quadrats

Description
Given a point pattern dataset, create a resampled point pattern by dividing the window into rectangular quadrats and randomly resampling the list of quadrats.
Usage
quadratresample(X, nx, ny=nx, ...,
replace = FALSE, nsamples = 1,
verbose = (nsamples > 1))
Arguments
X

A point pattern dataset (object of class "ppp").

nx,ny

Numbers of quadrats in the x and y directions.

...

Ignored.

replace

Logical value. Specifies whether quadrats should be sampled with or without
replacement.

nsamples

Number of randomised point patterns to be generated.

verbose

Logical value indicating whether to print progress reports.

Details
This command implements a very simple bootstrap resampling procedure for spatial point patterns
X.
The dataset X must be a point pattern (object of class "ppp") and its observation window must be a
rectangle.
The window is first divided into N = nx * ny rectangular tiles (quadrats) of equal size and shape.
To generate one resampled point pattern, a random sample of N quadrats is selected from the list
of N quadrats, with replacement (if replace=TRUE) or without replacement (if replace=FALSE).
The ith quadrat in the original dataset is then replaced by the ith sampled quadrat, after the latter
is shifted so that it occupies the correct spatial position. The quadrats are then reconstituted into a
point pattern inside the same window as X.
If replace=FALSE, this procedure effectively involves a random permutation of the quadrats. The
resulting resampled point pattern has the same number of points as X. If replace=TRUE, the number
of points in the resampled point pattern is random.
Value
A point pattern (if nsamples = 1) or a list of point patterns (if nsamples > 1).
Author(s)
Adrian Baddeley 
and Rolf Turner 

1126

quadrats

See Also
quadrats, quadratcount.
See varblock to estimate the variance of a summary statistic by block resampling.
Examples
data(bei)
quadratresample(bei, 6, 3)

quadrats

Divide Region into Quadrats

Description
Divides window into rectangular quadrats and returns the quadrats as a tessellation.
Usage
quadrats(X, nx = 5, ny = nx, xbreaks = NULL, ybreaks = NULL, keepempty=FALSE)
Arguments
X

A window (object of class "owin") or anything that can be coerced to a window
using as.owin, such as a point pattern.

nx,ny

Numbers of quadrats in the x and y directions. Incompatible with xbreaks and
ybreaks.

xbreaks

Numeric vector giving the x coordinates of the boundaries of the quadrats. Incompatible with nx.

ybreaks

Numeric vector giving the y coordinates of the boundaries of the quadrats. Incompatible with ny.

keepempty

Logical value indicating whether to delete or retain empty quadrats. See Details.

Details
If the window X is a rectangle, it is divided into an nx * ny grid of rectangular tiles or ‘quadrats’.
If X is not a rectangle, then the bounding rectangle of X is first divided into an nx * ny grid of
rectangular tiles, and these tiles are then intersected with the window X.
The resulting tiles are returned as a tessellation (object of class "tess") which can be plotted and
used in other analyses.
If xbreaks is given, it should be a numeric vector giving the x coordinates of the quadrat boundaries. If it is not given, it defaults to a sequence of nx+1 values equally spaced over the range of x
coordinates in the window Window(X).
Similarly if ybreaks is given, it should be a numeric vector giving the y coordinates of the quadrat
boundaries. It defaults to a vector of ny+1 values equally spaced over the range of y coordinates in
the window. The lengths of xbreaks and ybreaks may be different.
By default (if keepempty=FALSE), any rectangular tile which does not intersect the window X is
ignored, and only the non-empty intersections are treated as quadrats, so the tessellation may consist of fewer than nx * ny tiles. If keepempty=TRUE, empty intersections are retained, and the
tessellation always contains exactly nx * ny tiles, some of which may be empty.

quadscheme

1127

Value
A tessellation (object of class "tess") as described under tess.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
tess, quadratcount, quadrat.test, quadratresample
Examples
W <- square(10)
Z <- quadrats(W, 4, 5)
plot(Z)
data(letterR)
plot(quadrats(letterR, 5, 7))

quadscheme

Generate a Quadrature Scheme from a Point Pattern

Description
Generates a quadrature scheme (an object of class "quad") from point patterns of data and dummy
points.
Usage
quadscheme(data, dummy, method="grid", ...)
Arguments
data

The observed data point pattern. An object of class "ppp" or in a format recognised by as.ppp()

dummy

The pattern of dummy points for the quadrature. An object of class "ppp" or in
a format recognised by as.ppp() Defaults to default.dummy(data, ...)

method

The name of the method for calculating quadrature weights: either "grid" or
"dirichlet".

...

Parameters of the weighting method (see below) and parameters for constructing
the dummy points if necessary.

1128

quadscheme

Details
This is the primary method for producing a quadrature schemes for use by ppm.
The function ppm fits a point process model to an observed point pattern using the Berman-Turner
quadrature approximation (Berman and Turner, 1992; Baddeley and Turner, 2000) to the pseudolikelihood of the model. It requires a quadrature scheme consisting of the original data point pattern,
an additional pattern of dummy points, and a vector of quadrature weights for all these points. Such
quadrature schemes are represented by objects of class "quad". See quad.object for a description
of this class.
Quadrature schemes are created by the function quadscheme. The arguments data and dummy
specify the data and dummy points, respectively. There is a sensible default for the dummy points
(provided by default.dummy). Alternatively the dummy points may be specified arbitrarily and
given in any format recognised by as.ppp. There are also functions for creating dummy patterns
including corners, gridcentres, stratrand and spokes.
The quadrature region is the region over which we are integrating, and approximating integrals by
finite sums. If dummy is a point pattern object (class "ppp") then the quadrature region is taken to
be Window(dummy). If dummy is just a list of x, y coordinates then the quadrature region defaults to
the observation window of the data pattern, Window(data).
If dummy is missing, then a pattern of dummy points will be generated using default.dummy, taking
account of the optional arguments .... By default, the dummy points are arranged in a rectangular
grid; recognised arguments include nd (the number of grid points in the horizontal and vertical
directions) and eps (the spacing between dummy points). If random=TRUE, a systematic random
pattern of dummy points is generated instead. See default.dummy for details.
If method = "grid" then the optional arguments (for ...) are (nd, ntile, eps). The quadrature
region (defined above) is divided into an ntile[1] by ntile[2] grid of rectangular tiles. The
weight for each quadrature point is the area of a tile divided by the number of quadrature points in
that tile.
If method="dirichlet" then the optional arguments are (exact=TRUE, nd, eps). The quadrature
points (both data and dummy) are used to construct the Dirichlet tessellation. The quadrature weight
of each point is the area of its Dirichlet tile inside the quadrature region. If exact == TRUE then
this area is computed exactly using the package deldir; otherwise it is computed approximately by
discretisation.
Value
An object of class "quad" describing the quadrature scheme (data points, dummy points, and
quadrature weights) suitable as the argument Q of the function ppm() for fitting a point process
model.
The quadrature scheme can be inspected using the print and plot methods for objects of class
"quad".
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Baddeley, A. and Turner, R. Practical maximum pseudolikelihood for spatial point patterns. Australian and New Zealand Journal of Statistics 42 (2000) 283–322.

quadscheme.logi

1129

Berman, M. and Turner, T.R. Approximating point process likelihoods with GLIM. Applied Statistics 41 (1992) 31–38.
See Also
ppm, as.ppp, quad.object, gridweights, dirichletWeights, corners, gridcentres, stratrand,
spokes
Examples
data(simdat)
#
Q
Q
Q

grid weights
<- quadscheme(simdat)
<- quadscheme(simdat, method="grid")
<- quadscheme(simdat, eps=0.5)

# dummy point spacing 0.5 units

Q <- quadscheme(simdat, nd=50)
# 1 dummy point per tile
Q <- quadscheme(simdat, ntile=25, nd=50) # 4 dummy points per tile
# Dirichlet weights
Q <- quadscheme(simdat, method="dirichlet", exact=FALSE)
# random dummy pattern
## Not run:
D <- runifpoint(250, Window(simdat))
Q <- quadscheme(simdat, D, method="dirichlet", exact=FALSE)
## End(Not run)
# polygonal window
data(demopat)
X <- unmark(demopat)
Q <- quadscheme(X)
# mask window
Window(X) <- as.mask(Window(X))
Q <- quadscheme(X)

quadscheme.logi

Generate a Logistic Regression Quadrature Scheme from a Point Pattern

Description
Generates a logistic regression quadrature scheme (an object of class "logiquad" inheriting from
"quad") from point patterns of data and dummy points.
Usage
quadscheme.logi(data, dummy, dummytype = "stratrand",
nd = NULL, mark.repeat = FALSE, ...)

1130

quadscheme.logi

Arguments
data

The observed data point pattern. An object of class "ppp" or in a format recognised by as.ppp()

dummy

The pattern of dummy points for the quadrature. An object of class "ppp" or in
a format recognised by as.ppp(). If missing a sensible default is generated.

dummytype

The name of the type of dummy points to use when "dummy" is missing. Currently available options are: "stratrand" (default), "binomial", "poisson",
"grid" and "transgrid".

nd

Integer, or integer vector of length 2 controlling the intensity of dummy points
when "dummy" is missing.

mark.repeat

Repeating the dummy points for each level of a marked data pattern when
"dummy" is missing. (See details.)

...

Ignored.

Details
This is the primary method for producing a quadrature schemes for use by ppm when the logistic
regression approximation (Baddeley et al. 2013) to the pseudolikelihood of the model is applied
(i.e. when method="logi" in ppm).
The function ppm fits a point process model to an observed point pattern. When used with the option
method="logi" it requires a quadrature scheme consisting of the original data point pattern and an
additional pattern of dummy points. Such quadrature schemes are represented by objects of class
"logiquad".
Quadrature schemes are created by the function quadscheme.logi. The arguments data and dummy
specify the data and dummy points, respectively. There is a sensible default for the dummy points.
Alternatively the dummy points may be specified arbitrarily and given in any format recognised by
as.ppp.
The quadrature region is the region over which we are integrating, and approximating integrals by
finite sums. If dummy is a point pattern object (class "ppp") then the quadrature region is taken to
be Window(dummy). If dummy is just a list of x, y coordinates then the quadrature region defaults to
the observation window of the data pattern, Window(data).
If dummy is missing, then a pattern of dummy points will be generated, taking account of the optional
arguments dummytype, nd, and mark.repeat.
The currently accepted values for dummytype are:
• "grid" where the frame of the window is divided into a nd * nd or nd[1] * nd[2] regular
grid of tiles and the centers constitutes the dummy points.
• "transgrid" where a regular grid as above is translated by a random vector.
• "stratrand" where each point of a regular grid as above is randomly translated within its
tile.
• "binomial" where nd * nd or nd[1] * nd[2] points are generated uniformly in the frame of
the window. "poisson" where a homogeneous Poisson point process with intensity nd * nd
or nd[1] * nd[2] is generated within the frame of observation window.
Then if the window is not rectangular, any dummy points lying outside it are deleted.
If data is a multitype point pattern the dummy points should also be marked (with the same levels of
the marks as data). If dummy is missing and the dummy pattern is generated by quadscheme.logi
the default behaviour is to attach a uniformly distributed mark (from the levels of the marks) to each

quantess

1131

dummy point. Alternatively, if mark.repeat=TRUE each dummy point is repeated as many times as
there are levels of the marks with a distinct mark value attached to it.
Finally, each point (data and dummy) is assigned the weight 1. The weights are never used and only
appear to be compatible with the class "quad" from which the "logiquad" object inherits.
Value
An object of class "logiquad" inheriting from "quad" describing the quadrature scheme (data
points, dummy points, and quadrature weights) suitable as the argument Q of the function ppm() for
fitting a point process model.
The quadrature scheme can be inspected using the print and plot methods for objects of class
"quad".
Author(s)
Adrian Baddeley 
, Rolf Turner  and Ege Rubak  .
References
Baddeley, A., Coeurjolly, J.-F., Rubak, E. and Waagepetersen, R. (2014) Logistic regression for
spatial Gibbs point processes. Biometrika 101 (2) 377–392.
See Also
ppm, as.ppp
Examples
data(simdat)
Q <- quadscheme.logi(simdat)

quantess

Quantile Tessellation

Description
Divide space into tiles which contain equal amounts of stuff.
Usage
quantess(M, Z, n, ...)
## S3 method for class 'owin'
quantess(M, Z, n, ..., type=2)
## S3 method for class 'ppp'
quantess(M, Z, n, ..., type=2)
## S3 method for class 'im'
quantess(M, Z, n, ..., type=2)

1132

quantess

Arguments
M

A spatial object (such as a window, point pattern or pixel image) determining
the weight or amount of stuff at each location.

Z

A spatial covariate (a pixel image or a function(x,y)) or one of the strings "x"
or "y" indicating the x or y coordinate. The range of values of Z will be broken
into n bands containing equal amounts of stuff.

n

Number of bands. A positive integer.

type

Integer specifying the rule for calculating quantiles. Passed to quantile.default.

...

Additional arguments passed to quadrats or tess defining another tessellation
which should be intersected with the quantile tessellation.

Details
A quantile tessellation is a division of space into pieces which contain equal amounts of stuff.
The function quantess computes a quantile tessellation and returns the tessellation itself. The
function quantess is generic, with methods for windows (class "owin"), point patterns ("ppp")
and pixel images ("im").
The first argument M (for mass) specifies the spatial distribution of stuff that is to be divided. If M
is a window, the area of the window is to be divided into n equal pieces. If M is a point pattern, the
number of points in the pattern is to be divided into n equal parts, as far as possible. If M is a pixel
image, the pixel values are interpreted as weights, and the total weight is to be divided into n equal
parts.
The second argument Z is a spatial covariate. The range of values of Z will be divided into n bands,
each containing the same total weight. That is, we determine the quantiles of Z with weights given
by M.
For convenience, additional arguments ... can be given, to further subdivide the tiles of the tessellation.
The result of quantess is a tessellation of as.owin(M) determined by the quantiles of Z.
Value
A tessellation (object of class "tess").
Author(s)
Original idea by Ute Hahn. Implemented in spatstat by Adrian Baddeley 
Rolf Turner 
and Ege Rubak  .
See Also
tess, quadrats, quantile, tilenames
Examples
plot(quantess(letterR, "x", 5))
plot(quantess(bronzefilter, "x", 6))
points(unmark(bronzefilter))

quantile.density

1133

opa <- par(mar=c(0,0,2,5))
A <- quantess(Window(bei), bei.extra$elev, 4)
plot(A, ribargs=list(las=1))
B <- quantess(bei, bei.extra$elev, 4)
tilenames(B) <- paste(spatstat.utils::ordinal(1:4), "quartile")
plot(B, ribargs=list(las=1))
points(bei, pch=".", cex=2, col="white")
par(opa)

quantile.density

Quantiles of a Density Estimate

Description
Given a kernel estimate of a probability density, compute quantiles.
Usage
## S3 method for class 'density'
quantile(x, probs = seq(0, 1, 0.25), names = TRUE,
..., warn = TRUE)
Arguments
x

Object of class "density" computed by a method for density

probs

Numeric vector of probabilities for which the quantiles are required.

names

Logical value indicating whether to attach names (based on probs) to the result.

...

Ignored.

warn

Logical value indicating whether to issue a warning if the density estimate x had
to be renormalised because it was computed in a restricted interval.

Details
This function calculates quantiles of the probability distribution whose probability density has been
estimated and stored in the object x. The object x must belong to the class "density", and would
typically have been obtained from a call to the function density.
The probability density is first normalised so that the total probability is equal to 1. A warning is
issued if the density estimate was restricted to an interval (i.e. if x was created by a call to density
which included either of the arguments from and to).
Next, the density estimate is numerically integrated to obtain an estimate of the cumulative distribution function F (x). Then for each desired probability p, the algorithm finds the corresponding
quantile q.
The quantile q corresponding to probability p satisfies F (q) = p up to the resolution of the grid of
values contained in x. The quantile is computed from the right, that is, q is the smallest available
value of x such that F (x) ≥ p.
Value
A numeric vector containing the quantiles.

1134

quantile.ewcdf

Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
quantile, quantile.ewcdf, quantile.im, CDF.
Examples
dd <- density(runif(10))
quantile(dd)

quantile.ewcdf

Quantiles of Weighted Empirical Cumulative Distribution Function

Description
Compute quantiles of a weighted empirical cumulative distribution function.
Usage
## S3 method for class 'ewcdf'
quantile(x, probs = seq(0, 1, 0.25),
names = TRUE, ...,
normalise = TRUE, type=1)
Arguments
x

A weighted empirical cumulative distribution function (object of class "ewcdf",
produced by ewcdf) for which the quantiles are desired.

probs

probabilities for which the quantiles are desired. A numeric vector of values
between 0 and 1.

names

Logical. If TRUE, the resulting vector of quantiles is annotated with names corresponding to probs.

...

Ignored.

normalise

Logical value indicating whether x should first be normalised so that it ranges
between 0 and 1.

type

Integer specifying the type of quantile to be calculated, as explained in quantile.default.
Only types 1 and 2 are currently implemented.

Details
This is a method for the generic quantile function for the class ewcdf of empirical weighted
cumulative distribution functions.
The quantile for a probability p is computed as the right-continuous inverse of the cumulative distribution function x (assuming type=1, the default).
If normalise=TRUE (the default), the weighted cumulative function x is first normalised to have
total mass 1 so that it can be interpreted as a cumulative probability distribution function.

quantile.im

1135

Value
Numeric vector of quantiles, of the same length as probs.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak  and Kevin Ummel.
See Also
ewcdf, quantile
Examples
z <- rnorm(50)
w <- runif(50)
Fun <- ewcdf(z, w)
quantile(Fun, c(0.95,0.99))

quantile.im

Sample Quantiles of Pixel Image

Description
Compute the sample quantiles of the pixel values of a given pixel image.
Usage
## S3 method for class 'im'
quantile(x, ...)
Arguments
x

A pixel image. An object of class "im".

...

Optional arguments passed to quantile.default. They determine the probabilities for which quantiles should be computed. See quantile.default.

Details
This simple function applies the generic quantile operation to the pixel values of the image x.
This function is a convenient way to inspect an image and to obtain summary statistics. See the
examples.
Value
A vector of quantiles.
Author(s)
Adrian Baddeley 
and Rolf Turner 

1136

quasirandom

See Also
quantile, cut.im, im.object
Examples
# artificial image data
Z <- setcov(square(1))
# find the quartiles
quantile(Z)
# find the deciles
quantile(Z, probs=(0:10)/10)

quasirandom

Quasirandom Patterns

Description
Generates quasirandom sequences of numbers and quasirandom spatial patterns of points in any
dimension.
Usage
vdCorput(n, base)
Halton(n, bases = c(2, 3), raw = FALSE, simplify = TRUE)
Hammersley(n, bases = 2, raw = FALSE, simplify = TRUE)
Arguments
n

Number of points to generate.

base

A prime number giving the base of the sequence.

bases

Vector of prime numbers giving the bases of the sequences for each coordinate
axis.

raw

Logical value indicating whether to return the coordinates as a matrix (raw=TRUE)
or as a spatial point pattern (raw=FALSE, the default).

simplify

Argument passed to ppx indicating whether point patterns of dimension 2 or 3
should be returned as objects of class "ppp" or "pp3" respectively (simplify=TRUE,
the default) or as objects of class "ppx" (simplify=FALSE).

Details
The function vdCorput generates the quasirandom sequence of Van der Corput (1935) of length
n with the given base. These are numbers between 0 and 1 which are in some sense uniformly
distributed over the interval.
The function Halton generates the Halton quasirandom sequence of points in d-dimensional space,
where d = length(bases). The values of the i-th coordinate of the points are generated using the
van der Corput sequence with base equal to bases[i].

rags

1137
The function Hammersley generates the Hammersley set of points in d+1-dimensional space, where
d = length(bases). The first d coordinates of the points are generated using the van der Corput
sequence with base equal to bases[i]. The d+1-th coordinate is the sequence 1/n, 2/n, ..., 1.
If raw=FALSE (the default) then the Halton and Hammersley sets are interpreted as spatial point patterns of the appropriate dimension. They are returned as objects of class "ppx" (multidimensional
point patterns) unless simplify=TRUE and d=2 or d=3 when they are returned as objects of class
"ppp" or "pp3". If raw=TRUE, the coordinates are returned as a matrix with n rows and D columns
where D is the spatial dimension.

Value
For vdCorput, a numeric vector.
For Halton and Hammersley, an object of class "ppp", "pp3" or "ppx"; or if raw=TRUE, a numeric
matrix.
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak .
References
Van der Corput, J. G. (1935) Verteilungsfunktionen. Proc. Ned. Akad. v. Wetensch. 38: 813–821.
Kuipers, L. and Niederreiter, H. (2005) Uniform distribution of sequences, Dover Publications.
See Also
rQuasi
Examples
vdCorput(10, 2)
plot(Halton(256, c(2,3)))
plot(Hammersley(256, 3))

rags

Alternating Gibbs Sampler for Multitype Point Processes

Description
Simulate a realisation of a point process model using the alternating Gibbs sampler.
Usage
rags(model, ..., ncycles = 100)

1138

ragsAreaInter

Arguments
model

Data specifying some kind of point process model.

...

Additional arguments passed to other code.

ncycles

Number of cycles of the alternating Gibbs sampler that should be performed.

Details
The Alternating Gibbs Sampler for a multitype point process is an iterative simulation procedure.
Each step of the sampler updates the pattern of points of a particular type i, by drawing a realisation
from the conditional distribution of points of type i given the points of all other types. Successive
steps of the sampler update the points of type 1, then type 2, type 3, and so on.
This is an experimental implementation which currently works only for multitype hard core processes (see MultiHard) in which there is no interaction between points of the same type.
The argument model should be an object describing a point process model. At the moment, the only
permitted format for model is of the form list(beta, hradii) where beta gives the first order
trend and hradii is the matrix of interaction radii. See ragsMultiHard for full details.
Value
A point pattern (object of class "ppp").
Author(s)
Adrian Baddeley 
See Also
ragsMultiHard, ragsAreaInter
Examples
mo <- list(beta=c(30, 20),
hradii = 0.05 * matrix(c(0,1,1,0), 2, 2))
rags(mo, ncycles=10)

ragsAreaInter

Alternating Gibbs Sampler for Area-Interaction Process

Description
Generate a realisation of the area-interaction process using the alternating Gibbs sampler. Applies
only when the interaction parameter eta is greater than 1.
Usage
ragsAreaInter(beta, eta, r, ...,
win = NULL, bmax = NULL, periodic = FALSE, ncycles = 100)

ragsAreaInter

1139

Arguments
beta
eta
r
...
win
bmax
periodic

ncycles

First order trend. A number, a pixel image (object of class "im"), or a function(x,y).
Interaction parameter (canonical form) as described in the help for AreaInter.
A number greater than 1.
Disc radius in the model. A number greater than 1.
Additional arguments for beta if it is a function.
Simulation window. An object of class "owin". (Ignored if beta is a pixel
image.)
Optional. The maximum possible value of beta, or a number larger than this.
Logical value indicating whether to treat opposite sides of the simulation window as being the same, so that points close to one side may interact with points
close to the opposite side. Feasible only when the window is a rectangle.
Number of cycles of the alternating Gibbs sampler to be performed.

Details
This function generates a simulated realisation of the area-interaction process (see AreaInter)
using the alternating Gibbs sampler (see rags).
It exploits a mathematical relationship between the (unmarked) area-interaction process and the
two-type hard core process (Baddeley and Van Lieshout, 1995; Widom and Rowlinson, 1970).
This relationship only holds when the interaction parameter eta is greater than 1 so that the areainteraction process is clustered.
The parameters beta,eta are the canonical parameters described in the help for AreaInter. The
first order trend beta may be a constant, a function, or a pixel image.
The simulation window is determined by beta if it is a pixel image, and otherwise by the argument
win (the default is the unit square).
Value
A point pattern (object of class "ppp").
Author(s)
Adrian Baddeley .
References
Baddeley, A.J. and Van Lieshout, M.N.M. (1995). Area-interaction point processes. Annals of the
Institute of Statistical Mathematics 47 (1995) 601–619.
Widom, B. and Rowlinson, J.S. (1970). New model for the study of liquid-vapor phase transitions.
The Journal of Chemical Physics 52 (1970) 1670–1684.
See Also
rags, ragsMultiHard
AreaInter
Examples
plot(ragsAreaInter(100, 2, 0.07, ncycles=15))

1140

ragsMultiHard

ragsMultiHard

Alternating Gibbs Sampler for Multitype Hard Core Process

Description
Generate a realisation of the multitype hard core point process using the alternating Gibbs sampler.
Usage
ragsMultiHard(beta, hradii, ..., types=NULL, bmax = NULL,
periodic=FALSE, ncycles = 100)
Arguments
beta

First order trend. A numeric vector, a pixel image, a function, a list of functions,
or a list of pixel images.

hradii

Matrix of hard core radii between each pair of types. Diagonal entries should be
0 or NA.

types

Vector of all possible types for the multitype point pattern.

...

Arguments passed to rmpoispp when generating random points.

bmax

Optional upper bound on beta.

periodic

Logical value indicating whether to measure distances in the periodic sense, so
that opposite sides of the (rectangular) window are treated as identical.

ncycles

Number of cycles of the sampler to be performed.

Details
The Alternating Gibbs Sampler for a multitype point process is an iterative simulation procedure.
Each step of the sampler updates the pattern of points of a particular type i, by drawing a realisation
from the conditional distribution of points of type i given the points of all other types. Successive
steps of the sampler update the points of type 1, then type 2, type 3, and so on.
This is an experimental implementation which currently works only for multitype hard core processes (see MultiHard) in which there is no interaction between points of the same type, and for
the area-interaction process (see ragsAreaInter).
The argument beta gives the first order trend for each possible type of point. It may be a single
number, a numeric vector, a function(x,y), a pixel image, a list of functions, a function(x,y,m),
or a list of pixel images.
The argument hradii is the matrix of hard core radii between each pair of possible types of points.
Two points of types i and j respectively are forbidden to lie closer than a distance hradii[i,j]
apart. The diagonal of this matrix must contain NA or 0 values, indicating that there is no hard core
constraint applying between points of the same type.
Value
A point pattern (object of class "ppp").
Author(s)
Adrian Baddeley 

ranef.mppm

1141

See Also
rags, ragsAreaInter
Examples
b <- c(30,20)
h <- 0.05 * matrix(c(0,1,1,0), 2, 2)
ragsMultiHard(b, h, ncycles=10)
ragsMultiHard(b, h, ncycles=5, periodic=TRUE)

ranef.mppm

Extract Random Effects from Point Process Model

Description
Given a point process model fitted to a list of point patterns, extract the fixed effects of the model.
A method for ranef.
Usage
## S3 method for class 'mppm'
ranef(object, ...)
Arguments
object

A fitted point process model (an object of class "mppm").

...

Ignored.

Details
This is a method for the generic function ranef.
The argument object must be a fitted point process model (object of class "mppm") produced by
the fitting algorithm mppm). This represents a point process model that has been fitted to a list of
several point pattern datasets. See mppm for information.
This function extracts the coefficients of the random effects of the model.
Value
A data frame, or list of data frames, as described in the help for ranef.lme.
Author(s)
Adrian Baddeley, Ida-Maria Sintorn and Leanne Bischoff. Implemented in spatstat by Adrian
Baddeley , Rolf Turner  and
Ege Rubak .
References
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. London: Chapman and Hall/CRC Press.

1142

range.fv

See Also
fixef.mppm, coef.mppm
Examples
H <- hyperframe(Y = waterstriders)
# Tweak data to exaggerate differences
H$Y[[1]] <- rthin(H$Y[[1]], 0.3)
m1 <- mppm(Y ~ id, data=H, Strauss(7))
ranef(m1)
m2 <- mppm(Y ~ 1, random=~1|id, data=H, Strauss(7))
ranef(m2)

range.fv

Range of Function Values

Description
Compute the range, maximum, or minimum of the function values in a summary function.
Usage
## S3 method for class 'fv'
range(..., na.rm = TRUE, finite = na.rm)
## S3 method for class 'fv'
max(..., na.rm = TRUE, finite = na.rm)
## S3 method for class 'fv'
min(..., na.rm = TRUE, finite = na.rm)
Arguments
...

One or more function value tables (objects of class "fv" representing summary
functions) or other data.

na.rm

Logical. Whether to ignore NA values.

finite

Logical. Whether to ignore values that are infinite, NaN or NA.

Details
These are methods for the generic range, max and min. They compute the range, maximum, and
minimum of the function values that would be plotted on the y axis by default.
For more complicated calculations, use with.fv.
Value
Numeric vector of length 2.

raster.x

1143

Author(s)
Adrian Baddeley 
, Rolf Turner  and Ege Rubak .
See Also
with.fv
Examples
G <- Gest(cells)
range(G)
max(G)
min(G)

raster.x

Cartesian Coordinates for a Pixel Raster

Description
Return the x and y coordinates of each pixel in a pixel image or binary mask.
Usage
raster.x(w, drop=FALSE)
raster.y(w, drop=FALSE)
raster.xy(w, drop=FALSE)
Arguments
w

A pixel image (object of class "im") or a mask window (object of class "owin"
of type "mask").

drop

Logical. If TRUE, then coordinates of pixels that lie outside the window are
removed. If FALSE (the default) then the coordinates of every pixel in the containing rectangle are retained.

Details
The argument w should be either a pixel image (object of class "im") or a mask window (an object
of class "owin" of type "mask").
If drop=FALSE (the default), the functions raster.x and raster.y return a matrix of the same
dimensions as the pixel image or mask itself, with entries giving the x coordinate (for raster.x)
or y coordinate (for raster.y) of each pixel in the pixel grid.
If drop=TRUE, pixels that lie outside the window w (or outside the domain of the image w) are
removed, and raster.x and raster.y return numeric vectors containing the coordinates of the
pixels that are inside the window w.
The function raster.xy returns a list with components x and y which are numeric vectors of equal
length containing the pixel coordinates.

1144

rat

Value
raster.xy returns a list with components x and y which are numeric vectors of equal length containing the pixel coordinates.
If drop=FALSE, raster.x and raster.y return a matrix of the same dimensions as the pixel grid
in w, and giving the value of the x (or y) coordinate of each pixel in the raster.
If drop=TRUE, raster.x and raster.y return numeric vectors.
Author(s)
Adrian Baddeley  , Rolf Turner 
and Ege Rubak 
See Also
owin, as.mask, pixelcentres
Examples
u <- owin(c(-1,1),c(-1,1)) # square of side 2
w <- as.mask(u, eps=0.01) # 200 x 200 grid
X <- raster.x(w)
Y <- raster.y(w)
disc <- owin(c(-1,1), c(-1,1), mask=(X^2 + Y^2 <= 1))
## Not run: plot(disc)
# approximation to the unit disc

rat

Ratio object

Description
Stores the numerator, denominator, and value of a ratio as a single object.

Usage
rat(ratio, numerator, denominator, check = TRUE)
Arguments
ratio,numerator,denominator
Three objects belonging to the same class.
check

Logical. Whether to check that the objects are compatible.

rCauchy

1145

Details
The class "rat" is a simple mechanism for keeping track of the numerator and denominator when
calculating a ratio. Its main purpose is simply to signal that the object is a ratio.
The function rat creates an object of class "rat" given the numerator, the denominator and the
ratio. No calculation is performed; the three objects are simply stored together.
The arguments ratio, numerator, denominator can be objects of any kind. They should belong
to the same class. It is assumed that the relationship
ratio =

numerator
denominator

holds in some version of arithmetic. However, no calculation is performed.
By default the algorithm checks whether the three arguments ratio, numerator, denominator are
compatible objects, according to compatible.
The result is equivalent to ratio except for the addition of extra information.
Value
An object equivalent to the object ratio except that it also belongs to the class "rat" and has
additional attributes numerator and denominator.
Author(s)
Adrian Baddeley  and Rolf Turner .
See Also
compatible, pool

rCauchy

Simulate Neyman-Scott Point Process with Cauchy cluster kernel

Description
Generate a random point pattern, a simulated realisation of the Neyman-Scott process with Cauchy
cluster kernel.
Usage
rCauchy(kappa, scale, mu, win = owin(), thresh = 0.001,
nsim=1, drop=TRUE,
saveLambda=FALSE, expand = NULL, ...,
poisthresh=1e-6, saveparents=TRUE)

1146

rCauchy

Arguments
kappa

Intensity of the Poisson process of cluster centres. A single positive number, a
function, or a pixel image.

scale

Scale parameter for cluster kernel. Determines the size of clusters. A positive
number, in the same units as the spatial coordinates.

mu

Mean number of points per cluster (a single positive number) or reference intensity for the cluster points (a function or a pixel image).

win

Window in which to simulate the pattern. An object of class "owin" or something acceptable to as.owin.

thresh

Threshold relative to the cluster kernel value at the origin (parent location) determining when the cluster kernel will be treated as zero for simulation purposes.
Will be overridden by argument expand if that is given.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

saveLambda

Logical. If TRUE then the random intensity corresponding to the simulated parent
points will also be calculated and saved, and returns as an attribute of the point
pattern.

expand

Numeric. Size of window expansion for generation of parent points. By default
determined by calling clusterradius with the numeric threshold value given
in thresh.

...

Passed to clusterfield to control the image resolution when saveLambda=TRUE
and to clusterradius when expand is missing or NULL.

poisthresh

Numerical threshold below which the model will be treated as a Poisson process.
See Details.

saveparents

Logical value indicating whether to save the locations of the parent points as an
attribute.

Details
This algorithm generates a realisation of the Neyman-Scott process with Cauchy cluster kernel,
inside the window win.
The process is constructed by first generating a Poisson point process of “parent” points with intensity kappa. Then each parent point is replaced by a random cluster of points, the number of
points in each cluster being random with a Poisson (mu) distribution, and the points being placed
independently and uniformly according to a Cauchy kernel.
In this implementation, parent points are not restricted to lie in the window; the parent process is
effectively the uniform Poisson process on the infinite plane.
This model can be fitted to data by the method of minimum contrast, maximum composite likelihood or Palm likelihood using kppm.
The algorithm can also generate spatially inhomogeneous versions of the cluster process:
• The parent points can be spatially inhomogeneous. If the argument kappa is a function(x,y)
or a pixel image (object of class "im"), then it is taken as specifying the intensity function of
an inhomogeneous Poisson process that generates the parent points.
• The offspring points can be inhomogeneous. If the argument mu is a function(x,y) or a
pixel image (object of class "im"), then it is interpreted as the reference density for offspring
points, in the sense of Waagepetersen (2006).

rcell

1147
When the parents are homogeneous (kappa is a single number) and the offspring are inhomogeneous
(mu is a function or pixel image), the model can be fitted to data using kppm.
If the pair correlation function of the model is very close to that of a Poisson process, deviating by
less than poisthresh, then the model is approximately a Poisson process, and will be simulated as
a Poisson process with intensity kappa * mu, using rpoispp. This avoids computations that would
otherwise require huge amounts of memory.

Value
A point pattern (an object of class "ppp") if nsim=1, or a list of point patterns if nsim > 1.
Additionally, some intermediate results of the simulation are returned as attributes of this point pattern (see rNeymanScott). Furthermore, the simulated intensity function is returned as an attribute
"Lambda", if saveLambda=TRUE.
Author(s)

Abdollah Jalilian and Rasmus Waagepetersen. Adapted for spatstat by Adrian Baddeley  1.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Baddeley, A.J. and Silverman, B.W. (1984) A cautionary example on the use of second-order methods for analyzing point patterns. Biometrics 40, 1089-1094.
See Also
rcellnumber, rstrat, rsyst, runifpoint, Kest

rcellnumber

1149

Examples
X <- rcell(nx=15)
plot(X)
plot(Kest(X))

rcellnumber

Generate Random Numbers of Points for Cell Process

Description
Generates random integers for the Baddeley-Silverman counterexample.
Usage
rcellnumber(n, N = 10, mu=1)
Arguments
n

Number of random integers to be generated.

N

Distributional parameter: the largest possible value (when mu <= 1). An integer
greater than 1.

mu

Mean of the distribution (equals the variance). Any positive real number.

Details
If mu = 1 (the default), this function generates random integers which have mean and variance
equal to 1, but which do not have a Poisson distribution. The random integers take the values 0, 1
and N with probabilities 1/N , (N − 2)/(N − 1) and 1/(N (N − 1)) respectively. See Baddeley
and Silverman (1984).
If mu is another positive number, the random integers will have mean and variance equal to mu.
They are obtained by generating the one-dimensional counterpart of the cell process and counting
the number of points in the interval from 0 to mu. The maximum possible value of each random
integer is N * ceiling(mu).
Value
An integer vector of length n.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
References
Baddeley, A.J. and Silverman, B.W. (1984) A cautionary example on the use of second-order methods for analyzing point patterns. Biometrics 40, 1089-1094.
See Also
rcell

1150

rDGS

Examples
rcellnumber(30, 3)

rDGS

Perfect Simulation of the Diggle-Gates-Stibbard Process

Description
Generate a random pattern of points, a simulated realisation of the Diggle-Gates-Stibbard process,
using a perfect simulation algorithm.
Usage
rDGS(beta, rho, W = owin(), expand=TRUE, nsim=1, drop=TRUE)
Arguments
beta

intensity parameter (a positive number).

rho

interaction range (a non-negative number).

W

window (object of class "owin") in which to generate the random pattern.

expand

Logical. If FALSE, simulation is performed in the window W, which must be
rectangular. If TRUE (the default), simulation is performed on a larger window,
and the result is clipped to the original window W. Alternatively expand can
be an object of class "rmhexpand" (see rmhexpand) determining the expansion
method.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

Details
This function generates a realisation of the Diggle-Gates-Stibbard point process in the window W
using a ‘perfect simulation’ algorithm.
Diggle, Gates and Stibbard (1987) proposed a pairwise interaction point process in which each pair
of points separated by a distance d contributes a factor e(d) to the probability density, where
 
πd
e(d) = sin2
2ρ
for d < ρ, and e(d) is equal to 1 for d ≥ ρ.
The simulation algorithm used to generate the point pattern is ‘dominated coupling from the past’
as implemented by Berthelsen and Møller (2002, 2003). This is a ‘perfect simulation’ or ‘exact
simulation’ algorithm, so called because the output of the algorithm is guaranteed to have the correct probability distribution exactly (unlike the Metropolis-Hastings algorithm used in rmh, whose
output is only approximately correct).
There is a tiny chance that the algorithm will run out of space before it has terminated. If this occurs,
an error message will be generated.

rDiggleGratton

1151

Value
If nsim = 1, a point pattern (object of class "ppp"). If nsim > 1, a list of point patterns.
Author(s)
Adrian Baddeley , based on original code for the Strauss process by Kasper Klitgaard Berthelsen.
References
Berthelsen, K.K. and Møller, J. (2002) A primer on perfect simulation for spatial point processes.
Bulletin of the Brazilian Mathematical Society 33, 351-367.
Berthelsen, K.K. and Møller, J. (2003) Likelihood and non-parametric Bayesian MCMC inference
for spatial point processes based on perfect simulation and path sampling. Scandinavian Journal of
Statistics 30, 549-564.
Diggle, P.J., Gates, D.J., and Stibbard, A. (1987) A nonparametric estimator for pairwise-interaction
point processes. Biometrika 74, 763 – 770. Scandinavian Journal of Statistics 21, 359–373.
Møller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point
Processes. Chapman and Hall/CRC.
See Also
rmh, DiggleGatesStibbard.
rStrauss, rHardcore, rStraussHard, rDiggleGratton, rPenttinen.
Examples
X <- rDGS(50, 0.05)

rDiggleGratton

Perfect Simulation of the Diggle-Gratton Process

Description
Generate a random pattern of points, a simulated realisation of the Diggle-Gratton process, using a
perfect simulation algorithm.
Usage
rDiggleGratton(beta, delta, rho, kappa=1, W = owin(),
expand=TRUE, nsim=1, drop=TRUE)
Arguments
beta

intensity parameter (a positive number).

delta

hard core distance (a non-negative number).

rho

interaction range (a number greater than delta).

kappa

interaction exponent (a non-negative number).

1152

rDiggleGratton

W

window (object of class "owin") in which to generate the random pattern. Currently this must be a rectangular window.

expand

Logical. If FALSE, simulation is performed in the window W, which must be
rectangular. If TRUE (the default), simulation is performed on a larger window,
and the result is clipped to the original window W. Alternatively expand can
be an object of class "rmhexpand" (see rmhexpand) determining the expansion
method.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

Details
This function generates a realisation of the Diggle-Gratton point process in the window W using a
‘perfect simulation’ algorithm.
Diggle and Gratton (1984, pages 208-210) introduced the pairwise interaction point process with
pair potential h(t) of the form
κ

t−δ
if δ ≤ t ≤ ρ
h(t) =
ρ−δ
with h(t) = 0 for t < δ and h(t) = 1 for t > ρ. Here δ, ρ and κ are parameters.
Note that we use the symbol κ where Diggle and Gratton (1984) use β, since in spatstat we reserve
the symbol β for an intensity parameter.
The parameters must all be nonnegative, and must satisfy δ ≤ ρ.
The simulation algorithm used to generate the point pattern is ‘dominated coupling from the past’
as implemented by Berthelsen and Møller (2002, 2003). This is a ‘perfect simulation’ or ‘exact
simulation’ algorithm, so called because the output of the algorithm is guaranteed to have the correct probability distribution exactly (unlike the Metropolis-Hastings algorithm used in rmh, whose
output is only approximately correct).
There is a tiny chance that the algorithm will run out of space before it has terminated. If this occurs,
an error message will be generated.
Value
If nsim = 1, a point pattern (object of class "ppp"). If nsim > 1, a list of point patterns.
Author(s)
Adrian Baddeley 
based on original code for the Strauss process by Kasper Klitgaard Berthelsen.
References
Berthelsen, K.K. and Møller, J. (2002) A primer on perfect simulation for spatial point processes.
Bulletin of the Brazilian Mathematical Society 33, 351-367.
Berthelsen, K.K. and Møller, J. (2003) Likelihood and non-parametric Bayesian MCMC inference
for spatial point processes based on perfect simulation and path sampling. Scandinavian Journal of
Statistics 30, 549-564.
Diggle, P.J. and Gratton, R.J. (1984) Monte Carlo methods of inference for implicit statistical models. Journal of the Royal Statistical Society, series B 46, 193 – 212.

rdpp

1153

Møller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point
Processes. Chapman and Hall/CRC.
See Also
rmh, DiggleGratton.
rStrauss, rHardcore, rStraussHard, rDGS, rPenttinen.
Examples
X <- rDiggleGratton(50, 0.02, 0.07)

rdpp

Simulation of a Determinantal Point Process

Description
Generates simulated realisations from a determinantal point process.
Usage
rdpp(eig, index, basis = "fourierbasis",
window = boxx(rep(list(0:1), ncol(index))),
reject_max = 10000, progress = 0, debug = FALSE, ...)
Arguments
eig

vector of values between 0 and 1 specifying the non-zero eigenvalues for the
process.

index

data.frame or matrix (or something acceptable to as.matrix) specifying indices of the basis functions.

basis

character string giving the name of the basis.

window

window (of class "owin", "box3" or "boxx") giving the domain of the point
process.

reject_max

integer giving the maximal number of trials for rejection sampling.

progress

integer giving the interval for making a progress report. The value zero turns
reporting off.

debug

logical value indicating whether debug informationb should be outputted.

...

Ignored.

Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 

1154

reach

Examples
index <- expand.grid(-2:2,-2:2)
eig <- exp(-rowSums(index^2))
X <- rdpp(eig, index)
X
## To simulate a det. projection p. p. with the given indices set eig=1:
XX <- rdpp(1, index)
XX

reach

Interaction Distance of a Point Process

Description
Computes the interaction distance of a point process.
Usage
reach(x, ...)
## S3 method for class 'ppm'
reach(x, ..., epsilon=0)
## S3 method for class 'interact'
reach(x, ...)
## S3 method for class 'rmhmodel'
reach(x, ...)
## S3 method for class 'fii'
reach(x, ..., epsilon)
Arguments
x

Either a fitted point process model (object of class "ppm"), an interpoint interaction (object of class "interact"), a fitted interpoint interaction (object of class
"fii") or a point process model for simulation (object of class "rmhmodel").

epsilon

Numerical threshold below which interaction is treated as zero. See details.

...

Other arguments are ignored.

Details
The ‘interaction distance’ or ‘interaction range’ of a point process model is the smallest distance
D such that any two points in the process which are separated by a distance greater than D do not
interact with each other.
For example, the interaction range of a Strauss process (see Strauss) with parameters β, γ, r is
equal to r, unless γ = 1 in which case the model is Poisson and the interaction range is 0. The
interaction range of a Poisson process is zero. The interaction range of the Ord threshold process
(see OrdThresh) is infinite, since two points may interact at any distance apart.
The function reach(x) is generic, with methods for the case where x is

reach

1155
• a fitted point process model (object of class "ppm", usually obtained from the model-fitting
function ppm);
• an interpoint interaction structure (object of class "interact"), created by one of the functions Poisson, Strauss, StraussHard, MultiStrauss, MultiStraussHard, Softcore, DiggleGratton,
Pairwise, PairPiece, Geyer, LennardJones, Saturated, OrdThresh or Ord;
• a fitted interpoint interaction (object of class "fii") extracted from a fitted point process
model by the command fitin;
• a point process model for simulation (object of class "rmhmodel"), usually obtained from
rmhmodel.

When x is an "interact" object, reach(x) returns the maximum possible interaction range for any
point process model with interaction structure given by x. For example, reach(Strauss(0.2))
returns 0.2.
When x is a "ppm" object, reach(x) returns the interaction range for the point process model
represented by x. For example, a fitted Strauss process model with parameters beta,gamma,r will
return either 0 or r, depending on whether the fitted interaction parameter gamma is equal or not
equal to 1.
For some point process models, such as the soft core process (see Softcore), the interaction distance is infinite, because the interaction terms are positive for all pairs of points. A practical solution
is to compute the distance at which the interaction contribution from a pair of points falls below a
threshold epsilon, on the scale of the log conditional intensity. This is done by setting the argument
epsilon to a positive value.
Value
The interaction distance, or NA if this cannot be computed from the information given.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
ppm, Poisson, Strauss, StraussHard, MultiStrauss, MultiStraussHard, Softcore, DiggleGratton,
Pairwise, PairPiece, Geyer, LennardJones, Saturated, OrdThresh, Ord, rmhmodel
Examples
reach(Poisson())
# returns 0
reach(Strauss(r=7))
# returns 7
fit <- ppm(swedishpines ~ 1, Strauss(r=7))
reach(fit)
# returns 7
reach(OrdThresh(42))
# returns Inf
reach(MultiStrauss(matrix(c(1,3,3,1),2,2)))
# returns 3

1156

reach.dppm

reach.dppm

Range of Interaction for a Determinantal Point Process Model

Description
Returns the range of interaction for a determinantal point process model.

Usage
## S3 method for class 'dppm'
reach(x, ...)
## S3 method for class 'detpointprocfamily'
reach(x, ...)
Arguments
x

Model of class "detpointprocfamily" or "dppm".

...

Additional arguments passed to the range function of the given model.

Details
The range of interaction for a determinantal point process model may defined as the smallest number
R such that g(r) = 1 for all r ≥ R, where g is the pair correlation function. For many models the
range is infinite, but one may instead use a value where the pair correlation function is sufficiently
close to 1. For example in the Matern model this defaults to finding R such that g(R) = 0.99.

Value
Numeric

Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
Examples
reach(dppMatern(lambda=100, alpha=.01, nu=1, d=2))

reduced.sample

reduced.sample

1157

Reduced Sample Estimator using Histogram Data

Description
Compute the Reduced Sample estimator of a survival time distribution function, from histogram
data
Usage
reduced.sample(nco, cen, ncc, show=FALSE, uppercen=0)
Arguments
nco

vector of counts giving the histogram of uncensored observations (those survival
times that are less than or equal to the censoring time)

cen

vector of counts giving the histogram of censoring times

ncc

vector of counts giving the histogram of censoring times for the uncensored
observations only

uppercen

number of censoring times greater than the rightmost histogram breakpoint (if
there are any)

show

Logical value controlling the amount of detail returned by the function value
(see below)

Details
This function is needed mainly for internal use in spatstat, but may be useful in other applications
where you want to form the reduced sample estimator from a huge dataset.
Suppose Ti are the survival times of individuals i = 1, . . . , M with unknown distribution function
F (t) which we wish to estimate. Suppose these times are right-censored by random censoring
times Ci . Thus the observations consist of right-censored survival times T̃i = min(Ti , Ci ) and
non-censoring indicators Di = 1{Ti ≤ Ci } for each i.
If the number of observations M is large, it is efficient to use histograms. Form the histogram
cen of all censoring times Ci . That is, obs[k] counts the number of values Ci in the interval
(breaks[k],breaks[k+1]] for k > 1 and [breaks[1],breaks[2]] for k = 1. Also form the
histogram nco of all uncensored times, i.e. those T̃i such that Di = 1, and the histogram of all
censoring times for which the survival time is uncensored, i.e. those Ci such that Di = 1. These
three histograms are the arguments passed to kaplan.meier.
The return value rs is the reduced-sample estimator of the distribution function F (t). Specifically,
rs[k] is the reduced sample estimate of F(breaks[k+1]). The value is exact, i.e. the use of
histograms does not introduce any approximation error.
Note that, for the results to be valid, either the histogram breaks must span the censoring times,
or the number of censoring times that do not fall in a histogram cell must have been counted in
uppercen.

1158

reflect

Value
If show = FALSE, a numeric vector giving the values of the reduced sample estimator. If show=TRUE,
a list with three components which are vectors of equal length,
rs

Reduced sample estimate of the survival time c.d.f. F (t)

numerator

numerator of the reduced sample estimator

denominator

denominator of the reduced sample estimator

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
kaplan.meier, km.rs

reflect

Reflect In Origin

Description
Reflects a geometrical object through the origin.
Usage
reflect(X)
## S3 method for class 'im'
reflect(X)
## Default S3 method:
reflect(X)
Arguments
X

Any suitable dataset representing a two-dimensional object, such as a point pattern (object of class "ppp"), or a window (object of class "owin").

Details
The object X is reflected through the origin. That is, each point in X with coordinates (x, y) is
mapped to the position (−x, −y).
This is equivalent to applying the affine transformation with matrix diag(c(-1,-1)). It is also
equivalent to rotation about the origin by 180 degrees.
The command reflect is generic, with a method for pixel images and a default method.
Value
Another object of the same type, representing the result of reflection.

regularpolygon

1159

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
affine, flipxy
Examples
plot(reflect(as.im(letterR)))
plot(reflect(letterR), add=TRUE)

regularpolygon

Create A Regular Polygon

Description
Create a window object representing a regular (equal-sided) polygon.
Usage
regularpolygon(n, edge = 1, centre = c(0, 0), ...,
align = c("bottom", "top", "left", "right", "no"))
hexagon(edge = 1, centre = c(0,0), ...,
align = c("bottom", "top", "left", "right", "no"))
Arguments
n

Number of edges in the polygon.

edge

Length of each edge in the polygon. A single positive number.

centre

Coordinates of the centre of the polygon. A numeric vector of length 2, or a
list(x,y) giving the coordinates of exactly one point, or a point pattern (object
of class "ppp") containing exactly one point.

align

Character string specifying whether to align one of the edges with a vertical or
horizontal boundary.

...

Ignored.

Details
The function regularpolygon creates a regular (equal-sided) polygon with n sides, centred at
centre, with sides of equal length edge. The function hexagon is the special case n=6.
The orientation of the polygon is determined by the argument align. If align="no", one vertex of
the polygon is placed on the x-axis. Otherwise, an edge of the polygon is aligned with one side of
the frame, specified by the value of align.
Value
A window (object of class "owin").

1160

relevel.im

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
disc, ellipse, owin.
hextess for hexagonal tessellations.
Examples
plot(hexagon())
plot(regularpolygon(7))
plot(regularpolygon(7, align="left"))

relevel.im

Reorder Levels of a Factor-Valued Image or Pattern

Description
For a pixel image with factor values, or a point pattern with factor-valued marks, the levels of the
factor are re-ordered so that the level ref is first and the others are moved down.
Usage
## S3 method for class 'im'
relevel(x, ref, ...)
## S3 method for class 'ppp'
relevel(x, ref, ...)
## S3 method for class 'ppx'
relevel(x, ref, ...)
Arguments
x

A pixel image (object of class "im") with factor values, or a point pattern (object
of class "ppp", "ppx", "lpp" or "pp3") with factor-valued marks.

ref

The reference level.

...

Ignored.

Details
These functions are methods for the generic relevel.
If x is a pixel image (object of class "im") with factor values, or a point pattern (object of class
"ppp", "ppx", "lpp" or "pp3") with factor-valued marks, the levels of the factor are changed so
that the level specified by ref comes first.
Value
Object of the same kind as x.

reload.or.compute

1161

Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
mergeLevels
Examples
amacrine
relevel(amacrine, "on")

reload.or.compute

Compute Unless Previously Saved

Description
If the designated file does not yet exist, evaluate the expression and save the results in the file. If
the file already exists, re-load the results from the file.
Usage
reload.or.compute(filename, expr, objects = NULL,
destination = parent.frame(), force=FALSE)
Arguments
filename

Name of data file. A character string.

expr

R language expression to be evaluated.

objects

Optional character vector of names of objects to be saved in filename after
evaluating expr, or names of objects that should be present in filename when
loaded.

destination

Environment in which the resulting objects should be assigned.

force

Logical value indicating whether to perform the computation in any case.

Details
This facility is useful for saving, and later re-loading, the results of time-consuming computations.
It would typically be used in an R script file or an Sweave document.
If the file called filename does not yet exist, then expr will be evaluated and the results will be
saved in filename. The optional argument objects specifies which results should be saved to the
file: the default is to save all objects that were created by evaluating the expression.
If the file called filename already exists, then it will be loaded. The optional argument objects
specifies the names of objects that should be present in the file; a warning is issued if any of them
are missing.
The resulting objects can be assigned into any desired destination. The default behaviour is
equivalent to evaluating expr in the current environment.

1162

relrisk

If force=TRUE then expr will be evaluated (regardless of whether the file already exists or not)
and the results will be saved in filename, overwriting any previously-existing file with that name.
This is a convenient way to force the code to re-compute everything in an R script file or Sweave
document.
Value
Character vector (invisible) giving the names of the objects computed or loaded.
Author(s)
Adrian Baddeley 
and Rolf Turner 
Examples
## Not run:
if(FALSE) {
reload.or.compute("mydata.rda", {
x <- very.long.computation()
y <- 42
})
}
## End(Not run)

relrisk

Estimate of Spatially-Varying Relative Risk

Description
Generic command to estimate the spatially-varying probability of each type of point, or the ratios
of such probabilities.
Usage
relrisk(X, ...)
Arguments
X

Either a point pattern (class "ppp") or a fitted point process model (class "ppm")
from which the probabilities will be estimated.

...

Additional arguments appropriate to the method.

Details
In a point pattern containing several different types of points, we may be interested in the spatiallyvarying probability of each possible type, or the relative risks which are the ratios of such probabilities.
The command relrisk is generic and can be used to estimate relative risk in different ways.
The function relrisk.ppp is the method for point pattern datasets. It computes nonparametric
estimates of relative risk by kernel smoothing.

relrisk.ppm

1163

The function relrisk.ppm is the method for fitted point process models (class "ppm"). It computes
parametric estimates of relative risk, using the fitted model.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
relrisk.ppp, relrisk.ppm.

relrisk.ppm

Parametric Estimate of Spatially-Varying Relative Risk

Description
Given a point process model fitted to a multitype point pattern, this function computes the fitted
spatially-varying probability of each type of point, or the ratios of such probabilities, according to
the fitted model. Optionally the standard errors of the estimates are also computed.
Usage
## S3 method for class 'ppm'
relrisk(X, ...,
at = c("pixels", "points"),
relative = FALSE, se = FALSE,
casecontrol = TRUE, control = 1, case,
ngrid = NULL, window = NULL)
Arguments
X

A fitted point process model (object of class "ppm").

...

Ignored.

at

String specifying whether to compute the probability values at a grid of pixel
locations (at="pixels") or only at the points of X (at="points").

relative

Logical. If FALSE (the default) the algorithm computes the probabilities of each
type of point. If TRUE, it computes the relative risk, the ratio of probabilities of
each type relative to the probability of a control.

se

Logical value indicating whether to compute standard errors as well.

casecontrol

Logical. Whether to treat a bivariate point pattern as consisting of cases and
controls, and return only the probability or relative risk of a case. Ignored if
there are more than 2 types of points. See Details.

control

Integer, or character string, identifying which mark value corresponds to a control.

case

Integer, or character string, identifying which mark value corresponds to a case
(rather than a control) in a bivariate point pattern. This is an alternative to the
argument control in a bivariate point pattern. Ignored if there are more than 2
types of points.

1164

relrisk.ppm

ngrid

Optional. Dimensions of a rectangular grid of locations inside window where
the predictions should be computed. An integer, or an integer vector of length
2, specifying the number of grid points in the y and x directions. (Applies only
when at="pixels".)

window

Optional. A window (object of class "owin") delimiting the locations where
predictions should be computed. Defaults to the window of the original data
used to fit the model object. (Applies only when at="pixels".)

Details
The command relrisk is generic and can be used to estimate relative risk in different ways.
This function relrisk.ppm is the method for fitted point process models (class "ppm"). It computes
parametric estimates of relative risk, using the fitted model.
If X is a bivariate point pattern (a multitype point pattern consisting of two types of points) then by
default, the points of the first type (the first level of marks(X)) are treated as controls or non-events,
and points of the second type are treated as cases or events. Then by default this command computes
the spatially-varying probability of a case, i.e. the probability p(u) that a point at spatial location u
will be a case. If relative=TRUE, it computes the spatially-varying relative risk of a case relative
to a control, r(u) = p(u)/(1 − p(u)).
If X is a multitype point pattern with m > 2 types, or if X is a bivariate point pattern and casecontrol=FALSE,
then by default this command computes, for each type j, a nonparametric estimate of the spatiallyvarying probability of an event of type j. This is the probability pj (u) that a point at spatial location
u will belong to type j. If relative=TRUE, the command computes the relative risk of an event of
type j relative to a control, rj (u) = pj (u)/pk (u), where events of type k are treated as controls.
The argument control determines which type k is treated as a control.
If at = "pixels" the calculation is performed for every spatial location u on a fine pixel grid, and
the result is a pixel image representing the function p(u) or a list of pixel images representing the
functions pj (u) or rj (u) for j = 1, . . . , m. An infinite value of relative risk (arising because the
probability of a control is zero) will be returned as NA.
If at = "points" the calculation is performed only at the data points xi . By default the result is
a vector of values p(xi ) giving the estimated probability of a case at each data point, or a matrix
of values pj (xi ) giving the estimated probability of each possible type j at each data point. If
relative=TRUE then the relative risks r(xi ) or rj (xi ) are returned. An infinite value of relative risk
(arising because the probability of a control is zero) will be returned as Inf.
Probabilities and risks are computed from the fitted intensity of the model, using predict.ppm. If
se=TRUE then standard errors will also be computed, based on asymptotic theory, using vcov.ppm.
Value
If se=FALSE (the default), the format is described below. If se=TRUE, the result is a list of two
entries, estimate and SE, each having the format described below.
If X consists of only two types of points, and if casecontrol=TRUE, the result is a pixel image (if
at="pixels") or a vector (if at="points"). The pixel values or vector values are the probabilities
of a case if relative=FALSE, or the relative risk of a case (probability of a case divided by the
probability of a control) if relative=TRUE.
If X consists of more than two types of points, or if casecontrol=FALSE, the result is:
• (if at="pixels") a list of pixel images, with one image for each possible type of point. The
result also belongs to the class "solist" so that it can be printed and plotted.

relrisk.ppp

1165

• (if at="points") a matrix of probabilities, with rows corresponding to data points xi , and
columns corresponding to types j.
The pixel values or matrix entries are the probabilities of each type of point if relative=FALSE,
or the relative risk of each type (probability of each type divided by the probability of a control) if
relative=TRUE.
If relative=FALSE, the resulting values always lie between 0 and 1. If relative=TRUE, the results
are either non-negative numbers, or the values Inf or NA.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
There is another method relrisk.ppp for point pattern datasets which computes nonparametric
estimates of relative risk by kernel smoothing.
See also relrisk, relrisk.ppp, ppm
Examples
fit <- ppm(chorley ~ marks * (x+y))
rr <- relrisk(fit, relative=TRUE, control="lung", se=TRUE)
plot(rr$estimate)
plot(rr$SE)
rrX <- relrisk(fit, at="points", relative=TRUE, control="lung")

relrisk.ppp

Nonparametric Estimate of Spatially-Varying Relative Risk

Description
Given a multitype point pattern, this function estimates the spatially-varying probability of each
type of point, or the ratios of such probabilities, using kernel smoothing. The default smoothing
bandwidth is selected by cross-validation.
Usage
## S3 method for class 'ppp'
relrisk(X, sigma = NULL, ..., varcov = NULL, at = "pixels",
relative=FALSE,
se=FALSE,
casecontrol=TRUE, control=1, case)

1166

relrisk.ppp

Arguments
X

A multitype point pattern (object of class "ppp" which has factor valued marks).

sigma

Optional. The numeric value of the smoothing bandwidth (the standard deviation of isotropic Gaussian smoothing kernel). Alternatively sigma may be a
function which can be used to select a different bandwidth for each type of
point. See Details.

...

Arguments passed to bw.relrisk to select the bandwidth, or passed to density.ppp
to control the pixel resolution.

varcov

Optional. Variance-covariance matrix of anisotopic Gaussian smoothing kernel.
Incompatible with sigma.

at

String specifying whether to compute the probability values at a grid of pixel
locations (at="pixels") or only at the points of X (at="points").

relative

Logical. If FALSE (the default) the algorithm computes the probabilities of each
type of point. If TRUE, it computes the relative risk, the ratio of probabilities of
each type relative to the probability of a control.

se

Logical value indicating whether to compute standard errors as well.

casecontrol

Logical. Whether to treat a bivariate point pattern as consisting of cases and
controls, and return only the probability or relative risk of a case. Ignored if
there are more than 2 types of points. See Details.

control

Integer, or character string, identifying which mark value corresponds to a control.

case

Integer, or character string, identifying which mark value corresponds to a case
(rather than a control) in a bivariate point pattern. This is an alternative to the
argument control in a bivariate point pattern. Ignored if there are more than 2
types of points.

Details
The command relrisk is generic and can be used to estimate relative risk in different ways.
This function relrisk.ppp is the method for point pattern datasets. It computes nonparametric
estimates of relative risk by kernel smoothing.
If X is a bivariate point pattern (a multitype point pattern consisting of two types of points) then by
default, the points of the first type (the first level of marks(X)) are treated as controls or non-events,
and points of the second type are treated as cases or events. Then by default this command computes
the spatially-varying probability of a case, i.e. the probability p(u) that a point at spatial location u
will be a case. If relative=TRUE, it computes the spatially-varying relative risk of a case relative
to a control, r(u) = p(u)/(1 − p(u)).
If X is a multitype point pattern with m > 2 types, or if X is a bivariate point pattern and casecontrol=FALSE,
then by default this command computes, for each type j, a nonparametric estimate of the spatiallyvarying probability of an event of type j. This is the probability pj (u) that a point at spatial location
u will belong to type j. If relative=TRUE, the command computes the relative risk of an event of
type j relative to a control, rj (u) = pj (u)/pk (u), where events of type k are treated as controls.
The argument control determines which type k is treated as a control.
If at = "pixels" the calculation is performed for every spatial location u on a fine pixel grid, and
the result is a pixel image representing the function p(u) or a list of pixel images representing the
functions pj (u) or rj (u) for j = 1, . . . , m. An infinite value of relative risk (arising because the
probability of a control is zero) will be returned as NA.

relrisk.ppp

1167

If at = "points" the calculation is performed only at the data points xi . By default the result is
a vector of values p(xi ) giving the estimated probability of a case at each data point, or a matrix
of values pj (xi ) giving the estimated probability of each possible type j at each data point. If
relative=TRUE then the relative risks r(xi ) or rj (xi ) are returned. An infinite value of relative risk
(arising because the probability of a control is zero) will be returned as Inf.
Estimation is performed by a simple Nadaraja-Watson type kernel smoother (Diggle, 2003). The
smoothing bandwidth can be specified in any of the following ways:
• sigma is a single numeric value, giving the standard deviation of the isotropic Gaussian kernel.
• sigma is a numeric vector of length 2, giving the standard deviations in the x and y directions
of a Gaussian kernel.
• varcov is a 2 by 2 matrix giving the variance-covariance matrix of the Gaussian kernel.
• sigma is a function which selects the bandwidth. Bandwidth selection will be applied separately to each type of point. An example of such a function is bw.diggle.
• sigma and varcov are both missing or null. Then a common smoothing bandwidth sigma
will be selected by cross-validation using bw.relrisk.
If se=TRUE then standard errors will also be computed, based on asymptotic theory, assuming a
Poisson process.
Value
If se=FALSE (the default), the format is described below. If se=TRUE, the result is a list of two
entries, estimate and SE, each having the format described below.
If X consists of only two types of points, and if casecontrol=TRUE, the result is a pixel image (if
at="pixels") or a vector (if at="points"). The pixel values or vector values are the probabilities
of a case if relative=FALSE, or the relative risk of a case (probability of a case divided by the
probability of a control) if relative=TRUE.
If X consists of more than two types of points, or if casecontrol=FALSE, the result is:
• (if at="pixels") a list of pixel images, with one image for each possible type of point. The
result also belongs to the class "solist" so that it can be printed and plotted.
• (if at="points") a matrix of probabilities, with rows corresponding to data points xi , and
columns corresponding to types j.
The pixel values or matrix entries are the probabilities of each type of point if relative=FALSE,
or the relative risk of each type (probability of each type divided by the probability of a control) if
relative=TRUE.
If relative=FALSE, the resulting values always lie between 0 and 1. If relative=TRUE, the results
are either non-negative numbers, or the values Inf or NA.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Diggle, P.J. (2003) Statistical analysis of spatial point patterns, Second edition. Arnold.

1168

Replace.im

See Also
There is another method relrisk.ppm for point process models which computes parametric estimates of relative risk, using the fitted model.
See also bw.relrisk, density.ppp, Smooth.ppp, eval.im
Examples
p.oak <- relrisk(urkiola, 20)
if(interactive()) {
plot(p.oak, main="proportion of oak")
plot(eval.im(p.oak > 0.3), main="More than 30 percent oak")
plot(split(lansing), main="Lansing Woods")
p.lan <- relrisk(lansing, 0.05, se=TRUE)
plot(p.lan$estimate, main="Lansing Woods species probability")
plot(p.lan$SE, main="Lansing Woods standard error")
wh <- im.apply(p.lan$estimate, which.max)
types <- levels(marks(lansing))
wh <- eval.im(types[wh])
plot(wh, main="Most common species")
}

Replace.im

Reset Values in Subset of Image

Description
Reset the values in a subset of a pixel image.
Usage
## S3 replacement method for class 'im'
x[i, j] <- value
Arguments
x

A two-dimensional pixel image. An object of class "im".

i

Object defining the subregion or subset to be replaced. Either a spatial window
(an object of class "owin"), or a pixel image with logical values, or a point
pattern (an object of class "ppp"), or any type of index that applies to a matrix, or
something that can be converted to a point pattern by as.ppp (using the window
of x).

j

An integer or logical vector serving as the column index if matrix indexing is
being used. Ignored if i is appropriate to some sort of replacement other than
matrix indexing.

value

Vector, matrix, factor or pixel image containing the replacement values. Short
vectors will be recycled.

Replace.im

1169

Details
This function changes some of the pixel values in a pixel image. The image x must be an object
of class "im" representing a pixel image defined inside a rectangle in two-dimensional space (see
im.object).
The subset to be changed is determined by the arguments i,j according to the following rules
(which are checked in this order):
1. i is a spatial object such as a window, a pixel image with logical values, or a point pattern; or
2. i,j are indices for the matrix as.matrix(x); or
3. i can be converted to a point pattern by as.ppp(i, W=Window(x)), and i is not a matrix.
If i is a spatial window (an object of class "owin"), the values of the image inside this window are
changed.
If i is a point pattern (an object of class "ppp"), then the values of the pixel image at the points of
this pattern are changed.
If i does not satisfy any of the conditions above, then the algorithm tries to interpret i,j as indices
for the matrix as.matrix(x). Either i or j may be missing or blank.
If none of the conditions above are met, and if i is not a matrix, then i is converted into a point
pattern by as.ppp(i, W=Window(x)). Again the values of the pixel image at the points of this
pattern are changed.
Value
The image x with the values replaced.
Warning
If you have a 2-column matrix containing the x, y coordinates of point locations, then to prevent
this being interpreted as an array index, you should convert it to a data.frame or to a point pattern.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
im.object, [.im, [, ppp.object, as.ppp, owin.object
Examples
# make up an image
X <- setcov(unit.square())
plot(X)
# a rectangular subset
W <- owin(c(0,0.5),c(0.2,0.8))
X[W] <- 2
plot(X)
# a polygonal subset
data(letterR)

1170

Replace.linim
R <- affine(letterR, diag(c(1,1)/2), c(-2,-0.7))
X[R] <- 3
plot(X)
# a point pattern
P <- rpoispp(20)
X[P] <- 10
plot(X)
# change pixel value at a specific location
X[list(x=0.1,y=0.2)] <- 7
# matrix indexing --- single vector index
X[1:2570] <- 10
plot(X)
# matrix indexing using double indices
X[1:257,1:10] <- 5
plot(X)
# matrix indexing using a matrix of indices
X[cbind(1:257,1:257)] <- 10
X[cbind(257:1,1:257)] <- 10
plot(X)

Replace.linim

Reset Values in Subset of Image on Linear Network

Description
Reset the values in a subset of a pixel image on a linear network.
Usage
## S3 replacement method for class 'linim'
x[i, j] <- value
Arguments
x

A pixel image on a linear network. An object of class "linim".

i

Object defining the subregion or subset to be replaced. Either a spatial window
(an object of class "owin"), or a pixel image with logical values, or a point
pattern (an object of class "ppp"), or any type of index that applies to a matrix, or
something that can be converted to a point pattern by as.ppp (using the window
of x).

j

An integer or logical vector serving as the column index if matrix indexing is
being used. Ignored if i is appropriate to some sort of replacement other than
matrix indexing.

value

Vector, matrix, factor or pixel image containing the replacement values. Short
vectors will be recycled.

requireversion

1171

Details
This function changes some of the pixel values in a pixel image. The image x must be an object of
class "linim" representing a pixel image on a linear network.
The pixel values are replaced according to the rules described in the help for [<-.im. Then the
auxiliary data are updated.
Value
The image x with the values replaced.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
[<-.im.
Examples
# make a function
Y <- as.linim(distfun(runiflpp(5, simplenet)))
# replace some values
B <- square(c(0.25, 0.55))
Y[B] <- 2
plot(Y, main="")
plot(B, add=TRUE, lty=3)
X <- runiflpp(4, simplenet)
Y[X] <- 5

requireversion

Require a Specific Version of a Package

Description
Checks that the version number of a specified package is greater than or equal to the specified
version number. For use in stand-alone R scripts.
Usage
requireversion(pkg, ver)
Arguments
pkg

Package name.

ver

Character string containing version number.

1172

rescale

Details
This function checks whether the installed version of the package pkg is greater than or equal to
ver.
It is useful in stand-alone R scripts, which often require a particular version of a package in order
to work correctly.
This function should not be used inside a package: for that purpose, the dependence on packages
and versions should be specified in the package description file.
Value
Null.
Author(s)
Adrian Baddeley 
Examples
## Not run:
requireversion(spatstat, "1.42-0")
## End(Not run)

rescale

Convert dataset to another unit of length

Description
Converts between different units of length in a spatial dataset, such as a point pattern or a window.
Usage
rescale(X, s, unitname)
Arguments
X

Any suitable dataset representing a two-dimensional object, such as a point pattern (object of class "ppp"), or a window (object of class "owin").

s

Conversion factor: the new units are s times the old units.

unitname

Optional. New name for the unit of length. See unitname.

Details
This is generic. Methods are provided for many spatial objects.
The spatial coordinates in the dataset X will be re-expressed in terms of a new unit of length that is
s times the current unit of length given in X. The name of the unit of length will also be adjusted.
The result is an object of the same type, representing the same data, but expressed in the new units.
For example if X is a dataset giving coordinates in metres, then rescale(X,1000) will take the new
unit of length to be 1000 metres. To do this, it will divide the old coordinate values by 1000 to obtain

rescale.im

1173

coordinates expressed in kilometres, and change the name of the unit of length from "metres" to
"1000 metres".
If unitname is given, it will be taken as the new name of the unit of length. It should be a valid
name for the unit of length, as described in the help for unitname. For example if X is a dataset
giving coordinates in metres, rescale(X, 1000, "km") will divide the coordinate values by 1000
to obtain coordinates in kilometres, and the unit name will be changed to "km".
Value
Another object of the same type, representing the same data, but expressed in the new units.
Note
The result of this operation is equivalent to the original dataset. If you want to actually change the
coordinates by a linear transformation, producing a dataset that is not equivalent to the original one,
use affine.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
Available methods: rescale.im, rescale.layered, rescale.linnet, rescale.lpp, rescale.owin,
rescale.ppp, rescale.psp and rescale.unitname.
Other generics: unitname, affine, rotate, shift.

rescale.im

Convert Pixel Image to Another Unit of Length

Description
Converts a pixel image to another unit of length.
Usage
## S3 method for class 'im'
rescale(X, s, unitname)
Arguments
X

Pixel image (object of class "im").

s

Conversion factor: the new units are s times the old units.

unitname

Optional. New name for the unit of length. See unitname.

1174

rescale.owin

Details
This is a method for the generic function rescale.
The spatial coordinates of the pixels in X will be re-expressed in terms of a new unit of length that
is s times the current unit of length given in X. (Thus, the coordinate values are divided by s, while
the unit value is multiplied by s).
If s is missing, then the coordinates will be re-expressed in ‘native’ units; for example if the current
unit is equal to 0.1 metres, then the coordinates will be re-expressed in metres.
The result is a pixel image representing the same data but re-expressed in a different unit.
Pixel values are unchanged. This may not be what you intended!
Value
Another pixel image (of class "im"), containing the same pixel values, but with pixel coordinates
expressed in the new units.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
im, rescale, unitname, eval.im
Examples
# Bramble Canes data: 1 unit = 9 metres
data(bramblecanes)
# distance transform
Z <- distmap(bramblecanes)
# convert to metres
# first alter the pixel values
Zm <- eval.im(9 * Z)
# now rescale the pixel coordinates
Z <- rescale(Zm, 1/9)
# or equivalently
Z <- rescale(Zm)

rescale.owin

Convert Window to Another Unit of Length

Description
Converts a window to another unit of length.
Usage
## S3 method for class 'owin'
rescale(X, s, unitname)

rescale.owin

1175

Arguments
X

Window (object of class "owin").

s

Conversion factor: the new units are s times the old units.

unitname

Optional. New name for the unit of length. See unitname.

Details
This is a method for the generic function rescale.
The spatial coordinates in the window X (and its window) will be re-expressed in terms of a new
unit of length that is s times the current unit of length given in X. (Thus, the coordinate values are
divided by s, while the unit value is multiplied by s).
The result is a window representing the same region of space, but re-expressed in a different unit.
If s is missing, then the coordinates will be re-expressed in ‘native’ units; for example if the current
unit is equal to 0.1 metres, then the coordinates will be re-expressed in metres.
Value
Another window object (of class "owin") representing the same window, but expressed in the new
units.
Note
The result of this operation is equivalent to the original window. If you want to actually change
the coordinates by a linear transformation, producing a window that is larger or smaller than the
original one, use affine.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
unitname, rescale, rescale.owin, affine, rotate, shift
Examples
data(swedishpines)
W <- Window(swedishpines)
W
# coordinates are in decimetres (0.1 metre)
# convert to metres:
rescale(W, 10)
# or equivalently
rescale(W)

1176

rescale.ppp

rescale.ppp

Convert Point Pattern to Another Unit of Length

Description
Converts a point pattern dataset to another unit of length.
Usage
## S3 method for class 'ppp'
rescale(X, s, unitname)
Arguments
X

Point pattern (object of class "ppp").

s

Conversion factor: the new units are s times the old units.

unitname

Optional. New name for the unit of length. See unitname.

Details
This is a method for the generic function rescale.
The spatial coordinates in the point pattern X (and its window) will be re-expressed in terms of a
new unit of length that is s times the current unit of length given in X. (Thus, the coordinate values
are divided by s, while the unit value is multiplied by s).
The result is a point pattern representing the same data but re-expressed in a different unit.
Mark values are unchanged.
If s is missing, then the coordinates will be re-expressed in ‘native’ units; for example if the current
unit is equal to 0.1 metres, then the coordinates will be re-expressed in metres.
Value
Another point pattern (of class "ppp"), representing the same data, but expressed in the new units.
Note
The result of this operation is equivalent to the original point pattern. If you want to actually change
the coordinates by a linear transformation, producing a point pattern that is not equivalent to the
original one, use affine.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
unitname, rescale, rescale.owin, affine, rotate, shift

rescale.psp

1177

Examples
# Bramble Canes data: 1 unit = 9 metres
data(bramblecanes)
# convert to metres
bram <- rescale(bramblecanes, 1/9)
# or equivalently
bram <- rescale(bramblecanes)

rescale.psp

Convert Line Segment Pattern to Another Unit of Length

Description
Converts a line segment pattern dataset to another unit of length.
Usage
## S3 method for class 'psp'
rescale(X, s, unitname)
Arguments
X

Line segment pattern (object of class "psp").

s

Conversion factor: the new units are s times the old units.

unitname

Optional. New name for the unit of length. See unitname.

Details
This is a method for the generic function rescale.
The spatial coordinates in the line segment pattern X (and its window) will be re-expressed in terms
of a new unit of length that is s times the current unit of length given in X. (Thus, the coordinate
values are divided by s, while the unit value is multiplied by s).
The result is a line segment pattern representing the same data but re-expressed in a different unit.
Mark values are unchanged.
If s is missing, then the coordinates will be re-expressed in ‘native’ units; for example if the current
unit is equal to 0.1 metres, then the coordinates will be re-expressed in metres.
Value
Another line segment pattern (of class "psp"), representing the same data, but expressed in the new
units.
Note
The result of this operation is equivalent to the original segment pattern. If you want to actually
change the coordinates by a linear transformation, producing a segment pattern that is not equivalent
to the original one, use affine.

1178

rescue.rectangle

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
units, affine, rotate, shift
Examples
data(copper)
X <- copper$Lines
X
# data are in km
# convert to metres
rescale(X, 1/1000)
# convert data and rename unit
rescale(X, 1/1000, c("metre", "metres"))

rescue.rectangle

Convert Window Back To Rectangle

Description
Determines whether the given window is really a rectangle aligned with the coordinate axes, and if
so, converts it to a rectangle object.
Usage
rescue.rectangle(W)
Arguments
W

A window (object of class "owin").

Details
This function decides whether the window W is actually a rectangle aligned with the coordinate axes.
This will be true if W is
• a rectangle (window object of type "rectangle");
• a polygon (window object of type "polygonal" with a single polygonal boundary) that is a
rectangle aligned with the coordinate axes;
• a binary mask (window object of type "mask") in which all the pixel entries are TRUE.
If so, the function returns this rectangle, a window object of type "rectangle". If not, the function
returns W.
Value
Another object of class "owin" representing the same window.

residuals.dppm

1179

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
as.owin, owin.object
Examples
w <- owin(poly=list(x=c(0,1,1,0),y=c(0,0,1,1)))
rw <- rescue.rectangle(w)
w <- as.mask(unit.square())
rw <- rescue.rectangle(w)

residuals.dppm

Residuals for Fitted Determinantal Point Process Model

Description
Given a determinantal point process model fitted to a point pattern, compute residuals.
Usage
## S3 method for class 'dppm'
residuals(object, ...)
Arguments
object

The fitted determinatal point process model (an object of class "dppm") for
which residuals should be calculated.

...

Arguments passed to residuals.ppm.

Details
This function extracts the intensity component of the model using as.ppm and then applies residuals.ppm
to compute the residuals.
Use plot.msr to plot the residuals directly.
Value
An object of class "msr" representing a signed measure or vector-valued measure (see msr). This
object can be plotted.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 

1180

residuals.kppm

See Also
msr, dppm
Examples
fit <- dppm(swedishpines ~ x, dppGauss())
rr <- residuals(fit)

residuals.kppm

Residuals for Fitted Cox or Cluster Point Process Model

Description
Given a Cox or cluster point process model fitted to a point pattern, compute residuals.
Usage
## S3 method for class 'kppm'
residuals(object, ...)
Arguments
object

The fitted point process model (an object of class "kppm") for which residuals
should be calculated.

...

Arguments passed to residuals.ppm.

Details
This function extracts the intensity component of the model using as.ppm and then applies residuals.ppm
to compute the residuals.
Use plot.msr to plot the residuals directly.
Value
An object of class "msr" representing a signed measure or vector-valued measure (see msr). This
object can be plotted.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
msr, kppm
Examples
fit <- kppm(redwood ~ x, "Thomas")
rr <- residuals(fit)

residuals.mppm

residuals.mppm

1181

Residuals for Point Process Model Fitted to Multiple Point Patterns

Description
Given a point process model fitted to multiple point patterns, compute residuals for each pattern.
Usage
## S3 method for class 'mppm'
residuals(object, type = "raw", ...,
fittedvalues = fitted.mppm(object))
Arguments
object

Fitted point process model (object of class "mppm").

...

Ignored.

type

Type of residuals: either "raw", "pearson" or "inverse". Partially matched.

fittedvalues

Advanced use only. Fitted values of the model to be used in the calculation.

Details
Baddeley et al (2005) defined residuals for the fit of a point process model to spatial point pattern
data. For an explanation of these residuals, see the help file for residuals.ppm.
This function computes the residuals for a point process model fitted to multiple point patterns. The
object should be an object of class "mppm" obtained from mppm.
The return value is a list. The number of entries in the list equals the number of point patterns in
the original data. Each entry in the list has the same format as the output of residuals.ppm. That
is, each entry in the list is a signed measure (object of class "msr") giving the residual measure for
the corresponding point pattern.
Value
A list of signed measures (objects of class "msr") giving the residual measure for each of the
original point patterns. See Details.
Author(s)
Adrian Baddeley , Ida-Maria Sintorn and Leanne Bischoff.
Implemented by Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Baddeley, A., Turner, R., Moller, J. and Hazelton, M. (2005) Residual analysis for spatial point
processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. London: Chapman and Hall/CRC Press.

1182

residuals.ppm

See Also
mppm, residuals.mppm
Examples
fit <- mppm(Bugs ~ x, hyperframe(Bugs=waterstriders))
r <- residuals(fit)
# compute total residual for each point pattern
rtot <- sapply(r, integral.msr)
# standardise the total residuals
areas <- sapply(windows.mppm(fit), area.owin)
rtot/sqrt(areas)

residuals.ppm

Residuals for Fitted Point Process Model

Description
Given a point process model fitted to a point pattern, compute residuals.
Usage
## S3 method for class 'ppm'
residuals(object, type="raw", ...,
check=TRUE, drop=FALSE,
fittedvalues=NULL,
new.coef=NULL, dropcoef=FALSE,
quad=NULL)
Arguments
object

The fitted point process model (an object of class "ppm") for which residuals
should be calculated.

type

String indicating the type of residuals to be calculated. Current options are
"raw", "inverse", "pearson" and "score". A partial match is adequate.

...

Ignored.

check

Logical value indicating whether to check the internal format of object. If
there is any possibility that this object has been restored from a dump file, or
has otherwise lost track of the environment where it was originally computed,
set check=TRUE.

drop

Logical value determining whether to delete quadrature points that were not
used to fit the model. See quad.ppm for explanation.

fittedvalues

Vector of fitted values for the conditional intensity at the quadrature points, from
which the residuals will be computed. For expert use only.

new.coef

Optional. Numeric vector of coefficients for the model, replacing coef(object).
See the section on Modified Residuals below.

dropcoef

Internal use only.

quad

Optional. Data specifying how to re-fit the model. A list of arguments passed to
quadscheme. See the section on Modified Residuals below.

residuals.ppm

1183

Details
This function computes several kinds of residuals for the fit of a point process model to a spatial point pattern dataset (Baddeley et al, 2005). Use plot.msr to plot the residuals directly, or
diagnose.ppm to produce diagnostic plots based on these residuals.
The argument object must be a fitted point process model (object of class "ppm"). Such objects
are produced by the maximum pseudolikelihood fitting algorithm ppm. This fitted model object
contains complete information about the original data pattern.
Residuals are attached both to the data points and to some other points in the window of observation
(namely, to the dummy points of the quadrature scheme used to fit the model). If the fitted model
is correct, then the sum of the residuals over all (data and dummy) points in a spatial region B has
mean zero. For further explanation, see Baddeley et al (2005).
The type of residual is chosen by the argument type. Current options are
"raw": the raw residuals
rj = zj − wj λj
at the quadrature points uj , where zj is the indicator equal to 1 if uj is a data point and 0 if uj
is a dummy point; wj is the quadrature weight attached to uj ; and
λj = λ̂(uj , x)
is the conditional intensity of the fitted model at uj . These are the spatial analogue of the
martingale residuals of a one-dimensional counting process.
"inverse": the ‘inverse-lambda’ residuals (Baddeley et al, 2005)
(I)

rj

=

zj
rj
=
− wj
λj
λj

obtained by dividing the raw residuals by the fitted conditional intensity. These are a counterpart of the exponential energy marks (see eem).
"pearson": the Pearson residuals (Baddeley et al, 2005)
(P )

rj

p
zj
rj
= p = p − wj λj
λj
λj

obtained by dividing the raw residuals by the square root of the fitted conditional intensity. The
Pearson residuals are standardised, in the sense that if the model (true and fitted) is Poisson,
then the sum of the Pearson residuals in a spatial region B has variance equal to the area of B.
"score": the score residuals (Baddeley et al, 2005)
rj = (zj − wj λj )xj
obtained by multiplying the raw residuals rj by the covariates xj for quadrature point j. The
score residuals always sum to zero.
The result of residuals.ppm is a measure (object of class "msr"). Use plot.msr to plot the
residuals directly, or diagnose.ppm to produce diagnostic plots based on these residuals. Use
integral.msr to compute the total residual.
By default, the window of the measure is the same as the original window of the data. If drop=TRUE
then the window is the domain of integration of the pseudolikelihood or composite likelihood.
This only matters when the model object was fitted using the border correction: in that case, if
drop=TRUE the window of the residuals is the erosion of the original data window by the border
correction distance rbord.

1184

residuals.ppm

Value
An object of class "msr" representing a signed measure or vector-valued measure (see msr). This
object can be plotted.
Modified Residuals
Sometimes we want to modify the calculation of residuals by using different values for the model
parameters. This capability is provided by the arguments new.coef and quad.
If new.coef is given, then the residuals will be computed by taking the model parameters to be
new.coef. This should be a numeric vector of the same length as the vector of fitted model parameters coef(object).
If new.coef is missing and quad is given, then the model parameters will be determined by refitting the model using a new quadrature scheme specified by quad. Residuals will be computed for
the original model object using these new parameter values.
The argument quad should normally be a list of arguments in name=value format that will be passed
to quadscheme (together with the original data points) to determine the new quadrature scheme. It
may also be a quadrature scheme (object of class "quad") to which the model should be fitted, or a
point pattern (object of class "ppp") specifying the dummy points in a new quadrature scheme.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Baddeley, A., Turner, R., Møller, J. and Hazelton, M. (2005) Residual analysis for spatial point
processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
Baddeley, A., Møller, J. and Pakes, A.G. (2008) Properties of residuals for spatial point processes.
Annals of the Institute of Statistical Mathematics 60, 627–649.
See Also
msr, diagnose.ppm, ppm.object, ppm
Examples
fit <- ppm(cells, ~x, Strauss(r=0.15))
# Pearson residuals
rp <- residuals(fit, type="pe")
rp
# simulated data
X <- rStrauss(100,0.7,0.05)
# fit Strauss model
fit <- ppm(X, ~1, Strauss(0.05))
res.fit <- residuals(fit)
# check that total residual is 0
integral.msr(residuals(fit, drop=TRUE))
# true model parameters

rex

1185
truecoef <- c(log(100), log(0.7))
res.true <- residuals(fit, new.coef=truecoef)

rex

Richardson Extrapolation

Description
Performs Richardson Extrapolation on a sequence of approximate values.
Usage
rex(x, r = 2, k = 1, recursive = FALSE)
Arguments
x

A numeric vector or matrix, whose columns are successive estimates or approximations to a vector of parameters.

r

A number greater than 1. The ratio of successive step sizes. See Details.

k

Integer. The order of convergence assumed. See Details.

recursive

Logical value indicating whether to perform one step of Richardson extrapolation (recursive=FALSE, the default) or repeat the extrapolation procedure until
a best estimate is obtained (recursive=TRUE.

Details
Richardson extrapolation is a general technique for improving numerical approximations, often
used in numerical integration (Brezinski and Zaglia, 1991). It can also be used to improve parameter
estimates in statistical models (Baddeley and Turner, 2014).
The successive columns of x are assumed to have been obtained using approximations with step
sizes a, a/r, a/r2 , . . . where a is the initial step size (which does not need to be specified).
Estimates based on a step size s are assumed to have an error of order sk .
Thus, the default values r=2 and k=1 imply that the errors in the second column of x should be
roughly (1/r)k = 1/2 as large as the errors in the first column, and so on.
Value
A matrix whose columns contain a sequence of improved estimates.
Author(s)
Adrian Baddeley  and Rolf Turner .
References
Baddeley, A. and Turner, R. (2014) Bias correction for parameter estimates of spatial point process
models. Journal of Statistical Computation and Simulation 84, 1621–1643. DOI: 10.1080/00949655.2012.755976
Brezinski, C. and Zaglia, M.R. (1991) Extrapolation Methods. Theory and Practice. NorthHolland.

1186

rGaussPoisson

See Also
bc
Examples
# integrals of sin(x) and cos(x) from 0 to pi
# correct answers: 2, 0
est <- function(nsteps) {
xx <- seq(0, pi, length=nsteps)
ans <- pi * c(mean(sin(xx)), mean(cos(xx)))
names(ans) <- c("sin", "cos")
ans
}
X <- cbind(est(10), est(20), est(40))
X
rex(X)
rex(X, recursive=TRUE)
# fitted Gibbs point process model
fit0 <- ppm(cells ~ 1, Strauss(0.07), nd=16)
fit1 <- update(fit0, nd=32)
fit2 <- update(fit0, nd=64)
co <- cbind(coef(fit0), coef(fit1), coef(fit2))
co
rex(co, k=2, recursive=TRUE)

rGaussPoisson

Simulate Gauss-Poisson Process

Description
Generate a random point pattern, a simulated realisation of the Gauss-Poisson Process.
Usage
rGaussPoisson(kappa, r, p2, win = owin(c(0,1),c(0,1)),
..., nsim=1, drop=TRUE)
Arguments
kappa

Intensity of the Poisson process of cluster centres. A single positive number, a
function, or a pixel image.

r

Diameter of each cluster that consists of exactly 2 points.

p2

Probability that a cluster contains exactly 2 points.

win

Window in which to simulate the pattern. An object of class "owin" or something acceptable to as.owin.

...

Ignored.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

rgbim

1187

Details
This algorithm generates a realisation of the Gauss-Poisson point process inside the window win.
The process is constructed by first generating a Poisson point process of parent points with intensity
kappa. Then each parent point is either retained (with probability 1 - p2) or replaced by a pair of
points at a fixed distance r apart (with probability p2). In the case of clusters of 2 points, the line
joining the two points has uniform random orientation.
In this implementation, parent points are not restricted to lie in the window; the parent process is
effectively the uniform Poisson process on the infinite plane.
Value
A point pattern (an object of class "ppp") if nsim=1, or a list of point patterns if nsim > 1.
Additionally, some intermediate results of the simulation are returned as attributes of the point
pattern. See rNeymanScott.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
rpoispp, rThomas, rMatClust, rNeymanScott
Examples
pp <- rGaussPoisson(30, 0.07, 0.5)

rgbim

Create Colour-Valued Pixel Image

Description
Creates an object of class "im" representing a two-dimensional pixel image whose pixel values are
colours.
Usage
rgbim(R, G, B, A, maxColorValue=255, autoscale=FALSE)
hsvim(H, S, V, A, autoscale=FALSE)
Arguments
R,G,B

Pixel images (objects of class "im") or constants giving the red, green, and blue
components of a colour, respectively.

A

Optional. Pixel image or constant value giving the alpha (transparency) component of a colour.

maxColorValue

Maximum colour channel value for R,G,B,A.

H,S,V

Pixel images (objects of class "im") or constants giving the hue, saturation, and
value components of a colour, respectively.

1188

rHardcore

autoscale

Logical. If TRUE, input values are automatically rescaled to fit the permitted
range. RGB values are scaled to lie between 0 and maxColorValue. HSV values
are scaled to lie between 0 and 1.

Details
These functions take three pixel images, with real or integer pixel values, and create a single pixel
image whose pixel values are colours recognisable to R.
Some of the arguments may be constant numeric values, but at least one of the arguments must
be a pixel image. The image arguments should be compatible (in array dimension and in spatial
position).
rgbim calls rgb to compute the colours, while hsvim calls hsv. See the help for the relevant function
for more information about the meaning of the colour channels.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
im.object, rgb, hsv.
See colourtools for additional colour tools.
Examples
# create three images with values in [0,1]
X <- setcov(owin())
X <- eval.im(pmin(1,X))
M <- Window(X)
Y <- as.im(function(x,y){(x+1)/2}, W=M)
Z <- as.im(function(x,y){(y+1)/2}, W=M)
RGB <- rgbim(X, Y, Z, maxColorValue=1)
HSV <- hsvim(X, Y, Z)
plot(RGB, valuesAreColours=TRUE)
plot(HSV, valuesAreColours=TRUE)

rHardcore

Perfect Simulation of the Hardcore Process

Description
Generate a random pattern of points, a simulated realisation of the Hardcore process, using a perfect
simulation algorithm.
Usage
rHardcore(beta, R = 0, W = owin(), expand=TRUE, nsim=1, drop=TRUE)

rHardcore

1189

Arguments
beta

intensity parameter (a positive number).

R

hard core distance (a non-negative number).

W

window (object of class "owin") in which to generate the random pattern. Currently this must be a rectangular window.

expand

Logical. If FALSE, simulation is performed in the window W, which must be
rectangular. If TRUE (the default), simulation is performed on a larger window,
and the result is clipped to the original window W. Alternatively expand can
be an object of class "rmhexpand" (see rmhexpand) determining the expansion
method.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

Details
This function generates a realisation of the Hardcore point process in the window W using a ‘perfect
simulation’ algorithm.
The Hardcore process is a model for strong spatial inhibition. Two points of the process are forbidden to lie closer than R units apart. The Hardcore process is the special case of the Strauss process
(see rStrauss) with interaction parameter γ equal to zero.
The simulation algorithm used to generate the point pattern is ‘dominated coupling from the past’
as implemented by Berthelsen and Møller (2002, 2003). This is a ‘perfect simulation’ or ‘exact
simulation’ algorithm, so called because the output of the algorithm is guaranteed to have the correct probability distribution exactly (unlike the Metropolis-Hastings algorithm used in rmh, whose
output is only approximately correct).
There is a tiny chance that the algorithm will run out of space before it has terminated. If this occurs,
an error message will be generated.
Value
If nsim = 1, a point pattern (object of class "ppp"). If nsim > 1, a list of point patterns.
Author(s)
Adrian Baddeley 
based on original code for the Strauss process by Kasper Klitgaard Berthelsen.
References
Berthelsen, K.K. and Møller, J. (2002) A primer on perfect simulation for spatial point processes.
Bulletin of the Brazilian Mathematical Society 33, 351-367.
Berthelsen, K.K. and Møller, J. (2003) Likelihood and non-parametric Bayesian MCMC inference
for spatial point processes based on perfect simulation and path sampling. Scandinavian Journal of
Statistics 30, 549-564.
Møller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point
Processes. Chapman and Hall/CRC.

1190

rho2hat

See Also
rmh, Hardcore, rStrauss, rStraussHard, rDiggleGratton. rDGS, rPenttinen.
Examples
X <- rHardcore(0.05,1.5,square(141.4))
Z <- rHardcore(100,0.05)

rho2hat

Smoothed Relative Density of Pairs of Covariate Values

Description
Given a point pattern and two spatial covariates Z1 and Z2 , construct a smooth estimate of the
relative risk of the pair (Z1 , Z2 ).
Usage
rho2hat(object, cov1, cov2, ..., method=c("ratio", "reweight"))
Arguments
object
cov1,cov2

...
method

A point pattern (object of class "ppp"), a quadrature scheme (object of class
"quad") or a fitted point process model (object of class "ppm").
The two covariates. Each argument is either a function(x,y) or a pixel image
(object of class "im") providing the values of the covariate at any location, or
one of the strings "x" or "y" signifying the Cartesian coordinates.
Additional arguments passed to density.ppp to smooth the scatterplots.
Character string determining the smoothing method. See Details.

Details
This is a bivariate version of rhohat.
If object is a point pattern, this command produces a smoothed version of the scatterplot of the
values of the covariates cov1 and cov2 observed at the points of the point pattern.
The covariates cov1,cov2 must have continuous values.
If object is a fitted point process model, suppose X is the original data point pattern to which the
model was fitted. Then this command assumes X is a realisation of a Poisson point process with
intensity function of the form
λ(u) = ρ(Z1 (u), Z2 (u))κ(u)
where κ(u) is the intensity of the fitted model object, and ρ(z1 , z2 ) is a function to be estimated.
The algorithm computes a smooth estimate of the function ρ.
The method determines how the density estimates will be combined to obtain an estimate of ρ(z1 , z2 ):
• If method="ratio", then ρ(z1 , z2 ) is estimated by the ratio of two density estimates. The
numerator is a (rescaled) density estimate obtained by smoothing the points (Z1 (yi ), Z2 (yi ))
obtained by evaluating the two covariate Z1 , Z2 at the data points yi . The denominator is a
density estimate of the reference distribution of (Z1 , Z2 ).
• If method="reweight", then ρ(z1 , z2 ) is estimated by applying density estimation to the
points (Z1 (yi ), Z2 (yi )) obtained by evaluating the two covariate Z1 , Z2 at the data points
yi , with weights inversely proportional to the reference density of (Z1 , Z2 ).

rhohat

1191

Value
A pixel image (object of class "im"). Also belongs to the special class "rho2hat" which has a plot
method.
Author(s)
Adrian Baddeley 
References
Baddeley, A., Chang, Y.-M., Song, Y. and Turner, R. (2012) Nonparametric estimation of the dependence of a point process on spatial covariates. Statistics and Its Interface 5 (2), 221–236.
See Also
rhohat, methods.rho2hat
Examples
data(bei)
attach(bei.extra)
plot(rho2hat(bei, elev, grad))
fit <- ppm(bei, ~elev, covariates=bei.extra)
## Not run:
plot(rho2hat(fit, elev, grad))
## End(Not run)
plot(rho2hat(fit, elev, grad, method="reweight"))

rhohat

Smoothing Estimate of Intensity as Function of a Covariate

Description
Computes a smoothing estimate of the intensity of a point process, as a function of a (continuous)
spatial covariate.
Usage
rhohat(object, covariate, ...)
## S3 method for class 'ppp'
rhohat(object, covariate, ...,
baseline=NULL, weights=NULL,
method=c("ratio", "reweight", "transform"),
horvitz=FALSE,
smoother=c("kernel", "local"),
subset=NULL,
dimyx=NULL, eps=NULL,
n = 512, bw = "nrd0", adjust=1, from = NULL, to = NULL,
bwref=bw,
covname, confidence=0.95)

1192

rhohat

## S3 method for class 'quad'
rhohat(object, covariate, ...,
baseline=NULL, weights=NULL,
method=c("ratio", "reweight", "transform"),
horvitz=FALSE,
smoother=c("kernel", "local"),
subset=NULL,
dimyx=NULL, eps=NULL,
n = 512, bw = "nrd0", adjust=1, from = NULL, to = NULL,
bwref=bw,
covname, confidence=0.95)
## S3 method for class 'ppm'
rhohat(object, covariate, ...,
weights=NULL,
method=c("ratio", "reweight", "transform"),
horvitz=FALSE,
smoother=c("kernel", "local"),
subset=NULL,
dimyx=NULL, eps=NULL,
n = 512, bw = "nrd0", adjust=1, from = NULL, to = NULL,
bwref=bw,
covname, confidence=0.95)
## S3 method for class 'lpp'
rhohat(object, covariate, ...,
weights=NULL,
method=c("ratio", "reweight", "transform"),
horvitz=FALSE,
smoother=c("kernel", "local"),
subset=NULL,
nd=1000, eps=NULL, random=TRUE,
n = 512, bw = "nrd0", adjust=1, from = NULL, to = NULL,
bwref=bw,
covname, confidence=0.95)
## S3 method for class 'lppm'
rhohat(object, covariate, ...,
weights=NULL,
method=c("ratio", "reweight", "transform"),
horvitz=FALSE,
smoother=c("kernel", "local"),
subset=NULL,
nd=1000, eps=NULL, random=TRUE,
n = 512, bw = "nrd0", adjust=1, from = NULL, to = NULL,
bwref=bw,
covname, confidence=0.95)
Arguments
object

A point pattern (object of class "ppp" or "lpp"), a quadrature scheme (object of
class "quad") or a fitted point process model (object of class "ppm" or "lppm").

rhohat

1193

covariate

Either a function(x,y) or a pixel image (object of class "im") providing the
values of the covariate at any location. Alternatively one of the strings "x" or
"y" signifying the Cartesian coordinates.

weights

Optional weights attached to the data points. Either a numeric vector of weights
for each data point, or a pixel image (object of class "im") or a function(x,y)
providing the weights.

baseline

Optional baseline for intensity function. A function(x,y) or a pixel image
(object of class "im") providing the values of the baseline at any location.

method

Character string determining the smoothing method. See Details.

horvitz

Logical value indicating whether to use Horvitz-Thompson weights. See Details.

smoother

Character string determining the smoothing algorithm. See Details.

subset

Optional. A spatial window (object of class "owin") specifying a subset of the
data, from which the estimate should be calculated.
dimyx,eps,nd,random
Arguments controlling the pixel resolution at which the covariate will be evaluated. See Details.
bw

Smoothing bandwidth or bandwidth rule (passed to density.default).

adjust

Smoothing bandwidth adjustment factor (passed to density.default).

n, from, to

Arguments passed to density.default to control the number and range of values at which the function will be estimated.

bwref

Optional. An alternative value of bw to use when smoothing the reference density (the density of the covariate values observed at all locations in the window).

...

Additional arguments passed to density.default or locfit.

covname

Optional. Character string to use as the name of the covariate.

confidence

Confidence level for confidence intervals. A number between 0 and 1.

Details
This command estimates the relationship between point process intensity and a given spatial covariate. Such a relationship is sometimes called a resource selection function (if the points are
organisms and the covariate is a descriptor of habitat) or a prospectivity index (if the points are
mineral deposits and the covariate is a geological variable). This command uses a nonparametric
smoothing method which does not assume a particular form for the relationship.
If object is a point pattern, and baseline is missing or null, this command assumes that object
is a realisation of a Poisson point process with intensity function λ(u) of the form
λ(u) = ρ(Z(u))
where Z is the spatial covariate function given by covariate, and ρ(z) is a function to be estimated.
This command computes estimators of ρ(z) proposed by Baddeley and Turner (2005) and Baddeley
et al (2012).
The covariate Z must have continuous values.
If object is a point pattern, and baseline is given, then the intensity function is assumed to be
λ(u) = ρ(Z(u))B(u)
where B(u) is the baseline intensity at location u. A smoothing estimator of the relative intensity
ρ(z) is computed.

1194

rhohat

If object is a fitted point process model, suppose X is the original data point pattern to which the
model was fitted. Then this command assumes X is a realisation of a Poisson point process with
intensity function of the form
λ(u) = ρ(Z(u))κ(u)
where κ(u) is the intensity of the fitted model object. A smoothing estimator of ρ(z) is computed.
The estimation procedure is determined by the character strings method and smoother and the
argument horvitz. The estimation procedure involves computing several density estimates and
combining them. The algorithm used to compute density estimates is determined by smoother:
• If smoother="kernel", each the smoothing procedure is based on fixed-bandwidth kernel
density estimation, performed by density.default.
• If smoother="local", the smoothing procedure is based on local likelihood density estimation, performed by locfit.
The method determines how the density estimates will be combined to obtain an estimate of ρ(z):
• If method="ratio", then ρ(z) is estimated by the ratio of two density estimates. The numerator is a (rescaled) density estimate obtained by smoothing the values Z(yi ) of the covariate
Z observed at the data points yi . The denominator is a density estimate of the reference
distribution of Z.
• If method="reweight", then ρ(z) is estimated by applying density estimation to the values
Z(yi ) of the covariate Z observed at the data points yi , with weights inversely proportional to
the reference density of Z.
• If method="transform", the smoothing method is variable-bandwidth kernel smoothing, implemented by applying the Probability Integral Transform to the covariate values, yielding
values in the range 0 to 1, then applying edge-corrected density estimation on the interval
[0, 1], and back-transforming.
If horvitz=TRUE, then the calculations described above are modified by using Horvitz-Thompson
weighting. The contribution to the numerator from each data point is weighted by the reciprocal
of the baseline value or fitted intensity value at that data point; and a corresponding adjustment is
made to the denominator.
The covariate will be evaluated on a fine grid of locations, with spatial resolution controlled by
the arguments dimyx,eps,nd,random. In two dimensions (i.e. if object is of class "ppp", "ppm"
or "quad") the arguments dimyx, eps are passed to as.mask to control the pixel resolution. On
a linear network (i.e. if object is of class "lpp") the argument nd specifies the total number of
test locations on the linear network, eps specifies the linear separation between test locations, and
random specifies whether the test locations have a randomised starting position.
If the argument weights is present, then the contribution from each data point X[i] to the estimate
of ρ is multiplied by weights[i].
If the argument subset is present, then the calculations are performed using only the data inside
this spatial region.
Value
A function value table (object of class "fv") containing the estimated values of ρ for a sequence of
values of Z. Also belongs to the class "rhohat" which has special methods for print, plot and
predict.

ripras

1195

Categorical and discrete covariates
This technique assumes that the covariate has continuous values. It is not applicable to covariates
with categorical (factor) values or discrete values such as small integers. For a categorical covariate,
use intensity.quadratcount applied to the result of quadratcount(X, tess=covariate).
Author(s)
Adrian Baddeley , Ya-Mei Chang, Yong Song, and Rolf Turner
.
References
Baddeley, A., Chang, Y.-M., Song, Y. and Turner, R. (2012) Nonparametric estimation of the dependence of a point process on spatial covariates. Statistics and Its Interface 5 (2), 221–236.
Baddeley, A. and Turner, R. (2005) Modelling spatial point patterns in R. In: A. Baddeley, P. Gregori, J. Mateu, R. Stoica, and D. Stoyan, editors, Case Studies in Spatial Point Pattern Modelling,
Lecture Notes in Statistics number 185. Pages 23–74. Springer-Verlag, New York, 2006. ISBN:
0-387-28311-0.
See Also
rho2hat, methods.rhohat, parres.
See ppm for a parametric method for the same problem.
Examples
X <- rpoispp(function(x,y){exp(3+3*x)})
rho <- rhohat(X, "x")
rho <- rhohat(X, function(x,y){x})
plot(rho)
curve(exp(3+3*x), lty=3, col=2, add=TRUE)
rhoB <- rhohat(X, "x", method="reweight")
rhoC <- rhohat(X, "x", method="transform")

fit <- ppm(X, ~x)
rr <- rhohat(fit, "y")
# linear network
Y <- runiflpp(30, simplenet)
rhoY <- rhohat(Y, "y")

ripras

Estimate window from points alone

Description
Given an observed pattern of points, computes the Ripley-Rasson estimate of the spatial domain
from which they came.

1196

ripras

Usage
ripras(x, y=NULL, shape="convex", f)
Arguments
x

vector of x coordinates of observed points, or a 2-column matrix giving x,y
coordinates, or a list with components x,y giving coordinates (such as a point
pattern object of class "ppp".)

y

(optional) vector of y coordinates of observed points, if x is a vector.

shape

String indicating the type of window to be estimated: either "convex" or "rectangle".

f

(optional) scaling factor. See Details.

Details
Given an observed pattern of points with coordinates given by x and y, this function computes an
estimate due to Ripley and Rasson (1977) of the spatial domain from which the points came.
The points are assumed to have been generated independently and uniformly distributed inside an
unknown domain D.
If shape="convex" (the default), the domain D is assumed to be a convex set. The maximum likelihood estimate of D is the convex hull of the points (computed by convexhull.xy). Analogously
to the problems of estimating the endpoint of a uniform distribution, the MLE is not optimal. Ripley
and Rasson’s estimator is a rescaled copy of the convex hull, centred at the centroid of the convex
hull. The scaling factor is 1/sqrt(1 − m/n) where n is the number of data points and m the number
of vertices of the convex hull. The scaling factor may be overridden using the argument f.
If shape="rectangle", the domain D is assumed to be a rectangle with sides parallel to the coordinate axes. The maximum likelihood estimate of D is the bounding box of the points (computed
by bounding.box.xy). The Ripley-Rasson estimator is a rescaled copy of the bounding box, with
scaling factor (n + 1)/(n − 1) where n is the number of data points, centred at the centroid of the
bounding box. The scaling factor may be overridden using the argument f.
Value
A window (an object of class "owin").
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Ripley, B.D. and Rasson, J.-P. (1977) Finding the edge of a Poisson forest. Journal of Applied
Probability, 14, 483 – 491.
See Also
owin, as.owin, bounding.box.xy, convexhull.xy

rjitter

1197

Examples
x <- runif(30)
y <- runif(30)
w <- ripras(x,y)
plot(owin(), main="ripras(x,y)")
plot(w, add=TRUE)
points(x,y)
X <- rpoispp(15)
plot(X, main="ripras(X)")
plot(ripras(X), add=TRUE)
# two points insufficient
ripras(c(0,1),c(0,0))
# triangle
ripras(c(0,1,0.5), c(0,0,1))
# three collinear points
ripras(c(0,0,0), c(0,1,2))

rjitter

Random Perturbation of a Point Pattern

Description
Applies independent random displacements to each point in a point pattern.
Usage
rjitter(X, radius, retry=TRUE, giveup = 10000, ..., nsim=1, drop=TRUE)
Arguments
X

A point pattern (object of class "ppp").

radius

Scale of perturbations. A positive numerical value. The displacement vectors
will be uniformly distributed in a circle of this radius. There is a sensible default.

retry

What to do when a perturbed point lies outside the window of the original point
pattern. If retry=FALSE, the point will be lost; if retry=TRUE, the algorithm
will try again.

giveup

Maximum number of unsuccessful attempts.

...

Ignored.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

Details
Each of the points in the point pattern X is subjected to an independent random displacement. The
displacement vectors are uniformly distributed in a circle of radius radius.
If a displaced point lies outside the window, then if retry=FALSE the point will be lost.

1198

rknn

However if retry=TRUE, the algorithm will try again: each time a perturbed point lies outside the
window, the algorithm will reject it and generate another proposed perturbation of the original point,
until one lies inside the window, or until giveup unsuccessful attempts have been made. In the latter
case, any unresolved points will be included without any perturbation. The return value will always
be a point pattern with the same number of points as X.
Value
A point pattern (an object of class "ppp") if nsim=1, or a list of point patterns if nsim > 1, in the
same window as X.
Author(s)
Adrian Baddeley 
and Rolf Turner 
Examples
X <- rsyst(owin(), 10, 10)
Y <- rjitter(X, 0.02)
plot(Y)
Z <- rjitter(X)

rknn

Theoretical Distribution of Nearest Neighbour Distance

Description
Density, distribution function, quantile function and random generation for the random distance to
the kth nearest neighbour in a Poisson point process in d dimensions.
Usage
dknn(x,
pknn(q,
qknn(p,
rknn(n,

k
k
k
k

=
=
=
=

1,
1,
1,
1,

d
d
d
d

=
=
=
=

2,
2,
2,
2,

lambda
lambda
lambda
lambda

=
=
=
=

1)
1)
1)
1)

Arguments
x,q

vector of quantiles.

p

vector of probabilities.

n

number of observations to be generated.

k

order of neighbour.

d

dimension of space.

lambda

intensity of Poisson point process.

rlabel

1199

Details
In a Poisson point process in d-dimensional space, let the random variable R be the distance from a
fixed point to the k-th nearest random point, or the distance from a random point to the k-th nearest
other random point.
Then Rd has a Gamma distribution with shape parameter k and rate λ ∗ α where α is a constant
(equal to the volume of the unit ball in d-dimensional space). See e.g. Cressie (1991, page 61).
These functions support calculation and simulation for the distribution of R.
Value
A numeric vector: dknn returns the probability density, pknn returns cumulative probabilities (distribution function), qknn returns quantiles, and rknn generates random deviates.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Cressie, N.A.C. (1991) Statistics for spatial data. John Wiley and Sons, 1991.
Examples
x <- seq(0, 5, length=20)
densities <- dknn(x, k=3,
cdfvalues <- pknn(x, k=3,
randomvalues <- rknn(100,
deciles <- qknn((1:9)/10,

rlabel

d=2)
d=2)
k=3, d=2)
k=3, d=2)

Random Re-Labelling of Point Pattern

Description
Randomly allocates marks to a point pattern, or permutes the existing marks, or resamples from the
existing marks.
Usage
rlabel(X, labels=marks(X), permute=TRUE)
Arguments
X

Point pattern (object of class "ppp", "lpp", "pp3" or "ppx").

labels

Vector of values from which the new marks will be drawn at random. Defaults
to the vector of existing marks.

permute

Logical value indicating whether to generate new marks by randomly permuting
labels or by drawing a random sample with replacement.

1200

rLGCP

Details
This very simple function allocates random marks to an existing point pattern X. It is useful for
hypothesis testing purposes.
In the simplest case, the command rlabel(X) yields a point pattern obtained from X by randomly
permuting the marks of the points.
If permute=TRUE, then labels should be a vector of length equal to the number of points in X. The
result of rlabel will be a point pattern with locations given by X and marks given by a random
permutation of labels (i.e. a random sample without replacement).
If permute=FALSE, then labels may be a vector of any length. The result of rlabel will be a
point pattern with locations given by X and marks given by a random sample from labels (with
replacement).
Value
A marked point pattern (of the same class as X).
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
marks<- to assign arbitrary marks.
Examples
data(amacrine)
# Randomly permute the marks "on" and "off"
# Result always has 142 "off" and 152 "on"
Y <- rlabel(amacrine)
# randomly allocate marks "on" and "off"
# with probabilities p(off) = 0.48, p(on) = 0.52
Y <- rlabel(amacrine, permute=FALSE)
# randomly allocate marks "A" and "B" with equal probability
data(cells)
Y <- rlabel(cells, labels=factor(c("A", "B")), permute=FALSE)

rLGCP

Simulate Log-Gaussian Cox Process

Description
Generate a random point pattern, a realisation of the log-Gaussian Cox process.

rLGCP

1201

Usage
rLGCP(model="exp", mu = 0, param = NULL,
...,
win=NULL, saveLambda=TRUE, nsim=1, drop=TRUE)
Arguments
model

character string: the short name of a covariance model for the Gaussian random
field. After adding the prefix "RM", the code will search for a function of this
name in the RandomFields package.

mu

mean function of the Gaussian random field. Either a single number, a function(x,y, ...)
or a pixel image (object of class "im").

param

List of parameters for the covariance. Standard arguments are var and scale.

...

Additional parameters for the covariance, or arguments passed to as.mask to
determine the pixel resolution.

win

Window in which to simulate the pattern. An object of class "owin".

saveLambda

Logical. If TRUE (the default) then the simulated random intensity will also be
saved, and returns as an attribute of the point pattern.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

Details
This function generates a realisation of a log-Gaussian Cox process (LGCP). This is a Cox point
process in which the logarithm of the random intensity is a Gaussian random field with mean function µ and covariance function c(r). Conditional on the random intensity, the point process is a
Poisson process with this intensity.
The string model specifies the covariance function of the Gaussian random field, and the parameters
of the covariance are determined by param and ....
To determine the covariance model, the string model is prefixed by "RM", and a function of this
name is sought in the RandomFields package. For a list of available models see RMmodel in the
RandomFields package. For example the Matérn covariance is specified by model="matern",
corresponding to the function RMmatern in the RandomFields package.
Standard variance parameters (for all functions beginning with "RM" in the RandomFields package)
are var for the variance at distance zero, and scale for the scale parameter. Other parameters are
specified in the help files for the individual functions beginning with "RM". For example the help
file for RMmatern states that nu is a parameter for this model.
This algorithm uses the function RFsimulate in the RandomFields package to generate values of
a Gaussian random field, with the specified mean function mu and the covariance specified by the
arguments model and param, on the points of a regular grid. The exponential of this random field
is taken as the intensity of a Poisson point process, and a realisation of the Poisson process is then
generated by the function rpoispp in the spatstat package.
If the simulation window win is missing or NULL, then it defaults to Window(mu) if mu is a pixel
image, and it defaults to the unit square otherwise.
The LGCP model can be fitted to data using kppm.

1202

rlinegrid

Value
A point pattern (object of class "ppp") or a list of point patterns.
Additionally, the simulated intensity function for each point pattern is returned as an attribute
"Lambda" of the point pattern, if saveLambda=TRUE.
Author(s)
Abdollah Jalilian and Rasmus Waagepetersen. Modified by Adrian Baddeley ,
Rolf Turner  and Ege Rubak .
References
Møller, J., Syversveen, A. and Waagepetersen, R. (1998) Log Gaussian Cox Processes. Scandinavian Journal of Statistics 25, 451–482.
See Also
rpoispp, rMatClust, rGaussPoisson, rNeymanScott, lgcp.estK, kppm
Examples
if(require(RandomFields)) {
# homogeneous LGCP with exponential covariance function
X <- rLGCP("exp", 3, var=0.2, scale=.1)
# inhomogeneous LGCP with Gaussian covariance function
m <- as.im(function(x, y){5 - 1.5 * (x - 0.5)^2 + 2 * (y - 0.5)^2}, W=owin())
X <- rLGCP("gauss", m, var=0.15, scale =0.5)
plot(attr(X, "Lambda"))
points(X)
# inhomogeneous LGCP with Matern covariance function
X <- rLGCP("matern", function(x, y){ 1 - 0.4 * x},
var=2, scale=0.7, nu=0.5,
win = owin(c(0, 10), c(0, 10)))
plot(X)
}

rlinegrid

Generate grid of parallel lines with random displacement

Description
Generates a grid of parallel lines, equally spaced, inside the specified window.
Usage
rlinegrid(angle = 45, spacing = 0.1, win = owin())

rlpp

1203

Arguments
angle

Common orientation of the lines, in degrees anticlockwise from the x axis.

spacing

Spacing between successive lines.

win

Window in which to generate the lines. An object of class "owin" or something
acceptable to as.owin.

Details
The grid is randomly displaced from the origin.
Value
A line segment pattern (object of class "psp").
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
psp, rpoisline
Examples
plot(rlinegrid(30, 0.05))

rlpp

Random Points on a Linear Network

Description
Generates n independent random points on a linear network with a specified probability density.
Usage
rlpp(n, f, ..., nsim=1, drop=TRUE)
Arguments
n

Number of random points to generate. A nonnegative integer giving the number
of points, or an integer vector giving the numbers of points of each type.

f

Probability density (not necessarily normalised). A pixel image on a linear network (object of class "linim") or a function on a linear network (object of class
"linfun"). Alternatively, f can be a list of functions or pixel images, giving the
densities of points of each type.

...

Additional arguments passed to f if it is a function or a list of functions.

nsim

Number of simulated realisations to generate.

drop

Logical value indicating what to do when nsim=1. If drop=TRUE (the default),
the result is a point pattern. If drop=FALSE, the result is a list with one entry
which is a point pattern.

1204

rMatClust

Details
The linear network L, on which the points will be generated, is determined by the argument f.
If f is a function, it is converted to a pixel image on the linear network, using any additional function
arguments ....
If n is a single integer and f is a function or pixel image, then independent random points are
generated on L with probability density proportional to f.
If n is an integer vector and f is a list of functions or pixel images, where n and f have the same
length, then independent random points of several types are generated on L, with n[i] points of
type i having probability density proportional to f[[i]].
Value
If nsim = 1 and drop=TRUE, a point pattern on the linear network, i.e.\ an object of class "lpp".
Otherwise, a list of such point patterns.
Author(s)
Adrian Baddeley 
See Also
runiflpp
Examples
g <- function(x, y, seg, tp) { exp(x + 3*y) }
f <- linfun(g, simplenet)
rlpp(20, f)
plot(rlpp(20, f, nsim=3))

rMatClust

Simulate Matern Cluster Process

Description
Generate a random point pattern, a simulated realisation of the Matérn Cluster Process.
Usage
rMatClust(kappa, scale, mu, win = owin(c(0,1),c(0,1)),
nsim=1, drop=TRUE,
saveLambda=FALSE, expand = scale, ...,
poisthresh=1e-6, saveparents=TRUE)

rMatClust

1205

Arguments
kappa

Intensity of the Poisson process of cluster centres. A single positive number, a
function, or a pixel image.

scale

Radius parameter of the clusters.

mu

Mean number of points per cluster (a single positive number) or reference intensity for the cluster points (a function or a pixel image).

win

Window in which to simulate the pattern. An object of class "owin" or something acceptable to as.owin.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

saveLambda

Logical. If TRUE then the random intensity corresponding to the simulated parent
points will also be calculated and saved, and returns as an attribute of the point
pattern.

expand

Numeric. Size of window expansion for generation of parent points. Defaults to
scale which is the cluster radius.

...

Passed to clusterfield to control the image resolution when saveLambda=TRUE.

poisthresh

Numerical threshold below which the model will be treated as a Poisson process.
See Details.

saveparents

Logical value indicating whether to save the locations of the parent points as an
attribute.

Details
This algorithm generates a realisation of Matérn’s cluster process, a special case of the NeymanScott process, inside the window win.
In the simplest case, where kappa and mu are single numbers, the algorithm generates a uniform
Poisson point process of “parent” points with intensity kappa. Then each parent point is replaced
by a random cluster of “offspring” points, the number of points per cluster being Poisson (mu)
distributed, and their positions being placed and uniformly inside a disc of radius scale centred
on the parent point. The resulting point pattern is a realisation of the classical “stationary Matern
cluster process” generated inside the window win. This point process has intensity kappa * mu.
The algorithm can also generate spatially inhomogeneous versions of the Matérn cluster process:
• The parent points can be spatially inhomogeneous. If the argument kappa is a function(x,y)
or a pixel image (object of class "im"), then it is taken as specifying the intensity function of
an inhomogeneous Poisson process that generates the parent points.
• The offspring points can be inhomogeneous. If the argument mu is a function(x,y) or a pixel
image (object of class "im"), then it is interpreted as the reference density for offspring points,
in the sense of Waagepetersen (2007). For a given parent point, the offspring constitute a
Poisson process with intensity function equal to mu/(pi * scale^2) inside the disc of radius
scale centred on the parent point, and zero intensity outside this disc. Equivalently we first
generate, for each parent point, a Poisson (M ) random number of offspring (where M is the
maximum value of mu) placed independently and uniformly in the disc of radius scale centred
on the parent location, and then randomly thin the offspring points, with retention probability
mu/M.
• Both the parent points and the offspring points can be inhomogeneous, as described above.

1206

rMatClust

Note that if kappa is a pixel image, its domain must be larger than the window win. This is because
an offspring point inside win could have its parent point lying outside win. In order to allow this,
the simulation algorithm first expands the original window win by a distance expand and generates
the Poisson process of parent points on this larger window. If kappa is a pixel image, its domain
must contain this larger window.
The intensity of the Matérn cluster process is kappa * mu if either kappa or mu is a single number.
In the general case the intensity is an integral involving kappa, mu and scale.
The Matérn cluster process model with homogeneous parents (i.e. where kappa is a single number)
can be fitted to data using kppm. Currently it is not possible to fit the Matérn cluster process model
with inhomogeneous parents.
If the pair correlation function of the model is very close to that of a Poisson process, deviating by
less than poisthresh, then the model is approximately a Poisson process, and will be simulated as
a Poisson process with intensity kappa * mu, using rpoispp. This avoids computations that would
otherwise require huge amounts of memory.
Value
A point pattern (an object of class "ppp") if nsim=1, or a list of point patterns if nsim > 1.
Additionally, some intermediate results of the simulation are returned as attributes of this point pattern (see rNeymanScott). Furthermore, the simulated intensity function is returned as an attribute
"Lambda", if saveLambda=TRUE.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Matérn, B. (1960) Spatial Variation. Meddelanden från Statens Skogsforskningsinstitut, volume
59, number 5. Statens Skogsforskningsinstitut, Sweden.
Matérn, B. (1986) Spatial Variation. Lecture Notes in Statistics 36, Springer-Verlag, New York.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous NeymanScott processes. Biometrics 63, 252–258.
See Also
rpoispp, rThomas, rCauchy, rVarGamma, rNeymanScott, rGaussPoisson, kppm, clusterfit.
Examples
# homogeneous
X <- rMatClust(10, 0.05, 4)
# inhomogeneous
ff <- function(x,y){ 4 * exp(2 * abs(x) - 1) }
Z <- as.im(ff, owin())
Y <- rMatClust(10, 0.05, Z)
YY <- rMatClust(ff, 0.05, 3)

rMaternI

rMaternI

1207

Simulate Matern Model I

Description
Generate a random point pattern, a simulated realisation of the Matérn Model I inhibition process
model.
Usage
rMaternI(kappa, r, win = owin(c(0,1),c(0,1)), stationary=TRUE, ...,
nsim=1, drop=TRUE)
Arguments
kappa

Intensity of the Poisson process of proposal points. A single positive number.

r

Inhibition distance.

win

Window in which to simulate the pattern. An object of class "owin" or something acceptable to as.owin. Alternatively a higher-dimensional box of class
"box3" or "boxx".

stationary

Logical. Whether to start with a stationary process of proposal points (stationary=TRUE)
or to generate the proposal points only inside the window (stationary=FALSE).

...

Ignored.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

Details
This algorithm generates one or more realisations of Matérn’s Model I inhibition process inside the
window win.
The process is constructed by first generating a uniform Poisson point process of “proposal” points
with intensity kappa. If stationary = TRUE (the default), the proposal points are generated in a
window larger than win that effectively means the proposals are stationary. If stationary=FALSE
then the proposal points are only generated inside the window win.
A proposal point is then deleted if it lies within r units’ distance of another proposal point. Otherwise it is retained.
The retained points constitute Matérn’s Model I.
Value
A point pattern if nsim=1, or a list of point patterns if nsim > 1. Each point pattern is normally an
object of class "ppp", but may be of class "pp3" or "ppx" depending on the window.
Author(s)
Adrian Baddeley 
, Ute Hahn, Rolf Turner 
and Ege Rubak 

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rMaternII

See Also
rpoispp, rMatClust
Examples
X <- rMaternI(20, 0.05)
Y <- rMaternI(20, 0.05, stationary=FALSE)

rMaternII

Simulate Matern Model II

Description
Generate a random point pattern, a simulated realisation of the Matérn Model II inhibition process.
Usage
rMaternII(kappa, r, win = owin(c(0,1),c(0,1)), stationary=TRUE, ...,
nsim=1, drop=TRUE)
Arguments
kappa

Intensity of the Poisson process of proposal points. A single positive number.

r

Inhibition distance.

win

Window in which to simulate the pattern. An object of class "owin" or something acceptable to as.owin. Alternatively a higher-dimensional box of class
"box3" or "boxx".

stationary

Logical. Whether to start with a stationary process of proposal points (stationary=TRUE)
or to generate the proposal points only inside the window (stationary=FALSE).

...

Ignored.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

Details
This algorithm generates one or more realisations of Matérn’s Model II inhibition process inside
the window win.
The process is constructed by first generating a uniform Poisson point process of “proposal” points
with intensity kappa. If stationary = TRUE (the default), the proposal points are generated in a
window larger than win that effectively means the proposals are stationary. If stationary=FALSE
then the proposal points are only generated inside the window win.
Then each proposal point is marked by an “arrival time”, a number uniformly distributed in [0, 1]
independently of other variables.
A proposal point is deleted if it lies within r units’ distance of another proposal point that has an
earlier arrival time. Otherwise it is retained. The retained points constitute Matérn’s Model II.
The difference between Matérn’s Model I and II is the italicised statement above. Model II has a
higher intensity for the same parameter values.

rmh

1209

Value
A point pattern if nsim=1, or a list of point patterns if nsim > 1. Each point pattern is normally an
object of class "ppp", but may be of class "pp3" or "ppx" depending on the window.
Author(s)
Adrian Baddeley 
, Ute Hahn, Rolf Turner 
and Ege Rubak 
See Also
rpoispp, rMatClust, rMaternI
Examples
X <- rMaternII(20, 0.05)
Y <- rMaternII(20, 0.05, stationary=FALSE)

rmh

Simulate point patterns using the Metropolis-Hastings algorithm.

Description
Generic function for running the Metropolis-Hastings algorithm to produce simulated realisations
of a point process model.
Usage
rmh(model, ...)
Arguments
model

The point process model to be simulated.

...

Further arguments controlling the simulation.

Details
The Metropolis-Hastings algorithm can be used to generate simulated realisations from a wide range
of spatial point processes. For caveats, see below.
The function rmh is generic; it has methods rmh.ppm (for objects of class "ppm") and rmh.default
(the default). The actual implementation of the Metropolis-Hastings algorithm is contained in
rmh.default. For details of its use, see rmh.ppm or rmh.default.
[If the model is a Poisson process, then Metropolis-Hastings is not used; the Poisson model is
generated directly using rpoispp or rmpoispp.]
In brief, the Metropolis-Hastings algorithm is a Markov Chain, whose states are spatial point patterns, and whose limiting distribution is the desired point process. After running the algorithm for
a very large number of iterations, we may regard the state of the algorithm as a realisation from the
desired point process.

1210

rmh.default

However, there are difficulties in deciding whether the algorithm has run for “long enough”. The
convergence of the algorithm may indeed be extremely slow. No guarantees of convergence are
given!
While it is fashionable to decry the Metropolis-Hastings algorithm for its poor convergence and
other properties, it has the advantage of being easy to implement for a wide range of models.
Value
A point pattern, in the form of an object of class "ppp". See rmh.default for details.
Warning
As of version 1.22-1 of spatstat a subtle change was made to rmh.default(). We had noticed
that the results produced were sometimes not “scalable” in that two models, differing in effect only
by the units in which distances are measured and starting from the same seed, gave different results.
This was traced to an idiosyncracy of floating point arithmetic. The code of rmh.default() has
been changed so that the results produced by rmh are now scalable. The downside of this is that
code which users previously ran may now give results which are different from what they formerly
were.

In order to recover former behaviour (so that previous results can be reproduced) set spatstat.options(scalable=FALS
See the last example in the help for rmh.default.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
rmh.default
Examples
# See examples in rmh.default and rmh.ppm

rmh.default

Simulate Point Process Models using the Metropolis-Hastings Algorithm.

Description
Generates a random point pattern, simulated from a chosen point process model, using the MetropolisHastings algorithm.
Usage
## Default S3 method:
rmh(model, start=NULL,
control=default.rmhcontrol(model),
...,
nsim=1, drop=TRUE, saveinfo=TRUE,
verbose=TRUE, snoop=FALSE)

rmh.default

1211

Arguments
model

Data specifying the point process model that is to be simulated.

start

Data determining the initial state of the algorithm.

control

Data controlling the iterative behaviour and termination of the algorithm.

...

Further arguments passed to rmhcontrol or to trend functions in model.

nsim

Number of simulated point patterns that should be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a single point pattern.

saveinfo

Logical value indicating whether to save auxiliary information.

verbose

Logical value indicating whether to print progress reports.

snoop

Logical. If TRUE, activate the visual debugger.

Details
This function generates simulated realisations from any of a range of spatial point processes, using
the Metropolis-Hastings algorithm. It is the default method for the generic function rmh.
This function executes a Metropolis-Hastings algorithm with birth, death and shift proposals as
described in Geyer and Møller (1994).
The argument model specifies the point process model to be simulated. It is either a list, or an object
of class "rmhmodel", with the following components:
cif A character string specifying the choice of interpoint interaction for the point process.
par Parameter values for the conditional intensity function.
w (Optional) window in which the pattern is to be generated. An object of class "owin", or data
acceptable to as.owin.
trend Data specifying the spatial trend in the model, if it has a trend. This may be a function, a
pixel image (of class "im"), (or a list of functions or images if the model is multitype).
If the trend is a function or functions, any auxiliary arguments ... to rmh.default will be
passed to these functions, which should be of the form function(x, y, ...).
types List of possible types, for a multitype point process.
For full details of these parameters, see rmhmodel.default.
The argument start determines the initial state of the Metropolis-Hastings algorithm. It is either
NULL, or an object of class "rmhstart", or a list with the following components:
n.start Number of points in the initial point pattern. A single integer, or a vector of integers giving
the numbers of points of each type in a multitype point pattern. Incompatible with x.start.
x.start Initial point pattern configuration. Incompatible with n.start.
x.start may be a point pattern (an object of class "ppp"), or data which can be coerced to
this class by as.ppp, or an object with components x and y, or a two-column matrix. In the
last two cases, the window for the pattern is determined by model$w. In the first two cases,
if model$w is also present, then the final simulated pattern will be clipped to the window
model$w.
For full details of these parameters, see rmhstart.
The third argument control controls the simulation procedure (including conditional simulation),
iterative behaviour, and termination of the Metropolis-Hastings algorithm. It is either NULL, or a
list, or an object of class "rmhcontrol", with components:

1212

rmh.default

p The probability of proposing a “shift” (as opposed to a birth or death) in the Metropolis-Hastings
algorithm.
q The conditional probability of proposing a death (rather than a birth) given that birth/death has
been chosen over shift.
nrep The number of repetitions or iterations to be made by the Metropolis-Hastings algorithm. It
should be large.
expand Either a numerical expansion factor, or a window (object of class "owin"). Indicates that
the process is to be simulated on a larger domain than the original data window w, then clipped
to w when the algorithm has finished.
The default is to expand the simulation window if the model is stationary and non-Poisson
(i.e. it has no trend and the interaction is not Poisson) and not to expand in all other cases.
If the model has a trend, then in order for expansion to be feasible, the trend must be given
either as a function, or an image whose bounding box is large enough to contain the expanded
window.
periodic A logical scalar; if periodic is TRUE we simulate a process on the torus formed by
identifying opposite edges of a rectangular window.
ptypes A vector of probabilities (summing to 1) to be used in assigning a random type to a new
point.
fixall A logical scalar specifying whether to condition on the number of points of each type.
nverb An integer specifying how often “progress reports” (which consist simply of the number of
repetitions completed) should be printed out. If nverb is left at 0, the default, the simulation
proceeds silently.
x.cond If this argument is present, then conditional simulation will be performed, and x.cond
specifies the conditioning points and the type of conditioning.
nsave,nburn If these values are specified, then intermediate states of the simulation algorithm will
be saved every nsave iterations, after an initial burn-in period of nburn iterations.
track Logical flag indicating whether to save the transition history of the simulations.
For full details of these parameters, see rmhcontrol. The control parameters can also be given in
the ... arguments.
Value
A point pattern (an object of class "ppp", see ppp.object) or a list of point patterns.
The returned value has an attribute info containing modified versions of the arguments model,
start, and control which together specify the exact simulation procedure. The info attribute can
be printed (and is printed automatically by summary.ppp). For computational efficiency, the info
attribute can be omitted by setting saveinfo=FALSE.
The value of .Random.seed at the start of the simulations is also saved and returned as an attribute
seed.
If the argument track=TRUE was given (see rmhcontrol), the transition history of the algorithm
is saved, and returned as an attribute history. The transition history is a data frame containing
a factor proposaltype identifying the proposal type (Birth, Death or Shift) and a logical vector accepted indicating whether the proposal was accepted. The data frame also has columns
numerator, denominator which give the numerator and denominator of the Hastings ratio for the
proposal.
If the argument nsave was given (see rmhcontrol), the return value has an attribute saved which
is a list of point patterns, containing the intermediate states of the algorithm.

rmh.default

1213

Conditional Simulation
There are several kinds of conditional simulation.
• Simulation conditional upon the number of points, that is, holding the number of points fixed.
To do this, set control$p (the probability of a shift) equal to 1. The number of points is then
determined by the starting state, which may be specified either by setting start$n.start to
be a scalar, or by setting the initial pattern start$x.start.
• In the case of multitype processes, it is possible to simulate the model conditionally upon the
number of points of each type, i.e. holding the number of points of each type to be fixed. To do
this, set control$p equal to 1 and control$fixall to be TRUE. The number of points is then
determined by the starting state, which may be specified either by setting start$n.start to
be an integer vector, or by setting the initial pattern start$x.start.
• Simulation conditional on the configuration observed in a sub-window, that is, requiring that,
inside a specified sub-window V , the simulated pattern should agree with a specified point
pattern y.To do this, set control$x.cond to equal the specified point pattern y, making sure
that it is an object of class "ppp" and that the window Window(control$x.cond) is the conditioning window V .
• Simulation conditional on the presence of specified points, that is, requiring that the simulated
pattern should include a specified set of points. This is simulation from the Palm distribution
of the point process given a pattern y. To do this, set control$x.cond to be a data.frame
containing the coordinates (and marks, if appropriate) of the specified points.
For further information, see rmhcontrol.
Note that, when we simulate conditionally on the number of points, or conditionally on the number
of points of each type, no expansion of the window is possible.
Visual Debugger
If snoop = TRUE, an interactive debugger is activated. On the current plot device, the debugger
displays the current state of the Metropolis-Hastings algorithm together with the proposed transition
to the next state. Clicking on this graphical display (using the left mouse button) will re-centre the
display at the clicked location. Surrounding this graphical display is an array of boxes representing
different actions. Clicking on one of the action boxes (using the left mouse button) will cause the
action to be performed. Debugger actions include:
• Zooming in or out
• Panning (shifting the field of view) left, right, up or down
• Jumping to the next iteration
• Skipping 10, 100, 1000, 10000 or 100000 iterations
• Jumping to the next Birth proposal (etc)
• Changing the fate of the proposal (i.e. changing whether the proposal is accepted or rejected)
• Dumping the current state and proposal to a file
• Printing detailed information at the terminal
• Exiting the debugger (so that the simulation algorithm continues without further interruption).
Right-clicking the mouse will also cause the debugger to exit.

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rmh.default

Warnings
There is never a guarantee that the Metropolis-Hastings algorithm has converged to its limiting
distribution.
If start$x.start is specified then expand is set equal to 1 and simulation takes place in Window(x.start).
Any specified value for expand is simply ignored.
The presence of both a component w of model and a non-null value for Window(x.start) makes
sense ONLY if w is contained in Window(x.start).
For multitype processes make sure that, even if there is to be no trend corresponding to a particular
type, there is still a component (a NULL component) for that type, in the list.
Other models
In theory, any finite point process model can be simulated using the Metropolis-Hastings algorithm,
provided the conditional intensity is uniformly bounded.
In practice, the list of point process models that can be simulated using rmh.default is limited to
those that have been implemented in the package’s internal C code. More options will be added in
the future.
Note that the lookup conditional intensity function permits the simulation (in theory, to any desired
degree of approximation) of any pairwise interaction process for which the interaction depends only
on the distance between the pair of points.
Reproducible simulations
If the user wants the simulation to be exactly reproducible (e.g. for a figure in a journal article,
where it is useful to have the figure consistent from draft to draft) then the state of the random
number generator should be set before calling rmh.default. This can be done either by calling
set.seed or by assigning a value to .Random.seed. In the examples below, we use set.seed.
If a simulation has been performed and the user now wants to repeat it exactly, the random seed
should be extracted from the simulated point pattern X by seed <- attr(x, "seed"), then assigned to the system random nunber state by .Random.seed <- seed before calling rmh.default.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns.
Australian and New Zealand Journal of Statistics 42, 283 – 322.
Diggle, P. J. (2003) Statistical Analysis of Spatial Point Patterns (2nd ed.) Arnold, London.
Diggle, P.J. and Gratton, R.J. (1984) Monte Carlo methods of inference for implicit statistical models. Journal of the Royal Statistical Society, series B 46, 193 – 212.
Diggle, P.J., Gates, D.J., and Stibbard, A. (1987) A nonparametric estimator for pairwise-interaction
point processes. Biometrika 74, 763 – 770.
Geyer, C.J. and Møller, J. (1994) Simulation procedures and likelihood inference for spatial point
processes. Scandinavian Journal of Statistics 21, 359–373.

rmh.default

1215

Geyer, C.J. (1999) Likelihood Inference for Spatial Point Processes. Chapter 3 in O.E. BarndorffNielsen, W.S. Kendall and M.N.M. Van Lieshout (eds) Stochastic Geometry: Likelihood and Computation, Chapman and Hall / CRC, Monographs on Statistics and Applied Probability, number 80.
Pages 79–140.
See Also
rmh, rmh.ppm, rStrauss, ppp, ppm, AreaInter, BadGey, DiggleGatesStibbard, DiggleGratton,
Fiksel, Geyer, Hardcore, LennardJones, MultiHard, MultiStrauss, MultiStraussHard, PairPiece,
Poisson, Softcore, Strauss, StraussHard, Triplets
Examples
if(interactive()) {
nr
<- 1e5
nv <- 5000
ns <- 200
} else {
nr <- 10
nv <- 5
ns <- 20
oldopt <- spatstat.options()
spatstat.options(expand=1.1)
}
set.seed(961018)
# Strauss process.
mod01 <- list(cif="strauss",par=list(beta=2,gamma=0.2,r=0.7),
w=c(0,10,0,10))
X1.strauss <- rmh(model=mod01,start=list(n.start=ns),
control=list(nrep=nr,nverb=nv))
if(interactive()) plot(X1.strauss)
# Strauss process, conditioning on n = 42:
X2.strauss <- rmh(model=mod01,start=list(n.start=42),
control=list(p=1,nrep=nr,nverb=nv))
# Tracking algorithm progress:
X <- rmh(model=mod01,start=list(n.start=ns),
control=list(nrep=nr, nsave=nr/5, nburn=nr/2, track=TRUE))
History <- attr(X, "history")
Saved <- attr(X, "saved")
head(History)
plot(Saved)
# Hard core process:
mod02 <- list(cif="hardcore",par=list(beta=2,hc=0.7),w=c(0,10,0,10))
X3.hardcore <- rmh(model=mod02,start=list(n.start=ns),
control=list(nrep=nr,nverb=nv))
if(interactive()) plot(X3.hardcore)
# Strauss process equal to pure hardcore:
mod02s <- list(cif="strauss",par=list(beta=2,gamma=0,r=0.7),w=c(0,10,0,10))
X3.strauss <- rmh(model=mod02s,start=list(n.start=ns),
control=list(nrep=nr,nverb=nv))

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rmh.default

# Strauss process in a polygonal window.
x
<- c(0.55,0.68,0.75,0.58,0.39,0.37,0.19,0.26,0.42)
y
<- c(0.20,0.27,0.68,0.99,0.80,0.61,0.45,0.28,0.33)
mod03 <- list(cif="strauss",par=list(beta=2000,gamma=0.6,r=0.07),
w=owin(poly=list(x=x,y=y)))
X4.strauss <- rmh(model=mod03,start=list(n.start=ns),
control=list(nrep=nr,nverb=nv))
if(interactive()) plot(X4.strauss)
# Strauss process in a polygonal window, conditioning on n = 80.
X5.strauss <- rmh(model=mod03,start=list(n.start=ns),
control=list(p=1,nrep=nr,nverb=nv))
# Strauss process, starting off from X4.strauss, but with the
# polygonal window replace by a rectangular one. At the end,
# the generated pattern is clipped to the original polygonal window.
xxx <- X4.strauss
Window(xxx) <- as.owin(c(0,1,0,1))
X6.strauss <- rmh(model=mod03,start=list(x.start=xxx),
control=list(nrep=nr,nverb=nv))
# Strauss with hardcore:
mod04 <- list(cif="straush",par=list(beta=2,gamma=0.2,r=0.7,hc=0.3),
w=c(0,10,0,10))
X1.straush <- rmh(model=mod04,start=list(n.start=ns),
control=list(nrep=nr,nverb=nv))
# Another Strauss with hardcore (with a perhaps surprising result):
mod05 <- list(cif="straush",par=list(beta=80,gamma=0.36,r=45,hc=2.5),
w=c(0,250,0,250))
X2.straush <- rmh(model=mod05,start=list(n.start=ns),
control=list(nrep=nr,nverb=nv))
# Pure hardcore (identical to X3.strauss).
mod06 <- list(cif="straush",par=list(beta=2,gamma=1,r=1,hc=0.7),
w=c(0,10,0,10))
X3.straush <- rmh(model=mod06,start=list(n.start=ns),
control=list(nrep=nr,nverb=nv))
# Soft core:
w
<- c(0,10,0,10)
mod07 <- list(cif="sftcr",par=list(beta=0.8,sigma=0.1,kappa=0.5),
w=c(0,10,0,10))
X.sftcr <- rmh(model=mod07,start=list(n.start=ns),
control=list(nrep=nr,nverb=nv))
if(interactive()) plot(X.sftcr)
# Area-interaction process:
mod42 <- rmhmodel(cif="areaint",par=list(beta=2,eta=1.6,r=0.7),
w=c(0,10,0,10))
X.area <- rmh(model=mod42,start=list(n.start=ns),
control=list(nrep=nr,nverb=nv))
if(interactive()) plot(X.area)
# Triplets process
modtrip <- list(cif="triplets",par=list(beta=2,gamma=0.2,r=0.7),

rmh.default
w=c(0,10,0,10))
X.triplets <- rmh(model=modtrip,
start=list(n.start=ns),
control=list(nrep=nr,nverb=nv))
if(interactive()) plot(X.triplets)
# Multitype Strauss:
beta <- c(0.027,0.008)
gmma <- matrix(c(0.43,0.98,0.98,0.36),2,2)
r
<- matrix(c(45,45,45,45),2,2)
mod08 <- list(cif="straussm",par=list(beta=beta,gamma=gmma,radii=r),
w=c(0,250,0,250))
X1.straussm <- rmh(model=mod08,start=list(n.start=ns),
control=list(ptypes=c(0.75,0.25),nrep=nr,nverb=nv))
if(interactive()) plot(X1.straussm)
# Multitype Strauss conditioning upon the total number
# of points being 80:
X2.straussm <- rmh(model=mod08,start=list(n.start=ns),
control=list(p=1,ptypes=c(0.75,0.25),nrep=nr,
nverb=nv))
# Conditioning upon the number of points of type 1 being 60
# and the number of points of type 2 being 20:
X3.straussm <- rmh(model=mod08,start=list(n.start=c(60,20)),
control=list(fixall=TRUE,p=1,ptypes=c(0.75,0.25),
nrep=nr,nverb=nv))
# Multitype Strauss hardcore:
rhc <- matrix(c(9.1,5.0,5.0,2.5),2,2)
mod09 <- list(cif="straushm",par=list(beta=beta,gamma=gmma,
iradii=r,hradii=rhc),w=c(0,250,0,250))
X.straushm <- rmh(model=mod09,start=list(n.start=ns),
control=list(ptypes=c(0.75,0.25),nrep=nr,nverb=nv))
# Multitype Strauss hardcore with trends for each type:
beta <- c(0.27,0.08)
tr3
<- function(x,y){x <- x/250; y <- y/250;
exp((6*x + 5*y - 18*x^2 + 12*x*y - 9*y^2)/6)
}
# log quadratic trend
tr4
<- function(x,y){x <- x/250; y <- y/250;
exp(-0.6*x+0.5*y)}
# log linear trend
mod10 <- list(cif="straushm",par=list(beta=beta,gamma=gmma,
iradii=r,hradii=rhc),w=c(0,250,0,250),
trend=list(tr3,tr4))
X1.straushm.trend <- rmh(model=mod10,start=list(n.start=ns),
control=list(ptypes=c(0.75,0.25),
nrep=nr,nverb=nv))
if(interactive()) plot(X1.straushm.trend)
# Multitype Strauss hardcore with trends for each type, given as images:
bigwin <- square(250)
i1 <- as.im(tr3, bigwin)
i2 <- as.im(tr4, bigwin)
mod11 <- list(cif="straushm",par=list(beta=beta,gamma=gmma,

1217

1218

rmh.default
iradii=r,hradii=rhc),w=bigwin,
trend=list(i1,i2))
X2.straushm.trend <- rmh(model=mod11,start=list(n.start=ns),
control=list(ptypes=c(0.75,0.25),expand=1,
nrep=nr,nverb=nv))
# Diggle, Gates, and Stibbard:
mod12 <- list(cif="dgs",par=list(beta=3600,rho=0.08),w=c(0,1,0,1))
X.dgs <- rmh(model=mod12,start=list(n.start=ns),
control=list(nrep=nr,nverb=nv))
if(interactive()) plot(X.dgs)
# Diggle-Gratton:
mod13 <- list(cif="diggra",
par=list(beta=1800,kappa=3,delta=0.02,rho=0.04),
w=square(1))
X.diggra <- rmh(model=mod13,start=list(n.start=ns),
control=list(nrep=nr,nverb=nv))
if(interactive()) plot(X.diggra)
# Fiksel:
modFik <- list(cif="fiksel",
par=list(beta=180,r=0.15,hc=0.07,kappa=2,a= -1.0),
w=square(1))
X.fiksel <- rmh(model=modFik,start=list(n.start=ns),
control=list(nrep=nr,nverb=nv))
if(interactive()) plot(X.fiksel)
# Geyer:
mod14 <- list(cif="geyer",par=list(beta=1.25,gamma=1.6,r=0.2,sat=4.5),
w=c(0,10,0,10))
X1.geyer <- rmh(model=mod14,start=list(n.start=ns),
control=list(nrep=nr,nverb=nv))
if(interactive()) plot(X1.geyer)
# Geyer; same as a Strauss process with parameters
# (beta=2.25,gamma=0.16,r=0.7):
mod15 <- list(cif="geyer",par=list(beta=2.25,gamma=0.4,r=0.7,sat=10000),
w=c(0,10,0,10))
X2.geyer <- rmh(model=mod15,start=list(n.start=ns),
control=list(nrep=nr,nverb=nv))
mod16 <- list(cif="geyer",par=list(beta=8.1,gamma=2.2,r=0.08,sat=3))
data(redwood)
X3.geyer <- rmh(model=mod16,start=list(x.start=redwood),
control=list(periodic=TRUE,nrep=nr,nverb=nv))
# Geyer, starting from the redwood data set, simulating
# on a torus, and conditioning on n:
X4.geyer <- rmh(model=mod16,start=list(x.start=redwood),
control=list(p=1,periodic=TRUE,nrep=nr,nverb=nv))
# Lookup (interaction function h_2 from page 76, Diggle (2003)):
r <- seq(from=0,to=0.2,length=101)[-1] # Drop 0.
h <- 20*(r-0.05)
h[r<0.05] <- 0

rmh.default
h[r>0.10] <- 1
mod17 <- list(cif="lookup",par=list(beta=4000,h=h,r=r),w=c(0,1,0,1))
X.lookup <- rmh(model=mod17,start=list(n.start=ns),
control=list(nrep=nr,nverb=nv))
if(interactive()) plot(X.lookup)
# Strauss with trend
tr <- function(x,y){x <- x/250; y <- y/250;
exp((6*x + 5*y - 18*x^2 + 12*x*y - 9*y^2)/6)
}
beta <- 0.3
gmma <- 0.5
r
<- 45
modStr <- list(cif="strauss",par=list(beta=beta,gamma=gmma,r=r),
w=square(250), trend=tr)
X1.strauss.trend <- rmh(model=modStr,start=list(n.start=ns),
control=list(nrep=nr,nverb=nv))
# Baddeley-Geyer
r <- seq(0,0.2,length=8)[-1]
gmma <- c(0.5,0.6,0.7,0.8,0.7,0.6,0.5)
mod18 <- list(cif="badgey",par=list(beta=4000, gamma=gmma,r=r,sat=5),
w=square(1))
X1.badgey <- rmh(model=mod18,start=list(n.start=ns),
control=list(nrep=nr,nverb=nv))
mod19 <- list(cif="badgey",
par=list(beta=4000, gamma=gmma,r=r,sat=1e4),
w=square(1))
set.seed(1329)
X2.badgey <- rmh(model=mod18,start=list(n.start=ns),
control=list(nrep=nr,nverb=nv))
# Check:
h <- ((prod(gmma)/cumprod(c(1,gmma)))[-8])^2
hs <- stepfun(r,c(h,1))
mod20 <- list(cif="lookup",par=list(beta=4000,h=hs),w=square(1))
set.seed(1329)
X.check <- rmh(model=mod20,start=list(n.start=ns),
control=list(nrep=nr,nverb=nv))
# X2.badgey and X.check will be identical.
mod21 <- list(cif="badgey",par=list(beta=300,gamma=c(1,0.4,1),
r=c(0.035,0.07,0.14),sat=5), w=square(1))
X3.badgey <- rmh(model=mod21,start=list(n.start=ns),
control=list(nrep=nr,nverb=nv))
# Same result as Geyer model with beta=300, gamma=0.4, r=0.07,
# sat = 5 (if seeds and control parameters are the same)
# Or more simply:
mod22 <- list(cif="badgey",
par=list(beta=300,gamma=0.4,r=0.07, sat=5),
w=square(1))
X4.badgey <- rmh(model=mod22,start=list(n.start=ns),
control=list(nrep=nr,nverb=nv))
# Same again --- i.e. the BadGey model includes the Geyer model.
# Illustrating scalability.

1219

1220

rmh.ppm
## Not run:
M1 <- rmhmodel(cif="strauss",par=list(beta=60,gamma=0.5,r=0.04),w=owin())
set.seed(496)
X1 <- rmh(model=M1,start=list(n.start=300))
M2 <- rmhmodel(cif="strauss",par=list(beta=0.6,gamma=0.5,r=0.4),
w=owin(c(0,10),c(0,10)))
set.seed(496)
X2 <- rmh(model=M2,start=list(n.start=300))
chk <- affine(X1,mat=diag(c(10,10)))
all.equal(chk,X2,check.attributes=FALSE)
# Under the default spatstat options the foregoing all.equal()
# will yield TRUE. Setting spatstat.options(scalable=FALSE) and
# re-running the code will reveal differences between X1 and X2.

## End(Not run)
if(!interactive()) spatstat.options(oldopt)

rmh.ppm

Simulate from a Fitted Point Process Model

Description
Given a point process model fitted to data, generate a random simulation of the model, using the
Metropolis-Hastings algorithm.
Usage
## S3 method for class 'ppm'
rmh(model, start=NULL,
control=default.rmhcontrol(model, w=w),
...,
w = NULL,
project=TRUE,
nsim=1, drop=TRUE, saveinfo=TRUE,
verbose=TRUE, new.coef=NULL)
Arguments
model

A fitted point process model (object of class "ppm", see ppm.object) which it
is desired to simulate. This fitted model is usually the result of a call to ppm. See
Details below.

start

Data determining the initial state of the Metropolis-Hastings algorithm. See
rmhstart for description of these arguments. Defaults to list(x.start=data.ppm(model))

control

Data controlling the iterative behaviour of the Metropolis-Hastings algorithm.
See rmhcontrol for description of these arguments.

...

Further arguments passed to rmhcontrol, or to rmh.default, or to covariate
functions in the model.

w

Optional. Window in which the simulations should be generated. Default is the
window of the original data.

rmh.ppm

1221

project

Logical flag indicating what to do if the fitted model is invalid (in the sense
that the values of the fitted coefficients do not specify a valid point process). If
project=TRUE the closest valid model will be simulated; if project=FALSE an
error will occur.

nsim

Number of simulated point patterns that should be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a single point pattern.

saveinfo

Logical value indicating whether to save auxiliary information.

verbose

Logical flag indicating whether to print progress reports.

new.coef

New values for the canonical parameters of the model. A numeric vector of the
same length as coef(model).

Details
This function generates simulated realisations from a point process model that has been fitted to
point pattern data. It is a method for the generic function rmh for the class "ppm" of fitted point
process models. To simulate other kinds of point process models, see rmh or rmh.default.
The argument model describes the fitted model. It must be an object of class "ppm" (see ppm.object),
and will typically be the result of a call to the point process model fitting function ppm.
The current implementation enables simulation from any fitted model involving the interactions
AreaInter, DiggleGratton, DiggleGatesStibbard, Geyer, Hardcore, MultiStrauss, MultiStraussHard,
PairPiece, Poisson, Strauss, StraussHard and Softcore, including nonstationary models. See
the examples.
It is also possible to simulate hybrids of several such models. See Hybrid and the examples.
It is possible that the fitted coefficients of a point process model may be “illegal”, i.e. that there may
not exist a mathematically well-defined point process with the given parameter values. For example,
a Strauss process with interaction parameter γ > 1 does not exist, but the model-fitting procedure
used in ppm will sometimes produce values of γ greater than 1. In such cases, if project=FALSE
then an error will occur, while if project=TRUE then rmh.ppm will find the nearest legal model
and simulate this model instead. (The nearest legal model is obtained by projecting the vector of
coefficients onto the set of valid coefficient vectors. The result is usually the Poisson process with
the same fitted intensity.)
The arguments start and control are lists of parameters determining the initial state and the
iterative behaviour, respectively, of the Metropolis-Hastings algorithm.
The argument start is passed directly to rmhstart. See rmhstart for details of the parameters of
the initial state, and their default values.
The argument control is first passed to rmhcontrol. Then if any additional arguments ... are
given, update.rmhcontrol is called to update the parameter values. See rmhcontrol for details of
the iterative behaviour parameters, and default.rmhcontrol for their default values.
Note that if you specify expansion of the simulation window using the parameter expand (so that
the model will be simulated on a window larger than the original data window) then the model must
be capable of extrapolation to this larger window. This is usually not possible for models which
depend on external covariates, because the domain of a covariate image is usually the same as the
domain of the fitted model.
After extracting the relevant information from the fitted model object model, rmh.ppm invokes the
default rmh algorithm rmh.default, unless the model is Poisson. If the model is Poisson then
the Metropolis-Hastings algorithm is not needed, and the model is simulated directly, using one of
rpoispp, rmpoispp, rpoint or rmpoint.

1222

rmh.ppm

See rmh.default for further information about the implementation, or about the Metropolis-Hastings
algorithm.
Value
A point pattern (an object of class "ppp"; see ppp.object) or a list of point patterns.
Warnings
See Warnings in rmh.default.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
simulate.ppm, rmh, rmhmodel, rmhcontrol, default.rmhcontrol, update.rmhcontrol, rmhstart,
rmh.default, ppp.object, ppm,
Interactions: AreaInter, DiggleGratton, DiggleGatesStibbard, Geyer, Hardcore, Hybrid,
MultiStrauss, MultiStraussHard, PairPiece, Poisson, Strauss, StraussHard, Softcore
Examples
live <- interactive()
op <- spatstat.options()
spatstat.options(rmh.nrep=1e5)
Nrep <- 1e5
X <- swedishpines
if(live) plot(X, main="Swedish Pines data")
# Poisson process
fit <- ppm(X, ~1, Poisson())
Xsim <- rmh(fit)
if(live) plot(Xsim, main="simulation from fitted Poisson model")
# Strauss process
fit <- ppm(X, ~1, Strauss(r=7))
Xsim <- rmh(fit)
if(live) plot(Xsim, main="simulation from fitted Strauss model")
## Not run:
# Strauss process simulated on a larger window
# then clipped to original window
Xsim <- rmh(fit, control=list(nrep=Nrep, expand=1.1, periodic=TRUE))
Xsim <- rmh(fit, nrep=Nrep, expand=2, periodic=TRUE)
## End(Not run)
## Not run:
X <- rSSI(0.05, 100)
# piecewise-constant pairwise interaction function
fit <- ppm(X, ~1, PairPiece(seq(0.02, 0.1, by=0.01)))

rmhcontrol

1223

Xsim <- rmh(fit)
## End(Not run)
# marked point pattern
Y <- amacrine
## Not run:
# marked Poisson models
fit <- ppm(Y)
fit <- ppm(Y,~marks)
fit <- ppm(Y,~polynom(x,2))
fit <- ppm(Y,~marks+polynom(x,2))
fit <- ppm(Y,~marks*polynom(x,y,2))
Ysim <- rmh(fit)
## End(Not run)
# multitype Strauss models
MS <- MultiStrauss(radii=matrix(0.07, ncol=2, nrow=2),
types = levels(Y$marks))
## Not run:
fit <- ppm(Y ~marks, MS)
Ysim <- rmh(fit)
## End(Not run)
fit <- ppm(Y ~ marks*polynom(x,y,2), MS)
Ysim <- rmh(fit)
if(live) plot(Ysim, main="simulation from fitted inhomogeneous Multitype Strauss")
spatstat.options(op)
## Not run:
# Hybrid model
fit <- ppm(redwood, ~1, Hybrid(A=Strauss(0.02), B=Geyer(0.1, 2)))
Y <- rmh(fit)
## End(Not run)

rmhcontrol

Set Control Parameters for Metropolis-Hastings Algorithm.

Description
Sets up a list of parameters controlling the iterative behaviour of the Metropolis-Hastings algorithm.
Usage
rmhcontrol(...)
## Default S3 method:
rmhcontrol(..., p=0.9, q=0.5, nrep=5e5,
expand=NULL, periodic=NULL, ptypes=NULL,

1224

rmhcontrol
x.cond=NULL, fixall=FALSE, nverb=0,
nsave=NULL, nburn=nsave, track=FALSE,
pstage=c("block", "start"))

Arguments
...

Arguments passed to methods.

p

Probability of proposing a shift (as against a birth/death).

q

Conditional probability of proposing a death given that a birth or death will be
proposed.

nrep

Total number of steps (proposals) of Metropolis-Hastings algorithm that should
be run.

expand

Simulation window or expansion rule. Either a window (object of class "owin")
or a numerical expansion factor, specifying that simulations are to be performed
in a domain other than the original data window, then clipped to the original data
window. This argument is passed to rmhexpand. A numerical expansion factor
can be in several formats: see rmhexpand.

periodic

Logical value (or NULL) indicating whether to simulate “periodically”, i.e. identifying opposite edges of the rectangular simulation window. A NULL value
means “undecided.”

ptypes

For multitype point processes, the distribution of the mark attached to a new
random point (when a birth is proposed)

x.cond

Conditioning points for conditional simulation.

fixall

(Logical) for multitype point processes, whether to fix the number of points of
each type.

nverb

Progress reports will be printed every nverb iterations

nsave,nburn

If these values are specified, then intermediate states of the simulation algorithm
will be saved every nsave iterations, after an initial burn-in period of nburn
iterations.

track

Logical flag indicating whether to save the transition history of the simulations.

pstage

Character string specifying when to generate proposal points. Either "start"
or "block".

Details
The Metropolis-Hastings algorithm, implemented as rmh, generates simulated realisations of point
process models. The function rmhcontrol sets up a list of parameters which control the iterative
behaviour and termination of the Metropolis-Hastings algorithm, for use in a subsequent call to rmh.
It also checks that the parameters are valid.
(A separate function rmhstart determines the initial state of the algorithm, and rmhmodel determines the model to be simulated.)
The parameters are as follows:
p The probability of proposing a “shift” (as opposed to a birth or death) in the Metropolis-Hastings
algorithm.
If p = 1 then the algorithm only alters existing points, so the number of points never changes,
i.e. we are simulating conditionally upon the number of points. The number of points is
determined by the initial state (specified by rmhstart).

rmhcontrol

1225

If p = 1 and fixall=TRUE and the model is a multitype point process model, then the algorithm only shifts the locations of existing points and does not alter their marks (types). This is
equivalent to simulating conditionally upon the number of points of each type. These numbers
are again specified by the initial state.
If p = 1 then no expansion of the simulation window is allowed (see expand below).
The default value of p can be changed by setting the parameter rmh.p in spatstat.options.
q The conditional probability of proposing a death (rather than a birth) given that a shift is not
proposed. This is of course ignored if p is equal to 1.
The default value of q can be changed by setting the parameter rmh.q in spatstat.options.
nrep The number of repetitions or iterations to be made by the Metropolis-Hastings algorithm. It
should be large.
The default value of nrep can be changed by setting the parameter rmh.nrep in spatstat.options.
expand Either a number or a window (object of class "owin"). Indicates that the process is to
be simulated on a domain other than the original data window w, then clipped to w when the
algorithm has finished. This would often be done in order to approximate the simulation of
a stationary process (Geyer, 1999) or more generally a process existing in the whole plane,
rather than just in the window w.
If expand is a window object, it is taken as the larger domain in which simulation is performed.
If expand is numeric, it is interpreted as an expansion factor or expansion distance for determining the simulation domain from the data window. It should be a named scalar, such as
expand=c(area=2), expand=c(distance=0.1), expand=c(length=1.2). See rmhexpand()
for more details. If the name is omitted, it defaults to area.
Expansion is not permitted if the number of points has been fixed by setting p = 1 or if the
starting configuration has been specified via the argument x.start in rmhstart.
If expand is NULL, this is interpreted to mean “not yet decided”. An expansion rule will be
determined at a later stage, using appropriate defaults. See rmhexpand.
periodic A logical value (or NULL) determining whether to simulate “periodically”. If periodic is
TRUE, and if the simulation window is a rectangle, then the simulation algorithm effectively
identifies opposite edges of the rectangle. Points near the right-hand edge of the rectangle are
deemed to be close to points near the left-hand edge. Periodic simulation usually gives a better
approximation to a stationary point process. For periodic simulation, the simulation window
must be a rectangle. (The simulation window is determined by expand as described above.)
The value NULL means ‘undecided’. The decision is postponed until rmh is called. Depending
on the point process model to be simulated, rmh will then set periodic=TRUE if the simulation window is expanded and the expanded simulation window is rectangular; otherwise
periodic=FALSE.
Note that periodic=TRUE is only permitted when the simulation window (i.e. the expanded
window) is rectangular.
ptypes A vector of probabilities (summing to 1) to be used in assigning a random type to a new
point. Defaults to a vector each of whose entries is 1/nt where nt is the number of types for
the process. Convergence of the simulation algorithm should be improved if ptypes is close
to the relative frequencies of the types which will result from the simulation.
x.cond If this argument is given, then conditional simulation will be performed, and x.cond specifies the location of the fixed points as well as the type of conditioning. It should be either
a point pattern (object of class "ppp") or a list(x,y) or a data.frame. See the section on
Conditional Simulation.
fixall A logical scalar specifying whether to condition on the number of points of each type. Meaningful only if a marked process is being simulated, and if p = 1. A warning message is given
if fixall is set equal to TRUE when it is not meaningful.

1226

rmhcontrol

nverb An integer specifying how often “progress reports” (which consist simply of the number of
repetitions completed) should be printed out. If nverb is left at 0, the default, the simulation
proceeds silently.
nsave,nburn If these integers are given, then the current state of the simulation algorithm (i.e.
the current random point pattern) will be saved every nsave iterations, starting from iteration
nburn.
track Logical flag indicating whether to save the transition history of the simulations (i.e. information specifying what type of proposal was made, and whether it was accepted or rejected,
for each iteration).
pstage Character string specifying the stage of the algorithm at which the randomised proposal
points should be generated. If pstage="start" or if nsave=0, the entire sequence of nrep
random proposal points is generated at the start of the algorithm. This is the original behaviour
of the code, and should be used in order to maintain consistency with older versions of spatstat. If pstage="block" and nsave > 0, then a set of nsave random proposal points will be
generated before each block of nsave iterations. This is much more efficient. The default is
pstage="block".
Value
An object of class "rmhcontrol", which is essentially a list of parameter values for the algorithm.
There is a print method for this class, which prints a sensible description of the parameters chosen.
Conditional Simulation
For a Gibbs point process X, the Metropolis-Hastings algorithm easily accommodates several kinds
of conditional simulation:
conditioning on the total number of points: We fix the total number of points N (X) to be equal
to n. We simulate from the conditional distribution of X given N (X) = n.
conditioning on the number of points of each type: In a multitype point process, where Yj denotes the process of points of type j, we fix the number N (Yj ) of points of type j to be
equal to nj , for j = 1, 2, . . . , m. We simulate from the conditional distribution of X given
N (Yj ) = nj for j = 1, 2, . . . , m.
conditioning on the realisation in a subwindow: We require that the point process X should, within
a specified sub-window V , coincide with a specified point pattern y. We simulate from the
conditional distribution of X given X ∩ V = y.
Palm conditioning: We require that the point process X include a specified list of points y. We
simulate from the point process with probability density g(x) = cf (x ∪ y) where f is the
probability density of the original process X, and c is a normalising constant.
To achieve each of these types of conditioning we do as follows:
conditioning on the total number of points: Set p=1. The number of points is determined by the
initial state of the simulation: see rmhstart.
conditioning on the number of points of each type: Set p=1 and fixall=TRUE. The number of
points of each type is determined by the initial state of the simulation: see rmhstart.
conditioning on the realisation in a subwindow: Set x.cond to be a point pattern (object of class
"ppp"). Its window V=Window(x.cond) becomes the conditioning subwindow V .
Palm conditioning: Set x.cond to be a list(x,y) or data.frame with two columns containing
the coordinates of the points, or a list(x,y,marks) or data.frame with three columns containing the coordinates and marks of the points.
The arguments x.cond, p and fixall can be combined.

rmhexpand

1227

Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Geyer, C.J. (1999) Likelihood Inference for Spatial Point Processes. Chapter 3 in O.E. BarndorffNielsen, W.S. Kendall and M.N.M. Van Lieshout (eds) Stochastic Geometry: Likelihood and Computation, Chapman and Hall / CRC, Monographs on Statistics and Applied Probability, number 80.
Pages 79–140.
See Also
rmh, rmhmodel, rmhstart, rmhexpand, spatstat.options
Examples
# parameters given as named arguments
c1 <- rmhcontrol(p=0.3,periodic=TRUE,nrep=1e6,nverb=1e5)
# parameters given as a list
liz <- list(p=0.9, nrep=1e4)
c2 <- rmhcontrol(liz)
# parameters given in rmhcontrol object
c3 <- rmhcontrol(c1)

rmhexpand

Specify Simulation Window or Expansion Rule

Description
Specify a spatial domain in which point process simulations will be performed. Alternatively, specify a rule which will be used to determine the simulation window.
Usage
rmhexpand(x = NULL, ..., area = NULL, length = NULL, distance = NULL)
Arguments
x

Any kind of data determining the simulation window or the expansion rule. A
window (object of class "owin") specifying the simulation window, a numerical
value specifying an expansion factor or expansion distance, a list containing one
numerical value, an object of class "rmhexpand", or NULL.

...

Ignored.

area

Area expansion factor. Incompatible with other arguments.

length

Length expansion factor. Incompatible with other arguments.

distance

Expansion distance (buffer width). Incompatible with other arguments.

1228

rmhexpand

Details
In the Metropolis-Hastings algorithm rmh for simulating spatial point processes, simulations are
usually carried out on a spatial domain that is larger than the original window of the point process
model, then subsequently clipped to the original window.
The command rmhexpand can be used to specify the simulation window, or to specify a rule which
will later be used to determine the simulation window from data.
The arguments are all incompatible: at most one of them should be given.
If the first argument x is given, it may be any of the following:
• a window (object of class "owin") specifying the simulation window.
• an object of class "rmhexpand" specifying the expansion rule.
• a single numerical value, without attributes. This will be interpreted as the value of the argument area.
• either c(area=v) or list(area=v), where v is a single numeric value. This will be interpreted as the value of the argument area.
• either c(length=v) or list(length=v), where v is a single numeric value. This will be
interpreted as the value of the argument length.
• either c(distance=v) or list(distance=v), where v is a single numeric value. This will be
interpreted as the value of the argument distance.
• NULL, meaning that the expansion rule is not yet determined.
If one of the arguments area, length or distance is given, then the simulation window is determined from the original data window as follows.
area The bounding box of the original data window will be extracted, and the simulation window
will be a scalar dilation of this rectangle. The argument area should be a numerical value,
greater than or equal to 1. It specifies the area expansion factor, i.e. the ratio of the area of the
simulation window to the area of the original point process window’s bounding box.
length The bounding box of the original data window will be extracted, and the simulation window
will be a scalar dilation of this rectangle. The argument length should be a numerical value,
greater than or equal to 1. It specifies the length expansion factor, i.e. the ratio of the width
(height) of the simulation window to the width (height) of the original point process window’s
bounding box.
distance The argument distance should be a numerical value, greater than or equal to 0. It specifies the width of a buffer region around the original data window. If the original data window
is a rectangle, then this window is extended by a margin of width equal to distance around
all sides of the original rectangle. The result is a rectangle. If the original data window is
not a rectangle, then morphological dilation is applied using dilation.owin so that a margin
or buffer of width equal to distance is created around all sides of the original window. The
result is a non-rectangular window, typically of a different shape.
Value
An object of class "rmhexpand" specifying the expansion rule. There is a print method for this
class.

rmhmodel

1229

Undetermined expansion
If expand=NULL, this is interpreted to mean that the expansion rule is “not yet decided”. Expansion
will be decided later, by the simulation algorithm rmh. If the model cannot be expanded (for example
if the covariate data in the model are not available on a larger domain) then expansion will not occur.
If the model can be expanded, then if the point process model has a finite interaction range r, the
default is rmhexpand(distance=2*r), and otherwise rmhexpand(area=2).
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
expand.owin to apply the rule to a window.
will.expand to test whether expansion will occur.
rmh, rmhcontrol for background details.
Examples
rmhexpand()
rmhexpand(2)
rmhexpand(1)
rmhexpand(length=1.5)
rmhexpand(distance=0.1)
rmhexpand(letterR)

rmhmodel

Define Point Process Model for Metropolis-Hastings Simulation.

Description
Builds a description of a point process model for use in simulating the model by the MetropolisHastings algorithm.
Usage
rmhmodel(...)
Arguments
...

Arguments specifying the point process model in some format.

Details
Simulated realisations of many point process models can be generated using the Metropolis-Hastings
algorithm rmh. The algorithm requires the model to be specified in a particular format: an object of
class "rmhmodel".
The function rmhmodel takes a description of a point process model in some other format, and
converts it into an object of class "rmhmodel". It also checks that the parameters of the model are
valid.
The function rmhmodel is generic, with methods for

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rmhmodel.default

fitted point process models: an object of class "ppm", obtained by a call to the model-fitting function ppm. See rmhmodel.ppm.
lists: a list of parameter values in a certain format. See rmhmodel.list.
default: parameter values specified as separate arguments to .... See rmhmodel.default.
Value
An object of class "rmhmodel", which is essentially a list of parameter values for the model.
There is a print method for this class, which prints a sensible description of the model chosen.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Diggle, P. J. (2003) Statistical Analysis of Spatial Point Patterns (2nd ed.) Arnold, London.
Diggle, P.J. and Gratton, R.J. (1984) Monte Carlo methods of inference for implicit statistical models. Journal of the Royal Statistical Society, series B 46, 193 – 212.
Diggle, P.J., Gates, D.J., and Stibbard, A. (1987) A nonparametric estimator for pairwise-interaction
point processes. Biometrika 74, 763 – 770. Scandinavian Journal of Statistics 21, 359–373.
Geyer, C.J. (1999) Likelihood Inference for Spatial Point Processes. Chapter 3 in O.E. BarndorffNielsen, W.S. Kendall and M.N.M. Van Lieshout (eds) Stochastic Geometry: Likelihood and Computation, Chapman and Hall / CRC, Monographs on Statistics and Applied Probability, number 80.
Pages 79–140.
See Also
rmhmodel.ppm, rmhmodel.default, rmhmodel.list, rmh, rmhcontrol, rmhstart, ppm, Strauss,
Softcore, StraussHard, Triplets, MultiStrauss, MultiStraussHard, DiggleGratton, PairPiece

rmhmodel.default

Build Point Process Model for Metropolis-Hastings Simulation.

Description
Builds a description of a point process model for use in simulating the model by the MetropolisHastings algorithm.
Usage
## Default S3 method:
rmhmodel(...,
cif=NULL, par=NULL, w=NULL, trend=NULL, types=NULL)

rmhmodel.default

1231

Arguments
...

Ignored.

cif

Character string specifying the choice of model

par

Parameters of the model

w

Spatial window in which to simulate

trend

Specification of the trend in the model

types

A vector of factor levels defining the possible marks, for a multitype process.

Details
The generic function rmhmodel takes a description of a point process model in some format, and
converts it into an object of class "rmhmodel" so that simulations of the model can be generated
using the Metropolis-Hastings algorithm rmh.
This function rmhmodel.default is the default method. It builds a description of the point process
model from the simple arguments listed.
The argument cif is a character string specifying the choice of interpoint interaction for the point
process. The current options are
’areaint’ Area-interaction process.
’badgey’ Baddeley-Geyer (hybrid Geyer) process.
’dgs’ Diggle, Gates and Stibbard (1987) process
’diggra’ Diggle and Gratton (1984) process
’fiksel’ Fiksel double exponential process (Fiksel, 1984).
’geyer’ Saturation process (Geyer, 1999).
’hardcore’ Hard core process
’lennard’ Lennard-Jones process
’lookup’ General isotropic pairwise interaction process, with the interaction function specified
via a “lookup table”.
’multihard’ Multitype hardcore process
’penttinen’ The Penttinen process
’strauss’ The Strauss process
’straush’ The Strauss process with hard core
’sftcr’ The Softcore process
’straussm’ The multitype Strauss process
’straushm’ Multitype Strauss process with hard core
’triplets’ Triplets process (Geyer, 1999).
It is also possible to specify a hybrid of these interactions in the sense of Baddeley et al (2013). In
this case, cif is a character vector containing names from the list above. For example, cif=c('strauss', 'geyer')
would specify a hybrid of the Strauss and Geyer models.
The argument par supplies parameter values appropriate to the conditional intensity function being
invoked. For the interactions listed above, these parameters are:
areaint: (Area-interaction process.) A named list with components beta,eta,r which are respectively the “base” intensity, the scaled interaction parameter and the interaction radius.

1232

rmhmodel.default

badgey: (Baddeley-Geyer process.) A named list with components beta (the “base” intensity),
gamma (a vector of non-negative interaction parameters), r (a vector of interaction radii, of the
same length as gamma, in increasing order), and sat (the saturation parameter(s); this may be
a scalar, or a vector of the same length as gamma and r; all values should be at least 1). Note
that because of the presence of “saturation” the gamma values are permitted to be larger than
1.
dgs: (Diggle, Gates, and Stibbard process. See Diggle, Gates, and Stibbard (1987)) A named list
with components beta and rho. This process has pairwise interaction function equal to
 
πt
2
e(t) = sin
2ρ
for t < ρ, and equal to 1 for t ≥ ρ.
diggra: (Diggle-Gratton process. See Diggle and Gratton (1984) and Diggle, Gates and Stibbard
(1987).) A named list with components beta, kappa, delta and rho. This process has
pairwise interaction function e(t) equal to 0 for t < δ, equal to


t−δ
ρ−δ

κ

for δ ≤ t < ρ, and equal to 1 for t ≥ ρ. Note that here we use the symbol κ where Diggle,
Gates, and Stibbard use β since we reserve the symbol β for an intensity parameter.
fiksel: (Fiksel double exponential process, see Fiksel (1984)) A named list with components beta,
r, hc, kappa and a. This process has pairwise interaction function e(t) equal to 0 for t < hc,
equal to
exp(a exp(−κt))
for hc ≤ t < r, and equal to 1 for t ≥ r.
geyer: (Geyer’s saturation process. See Geyer (1999).) A named list with components beta,
gamma, r, and sat. The components beta, gamma, r are as for the Strauss model, and sat is the
“saturation” parameter. The model is Geyer’s “saturation” point process model, a modification
of the Strauss process in which we effectively impose an upper limit (sat) on the number of
neighbours which will be counted as close to a given point.
Explicitly, a saturation point process with interaction radius r, saturation threshold s, and
parameters β and γ, is the point process in which each point xi in the pattern X contributes a
factor
βγ min(s,t(xi ,X))
to the probability density of the point pattern, where t(xi , X) denotes the number of “r-close
neighbours” of xi in the pattern X.
If the saturation threshold s is infinite, the Geyer process reduces to a Strauss process with
interaction parameter γ 2 rather than γ.
hardcore: (Hard core process.) A named list with components beta and hc where beta is the
base intensity and hc is the hard core distance. This process has pairwise interaction function
e(t) equal to 1 if t > hc and 0 if t <= hc.
lennard: (Lennard-Jones process.) A named list with components sigma and epsilon, where
sigma is the characteristic diameter and epsilon is the well depth. See LennardJones for
explanation.
multihard: (Multitype hard core process.) A named list with components beta and hradii, where
beta is a vector of base intensities for each type of point, and hradii is a matrix of hard core
radii between each pair of types.

rmhmodel.default

1233

penttinen: (Penttinen process.) A named list with components beta,gamma,r which are respectively the “base” intensity, the pairwise interaction parameter, and the disc radius. Note that
gamma must be less than or equal to 1. See Penttinen for explanation. (Note that there is also
an algorithm for perfect simulation of the Penttinen process, rPenttinen)
strauss: (Strauss process.) A named list with components beta,gamma,r which are respectively
the “base” intensity, the pairwise interaction parameter and the interaction radius. Note that
gamma must be less than or equal to 1. (Note that there is also an algorithm for perfect simulation of the Strauss process, rStrauss)
straush: (Strauss process with hardcore.) A named list with entries beta,gamma,r,hc where
beta, gamma, and r are as for the Strauss process, and hc is the hardcore radius. Of course hc
must be less than r.
sftcr: (Softcore process.) A named list with components beta,sigma,kappa. Again beta is a
“base” intensity. The pairwise interaction between two points u 6= v is
( 
2/κ )
σ
exp −
||u − v||
Note that it is necessary that 0 < κ < 1.
straussm: (Multitype Strauss process.) A named list with components
• beta: A vector of “base” intensities, one for each possible type.
• gamma: A symmetric matrix of interaction parameters, with γij pertaining to the interaction between type i and type j.
• radii: A symmetric matrix of interaction radii, with entries rij pertaining to the interaction between type i and type j.
straushm: (Multitype Strauss process with hardcore.) A named list with components beta and
gamma as for straussm and two “radii” components:
• iradii: the interaction radii
• hradii: the hardcore radii
which are both symmetric matrices of nonnegative numbers. The entries of hradii must be
less than the corresponding entries of iradii.
triplets: (Triplets process.) A named list with components beta,gamma,r which are respectively
the “base” intensity, the triplet interaction parameter and the interaction radius. Note that
gamma must be less than or equal to 1.
lookup: (Arbitrary pairwise interaction process with isotropic interaction.) A named list with
components beta, r, and h, or just with components beta and h.
This model is the pairwise interaction process with an isotropic interaction given by any chosen function H. Each pair of points xi , xj in the point pattern contributes a factor H(d(xi , xj ))
to the probability density, where d denotes distance and H is the pair interaction function.
The component beta is a (positive) scalar which determines the “base” intensity of the process.
In this implementation, H must be a step function. It is specified by the user in one of two
ways.
• as a vector of values: If r is present, then r is assumed to give the locations of jumps in
the function H, while the vector h gives the corresponding values of the function.
Specifically, the interaction function H(t) takes the value h[1] for distances t in the
interval [0, r[1]); takes the value h[i] for distances t in the interval [r[i-1], r[i])
where i = 2, . . . , n; and takes the value 1 for t ≥ r[n]. Here n denotes the length of r.
The components r and h must be numeric vectors of equal length. The r values must be
strictly positive, and sorted in increasing order.

1234

rmhmodel.default
The entries of h must be non-negative. If any entry of h is greater than 1, then the entry
h[1] must be 0 (otherwise the specified process is non-existent).
Greatest efficiency is achieved if the values of r are equally spaced.
[Note: The usage of r and h has changed from the previous usage in spatstat versions
1.4-7 to 1.5-1, in which ascending order was not required, and in which the first entry of
r had to be 0.]
• as a stepfun object: If r is absent, then h must be an object of class "stepfun" specifying
a step function. Such objects are created by stepfun.
The stepfun object h must be right-continuous (which is the default using stepfun.)
The values of the step function must all be nonnegative. The values must all be less than 1
unless the function is identically zero on some initial interval [0, r). The rightmost value
(the value of h(t) for large t) must be equal to 1.
Greatest efficiency is achieved if the jumps (the “knots” of the step function) are equally
spaced.

For a hybrid model, the argument par should be a list, of the same length as cif, such that par[[i]]
is a list of the parameters required for the interaction cif[i]. See the Examples.
The optional argument trend determines the spatial trend in the model, if it has one. It should be a
function or image (or a list of such, if the model is multitype) to provide the value of the trend at an
arbitrary point.
trend given as a function: A trend function may be a function of any number of arguments, but
the first two must be the x, y coordinates of a point. Auxiliary arguments may be passed to
the trend function at the time of simulation, via the ... argument to rmh.
The function must be vectorized. That is, it must be capable of accepting vector valued x and
y arguments. Put another way, it must be capable of calculating the trend value at a number of
points, simultaneously, and should return the vector of corresponding trend values.
trend given as an image: An image (see im.object) provides the trend values at a grid of points
in the observation window and determines the trend value at other points as the value at the
nearest grid point.
Note that the trend or trends must be non-negative; no checking is done for this.
The optional argument w specifies the window in which the pattern is to be generated. If specified,
it must be in a form which can be coerced to an object of class owin by as.owin.
The optional argument types specifies the possible types in a multitype point process. If the model
being simulated is multitype, and types is not specified, then this vector defaults to 1:ntypes
where ntypes is the number of types.
Value
An object of class "rmhmodel", which is essentially a list of parameter values for the model.
There is a print method for this class, which prints a sensible description of the model chosen.
Warnings in Respect of “lookup”
For the lookup cif, the entries of the r component of par must be strictly positive and sorted into
ascending order.
Note that if you specify the lookup pairwise interaction function via stepfun() the arguments x
and y which are passed to stepfun() are slightly different from r and h: length(y) is equal to
1+length(x); the final entry of y must be equal to 1 — i.e. this value is explicitly supplied by the
user rather than getting tacked on internally.
The step function returned by stepfun() must be right continuous (this is the default behaviour of
stepfun()) otherwise an error is given.

rmhmodel.default

1235

Author(s)
Adrian Baddeley  and Rolf Turner 
References
Baddeley, A., Turner, R., Mateu, J. and Bevan, A. (2013) Hybrids of Gibbs point process models
and their implementation. Journal of Statistical Software 55:11, 1–43. http://www.jstatsoft.
org/v55/i11/
Diggle, P. J. (2003) Statistical Analysis of Spatial Point Patterns (2nd ed.) Arnold, London.
Diggle, P.J. and Gratton, R.J. (1984) Monte Carlo methods of inference for implicit statistical models. Journal of the Royal Statistical Society, series B 46, 193 – 212.
Diggle, P.J., Gates, D.J., and Stibbard, A. (1987) A nonparametric estimator for pairwise-interaction
point processes. Biometrika 74, 763 – 770. Scandinavian Journal of Statistics 21, 359–373.
Fiksel, T. (1984) Estimation of parameterized pair potentials of marked and non-marked Gibbsian
point processes. Electronische Informationsverabeitung und Kybernetika 20, 270–278.
Geyer, C.J. (1999) Likelihood Inference for Spatial Point Processes. Chapter 3 in O.E. BarndorffNielsen, W.S. Kendall and M.N.M. Van Lieshout (eds) Stochastic Geometry: Likelihood and Computation, Chapman and Hall / CRC, Monographs on Statistics and Applied Probability, number 80.
Pages 79–140.
See Also
rmh, rmhcontrol, rmhstart, ppm, AreaInter, BadGey, DiggleGatesStibbard, DiggleGratton,
Fiksel, Geyer, Hardcore, Hybrid, LennardJones, MultiStrauss, MultiStraussHard, PairPiece,
Penttinen, Poisson, Softcore, Strauss, StraussHard and Triplets.
Examples
# Strauss process:
mod01 <- rmhmodel(cif="strauss",par=list(beta=2,gamma=0.2,r=0.7),
w=c(0,10,0,10))
# The above could also be simulated using 'rStrauss'
# Strauss with hardcore:
mod04 <- rmhmodel(cif="straush",par=list(beta=2,gamma=0.2,r=0.7,hc=0.3),
w=owin(c(0,10),c(0,5)))
# Hard core:
mod05 <- rmhmodel(cif="hardcore",par=list(beta=2,hc=0.3),
w=square(5))
# Soft core:
w
<- square(10)
mod07 <- rmhmodel(cif="sftcr",
par=list(beta=0.8,sigma=0.1,kappa=0.5),
w=w)
# Area-interaction process:
mod42 <- rmhmodel(cif="areaint",par=list(beta=2,eta=1.6,r=0.7),
w=c(0,10,0,10))
# Baddeley-Geyer process:
mod99 <- rmhmodel(cif="badgey",par=list(beta=0.3,

1236

rmhmodel.default
gamma=c(0.2,1.8,2.4),r=c(0.035,0.07,0.14),sat=5),
w=unit.square())
# Multitype Strauss:
beta <- c(0.027,0.008)
gmma <- matrix(c(0.43,0.98,0.98,0.36),2,2)
r
<- matrix(c(45,45,45,45),2,2)
mod08 <- rmhmodel(cif="straussm",
par=list(beta=beta,gamma=gmma,radii=r),
w=square(250))
# specify types
mod09 <- rmhmodel(cif="straussm",
par=list(beta=beta,gamma=gmma,radii=r),
w=square(250),
types=c("A", "B"))
# Multitype Hardcore:
rhc <- matrix(c(9.1,5.0,5.0,2.5),2,2)
mod08hard <- rmhmodel(cif="multihard",
par=list(beta=beta,hradii=rhc),
w=square(250),
types=c("A", "B"))
# Multitype Strauss hardcore with trends for each type:
beta <- c(0.27,0.08)
ri
<- matrix(c(45,45,45,45),2,2)
rhc <- matrix(c(9.1,5.0,5.0,2.5),2,2)
tr3
<- function(x,y){x <- x/250; y <- y/250;
exp((6*x + 5*y - 18*x^2 + 12*x*y - 9*y^2)/6)
}
# log quadratic trend
tr4
<- function(x,y){x <- x/250; y <- y/250;
exp(-0.6*x+0.5*y)}
# log linear trend
mod10 <- rmhmodel(cif="straushm",par=list(beta=beta,gamma=gmma,
iradii=ri,hradii=rhc),w=c(0,250,0,250),
trend=list(tr3,tr4))
# Triplets process:
mod11 <- rmhmodel(cif="triplets",par=list(beta=2,gamma=0.2,r=0.7),
w=c(0,10,0,10))
# Lookup (interaction function h_2 from page 76, Diggle (2003)):
r <- seq(from=0,to=0.2,length=101)[-1] # Drop 0.
h <- 20*(r-0.05)
h[r<0.05] <- 0
h[r>0.10] <- 1
mod17 <- rmhmodel(cif="lookup",par=list(beta=4000,h=h,r=r),w=c(0,1,0,1))
# hybrid model
modhy <- rmhmodel(cif=c('strauss', 'geyer'),
par=list(list(beta=100,gamma=0.5,r=0.05),
list(beta=1, gamma=0.7,r=0.1, sat=2)),
w=square(1))

rmhmodel.list

rmhmodel.list

1237

Define Point Process Model for Metropolis-Hastings Simulation.

Description
Given a list of parameters, builds a description of a point process model for use in simulating the
model by the Metropolis-Hastings algorithm.
Usage
## S3 method for class 'list'
rmhmodel(model, ...)
Arguments
model

A list of parameters. See Details.

...

Optional list of additional named parameters.

Details
The generic function rmhmodel takes a description of a point process model in some format, and
converts it into an object of class "rmhmodel" so that simulations of the model can be generated
using the Metropolis-Hastings algorithm rmh.
This function rmhmodel.list is the method for lists. The argument model should be a named list
of parameters of the form
list(cif, par, w, trend, types)
where cif and par are required and the others are optional. For details about these components,
see rmhmodel.default.
The subsequent arguments ... (if any) may also have these names, and they will take precedence
over elements of the list model.
Value
An object of class "rmhmodel", which is essentially a validated list of parameter values for the
model.
There is a print method for this class, which prints a sensible description of the model chosen.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Diggle, P. J. (2003) Statistical Analysis of Spatial Point Patterns (2nd ed.) Arnold, London.
Diggle, P.J. and Gratton, R.J. (1984) Monte Carlo methods of inference for implicit statistical models. Journal of the Royal Statistical Society, series B 46, 193 – 212.
Diggle, P.J., Gates, D.J., and Stibbard, A. (1987) A nonparametric estimator for pairwise-interaction
point processes. Biometrika 74, 763 – 770. Scandinavian Journal of Statistics 21, 359–373.

1238

rmhmodel.list

Geyer, C.J. (1999) Likelihood Inference for Spatial Point Processes. Chapter 3 in O.E. BarndorffNielsen, W.S. Kendall and M.N.M. Van Lieshout (eds) Stochastic Geometry: Likelihood and Computation, Chapman and Hall / CRC, Monographs on Statistics and Applied Probability, number 80.
Pages 79–140.
See Also
rmhmodel, rmhmodel.default, rmhmodel.ppm, rmh, rmhcontrol, rmhstart, ppm, Strauss, Softcore,
StraussHard, MultiStrauss, MultiStraussHard, DiggleGratton, PairPiece
Examples
# Strauss process:
mod01 <- list(cif="strauss",par=list(beta=2,gamma=0.2,r=0.7),
w=c(0,10,0,10))
mod01 <- rmhmodel(mod01)
# Strauss with hardcore:
mod04 <- list(cif="straush",par=list(beta=2,gamma=0.2,r=0.7,hc=0.3),
w=owin(c(0,10),c(0,5)))
mod04 <- rmhmodel(mod04)
# Soft core:
w
<- square(10)
mod07 <- list(cif="sftcr",
par=list(beta=0.8,sigma=0.1,kappa=0.5),
w=w)
mod07 <- rmhmodel(mod07)
# Multitype Strauss:
beta <- c(0.027,0.008)
gmma <- matrix(c(0.43,0.98,0.98,0.36),2,2)
r
<- matrix(c(45,45,45,45),2,2)
mod08 <- list(cif="straussm",
par=list(beta=beta,gamma=gmma,radii=r),
w=square(250))
mod08 <- rmhmodel(mod08)
# specify types
mod09 <- rmhmodel(list(cif="straussm",
par=list(beta=beta,gamma=gmma,radii=r),
w=square(250),
types=c("A", "B")))
# Multitype Strauss hardcore with trends for each type:
beta <- c(0.27,0.08)
ri
<- matrix(c(45,45,45,45),2,2)
rhc <- matrix(c(9.1,5.0,5.0,2.5),2,2)
tr3
<- function(x,y){x <- x/250; y <- y/250;
exp((6*x + 5*y - 18*x^2 + 12*x*y - 9*y^2)/6)
}
# log quadratic trend
tr4
<- function(x,y){x <- x/250; y <- y/250;
exp(-0.6*x+0.5*y)}
# log linear trend
mod10 <- list(cif="straushm",par=list(beta=beta,gamma=gmma,
iradii=ri,hradii=rhc),w=c(0,250,0,250),

rmhmodel.ppm

1239

trend=list(tr3,tr4))
mod10 <- rmhmodel(mod10)
# Lookup (interaction function h_2 from page 76, Diggle (2003)):
r <- seq(from=0,to=0.2,length=101)[-1] # Drop 0.
h <- 20*(r-0.05)
h[r<0.05] <- 0
h[r>0.10] <- 1
mod17 <- list(cif="lookup",par=list(beta=4000,h=h,r=r),w=c(0,1,0,1))
mod17 <- rmhmodel(mod17)

rmhmodel.ppm

Interpret Fitted Model for Metropolis-Hastings Simulation.

Description
Converts a fitted point process model into a format that can be used to simulate the model by the
Metropolis-Hastings algorithm.
Usage
## S3 method for class 'ppm'
rmhmodel(model, w, ..., verbose=TRUE, project=TRUE,
control=rmhcontrol(),
new.coef=NULL)
Arguments
model

Fitted point process model (object of class "ppm").

w

Optional. Window in which the simulations should be generated.

...

Ignored.

verbose

Logical flag indicating whether to print progress reports while the model is being
converted.

project

Logical flag indicating what to do if the fitted model does not correspond to a
valid point process. See Details.

control

Parameters determining the iterative behaviour of the simulation algorithm. Passed
to rmhcontrol.

new.coef

New values for the canonical parameters of the model. A numeric vector of the
same length as coef(model).

Details
The generic function rmhmodel takes a description of a point process model in some format, and
converts it into an object of class "rmhmodel" so that simulations of the model can be generated
using the Metropolis-Hastings algorithm rmh.
This function rmhmodel.ppm is the method for the class "ppm" of fitted point process models.
The argument model should be a fitted point process model (object of class "ppm") typically obtained from the model-fitting function ppm. This will be converted into an object of class "rmhmodel".
The optional argument w specifies the window in which the pattern is to be generated. If specified,
it must be in a form which can be coerced to an object of class owin by as.owin.

1240

rmhmodel.ppm

Not all fitted point process models obtained from ppm can be simulated. We have not yet implemented simulation code for the LennardJones and OrdThresh models.
It is also possible that a fitted point process model obtained from ppm may not correspond to a valid
point process. For example a fitted model with the Strauss interpoint interaction may have any
value of the interaction parameter γ; however the Strauss process is not well-defined for γ > 1
(Kelly and Ripley, 1976).
The argument project determines what to do in such cases. If project=FALSE, a fatal error will
occur. If project=TRUE, the fitted model parameters will be adjusted to the nearest values which
do correspond to a valid point process. For example a Strauss process with γ > 1 will be projected
to a Strauss process with γ = 1, equivalent to a Poisson process.
Value
An object of class "rmhmodel", which is essentially a list of parameter values for the model.
There is a print method for this class, which prints a sensible description of the model chosen.
Author(s)
Adrian Baddeley  and Rolf Turner 
References
Diggle, P. J. (2003) Statistical Analysis of Spatial Point Patterns (2nd ed.) Arnold, London.
Diggle, P.J. and Gratton, R.J. (1984) Monte Carlo methods of inference for implicit statistical models. Journal of the Royal Statistical Society, series B 46, 193 – 212.
Geyer, C.J. (1999) Likelihood Inference for Spatial Point Processes. Chapter 3 in O.E. BarndorffNielsen, W.S. Kendall and M.N.M. Van Lieshout (eds) Stochastic Geometry: Likelihood and Computation, Chapman and Hall / CRC, Monographs on Statistics and Applied Probability, number 80.
Pages 79–140.
Kelly, F.P. and Ripley, B.D. (1976) On Strauss’s model for clustering. Biometrika 63, 357–360.
See Also
rmhmodel, rmhmodel.list, rmhmodel.default, rmh, rmhcontrol, rmhstart, ppm, AreaInter,
BadGey, DiggleGatesStibbard, DiggleGratton, Fiksel, Geyer, Hardcore, Hybrid, LennardJones,
MultiStrauss, MultiStraussHard, PairPiece, Penttinen, Poisson, Softcore, Strauss, StraussHard
and Triplets.
Examples
fit1 <- ppm(cells ~1, Strauss(0.07))
mod1 <- rmhmodel(fit1)
fit2 <- ppm(cells ~x, Geyer(0.07, 2))
mod2 <- rmhmodel(fit2)
fit3 <- ppm(cells ~x, Hardcore(0.07))
mod3 <- rmhmodel(fit3)
# Then rmh(mod1), etc

rmhstart

rmhstart

1241

Determine Initial State for Metropolis-Hastings Simulation.

Description
Builds a description of the initial state for the Metropolis-Hastings algorithm.
Usage
rmhstart(start, ...)
## Default S3 method:
rmhstart(start=NULL, ..., n.start=NULL, x.start=NULL)
Arguments
start
...
n.start
x.start

An existing description of the initial state in some format. Incompatible with
the arguments listed below.
There should be no other arguments.
Number of initial points (to be randomly generated). Incompatible with x.start.
Initial point pattern configuration. Incompatible with n.start.

Details
Simulated realisations of many point process models can be generated using the Metropolis-Hastings
algorithm implemented in rmh.
This function rmhstart creates a full description of the initial state of the Metropolis-Hastings algorithm, including possibly the initial state of the random number generator, for use in a subsequent
call to rmh. It also checks that the initial state is valid.
The initial state should be specified either by the first argument start or by the other arguments
n.start, x.start etc.
If start is a list, then it should have components named n.start or x.start, with the same
interpretation as described below.
The arguments are:
n.start The number of “initial” points to be randomly (uniformly) generated in the simulation
window w. Incompatible with x.start.
For a multitype point process, n.start may be a vector (of length equal to the number of
types) giving the number of points of each type to be generated.
If expansion of the simulation window is selected (see the argument expand to rmhcontrol),
then the actual number of starting points in the simulation will be n.start multiplied by the
expansion factor (ratio of the areas of the expanded window and original window).
For faster convergence of the Metropolis-Hastings algorithm, the value of n.start should be
roughly equal to (an educated guess at) the expected number of points for the point process
inside the window.
x.start Initial point pattern configuration. Incompatible with n.start.
x.start may be a point pattern (an object of class ppp), or an object which can be coerced to
this class by as.ppp, or a dataset containing vectors x and y.
If x.start is specified, then expansion of the simulation window (the argument expand of
rmhcontrol) is not permitted.
The parameters n.start and x.start are incompatible.

1242

rMosaicField

Value
An object of class "rmhstart", which is essentially a list of parameters describing the initial point
pattern and (optionally) the initial state of the random number generator.
There is a print method for this class, which prints a sensible description of the initial state.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
rmh, rmhcontrol, rmhmodel
Examples
# 30 random points
a <- rmhstart(n.start=30)
# a particular point pattern
data(cells)
b <- rmhstart(x.start=cells)

rMosaicField

Mosaic Random Field

Description
Generate a realisation of a random field which is piecewise constant on the tiles of a given tessellation.
Usage
rMosaicField(X,
rgen = function(n) { sample(0:1, n, replace = TRUE)},
...,
rgenargs=NULL)
Arguments
X

A tessellation (object of class "tess").

...

Arguments passed to as.mask determining the pixel resolution.

rgen

Function that generates random values for the tiles of the tessellation.

rgenargs

List containing extra arguments that should be passed to rgen (typically specifying parameters of the distribution of the values).

Details
This function generates a realisation of a random field which is piecewise constant on the tiles of
the given tessellation X. The values in each tile are independent and identically distributed.

rMosaicSet

1243

Value
A pixel image (object of class "im").
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
rpoislinetess, rMosaicSet
Examples
X <- rpoislinetess(3)
plot(rMosaicField(X, runif))
plot(rMosaicField(X, runif, dimyx=256))
plot(rMosaicField(X, rnorm, rgenargs=list(mean=10, sd=2)))
plot(rMosaicField(dirichlet(runifpoint(30)), rnorm))

rMosaicSet

Mosaic Random Set

Description
Generate a random set by taking a random selection of tiles of a given tessellation.
Usage
rMosaicSet(X, p=0.5)
Arguments
X

A tessellation (object of class "tess").

p

Probability of including a given tile. A number strictly between 0 and 1.

Details
Given a tessellation X, this function randomly selects some of the tiles of X, including each tile with
probability p independently of the other tiles. The selected tiles are then combined to form a set in
the plane.
One application of this is Switzer’s (1965) example of a random set which has a Markov property.
It is constructed by generating X according to a Poisson line tessellation (see rpoislinetess).
Value
A window (object of class "owin").

1244

rmpoint

Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Switzer, P. A random set process in the plane with a Markovian property. Annals of Mathematical
Statistics 36 (1965) 1859–1863.
See Also
rpoislinetess, rMosaicField
Examples
# Switzer's random set
X <- rpoislinetess(3)
plot(rMosaicSet(X, 0.5), col="green", border=NA)
# another example
plot(rMosaicSet(dirichlet(runifpoint(30)), 0.4))

rmpoint

Generate N Random Multitype Points

Description
Generate a random multitype point pattern with a fixed number of points, or a fixed number of
points of each type.
Usage
rmpoint(n, f=1, fmax=NULL, win=unit.square(),
types, ptypes,
..., giveup=1000, verbose=FALSE,
nsim=1, drop=TRUE)
Arguments
n

Number of marked points to generate. Either a single number specifying the
total number of points, or a vector specifying the number of points of each type.

f

The probability density of the multitype points, usually un-normalised. Either a
constant, a vector, a function f(x,y,m, ...), a pixel image, a list of functions
f(x,y,...) or a list of pixel images.

fmax

An upper bound on the values of f. If missing, this number will be estimated.

win

Window in which to simulate the pattern. Ignored if f is a pixel image or list of
pixel images.

types

All the possible types for the multitype pattern.

ptypes

Optional vector of probabilities for each type.

...

Arguments passed to f if it is a function.

rmpoint
giveup
verbose
nsim
drop

1245
Number of attempts in the rejection method after which the algorithm should
stop trying to generate new points.
Flag indicating whether to report details of performance of the simulation algorithm.
Number of simulated realisations to be generated.
Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

Details
This function generates random multitype point patterns consisting of a fixed number of points.
Three different models are available:
I. Random location and type: If n is a single number and the argument ptypes is missing, then
n independent, identically distributed random multitype points are generated. Their locations
(x[i],y[i]) and types m[i] have joint probability density proportional to f (x, y, m).
II. Random type, and random location given type: If n is a single number and ptypes is given,
then n independent, identically distributed random multitype points are generated. Their types
m[i] have probability distribution ptypes. Given the types, the locations (x[i],y[i]) have
conditional probability density proportional to f (x, y, m).
III. Fixed types, and random location given type: If n is a vector, then we generate n[i] independent, identically distributed random points of type types[i]. For points of type m the
conditional probability density of location (x, y) is proportional to f (x, y, m).
Note that the density f is normalised in different ways in Model I and Models II and III. In Model I
the normalised joint density is g(x, y, m) = f (x, y, m)/Z where
XZ Z
λ(x, y, m)dx dy
Z=
m

while in Models II and III the normalised conditional density is g(x, y | m) = f (x, y, m)/Zm
where
Z Z
Zm =
λ(x, y, m)dx dy.
In Model I, the marginal distribution of types is pm = Zm /Z.
The unnormalised density f may be specified in any of the following ways.
single number: If f is a single number, the conditional density of location given type is uniform.
That is, the points of each type are uniformly distributed. In Model I, the marginal distribution
of types is also uniform (all possible types have equal probability).
vector: If f is a numeric vector, the conditional density of location given type is uniform. That
is, the points of each type are uniformly distributed. In Model I, the marginal distribution of
types is proportional to the vector f. In Model II, the marginal distribution of types is ptypes,
that is, the values in f are ignored. The argument types defaults to names(f), or if that is
null, 1:length(f).
function: If f is a function, it will be called in the form f(x,y,m,...) at spatial location (x,y)
for points of type m. In Model I, the joint probability density of location and type is proportional to f(x,y,m,...). In Models II and III, the conditional probability density of location
(x,y) given type m is proportional to f(x,y,m,...). The function f must work correctly
with vectors x, y and m, returning a vector of function values. (Note that m will be a factor with
levels types.) The value fmax must be given and must be an upper bound on the values of
f(x,y,m,...) for all locations (x, y) inside the window win and all types m. The argument
types must be given.

1246

rmpoint

list of functions: If f is a list of functions, then the functions will be called in the form f[[i]](x,y,...)
at spatial location (x,y) for points of type types[i]. In Model I, the joint probability density
of location and type is proportional to f[[m]](x,y,...). In Models II and III, the conditional probability density of location (x,y) given type m is proportional to f[[m]](x,y,...).
The function f[[i]] must work correctly with vectors x and y, returning a vector of function values. The value fmax must be given and must be an upper bound on the values of
f[[i]](x,y,...) for all locations (x, y) inside the window win. The argument types
defaults to names(f), or if that is null, 1:length(f).
pixel image: If f is a pixel image object of class "im" (see im.object), the unnormalised density
at a location (x,y) for points of any type is equal to the pixel value of f for the pixel nearest
to (x,y). In Model I, the marginal distribution of types is uniform. The argument win is
ignored; the window of the pixel image is used instead. The argument types must be given.
list of pixel images: If f is a list of pixel images, then the image f[[i]] determines the density values of points of type types[i]. The argument win is ignored; the window of the pixel image
is used instead. The argument types defaults to names(f), or if that is null, 1:length(f).
The implementation uses the rejection method. For Model I, rmpoispp is called repeatedly until
n points have been generated. It gives up after giveup calls if there are still fewer than n points.
For Model II, the types are first generated according to ptypes, then the locations of the points of
each type are generated using rpoint. For Model III, the locations of the points of each type are
generated using rpoint.
Value
A point pattern (an object of class "ppp") if nsim=1, or a list of point patterns if nsim > 1.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
ppp.object, owin.object
Examples
abc <- c("a","b","c")
##### Model I
rmpoint(25, types=abc)
rmpoint(25, 1, types=abc)
# 25 points, equal probability for each type, uniformly distributed locations
rmpoint(25, function(x,y,m) {rep(1, length(x))}, types=abc)
# same as above
rmpoint(25, list(function(x,y){rep(1, length(x))},
function(x,y){rep(1, length(x))},
function(x,y){rep(1, length(x))}),
types=abc)
# same as above

rmpoint
rmpoint(25, function(x,y,m) { x }, types=abc)
# 25 points, equal probability for each type,
# locations nonuniform with density proportional to x
rmpoint(25, function(x,y,m) { ifelse(m == "a", 1, x) }, types=abc)
rmpoint(25, list(function(x,y) { rep(1, length(x)) },
function(x,y) { x },
function(x,y) { x }),
types=abc)
# 25 points, UNEQUAL probabilities for each type,
# type "a" points uniformly distributed,
# type "b" and "c" points nonuniformly distributed.
##### Model II
rmpoint(25, 1, types=abc, ptypes=rep(1,3)/3)
rmpoint(25, 1, types=abc, ptypes=rep(1,3))
# 25 points, equal probability for each type,
# uniformly distributed locations
rmpoint(25, function(x,y,m) {rep(1, length(x))}, types=abc, ptypes=rep(1,3))
# same as above
rmpoint(25, list(function(x,y){rep(1, length(x))},
function(x,y){rep(1, length(x))},
function(x,y){rep(1, length(x))}),
types=abc, ptypes=rep(1,3))
# same as above
rmpoint(25, function(x,y,m) { x }, types=abc, ptypes=rep(1,3))
# 25 points, equal probability for each type,
# locations nonuniform with density proportional to x
rmpoint(25, function(x,y,m) { ifelse(m == "a", 1, x) }, types=abc, ptypes=rep(1,3))
# 25 points, EQUAL probabilities for each type,
# type "a" points uniformly distributed,
# type "b" and "c" points nonuniformly distributed.
###### Model III
rmpoint(c(12, 8, 4), 1, types=abc)
# 12 points of type "a",
# 8 points of type "b",
# 4 points of type "c",
# each uniformly distributed
rmpoint(c(12, 8, 4), function(x,y,m) { ifelse(m=="a", 1, x)}, types=abc)
rmpoint(c(12, 8, 4), list(function(x,y) { rep(1, length(x)) },
function(x,y) { x },
function(x,y) { x }),
types=abc)
# 12 points of type "a", uniformly distributed
# 8 points of type "b", nonuniform
# 4 points of type "c", nonuniform
#########

1247

1248

rmpoispp

## Randomising an existing point pattern:
# same numbers of points of each type, uniform random locations (Model III)
rmpoint(table(marks(demopat)), 1, win=Window(demopat))
# same total number of points, distribution of types estimated from X,
# uniform random locations (Model II)
rmpoint(npoints(demopat), 1, types=levels(marks(demopat)), win=Window(demopat),
ptypes=table(marks(demopat)))

rmpoispp

Generate Multitype Poisson Point Pattern

Description
Generate a random point pattern, a realisation of the (homogeneous or inhomogeneous) multitype
Poisson process.
Usage
rmpoispp(lambda, lmax=NULL, win, types, ...,
nsim=1, drop=TRUE, warnwin=!missing(win))
Arguments
lambda

Intensity of the multitype Poisson process. Either a single positive number, a
vector, a function(x,y,m, ...), a pixel image, a list of functions function(x,y, ...),
or a list of pixel images.

lmax

An upper bound for the value of lambda. May be omitted

win

Window in which to simulate the pattern. An object of class "owin" or something acceptable to as.owin. Ignored if lambda is a pixel image or list of images.

types

All the possible types for the multitype pattern.

...

Arguments passed to lambda if it is a function.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

warnwin

Logical value specifying whether to issue a warning when win is ignored.

Details
This function generates a realisation of the marked Poisson point process with intensity lambda.
Note that the intensity function λ(x, y, m) is the average number of points of type m per unit area
near the location (x, y). Thus a marked point process with a constant intensity of 10 and three
possible types will have an average of 30 points per unit area, with 10 points of each type on
average.
The intensity function may be specified in any of the following ways.

rmpoispp

1249

single number: If lambda is a single number, then this algorithm generates a realisation of the
uniform marked Poisson process inside the window win with intensity lambda for each type.
The total intensity of points of all types is lambda * length(types). The argument types
must be given and determines the possible types in the multitype pattern.
vector: If lambda is a numeric vector, then this algorithm generates a realisation of the stationary
marked Poisson process inside the window win with intensity lambda[i] for points of type
types[i]. The total intensity of points of all types is sum(lambda). The argument types
defaults to names(lambda), or if that is null, 1:length(lambda).
function: If lambda is a function, the process has intensity lambda(x,y,m,...) at spatial location
(x,y) for points of type m. The function lambda must work correctly with vectors x, y and m,
returning a vector of function values. (Note that m will be a factor with levels equal to types.)
The value lmax, if present, must be an upper bound on the values of lambda(x,y,m,...) for
all locations (x, y) inside the window win and all types m. The argument types must be
given.
list of functions: If lambda is a list of functions, the process has intensity lambda[[i]](x,y,...)
at spatial location (x,y) for points of type types[i]. The function lambda[[i]] must work
correctly with vectors x and y, returning a vector of function values. The value lmax, if
given, must be an upper bound on the values of lambda(x,y,...) for all locations (x, y)
inside the window win. The argument types defaults to names(lambda), or if that is null,
1:length(lambda).
pixel image: If lambda is a pixel image object of class "im" (see im.object), the intensity at a
location (x,y) for points of any type is equal to the pixel value of lambda for the pixel nearest
to (x,y). The argument win is ignored; the window of the pixel image is used instead. The
argument types must be given.
list of pixel images: If lambda is a list of pixel images, then the image lambda[[i]] determines
the intensity of points of type types[i]. The argument win is ignored; the window of the
pixel image is used instead. The argument types defaults to names(lambda), or if that is null,
1:length(lambda).
If lmax is missing, an approximate upper bound will be calculated.
To generate an inhomogeneous Poisson process the algorithm uses “thinning”: it first generates a
uniform Poisson process of intensity lmax for points of each type m, then randomly deletes or retains
each point independently, with retention probability p(x, y, m) = λ(x, y, m)/lmax.
Value
A point pattern (an object of class "ppp") if nsim=1, or a list of point patterns if nsim > 1. Each
point pattern is multitype (it carries a vector of marks which is a factor).
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
rpoispp for unmarked Poisson point process; rmpoint for a fixed number of random marked
points; ppp.object, owin.object.

1250

rmpoispp

Examples
# uniform bivariate Poisson process with total intensity 100 in unit square
pp <- rmpoispp(50, types=c("a","b"))
# stationary bivariate Poisson process with intensity A = 30, B = 70
pp <- rmpoispp(c(30,70), types=c("A","B"))
pp <- rmpoispp(c(30,70))
# works in any window
data(letterR)
pp <- rmpoispp(c(30,70), win=letterR, types=c("A","B"))
# inhomogeneous lambda(x,y,m)
# note argument 'm' is a factor
lam <- function(x,y,m) { 50 * (x^2 + y^3) * ifelse(m=="A", 2, 1)}
pp <- rmpoispp(lam, win=letterR, types=c("A","B"))
# extra arguments
lam <- function(x,y,m,scal) { scal * (x^2 + y^3) * ifelse(m=="A", 2, 1)}
pp <- rmpoispp(lam, win=letterR, types=c("A","B"), scal=50)
# list of functions lambda[[i]](x,y)
lams <- list(function(x,y){50 * x^2}, function(x,y){20 * abs(y)})
pp <- rmpoispp(lams, win=letterR, types=c("A","B"))
pp <- rmpoispp(lams, win=letterR)
# functions with extra arguments
lams <- list(function(x,y,scal){5 * scal * x^2},
function(x,y, scal){2 * scal * abs(y)})
pp <- rmpoispp(lams, win=letterR, types=c("A","B"), scal=10)
pp <- rmpoispp(lams, win=letterR, scal=10)
# florid example
lams <- list(function(x,y){
100*exp((6*x + 5*y - 18*x^2 + 12*x*y - 9*y^2)/6)
}
# log quadratic trend
,
function(x,y){
100*exp(-0.6*x+0.5*y)
}
# log linear trend
)
X <- rmpoispp(lams, win=unit.square(), types=c("on", "off"))
# pixel image
Z <- as.im(function(x,y){30 * (x^2 + y^3)}, letterR)
pp <- rmpoispp(Z, types=c("A","B"))
# list of pixel images
ZZ <- list(
as.im(function(x,y){20 * (x^2 + y^3)}, letterR),
as.im(function(x,y){40 * (x^3 + y^2)}, letterR))
pp <- rmpoispp(ZZ, types=c("A","B"))
pp <- rmpoispp(ZZ)
# randomising an existing point pattern
rmpoispp(intensity(amacrine), win=Window(amacrine))

rNeymanScott

rNeymanScott

1251

Simulate Neyman-Scott Process

Description
Generate a random point pattern, a realisation of the Neyman-Scott cluster process.
Usage
rNeymanScott(kappa, expand, rcluster, win = owin(c(0,1),c(0,1)),
..., lmax=NULL, nsim=1, drop=TRUE,
nonempty=TRUE, saveparents=TRUE)
Arguments
kappa

Intensity of the Poisson process of cluster centres. A single positive number, a
function, or a pixel image.

expand

Size of the expansion of the simulation window for generating parent points. A
single non-negative number.

rcluster

A function which generates random clusters, or other data specifying the random
cluster mechanism. See Details.

win

Window in which to simulate the pattern. An object of class "owin" or something acceptable to as.owin.

...

Arguments passed to rcluster.

lmax

Optional. Upper bound on the values of kappa when kappa is a function or pixel
image.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

nonempty

Logical. If TRUE (the default), a more efficient algorithm is used, in which parents are generated conditionally on having at least one offspring point. If FALSE,
parents are generated even if they have no offspring. Both choices are valid;
the default is recommended unless you need to simulate all the parent points for
some other purpose.

saveparents

Logical value indicating whether to save the locations of the parent points as an
attribute.

Details
This algorithm generates a realisation of the general Neyman-Scott process, with the cluster mechanism given by the function rcluster.
First, the algorithm generates a Poisson point process of “parent” points with intensity kappa in
an expanded window as explained below. Here kappa may be a single positive number, a function
kappa(x,y), or a pixel image object of class "im" (see im.object). See rpoispp for details.
Second, each parent point is replaced by a random cluster of points. These clusters are combined
together to yield a single point pattern, and the restriction of this pattern to the window win is then
returned as the result of rNeymanScott.

1252

rNeymanScott

The expanded window consists of as.rectangle(win) extended by the amount expand in each
direction. The size of the expansion is saved in the attribute "expand" and may be extracted by
attr(X, "expand") where X is the generated point pattern.
The argument rcluster specifies the cluster mechanism. It may be either:
• A function which will be called to generate each random cluster (the offspring points of each
parent point). The function should expect to be called in the form rcluster(x0,y0,...)
for a parent point at a location (x0,y0). The return value of rcluster should specify the
coordinates of the points in the cluster; it may be a list containing elements x,y, or a point
pattern (object of class "ppp"). If it is a marked point pattern then the result of rNeymanScott
will be a marked point pattern.
• A list(mu, f) where mu specifies the mean number of offspring points in each cluster, and
f generates the random displacements (vectors pointing from the parent to the offspring).
In this case, the number of offspring in a cluster is assumed to have a Poisson distribution,
implying that the Neyman-Scott process is also a Cox process. The first element mu should
be either a single nonnegative number (interpreted as the mean of the Poisson distribution of
cluster size) or a pixel image or a function(x,y) giving a spatially varying mean cluster size
(interpreted in the sense of Waagepetersen, 2007). The second element f should be a function
that will be called once in the form f(n) to generate n independent and identically distributed
displacement vectors (i.e. as if there were a cluster of size n with a parent at the origin (0,0)).
The function should return a point pattern (object of class "ppp") or something acceptable to
xy.coords that specifies the coordinates of n points.
If required, the intermediate stages of the simulation (the parents and the individual clusters) can
also be extracted from the return value of rNeymanScott through the attributes "parents" and
"parentid". The attribute "parents" is the point pattern of parent points. The attribute "parentid"
is an integer vector specifying the parent for each of the points in the simulated pattern.
Neyman-Scott models where kappa is a single number and rcluster = list(mu,f) can be fitted
to data using the function kppm.
Value
A point pattern (an object of class "ppp") if nsim=1, or a list of point patterns if nsim > 1.
Additionally, some intermediate results of the simulation are returned as attributes of this point
pattern: see Details.
Inhomogeneous Neyman-Scott Processes
There are several different ways of specifying a spatially inhomogeneous Neyman-Scott process:
• The point process of parent points can be inhomogeneous. If the argument kappa is a function(x,y)
or a pixel image (object of class "im"), then it is taken as specifying the intensity function of
an inhomogeneous Poisson process according to which the parent points are generated.
• The number of points in a typical cluster can be spatially varying. If the argument rcluster
is a list of two elements mu, f and the first entry mu is a function(x,y) or a pixel image
(object of class "im"), then mu is interpreted as the reference intensity for offspring points,
in the sense of Waagepetersen (2007). For a given parent point, the offspring constitute a
Poisson process with intensity function equal to mu(x, y) * g(x-x0, y-y0) where g is the
probability density of the offspring displacements generated by the function f.
Equivalently, clusters are first generated with a constant expected number of points per cluster:
the constant is mumax, the maximum of mu. Then the offspring are randomly thinned (see
rthin) with spatially-varying retention probabilities given by mu/mumax.

rNeymanScott

1253

• The entire mechanism for generating a cluster can be dependent on the location of the parent
point. If the argument rcluster is a function, then the cluster associated with a parent point
at location (x0,y0) will be generated by calling rcluster(x0, y0, ...). The behaviour of
this function could depend on the location (x0,y0) in any fashion.
Note that if kappa is an image, the spatial domain covered by this image must be large enough
to include the expanded window in which the parent points are to be generated. This requirement
means that win must be small enough so that the expansion of as.rectangle(win) is contained
in the spatial domain of kappa. As a result, one may wind up having to simulate the process in a
window smaller than what is really desired.
In the first two cases, the intensity of the Neyman-Scott process is equal to kappa * mu if at least
one of kappa or mu is a single number, and is otherwise equal to an integral involving kappa, mu
and f.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Neyman, J. and Scott, E.L. (1958) A statistical approach to problems of cosmology. Journal of the
Royal Statistical Society, Series B 20, 1–43.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous NeymanScott processes. Biometrics 63, 252–258.

See Also
rpoispp, rThomas, rGaussPoisson, rMatClust, rCauchy, rVarGamma
Examples
# each cluster consist of 10 points in a disc of radius 0.2
nclust <- function(x0, y0, radius, n) {
return(runifdisc(n, radius, centre=c(x0, y0)))
}
plot(rNeymanScott(10, 0.2, nclust, radius=0.2, n=5))
# multitype Neyman-Scott process (each cluster is a multitype process)
nclust2 <- function(x0, y0, radius, n, types=c("a", "b")) {
X <- runifdisc(n, radius, centre=c(x0, y0))
M <- sample(types, n, replace=TRUE)
marks(X) <- M
return(X)
}
plot(rNeymanScott(15,0.1,nclust2, radius=0.1, n=5))

1254

rnoise

rnoise

Random Pixel Noise

Description
Generate a pixel image whose pixel values are random numbers following a specified probability
distribution.
Usage
rnoise(rgen = runif, w = square(1), ...)
Arguments
rgen

Random generator for the pixel values. A function in the R language.

w

Window (region or pixel raster) in which to generate the image. Any data acceptable to as.mask.

...

Arguments, matched by name, to be passed to rgen to specify the parameters of
the probability distribution, or passed to as.mask to control the pixel resolution.

Details
The argument w could be a window (class "owin"), a pixel image (class "im") or other data. It is
first converted to a binary mask by as.mask using any relevant arguments in ....
Then each pixel inside the window (i.e. with logical value TRUE in the mask) is assigned a random
numerical value by calling the function rgen.
The function rgen would typically be one of the standard random variable generators like runif
(uniformly distributed random values) or rnorm (Gaussian random values). Its first argument n is
the number of values to be generated. Other arguments to rgen must be matched by name.
Value
A pixel image (object of class "im").
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
as.mask, as.im, Distributions.
Examples
plot(rnoise(), main="Uniform noise")
plot(rnoise(rnorm, dimyx=32, mean=2, sd=1),
main="White noise")

roc

1255

roc

Receiver Operating Characteristic

Description
Computes the Receiver Operating Characteristic curve for a point pattern or a fitted point process
model.
Usage
roc(X, ...)
## S3 method for class 'ppp'
roc(X, covariate, ..., high = TRUE)
## S3 method for class 'ppm'
roc(X, ...)
## S3 method for class 'kppm'
roc(X, ...)
## S3 method for class 'lpp'
roc(X, covariate, ..., high = TRUE)
## S3 method for class 'lppm'
roc(X, ...)
Arguments
X

Point pattern (object of class "ppp" or "lpp") or fitted point process model
(object of class "ppm" or "kppm" or "lppm").

covariate

Spatial covariate. Either a function(x,y), a pixel image (object of class "im"),
or one of the strings "x" or "y" indicating the Cartesian coordinates.

...

Arguments passed to as.mask controlling the pixel resolution for calculations.

high

Logical value indicating whether the threshold operation should favour high or
low values of the covariate.

Details
This command computes Receiver Operating Characteristic curve. The area under the ROC is
computed by auc.
For a point pattern X and a covariate Z, the ROC is a plot showing the ability of the covariate to
separate the spatial domain into areas of high and low density of points. For each possible threshold
z, the algorithm calculates the fraction a(z) of area in the study region where the covariate takes
a value greater than z, and the fraction b(z) of data points for which the covariate value is greater
than z. The ROC is a plot of b(z) against a(z) for all thresholds z.
For a fitted point process model, the ROC shows the ability of the fitted model intensity to separate
the spatial domain into areas of high and low density of points. The ROC is not a diagnostic for the
goodness-of-fit of the model (Lobo et al, 2007).

1256

rose

Value
Function value table (object of class "fv") which can be plotted to show the ROC curve.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Lobo, J.M., Jiménez-Valverde, A. and Real, R. (2007) AUC: a misleading measure of the performance of predictive distribution models. Global Ecology and Biogeography 17(2) 145–151.
Nam, B.-H. and D’Agostino, R. (2002) Discrimination index, the area under the ROC curve. Pages
267–279 in Huber-Carol, C., Balakrishnan, N., Nikulin, M.S. and Mesbah, M., Goodness-of-fit tests
and model validity, Birkhäuser, Basel.
See Also
auc
Examples
plot(roc(swedishpines, "x"))
fit <- ppm(swedishpines ~ x+y)
plot(roc(fit))

rose

Rose Diagram

Description
Plots a rose diagram (rose of directions), the analogue of a histogram or density plot for angular
data.
Usage
rose(x, ...)
## Default S3 method:
rose(x, breaks = NULL, ...,
weights=NULL,
nclass = NULL,
unit = c("degree", "radian"),
start=0, clockwise=FALSE,
main)
## S3 method for class 'histogram'
rose(x, ...,
unit = c("degree", "radian"),
start=0, clockwise=FALSE,

rose

1257
main, labels=TRUE, at=NULL, do.plot = TRUE)
## S3 method for class 'density'
rose(x, ...,
unit = c("degree", "radian"),
start=0, clockwise=FALSE,
main, labels=TRUE, at=NULL, do.plot = TRUE)
## S3 method for class 'fv'
rose(x, ...,
unit = c("degree", "radian"),
start=0, clockwise=FALSE,
main, labels=TRUE, at=NULL, do.plot = TRUE)

Arguments
x

Data to be plotted. A numeric vector containing angles, or a histogram object containing a histogram of angular values, or a density object containing a
smooth density estimate for angular data, or an fv object giving a function of an
angular argument.

breaks, nclass Arguments passed to hist to determine the histogram breakpoints.
...

Additional arguments passed to polygon controlling the appearance of the plot
(or passed from rose.default to hist to control the calculation of the histogram).

unit

The unit in which the angles are expressed.

start

The starting direction for measurement of angles, that is, the spatial direction
which corresponds to a measured angle of zero. Either a character string giving
a compass direction ("N" for north, "S" for south, "E" for east, or "W" for west)
or a number giving the angle from the the horizontal (East) axis to the starting direction. For example, if unit="degree" and clockwise=FALSE, then start=90
and start="N" are equivalent. The default is to measure angles anti-clockwise
from the horizontal axis (East direction).

clockwise

Logical value indicating whether angles increase in the clockwise direction (clockwise=TRUE)
or anti-clockwise, counter-clockwise direction (clockwise=FALSE, the default).

weights

Optional vector of numeric weights associated with x.

main

Optional main title for the plot.

labels

Either a logical value indicating whether to plot labels next to the tick marks, or
a vector of labels for the tick marks.

at

Optional vector of angles at which tick marks should be plotted. Set at=numeric(0)
to suppress tick marks.

do.plot

Logical value indicating whether to really perform the plot.

Details
A rose diagram or rose of directions is the analogue of a histogram or bar chart for data which
represent angles in two dimensions. The bars of the bar chart are replaced by circular sectors in the
rose diagram.
The function rose is generic, with a default method for numeric data, and methods for histograms
and function tables.

1258

rotate

If x is a numeric vector, it must contain angular values in the range 0 to 360 (if unit="degree") or
in the range 0 to 2 * pi (if unit="radian"). A histogram of the data will first be computed using
hist. Then the rose diagram of this histogram will be plotted by rose.histogram.
If x is an object of class "histogram" produced by the function hist, representing the histogram of
angular data, then the rose diagram of the densities (rather than the counts) in this histogram object
will be plotted.
If x is an object of class "density" produced by circdensity or density.default, representing
a kernel smoothed density estimate of angular data, then the rose diagram of the density estimate
will be plotted.
If x is a function value table (object of class "fv") then the argument of the function will be interpreted as an angle, and the value of the function will be interpreted as the radius.
By default, angles are interpreted using the mathematical convention where the zero angle is the
horizontal x axis, and angles increase anti-clockwise. Other conventions can be specified using the arguments start and clockwise. Standard compass directions are obtained by setting
unit="degree", start="N" and clockwise=TRUE.
Value
A window (class "owin") containing the plotted region.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
fv, hist, circdensity, density.default.
Examples
ang <- runif(1000, max=360)
rose(ang, col="grey")
rose(ang, col="grey", start="N", clockwise=TRUE)

rotate

Rotate

Description
Applies a rotation to any two-dimensional object, such as a point pattern or a window.
Usage
rotate(X, ...)
Arguments
X
...

Any suitable dataset representing a two-dimensional object, such as a point pattern (object of class "ppp"), or a window (object of class "owin").
Data specifying the rotation.

rotate.im

1259

Details
This is generic. Methods are provided for point patterns (rotate.ppp) and windows (rotate.owin).
Value
Another object of the same type, representing the result of rotating X through the specified angle.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
rotate.ppp, rotate.owin

rotate.im

Rotate a Pixel Image

Description
Rotates a pixel image
Usage
## S3 method for class 'im'
rotate(X, angle=pi/2, ..., centre=NULL)
Arguments
X

A pixel image (object of class "im").

angle

Angle of rotation, in radians.

...

Ignored.

centre

Centre of rotation. Either a vector of length 2, or a character string (partially
matched to "centroid", "midpoint" or "bottomleft"). The default is the
coordinate origin c(0,0).

Details
The image is rotated by the angle specified. Angles are measured in radians, anticlockwise. The
default is to rotate the image 90 degrees anticlockwise.
Value
Another object of class "im" representing the rotated pixel image.
Author(s)
Adrian Baddeley 
and Rolf Turner 

1260

rotate.infline

See Also
affine.im, shift.im, rotate
Examples
Z <- distmap(letterR)
X <- rotate(Z)
## Not run:
plot(X)
## End(Not run)
Y <- rotate(X, centre="midpoint")

rotate.infline

Rotate or Shift Infinite Lines

Description
Given the coordinates of one or more infinite straight lines in the plane, apply a rotation or shift.
Usage
## S3 method for class 'infline'
rotate(X, angle = pi/2, ...)
## S3 method for class 'infline'
shift(X, vec = c(0,0), ...)
## S3 method for class 'infline'
reflect(X)
## S3 method for class 'infline'
flipxy(X)
Arguments
X

Object of class "infline" representing one or more infinite straight lines in the
plane.

angle

Angle of rotation, in radians.

vec

Translation (shift) vector: a numeric vector of length 2, or a list(x,y), or a
point pattern containing one point.

...

Ignored.

Details
These functions are methods for the generic shift, rotate, reflect and flipxy for the class
"infline".
An object of class "infline" represents one or more infinite lines in the plane.

rotate.owin

1261

Value
Another "infline" object representing the result of the transformation.
Author(s)
Adrian Baddeley .
See Also
infline
Examples
L <- infline(v=0.5)
plot(square(c(-1,1)), main="rotate lines", type="n")
points(0, 0, pch=3)
plot(L, col="green")
plot(rotate(L, pi/12), col="red")
plot(rotate(L, pi/6), col="red")
plot(rotate(L, pi/4), col="red")
L <- infline(p=c(0.4, 0.9), theta=pi* c(0.2, 0.6))
plot(square(c(-1,1)), main="shift lines", type="n")
L <- infline(p=c(0.7, 0.8), theta=pi* c(0.2, 0.6))
plot(L, col="green")
plot(shift(L, c(-0.5, -0.4)), col="red")
plot(square(c(-1,1)), main="reflect lines", type="n")
points(0, 0, pch=3)
L <- infline(p=c(0.7, 0.8), theta=pi* c(0.2, 0.6))
plot(L, col="green")
plot(reflect(L), col="red")

rotate.owin

Rotate a Window

Description
Rotates a window
Usage
## S3 method for class 'owin'
rotate(X, angle=pi/2, ..., rescue=TRUE, centre=NULL)

1262

rotate.ppp

Arguments
X

A window (object of class "owin").

angle

Angle of rotation.

rescue

Logical. If TRUE, the rotated window will be processed by rescue.rectangle.

...

Optional arguments passed to as.mask controlling the resolution of the rotated
window, if X is a binary pixel mask. Ignored if X is not a binary mask.

centre

Centre of rotation. Either a vector of length 2, or a character string (partially
matched to "centroid", "midpoint" or "bottomleft"). The default is the
coordinate origin c(0,0).

Details
Rotates the window by the specified angle. Angles are measured in radians, anticlockwise. The
default is to rotate the window 90 degrees anticlockwise. The centre of rotation is the origin, by
default, unless centre is specified.
Value
Another object of class "owin" representing the rotated window.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
owin.object
Examples
w <- owin(c(0,1),c(0,1))
v <- rotate(w, pi/3)
e <- rotate(w, pi/2, centre="midpoint")
## Not run:
plot(v)
## End(Not run)
w <- as.mask(letterR)
v <- rotate(w, pi/5)

rotate.ppp

Rotate a Point Pattern

Description
Rotates a point pattern
Usage
## S3 method for class 'ppp'
rotate(X, angle=pi/2, ..., centre=NULL)

rotate.psp

1263

Arguments
X

A point pattern (object of class "ppp").

angle

Angle of rotation.

...

Arguments passed to rotate.owin affecting the handling of the observation
window, if it is a binary pixel mask.

centre

Centre of rotation. Either a vector of length 2, or a character string (partially
matched to "centroid", "midpoint" or "bottomleft"). The default is the
coordinate origin c(0,0).

Details
The points of the pattern, and the window of observation, are rotated about the origin by the angle
specified. Angles are measured in radians, anticlockwise. The default is to rotate the pattern 90
degrees anticlockwise. If the points carry marks, these are preserved.
Value
Another object of class "ppp" representing the rotated point pattern.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
ppp.object, rotate.owin
Examples
data(cells)
X <- rotate(cells, pi/3)
## Not run:
plot(X)
## End(Not run)

rotate.psp

Rotate a Line Segment Pattern

Description
Rotates a line segment pattern
Usage
## S3 method for class 'psp'
rotate(X, angle=pi/2, ..., centre=NULL)

1264

rotmean

Arguments
X

A line segment pattern (object of class "psp").

angle

Angle of rotation.

...

Arguments passed to rotate.owin affecting the handling of the observation
window, if it is a binary pixel mask.

centre

Centre of rotation. Either a vector of length 2, or a character string (partially
matched to "centroid", "midpoint" or "bottomleft"). The default is the
coordinate origin c(0,0).

Details
The line segments of the pattern, and the window of observation, are rotated about the origin by the
angle specified. Angles are measured in radians, anticlockwise. The default is to rotate the pattern
90 degrees anticlockwise. If the line segments carry marks, these are preserved.
Value
Another object of class "psp" representing the rotated line segment pattern.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
psp.object, rotate.owin, rotate.ppp
Examples
oldpar <- par(mfrow=c(2,1))
X <- psp(runif(10), runif(10), runif(10), runif(10), window=owin())
plot(X, main="original")
Y <- rotate(X, pi/4)
plot(Y, main="rotated")
par(oldpar)

rotmean

Rotational Average of a Pixel Image

Description
Compute the average pixel value over all rotations of the image about the origin, as a function of
distance from the origin.
Usage
rotmean(X, ..., origin, padzero=TRUE, Xname, result=c("fv", "im"))

rotmean

1265

Arguments
X

A pixel image.

...

Ignored.

origin

Optional. Origin about which the rotations should be performed. Either a numeric vector or a character string as described in the help for shift.owin.

padzero

Logical. If TRUE (the default), the value of X is assumed to be zero outside
the window of X. If FALSE, the value of X is taken to be undefined outside the
window of X.

Xname

Optional name for X to be used in the function labels.

result

Character string specifying the kind of result required: either a function object
or a pixel image.

Details
This command computes, for each possible distance r, the average pixel value of the pixels lying at
distance r from the origin. Kernel smoothing is used to obtain a smooth function of r.
If result="fv" (the default) the result is a function object of class "fv" giving the mean pixel value
of X as a function of distance from the origin.
If result="im" the result is a pixel image, with the same dimensions as X, giving the mean value
of X over all pixels lying at the same distance from the origin as the current pixel.
If padzero=TRUE (the default), the value of X is assumed to be zero outside the window of X. The
rotational mean at a given distance r is the average value of the image X over the entire circle of
radius r, including zero values outside the window if the circle lies partly outside the window.
If padzero=FALSE, the value of X is taken to be undefined outside the window of X. The rotational
mean is the average of the X values over the subset of the circle of radius r that lies entirely inside
the window.
Value
An object of class "fv" or "im".
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
Examples
if(interactive()) {
Z <- setcov(square(1))
plot(rotmean(Z))
plot(rotmean(Z, result="im"))
} else {
Z <- setcov(square(1), dimyx=32)
f <- rotmean(Z)
}

1266

round.ppp

round.ppp

Apply Numerical Rounding to Spatial Coordinates

Description
Apply numerical rounding to the spatial coordinates of a point pattern.
Usage
## S3 method for class 'ppp'
round(x, digits = 0)
## S3 method for class 'pp3'
round(x, digits = 0)
## S3 method for class 'ppx'
round(x, digits = 0)
Arguments
x

A spatial point pattern in any dimension (object of class "ppp", "pp3" or "ppx").

digits

integer indicating the number of decimal places.

Details
These functions are methods for the generic function round. They apply numerical rounding to the
spatial coordinates of the point pattern x.
Value
A point pattern object, of the same class as x.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
rounding to determine whether numbers have been rounded.
round in the Base package.
Examples
round(cells, 1)

rounding

1267

rounding

Detect Numerical Rounding

Description
Given a numeric vector, or an object containing numeric spatial coordinates, determine whether the
values have been rounded to a certain number of decimal places.
Usage
rounding(x)
## Default S3 method:
rounding(x)
## S3 method for class 'ppp'
rounding(x)
## S3 method for class 'pp3'
rounding(x)
## S3 method for class 'ppx'
rounding(x)
Arguments
x

A numeric vector, or an object containing numeric spatial coordinates.

Details
For a numeric vector x, this function determines whether the values have been rounded to a certain
number of decimal places.
• If the entries of x are not all integers, then rounding(x) returns the smallest number of digits
d after the decimal point such that round(x, digits=d) is identical to x. For example if
rounding(x) = 2 then the entries of x are rounded to 2 decimal places, and are multiples of
0.01.
• If all the entries of x are integers, then rounding(x) returns -d, where d is the smallest number
of digits before the decimal point such that round(x, digits=-d) is identical to x. For
example if rounding(x) = -3 then the entries of x are multiples of 1000. If rounding(x) = 0
then the entries of x are integers but not multiples of 10.
• If all entries of x are equal to 0, the rounding is not determined, and a value of NULL is returned.
For a point pattern (object of class "ppp") or similar object x containing numeric spatial coordinates,
this procedure is applied to the spatial coordinates.
Value
An integer.

1268

rPenttinen

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
round.ppp
Examples
rounding(c(0.1, 0.3, 1.2))
rounding(c(1940, 1880, 2010))
rounding(0)
rounding(cells)

rPenttinen

Perfect Simulation of the Penttinen Process

Description
Generate a random pattern of points, a simulated realisation of the Penttinen process, using a perfect
simulation algorithm.
Usage
rPenttinen(beta, gamma=1, R, W = owin(), expand=TRUE, nsim=1, drop=TRUE)
Arguments
beta

intensity parameter (a positive number).

gamma

Interaction strength parameter (a number between 0 and 1).

R

disc radius (a non-negative number).

W

window (object of class "owin") in which to generate the random pattern.

expand

Logical. If FALSE, simulation is performed in the window W, which must be
rectangular. If TRUE (the default), simulation is performed on a larger window,
and the result is clipped to the original window W. Alternatively expand can
be an object of class "rmhexpand" (see rmhexpand) determining the expansion
method.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

rPenttinen

1269

Details
This function generates a realisation of the Penttinen point process in the window W using a ‘perfect
simulation’ algorithm.
Penttinen (1984, Example 2.1, page 18), citing Cormack (1979), described the pairwise interaction
point process with interaction factor
h(d) = eθA(d) = γ A(d)
between each pair of points separated by a distance $d$. Here A(d) is the area of intersection
between two discs of radius R separated by a distance d, normalised so that A(0) = 1.
The simulation algorithm used to generate the point pattern is ‘dominated coupling from the past’
as implemented by Berthelsen and Møller (2002, 2003). This is a ‘perfect simulation’ or ‘exact
simulation’ algorithm, so called because the output of the algorithm is guaranteed to have the correct probability distribution exactly (unlike the Metropolis-Hastings algorithm used in rmh, whose
output is only approximately correct).
There is a tiny chance that the algorithm will run out of space before it has terminated. If this occurs,
an error message will be generated.
Value
If nsim = 1, a point pattern (object of class "ppp"). If nsim > 1, a list of point patterns.
Author(s)
Adrian Baddeley , based on original code for the Strauss process by Kasper Klitgaard Berthelsen.
References
Berthelsen, K.K. and Møller, J. (2002) A primer on perfect simulation for spatial point processes.
Bulletin of the Brazilian Mathematical Society 33, 351-367.
Berthelsen, K.K. and Møller, J. (2003) Likelihood and non-parametric Bayesian MCMC inference
for spatial point processes based on perfect simulation and path sampling. Scandinavian Journal of
Statistics 30, 549-564.
Cormack, R.M. (1979) Spatial aspects of competition between individuals. Pages 151–212 in Spatial and Temporal Analysis in Ecology, eds. R.M. Cormack and J.K. Ord, International Co-operative
Publishing House, Fairland, MD, USA.
Møller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point
Processes. Chapman and Hall/CRC.
Penttinen, A. (1984) Modelling Interaction in Spatial Point Patterns: Parameter Estimation by the
Maximum Likelihood Method. Jyväskylä Studies in Computer Science, Economics and Statistics 7,
University of Jyväskylä, Finland.
See Also
rmh, Penttinen.
rStrauss, rHardcore, rStraussHard, rDiggleGratton, rDGS.
Examples
X <- rPenttinen(50, 0.5, 0.02)

1270

rpoint

rpoint

Generate N Random Points

Description
Generate a random point pattern containing n independent, identically distributed random points
with any specified distribution.
Usage
rpoint(n, f, fmax=NULL, win=unit.square(),
..., giveup=1000, verbose=FALSE,
nsim=1, drop=TRUE)
Arguments
n

Number of points to generate.

f

The probability density of the points, possibly un-normalised. Either a constant,
a function f(x,y,...), or a pixel image object.

fmax

An upper bound on the values of f. If missing, this number will be estimated.

win

Window in which to simulate the pattern. Ignored if f is a pixel image.

...

Arguments passed to the function f.

giveup

Number of attempts in the rejection method after which the algorithm should
stop trying to generate new points.

verbose

Flag indicating whether to report details of performance of the simulation algorithm.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

Details
This function generates n independent, identically distributed random points with common probability density proportional to f.
The argument f may be
a numerical constant: uniformly distributed random points will be generated.
a function: random points will be generated in the window win with probability density proportional to f(x,y,...) where x and y are the cartesian coordinates. The function f must accept
two vectors of coordinates x,y and return the corresponding vector of function values. Additional arguments ... of any kind may be passed to the function.
a pixel image: if f is a pixel image object of class "im" (see im.object) then random points will
be generated in the window of this pixel image, with probability density proportional to the
pixel values of f.
The algorithm is as follows:
• If f is a constant, we invoke runifpoint.

rpoisline

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• If f is a function, then we use the rejection method. Proposal points are generated from the
uniform distribution. A proposal point (x, y) is accepted with probability f(x,y,...)/fmax
and otherwise rejected. The algorithm continues until n points have been accepted. It gives up
after giveup * n proposals if there are still fewer than n points.
• If f is a pixel image, then a random sequence of pixels is selected (using sample) with probabilities proportional to the pixel values of f. Then for each pixel in the sequence we generate
a uniformly distributed random point in that pixel.
The algorithm for pixel images is more efficient than that for functions.
Value
A point pattern (an object of class "ppp") if nsim=1, or a list of point patterns if nsim > 1.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
ppp.object, owin.object, runifpoint
Examples
# 100 uniform random points in the unit square
X <- rpoint(100)
# 100 random points with probability density proportional to x^2 + y^2
X <- rpoint(100, function(x,y) { x^2 + y^2}, 1)
# `fmax' may be omitted
X <- rpoint(100, function(x,y) { x^2 + y^2})
# irregular window
data(letterR)
X <- rpoint(100, function(x,y) { x^2 + y^2}, win=letterR)
#
Z
#
X

make a pixel image
<- setcov(letterR)
100 points with density proportional to pixel values
<- rpoint(100, Z)

rpoisline

Generate Poisson Random Line Process

Description
Generate a random pattern of line segments obtained from the Poisson line process.
Usage
rpoisline(lambda, win=owin())

1272

rpoislinetess

Arguments
lambda

Intensity of the Poisson line process. A positive number.

win

Window in which to simulate the pattern. An object of class "owin" or something acceptable to as.owin.

Details
This algorithm generates a realisation of the uniform Poisson line process, and clips it to the window
win.
The argument lambda must be a positive number. It controls the intensity of the process. The
expected number of lines intersecting a convex region of the plane is equal to lambda times the
perimeter length of the region. The expected total length of the lines crossing a region of the plane
is equal to lambda * pi times the area of the region.
Value
A line segment pattern (an object of class "psp").
The result also has an attribute called "lines" (an object of class "infline" specifying the original
infinite random lines) and an attribute "linemap" (an integer vector mapping the line segments to
their parent lines).
Author(s)
Adrian Baddeley  and Rolf Turner 
See Also
psp
Examples
# uniform Poisson line process with intensity 10,
# clipped to the unit square
rpoisline(10)

rpoislinetess

Poisson Line Tessellation

Description
Generate a tessellation delineated by the lines of the Poisson line process
Usage
rpoislinetess(lambda, win = owin())
Arguments
lambda

Intensity of the Poisson line process. A positive number.

win

Window in which to simulate the pattern. An object of class "owin" or something acceptable to as.owin. Currently, the window must be a rectangle.

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Details
This algorithm generates a realisation of the uniform Poisson line process, and divides the window
win into tiles separated by these lines.
The argument lambda must be a positive number. It controls the intensity of the process. The
expected number of lines intersecting a convex region of the plane is equal to lambda times the
perimeter length of the region. The expected total length of the lines crossing a region of the plane
is equal to lambda * pi times the area of the region.
Value
A tessellation (object of class "tess").
Also has an attribute "lines" containing the realisation of the Poisson line process, as an object of
class "infline".
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
rpoisline to generate the lines only.
Examples
X <- rpoislinetess(3)
plot(as.im(X), main="rpoislinetess(3)")
plot(X, add=TRUE)

rpoislpp

Poisson Point Process on a Linear Network

Description
Generates a realisation of the Poisson point process with specified intensity on the given linear
network.
Usage
rpoislpp(lambda, L, ..., nsim=1, drop=TRUE)
Arguments
lambda

Intensity of the Poisson process. A single number, a function(x,y), a pixel
image (object of class "im"), or a vector of numbers, a list of functions, or a list
of images.

L

A linear network (object of class "linnet", see linnet). Can be omitted in
some cases: see Details.

...

Arguments passed to rpoisppOnLines.

nsim

Number of simulated realisations to generate.

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rpoispp

drop

Logical value indicating what to do when nsim=1. If drop=TRUE (the default),
the result is a point pattern. If drop=FALSE, the result is a list with one entry
which is a point pattern.

Details
This function uses rpoisppOnLines to generate the random points.
Argument L can be omitted, and defaults to as.linnet(lambda), when lambda is a function on a
linear network (class "linfun") or a pixel image on a linear network ("linim").
Value
If nsim = 1 and drop=TRUE, a point pattern on the linear network, i.e.\ an object of class "lpp".
Otherwise, a list of such point patterns.
Author(s)
Ang Qi Wei  and Adrian Baddeley 
See Also
runiflpp, rlpp, lpp, linnet
Examples
X <- rpoislpp(5, simplenet)
plot(X)
# multitype
X <- rpoislpp(c(a=5, b=5), simplenet)

rpoispp

Generate Poisson Point Pattern

Description
Generate a random point pattern using the (homogeneous or inhomogeneous) Poisson process. Includes CSR (complete spatial randomness).
Usage
rpoispp(lambda, lmax=NULL, win=owin(), ...,
nsim=1, drop=TRUE, ex=NULL, warnwin=TRUE)
Arguments
lambda

Intensity of the Poisson process. Either a single positive number, a function(x,y, ...),
or a pixel image.

lmax

Optional. An upper bound for the value of lambda(x,y), if lambda is a function.

win

Window in which to simulate the pattern. An object of class "owin" or something acceptable to as.owin. Ignored if lambda is a pixel image.

...

Arguments passed to lambda if it is a function.

rpoispp

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nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

ex

Optional. A point pattern to use as the example. If ex is given and lambda,lmax,win
are missing, then lambda and win will be calculated from the point pattern ex.

warnwin

Logical value specifying whether to issue a warning when win is ignored (which
occurs when lambda is an image and win is present).

Details
If lambda is a single number, then this algorithm generates a realisation of the uniform Poisson
process (also known as Complete Spatial Randomness, CSR) inside the window win with intensity
lambda (points per unit area).
If lambda is a function, then this algorithm generates a realisation of the inhomogeneous Poisson
process with intensity function lambda(x,y,...) at spatial location (x,y) inside the window win.
The function lambda must work correctly with vectors x and y.
If lmax is given, it must be an upper bound on the values of lambda(x,y,...) for all locations
(x, y) inside the window win. That is, we must have lambda(x,y,...) <= lmax for all locations
(x,y). If this is not true then the results of the algorithm will be incorrect.
If lmax is missing or NULL, an approximate upper bound is computed by finding the maximum value
of lambda(x,y,...) on a grid of locations (x,y) inside the window win, and adding a safety
margin equal to 5 percent of the range of lambda values. This can be computationally intensive, so
it is advisable to specify lmax if possible.
If lambda is a pixel image object of class "im" (see im.object), this algorithm generates a realisation of the inhomogeneous Poisson process with intensity equal to the pixel values of the image.
(The value of the intensity function at an arbitrary location is the pixel value of the nearest pixel.)
The argument win is ignored; the window of the pixel image is used instead. It will be converted to
a rectangle if possible, using rescue.rectangle.
To generate an inhomogeneous Poisson process the algorithm uses “thinning”: it first generates a
uniform Poisson process of intensity lmax, then randomly deletes or retains each point, independently of other points, with retention probability p(x, y) = λ(x, y)/lmax.
For marked point patterns, use rmpoispp.
Value
A point pattern (an object of class "ppp") if nsim=1, or a list of point patterns if nsim > 1.
Warning
Note that lambda is the intensity, that is, the expected number of points per unit area. The total
number of points in the simulated pattern will be random with expected value mu = lambda * a
where a is the area of the window win.
Reproducibility
The simulation algorithm, for the case where lambda is a pixel image, was changed in spatstat
version 1.42-3. Set spatstat.options(fastpois=FALSE) to use the previous, slower algorithm,
if it is desired to reproduce results obtained with earlier versions.

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rpoispp3

Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
rmpoispp for Poisson marked point patterns, runifpoint for a fixed number of independent uniform random points; rpoint, rmpoint for a fixed number of independent random points with any
distribution; rMaternI, rMaternII, rSSI, rStrauss, rstrat for random point processes with spatial inhibition or regularity; rThomas, rGaussPoisson, rMatClust, rcell for random point processes exhibiting clustering; rmh.default for Gibbs processes. See also ppp.object, owin.object.
Examples
# uniform Poisson process with intensity 100 in the unit square
pp <- rpoispp(100)
# uniform Poisson process with intensity 1 in a 10 x 10 square
pp <- rpoispp(1, win=owin(c(0,10),c(0,10)))
# plots should look similar !
# inhomogeneous Poisson process in unit square
# with intensity lambda(x,y) = 100 * exp(-3*x)
# Intensity is bounded by 100
pp <- rpoispp(function(x,y) {100 * exp(-3*x)}, 100)
# How to tune the coefficient of x
lamb <- function(x,y,a) { 100 * exp( - a * x)}
pp <- rpoispp(lamb, 100, a=3)
# pixel image
Z <- as.im(function(x,y){100 * sqrt(x+y)}, unit.square())
pp <- rpoispp(Z)
# randomising an existing point pattern
rpoispp(intensity(cells), win=Window(cells))
rpoispp(ex=cells)

rpoispp3

Generate Poisson Point Pattern in Three Dimensions

Description
Generate a random three-dimensional point pattern using the homogeneous Poisson process.
Usage
rpoispp3(lambda, domain = box3(), nsim=1, drop=TRUE)

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Arguments
lambda

Intensity of the Poisson process. A single positive number.

domain

Three-dimensional box in which the process should be generated. An object of
class "box3".

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

Details
This function generates a realisation of the homogeneous Poisson process in three dimensions, with
intensity lambda (points per unit volume).
The realisation is generated inside the three-dimensional region domain which currently must be a
rectangular box (object of class "box3").
Value
If nsim = 1 and drop=TRUE, a point pattern in three dimensions (an object of class "pp3"). If
nsim > 1, a list of such point patterns.
Note
The intensity lambda is the expected number of points per unit volume.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
runifpoint3, pp3, box3
Examples
X <- rpoispp3(50)

rpoisppOnLines

Generate Poisson Point Pattern on Line Segments

Description
Given a line segment pattern, generate a Poisson random point pattern on the line segments.
Usage
rpoisppOnLines(lambda, L, lmax = NULL, ..., nsim=1)

1278

rpoisppOnLines

Arguments
lambda

Intensity of the Poisson process. A single number, a function(x,y), a pixel
image (object of class "im"), or a vector of numbers, a list of functions, or a list
of images.

L

Line segment pattern (object of class "psp") on which the points should be
generated.

lmax

Optional upper bound (for increased computational efficiency). A known upper
bound for the values of lambda, if lambda is a function or a pixel image. That
is, lmax should be a number which is known to be greater than or equal to all
values of lambda.

...

Additional arguments passed to lambda if it is a function.

nsim

Number of simulated realisations to be generated.

Details
This command generates a Poisson point process on the one-dimensional system of line segments
in L. The result is a point pattern consisting of points lying on the line segments in L. The number
of random points falling on any given line segment follows a Poisson distribution. The patterns of
points on different segments are independent.
The intensity lambda is the expected number of points per unit length of line segment. It may be
constant, or it may depend on spatial location.
In order to generate an unmarked Poisson process, the argument lambda may be a single number,
or a function(x,y), or a pixel image (object of class "im").
In order to generate a marked Poisson process, lambda may be a numeric vector, a list of functions,
or a list of images, each entry giving the intensity for a different mark value.
If lambda is not numeric, then the (Lewis-Shedler) rejection method is used. The rejection method
requires knowledge of lmax, the maximum possible value of lambda. This should be either a single
number, or a numeric vector of the same length as lambda. If lmax is not given, it will be computed
approximately, by sampling many values of lambda.
If lmax is given, then it must be larger than any possible value of lambda, otherwise the results of
the algorithm will be incorrect.
Value
If nsim = 1, a point pattern (object of class "ppp") in the same window as L. If nsim > 1, a list of
such point patterns.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
psp, ppp, runifpointOnLines, rpoispp

rpoisppx

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Examples
live <- interactive()
L <- psp(runif(10), runif(10), runif(10), runif(10),
if(live) plot(L, main="")

window=owin())

# uniform intensity
Y <- rpoisppOnLines(4, L)
if(live) plot(Y, add=TRUE, pch="+")
# uniform MARKED process with types 'a' and 'b'
Y <- rpoisppOnLines(c(a=4, b=5), L)
if(live) {
plot(L, main="")
plot(Y, add=TRUE, pch="+")
}
# intensity is a function
Y <- rpoisppOnLines(function(x,y){ 10 * x^2}, L, 10)
if(live) {
plot(L, main="")
plot(Y, add=TRUE, pch="+")
}
# intensity is an image
Z <- as.im(function(x,y){10 * sqrt(x+y)}, unit.square())
Y <- rpoisppOnLines(Z, L, 15)
if(live) {
plot(L, main="")
plot(Y, add=TRUE, pch="+")
}

rpoisppx

Generate Poisson Point Pattern in Any Dimensions

Description
Generate a random multi-dimensional point pattern using the homogeneous Poisson process.
Usage
rpoisppx(lambda, domain, nsim=1, drop=TRUE)
Arguments
lambda

Intensity of the Poisson process. A single positive number.

domain

Multi-dimensional box in which the process should be generated. An object of
class "boxx".

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a single point pattern.

1280

rPoissonCluster

Details
This function generates a realisation of the homogeneous Poisson process in multi dimensions, with
intensity lambda (points per unit volume).
The realisation is generated inside the multi-dimensional region domain which currently must be a
rectangular box (object of class "boxx").
Value
If nsim = 1 and drop=TRUE, a point pattern (an object of class "ppx"). If nsim > 1 or drop=FALSE,
a list of such point patterns.
Note
The intensity lambda is the expected number of points per unit volume.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
runifpointx, ppx, boxx
Examples
w <- boxx(x=c(0,1), y=c(0,1), z=c(0,1), t=c(0,3))
X <- rpoisppx(10, w)

rPoissonCluster

Simulate Poisson Cluster Process

Description
Generate a random point pattern, a realisation of the general Poisson cluster process.
Usage
rPoissonCluster(kappa, expand, rcluster, win = owin(c(0,1),c(0,1)),
..., lmax=NULL, nsim=1, drop=TRUE, saveparents=TRUE)
Arguments
kappa

Intensity of the Poisson process of cluster centres. A single positive number, a
function, or a pixel image.

expand

Size of the expansion of the simulation window for generating parent points. A
single non-negative number.

rcluster

A function which generates random clusters.

win

Window in which to simulate the pattern. An object of class "owin" or something acceptable to as.owin.

rPoissonCluster

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...

Arguments passed to rcluster

lmax

Optional. Upper bound on the values of kappa when kappa is a function or pixel
image.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

saveparents

Logical value indicating whether to save the locations of the parent points as an
attribute.

Details
This algorithm generates a realisation of the general Poisson cluster process, with the cluster mechanism given by the function rcluster.
First, the algorithm generates a Poisson point process of “parent” points with intensity kappa in an
expanded window as explained below.. Here kappa may be a single positive number, a function
kappa(x, y), or a pixel image object of class "im" (see im.object). See rpoispp for details.
Second, each parent point is replaced by a random cluster of points, created by calling the function
rcluster. These clusters are combined together to yield a single point pattern, and the restriction
of this pattern to the window win is then returned as the result of rPoissonCluster.
The expanded window consists of as.rectangle(win) extended by the amount expand in each
direction. The size of the expansion is saved in the attribute "expand" and may be extracted by
attr(X, "expand") where X is the generated point pattern.
The function rcluster should expect to be called as rcluster(xp[i],yp[i],...) for each parent
point at a location (xp[i],yp[i]). The return value of rcluster should be a list with elements
x,y which are vectors of equal length giving the absolute x and y coordinates of the points in the
cluster.
If the return value of rcluster is a point pattern (object of class "ppp") then it may have marks.
The result of rPoissonCluster will then be a marked point pattern.
If required, the intermediate stages of the simulation (the parents and the individual clusters) can
also be extracted from the return value of rPoissonCluster through the attributes "parents"
and "parentid". The attribute "parents" is the point pattern of parent points. The attribute
"parentid" is an integer vector specifying the parent for each of the points in the simulated pattern.
(If these data are not required, it is more efficient to set saveparents=FALSE.)
Value
A point pattern (an object of class "ppp") if nsim=1, or a list of point patterns if nsim > 1.
Additionally, some intermediate results of the simulation are returned as attributes of the point
pattern: see Details.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
rpoispp, rMatClust, rThomas, rCauchy, rVarGamma, rNeymanScott, rGaussPoisson.

1282

rppm

Examples
# each cluster consist of 10 points in a disc of radius 0.2
nclust <- function(x0, y0, radius, n) {
return(runifdisc(n, radius, centre=c(x0, y0)))
}
plot(rPoissonCluster(10, 0.2, nclust, radius=0.2, n=5))
# multitype Neyman-Scott process (each cluster is a multitype process)
nclust2 <- function(x0, y0, radius, n, types=c("a", "b")) {
X <- runifdisc(n, radius, centre=c(x0, y0))
M <- sample(types, n, replace=TRUE)
marks(X) <- M
return(X)
}
plot(rPoissonCluster(15,0.1,nclust2, radius=0.1, n=5))

rppm

Recursively Partitioned Point Process Model

Description
Fits a recursive partition model to point pattern data.
Usage
rppm(..., rpargs=list())
Arguments
...

Arguments passed to ppm specifying the point pattern data and the explanatory
covariates.

rpargs

Optional list of arguments passed to rpart controlling the recursive partitioning
procedure.

Details
This function attempts to find a simple rule for predicting low and high intensity regions of points
in a point pattern, using explanatory covariates.
The arguments ... specify the point pattern data and explanatory covariates in the same way as
they would be in the function ppm.
The recursive partitioning algorithm rpart is then used to find a partitioning rule.
Value
An object of class "rppm". There are methods for print, plot, fitted, predict and prune for
this class.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .

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References
Breiman, L., Friedman, J. H., Olshen, R. A., and Stone, C. J. (1984) Classification and Regression
Trees. Wadsworth.
See Also
plot.rppm, predict.rppm, prune.rppm.
Examples
# New Zealand trees data: trees planted along border
# Use covariates 'x', 'y'
nzfit <- rppm(nztrees ~ x + y)
nzfit
prune(nzfit, cp=0.035)
# Murchison gold data: numeric and logical covariates
mur <- solapply(murchison, rescale, s=1000, unitname="km")
mur$dfault <- distfun(mur$faults)
#
mfit <- rppm(gold ~ dfault + greenstone, data=mur)
mfit
# Gorillas data: factor covariates
#
(symbol '.' indicates 'all variables')
gfit <- rppm(unmark(gorillas) ~ . , data=gorillas.extra)
gfit

rQuasi

Generate Quasirandom Point Pattern in Given Window

Description
Generates a quasirandom pattern of points in any two-dimensional window.
Usage
rQuasi(n, W, type = c("Halton", "Hammersley"), ...)
Arguments
n

Maximum number of points to be generated.

W

Window (object of class "owin") in which to generate the points.

type

String identifying the quasirandom generator.

...

Arguments passed to the quasirandom generator.

Details
This function generates a quasirandom point pattern, using the quasirandom sequence generator
Halton or Hammersley as specified.
If W is a rectangle, exactly n points will be generated.
If W is not a rectangle, n points will be generated in the containing rectangle as.rectangle(W), and
only the points lying inside W will be retained.

1284

rshift

Value
Point pattern (object of class "ppp") inside the window W.
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak .
See Also
Halton
Examples
plot(rQuasi(256, letterR))

rshift

Random Shift

Description
Randomly shifts the points of a point pattern or line segment pattern. Generic.
Usage
rshift(X, ...)
Arguments
X

Pattern to be subjected to a random shift. A point pattern (class "ppp"), a line
segment pattern (class "psp") or an object of class "splitppp".

...

Arguments controlling the generation of the random shift vector, or specifying
which parts of the pattern will be shifted.

Details
This operation applies a random shift (vector displacement) to the points in a point pattern, or to the
segments in a line segment pattern.
The argument X may be
• a point pattern (an object of class "ppp")
• a line segment pattern (an object of class "psp")
• an object of class "splitppp" (basically a list of point patterns, obtained from split.ppp).
The function rshift is generic, with methods for the three classes "ppp", "psp" and "splitppp".
See the help pages for these methods, rshift.ppp, rshift.psp and rshift.splitppp, for further
information.

rshift.ppp

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Value
An object of the same type as X.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
rshift.ppp, rshift.psp, rshift.splitppp

rshift.ppp

Randomly Shift a Point Pattern

Description
Randomly shifts the points of a point pattern.
Usage
## S3 method for class 'ppp'
rshift(X, ..., which=NULL, group)
Arguments
X

Point pattern to be subjected to a random shift. An object of class "ppp"

...

Arguments that determine the random shift. See Details.

group

Optional. Factor specifying a grouping of the points of X, or NULL indicating
that all points belong to the same group. Each group will be shifted together,
and separately from other groups. By default, points in a marked point pattern
are grouped according to their mark values, while points in an unmarked point
pattern are treated as a single group.

which

Optional. Identifies which groups of the pattern will be shifted, while other
groups are not shifted. A vector of levels of group.

Details
This operation randomly shifts the locations of the points in a point pattern.
The function rshift is generic. This function rshift.ppp is the method for point patterns.
The most common use of this function is to shift the points in a multitype point pattern. By default, points of the same type are shifted in parallel (i.e. points of a common type are shifted by
a common displacement vector), and independently of other types. This is useful for testing the
hypothesis of independence of types (the null hypothesis that the sub-patterns of points of each type
are independent point processes).
In general the points of X are divided into groups, then the points within a group are shifted by a
common random displacement vector. Different groups of points are shifted independently. The
grouping is determined as follows:

1286

rshift.ppp
• If the argument group is present, then this determines the grouping.
• Otherwise, if X is a multitype point pattern, the marks determine the grouping.
• Otherwise, all points belong to a single group.

The argument group should be a factor, of length equal to the number of points in X. Alternatively
group may be NULL, which specifies that all points of X belong to a single group.
By default, every group of points will be shifted. The argument which indicates that only some of
the groups should be shifted, while other groups should be left unchanged. which must be a vector
of levels of group (for example, a vector of types in a multitype pattern) indicating which groups
are to be shifted.
The displacement vector, i.e. the vector by which the data points are shifted, is generated at random.
Parameters that control the randomisation and the handling of edge effects are passed through the
... argument. They are
radius,width,height Parameters of the random shift vector.
edge String indicating how to deal with edges of the pattern. Options are "torus", "erode" and
"none".
clip Optional. Window to which the final point pattern should be clipped.
If the window is a rectangle, the default behaviour is to generate a displacement vector at random
with equal probability for all possible displacements. This means that the x and y coordinates of
the displacement vector are independent random variables, uniformly distributed over the range of
possible coordinates.
Alternatively, the displacement vector can be generated by another random mechanism, controlled
by the arguments radius, width and height.
rectangular: if width and height are given, then the displacement vector is uniformly distributed
in a rectangle of these dimensions, centred at the origin. The maximum possible displacement
in the x direction is width/2. The maximum possible displacement in the y direction is
height/2. The x and y displacements are independent. (If width and height are actually
equal to the dimensions of the observation window, then this is equivalent to the default.)
radial: if radius is given, then the displacement vector is generated by choosing a random point
inside a disc of the given radius, centred at the origin, with uniform probability density over
the disc. Thus the argument radius determines the maximum possible displacement distance.
The argument radius is incompatible with the arguments width and height.
The argument edge controls what happens when a shifted point lies outside the window of X. Options are:
"none": Points shifted outside the window of X simply disappear.
"torus": Toroidal or periodic boundary. Treat opposite edges of the window as identical, so that
a point which disappears off the right-hand edge will re-appear at the left-hand edge. This is
called a “toroidal shift” because it makes the rectangle topologically equivalent to the surface
of a torus (doughnut).
The window must be a rectangle. Toroidal shifts are undefined if the window is non-rectangular.
"erode": Clip the point pattern to a smaller window.
If the random displacements are generated by a radial mechanism (see above), then the window of X is eroded by a distance equal to the value of the argument radius, using erosion.
If the random displacements are generated by a rectangular mechanism, then the window of X
is (if it is not rectangular) eroded by a distance max(height,width) using erosion; or (if it

rshift.psp

1287
is rectangular) trimmed by a margin of width width at the left and right sides and trimmed by
a margin of height height at the top and bottom.
The rationale for this is that the clipping window is the largest window for which edge effects
can be ignored.

The optional argument clip specifies a smaller window to which the pattern should be restricted.
Value
A point pattern (object of class "ppp").
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
rshift, rshift.psp
Examples
data(amacrine)
# random toroidal shift
# shift "on" and "off" points separately
X <- rshift(amacrine)
# shift "on" points and leave "off" points fixed
X <- rshift(amacrine, which="on")
# shift all points simultaneously
X <- rshift(amacrine, group=NULL)
# maximum displacement distance 0.1 units
X <- rshift(amacrine, radius=0.1)
# shift with erosion
X <- rshift(amacrine, radius=0.1, edge="erode")

rshift.psp

Randomly Shift a Line Segment Pattern

Description
Randomly shifts the segments in a line segment pattern.
Usage
## S3 method for class 'psp'
rshift(X, ..., group=NULL, which=NULL)

1288

rshift.psp

Arguments
X

Line segment pattern to be subjected to a random shift. An object of class "psp".

...

Arguments controlling the randomisation and the handling of edge effects. See
rshift.ppp.

group

Optional. Factor specifying a grouping of the line segments of X, or NULL indicating that all line segments belong to the same group. Each group will be
shifted together, and separately from other groups.

which

Optional. Identifies which groups of the pattern will be shifted, while other
groups are not shifted. A vector of levels of group.

Details
This operation randomly shifts the locations of the line segments in a line segment pattern.
The function rshift is generic. This function rshift.psp is the method for line segment patterns.
The line segments of X are first divided into groups, then the line segments within a group are
shifted by a common random displacement vector. Different groups of line segments are shifted
independently. If the argument group is present, then this determines the grouping. Otherwise, all
line segments belong to a single group.
The argument group should be a factor, of length equal to the number of line segments in X. Alternatively group may be NULL, which specifies that all line segments of X belong to a single group.
By default, every group of line segments will be shifted. The argument which indicates that only
some of the groups should be shifted, while other groups should be left unchanged. which must be
a vector of levels of group indicating which groups are to be shifted.
The displacement vector, i.e. the vector by which the data line segments are shifted, is generated
at random. The default behaviour is to generate a displacement vector at random with equal probability for all possible displacements. This means that the x and y coordinates of the displacement
vector are independent random variables, uniformly distributed over the range of possible coordinates.
Alternatively, the displacement vector can be generated by another random mechanism, controlled
by the arguments radius, width and height.
rectangular: if width and height are given, then the displacement vector is uniformly distributed
in a rectangle of these dimensions, centred at the origin. The maximum possible displacement
in the x direction is width/2. The maximum possible displacement in the y direction is
height/2. The x and y displacements are independent. (If width and height are actually
equal to the dimensions of the observation window, then this is equivalent to the default.)
radial: if radius is given, then the displacement vector is generated by choosing a random line
segment inside a disc of the given radius, centred at the origin, with uniform probability density over the disc. Thus the argument radius determines the maximum possible displacement
distance. The argument radius is incompatible with the arguments width and height.
The argument edge controls what happens when a shifted line segment lies partially or completely
outside the window of X. Currently the only option is "erode" which specifies that the segments
will be clipped to a smaller window.
The optional argument clip specifies a smaller window to which the pattern should be restricted.
Value
A line segment pattern (object of class "psp").

rshift.splitppp

1289

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
rshift, rshift.ppp
Examples
X <- psp(runif(20), runif(20), runif(20), runif(20),
Y <- rshift(X, radius=0.1)

rshift.splitppp

window=owin())

Randomly Shift a List of Point Patterns

Description
Randomly shifts each point pattern in a list of point patterns.
Usage
## S3 method for class 'splitppp'
rshift(X, ..., which=seq_along(X))
Arguments
X

An object of class "splitppp". Basically a list of point patterns.

...

Parameters controlling the generation of the random shift vector and the handling of edge effects. See rshift.ppp.

which

Optional. Identifies which patterns will be shifted, while other patterns are not
shifted. Any valid subset index for X.

Details
This operation applies a random shift to each of the point patterns in the list X.
The function rshift is generic. This function rshift.splitppp is the method for objects of class
"splitppp", which are essentially lists of point patterns, created by the function split.ppp.
By default, every pattern in the list X will be shifted. The argument which indicates that only some
of the patterns should be shifted, while other groups should be left unchanged. which can be any
valid subset index for X.
Each point pattern in the list X (or each pattern in X[which]) is shifted by a random displacement
vector. The shifting is performed by rshift.ppp.
See the help page for rshift.ppp for details of the other arguments.
Value
Another object of class "splitppp".

1290

rSSI

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
rshift, rshift.ppp
Examples
data(amacrine)
Y <- split(amacrine)
# random toroidal shift
# shift "on" and "off" points separately
X <- rshift(Y)
# shift "on" points and leave "off" points fixed
X <- rshift(Y, which="on")
# maximum displacement distance 0.1 units
X <- rshift(Y, radius=0.1)
# shift with erosion
X <- rshift(Y, radius=0.1, edge="erode")

rSSI

Simulate Simple Sequential Inhibition

Description
Generate a random point pattern, a realisation of the Simple Sequential Inhibition (SSI) process.
Usage
rSSI(r, n=Inf, win = square(1), giveup = 1000, x.init=NULL, ...,
f=NULL, fmax=NULL, nsim=1, drop=TRUE)
Arguments
r

Inhibition distance.

n

Maximum number of points allowed. If n is finite, stop when the total number
of points in the point pattern reaches n. If n is infinite (the default), stop only
when it is apparently impossible to add any more points. See Details.

win

Window in which to simulate the pattern. An object of class "owin" or something acceptable to as.owin. The default window is the unit square, unless
x.init is specified, when the default window is the window of x.init.

giveup

Number of rejected proposals after which the algorithm should terminate.

x.init

Optional. Initial configuration of points. A point pattern (object of class "ppp").
The pattern returned by rSSI consists of this pattern together with the points
added via simple sequential inhibition. See Details.

rSSI

1291
...

Ignored.

f,fmax

Optional arguments passed to rpoint to specify a non-uniform probability density for the random points.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

Details
This algorithm generates one or more realisations of the Simple Sequential Inhibition point process
inside the window win.
Starting with an empty window (or with the point pattern x.init if specified), the algorithm adds
points one-by-one. Each new point is generated uniformly in the window and independently of
preceding points. If the new point lies closer than r units from an existing point, then it is rejected
and another random point is generated. The algorithm terminates when either
(a) the desired number n of points is reached, or
(b) the current point configuration has not changed for giveup iterations, suggesting that it is no
longer possible to add new points.
If n is infinite (the default) then the algorithm terminates only when (b) occurs. The result is sometimes called a Random Sequential Packing.
Note that argument n specifies the maximum permitted total number of points in the pattern returned by rSSI(). If x.init is not NULL then the number of points that are added is at most
n - npoints(x.init) if n is finite.
Thus if x.init is not NULL then argument n must be at least as large as npoints(x.init), otherwise
an error is given. If n==npoints(x.init) then a warning is given and the call to rSSI() has no
real effect; x.init is returned.
There is no requirement that the points of x.init be at a distance at least r from each other. All of
the added points will be at a distance at least r from each other and from any point of x.init.
The points will be generated inside the window win and the result will be a point pattern in the same
window.
The default window is the unit square, win = square(1), unless x.init is specified, when the
default is win=Window(x.init), the window of x.init.
If both win and x.init are specified, and if the two windows are different, then a warning will be
issued. Any points of x.init lying outside win will be removed, with a warning.
Value
A point pattern (an object of class "ppp") if nsim=1, or a list of point patterns if nsim > 1.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
rpoispp, rMaternI, rMaternII.

1292

rstrat

Examples
Vinf <- rSSI(0.07)
V100 <- rSSI(0.07, 100)
X <- runifpoint(100)
Y <- rSSI(0.03,142,x.init=X) # Y consists of X together with
# 42 added points.
plot(Y, main="rSSI")
plot(X,add=TRUE,chars=20,cols="red")
## inhomogeneous
Z <- rSSI(0.07, 50, f=function(x,y){x})
plot(Z)

rstrat

Simulate Stratified Random Point Pattern

Description
Generates a “stratified random” pattern of points in a window, by dividing the window into rectangular tiles and placing k random points independently in each tile.
Usage
rstrat(win=square(1), nx, ny=nx, k = 1, nsim=1, drop=TRUE)
Arguments
win

A window. An object of class owin, or data in any format acceptable to as.owin().

nx

Number of tiles in each column.

ny

Number of tiles in each row.

k

Number of random points to generate in each tile.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

Details
This function generates a random pattern of points in a “stratified random” sampling design. It can
be useful for generating random spatial sampling points.
The bounding rectangle of win is divided into a regular nx × ny grid of rectangular tiles. In each
tile, k random points are generated independently with a uniform distribution in that tile.
Some of these grid points may lie outside the window win: if they do, they are deleted.
The result is a point pattern inside the window win.
This function is useful in creating dummy points for quadrature schemes (see quadscheme) as well
as in simulating random point patterns.

rStrauss

1293

Value
A point pattern (an object of class "ppp") if nsim=1, or a list of point patterns if nsim > 1.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
rsyst, runifpoint, quadscheme
Examples
X <- rstrat(nx=10)
plot(X)
# polygonal boundary
data(letterR)
X <- rstrat(letterR, 5, 10, k=3)
plot(X)

rStrauss

Perfect Simulation of the Strauss Process

Description
Generate a random pattern of points, a simulated realisation of the Strauss process, using a perfect
simulation algorithm.
Usage
rStrauss(beta, gamma = 1, R = 0, W = owin(), expand=TRUE, nsim=1, drop=TRUE)
Arguments
beta

intensity parameter (a positive number).

gamma

interaction parameter (a number between 0 and 1, inclusive).

R

interaction radius (a non-negative number).

W

window (object of class "owin") in which to generate the random pattern.

expand

Logical. If FALSE, simulation is performed in the window W, which must be
rectangular. If TRUE (the default), simulation is performed on a larger window,
and the result is clipped to the original window W. Alternatively expand can
be an object of class "rmhexpand" (see rmhexpand) determining the expansion
method.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

1294

rStrauss

Details
This function generates a realisation of the Strauss point process in the window W using a ‘perfect
simulation’ algorithm.
The Strauss process (Strauss, 1975; Kelly and Ripley, 1976) is a model for spatial inhibition, ranging
from a strong ‘hard core’ inhibition to a completely random pattern according to the value of gamma.
The Strauss process with interaction radius R and parameters β and γ is the pairwise interaction
point process with probability density
f (x1 , . . . , xn ) = αβ n(x) γ s(x)
where x1 , . . . , xn represent the points of the pattern, n(x) is the number of points in the pattern,
s(x) is the number of distinct unordered pairs of points that are closer than R units apart, and α is the
normalising constant. Intuitively, each point of the pattern contributes a factor β to the probability
density, and each pair of points closer than r units apart contributes a factor γ to the density.
The interaction parameter γ must be less than or equal to 1 in order that the process be well-defined
(Kelly and Ripley, 1976). This model describes an “ordered” or “inhibitive” pattern. If γ = 1 it
reduces to a Poisson process (complete spatial randomness) with intensity β. If γ = 0 it is called a
“hard core process” with hard core radius R/2, since no pair of points is permitted to lie closer than
R units apart.
The simulation algorithm used to generate the point pattern is ‘dominated coupling from the past’
as implemented by Berthelsen and Møller (2002, 2003). This is a ‘perfect simulation’ or ‘exact
simulation’ algorithm, so called because the output of the algorithm is guaranteed to have the correct probability distribution exactly (unlike the Metropolis-Hastings algorithm used in rmh, whose
output is only approximately correct).
There is a tiny chance that the algorithm will run out of space before it has terminated. If this occurs,
an error message will be generated.
Value
If nsim = 1, a point pattern (object of class "ppp"). If nsim > 1, a list of point patterns.
Author(s)
Kasper Klitgaard Berthelsen, adapted for spatstat by Adrian Baddeley 
References
Berthelsen, K.K. and Møller, J. (2002) A primer on perfect simulation for spatial point processes.
Bulletin of the Brazilian Mathematical Society 33, 351-367.
Berthelsen, K.K. and Møller, J. (2003) Likelihood and non-parametric Bayesian MCMC inference
for spatial point processes based on perfect simulation and path sampling. Scandinavian Journal of
Statistics 30, 549-564.
Kelly, F.P. and Ripley, B.D. (1976) On Strauss’s model for clustering. Biometrika 63, 357–360.
Møller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point
Processes. Chapman and Hall/CRC.
Strauss, D.J. (1975) A model for clustering. Biometrika 62, 467–475.
See Also
rmh, Strauss, rHardcore, rStraussHard, rDiggleGratton, rDGS, rPenttinen.

rStraussHard

1295

Examples
X <- rStrauss(0.05,0.2,1.5,square(141.4))
Z <- rStrauss(100,0.7,0.05)

rStraussHard

Perfect Simulation of the Strauss-Hardcore Process

Description
Generate a random pattern of points, a simulated realisation of the Strauss-Hardcore process, using
a perfect simulation algorithm.
Usage
rStraussHard(beta, gamma = 1, R = 0, H = 0, W = owin(),
expand=TRUE, nsim=1, drop=TRUE)
Arguments
beta

intensity parameter (a positive number).

gamma

interaction parameter (a number between 0 and 1, inclusive).

R

interaction radius (a non-negative number).

H

hard core distance (a non-negative number smaller than R).

W

window (object of class "owin") in which to generate the random pattern. Currently this must be a rectangular window.

expand

Logical. If FALSE, simulation is performed in the window W, which must be
rectangular. If TRUE (the default), simulation is performed on a larger window,
and the result is clipped to the original window W. Alternatively expand can
be an object of class "rmhexpand" (see rmhexpand) determining the expansion
method.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

Details
This function generates a realisation of the Strauss-Hardcore point process in the window W using a
‘perfect simulation’ algorithm.
The Strauss-Hardcore process is described in StraussHard.
The simulation algorithm used to generate the point pattern is ‘dominated coupling from the past’
as implemented by Berthelsen and Møller (2002, 2003). This is a ‘perfect simulation’ or ‘exact
simulation’ algorithm, so called because the output of the algorithm is guaranteed to have the correct probability distribution exactly (unlike the Metropolis-Hastings algorithm used in rmh, whose
output is only approximately correct).
A limitation of the perfect simulation algorithm is that the interaction parameter γ must be less than
or equal to 1. To simulate a Strauss-hardcore process with γ > 1, use rmh.
There is a tiny chance that the algorithm will run out of space before it has terminated. If this occurs,
an error message will be generated.

1296

rsyst

Value
If nsim = 1, a point pattern (object of class "ppp"). If nsim > 1, a list of point patterns.
Author(s)
Kasper Klitgaard Berthelsen and Adrian Baddeley 
References
Berthelsen, K.K. and Møller, J. (2002) A primer on perfect simulation for spatial point processes.
Bulletin of the Brazilian Mathematical Society 33, 351-367.
Berthelsen, K.K. and Møller, J. (2003) Likelihood and non-parametric Bayesian MCMC inference
for spatial point processes based on perfect simulation and path sampling. Scandinavian Journal of
Statistics 30, 549-564.
Møller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point
Processes. Chapman and Hall/CRC.
See Also
rmh, StraussHard.
rHardcore, rStrauss, rDiggleGratton, rDGS, rPenttinen.
Examples
Z <- rStraussHard(100,0.7,0.05,0.02)

rsyst

Simulate systematic random point pattern

Description
Generates a “systematic random” pattern of points in a window, consisting of a grid of equallyspaced points with a random common displacement.
Usage
rsyst(win=square(1), nx=NULL, ny=nx, ..., dx=NULL, dy=dx,
nsim=1, drop=TRUE)
Arguments
win

A window. An object of class owin, or data in any format acceptable to as.owin().

nx

Number of columns of grid points in the window. Incompatible with dx.

ny

Number of rows of grid points in the window. Incompatible with dy.

...

Ignored.

dx

Spacing of grid points in x direction. Incompatible with nx.

dy

Spacing of grid points in y direction. Incompatible with ny.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

rtemper

1297

Details
This function generates a “systematic random” pattern of points in the window win. The pattern
consists of a rectangular grid of points with a random common displacement.
The grid spacing in the x direction is determined either by the number of columns nx or by the
horizontal spacing dx. The grid spacing in the y direction is determined either by the number of
rows ny or by the vertical spacing dy.
The grid is then given a random displacement (the common displacement of the grid points is a
uniformly distributed random vector in the tile of dimensions dx, dy).
Some of the resulting grid points may lie outside the window win: if they do, they are deleted. The
result is a point pattern inside the window win.
This function is useful in creating dummy points for quadrature schemes (see quadscheme) as well
as in simulating random point patterns.
Value
A point pattern (an object of class "ppp") if nsim=1, or a list of point patterns if nsim > 1.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
rstrat, runifpoint, quadscheme
Examples
X <- rsyst(nx=10)
plot(X)
# polygonal boundary
data(letterR)
X <- rsyst(letterR, 5, 10)
plot(X)

rtemper

Simulated Annealing or Simulated Tempering for Gibbs Point Processes

Description
Performs simulated annealing or simulated tempering for a Gibbs point process model using a
specified annealing schedule.
Usage
rtemper(model, invtemp, nrep, ..., start = NULL, verbose = FALSE)

1298

rtemper

Arguments
model

A Gibbs point process model: a fitted Gibbs point process model (object of class
"ppm"), or any data acceptable to rmhmodel.

invtemp

A numeric vector of positive numbers. The sequence of values of inverse temperature that will be used.

nrep

An integer vector of the same length as invtemp. The value nrep[i] specifies
the number of steps of the Metropolis-Hastings algorithm that will be performed
at inverse temperature invtemp[i].

start

Initial starting state for the simulation. Any data acceptable to rmhstart.

...

Additional arguments passed to rmh.default.

verbose

Logical value indicating whether to print progress reports.

Details
The Metropolis-Hastings simulation algorithm rmh is run for nrep[1] steps at inverse temperature
invtemp[1], then for nrep[2] steps at inverse temperature invtemp[2], and so on.
Setting the inverse temperature to a value α means that the probability density of the Gibbs model,
f (x), is replaced by g(x) = C f (x)α where C is a normalising constant depending on α. Larger
values of α exaggerate the high and low values of probability density, while smaller values of α
flatten out the probability density.
For example if the original model is a Strauss process, the modified model is close to a hard core
process for large values of inverse temperature, and close to a Poisson process for small values of
inverse temperature.
Value
A point pattern (object of class "ppp").
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
rmh.default, rmh.
Examples
stra <- rmhmodel(cif="strauss",
par=list(beta=2,gamma=0.2,r=0.7),
w=square(10))
nr <- if(interactive()) 1e5 else 1e4
Y <- rtemper(stra, c(1, 2, 4, 8), nr * (1:4), verbose=TRUE)

rthin

rthin

1299

Random Thinning

Description
Applies independent random thinning to a point pattern.
Usage
rthin(X, P, ..., nsim=1, drop=TRUE)
Arguments
X

A point pattern (object of class "ppp" or "lpp") that will be thinned.

P

Data giving the retention probabilities, i.e. the probability that each point in X
will be retained. Either a single number, or a vector of numbers, or a function(x,y)
in the R language, or a function object (class "funxy" or "linfun"), or a pixel
image (object of class "im" or "linim").

...

Additional arguments passed to P, if it is a function.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

Details
In a random thinning operation, each point of the pattern X is randomly either deleted or retained
(i.e. not deleted). The result is a point pattern, consisting of those points of X that were retained.
Independent random thinning means that the retention/deletion of each point is independent of other
points.
The argument P determines the probability of retaining each point. It may be
a single number, so that each point will be retained with the same probability P;
a vector of numbers, so that the ith point of X will be retained with probability P[i];
a function P(x,y), so that a point at a location (x,y) will be retained with probability P(x,y);
an object of class "funxy" or "linfun", so that points in the pattern X will be retained with probabilities P(X);
a pixel image, containing values of the retention probability for all locations in a region encompassing the point pattern.
If P is a function P(x,y), it should be ‘vectorised’, that is, it should accept vector arguments x,y and
should yield a numeric vector of the same length. The function may have extra arguments which
are passed through the ... argument.
Value
A point pattern (object of class "ppp" or "lpp") if nsim=1, or a list of point patterns if nsim > 1.

1300

rThomas

Reproducibility
The algorithm for random thinning was changed in spatstat version 1.42-3. Set spatstat.options(fastthin=FALSE)
to use the previous, slower algorithm, if it is desired to reproduce results obtained with earlier versions.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
Examples
plot(redwood, main="thinning")
# delete 20% of points
Y <- rthin(redwood, 0.8)
points(Y, col="green", cex=1.4)
# function
f <- function(x,y) { ifelse(x < 0.4, 1, 0.5) }
Y <- rthin(redwood, f)
# pixel image
Z <- as.im(f, Window(redwood))
Y <- rthin(redwood, Z)
#
A
B
g
B

pattern on a linear network
<- runiflpp(30, simplenet)
<- rthin(A, 0.2)
<- function(x,y,seg,tp) { ifelse(y < 0.4, 1, 0.5) }
<- rthin(A, linfun(g, simplenet))

rThomas

Simulate Thomas Process

Description
Generate a random point pattern, a realisation of the Thomas cluster process.
Usage
rThomas(kappa, scale, mu, win = owin(c(0,1),c(0,1)),
nsim=1, drop=TRUE,
saveLambda=FALSE, expand = 4*scale, ...,
poisthresh=1e-6, saveparents=TRUE)
Arguments
kappa

Intensity of the Poisson process of cluster centres. A single positive number, a
function, or a pixel image.

scale

Standard deviation of random displacement (along each coordinate axis) of a
point from its cluster centre.

rThomas

1301

mu

Mean number of points per cluster (a single positive number) or reference intensity for the cluster points (a function or a pixel image).

win

Window in which to simulate the pattern. An object of class "owin" or something acceptable to as.owin.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

saveLambda

Logical. If TRUE then the random intensity corresponding to the simulated parent
points will also be calculated and saved, and returns as an attribute of the point
pattern.

expand

Numeric. Size of window expansion for generation of parent points. Has a
sensible default.

...

Passed to clusterfield to control the image resolution when saveLambda=TRUE
and to clusterradius when expand is missing.

poisthresh

Numerical threshold below which the model will be treated as a Poisson process.
See Details.

saveparents

Logical value indicating whether to save the locations of the parent points as an
attribute.

Details
This algorithm generates a realisation of the (‘modified’) Thomas process, a special case of the
Neyman-Scott process, inside the window win.
In the simplest case, where kappa and mu are single numbers, the algorithm generates a uniform
Poisson point process of “parent” points with intensity kappa. Then each parent point is replaced
by a random cluster of “offspring” points, the number of points per cluster being Poisson (mu) distributed, and their positions being isotropic Gaussian displacements from the cluster parent location.
The resulting point pattern is a realisation of the classical “stationary Thomas process” generated
inside the window win. This point process has intensity kappa * mu.
The algorithm can also generate spatially inhomogeneous versions of the Thomas process:
• The parent points can be spatially inhomogeneous. If the argument kappa is a function(x,y)
or a pixel image (object of class "im"), then it is taken as specifying the intensity function of
an inhomogeneous Poisson process that generates the parent points.
• The offspring points can be inhomogeneous. If the argument mu is a function(x,y) or a
pixel image (object of class "im"), then it is interpreted as the reference density for offspring
points, in the sense of Waagepetersen (2007). For a given parent point, the offspring constitute
a Poisson process with intensity function equal to mu * f, where f is the Gaussian probability
density centred at the parent point. Equivalently we first generate, for each parent point, a
Poisson (mumax) random number of offspring (where M is the maximum value of mu) with
independent Gaussian displacements from the parent location, and then randomly thin the
offspring points, with retention probability mu/M.
• Both the parent points and the offspring points can be spatially inhomogeneous, as described
above.
Note that if kappa is a pixel image, its domain must be larger than the window win. This is because
an offspring point inside win could have its parent point lying outside win. In order to allow this,
the simulation algorithm first expands the original window win by a distance expand and generates
the Poisson process of parent points on this larger window. If kappa is a pixel image, its domain
must contain this larger window.

1302

run.simplepanel

The intensity of the Thomas process is kappa * mu if either kappa or mu is a single number. In the
general case the intensity is an integral involving kappa, mu and f.
The Thomas process with homogeneous parents (i.e. where kappa is a single number) can be fitted
to data using kppm. Currently it is not possible to fit the Thomas model with inhomogeneous parents.
If the pair correlation function of the model is very close to that of a Poisson process, deviating by
less than poisthresh, then the model is approximately a Poisson process, and will be simulated as
a Poisson process with intensity kappa * mu, using rpoispp. This avoids computations that would
otherwise require huge amounts of memory.
Value
A point pattern (an object of class "ppp") if nsim=1, or a list of point patterns if nsim > 1.
Additionally, some intermediate results of the simulation are returned as attributes of this point pattern (see rNeymanScott). Furthermore, the simulated intensity function is returned as an attribute
"Lambda", if saveLambda=TRUE.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Diggle, P. J., Besag, J. and Gleaves, J. T. (1976) Statistical analysis of spatial point patterns by
means of distance methods. Biometrics 32 659–667.
Thomas, M. (1949) A generalisation of Poisson’s binomial limit for use in ecology. Biometrika 36,
18–25.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous NeymanScott processes. Biometrics 63, 252–258.
See Also
rpoispp, rMatClust, rCauchy, rVarGamma, rNeymanScott, rGaussPoisson, kppm, clusterfit.
Examples
#homogeneous
X <- rThomas(10, 0.2, 5)
#inhomogeneous
Z <- as.im(function(x,y){ 5 * exp(2 * x - 1) }, owin())
Y <- rThomas(10, 0.2, Z)

run.simplepanel

Run Point-and-Click Interface

Description
Execute various operations in a simple point-and-click user interface.

run.simplepanel

1303

Usage
run.simplepanel(P, popup=TRUE, verbose = FALSE)
clear.simplepanel(P)
redraw.simplepanel(P, verbose = FALSE)
Arguments
P

An interaction panel (object of class "simplepanel", created by simplepanel
or grow.simplepanel).

popup

Logical. If popup=TRUE (the default), the panel will be displayed in a new popup
window. If popup=FALSE, the panel will be displayed on the current graphics
window if it already exists, and on a new window otherwise.

verbose

Logical. If TRUE, debugging information will be printed.

Details
These commands enable the user to run a simple, robust, point-and-click interface to any R code.
The interface is implemented using only the basic graphics package in R.
The argument P is an object of class "simplepanel", created by simplepanel or grow.simplepanel,
which specifies the graphics to be displayed and the actions to be performed when the user interacts
with the panel.
The command run.simplepanel(P) activates the panel: the display is initialised and the graphics
system waits for the user to click the panel. While the panel is active, the user can only interact with
the panel; the R command line interface and the R GUI cannot be used. When the panel terminates
(typically because the user clicked a button labelled Exit), control returns to the R command line
interface and the R GUI.
The command clear.simplepanel(P) clears all the display elements in the panel, resulting in a
blank display except for the title of the panel.
The command redraw.simplepanel(P) redraws all the buttons of the panel, according to the
redraw functions contained in the panel.
If popup=TRUE (the default), run.simplepanel begins by calling dev.new so that a new popup
window is created; this window is closed using dev.off when run.simplepanel terminates. If
popup=FALSE, the panel will be displayed on the current graphics window if it already exists, and
on a new window otherwise; this window is not closed when run.simplepanel terminates.
For more sophisticated control of the graphics focus (for example, to use the panel to control
the display on another window), initialise the graphics devices yourself using dev.new or similar commands; save these devices in the shared environment env of the panel P; and write the
click/redraw functions of P in such a way that they access these devices using dev.set. Then use
run.simplepanel with popup=FALSE.
Value
The return value of run.simplepanel(P) is the value returned by the exit function of P. See
simplepanel.
The functions clear.simplepanel and redraw.simplepanel return NULL.
Author(s)
Adrian Baddeley 
and Rolf Turner 

1304

run.simplepanel

See Also
simplepanel
Examples
if(interactive()) {
# make boxes (alternatively use layout.boxes())
Bminus <- square(1)
Bvalue <- shift(Bminus, c(1.2, 0))
Bplus <- shift(Bvalue, c(1.2, 0))
Bdone <- shift(Bplus, c(1.2, 0))
myboxes <- list(Bminus, Bvalue, Bplus, Bdone)
myB <- do.call(boundingbox,myboxes)
# make environment containing an integer count
myenv <- new.env()
assign("answer", 0, envir=myenv)
# what to do when finished: return the count.
myexit <- function(e) { return(get("answer", envir=e)) }
# button clicks
# decrement the count
Cminus <- function(e, xy) {
ans <- get("answer", envir=e)
assign("answer", ans - 1, envir=e)
return(TRUE)
}
# display the count (clicking does nothing)
Cvalue <- function(...) { TRUE }
# increment the count
Cplus <- function(e, xy) {
ans <- get("answer", envir=e)
assign("answer", ans + 1, envir=e)
return(TRUE)
}
# quit button
Cdone <- function(e, xy) { return(FALSE) }
myclicks <- list("-"=Cminus,
value=Cvalue,
"+"=Cplus,
done=Cdone)
# redraw the button that displays the current value of the count
Rvalue <- function(button, nam, e) {
plot(button, add=TRUE)
ans <- get("answer", envir=e)
text(centroid.owin(button), labels=ans)
return(TRUE)
}
# make the panel
P <- simplepanel("Counter",
B=myB, boxes=myboxes,
clicks=myclicks,

runifdisc

P

1305
redraws = list(NULL, Rvalue, NULL, NULL),
exit=myexit, env=myenv)

run.simplepanel(P)
}

runifdisc

Generate N Uniform Random Points in a Disc

Description
Generate a random point pattern containing n independent uniform random points in a circular disc.
Usage
runifdisc(n, radius=1, centre=c(0,0), ..., nsim=1, drop=TRUE)
Arguments
n

Number of points.

radius

Radius of the circle.

centre

Coordinates of the centre of the circle.

...

Arguments passed to disc controlling the accuracy of approximation to the circle.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

Details
This function generates n independent random points, uniformly distributed in a circular disc.
It is faster (for a circular window) than the general code used in runifpoint.
To generate random points in an ellipse, first generate points in a circle using runifdisc, then
transform to an ellipse using affine, as shown in the examples.
To generate random points in other windows, use runifpoint. To generate non-uniform random
points, use rpoint.
Value
A point pattern (an object of class "ppp") if nsim=1, or a list of point patterns if nsim > 1.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
disc, runifpoint, rpoint

1306

runiflpp

Examples
# 100 random points in the unit disc
plot(runifdisc(100))
# 42 random points in the ellipse with major axis 3 and minor axis 1
X <- runifdisc(42)
Y <- affine(X, mat=diag(c(3,1)))
plot(Y)

runiflpp

Uniform Random Points on a Linear Network

Description
Generates n random points, independently and uniformly distributed, on a linear network.
Usage
runiflpp(n, L, nsim=1, drop=TRUE)
Arguments
n

Number of random points to generate. A nonnegative integer, or a vector of
integers specifying the number of points of each type.

L

A linear network (object of class "linnet", see linnet).

nsim

Number of simulated realisations to generate.

drop

Logical value indicating what to do when nsim=1. If drop=TRUE (the default),
the result is a point pattern. If drop=FALSE, the result is a list with one entry
which is a point pattern.

Details
This function uses runifpointOnLines to generate the random points.
Value
If nsim = 1 and drop=TRUE, a point pattern on the linear network, i.e.\ an object of class "lpp".
Otherwise, a list of such point patterns.
Author(s)
Ang Qi Wei  and Adrian Baddeley 
See Also
rlpp for non-uniform random points; rpoislpp for Poisson point process;
lpp, linnet

runifpoint

1307

Examples
data(simplenet)
X <- runiflpp(10, simplenet)
plot(X)
# marked
Z <- runiflpp(c(a=10, b=3), simplenet)

runifpoint

Generate N Uniform Random Points

Description
Generate a random point pattern containing n independent uniform random points.
Usage
runifpoint(n, win=owin(c(0,1),c(0,1)), giveup=1000, warn=TRUE, ...,
nsim=1, drop=TRUE, ex=NULL)
Arguments
n

Number of points.

win

Window in which to simulate the pattern. An object of class "owin" or something acceptable to as.owin.

giveup

Number of attempts in the rejection method after which the algorithm should
stop trying to generate new points.

warn

Logical. Whether to issue a warning if n is very large. See Details.

...

Ignored.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

ex

Optional. A point pattern to use as the example. If ex is given and n and win are
missing, then n and win will be calculated from the point pattern ex.

Details
This function generates n independent random points, uniformly distributed in the window win.
(For nonuniform distributions, see rpoint.)
The algorithm depends on the type of window, as follows:
• If win is a rectangle then n independent random points, uniformly distributed in the rectangle,
are generated by assigning uniform random values to their cartesian coordinates.
• If win is a binary image mask, then a random sequence of pixels is selected (using sample)
with equal probabilities. Then for each pixel in the sequence we generate a uniformly distributed random point in that pixel.
• If win is a polygonal window, the algorithm uses the rejection method. It finds a rectangle
enclosing the window, generates points in this rectangle, and tests whether they fall in the
desired window. It gives up when giveup * n tests have been performed without yielding n
successes.

1308

runifpoint3

The algorithm for binary image masks is faster than the rejection method but involves discretisation.
If warn=TRUE, then a warning will be issued if n is very large. The threshold is spatstat.options("huge.npoints").
This warning has no consequences, but it helps to trap a number of common errors.
Value
A point pattern (an object of class "ppp") if nsim=1, or a list of point patterns if nsim > 1.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
ppp.object, owin.object, rpoispp, rpoint
Examples
# 100 random points in the unit square
pp <- runifpoint(100)
# irregular window
data(letterR)
# polygonal
pp <- runifpoint(100, letterR)
# binary image mask
pp <- runifpoint(100, as.mask(letterR))
##
# randomising an existing point pattern
runifpoint(npoints(cells), win=Window(cells))
runifpoint(ex=cells)

runifpoint3

Generate N Uniform Random Points in Three Dimensions

Description
Generate a random point pattern containing n independent, uniform random points in three dimensions.
Usage
runifpoint3(n, domain = box3(), nsim=1, drop=TRUE)
Arguments
n

Number of points to be generated.

domain

Three-dimensional box in which the process should be generated. An object of
class "box3".

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

runifpointOnLines

1309

Details
This function generates n independent random points, uniformly distributed in the three-dimensional
box domain.
Value
If nsim = 1 and drop=TRUE, a point pattern in three dimensions (an object of class "pp3"). If
nsim > 1, a list of such point patterns.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
rpoispp3, pp3, box3
Examples
X <- runifpoint3(50)

runifpointOnLines

Generate N Uniform Random Points On Line Segments

Description
Given a line segment pattern, generate a random point pattern consisting of n points uniformly
distributed on the line segments.
Usage
runifpointOnLines(n, L, nsim=1)
Arguments
n
L
nsim

Number of points to generate.
Line segment pattern (object of class "psp") on which the points should lie.
Number of simulated realisations to be generated.

Details
This command generates a point pattern consisting of n independent random points, each point
uniformly distributed on the line segment pattern. This means that, for each random point,
• the probability of falling on a particular segment is proportional to the length of the segment;
and
• given that the point falls on a particular segment, it has uniform probability density along that
segment.
If n is a single integer, the result is an unmarked point pattern containing n points. If n is a vector of
integers, the result is a marked point pattern, with m different types of points, where m = length(n),
in which there are n[j] points of type j.

1310

runifpointx

Value
If nsim = 1, a point pattern (object of class "ppp") with the same window as L. If nsim > 1, a list
of point patterns.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
psp, ppp, pointsOnLines, runifpoint
Examples
X <- psp(runif(10), runif(10), runif(10), runif(10),
Y <- runifpointOnLines(20, X)
plot(X, main="")
plot(Y, add=TRUE)
Z <- runifpointOnLines(c(5,5), X)

runifpointx

window=owin())

Generate N Uniform Random Points in Any Dimensions

Description
Generate a random point pattern containing n independent, uniform random points in any number
of spatial dimensions.
Usage
runifpointx(n, domain, nsim=1, drop=TRUE)
Arguments
n

Number of points to be generated.

domain

Multi-dimensional box in which the process should be generated. An object of
class "boxx".

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a single point pattern.

Details
This function generates a pattern of n independent random points, uniformly distributed in the
multi-dimensional box domain.
Value
If nsim = 1 and drop=TRUE, a point pattern (an object of class "ppx"). If nsim > 1 or drop=FALSE,
a list of such point patterns.

rVarGamma

1311

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
rpoisppx, ppx, boxx
Examples
w <- boxx(x=c(0,1), y=c(0,1), z=c(0,1), t=c(0,3))
X <- runifpointx(50, w)

rVarGamma

Simulate Neyman-Scott Point Process with Variance Gamma cluster
kernel

Description
Generate a random point pattern, a simulated realisation of the Neyman-Scott process with Variance
Gamma (Bessel) cluster kernel.
Usage
rVarGamma(kappa, nu, scale, mu, win = owin(),
thresh = 0.001, nsim=1, drop=TRUE,
saveLambda=FALSE, expand = NULL, ...,
poisthresh=1e-6, saveparents=TRUE)
Arguments
kappa

Intensity of the Poisson process of cluster centres. A single positive number, a
function, or a pixel image.

nu

Shape parameter for the cluster kernel. A number greater than -1.

scale

Scale parameter for cluster kernel. Determines the size of clusters. A positive
number in the same units as the spatial coordinates.

mu

Mean number of points per cluster (a single positive number) or reference intensity for the cluster points (a function or a pixel image).

win

Window in which to simulate the pattern. An object of class "owin" or something acceptable to as.owin.

thresh

Threshold relative to the cluster kernel value at the origin (parent location) determining when the cluster kernel will be treated as zero for simulation purposes.
Will be overridden by argument expand if that is given.

nsim

Number of simulated realisations to be generated.

drop

Logical. If nsim=1 and drop=TRUE (the default), the result will be a point pattern, rather than a list containing a point pattern.

saveLambda

Logical. If TRUE then the random intensity corresponding to the simulated parent
points will also be calculated and saved, and returns as an attribute of the point
pattern.

1312

rVarGamma

expand

Numeric. Size of window expansion for generation of parent points. By default
determined by calling clusterradius with the numeric threshold value given
in thresh.

...

Passed to clusterfield to control the image resolution when saveLambda=TRUE
and to clusterradius when expand is missing or NULL.

poisthresh

Numerical threshold below which the model will be treated as a Poisson process.
See Details.

saveparents

Logical value indicating whether to save the locations of the parent points as an
attribute.

Details
This algorithm generates a realisation of the Neyman-Scott process with Variance Gamma (Bessel)
cluster kernel, inside the window win.
The process is constructed by first generating a Poisson point process of “parent” points with intensity kappa. Then each parent point is replaced by a random cluster of points, the number of
points in each cluster being random with a Poisson (mu) distribution, and the points being placed
independently and uniformly according to a Variance Gamma kernel.
The shape of the kernel is determined by the dimensionless index nu. This is the parameter ν 0 =
α/2 − 1 appearing in equation (12) on page 126 of Jalilian et al (2013).
The scale of the kernel is determined by the argument scale, which is the parameter η appearing
in equations (12) and (13) of Jalilian et al (2013). It is expressed in units of length (the same as the
unit of length for the window win).
In this implementation, parent points are not restricted to lie in the window; the parent process is
effectively the uniform Poisson process on the infinite plane.
This model can be fitted to data by the method of minimum contrast, maximum composite likelihood or Palm likelihood using kppm.
The algorithm can also generate spatially inhomogeneous versions of the cluster process:
• The parent points can be spatially inhomogeneous. If the argument kappa is a function(x,y)
or a pixel image (object of class "im"), then it is taken as specifying the intensity function of
an inhomogeneous Poisson process that generates the parent points.
• The offspring points can be inhomogeneous. If the argument mu is a function(x,y) or a
pixel image (object of class "im"), then it is interpreted as the reference density for offspring
points, in the sense of Waagepetersen (2006).
When the parents are homogeneous (kappa is a single number) and the offspring are inhomogeneous (mu is a function or pixel image), the model can be fitted to data using kppm, or using
vargamma.estK or vargamma.estpcf applied to the inhomogeneous K function.
If the pair correlation function of the model is very close to that of a Poisson process, deviating by
less than poisthresh, then the model is approximately a Poisson process, and will be simulated as
a Poisson process with intensity kappa * mu, using rpoispp. This avoids computations that would
otherwise require huge amounts of memory.
Value
A point pattern (an object of class "ppp") if nsim=1, or a list of point patterns if nsim > 1.
Additionally, some intermediate results of the simulation are returned as attributes of this point pattern (see rNeymanScott). Furthermore, the simulated intensity function is returned as an attribute
"Lambda", if saveLambda=TRUE.

SatPiece

1313

Author(s)

Abdollah Jalilian and Rasmus Waagepetersen. Adapted for spatstat by Adrian Baddeley 
and Rolf Turner 
in collaboration with Hao Wang and Jeff Picka
See Also
ppm, pairsat.family, Geyer, PairPiece, BadGey.
Examples
SatPiece(c(0.1,0.2), c(1,1))
# prints a sensible description of itself
SatPiece(c(0.1,0.2), 1)
data(cells)

Saturated

1315

ppm(cells, ~1, SatPiece(c(0.07, 0.1, 0.13), 2))
# fit a stationary piecewise constant Saturated pairwise interaction process
## Not run:
ppm(cells, ~polynom(x,y,3), SatPiece(c(0.07, 0.1, 0.13), 2))
# nonstationary process with log-cubic polynomial trend
## End(Not run)

Saturated

Saturated Pairwise Interaction model

Description
Experimental.
Usage
Saturated(pot, name)
Arguments
pot

An S language function giving the user-supplied pairwise interaction potential.

name

Character string.

Details
This is experimental. It constructs a member of the “saturated pairwise” family pairsat.family.
Value
An object of class "interact" describing the interpoint interaction structure of a point process.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
ppm, pairsat.family, Geyer, SatPiece, ppm.object

1316

scalardilate

scalardilate

Apply Scalar Dilation

Description
Applies scalar dilation to a plane geometrical object, such as a point pattern or a window, relative
to a specified origin.
Usage
scalardilate(X, f, ...)
## S3 method for class 'im'
scalardilate(X, f, ..., origin=NULL)
## S3 method for class 'owin'
scalardilate(X, f, ..., origin=NULL)
## S3 method for class 'ppp'
scalardilate(X, f, ..., origin=NULL)
## S3 method for class 'psp'
scalardilate(X, f, ..., origin=NULL)
## Default S3 method:
scalardilate(X, f, ...)
Arguments
X

Any suitable dataset representing a two-dimensional object, such as a point pattern (object of class "ppp"), a window (object of class "owin"), a pixel image
(class "im") and so on.

f

Scalar dilation factor. A finite number greater than zero.

...

Ignored by the methods.

origin

Origin for the scalar dilation. Either a vector of 2 numbers, or one of the character strings "centroid", "midpoint" or "bottomleft" (partially matched).

Details
This command performs scalar dilation of the object X by the factor f relative to the origin specified
by origin.
The function scalardilate is generic, with methods for windows (class "owin"), point patterns
(class "ppp"), pixel images (class "im"), line segment patterns (class "psp") and a default method.
If the argument origin is not given, then every spatial coordinate is multiplied by the factor f.
If origin is given, then scalar dilation is performed relative to the specified origin. Effectively, X
is shifted so that origin is moved to c(0,0), then scalar dilation is performed, then the result is
shifted so that c(0,0) is moved to origin.
This command is a special case of an affine transformation: see affine.

scaletointerval

1317

Value
Another object of the same type, representing the result of applying the scalar dilation.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
affine, shift
Examples
plot(letterR)
plot(scalardilate(letterR, 0.7, origin="bot"), col="red", add=TRUE)

scaletointerval

Rescale Data to Lie Between Specified Limits

Description
Rescales a dataset so that the values range exactly between the specified limits.
Usage
scaletointerval(x, from=0, to=1, xrange=range(x))
## Default S3 method:
scaletointerval(x, from=0, to=1, xrange=range(x))
## S3 method for class 'im'
scaletointerval(x, from=0, to=1, xrange=range(x))
Arguments
x

Data to be rescaled.

from,to

Lower and upper endpoints of the interval to which the values of x should be
rescaled.

xrange

Optional range of values of x that should be mapped to the new interval.

Details
These functions rescale a dataset x so that its values range exactly between the limits from and to.
The method for pixel images (objects of class "im") applies this scaling to the pixel values of x.
Rescaling cannot be performed if the values in x are not interpretable as numeric, or if the values in
x are all equal.
Value
An object of the same type as x.

1318

scan.test

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
scale
Examples
X <- as.im(function(x,y) {x+y+3}, unit.square())
summary(X)
Y <- scaletointerval(X)
summary(Y)

scan.test

Spatial Scan Test

Description
Performs the Spatial Scan Test for clustering in a spatial point pattern, or for clustering of one type
of point in a bivariate spatial point pattern.
Usage
scan.test(X, r, ...,
method = c("poisson", "binomial"),
nsim = 19,
baseline = NULL,
case = 2,
alternative = c("greater", "less", "two.sided"),
verbose = TRUE)
Arguments
X

A point pattern (object of class "ppp").

r

Radius of circle to use. A single number or a numeric vector.

...

Optional. Arguments passed to as.mask to determine the spatial resolution of
the computations.

method

Either "poisson" or "binomial" specifying the type of likelihood.

nsim

Number of simulations for computing Monte Carlo p-value.

baseline

Baseline for the Poisson intensity, if method="poisson". A pixel image or a
function.

case

Which type of point should be interpreted as a case, if method="binomial".
Integer or character string.

alternative

Alternative hypothesis: "greater" if the alternative postulates that the mean
number of points inside the circle will be greater than expected under the null.

verbose

Logical. Whether to print progress reports.

scan.test

1319

Details
The spatial scan test (Kulldorf, 1997) is applied to the point pattern X.
In a nutshell,
• If method="poisson" then a significant result would mean that there is a circle of radius r,
located somewhere in the spatial domain of the data, which contains a significantly higher
than expected number of points of X. That is, the pattern X exhibits spatial clustering.
• If method="binomial" then X must be a bivariate (two-type) point pattern. By default, the
first type of point is interpreted as a control (non-event) and the second type of point as a case
(event). A significant result would mean that there is a circle of radius r which contains a
significantly higher than expected number of cases. That is, the cases are clustered together,
conditional on the locations of all points.
Following is a more detailed explanation.
• If method="poisson" then the scan test based on Poisson likelihood is performed (Kulldorf,
1997). The dataset X is treated as an unmarked point pattern. By default (if baseline is not
specified) the null hypothesis is complete spatial randomness CSR (i.e. a uniform Poisson
process). The alternative hypothesis is a Poisson process with one intensity β1 inside some
circle of radius r and another intensity β0 outside the circle. If baseline is given, then it
should be a pixel image or a function(x,y). The null hypothesis is an inhomogeneous
Poisson process with intensity proportional to baseline. The alternative hypothesis is an
inhomogeneous Poisson process with intensity beta1 * baseline inside some circle of
radius r, and beta0 * baseline outside the circle.
• If method="binomial" then the scan test based on binomial likelihood is performed (Kulldorf,
1997). The dataset X must be a bivariate point pattern, i.e. a multitype point pattern with two
types. The null hypothesis is that all permutations of the type labels are equally likely. The
alternative hypothesis is that some circle of radius r has a higher proportion of points of the
second type, than expected under the null hypothesis.
The result of scan.test is a hypothesis test (object of class "htest") which can be plotted to report
the results. The component p.value contains the p-value.
The result of scan.test can also be plotted (using the plot method for the class "scan.test").
The plot is a pixel image of the Likelihood Ratio Test Statistic (2 times the log likelihood ratio) as a
function of the location of the centre of the circle. This pixel image can be extracted from the object
using as.im.scan.test. The Likelihood Ratio Test Statistic is computed by scanLRTS.
Value
An object of class "htest" (hypothesis test) which also belongs to the class "scan.test". Printing
this object gives the result of the test. Plotting this object displays the Likelihood Ratio Test Statistic
as a function of the location of the centre of the circle.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Kulldorff, M. (1997) A spatial scan statistic. Communications in Statistics — Theory and Methods
26, 1481–1496.

1320

scanLRTS

See Also
plot.scan.test, as.im.scan.test, relrisk, scanLRTS
Examples
nsim <- if(interactive()) 19 else 2
rr <- if(interactive()) seq(0.5, 1, by=0.1) else c(0.5, 1)
scan.test(redwood, 0.1 * rr, method="poisson", nsim=nsim)
scan.test(chorley, rr, method="binomial", case="larynx", nsim=nsim)

scanLRTS

Likelihood Ratio Test Statistic for Scan Test

Description
Calculate the Likelihood Ratio Test Statistic for the Scan Test, at each spatial location.

Usage
scanLRTS(X, r, ...,
method = c("poisson", "binomial"),
baseline = NULL, case = 2,
alternative = c("greater", "less", "two.sided"),
saveopt = FALSE,
Xmask = NULL)
Arguments
X

A point pattern (object of class "ppp").

r

Radius of circle to use. A single number or a numeric vector.

...

Optional. Arguments passed to as.mask to determine the spatial resolution of
the computations.

method

Either "poisson" or "binomial" specifying the type of likelihood.

baseline

Baseline for the Poisson intensity, if method="poisson". A pixel image or a
function.

case

Which type of point should be interpreted as a case, if method="binomial".
Integer or character string.

alternative

Alternative hypothesis: "greater" if the alternative postulates that the mean
number of points inside the circle will be greater than expected under the null.

saveopt

Logical value indicating to save the optimal value of r at each location.

Xmask

Internal use only.

scanLRTS

1321

Details
This command computes, for all spatial locations u, the Likelihood Ratio Test Statistic Λ(u) for a
test of homogeneity at the location u, as described below. The result is a pixel image giving the
values of Λ(u) at each pixel.
The maximum value of Λ(u) over all locations u is the scan statistic, which is the basis of the scan
test performed by scan.test.
• If method="poisson" then the test statistic is based on Poisson likelihood. The dataset X
is treated as an unmarked point pattern. By default (if baseline is not specified) the null
hypothesis is complete spatial randomness CSR (i.e. a uniform Poisson process). At the
spatial location u, the alternative hypothesis is a Poisson process with one intensity β1 inside
the circle of radius r centred at u, and another intensity β0 outside the circle. If baseline
is given, then it should be a pixel image or a function(x,y). The null hypothesis is an
inhomogeneous Poisson process with intensity proportional to baseline. The alternative
hypothesis is an inhomogeneous Poisson process with intensity beta1 * baseline inside the
circle, and beta0 * baseline outside the circle.
• If method="binomial" then the test statistic is based on binomial likelihood. The dataset X
must be a bivariate point pattern, i.e. a multitype point pattern with two types. The null hypothesis is that all permutations of the type labels are equally likely. The alternative hypothesis
is that the circle of radius r centred at u has a higher proportion of points of the second type,
than expected under the null hypothesis.
If r is a vector of more than one value for the radius, then the calculations described above are
performed for every value of r. Then the maximum over r is taken for each spatial location u. The
resulting pixel value of scanLRTS at a location u is the profile maximum of the Likelihood Ratio
Test Statistic, that is, the maximum of the Likelihood Ratio Test Statistic for circles of all radii,
centred at the same location u.
If you have already performed a scan test using scan.test, the Likelihood Ratio Test Statistic can
be extracted from the test result using the function as.im.scan.test.
Value
A pixel image (object of class "im") whose pixel values are the values of the (profile) Likelihood
Ratio Test Statistic at each spatial location.
Warning: window size
Note that the result of scanLRTS is a pixel image on a larger window than the original window of X.
The expanded window contains the centre of any circle of radius r that has nonempty intersection
with the original window.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Kulldorff, M. (1997) A spatial scan statistic. Communications in Statistics — Theory and Methods
26, 1481–1496.

1322

scanpp

See Also
scan.test, as.im.scan.test
Examples
plot(scanLRTS(redwood, 0.1, method="poisson"))
sc <- scanLRTS(chorley, 1, method="binomial", case="larynx")
plot(sc)
scanstatchorley <- max(sc)

scanpp

Read Point Pattern From Data File

Description
Reads a point pattern dataset from a text file.
Usage
scanpp(filename, window, header=TRUE, dir="", factor.marks=NULL, ...)
Arguments
filename

String name of the file containing the coordinates of the points in the point pattern, and their marks if any.

window

Window for the point pattern. An object of class "owin".

header

Logical flag indicating whether the first line of the file contains headings for the
columns. Passed to read.table.

dir

String containing the path name of the directory in which filename is to be
found. Default is the current directory.

factor.marks

Logical vector (or NULL) indicating whether marks are to be interpreted as
factors. Defaults to NULL which means that strings will be interpreted as factors
while numeric variables will not. See details.

...

Ignored.

Details
This simple function reads a point pattern dataset from a file containing the cartesian coordinates of
its points, and optionally the mark values for these points.
The file identified by filename in directory dir should be a text file that can be read using read.table.
Thus, each line of the file (except possibly the first line) contains data for one point in the point pattern. Data are arranged in columns. There should be either two columns (for an unmarked point
pattern) or more columns (for a marked point pattern).
If header=FALSE then the first two columns of data will be interpreted as the x and y coordinates of
points. Remaining columns, if present, will be interpreted as containing the marks for these points.
If header=TRUE then the first line of the file should contain string names for each of the columns of
data. If there are columns named x and y then these will be taken as the cartesian coordinates, and
any remaining columns will be taken as the marks. If there are no columns named x and y then the
first and second columns will be taken as the cartesian coordinates.

sdr

1323
If a logical vector is provided for factor.marks the length should equal the number of mark
columns (a shorter factor.marks is recycled to this length). This vector is then used to determine which mark columns should be interpreted as factors. Note: Strings will not be interpreted as
factors if the corresponding entry in factor.marks is FALSE.
Note that there is intentionally no default for window. The window of observation should be specified. If you really need to estimate the window, use the Ripley-Rasson estimator ripras.

Value
A point pattern (an object of class "ppp", see ppp.object).
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
ppp.object, ppp, as.ppp, ripras

sdr

Sufficient Dimension Reduction

Description
Given a point pattern and a set of predictors, find a minimal set of new predictors, each constructed
as a linear combination of the original predictors.
Usage
sdr(X, covariates, method = c("DR", "NNIR", "SAVE", "SIR", "TSE"),
Dim1 = 1, Dim2 = 1, predict=FALSE)
Arguments
X

A point pattern (object of class "ppp").

covariates

A list of pixel images (objects of class "im") to serve as predictor variables.

method

Character string indicating which method to use. See Details.

Dim1

Dimension of the first order Central Intensity Subspace (applicable when method
is "DR", "NNIR", "SAVE" or "TSE").

Dim2

Dimension of the second order Central Intensity Subspace (applicable when
method="TSE").

predict

Logical value indicating whether to compute the new predictors as well.

1324

sdr

Details
Given a point pattern X and predictor variables Z1 , . . . , Zp , Sufficient Dimension Reduction methods (Guan and Wang, 2010) attempt to find a minimal set of new predictor variables, each constructed by taking a linear combination of the original predictors, which explain the dependence
of X on Z1 , . . . , Zp . The methods do not assume any particular form of dependence of the point
pattern on the predictors. The predictors are assumed to be Gaussian random fields.
Available methods are:

sdr

1325
method="DR"
method="NNIR"
method="SAVE" & sliced average variance estimation
method="SIR" & sliced inverse regression
method="TSE" & two-step estimation

directional regression
nearest neighbour inverse regression

The result includes a matrix B whose columns are estimates of the basis vectors of the space of new
predictors. That is, the jth column of B expresses the jth new predictor as a linear combination of
the original predictors.
If predict=TRUE, the new predictors are also evaluated. They can also be evaluated using sdrPredict.
Value
A list with components B, M or B, M1, M2 where B is a matrix whose columns are estimates of the
basis vectors for the space, and M or M1,M2 are matrices containing estimates of the kernel.
If predict=TRUE, the result also includes a component Y which is a list of pixel images giving the
values of the new predictors.

Author(s)
Matlab original by Yongtao Guan, translated to R by Suman Rakshit.

References
Guan, Y. and Wang, H. (2010) Sufficient dimension reduction for spatial point processes directed
by Gaussian random fields. Journal of the Royal Statistical Society, Series B, 72, 367–387.

See Also
sdrPredict to compute the new predictors from the coefficient matrix.
dimhat to estimate the subspace dimension.
subspaceDistance
Examples
A <- sdr(bei, bei.extra, predict=TRUE)
A
Y1 <- A$Y[[1]]
plot(Y1)
points(bei, pch=".", cex=2)
# investigate likely form of dependence
plot(rhohat(bei, Y1))

1326

sdrPredict

sdrPredict

Compute Predictors from Sufficient Dimension Reduction

Description
Given the result of a Sufficient Dimension Reduction method, compute the new predictors.

Usage
sdrPredict(covariates, B)
Arguments
covariates

A list of pixel images (objects of class "im").

B

Either a matrix of coefficients for the covariates, or the result of a call to sdr.

Details
This function assumes that sdr has already been used to find a minimal set of predictors based
on the covariates. The argument B should be either the result of sdr or the coefficient matrix
returned as one of the results of sdr. The columns of this matrix define linear combinations of the
covariates. This function evaluates those linear combinations, and returns a list of pixel images
containing the new predictors.

Value
A list of pixel images (objects of class "im") with one entry for each column of B.
Author(s)
Adrian Baddeley 
See Also
sdr
Examples
A <- sdr(bei, bei.extra)
Y <- sdrPredict(bei.extra, A)
Y

segregation.test

segregation.test

1327

Test of Spatial Segregation of Types

Description
Performs a Monte Carlo test of spatial segregation of the types in a multitype point pattern.
Usage
segregation.test(X, ...)
## S3 method for class 'ppp'
segregation.test(X, ..., nsim = 19,
permute = TRUE, verbose = TRUE, Xname)
Arguments
X

Multitype point pattern (object of class "ppp" with factor-valued marks).

...

Additional arguments passed to relrisk.ppp to control the smoothing parameter or bandwidth selection.

nsim

Number of simulations for the Monte Carlo test.

permute

Argument passed to rlabel. If TRUE (the default), randomisation is performed
by randomly permuting the labels of X. If FALSE, randomisation is performing
by resampling the labels with replacement.

verbose

Logical value indicating whether to print progress reports.

Xname

Optional character string giving the name of the dataset X.

Details
The Monte Carlo test of spatial segregation of types, proposed by Kelsall and Diggle (1995) and
Diggle et al (2005), is applied to the point pattern X. The test statistic is
T =

XX
i

(b
p(m | xi ) − pm )

2

m

where pb(m | xi ) is the leave-one-out kernel smoothing estimate of the probability that the i-th data
point has type m, and pm is the average fraction of data points which are of type m. The statistic T
is evaluated for the data and for nsim randomised versions of X, generated by randomly permuting
or resampling the marks.
Note that, by default, automatic bandwidth selection will be performed separately for each randomised pattern. This computation can be very time-consuming but is necessary for the test to be
valid in most conditions. A short-cut is to specify the value of the smoothing bandwidth sigma as
shown in the examples.
Value
An object of class "htest" representing the result of the test.

1328

selfcrossing.psp

Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Kelsall, J.E. and Diggle, P.J. (1995) Kernel estimation of relative risk. Bernoulli 1, 3–16.
Diggle, P.J., Zheng, P. and Durr, P. (2005) Non-parametric estimation of spatial segregation in a
multivariate point process: bovine tuberculosis in Cornwall, UK. Applied Statistics 54, 645–658.
See Also
relrisk
Examples
segregation.test(hyytiala, 5)
if(interactive()) segregation.test(hyytiala, hmin=0.05)

selfcrossing.psp

Crossing Points in a Line Segment Pattern

Description
Finds any crossing points between the line segments in a line segment pattern.
Usage
selfcrossing.psp(A)
Arguments
A

Line segment pattern (object of class "psp").

Details
This function finds any crossing points between different line segments in the line segment pattern
A.
A crossing point occurs whenever one of the line segments in A intersects another line segment in
A, at a nonzero angle of intersection.
Value
Point pattern (object of class "ppp").
Author(s)
Adrian Baddeley 
and Rolf Turner 

selfcut.psp

1329

See Also
crossing.psp, psp.object, ppp.object.
Examples
a <- psp(runif(10), runif(10), runif(10), runif(10), window=owin())
plot(a, col="green", main="selfcrossing.psp")
P <- selfcrossing.psp(a)
plot(P, add=TRUE, col="red")

selfcut.psp

Cut Line Segments Where They Intersect

Description
Finds any crossing points between the line segments in a line segment pattern, and cuts the segments
into pieces at these crossing-points.
Usage
selfcut.psp(A, ..., eps)
Arguments
A

Line segment pattern (object of class "psp").

eps

Optional. Smallest permissible length of the resulting line segments. There is a
sensible default.

...

Ignored.

Details
This function finds any crossing points between different line segments in the line segment pattern
A, and cuts the line segments into pieces at these intersection points.
A crossing point occurs whenever one of the line segments in A intersects another line segment in
A, at a nonzero angle of intersection.
Value
Another line segment pattern (object of class "psp") in the same window as A with the same kind
of marks as A.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
selfcrossing.psp

1330

sessionLibs

Examples
X <- psp(runif(10), runif(10), runif(10), runif(10), window=owin())
Y <- selfcut.psp(X)
n <- nsegments(Y)
plot(Y %mark% factor(sample(seq_len(n), n, replace=TRUE)))

sessionLibs

Print Names and Version Numbers of Libraries Loaded

Description
Prints the names and version numbers of libraries currently loaded by the user.

Usage
sessionLibs()

Details
This function prints a list of the libraries loaded by the user in the current session, giving just their
name and version number. It obtains this information from sessionInfo.
This function is not needed in an interactive R session because the package startup messages will
usually provide this information.
Its main use is in an Sweave script, where it is needed because the package startup messages are not
printed.

Value
Null.

Author(s)
Adrian Baddeley 
and Rolf Turner 

See Also
sessionInfo

setcov

1331

setcov

Set Covariance of a Window

Description
Computes the set covariance function of a window.
Usage
setcov(W, V=W, ...)
Arguments
W
V
...

A window (object of class "owin".
Optional. Another window.
Optional arguments passed to as.mask to control the pixel resolution.

Details
The set covariance function of a region W in the plane is the function C(v) defined for each vector
v as the area of the intersection between W and W + v, where W + v is the set obtained by shifting
(translating) W by v.
We may interpret C(v) as the area of the set of all points x in W such that x + v also lies in W .
This command computes a discretised approximation to the set covariance function of any plane
region W represented as a window object (of class "owin", see owin.object). The return value
is a pixel image (object of class "im") whose greyscale values are values of the set covariance
function.
The set covariance is computed using the Fast Fourier Transform, unless W is a rectangle, when an
exact formula is used.
If the argument V is present, then setcov(W,V) computes the set cross-covariance function C(x)
defined for each vector x as the area of the intersection between W and V + x.
Value
A pixel image (an object of class "im") representing the set covariance function of W, or the crosscovariance of W and V.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
imcov, owin, as.owin, erosion
Examples
w <- owin(c(0,1),c(0,1))
v <- setcov(w)
plot(v)

1332

sharpen

sharpen

Data Sharpening of Point Pattern

Description
Performs Choi-Hall data sharpening of a spatial point pattern.
Usage
sharpen(X, ...)
## S3 method for class 'ppp'
sharpen(X, sigma=NULL, ...,
varcov=NULL, edgecorrect=FALSE)
Arguments
X

A marked point pattern (object of class "ppp").

sigma

Standard deviation of isotropic Gaussian smoothing kernel.

varcov

Variance-covariance matrix of anisotropic Gaussian kernel. Incompatible with
sigma.

edgecorrect

Logical value indicating whether to apply edge effect bias correction.

...

Arguments passed to density.ppp to control the pixel resolution of the result.

Details
Choi and Hall (2001) proposed a procedure for data sharpening of spatial point patterns. This procedure is appropriate for earthquake epicentres and other point patterns which are believed to exhibit
strong concentrations of points along a curve. Data sharpening causes such points to concentrate
more tightly along the curve.
If the original data points are X1 , . . . , Xn then the sharpened points are
P
j Xj k(Xj − Xi )
X̂i = P
j k(Xj − Xi )
where k is a smoothing kernel in two dimensions. Thus, the new point X̂i is a vector average of the
nearby points X[j].
The function sharpen is generic. It currently has only one method, for two-dimensional point
patterns (objects of class "ppp").
If sigma is given, the smoothing kernel is the isotropic two-dimensional Gaussian density with
standard deviation sigma in each axis. If varcov is given, the smoothing kernel is the Gaussian
density with variance-covariance matrix varcov.
The data sharpening procedure tends to cause the point pattern to contract away from the boundary
of the window. That is, points X_iX[i] that lie ‘quite close to the edge of the window of the point
pattern tend to be displaced inward. If edgecorrect=TRUE then the algorithm is modified to correct
this vector bias.
Value
A point pattern (object of class "ppp") in the same window as the original pattern X, and with the
same marks as X.

shift

1333

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
References
Choi, E. and Hall, P. (2001) Nonparametric analysis of earthquake point-process data. In M. de
Gunst, C. Klaassen and A. van der Vaart (eds.) State of the art in probability and statistics:
Festschrift for Willem R. van Zwet, Institute of Mathematical Statistics, Beachwood, Ohio. Pages
324–344.
See Also
density.ppp, Smooth.ppp.
Examples
data(shapley)
X <- unmark(shapley)
Y <- sharpen(X, sigma=0.5)
Z <- sharpen(X, sigma=0.5, edgecorrect=TRUE)
opa <- par(mar=rep(0.2, 4))
plot(solist(X, Y, Z), main= " ",
main.panel=c("data", "sharpen", "sharpen, correct"),
pch=".", equal.scales=TRUE, mar.panel=0.2)
par(opa)

shift

Apply Vector Translation

Description
Applies a vector shift of the plane to a geometrical object, such as a point pattern or a window.
Usage
shift(X, ...)
Arguments
X

Any suitable dataset representing a two-dimensional object, such as a point pattern (object of class "ppp"), or a window (object of class "owin").

...

Arguments determining the shift vector.

Details
This is generic. Methods are provided for point patterns (shift.ppp) and windows (shift.owin).
The object is translated by the vector vec.
Value
Another object of the same type, representing the result of applying the shift.

1334

shift.im

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
shift.ppp, shift.owin, rotate, affine, periodify

shift.im

Apply Vector Translation To Pixel Image

Description
Applies a vector shift to a pixel image
Usage
## S3 method for class 'im'
shift(X, vec=c(0,0), ..., origin=NULL)
Arguments
X

Pixel image (object of class "im").

vec

Vector of length 2 representing a translation.

...

Ignored

origin

Character string determining a location that will be shifted to the origin. Options
are "centroid", "midpoint" and "bottomleft". Partially matched.

Details
The spatial location of each pixel in the image is translated by the vector vec. This is a method for
the generic function shift.
If origin is given, then it should be one of the character strings "centroid", "midpoint" or
"bottomleft". The argument vec will be ignored; instead the shift will be performed so that the
specified geometric location is shifted to the origin. If origin="centroid" then the centroid of the
image window will be shifted to the origin. If origin="midpoint" then the centre of the bounding
rectangle of the image will be shifted to the origin. If origin="bottomleft" then the bottom left
corner of the bounding rectangle of the image will be shifted to the origin.
Value
Another pixel image (of class "im") representing the result of applying the vector shift.
Author(s)
Adrian Baddeley 
and Rolf Turner 

shift.owin

1335

See Also
shift
Examples
# make up an image
X <- setcov(unit.square())
plot(X)
Y <- shift(X, c(10,10))
plot(Y)
# no discernible difference except coordinates are different
shift(X, origin="mid")

shift.owin

Apply Vector Translation To Window

Description
Applies a vector shift to a window
Usage
## S3 method for class 'owin'
shift(X, vec=c(0,0), ..., origin=NULL)
Arguments
X

Window (object of class "owin").

vec

Vector of length 2 representing a translation.

...

Ignored

origin

Character string determining a location that will be shifted to the origin. Options
are "centroid", "midpoint" and "bottomleft". Partially matched.

Details
The window is translated by the vector vec. This is a method for the generic function shift.
If origin is given, then it should be one of the character strings "centroid", "midpoint" or
"bottomleft". The argument vec will be ignored; instead the shift will be performed so that the
specified geometric location is shifted to the origin. If origin="centroid" then the centroid of
the window will be shifted to the origin. If origin="midpoint" then the centre of the bounding
rectangle of the window will be shifted to the origin. If origin="bottomleft" then the bottom left
corner of the bounding rectangle of the window will be shifted to the origin.
Value
Another window (of class "owin") representing the result of applying the vector shift.

1336

shift.ppp

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
shift, shift.ppp, periodify, rotate, affine, centroid.owin
Examples
W <- owin(c(0,1),c(0,1))
X <- shift(W, c(2,3))
## Not run:
plot(W)
# no discernible difference except coordinates are different
## End(Not run)
shift(W, origin="mid")

shift.ppp

Apply Vector Translation To Point Pattern

Description
Applies a vector shift to a point pattern.
Usage
## S3 method for class 'ppp'
shift(X, vec=c(0,0), ..., origin=NULL)
Arguments
X

Point pattern (object of class "ppp").

vec

Vector of length 2 representing a translation.

...

Ignored

origin

Character string determining a location that will be shifted to the origin. Options
are "centroid", "midpoint" and "bottomleft". Partially matched.

Details
The point pattern, and its window, are translated by the vector vec.
This is a method for the generic function shift.
If origin is given, then it should be one of the character strings "centroid", "midpoint" or
"bottomleft". The argument vec will be ignored; instead the shift will be performed so that the
specified geometric location is shifted to the origin. If origin="centroid" then the centroid of
the window will be shifted to the origin. If origin="midpoint" then the centre of the bounding
rectangle of the window will be shifted to the origin. If origin="bottomleft" then the bottom left
corner of the bounding rectangle of the window will be shifted to the origin.

shift.psp

1337

Value
Another point pattern (of class "ppp") representing the result of applying the vector shift.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
shift, shift.owin, periodify, rotate, affine
Examples
data(cells)
X <- shift(cells, c(2,3))
## Not run:
plot(X)
# no discernible difference except coordinates are different
## End(Not run)
plot(cells, pch=16)
plot(shift(cells, c(0.03,0.03)), add=TRUE)
shift(cells, origin="mid")

shift.psp

Apply Vector Translation To Line Segment Pattern

Description
Applies a vector shift to a line segment pattern.
Usage
## S3 method for class 'psp'
shift(X, vec=c(0,0), ..., origin=NULL)
Arguments
X

Line Segment pattern (object of class "psp").

vec

Vector of length 2 representing a translation.

...

Ignored

origin

Character string determining a location that will be shifted to the origin. Options
are "centroid", "midpoint" and "bottomleft". Partially matched.

1338

sidelengths.owin

Details
The line segment pattern, and its window, are translated by the vector vec.
This is a method for the generic function shift.
If origin is given, then it should be one of the character strings "centroid", "midpoint" or
"bottomleft". The argument vec will be ignored; instead the shift will be performed so that the
specified geometric location is shifted to the origin. If origin="centroid" then the centroid of
the window will be shifted to the origin. If origin="midpoint" then the centre of the bounding
rectangle of the window will be shifted to the origin. If origin="bottomleft" then the bottom left
corner of the bounding rectangle of the window will be shifted to the origin.
Value
Another line segment pattern (of class "psp") representing the result of applying the vector shift.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
shift, shift.owin, shift.ppp, periodify, rotate, affine
Examples
X <- psp(runif(10), runif(10), runif(10), runif(10), window=owin())
plot(X, col="red")
Y <- shift(X, c(0.05,0.05))
plot(Y, add=TRUE, col="blue")
shift(Y, origin="mid")

sidelengths.owin

Side Lengths of Enclosing Rectangle of a Window

Description
Computes the side lengths of the (enclosing rectangle of) a window.
Usage
## S3 method for class 'owin'
sidelengths(x)
## S3 method for class 'owin'
shortside(x)
Arguments
x

A window whose side lengths will be computed. Object of class "owin".

simplepanel

1339

Details
The functions shortside and sidelengths are generic. The functions documented here are the
methods for the class "owin".
sidelengths.owin computes the side-lengths of the enclosing rectangle of the window x.
For safety, both functions give a warning if the window is not a rectangle. To suppress the warning,
first convert the window to a rectangle using as.rectangle.
shortside.owin computes the minimum of the two side-lengths.
Value
For sidelengths.owin, a numeric vector of length 2 giving the side-lengths (x then y) of the
enclosing rectangle. For shortside.owin, a numeric value.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
shortside, sidelengths for the generic functions.
area.owin, diameter.owin, perimeter for other geometric calculations on "owin" objects.
owin, as.owin.
Examples
w <- owin(c(0,2),c(-1,3))
sidelengths(w)
shortside(as.rectangle(letterR))

simplepanel

Simple Point-and-Click Interface Panels

Description
These functions enable the user to create a simple, robust, point-and-click interface to any R code.
Usage
simplepanel(title, B, boxes, clicks,
redraws=NULL, exit = NULL, env)
grow.simplepanel(P, side = c("right", "left", "top", "bottom"),
len = NULL, new.clicks, new.redraws=NULL, ..., aspect)

1340

simplepanel

Arguments
title

Character string giving the title of the interface panel.

B

Bounding box of the panel coordinates. A rectangular window (object of class
"owin")

boxes

A list of rectangular windows (objects of class "owin") specifying the placement
of the buttons and other interactive components of the panel.

clicks

A list of R functions, of the same length as boxes, specifying the operations to
be performed when each button is clicked. Entries can also be NULL indicating
that no action should occur. See Details.

redraws

Optional list of R functions, of the same length as boxes, specifying how to
redraw each button. Entries can also be NULL indicating a simple default. See
Details.

exit

An R function specifying actions to be taken when the interactive panel terminates.

env

An environment that will be passed as an argument to all the functions in
clicks, redraws and exit.

P

An existing interaction panel (object of class "simplepanel").

side

Character string identifying which side of the panel P should be grown to accommodate the new buttons.

len

Optional. Thickness of the new panel area that should be grown to accommodate
the new buttons. A single number in the same units as the coordinate system of
P.

new.clicks

List of R functions defining the operations to be performed when each of the
new buttons is clicked.

new.redraws

Optional. List of R functions, of the same length as new.clicks, defining how
to redraw each of the new buttons.

...

Arguments passed to layout.boxes to determine the layout of the new buttons.

aspect

Optional. Aspect ratio (height/width) of the new buttons.

Details
These functions enable the user to create a simple, robust, point-and-click interface to any R code.
The functions simplepanel and grow.simplepanel create an object of class "simplepanel".
Such an object defines the graphics to be displayed and the actions to be performed when the user
interacts with the panel.
The panel is activated by calling run.simplepanel.
The function simplepanel creates a panel object from basic data. The function grow.simplepanel
modifies an existing panel object P by growing an additional row or column of buttons.
For simplepanel,
• The spatial layout of the panel is determined by the rectangles B and boxes.
• The argument clicks must be a list of functions specifying the action to be taken when
each button is clicked (or NULL to indicate that no action should be taken). The list entries
should have names (but there are sensible defaults). Each function should be of the form
function(env, xy) where env is an environment that may contain shared data, and xy
gives the coordinates of the mouse click, in the format list(x, y). The function returns
TRUE if the panel should continue running, and FALSE if the panel should terminate.

simplepanel

1341

• The argument redraws, if given, must be a list of functions specifying the action to be taken
when each button is to be redrawn. Each function should be of the form function(button, name, env)
where button is a rectangle specifying the location of the button in the current coordinate system; name is a character string giving the name of the button; and env is the environment that
may contain shared data. The function returns TRUE if the panel should continue running,
and FALSE if the panel should terminate. If redraws is not given (or if one of the entries in
redraws is NULL), the default action is to draw a pink rectangle showing the button position,
draw the name of the button in the middle of this rectangle, and return TRUE.
• The argument exit, if given, must be a function specifying the action to be taken when the
panel terminates. (Termination occurs when one of the clicks functions returns FALSE). The
exit function should be of the form function(env) where env is the environment that may
contain shared data. Its return value will be used as the return value of run.simplepanel.
• The argument env should be an R environment. The panel buttons will have access to this
environment, and will be able to read and write data in it. This mechanism is used to exchange
data between the panel and other R code.
For grow.simplepanel,
• the spatial layout of the new boxes is determined by the arguments side, len, aspect and by
the additional ... arguments passed to layout.boxes.
• the argument new.clicks should have the same format as clicks. It implicitly specifies the
number of new buttons to be added, and the actions to be performed when they are clicked.
• the optional argument new.redraws, if given, should have the same format as redraws. It
specifies the actions to be performed when the new buttons are clicked.
Value
An object of class "simplepanel".
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
run.simplepanel, layout.boxes
Examples
# make boxes (alternatively use layout.boxes())
Bminus <- square(1)
Bvalue <- shift(Bminus, c(1.2, 0))
Bplus <- shift(Bvalue, c(1.2, 0))
Bdone <- shift(Bplus, c(1.2, 0))
myboxes <- list(Bminus, Bvalue, Bplus, Bdone)
myB <- do.call(boundingbox,myboxes)
# make environment containing an integer count
myenv <- new.env()
assign("answer", 0, envir=myenv)
# what to do when finished: return the count.
myexit <- function(e) { return(get("answer", envir=e)) }

1342

simplify.owin

# button clicks
# decrement the count
Cminus <- function(e, xy) {
ans <- get("answer", envir=e)
assign("answer", ans - 1, envir=e)
return(TRUE)
}
# display the count (clicking does nothing)
Cvalue <- function(...) { TRUE }
# increment the count
Cplus <- function(e, xy) {
ans <- get("answer", envir=e)
assign("answer", ans + 1, envir=e)
return(TRUE)
}
# 'Clear' button
Cclear <- function(e, xy) {
assign("answer", 0, envir=e)
return(TRUE)
}
# quit button
Cdone <- function(e, xy) { return(FALSE) }
myclicks <- list("-"=Cminus,
value=Cvalue,
"+"=Cplus,
done=Cdone)
# redraw the button that displays the current value of the count
Rvalue <- function(button, nam, e) {
plot(button, add=TRUE)
ans <- get("answer", envir=e)
text(centroid.owin(button), labels=ans)
return(TRUE)
}
# make the panel
P <- simplepanel("Counter",
B=myB, boxes=myboxes,
clicks=myclicks,
redraws = list(NULL, Rvalue, NULL, NULL),
exit=myexit, env=myenv)
P
# ( type run.simplepanel(P) to run the panel interactively )
# add another button to right
Pplus <- grow.simplepanel(P, "right", new.clicks=list(clear=Cclear))

simplify.owin

Approximate a Polygon by a Simpler Polygon

Description
Given a polygonal window, this function finds a simpler polygon that approximates it.

simulate.dppm

1343

Usage
simplify.owin(W, dmin)
Arguments
W

The polygon which is to be simplied. An object of class "owin".

dmin

Numeric value. The smallest permissible length of an edge.

Details
This function simplifies a polygon W by recursively deleting the shortest edge of W until all remaining
edges are longer than the specified minimum length dmin, or until there are only three edges left.
The argument W must be a window (object of class "owin"). It should be of type "polygonal". If
W is a rectangle, it is returned without alteration.
The simplification algorithm is not yet implemented for binary masks. If W is a mask, an error is
generated.
Value
Another window (object of class "owin") of type "polygonal".
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
owin
Examples
plot(letterR, col="red")
plot(simplify.owin(letterR, 0.3), col="blue", add=TRUE)
W <- Window(chorley)
plot(W)
WS <- simplify.owin(W, 2)
plot(WS, add=TRUE, border="green")
points(vertices(WS))

simulate.dppm

Simulation of Determinantal Point Process Model

Description
Generates simulated realisations from a determinantal point process model.

1344

simulate.dppm

Usage
## S3 method for class 'dppm'
simulate(object, nsim = 1, seed = NULL, ...,
W = NULL, trunc = 0.99, correction = "periodic", rbord = reach(object))
## S3 method for class 'detpointprocfamily'
simulate(object, nsim = 1, seed = NULL, ...,
W = NULL, trunc = 0.99, correction = "periodic", rbord = reach(object))
Arguments
object

Determinantal point process model. An object of class "detpointprocfamily"
or "dppm".

nsim

Number of simulated realisations.

seed

an object specifying whether and how to initialise the random number generator. Either NULL or an integer that will be used in a call to set.seed before
simulating the point patterns.

...

Arguments passed on to rdpp.

W

Object specifying the window of simulation (defaults to a unit box if nothing else
is sensible – see Details). Can be any single argument acceptable to as.boxx
(e.g. an "owin", "box3" or "boxx" object).

trunc

Numeric value specifying how the model truncation is preformed. See Details.

correction

Character string specifying the type of correction to use. The options are "periodic" (default) and "border". See Details.

rbord

Numeric value specifying the extent of the border correction if this correction is
used. See Details.

Details
These functions are methods for the generic function simulate for the classes "detpointprocfamily"
and "dppm" of determinantal point process models.
The return value is a list of nsim point patterns. It also carries an attribute "seed" that captures
the initial state of the random number generator. This follows the convention used in simulate.lm
(see simulate). It can be used to force a sequence of simulations to be repeated exactly, as shown
in the examples for simulate.
The exact simulation of a determinantal point process model involves an infinite series, which typically has no analytical solution. In the implementation a truncation is performed. The truncation
trunc can be specified either directly as a positive integer or as a fraction between 0 and 1. In the
latter case the truncation is chosen such that the expected number of points in a simulation is trunc
times the theoretical expected number of points in the model. The default is 0.99.
The window of the returned point pattern(s) can be specified via the argument W. For a fitted model
(of class "dppm") it defaults to the observation window of the data used to fit the model. For
inhomogeneous models it defaults to the window of the intensity image. Otherwise it defaults to
a unit box. For non-rectangular windows simulation is done in the containing rectangle and then
restricted to the window. For inhomogeneous models a stationary model is first simulated using the
maximum intensity and then the result is obtained by thinning.
The default is to use periodic edge correction for simulation such that opposite edges are glued
together. If border correction is used then the simulation is done in an extended window. Edge
effects are theoretically completely removed by doubling the size of the window in each spatial

simulate.kppm

1345

dimension, but for practical purposes much less extension may be sufficient. The numeric rbord
determines the extent of the extra space added to the window.
Value
A list of length nsim containing simulated point patterns. If the patterns are two-dimensional, then
they are objects of class "ppp", and the list has class "solist". Otherwise, the patterns are objects
of class "ppx" and the list has class "anylist".
The return value also carries an attribute "seed" that captures the initial state of the random number
generator. See Details.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Lavancier, F. Møller, J. and Rubak, E. (2015) Determinantal point process models and statistical
inference Journal of the Royal Statistical Society, Series B 77, 853–977.
See Also
rdpp, simulate
Examples
model <- dppGauss(lambda=100, alpha=.05, d=2)
simulate(model, 2)

simulate.kppm

Simulate a Fitted Cluster Point Process Model

Description
Generates simulated realisations from a fitted cluster point process model.
Usage
## S3 method for class 'kppm'
simulate(object, nsim = 1, seed=NULL, ...,
window=NULL, covariates=NULL, verbose=TRUE, retry=10,
drop=FALSE)

1346

simulate.kppm

Arguments
object

Fitted cluster point process model. An object of class "kppm".

nsim

Number of simulated realisations.

seed

an object specifying whether and how to initialise the random number generator. Either NULL or an integer that will be used in a call to set.seed before
simulating the point patterns.

...

Ignored.

window

Optional. Window (object of class "owin") in which the model should be simulated.

covariates

Optional. A named list containing new values for the covariates in the model.

verbose

Logical. Whether to print progress reports (when nsim > 1).

retry

Number of times to repeat the simulation if it fails (e.g. because of insufficient
memory).

drop

Logical. If nsim=1 and drop=TRUE, the result will be a point pattern, rather than
a list containing a point pattern.

Details
This function is a method for the generic function simulate for the class "kppm" of fitted cluster
point process models.
Simulations are performed by rThomas, rMatClust or rLGCP depending on the model.
The return value is a list of point patterns. It also carries an attribute "seed" that captures the
initial state of the random number generator. This follows the convention used in simulate.lm
(see simulate). It can be used to force a sequence of simulations to be repeated exactly, as shown
in the examples for simulate.
Value
A list of length nsim containing simulated point patterns (objects of class "ppp").
The return value also carries an attribute "seed" that captures the initial state of the random number
generator. See Details.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
kppm, rThomas, rMatClust, rLGCP, simulate.ppm, simulate
Examples
data(redwood)
fit <- kppm(redwood, ~1, "Thomas")
simulate(fit, 2)

simulate.lppm

simulate.lppm

1347

Simulate a Fitted Point Process Model on a Linear Network

Description
Generates simulated realisations from a fitted Poisson point process model on a linear network.
Usage
## S3 method for class 'lppm'
simulate(object, nsim=1, ...,
new.coef=NULL,
progress=(nsim > 1),
drop=FALSE)
Arguments
object

Fitted point process model on a linear network. An object of class "lppm".

nsim

Number of simulated realisations.

progress

Logical flag indicating whether to print progress reports for the sequence of
simulations.

new.coef

New values for the canonical parameters of the model. A numeric vector of the
same length as coef(object).

...

Arguments passed to predict.lppm to determine the spatial resolution of the
image of the fitted intensity used in the simulation.

drop

Logical. If nsim=1 and drop=TRUE, the result will be a point pattern, rather than
a list containing a point pattern.

Details
This function is a method for the generic function simulate for the class "lppm" of fitted point
process models on a linear network.
Only Poisson process models are supported so far.
Simulations are performed by rpoislpp.
Value
A list of length nsim containing simulated point patterns (objects of class "lpp") on the same linear
network as the original data used to fit the model. The result also belongs to the class "solist", so
that it can be plotted, and the class "timed", so that the total computation time is recorded.
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak 
See Also
lppm, rpoislpp, simulate

1348

simulate.mppm

Examples
fit <- lppm(unmark(chicago) ~ y)
simulate(fit)[[1]]

simulate.mppm

Simulate a Point Process Model Fitted to Several Point Patterns

Description
Generates simulated realisations from a point process model that was fitted to several point patterns.
Usage
## S3 method for class 'mppm'
simulate(object, nsim=1, ..., verbose=TRUE)
Arguments
object

Point process model fitted to several point patterns. An object of class "mppm".

nsim

Number of simulated realisations (of each original pattern).

...

Further arguments passed to simulate.ppm to control the simulation.

verbose

Logical value indicating whether to print progress reports.

Details
This function is a method for the generic function simulate for the class "mppm" of fitted point
process models for replicated point pattern data.
The result is a hyperframe with n rows and nsim columns, where n is the number of original point
pattern datasets to which the model was fitted. Each column of the hyperframe contains a simulated
version of the original data.
For each of the original point pattern datasets, the fitted model for this dataset is extracted using
subfits, then nsim simulated realisations of this model are generated using simulate.ppm, and
these are stored in the corresponding row of the output.
Value
A hyperframe.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
mppm, simulate.ppm.

simulate.ppm

1349

Examples
H <- hyperframe(Bugs=waterstriders)
fit <- mppm(Bugs ~ id, H)
y <- simulate(fit, nsim=2)
y
plot(y[1,,drop=TRUE], main="Simulations for Waterstriders pattern 1")
plot(y[,1,drop=TRUE], main="Simulation 1 for each Waterstriders pattern")

simulate.ppm

Simulate a Fitted Gibbs Point Process Model

Description
Generates simulated realisations from a fitted Gibbs or Poisson point process model.
Usage
## S3 method for class 'ppm'
simulate(object, nsim=1, ...,
singlerun = FALSE,
start = NULL,
control = default.rmhcontrol(object, w=w),
w = NULL,
project=TRUE, new.coef=NULL,
verbose=FALSE, progress=(nsim > 1),
drop=FALSE)
Arguments
object

Fitted point process model. An object of class "ppm".

nsim

Number of simulated realisations.

singlerun

Logical. Whether to generate the simulated realisations from a single long run
of the Metropolis-Hastings algorithm (singlerun=TRUE) or from separate, independent runs of the algorithm (singlerun=FALSE, the default).

start

Data determining the initial state of the Metropolis-Hastings algorithm. See
rmhstart for description of these arguments. Defaults to list(n.start=npoints(data.ppm(objec
meaning that the initial state of the algorithm has the same number of points as
the original dataset.

control

Data controlling the running of the Metropolis-Hastings algorithm. See rmhcontrol
for description of these arguments.

w

Optional. The window in which the model is defined. An object of class "owin".

...

Further arguments passed to rmhcontrol, or to rmh.default, or to covariate
functions in the model.

project

Logical flag indicating what to do if the fitted model is invalid (in the sense
that the values of the fitted coefficients do not specify a valid point process). If
project=TRUE the closest valid model will be simulated; if project=FALSE an
error will occur.

verbose

Logical flag indicating whether to print progress reports from rmh.ppm during
the simulation of each point pattern.

1350

simulate.ppm

progress

Logical flag indicating whether to print progress reports for the sequence of
simulations.

new.coef

New values for the canonical parameters of the model. A numeric vector of the
same length as coef(object).

drop

Logical. If nsim=1 and drop=TRUE, the result will be a point pattern, rather than
a list containing a point pattern.

Details
This function is a method for the generic function simulate for the class "ppm" of fitted point
process models.
Simulations are performed by rmh.ppm.
If singlerun=FALSE (the default), the simulated patterns are the results of independent runs of
the Metropolis-Hastings algorithm. If singlerun=TRUE, a single long run of the algorithm is performed, and the state of the simulation is saved every nsave iterations to yield the simulated patterns.
In the case of a single run, the behaviour is controlled by the parameters nsave,nburn,nrep. These
are described in rmhcontrol. They may be passed in the ... arguments or included in control. It
is sufficient to specify two of the three parameters nsave,nburn,nrep.

Value
A list of length nsim containing simulated point patterns (objects of class "ppp"). It also belongs
to the class "solist", so that it can be plotted, and the class "timed", so that the total computation
time is recorded.

Author(s)
Adrian Baddeley 
and Rolf Turner 

See Also
ppm, simulate.kppm, simulate

Examples
fit <- ppm(japanesepines, ~1, Strauss(0.1))
simulate(fit, 2)
simulate(fit, 2, singlerun=TRUE, nsave=1e4, nburn=1e4)

simulate.slrm

simulate.slrm

1351

Simulate a Fitted Spatial Logistic Regression Model

Description
Generates simulated realisations from a fitted spatial logistic regresson model
Usage
## S3 method for class 'slrm'
simulate(object, nsim = 1, seed=NULL, ...,
window=NULL, covariates=NULL, verbose=TRUE, drop=FALSE)
Arguments
object

Fitted spatial logistic regression model. An object of class "slrm".

nsim

Number of simulated realisations.

seed

an object specifying whether and how to initialise the random number generator. Either NULL or an integer that will be used in a call to set.seed before
simulating the point patterns.

...

Ignored.

window

Optional. Window (object of class "owin") in which the model should be simulated.

covariates

Optional. A named list containing new values for the covariates in the model.

verbose

Logical. Whether to print progress reports (when nsim > 1).

drop

Logical. If nsim=1 and drop=TRUE, the result will be a point pattern, rather than
a list containing a point pattern.

Details
This function is a method for the generic function simulate for the class "slrm" of fitted spatial
logistic regression models.
Simulations are performed by rpoispp after the intensity has been computed by predict.slrm.
The return value is a list of point patterns. It also carries an attribute "seed" that captures the
initial state of the random number generator. This follows the convention used in simulate.lm
(see simulate). It can be used to force a sequence of simulations to be repeated exactly, as shown
in the examples for simulate.
Value
A list of length nsim containing simulated point patterns (objects of class "ppp").
The return value also carries an attribute "seed" that captures the initial state of the random number
generator. See Details.
Author(s)
Adrian Baddeley 
and Rolf Turner 

1352

slrm

See Also
slrm, rpoispp, simulate.ppm, simulate.kppm, simulate
Examples
X <- copper$SouthPoints
fit <- slrm(X ~ 1)
simulate(fit, 2)
fitxy <- slrm(X ~ x+y)
simulate(fitxy, 2, window=square(2))

slrm

Spatial Logistic Regression

Description
Fits a spatial logistic regression model to a spatial point pattern.
Usage
slrm(formula, ..., data = NULL, offset = TRUE, link = "logit",
dataAtPoints=NULL, splitby=NULL)
Arguments
formula

The model formula. See Details.

...

Optional arguments passed to pixellate determining the pixel resolution for
the discretisation of the point pattern.

data

Optional. A list containing data required in the formula. The names of entries in
the list should correspond to variable names in the formula. The entries should
be point patterns, pixel images or windows.

offset

Logical flag indicating whether the model formula should be augmented by an
offset equal to the logarithm of the pixel area.

link

The link function for the regression model. A character string, specifying a link
function for binary regression.

dataAtPoints

Optional. Exact values of the covariates at the data points. A data frame, with
column names corresponding to variables in the formula, with one row for each
point in the point pattern dataset.

splitby

Optional. Character string identifying a window. The window will be used to
split pixels into sub-pixels.

Details
This function fits a Spatial Logistic Regression model (Tukey, 1972; Agterberg, 1974) to a spatial
point pattern dataset. The logistic function may be replaced by another link function.
The formula specifies the form of the model to be fitted, and the data to which it should be fitted.
The formula must be an R formula with a left and right hand side.
The left hand side of the formula is the name of the point pattern dataset, an object of class "ppp".

slrm

1353
The right hand side of the formula is an expression, in the usual R formula syntax, representing
the functional form of the linear predictor for the model.
Each variable name that appears in the formula may be
• one of the reserved names x and y, referring to the Cartesian coordinates;
• the name of an entry in the list data, if this argument is given;
• the name of an object in the parent environment, that is, in the environment where the call to
slrm was issued.
Each object appearing on the right hand side of the formula may be
• a pixel image (object of class "im") containing the values of a covariate;
• a window (object of class "owin"), which will be interpreted as a logical covariate which is
TRUE inside the window and FALSE outside it;
• a function in the R language, with arguments x,y, which can be evaluated at any location to
obtain the values of a covariate.
See the Examples below.
The fitting algorithm discretises the point pattern onto a pixel grid. The value in each pixel is 1 if
there are any points of the point pattern in the pixel, and 0 if there are no points in the pixel. The
dimensions of the pixel grid will be determined as follows:
• The pixel grid will be determined by the extra arguments ... if they are specified (for example
the argument dimyx can be used to specify the number of pixels).
• Otherwise, if the right hand side of the formula includes the names of any pixel images
containing covariate values, these images will determine the pixel grid for the discretisation.
The covariate image with the finest grid (the smallest pixels) will be used.
• Otherwise, the default pixel grid size is given by spatstat.options("npixel").
If link="logit" (the default), the algorithm fits a Spatial Logistic Regression model. This model
states that the probability p that a given pixel contains a data point, is related to the covariates
through
p
log
=η
1−p
where η is the linear predictor of the model (a linear combination of the covariates, whose form is
specified by the formula).
If link="cloglog" then the algorithm fits a model stating that
log(− log(1 − p)) = η
.
If offset=TRUE (the default), the model formula will be augmented by adding an offset term equal
to the logarithm of the pixel area. This ensures that the fitted parameters are approximately independent of pixel size. If offset=FALSE, the offset is not included, and the traditional form of Spatial
Logistic Regression is fitted.

Value
An object of class "slrm" representing the fitted model.
There are many methods for this class, including methods for print, fitted, predict, anova,
coef, logLik, terms, update, formula and vcov. Automated stepwise model selection is possible
using step. Confidence intervals for the parameters can be computed using confint.

1354

Smooth

Author(s)
Adrian Baddeley  
and Rolf Turner 
References
Agterberg, F.P. (1974) Automatic contouring of geological maps to detect target areas for mineral
exploration. Journal of the International Association for Mathematical Geology 6, 373–395.
Baddeley, A., Berman, M., Fisher, N.I., Hardegen, A., Milne, R.K., Schuhmacher, D., Shah, R.
and Turner, R. (2010) Spatial logistic regression and change-of-support for spatial Poisson point
processes. Electronic Journal of Statistics 4, 1151–1201. doi: 10.1214/10-EJS581
Tukey, J.W. (1972) Discussion of paper by F.P. Agterberg and S.C. Robinson. Bulletin of the International Statistical Institute 44 (1) p. 596. Proceedings, 38th Congress, International Statistical
Institute.
See Also
anova.slrm, coef.slrm, fitted.slrm, logLik.slrm, plot.slrm, predict.slrm, vcov.slrm
Examples
X <- copper$SouthPoints
slrm(X ~ 1)
slrm(X ~ x+y)
slrm(X ~ x+y, link="cloglog")
# specify a grid of 2-km-square pixels
slrm(X ~ 1, eps=2)
Y <- copper$SouthLines
Z <- distmap(Y)
slrm(X ~ Z)
slrm(X ~ Z, dataAtPoints=list(Z=nncross(X,Y,what="dist")))
dat <- list(A=X, V=Z)
slrm(A ~ V, data=dat)

Smooth

Spatial smoothing of data

Description
Generic function to perform spatial smoothing of spatial data.
Usage
Smooth(X, ...)
Arguments
X

Some kind of spatial data

...

Arguments passed to methods.

Smooth.fv

1355

Details
This generic function calls an appropriate method to perform spatial smoothing on the spatial dataset
X.
Methods for this function include
• Smooth.ppp for point patterns
• Smooth.msr for measures
• Smooth.fv for function value tables
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
Smooth.ppp, Smooth.im, Smooth.msr, Smooth.fv.

Smooth.fv

Apply Smoothing to Function Values

Description
Applies smoothing to the values in selected columns of a function value table.
Usage
## S3 method for class 'fv'
Smooth(X, which = "*", ...,
method=c("smooth.spline", "loess"),
xinterval=NULL)
Arguments
X

Values to be smoothed. A function value table (object of class "fv", see fv.object).

which

Character vector identifying which columns of the table should be smoothed.
Either a vector containing names of columns, or one of the wildcard strings "*"
or "." explained below.

...

Extra arguments passed to smooth.spline or loess to control the smoothing.

method

Smoothing algorithm. A character string, partially matched to either "smooth.spline"
or "loess".

xinterval

Optional. Numeric vector of length 2 specifying a range of x values. Smoothing
will be performed only on the part of the function corresponding to this range.

1356

Smooth.msr

Details
The command Smooth.fv applies smoothing to the function values in a function value table (object
of class "fv").
Smooth.fv is a method for the generic function Smooth.
The smoothing is performed either by smooth.spline or by loess.
Smoothing is applied to every column (or to each of the selected columns) of function values in
turn, using the function argument as the x coordinate and the selected column as the y coordinate.
The original function values are then replaced by the corresponding smooth interpolated function
values.
The optional argument which specifies which of the columns of function values in x will be smoothed.
The default (indicated by the wildcard which="*") is to smooth all function values, i.e.\ all columns
except the function argument. Alternatively which="." designates the subset of function values that
are displayed in the default plot. Alternatively which can be a character vector containing the names
of columns of x.
If the argument xinterval is given, then smoothing will be performed only in the specified range
of x values.
Value
Another function value table (object of class "fv") of the same format.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
Smooth, with.fv, fv.object, smooth.spline, smooth.spline
Examples
data(cells)
G <- Gest(cells)
plot(G)
plot(Smooth(G, df=9), add=TRUE)

Smooth.msr

Smooth a Signed or Vector-Valued Measure

Description
Apply kernel smoothing to a signed measure or vector-valued measure.
Usage
## S3 method for class 'msr'
Smooth(X, ..., drop=TRUE)

Smooth.msr

1357

Arguments
X

Object of class "msr" representing a signed measure or vector-valued measure.

...

Arguments passed to density.ppp controlling the smoothing bandwidth and
the pixel resolution.

drop

Logical. If TRUE (the default), the result of smoothing a scalar-valued measure
is a pixel image. If FALSE, the result of smoothing a scalar-valued measure is a
list containing one pixel image.

Details
This function applies kernel smoothing to a signed measure or vector-valued measure X. The Gaussian kernel is used.
The object X would typically have been created by residuals.ppm or msr.
Value
A pixel image or a list of pixel images. For scalar-valued measures, a pixel image (object of class
"im") provided drop=TRUE. For vector-valued measures (or if drop=FALSE), a list of pixel images;
the list also belongs to the class "solist" so that it can be printed and plotted.
Author(s)
Adrian Baddeley 
References
Baddeley, A., Turner, R., Møller, J. and Hazelton, M. (2005) Residual analysis for spatial point
processes. Journal of the Royal Statistical Society, Series B 67, 617–666.
Baddeley, A., Møller, J. and Pakes, A.G. (2008) Properties of residuals for spatial point processes.
Annals of the Institute of Statistical Mathematics 60, 627–649.
See Also
Smooth, msr, plot.msr
Examples
X <- rpoispp(function(x,y) { exp(3+3*x) })
fit <- ppm(X, ~x+y)
rp <- residuals(fit, type="pearson")
rs <- residuals(fit, type="score")
plot(Smooth(rp))
plot(Smooth(rs))

1358

Smooth.ppp

Smooth.ppp

Spatial smoothing of observations at irregular points

Description
Performs spatial smoothing of numeric values observed at a set of irregular locations. Uses Gaussian
kernel smoothing and least-squares cross-validated bandwidth selection.
Usage
## S3 method for class 'ppp'
Smooth(X, sigma=NULL,
...,
weights = rep(1, npoints(X)),
at="pixels",
edge=TRUE, diggle=FALSE, geometric=FALSE)
markmean(X, ...)
markvar(X, sigma=NULL, ..., weights=NULL, varcov=NULL)
Arguments
X

A marked point pattern (object of class "ppp").

sigma

Smoothing bandwidth. A single positive number, a numeric vector of length 2,
or a function that selects the bandwidth automatically. See density.ppp.

...

Further arguments passed to bw.smoothppp and density.ppp to control the
kernel smoothing and the pixel resolution of the result.

weights

Optional weights attached to the observations. A numeric vector, numeric matrix, an expression or a pixel image. See density.ppp.

at

String specifying whether to compute the smoothed values at a grid of pixel
locations (at="pixels") or only at the points of X (at="points").

edge,diggle

Arguments passed to density.ppp to determine the edge correction.

varcov

Variance-covariance matrix. An alternative to sigma. See density.ppp.

geometric

Logical value indicating whether to perform geometric mean smoothing instead
of arithmetic mean smoothing. See Details.

Details
The function Smooth.ppp performs spatial smoothing of numeric values observed at a set of irregular locations. The functions markmean and markvar are wrappers for Smooth.ppp which compute
the spatially-varying mean and variance of the marks of a point pattern.
Smooth.ppp is a method for the generic function Smooth for the class "ppp" of point patterns. Thus
you can type simply Smooth(X).
Smoothing is performed by Gaussian kernel weighting. If the observed values are v1 , . . . , vn at
locations x1 , . . . , xn respectively, then the smoothed value at a location u is (ignoring edge corrections)
P
k(u − xi )vi
g(u) = Pi
i k(u − xi )

Smooth.ppp

1359

where k is a Gaussian kernel. This is known as the Nadaraya-Watson smoother (Nadaraya, 1964,
1989; Watson, 1964). By default, the smoothing kernel bandwidth is chosen by least squares crossvalidation (see below).
The argument X must be a marked point pattern (object of class "ppp", see ppp.object). The points
of the pattern are taken to be the observation locations xi , and the marks of the pattern are taken to
be the numeric values vi observed at these locations.
The marks are allowed to be a data frame (in Smooth.ppp and markmean). Then the smoothing
procedure is applied to each column of marks.
The numerator and denominator are computed by density.ppp. The arguments ... control the
smoothing kernel parameters and determine whether edge correction is applied. The smoothing
kernel bandwidth can be specified by either of the arguments sigma or varcov which are passed to
density.ppp. If neither of these arguments is present, then by default the bandwidth is selected by
least squares cross-validation, using bw.smoothppp.
The optional argument weights allows numerical weights to be applied to the data. If a weight wi
is associated with location xi , then the smoothed function is (ignoring edge corrections)
P
k(u − xi )vi wi
g(u) = Pi
i k(u − xi )wi
If geometric=TRUE then geometric mean smoothing is performed instead of arithmetic mean smoothing. The mark values must be non-negative numbers. The logarithm of the mark values is computed;
these logarithmic values are kernel-smoothed as described above; then the exponential function is
applied to the smoothed values.
An alternative to kernel smoothing is inverse-distance weighting, which is performed by idw.
Value
If X has a single column of marks:
• If at="pixels" (the default), the result is a pixel image (object of class "im"). Pixel values
are values of the interpolated function.
• If at="points", the result is a numeric vector of length equal to the number of points in X.
Entries are values of the interpolated function at the points of X.
If X has a data frame of marks:
• If at="pixels" (the default), the result is a named list of pixel images (object of class "im").
There is one image for each column of marks. This list also belongs to the class "solist",
for which there is a plot method.
• If at="points", the result is a data frame with one row for each point of X, and one column
for each column of marks. Entries are values of the interpolated function at the points of X.
The return value has attributes "sigma" and "varcov" which report the smoothing bandwidth that
was used.
Very small bandwidth
If the chosen bandwidth sigma is very small, kernel smoothing is mathematically equivalent to
nearest-neighbour interpolation; the result will be computed by nnmark. This is unless at="points"
and leaveoneout=FALSE, when the original mark values are returned.

1360

Smooth.ssf

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
References
Nadaraya, E.A. (1964) On estimating regression. Theory of Probability and its Applications 9,
141–142.
Nadaraya, E.A. (1989) Nonparametric estimation of probability densities and regression curves.
Kluwer, Dordrecht.
Watson, G.S. (1964) Smooth regression analysis. Sankhya A 26, 359–372.
See Also
Smooth,
density.ppp, bw.smoothppp, nnmark, ppp.object, im.object.
See idw for inverse-distance weighted smoothing.
To perform interpolation, see also the akima package.
Examples
# Longleaf data - tree locations, marked by tree diameter
# Local smoothing of tree diameter (automatic bandwidth selection)
Z <- Smooth(longleaf)
# Kernel bandwidth sigma=5
plot(Smooth(longleaf, 5))
# mark variance
plot(markvar(longleaf, sigma=5))
# data frame of marks: trees marked by diameter and height
plot(Smooth(finpines, sigma=2))
head(Smooth(finpines, sigma=2, at="points"))

Smooth.ssf

Smooth a Spatially Sampled Function

Description
Applies kernel smoothing to a spatially sampled function.
Usage
## S3 method for class 'ssf'
Smooth(X, ...)
Arguments
X

Object of class "ssf".

...

Arguments passed to Smooth.ppp to control the smoothing.

Smoothfun.ppp

1361

Details
An object of class "ssf" represents a real-valued or vector-valued function that has been evaluated
or sampled at an irregular set of points.
The function values will be smoothed using a Gaussian kernel.
Value
A pixel image or a list of pixel images.
Author(s)
Adrian Baddeley .
See Also
ssf, Smooth.ppp
Examples
f <- ssf(redwood, nndist(redwood))
Smooth(f, sigma=0.1)

Smoothfun.ppp

Smooth Interpolation of Marks as a Spatial Function

Description
Perform spatial smoothing of numeric values observed at a set of irregular locations, and return the
result as a function of spatial location.
Usage
Smoothfun(X, ...)
## S3 method for class 'ppp'
Smoothfun(X, sigma = NULL, ...,
weights = NULL, edge = TRUE, diggle = FALSE)
Arguments
X

Marked point pattern (object of class "ppp").

sigma

Smoothing bandwidth, or bandwidth selection function, passed to Smooth.ppp.

...

Additional arguments passed to Smooth.ppp.

weights

Optional vector of weights associated with the points of X.

edge,diggle

Logical arguments controlling the edge correction. Arguments passed to Smooth.ppp.

1362

Softcore

Details
The commands Smoothfun and Smooth both perform kernel-smoothed spatial interpolation of numeric values observed at irregular spatial locations. The difference is that Smooth returns a pixel image, containing the interpolated values at a grid of locations, while Smoothfun returns a function(x,y)
which can be used to compute the interpolated value at any spatial location. For purposes such as
model-fitting it is more accurate to use Smoothfun to interpolate data.
Value
A function with arguments x,y. The function also belongs to the class "Smoothfun" which has
methods for print and as.im. It also belongs to the class "funxy" which has methods for plot,
contour and persp.
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak .
See Also
Smooth
Examples
f <- Smoothfun(longleaf)
f
f(120, 80)
plot(f)

Softcore

The Soft Core Point Process Model

Description
Creates an instance of the Soft Core point process model which can then be fitted to point pattern
data.
Usage
Softcore(kappa, sigma0=NA)
Arguments
kappa

The exponent κ of the Soft Core interaction

sigma0

Optional. Initial estimate of the parameter σ. A positive number.

Softcore

1363

Details
The (stationary) Soft Core point process with parameters β and σ and exponent κ is the pairwise
interaction point process in which each point contributes a factor β to the probability density of the
point pattern, and each pair of points contributes a factor
   
σ 2/κ
exp −
d
to the density, where d is the distance between the two points.
Thus the process has probability density


2/κ 
 X
σ
f (x1 , . . . , xn ) = αβ n(x) exp −


||xi − xj ||
i 0, the process is well-defined only for κ in (0, 1). The limit of the
model as κ → 0 is the hard core process with hard core distance h = σ.
The nonstationary Soft Core process is similar except that the contribution of each individual point
xi is a function β(xi ) of location, rather than a constant beta.
The function ppm(), which fits point process models to point pattern data, requires an argument
of class "interact" describing the interpoint interaction structure of the model to be fitted. The
appropriate description of the Soft Core process pairwise interaction is yielded by the function
Softcore(). See the examples below.
The main argument is the exponent kappa. When kappa is fixed, the model becomes an exponential
family with canonical parameters log β and
log γ =

2
log σ
κ

The canonical parameters are estimated by ppm(), not fixed in Softcore().
The optional argument sigma0 can be used to improve numerical stability. If sigma0 is given, it
should be a positive number, and it should be a rough estimate of the parameter σ.
Value
An object of class "interact" describing the interpoint interaction structure of the Soft Core process with exponent κ.
Author(s)
Adrian Baddeley 
and Rolf Turner 

1364

solapply

References
Ogata, Y, and Tanemura, M. (1981). Estimation of interaction potentials of spatial point patterns
through the maximum likelihood procedure. Annals of the Institute of Statistical Mathematics, B
33, 315–338.
Ogata, Y, and Tanemura, M. (1984). Likelihood analysis of spatial point patterns. Journal of the
Royal Statistical Society, series B 46, 496–518.
See Also
ppm, pairwise.family, ppm.object
Examples
data(cells)
ppm(cells, ~1, Softcore(kappa=0.5), correction="isotropic")
# fit the stationary Soft Core process to `cells'

solapply

Apply a Function Over a List and Obtain a List of Objects

Description
Applies the function FUN to each element of the list X, and returns the result as a list of class
"solist" or "anylist" as appropriate.
Usage
anylapply(X, FUN, ...)
solapply(X, FUN, ..., check = TRUE, promote = TRUE, demote = FALSE)
Arguments
X

A list.

FUN

Function to be applied to each element of X.

...
Additional arguments to FUN.
check,promote,demote
Arguments passed to solist which determine how to handle different classes
of objects.
Details
These convenience functions are similar to lapply except that they return a list of class "solist"
or "anylist".
In both functions, the result is computed by lapply(X, FUN, ...).
In anylapply the result is converted to a list of class "anylist" and returned.
In solapply the result is converted to a list of class "solist" if possible, using as.solist. If this
is not possible, then the behaviour depends on the argument demote. If demote=TRUE the result will
be returned as a list of class "anylist". If demote=FALSE (the default), an error occurs.

solist

1365

Value
A list, usually of class "solist".
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
solist, anylist.
Examples
solapply(waterstriders, density)

solist

List of Two-Dimensional Spatial Objects

Description
Make a list of two-dimensional spatial objects.
Usage
solist(..., check=TRUE, promote=TRUE, demote=FALSE)
Arguments
...

Any number of objects, each representing a two-dimensional spatial dataset.

check

Logical value. If TRUE, check that each of the objects is a 2D spatial object.

promote

Logical value. If TRUE, test whether all objects belong to the same class, and if
so, promote the list of objects to the appropriate class of list.

demote

Logical value determining what should happen if any of the objects is not a 2D
spatial object: if demote=FALSE (the default), a fatal error occurs; if demote=TRUE,
a list of class "anylist" is returned.

Details
This command creates an object of class "solist" (spatial object list) which represents a list of
two-dimensional spatial datasets. The datasets do not necessarily belong to the same class.
Typically the intention is that the datasets in the list should be treated in the same way, for example, they should be plotted side-by-side. The spatstat package provides a plotting function,
plot.solist, and many other functions for this class.
In the spatstat package, various functions produce an object of class "solist". For example,
when a point pattern is split into several point patterns by split.ppp, or an image is split into
several images by split.im, the result is of class "solist".

1366

solutionset

If check=TRUE then the code will check whether all objects in ... belong to the classes of twodimensional spatial objects defined in the spatstat package. They do not have to belong to the same
class. Set check=FALSE for efficiency, but only if you are sure that all the objects are valid.
If some of the objects in ... are not two-dimensional spatial objects, the action taken depends on
the argument demote. If demote=TRUE, the result will belong to the more general class "anylist"
instead of "solist". If demote=FALSE (the default), an error occurs.
If promote=TRUE then the code will check whether all the objects ... belong to the same class. If
they are all point patterns (class "ppp"), the result will also belong to the class "ppplist". If they
are all pixel images (class "im"), the result will also belong to the class "imlist".
Use as.solist to convert a list to a "solist".
Value
A list, usually belonging to the class "solist".
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
as.solist, anylist, solapply
Examples
solist(cells, density(cells))
solist(cells, japanesepines, redwood)

solutionset

Evaluate Logical Expression Involving Pixel Images and Return Region Where Expression is True

Description
Given a logical expression involving one or more pixel images, find all pixels where the expression
is true, and assemble these pixels into a window.
Usage
solutionset(..., envir)
Arguments
...

An expression in the R language, involving one or more pixel images.

envir

Optional. The environment in which to evaluate the expression.

solutionset

1367

Details
Given a logical expression involving one or more pixel images, this function will find all pixels
where the expression is true, and assemble these pixels into a spatial window.
Pixel images in spatstat are represented by objects of class "im" (see im.object). These are
essentially matrices of pixel values, with extra attributes recording the pixel dimensions, etc.
Suppose X is a pixel image. Then solutionset(abs(X) > 3) will find all the pixels in X for which
the pixel value is greater than 3 in absolute value, and return a window containing all these pixels.
If X and Y are two pixel images, solutionset(X > Y) will find all pixels for which the pixel value
of X is greater than the corresponding pixel value of Y, and return a window containing these pixels.
In general, ... can be any logical expression involving pixel images.
The code first tries to evaluate the expression using eval.im. This is successful if the expression
involves only (a) the names of pixel images, (b) scalar constants, and (c) functions which are vectorised. There must be at least one pixel image in the expression. The expression expr must be
vectorised. See the Examples.
If this is unsuccessful, the code then tries to evaluate the expression using pixel arithmetic. This is
successful if all the arithmetic operations in the expression are listed in Math.im.
Value
A spatial window (object of class "owin", see owin.object).
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
im.object, owin.object, eval.im, levelset
Examples
# test images
X <- as.im(function(x,y) { x^2 - y^2 }, unit.square())
Y <- as.im(function(x,y) { 3 * x + y - 1}, unit.square())
W <- solutionset(abs(X) > 0.1)
W <- solutionset(X > Y)
W <- solutionset(X + Y >= 1)
area(solutionset(X < Y))
solutionset(density(cells) > 20)

1368

spatdim

spatdim

Spatial Dimension of a Dataset

Description
Extracts the spatial dimension of an object in the spatstat package.
Usage
spatdim(X)
Arguments
X

Object belonging to any class defined in the spatstat package.

Details
This function returns the number of spatial coordinate dimensions of the dataset X. The results for
some of the more common types of objects are as follows:
object class
"ppp"
"lpp"
"pp3"
"ppx"
"owin"
"psp"
"ppm"

dimension
2
2
3
number of spatial dimensions
2
2
2

Note that time dimensions are not counted.
If X is not a recognised spatial object, the result is NA.
Value
An integer, or NA.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
Examples
spatdim(lansing)

spatialcdf

spatialcdf

1369

Spatial Cumulative Distribution Function

Description
Compute the spatial cumulative distribution function of a spatial covariate, optionally using spatiallyvarying weights.
Usage
spatialcdf(Z, weights = NULL, normalise = FALSE, ..., W = NULL, Zname = NULL)
Arguments
Z

Spatial covariate. A pixel image or a function(x,y,...)

weights

Spatial weighting for different locations. A pixel image, a function(x,y,...),
a window, a constant value, or a fitted point process model (object of class "ppm"
or "kppm").

normalise

Logical. Whether the weights should be normalised so that they sum to 1.

...

Arguments passed to as.mask to determine the pixel resolution, or extra arguments passed to Z if it is a function.

W

Optional window (object of class "owin") defining the spatial domain.

Zname

Optional character string for the name of the covariate Z used in plots.

Details
If weights is missing or NULL, it defaults to 1. The values of the covariate Z are computed on a
grid of pixels. The weighted cumulative distribution function of Z values is computed, taking each
value with weight equal to the pixel area. The resulting function F is such that F (t) is the area of
the region of space where Z ≤ t.
If weights is a pixel image or a function, then the values of weights and of the covariate Z are
computed on a grid of pixels. The weights are multiplied by the pixel area. Then the weighted
empirical cumulative distribution function of Z values is computed using ewcdf. The resulting
function F is such that F (t) is the total weight (or weighted area) of the region of space where
Z ≤ t.
If weights is a fitted point process model, then it should be a Poisson process. The fitted intensity
of the model, and the value of the covariate Z, are evaluated at the quadrature points used to fit the
model. The weights are multiplied by the weights of the quadrature points. Then the weighted
empirical cumulative distribution of Z values is computed using ewcdf. The resulting function F is
such that F (t) is the expected number of points in the point process that will fall in the region of
space where Z ≤ t.
If normalise=TRUE, the function is normalised so that its maximum value equals 1, so that it gives
the cumulative fraction of weight or cumulative fraction of points.
The result can be printed, plotted, and used as a function.
Value
A cumulative distribution function object belonging to the classes "spatialcdf", "ewcdf", "ecdf"
and "stepfun".

1370

spatstat.options

Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak 
See Also
ewcdf, cdf.test
Examples
with(bei.extra, {
plot(spatialcdf(grad))
fit <- ppm(bei ~ elev)
plot(spatialcdf(grad, predict(fit)))
plot(A <- spatialcdf(grad, fit))
A(0.1)
})

spatstat.options

Internal Options in Spatstat Package

Description
Allows the user to examine and reset the values of global parameters which control actions in the
spatstat package.
Usage
spatstat.options(...)
reset.spatstat.options()
Arguments
...

Either empty, or a succession of parameter names in quotes, or a succession of
name=value pairs. See below for the parameter names.

Details
The function spatstat.options allows the user to examine and reset the values of global parameters which control actions in the spatstat package. It is analogous to the system function options.
The function reset.spatstat.options resets all the global parameters in spatstat to their original, default values.
The global parameters of interest to the user are:
checkpolygons Logical flag indicating whether the functions owin and as.owin should apply very
strict checks on the validity of polygon data. These strict checks are no longer necessary, and
the default is checkpolygons=FALSE. See also fixpolygons below.
checksegments Logical flag indicating whether the functions psp and as.psp should check the
validity of line segment data (in particular, checking that the endpoints of the line segments
are inside the specified window). It is advisable to leave this flag set to TRUE.

spatstat.options

1371

eroded.intensity Logical flag affecting the behaviour of the score and pseudo-score residual functions Gcom, Gres Kcom, Kres, psstA, psstG, psst. The flag indicates whether to compute
intensity estimates on an eroded window (eroded.intensity=TRUE) or on the original data
window (eroded.intensity=FALSE, the default).
expand The default expansion factor (area inflation factor) for expansion of the simulation window
in rmh (see rmhcontrol). Initialised to 2.
expand.polynom Logical. Whether expressions involving polynom in a model formula should be
expanded, so that polynom(x,2) is replaced by x + I(x^2) and so on. Initialised to TRUE.
fastpois Logical. Whether to use a fast algorithm (introduced in spatstat 1.42-3) for simulating
the Poisson point process in rpoispp when the argument lambda is a pixel image. Initialised
to TRUE. Should be set to FALSE if needed to guarantee repeatability of results computed using
earlier versions of spatstat.
fastthin Logical. Whether to use a fast C language algorithm (introduced in spatstat 1.42-3) for
random thinning in rthin when the argument P is a single number. Initialised to TRUE. Should
be set to FALSE if needed to guarantee repeatability of results computed using earlier versions
of spatstat.
fastK.lgcp Logical. Whether to use fast or slow algorithm to compute the (theoretical) K-function
of a log-Gaussian Cox process for use in lgcp.estK or Kmodel. The slow algorithm uses accurate numerical integration; the fast algorithm uses Simpson’s Rule for numerical integration,
and is about two orders of magnitude faster. Initialised to FALSE.
fixpolygons Logical flag indicating whether the functions owin and as.owin should repair errors
in polygon data. For example, self-intersecting polygons and overlapping polygons will be
repaired. The default is fixpolygons=TRUE.
fftw Logical value indicating whether the two-dimensional Fast Fourier Transform should be computed using the package fftwtools, instead of the fft function in the stats package. This
affects the speed of density.ppp, density.psp, blur setcov and Smooth.ppp.
gpclib Defunct. This parameter was used to permit or forbid the use of the package gpclib, because
of its restricted software licence. This package is no longer needed.
huge.npoints The maximum value of n for which runif(n) will not generate an error (possible
errors include failure to allocate sufficient memory, and integer overflow of n). An attempt to
generate more than this number of random points triggers a warning from runifpoint and
other functions. Defaults to 1e6.
image.colfun Function determining the default colour map for plot.im. When called with one
integer argument n, this function should return a character vector of length n specifying n
different colours.
Kcom.remove.zeroes Logical value, determining whether the algorithm in Kcom and Kres removes
or retains the contributions to the function from pairs of points that are identical. If these are
retained then the function has a jump at r = 0. Initialised to TRUE.
maxedgewt Edge correction weights will be trimmed so as not to exceed this value. This applies
to the weights computed by edge.Trans or edge.Ripley and used in Kest and its relatives.
maxmatrix The maximum permitted size (rows times columns) of matrices generated by spatstat’s internal code. Used by ppm and predict.ppm (for example) to decide when to split a
large calculation into blocks. Defaults to 2^24=16777216.
monochrome Logical flag indicating whether graphics should be plotted in grey scale (monochrome=TRUE)
or in colour (monochrome=FALSE, the default).
n.bandwidth Integer. Number of trial values of smoothing bandwidth to use for cross-validation
in bw.relrisk and similar functions.

1372

spatstat.options

ndummy.min The minimum number of dummy points in a quadrature scheme created by default.dummy.
Either an integer or a pair of integers giving the minimum number of dummy points in the x
and y directions respectively.
ngrid.disc Number of points in the square grid used to compute a discrete approximation to the
areas of discs in areaLoss and areaGain when exact calculation is not available. A single
integer.
npixel Default number of pixels in a binary mask or pixel image. Either an integer, or a pair of
integers, giving the number of pixels in the x and y directions respectively.
nvoxel Default number of voxels in a 3D image, typically for calculating the distance transform in
F3est. Initialised to 4 megavoxels: nvoxel = 2^22 = 4194304.
par.binary List of arguments to be passed to the function image when displaying a binary image
mask (in plot.owin or plot.ppp). Typically used to reset the colours of foreground and
background.
par.contour List of arguments controlling contour plots of pixel images by contour.im.
par.fv List of arguments controlling the plotting of functions by plot.fv and its relatives.
par.persp List of arguments to be passed to the function persp when displaying a real-valued
image, such as the fitted surfaces in plot.ppm.
par.points List of arguments controlling the plotting of point patterns by plot.ppp.
par.pp3 List of arguments controlling the plotting of three-dimensional point patterns by plot.pp3.
print.ppm.SE Default rule used by print.ppm to decide whether to calculate and print standard
errors of the estimated coefficients of the model. One of the strings "always", "never"
or "poisson" (the latter indicating that standard errors will be calculated only for Poisson
models). The default is "poisson" because the calculation for non-Poisson models can take
a long time.
progress Character string determining the style of progress reports printed by progressreport.
Either "tty", "tk" or "txtbar". For explanation of these options, see progressreport.
project.fast Logical. If TRUE, the algorithm of project.ppm will be accelerated using a shorcut.
Initialised to FALSE.
psstA.ngrid Single integer, controlling the accuracy of the discrete approximation of areas computed in the function psstA. The area of a disc is approximated by counting points on an n×n
grid. Initialised to 32.
psstA.nr Single integer, determining the number of distances r at which the function psstA will
be evaluated (in the default case where argument r is absent). Initialised to 30.
psstG.remove.zeroes Logical value, determining whether the algorithm in psstG removes or retains the contributions to the function from pairs of points that are identical. If these are
retained then the function has a jump at r = 0. Initialised to TRUE.
rmh.p, rmh.q, rmh.nrep New default values for the parameters p, q and nrep in the MetropolisHastings simulation algorithm. These override the defaults in rmhcontrol.default.
scalable Logical flag indicating whether the new code in rmh.default which makes the results
scalable (invariant to change of units) should be used. In order to recover former behaviour
(so that previous results can be reproduced) set this option equal to FALSE. See the “Warning”
section in the help for rmh() for more detail.
terse Integer between 0 and 4. The level of terseness (brevity) in printed output from many functions in spatstat. Higher values mean shorter output. A rough guide is the following:
0
1
2
3
4

Full output
Avoid wasteful output
Remove space between paragraphs
Suppress extras such as standard errors
Compress text, suppress internal warnings

spatstat.options

1373

The value of terse is initialised to 0.
transparent Logical value indicating whether default colour maps are allowed to include semitransparent colours, where possible. Default is TRUE. Currently this only affects plot.ppp.
units.paren The kind of parenthesis which encloses the text that explains a unitname. This text is
seen in the text output of functions like print.ppp and in the graphics generated by plot.fv.
The value should be one of the character strings '(', '[', '{' or ''. The default is '('.
If no arguments are given, the current values of all parameters are returned, in a list.
If one parameter name is given, the current value of this parameter is returned (not in a list, just the
value).
If several parameter names are given, the current values of these parameters are returned, in a list.
If name=value pairs are given, the named parameters are reset to the given values, and the previous
values of these parameters are returned, in a list.
Value
Either a list of parameters and their values, or a single value. See Details.
Internal parameters
The following parameters may also be specified to spatstat.options but are intended for software
development or testing purposes.
closepairs.newcode Logical. Whether to use new version of the code for closepairs. Initialised
to TRUE.
crossing.psp.useCall Logical. Whether to use new version of the code for crossing.psp. Initialised to TRUE.
crosspairs.newcode Logical. Whether to use new version of the code for crosspairs. Initialised
to TRUE.
densityC Logical. Indicates whether to use accelerated C code (densityC=TRUE) or interpreted R
code (densityC=FALSE) to evaluate density.ppp(X, at="points"). Initialised to TRUE.

exactdt.checks.data Logical. Do not change this value, unless you are Adrian Baddeley 
and Rolf Turner 
See Also
options

1374

split.hyperframe

Examples
# save current values
oldopt <- spatstat.options()
spatstat.options("npixel")
spatstat.options(npixel=150)
spatstat.options(npixel=c(100,200))
spatstat.options(par.binary=list(col=grey(c(0.5,1))))
spatstat.options(par.persp=list(theta=-30,phi=40,d=4))
# see help(persp.default) for other options
# revert
spatstat.options(oldopt)

split.hyperframe

Divide Hyperframe Into Subsets and Reassemble

Description
split divides the data x into subsets defined by f. The replacement form replaces values corresponding to such a division.
Usage
## S3 method for class 'hyperframe'
split(x, f, drop = FALSE, ...)
## S3 replacement method for class 'hyperframe'
split(x, f, drop = FALSE, ...) <- value
Arguments
x

Hyperframe (object of class "hyperframe").

f

a factor in the sense that as.factor(f) defines the grouping, or a list of such
factors in which case their interaction is used for the grouping.

drop

logical value, indicating whether levels that do not occur should be dropped
from the result.

value

a list of hyperframes which arose (or could have arisen) from the command
split(x,f,drop=drop).

...

Ignored.

Details
These are methods for the generic functions split and split<- for hyperframes (objects of class
"hyperframe").
A hyperframe is like a data frame, except that its entries can be objects of any kind. The behaviour
of these methods is analogous to the corresponding methods for data frames.

split.im

1375

Value
The value returned from split.hyperframe is a list of hyperframe containing the values for the
groups. The components of the list are named by the levels of f (after converting to a factor, or if
already a factor and drop = TRUE, dropping unused levels).
The replacement method split<-.hyperframe returns a new hyperframe x for which split(x,f)
equals value.
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak 
See Also
hyperframe, [.hyperframe
Examples
split(pyramidal, pyramidal$group)

split.im

Divide Image Into Sub-images

Description
Divides a pixel image into several sub-images according to the value of a factor, or according to the
tiles of a tessellation.
Usage
## S3 method for class 'im'
split(x, f, ..., drop = FALSE)
Arguments
x

Pixel image (object of class "im").

f

Splitting criterion. Either a tessellation (object of class "tess") or a pixel image
with factor values.

...

Ignored.

drop

Logical value determining whether each subset should be returned as a pixel images (drop=FALSE) or as a one-dimensional vector of pixel values (drop=TRUE).

1376

split.msr

Details
This is a method for the generic function split for the class of pixel images. The image x will be
divided into subsets determined by the data f. The result is a list of these subsets.
The splitting criterion may be either
• a tessellation (object of class "tess"). Each tile of the tessellation delineates a subset of the
spatial domain.
• a pixel image (object of class "im") with factor values. The levels of the factor determine
subsets of the spatial domain.
If drop=FALSE (the default), the result is a list of pixel images, each one a subset of the pixel image
x, obtained by restricting the pixel domain to one of the subsets. If drop=TRUE, then the pixel values
are returned as numeric vectors.
Value
If drop=FALSE, a list of pixel images (objects of class "im"). It is also of class "solist" so that it
can be plotted immediately.
If drop=TRUE, a list of numeric vectors.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
by.im, tess, im
Examples
W <- square(1)
X <- as.im(function(x,y){sqrt(x^2+y^2)}, W)
Y <- dirichlet(runifpoint(12, W))
plot(split(X,Y))

split.msr

Divide a Measure into Parts

Description
Decomposes a measure into components, each component being a measure.
Usage
## S3 method for class 'msr'
split(x, f, drop = FALSE, ...)

split.msr

1377

Arguments
x

Measure (object of class "msr") to be decomposed.

f

Factor or tessellation determining the decomposition. Argument passed to split.ppp.
See Details.

drop

Logical value indicating whether empty components should be retained in the
list (drop=FALSE, the default) or deleted (drop=TRUE).

...

Ignored.

Details
An object of class "msr" represents a signed (i.e. real-valued) or vector-valued measure in the
spatstat package. See msr for explanation.
This function is a method for the generic split. It divides the measure x into components, each of
which is a measure.
A measure x is represented in spatstat by a finite set of sample points with values attached to them.
The function split.msr divides this pattern of sample points into several sub-patterns of points
using split.ppp. For each sub-pattern, the values attached to these points are extracted from x,
and these values and sample points determine a measure, which is a component or piece of the
original x.
The argument f can be missing, if the sample points of x are multitype points. In this case, x represents a measure associated with marked spatial locations, and the command split(x) separates x
into a list of component measures, one for each possible mark.
Otherwise the argument f is passed to split.ppp. It should be either a factor (of length equal to
the number of sample points of x) or a tessellation (object of class "tess" representing a division
of space into tiles) as documented under split.ppp.
Value
A list, each of whose entries is a measure (object of class "msr").
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
msr, [.msr, with.msr
Examples
## split by tessellation
a <- residuals(ppm(cells ~ x))
aa <- split(a, dirichlet(runifpoint(4)))
aa
sapply(aa, integral)
## split by type of point
b <- residuals(ppm(amacrine ~ marks + x))
bb <- split(b)
bb

1378

split.ppp

split.ppp

Divide Point Pattern into Sub-patterns

Description
Divides a point pattern into several sub-patterns, according to their marks, or according to any
user-specified grouping.
Usage
## S3 method for class 'ppp'
split(x, f = marks(x), drop=FALSE, un=NULL, reduce=FALSE, ...)
## S3 replacement method for class 'ppp'
split(x, f = marks(x), drop=FALSE, un=missing(f), ...) <- value
Arguments
x

A two-dimensional point pattern. An object of class "ppp".

f

Data determining the grouping. Either a factor, a logical vector, a pixel image
with factor values, a tessellation, a window, or the name of one of the columns
of marks.

drop

Logical. Determines whether empty groups will be deleted.

un

Logical. Determines whether the resulting subpatterns will be unmarked (i.e.
whether marks will be removed from the points in each subpattern).

reduce

Logical. Determines whether to delete the column of marks used to split the
pattern, when the marks are a data frame.

...

Other arguments are ignored.

value

List of point patterns.

Details
The function split.ppp divides up the points of the point pattern x into several sub-patterns according to the values of f. The result is a list of point patterns.
The argument f may be
• a factor, of length equal to the number of points in x. The levels of f determine the destination
of each point in x. The ith point of x will be placed in the sub-pattern split.ppp(x)$l where
l = f[i].
• a pixel image (object of class "im") with factor values. The pixel value of f at each point of x
will be used as the classifying variable.
• a tessellation (object of class "tess"). Each point of x will be classified according to the tile
of the tessellation into which it falls.
• a window (object of class "owin"). Each point of x will be classified according to whether it
falls inside or outside this window.
• a character string, matching the name of one of the columns of marks, if marks(x) is a data
frame. This column should be a factor.
If f is missing, then it will be determined by the marks of the point pattern. The pattern x can be
either

split.ppp

1379

• a multitype point pattern (a marked point pattern whose marks vector is a factor). Then f
is taken to be the marks vector. The effect is that the points of each type are separated into
different point patterns.
• a marked point pattern with a data frame of marks, containing at least one column that is a
factor. The first such column will be used to determine the splitting factor f.
Some of the sub-patterns created by the split may be empty. If drop=TRUE, then empty sub-patterns
will be deleted from the list. If drop=FALSE then they are retained.
The argument un determines how to handle marks in the case where x is a marked point pattern.
If un=TRUE then the marks of the points will be discarded when they are split into groups, while if
un=FALSE then the marks will be retained.
If f and un are both missing, then the default is un=TRUE for multitype point patterns and un=FALSE
for marked point patterns with a data frame of marks.
If the marks of x are a data frame, then split(x, reduce=TRUE) will discard only the column of
marks that was used to split the pattern. This applies only when the argument f is missing.
The result of split.ppp has class "splitppp" and can be plotted using plot.splitppp.
The assignment function split<-.ppp updates the point pattern x so that it satisfies split(x, f, drop, un) = value.
The argument value is expected to be a list of point patterns, one for each level of f. These point
patterns are expected to be compatible with the type of data in the original pattern x.
Splitting can also be undone by the function superimpose, but this typically changes the ordering
of the data.
Value
The value of split.ppp is a list of point patterns. The components of the list are named by the
levels of f. The list also has the class "splitppp".
The assignment form split<-.ppp returns the updated point pattern x.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
cut.ppp, plot.splitppp, superimpose, im, tess, ppp.object
Examples
# (1) Splitting by marks
# Multitype point pattern: separate into types
u <- split(amacrine)
# plot them
plot(split(amacrine))
# the
amon
amon
amon

following are equivalent:
<- split(amacrine)$on
<- unmark(amacrine[amacrine$marks == "on"])
<- subset(amacrine, marks == "on", -marks)

1380

split.ppx

# the following are equivalent:
amon <- split(amacrine, un=FALSE)$on
amon <- amacrine[amacrine$marks == "on"]
# Scramble the locations of the 'on' cells
X <- amacrine
u <- split(X)
u$on <- runifpoint(ex=amon)
split(X) <- u
# Point pattern with continuous marks
trees <- longleaf
# cut the range of tree diameters into three intervals
# using cut.ppp
long3 <- cut(trees, breaks=3)
# now split them
long3split <- split(long3)
# (2) Splitting by a factor
# Unmarked point pattern
swedishpines
# cut & split according to nearest neighbour distance
f <- cut(nndist(swedishpines), 3)
u <- split(swedishpines, f)
# (3) Splitting over a tessellation
tes <- tess(xgrid=seq(0,96,length=5),ygrid=seq(0,100,length=5))
v <- split(swedishpines, tes)
# (4) how to apply an operation to selected points:
# split into components, transform desired component, then un-split
# e.g. apply random jitter to 'on' points only
X <- amacrine
Y <- split(X)
Y$on <- rjitter(Y$on, 0.1)
split(X) <- Y

split.ppx

Divide Multidimensional Point Pattern into Sub-patterns

Description
Divides a multidimensional point pattern into several sub-patterns, according to their marks, or
according to any user-specified grouping.
Usage
## S3 method for class 'ppx'
split(x, f = marks(x), drop=FALSE, un=NULL, ...)

split.ppx

1381

Arguments
x
f
drop
un
...

A multi-dimensional point pattern. An object of class "ppx".
Data determining the grouping. Either a factor, or the name of one of the
columns of marks.
Logical. Determines whether empty groups will be deleted.
Logical. Determines whether the resulting subpatterns will be unmarked (i.e.
whether marks will be removed from the points in each subpattern).
Other arguments are ignored.

Details
The generic command split allows a dataset to be separated into subsets according to the value of
a grouping variable.
The function split.ppx is a method for the generic split for the class "ppx" of multidimensional
point patterns. It divides up the points of the point pattern x into several sub-patterns according to
the values of f. The result is a list of point patterns.
The argument f may be
• a factor, of length equal to the number of points in x. The levels of f determine the destination
of each point in x. The ith point of x will be placed in the sub-pattern split.ppx(x)$l where
l = f[i].
• a character string, matching the name of one of the columns of marks, if marks(x) is a data
frame. This column should be a factor.
If f is missing, then it will be determined by the marks of the point pattern. The pattern x can be
either
• a multitype point pattern (a marked point pattern whose marks vector is a factor). Then f
is taken to be the marks vector. The effect is that the points of each type are separated into
different point patterns.
• a marked point pattern with a data frame or hyperframe of marks, containing at least one
column that is a factor. The first such column will be used to determine the splitting factor f.
Some of the sub-patterns created by the split may be empty. If drop=TRUE, then empty sub-patterns
will be deleted from the list. If drop=FALSE then they are retained.
The argument un determines how to handle marks in the case where x is a marked point pattern.
If un=TRUE then the marks of the points will be discarded when they are split into groups, while if
un=FALSE then the marks will be retained.
If f and un are both missing, then the default is un=TRUE for multitype point patterns and un=FALSE
for marked point patterns with a data frame of marks.
The result of split.ppx has class "splitppx" and "anylist". There are methods for print,
summary and plot.
Value
A list of point patterns. The components of the list are named by the levels of f. The list also has
the class "splitppx" and "anylist".
Author(s)
Adrian Baddeley 
and Rolf Turner 

1382

spokes

See Also
ppx, plot.anylist
Examples
df <- data.frame(x=runif(4),y=runif(4),t=runif(4),
age=rep(c("old", "new"), 2),
size=runif(4))
X <- ppx(data=df, coord.type=c("s","s","t","m","m"))
X
split(X)

spokes

Spokes pattern of dummy points

Description
Generates a pattern of dummy points in a window, given a data point pattern. The dummy points
lie on the radii of circles emanating from each data point.
Usage
spokes(x, y, nrad = 3, nper = 3, fctr = 1.5, Mdefault = 1)
Arguments
x
y
nrad
nper
fctr
Mdefault

Vector of x coordinates of data points, or a list with components x and y, or a
point pattern (an object of class ppp).
Vector of y coordinates of data points. Ignored unless x is a vector.
Number of radii emanating from each data point.
Number of dummy points per radius.
Scale factor. Length of largest spoke radius is fctr * M where M is the mean
nearest neighbour distance for the data points.
Value of M to be used if x has length 1.

Details
This function is useful in creating dummy points for quadrature schemes (see quadscheme).
Given the data points, the function creates a collection of nrad * nper * length(x) dummy
points.
Around each data point (x[i],y[i]) there are nrad * nper dummy points, lying on nrad radii
emanating from (x[i],y[i]), with nper dummy points equally spaced along each radius.
The (equal) spacing of dummy points along each radius is controlled by the factor fctr. The
distance from a data point to the furthest of its associated dummy points is fctr * M where M is the
mean nearest neighbour distance for the data points.
If there is only one data point the nearest neighbour distance is infinite, so the value Mdefault will
be used in place of M.
If x is a point pattern, then the value returned is also a point pattern, which is clipped to the window
of x. Hence there may be fewer than nrad * nper * length(x) dummy points in the pattern
returned.

square

1383

Value
If argument x is a point pattern, a point pattern with window equal to that of x. Otherwise a list with
two components x and y. In either case the components x and y of the value are numeric vectors
giving the coordinates of the dummy points.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
quad.object, quadscheme, inside.owin, gridcentres, stratrand
Examples
dat <- runifrect(10)
dum <- spokes(dat$x, dat$y, 5, 3, 0.7)
plot(dum)
Q <- quadscheme(dat, dum, method="dirichlet")
plot(Q, tiles=TRUE)

square

Square Window

Description
Creates a square window
Usage
square(r=1, unitname=NULL)
unit.square()
Arguments
r

Numeric. The side length of the square, or a vector giving the minimum and
maximum coordinate values.

unitname

Optional. Name of unit of length. Either a single character string, or a vector of
two character strings giving the singular and plural forms, respectively.

Details
If r is a number, square(r) is a shortcut for creating a window object representing the square
[0, r] × [0, r]. It is equivalent to the command owin(c(0,r),c(0,r)).
If r is a vector of length 2, then square(r) creates the square with x and y coordinates ranging
from r[1] to r[2].
unit.square creates the unit square [0, 1] × [0, 1]. It is equivalent to square(1) or square() or
owin(c(0,1),c(0,1)).
These commands are included for convenience, and to improve the readability of some code.

1384

ssf

Value
An object of class "owin" (see owin.object) specifying a window.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
owin.object, owin
Examples
W <- square(10)
W <- square(c(-1,1))

ssf

Spatially Sampled Function

Description
Create an object that represents a spatial function which has been evaluated or sampled at an irregular set of points.
Usage
ssf(loc, val)
Arguments
loc

The spatial locations at which the function has been evaluated. A point pattern
(object of class "ppp").

val

The function values at these locations. A numeric vector with one entry for each
point of loc, or a data frame with one row for each point of loc.

Details
An object of class "ssf" represents a real-valued or vector-valued function that has been evaluated
or sampled at an irregular set of points. An example would be a spatial covariate that has only been
measured at certain locations.
An object of this class also inherits the class "ppp", and is essentially the same as a marked point
pattern, except for the class membership which enables it to be handled in a different way.
There are methods for plot, print etc; see plot.ssf and methods.ssf.
Use unmark to extract only the point locations, and marks.ssf to extract only the function values.
Value
Object of class "ssf".

stieltjes

1385

Author(s)
Adrian Baddeley 
See Also
plot.ssf, methods.ssf, Smooth.ssf, with.ssf, [.ssf.
Examples
ssf(cells, nndist(cells, k=1:3))

stieltjes

Compute Integral of Function Against Cumulative Distribution

Description
Computes the Stieltjes integral of a function f with respect to a function M .
Usage
stieltjes(f, M, ...)
Arguments
f

The integrand. A function in the R language.

M

The cumulative function against which f will be integrated. An object of class
"fv" or "stepfun".

...

Additional arguments passed to f.

Details
This command computes the Stieltjes integral
Z
I = f (x)dM (x)
of a real-valued function f (x) with respect to a nondecreasing function M (x).
One common use of the Stieltjes integral is to find the mean value of a random variable from its
cumulative distribution function F (x). The mean value is the Stieltjes integral of f (x) = x with
respect to F (x).
The argument f should be a function in the R language. It should accept a numeric vector argument x and should return a numeric vector of the same length.
The argument M should be either a step function (object of class "stepfun") or a function value
table (object of class "fv", see fv.object). Objects of class "stepfun" are returned by ecdf,
ewcdf, spatialcdf and other utilities. Objects of class "fv" are returned by the commands Kest,
Gest, etc.
Value
A list containing the value of the Stieltjes integral computed using each of the versions of the
function M.

1386

stienen

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
fv.object, Gest
Examples
# estimate cdf of nearest neighbour distance in redwood data
G <- Gest(redwood)
# compute estimate of mean nearest neighbour distance
stieltjes(function(x){x}, G)
# estimated probability of a distance in the interval [0.1,0.2]
stieltjes(function(x,a,b){ (x >= a) & (x <= b)}, G, a=0.1, b=0.2)
# stepfun example
H <- spatialcdf(bei.extra$elev, normalise=TRUE)
stieltjes(function(x){x}, H)

stienen

Stienen Diagram

Description
Draw the Stienen diagram of a point pattern, or compute the region covered by the Stienen diagram.

Usage
stienen(X, ..., bg = "grey", border = list(bg = NULL))
stienenSet(X, edge=TRUE)
Arguments
X

Point pattern (object of class "ppp").

...

Arguments passed to plot.ppp to control the plot.

bg

Fill colour for circles.

border

Either a list of arguments passed to plot.ppp to control the display of circles at
the border of the diagram, or the value FALSE indicating that the border circles
should not be plotted.

edge

Logical value indicating whether to include the circles at the border of the diagram.

stratrand

1387

Details
The Stienen diagram of a point pattern (Stienen, 1982) is formed by drawing a circle around each
point of the pattern, with diameter equal to the nearest-neighbour distance for that point. These
circles do not overlap. If two points are nearest neighbours of each other, then the corresponding
circles touch.
stienenSet(X) computes the union of these circles and returns it as a window (object of class
"owin").
stienen(X) generates a plot of the Stienen diagram of the point pattern X. By default, circles are
shaded in grey if they lie inside the window of X, and are not shaded otherwise.
Value
The plotting function stienen returns NULL.
The return value of stienenSet is a window (object of class "owin").
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Stienen, H. (1982) Die Vergroeberung von Karbiden in reinen Eisen-Kohlenstoff Staehlen. Dissertation, RWTH Aachen.
See Also
nndist, plot.ppp
Examples
Y <- stienenSet(cells)
stienen(redwood)
stienen(redwood, border=list(bg=NULL, lwd=2, cols="red"))

stratrand

Stratified random point pattern

Description
Generates a “stratified random” pattern of points in a window, by dividing the window into rectangular tiles and placing k random points in each tile.
Usage
stratrand(window, nx, ny, k = 1)

1388

stratrand

Arguments
window

A window. An object of class owin, or data in any format acceptable to as.owin().

nx

Number of tiles in each row.

ny

Number of tiles in each column.

k

Number of random points to generate in each tile.

Details
The bounding rectangle of window is divided into a regular nx × ny grid of rectangular tiles. In
each tile, k random points are generated independently with a uniform distribution in that tile.
Note that some of these grid points may lie outside the window, if window is not of type "rectangle".
The function inside.owin can be used to select those grid points which do lie inside the window.
See the examples.
This function is useful in creating dummy points for quadrature schemes (see quadscheme) as well
as in simulating random point patterns.
Value
A list with two components x and y, which are numeric vectors giving the coordinates of the random
points.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
quad.object, quadscheme, inside.owin, gridcentres
Examples
w <- unit.square()
xy <- stratrand(w, 10, 10)
## Not run:
plot(w)
points(xy)
## End(Not run)
# polygonal boundary
bdry <- list(x=c(0.1,0.3,0.7,0.4,0.2),
y=c(0.1,0.1,0.5,0.7,0.3))
w <- owin(c(0,1), c(0,1), poly=bdry)
xy <- stratrand(w, 10, 10, 3)
## Not run:
plot(w)
points(xy)
## End(Not run)
# determine which grid points are inside polygon
ok <- inside.owin(xy$x, xy$y, w)
## Not run:

Strauss

1389

plot(w)
points(xy$x[ok], xy$y[ok])
## End(Not run)

Strauss

The Strauss Point Process Model

Description
Creates an instance of the Strauss point process model which can then be fitted to point pattern data.
Usage
Strauss(r)
Arguments
r

The interaction radius of the Strauss process

Details
The (stationary) Strauss process with interaction radius r and parameters β and γ is the pairwise
interaction point process in which each point contributes a factor β to the probability density of the
point pattern, and each pair of points closer than r units apart contributes a factor γ to the density.
Thus the probability density is
f (x1 , . . . , xn ) = αβ n(x) γ s(x)
where x1 , . . . , xn represent the points of the pattern, n(x) is the number of points in the pattern,
s(x) is the number of distinct unordered pairs of points that are closer than r units apart, and α is
the normalising constant.
The interaction parameter γ must be less than or equal to 1 so that this model describes an “ordered”
or “inhibitive” pattern.
The nonstationary Strauss process is similar except that the contribution of each individual point xi
is a function β(xi ) of location, rather than a constant beta.
The function ppm(), which fits point process models to point pattern data, requires an argument of
class "interact" describing the interpoint interaction structure of the model to be fitted. The appropriate description of the Strauss process pairwise interaction is yielded by the function Strauss().
See the examples below.
Note the only argument is the interaction radius r. When r is fixed, the model becomes an exponential family. The canonical parameters log(β) and log(γ) are estimated by ppm(), not fixed in
Strauss().
Value
An object of class "interact" describing the interpoint interaction structure of the Strauss process
with interaction radius r.

1390

StraussHard

Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Kelly, F.P. and Ripley, B.D. (1976) On Strauss’s model for clustering. Biometrika 63, 357–360.
Strauss, D.J. (1975) A model for clustering. Biometrika 62, 467–475.
See Also
ppm, pairwise.family, ppm.object
Examples
Strauss(r=0.1)
# prints a sensible description of itself
data(cells)
## Not run:
ppm(cells, ~1, Strauss(r=0.07))
# fit the stationary Strauss process to `cells'
## End(Not run)
ppm(cells, ~polynom(x,y,3), Strauss(r=0.07))
# fit a nonstationary Strauss process with log-cubic polynomial trend

StraussHard

The Strauss / Hard Core Point Process Model

Description
Creates an instance of the “Strauss/ hard core” point process model which can then be fitted to point
pattern data.
Usage
StraussHard(r, hc=NA)
Arguments
r

The interaction radius of the Strauss interaction

hc

The hard core distance. Optional.

StraussHard

1391

Details
A Strauss/hard core process with interaction radius r, hard core distance h < r, and parameters β
and γ, is a pairwise interaction point process in which
• distinct points are not allowed to come closer than a distance h apart
• each pair of points closer than r units apart contributes a factor γ to the probability density.
This is a hybrid of the Strauss process and the hard core process.
The probability density is zero if any pair of points is closer than h units apart, and otherwise equals
f (x1 , . . . , xn ) = αβ n(x) γ s(x)
where x1 , . . . , xn represent the points of the pattern, n(x) is the number of points in the pattern,
s(x) is the number of distinct unordered pairs of points that are closer than r units apart, and α is
the normalising constant.
The interaction parameter γ may take any positive value (unlike the case for the Strauss process).
If γ < 1, the model describes an “ordered” or “inhibitive” pattern. If γ > 1, the model is “ordered”
or “inhibitive” up to the distance h, but has an “attraction” between points lying at distances in the
range between h and r.
If γ = 1, the process reduces to a classical hard core process with hard core distance h. If γ = 0,
the process reduces to a classical hard core process with hard core distance r.
The function ppm(), which fits point process models to point pattern data, requires an argument of
class "interact" describing the interpoint interaction structure of the model to be fitted. The appropriate description of the Strauss/hard core process pairwise interaction is yielded by the function
StraussHard(). See the examples below.
The canonical parameter log(γ) is estimated by ppm(), not fixed in StraussHard().
If the hard core distance argument hc is missing or NA, it will be estimated from the data when
ppm is called. The estimated value of hc is the minimum nearest neighbour distance multiplied by
n/(n + 1), where n is the number of data points.
Value
An object of class "interact" describing the interpoint interaction structure of the “Strauss/hard
core” process with Strauss interaction radius r and hard core distance hc.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns.
Australian and New Zealand Journal of Statistics 42, 283–322.
Ripley, B.D. (1981) Spatial statistics. John Wiley and Sons.
Strauss, D.J. (1975) A model for clustering. Biometrika 62, 467–475.
See Also
ppm, pairwise.family, ppm.object

1392

studpermu.test

Examples
StraussHard(r=1,hc=0.02)
# prints a sensible description of itself
data(cells)
## Not run:
ppm(cells, ~1, StraussHard(r=0.1, hc=0.05))
# fit the stationary Strauss/hard core process to `cells'
## End(Not run)
ppm(cells, ~ polynom(x,y,3), StraussHard(r=0.1, hc=0.05))
# fit a nonstationary Strauss/hard core process
# with log-cubic polynomial trend

studpermu.test

Studentised Permutation Test

Description
Perform a studentised permutation test for a difference between groups of point patterns.
Usage
studpermu.test(X, formula, summaryfunction = Kest,
..., rinterval = NULL, nperm = 999,
use.Tbar = FALSE, minpoints = 20, rsteps = 128,
r = NULL, arguments.in.data = FALSE)
Arguments
X

Data. Either a hyperframe or a list of lists of point patterns.

formula

Formula describing the grouping, when X is a hyperframe. The left side of the
formula identifies which column of X contains the point patterns. The right side
identifies the grouping factor. If the formula is missing, the grouping variable is
taken to be the first column of X that contains a factor, and the point patterns are
taken from the first column that contains point patterns.

summaryfunction
Summary function applicable to point patterns.
...

Additional arguments passed to summaryfunction.

rinterval

Interval of distance values r over which the summary function should be evaluated and over which the test statistic will be integrated. If NULL, the default
range of the summary statistic is used (taking the intersection of these ranges
over all patterns).

nperm

Number of random permutations for the test.

use.Tbar

Logical value indicating choice of test statistic. If TRUE, use the alternative test
statistic, which is appropriate for summary functions with roughly constant variance, such as K(r)/r or L(r).

studpermu.test

1393

minpoints

Minimum permissible number of points in a point pattern for inclusion in the
test calculation.

rsteps

Number of discretisation steps in the rinterval.

r

Optional vector of distance values as the argument for summaryfunction. Should
not usually be given. There is a sensible default.
arguments.in.data
Logical. If TRUE, individual extra arguments to summaryfunction will be taken
from X (which must be a hyperframe). This assumes that the first argument of
summaryfunction is the point pattern dataset.

Details
This function performs the studentized permutation test of Hahn (2012) for a difference between
groups of point patterns.
The first argument X should be either
a list of lists of point patterns. Each element of X will be interpreted as a group of point patterns,
assumed to be replicates of the same point process.
a hyperframe: One column of the hyperframe should contain point patterns, and another column
should contain a factor indicating the grouping. The argument formula should be a formula
in the R language specifying the grouping: it should be of the form P ~ G where P is the name
of the column of point patterns, and G is the name of the factor.
A group needs to contain at least two point patterns with at least minpoints points in each pattern.
The function returns an object of class "htest" and "studpermutest" that can be printed and
plotted. The printout shows the test result and p-value. The plot shows the summary functions for
the groups (and the group means if requested).
Value
Object of class "studpermutest".
Author(s)
Ute Hahn.
Modified for spatstat by Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
References
Hahn, U. (2012) A studentized permutation test for the comparison of spatial point patterns. Journal
of the American Statistical Association 107 (498), 754–764.
Examples
np <- if(interactive()) 99 else 19
testpyramidal <- studpermu.test(pyramidal, Neurons ~ group, nperm=np)
testpyramidal

1394

subfits

subfits

Extract List of Individual Point Process Models

Description
Takes a Gibbs point process model that has been fitted to several point patterns simultaneously, and
produces a list of fitted point process models for the individual point patterns.
Usage
subfits(object, what="models", verbose=FALSE)
subfits.old(object, what="models", verbose=FALSE)
subfits.new(object, what="models", verbose=FALSE)
Arguments
object

An object of class "mppm" representing a point process model fitted to several
point patterns.

what

What should be returned. Either "models" to return the fitted models, or "interactions"
to return the fitted interactions only.

verbose

Logical flag indicating whether to print progress reports.

Details
object is assumed to have been generated by mppm. It represents a point process model that has
been fitted to a list of several point patterns, with covariate data.
For each of the individual point pattern datasets, this function derives the corresponding fitted model
for that dataset only (i.e. a point process model for the ith point pattern, that is consistent with
object).
If what="models", the result is a list of point process models (a list of objects of class "ppm"), one
model for each point pattern dataset in the original fit. If what="interactions", the result is a list
of fitted interpoint interactions (a list of objects of class "fii").
Two different algorithms are provided, as subfits.old and subfits.new. Currently subfits is
the same as the old algorithm subfits.old because the newer algorithm is too memory-hungry.
Value
A list of point process models (a list of objects of class "ppm") or a list of fitted interpoint interactions (a list of objects of class "fii").
Author(s)
Adrian Baddeley, Ida-Maria Sintorn and Leanne Bischoff. Implemented in spatstat by Adrian
Baddeley , Rolf Turner  and
Ege Rubak .
References
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. London: Chapman and Hall/CRC Press.

subset.hyperframe

1395

See Also
mppm, ppm
Examples
H <- hyperframe(Wat=waterstriders)
fit <- mppm(Wat~x, data=H)
subfits(fit)
H$Wat[[3]] <- rthin(H$Wat[[3]], 0.1)
fit2 <- mppm(Wat~x, data=H, random=~1|id)
subfits(fit2)

subset.hyperframe

Subset of Hyperframe Satisfying A Condition

Description
Given a hyperframe, return the subset specified by imposing a condition on each row, and optionally
by choosing only some of the columns.
Usage
## S3 method for class 'hyperframe'
subset(x, subset, select, ...)
Arguments
x

A hyperframe pattern (object of class "hyperframe".

subset

Logical expression indicating which points are to be kept. The expression may
involve the names of columns of x and will be evaluated by with.hyperframe.

select

Expression indicating which columns of marks should be kept.

...

Arguments passed to [.hyperframe such as drop and strip.

Details
This is a method for the generic function subset. It extracts the subset of rows of x that satisfy the
logical expression subset, and retains only the columns of x that are specified by the expression
select. The result is always a hyperframe.
The argument subset determines the subset of rows that will be extracted. It should be a logical
expression. It may involve the names of columns of x. The default is to keep all points.
The argument select determines which columns of x will be retained. It should be an expression
involving the names of columns (which will be interpreted as integers representing the positions
of these columns). For example if there are columns named A to Z, then select=D:F is a valid
expression and means that columns D, E and F will be retained. Similarly select=-(A:C) is valid
and means that columns A to C will be deleted. The default is to retain all columns.
Setting subset=FALSE will remove all the rows. Setting select=FALSE will remove all the columns.
The result is always a hyperframe.

1396

subset.ppp

Value
A hyperframe.
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak 
See Also
subset, [.hyperframe
Examples
a <- subset(flu, virustype=="wt")
aa <- subset(flu, minnndist(pattern) > 10)
aaa <- subset(flu, virustype=="wt", select = -pattern)

subset.ppp

Subset of Point Pattern Satisfying A Condition

Description
Given a point pattern, return the subset of points which satisfy a specified condition.
Usage
## S3 method for class 'ppp'
subset(x, subset, select, drop=FALSE, ...)
## S3 method for class 'pp3'
subset(x, subset, select, drop=FALSE, ...)
## S3 method for class 'lpp'
subset(x, subset, select, drop=FALSE, ...)
## S3 method for class 'ppx'
subset(x, subset, select, drop=FALSE, ...)
Arguments
x

A point pattern (object of class "ppp", "lpp", "pp3" or "ppx").

subset

Logical expression indicating which points are to be kept. The expression may
involve the names of spatial coordinates (x, y, etc), the marks, and (if there is
more than one column of marks) the names of individual columns of marks.
Missing values are taken as false. See Details.

subset.ppp

1397

select

Expression indicating which columns of marks should be kept. The names of
columns of marks can be used in this expression, and will be treated as if they
were column indices. See Details.

drop

Logical value indicating whether to remove unused levels of the marks, if the
marks are a factor.

...

Ignored.

Details
This is a method for the generic function subset. It extracts the subset of points of x that satisfy
the logical expression subset, and retains only the columns of marks that are specified by the
expression select. The result is always a point pattern, with the same window as x.
The argument subset determines the subset of points that will be extracted. It should be a logical
expression. It may involve the variable names x and y representing the Cartesian coordinates; the
names of other spatial coordinates or local coordinates; the name marks representing the marks;
and (if there is more than one column of marks) the names of individual columns of marks. The
default is to keep all points.
The argument select determines which columns of marks will be retained (if there are several
columns of marks). It should be an expression involving the names of columns of marks (which
will be interpreted as integers representing the positions of these columns). For example if there are
columns of marks named A to Z, then select=D:F is a valid expression and means that columns D,
E and F will be retained. Similarly select=-(A:C) is valid and means that columns A to C will be
deleted. The default is to retain all columns.
Setting subset=FALSE will produce an empty point pattern (i.e. containing zero points) in the same
window as x. Setting select=FALSE or select= -marks will remove all the marks from x.
The argument drop determines whether to remove unused levels of a factor, if the resulting point
pattern is multitype (i.e. the marks are a factor) or if the marks are a data frame in which some of
the columns are factors.
The result is always a point pattern, of the same class as x. Spatial coordinates (and local coordinates) are always retained. To extract only some columns of marks or coordinates as a data frame,
use subset(as.data.frame(x), ...)
Value
A point pattern of the same class as x, in the same spatial window as x. The result is a subset of x,
possibly with some columns of marks removed.
Other kinds of subset arguments
Alternatively the argument subset can be any kind of subset index acceptable to [.ppp, [.pp3,
[.ppx. This argument selects which points of x will be retained.
Warning: if the argument subset is a window, this is interpreted as specifying the subset of points
that fall inside that window, but the resulting point pattern has the same window as the original
pattern x.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .

1398

subspaceDistance

See Also
subset,
[.ppp, [.pp3, [.lpp, [.ppx
Examples
plot(subset(cells, x > 0.5))
subset(amacrine, marks == "on")
subset(amacrine, marks == "on", drop=TRUE)
subset(redwood, nndist(redwood) > 0.04)
subset(finpines, select=height)
subset(finpines, diameter > 2, height)
subset(nbfires, year==1999 & ign.src == "campfire",
select=cause:fnl.size)
v <- subset(chicago, x + y > 1100 & marks == "assault")
vv <- subset(chicago, x + y > 1100 & marks == "assault", drop=TRUE)
a <- subset(rpoispp3(40), z > 0.5)

subspaceDistance

Distance Between Linear Spaces

Description
Evaluate the distance between two linear subspaces using the measure proposed by Li, Zha and
Chiaromonte (2005).
Usage
subspaceDistance(B0, B1)
Arguments
B0

Matrix whose columns are a basis for the first subspace.

B1

Matrix whose columns are a basis for the second subspace.

Details
This algorithm calculates the maximum absolute value of the eigenvalues of P 1−P 0 where P 0, P 1
are the projection matrices onto the subspaces generated by B0,B1. This measure of distance was
proposed by Li, Zha and Chiaromonte (2005). See also Xia (2007).
Value
A single numeric value.

suffstat

1399

Author(s)
Matlab original by Yongtao Guan, translated to R by Suman Rakshit.
References
Guan, Y. and Wang, H. (2010) Sufficient dimension reduction for spatial point processes directed
by Gaussian random fields. Journal of the Royal Statistical Society, Series B, 72, 367–387.
Li, B., Zha, H. and Chiaromonte, F. (2005) Contour regression: a general approach to dimension
reduction. Annals of Statistics 33, 1580–1616.
Xia, Y. (2007) A constructive approach to the estimation of dimension reduction directions. Annals
of Statistics 35, 2654–2690.

suffstat

Sufficient Statistic of Point Process Model

Description
The canonical sufficient statistic of a point process model is evaluated for a given point pattern.
Usage
suffstat(model, X=data.ppm(model))
Arguments
model

A fitted point process model (object of class "ppm").

X

A point pattern (object of class "ppp").

Details
The canonical sufficient statistic of model is evaluated for the point pattern X. This computation is
useful for various Monte Carlo methods.
Here model should be a point process model (object of class "ppm", see ppm.object), typically
obtained from the model-fitting function ppm. The argument X should be a point pattern (object of
class "ppp").
Every point process model fitted by ppm has a probability density of the form
f (x) = Z(θ) exp(θT S(x))
where x denotes a typical realisation (i.e. a point pattern), θ is the vector of model coefficients,
Z(θ) is a normalising constant, and S(x) is a function of the realisation x, called the “canonical
sufficient statistic” of the model.
For example, the stationary Poisson process has canonical sufficient statistic S(x) = n(x), the
number of points in x. The stationary Strauss process with interaction range r (and fitted with no
edge correction) has canonical sufficient statistic S(x) = (n(x), s(x)) where s(x) is the number of
pairs of points in x which are closer than a distance r to each other.
suffstat(model, X) returns the value of S(x), where S is the canonical sufficient statistic associated with model, evaluated when x is the given point pattern X. The result is a numeric vector, with
entries which correspond to the entries of the coefficient vector coef(model).

1400

suffstat

The sufficient statistic S does not depend on the fitted coefficients of the model. However it does
depend on the irregular parameters which are fixed in the original call to ppm, for example, the
interaction range r of the Strauss process.
The sufficient statistic also depends on the edge correction that was used to fit the model. For
example in a Strauss process,
• If the model is fitted with correction="none", the sufficient statistic is S(x) = (n(x), s(x))
where n(x) is the number of points and s(x) is the number of pairs of points which are closer
than r units apart.
• If the model is fitted with correction="periodic", the sufficient statistic is the same as
above, except that distances are measured in the periodic sense.
• If the model is fitted with correction="translate", then n(x) is unchanged but s(x) is
replaced by a weighted sum (the sum of the translation correction weights for all pairs of
points which are closer than r units apart).
• If the model is fitted with correction="border" (the default), then points lying less than r
units from the boundary of the observation window are treated as fixed. Thus n(x) is replaced
by the number nr (x) of points lying at least r units from the boundary of the observation window, and s(x) is replaced by the number sr (x) of pairs of points, which are closer than r units
apart, and at least one of which lies more than r units from the boundary of the observation
window.
Non-finite values of the sufficient statistic (NA or -Inf) may be returned if the point pattern X is not
a possible realisation of the model (i.e. if X has zero probability of occurring under model for all
values of the canonical coefficients θ).

Value
A numeric vector of sufficient statistics. The entries correspond to the model coefficients coef(model).
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
ppm
Examples
fitS <- ppm(swedishpines~1, Strauss(7))
X <- rpoispp(intensity(swedishpines), win=Window(swedishpines))
suffstat(fitS, X)

summary.anylist

summary.anylist

1401

Summary of a List of Things

Description
Prints a useful summary of each item in a list of things.

Usage
## S3 method for class 'anylist'
summary(object, ...)

Arguments
object

An object of class "anylist".

...

Ignored.

Details
This is a method for the generic function summary.
An object of the class "anylist" is effectively a list of things which are intended to be treated in a
similar way. See anylist.
This function extracts a useful summary of each of the items in the list.

Author(s)
Adrian Baddeley 
and Rolf Turner 

See Also
anylist, summary, plot.anylist

Examples
x <- anylist(A=runif(10), B=runif(10), C=runif(10))
summary(x)

1402

summary.im

summary.im

Summarizing a Pixel Image

Description
summary method for class "im".
Usage
## S3 method for class 'im'
summary(object, ...)
## S3 method for class 'summary.im'
print(x, ...)
Arguments
object

A pixel image.

...

Ignored.

x

Object of class "summary.im" as returned by summary.im.

Details
This is a method for the generic summary for the class "im". An object of class "im" describes a
pixel image. See im.object) for details of this class.
summary.im extracts information about the pixel image, and print.summary.im prints this information in a comprehensible format.
In normal usage, print.summary.im is invoked implicitly when the user calls summary.im without
assigning its value to anything. See the examples.
The information extracted by summary.im includes
range The range of the image values.
mean The mean of the image values.
integral The “integral” of the image values, calculated as the sum of the image values multiplied
by the area of one pixel.
dim The dimensions of the pixel array: dim[1] is the number of rows in the array, corresponding
to the y coordinate.
Value
summary.im returns an object of class "summary.im", while print.summary.im returns NULL.
Author(s)
Adrian Baddeley  and Rolf Turner 
See Also
mean.im, integral.im, anyNA.im

summary.kppm

1403

Examples
# make an image
X <- as.im(function(x,y) {x^2}, unit.square())
# summarize it
summary(X)
# save the summary
s <- summary(X)
# print it
print(X)
s
# extract stuff
X$dim
X$range
X$integral

summary.kppm

Summarizing a Fitted Cox or Cluster Point Process Model

Description
summary method for class "kppm".
Usage
## S3 method for class 'kppm'
summary(object, ..., quick=FALSE)
## S3 method for class 'summary.kppm'
print(x, ...)
Arguments
object

A fitted Cox or cluster point process model (object of class "kppm").

quick

Logical value controlling the scope of the summary.

...

Arguments passed to summary.ppm or print.summary.ppm controlling the treatment of the trend component of the model.

x

Object of class "summary.kppm" as returned by summary.kppm.

Details
This is a method for the generic summary for the class "kppm". An object of class "kppm" describes
a fitted Cox or cluster point process model. See kppm.
summary.kppm extracts information about the type of model that has been fitted, the data to which
the model was fitted, and the values of the fitted coefficients.
print.summary.kppm prints this information in a comprehensible format.
In normal usage, print.summary.kppm is invoked implicitly when the user calls summary.kppm
without assigning its value to anything. See the examples.
You can also type coef(summary(object)) to extract a table of the fitted coefficients of the point
process model object together with standard errors and confidence limits.

1404

summary.listof

Value
summary.kppm returns an object of class "summary.kppm", while print.summary.kppm returns
NULL.
The result of summary.kppm includes at least the following components:
Xname

character string name of the original point pattern data

stationary

logical value indicating whether the model is stationary

clusters

the clusters argument to kppm

modelname

character string describing the model

isPCP

TRUE if the model is a Poisson cluster process, FALSE if it is a log-Gaussian Cox
process

lambda

Estimated intensity: numeric value, or pixel image

mu

Mean cluster size: numeric value, pixel image, or NULL

clustpar

list of fitted parameters for the cluster model

clustargs

list of fixed parameters for the cluster model, if any

callstring

character string representing the original call to kppm

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
Examples
fit <- kppm(redwood ~ 1, "Thomas")
summary(fit)
coef(summary(fit))

summary.listof

Summary of a List of Things

Description
Prints a useful summary of each item in a list of things.
Usage
## S3 method for class 'listof'
summary(object, ...)
Arguments
object

An object of class "listof".

...

Ignored.

summary.owin

1405

Details
This is a method for the generic function summary.
An object of the class "listof" is effectively a list of things which are all of the same class.
This function extracts a useful summary of each of the items in the list.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
summary, plot.listof
Examples
x <- list(A=runif(10), B=runif(10), C=runif(10))
class(x) <- c("listof", class(x))
summary(x)

summary.owin

Summary of a Spatial Window

Description
Prints a useful description of a window object.
Usage
## S3 method for class 'owin'
summary(object, ...)
Arguments
object

Window (object of class "owin").

...

Ignored.

Details
A useful description of the window object is printed.
This is a method for the generic function summary.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
summary, summary.ppp, print.owin

1406

summary.ppm

Examples
summary(owin())

# the unit square

data(demopat)
W <- Window(demopat) # weird polygonal window
summary(W)
# describes it
summary(as.mask(W))

summary.ppm

# demonstrates current pixel resolution

Summarizing a Fitted Point Process Model

Description
summary method for class "ppm".
Usage
## S3 method for class 'ppm'
summary(object, ..., quick=FALSE, fine=FALSE)
## S3 method for class 'summary.ppm'
print(x, ...)
Arguments
object

A fitted point process model.

...

Ignored.

quick

Logical flag controlling the scope of the summary.

fine

Logical value passed to vcov.ppm determining whether to compute the quick,
coarse estimate of variance (fine=FALSE, the default) or the slower, finer estimate (fine=TRUE).

x

Object of class "summary.ppm" as returned by summary.ppm.

Details
This is a method for the generic summary for the class "ppm". An object of class "ppm" describes a
fitted point process model. See ppm.object) for details of this class.
summary.ppm extracts information about the type of model that has been fitted, the data to which the
model was fitted, and the values of the fitted coefficients. (If quick=TRUE then only the information
about the type of model is extracted.)
print.summary.ppm prints this information in a comprehensible format.
In normal usage, print.summary.ppm is invoked implicitly when the user calls summary.ppm without assigning its value to anything. See the examples.
You can also type coef(summary(object)) to extract a table of the fitted coefficients of the point
process model object together with standard errors and confidence limits.
Value
summary.ppm returns an object of class "summary.ppm", while print.summary.ppm returns NULL.

summary.ppp

1407

Author(s)
Adrian Baddeley 
and Rolf Turner 
Examples
# invent some data
X <- rpoispp(42)
# fit a model to it
fit <- ppm(X ~ x, Strauss(r=0.1))
# summarize the fitted model
summary(fit)
# `quick' option
summary(fit, quick=TRUE)
# coefficients with standard errors and CI
coef(summary(fit))
coef(summary(fit, fine=TRUE))
# save the full summary
s <- summary(fit)
# print it
print(s)
s
# extract stuff
names(s)
coef(s)
s$args$correction
s$name
s$trend$value
## Not run:
# multitype pattern
data(demopat)
fit <- ppm(demopat, ~marks, Poisson())
summary(fit)
## End(Not run)
# model with external covariates
fitX <- ppm(X, ~Z, covariates=list(Z=function(x,y){x+y}))
summary(fitX)

summary.ppp

Summary of a Point Pattern Dataset

Description
Prints a useful summary of a point pattern dataset.
Usage
## S3 method for class 'ppp'
summary(object, ..., checkdup=TRUE)

1408

summary.psp

Arguments
object

Point pattern (object of class "ppp").

...

Ignored.

checkdup

Logical value indicating whether to check for the presence of duplicate points.

Details
A useful summary of the point pattern object is printed.
This is a method for the generic function summary.
If checkdup=TRUE, the pattern will be checked for the presence of duplicate points, using duplicated.ppp.
This can be time-consuming if the pattern contains many points, so the checking can be disabled by
setting checkdup=FALSE.
If the point pattern was generated by simulation using rmh, the parameters of the algorithm are
printed.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
summary, summary.owin, print.ppp
Examples
summary(cells)

# plain vanilla point pattern

# multitype point pattern
woods <- lansing
summary(woods) # tabulates frequencies of each mark
# numeric marks
trees <- longleaf
summary(trees) # prints summary.default(marks(trees))
# weird polygonal window
summary(demopat) # describes it

summary.psp

Summary of a Line Segment Pattern Dataset

Description
Prints a useful summary of a line segment pattern dataset.
Usage
## S3 method for class 'psp'
summary(object, ...)

summary.quad

1409

Arguments
object

Line segment pattern (object of class "psp").

...

Ignored.

Details
A useful summary of the line segment pattern object is printed.
This is a method for the generic function summary.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
summary, summary.owin, print.psp
Examples
a <- psp(runif(10), runif(10), runif(10), runif(10), window=owin())
summary(a) # describes it

summary.quad

Summarizing a Quadrature Scheme

Description
summary method for class "quad".
Usage
## S3 method for class 'quad'
summary(object, ..., checkdup=FALSE)
## S3 method for class 'summary.quad'
print(x, ..., dp=3)
Arguments
object

A quadrature scheme.

...

Ignored.

checkdup

Logical value indicating whether to test for duplicated points.

dp

Number of significant digits to print.

x

Object of class "summary.quad" returned by summary.quad.

1410

summary.solist

Details
This is a method for the generic summary for the class "quad". An object of class "quad" describes
a quadrature scheme, used to fit a point process model. See quad.object) for details of this class.
summary.quad extracts information about the quadrature scheme, and print.summary.quad prints
this information in a comprehensible format.
In normal usage, print.summary.quad is invoked implicitly when the user calls summary.quad
without assigning its value to anything. See the examples.
Value
summary.quad returns an object of class "summary.quad", while print.summary.quad returns
NULL.
Author(s)
Adrian Baddeley 
and Rolf Turner 
Examples
# make a quadrature scheme
Q <- quadscheme(rpoispp(42))
# summarize it
summary(Q)
# save the summary
s <- summary(Q)
# print it
print(s)
s
# extract total quadrature weight
s$w$all$sum

summary.solist

Summary of a List of Spatial Objects

Description
Prints a useful summary of each entry in a list of two-dimensional spatial objects.
Usage
## S3 method for class 'solist'
summary(object, ...)
Arguments
object

An object of class "solist".

...

Ignored.

summary.splitppp

1411

Details
This is a method for the generic function summary.
An object of the class "solist" is effectively a list of two-dimensional spatial datasets. See solist.
This function extracts a useful summary of each of the datasets.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
solist, summary, plot.solist
Examples
x <- solist(cells, japanesepines, redwood)
summary(x)

summary.splitppp

Summary of a Split Point Pattern

Description
Prints a useful summary of a split point pattern.
Usage
## S3 method for class 'splitppp'
summary(object, ...)
Arguments
object

Split point pattern (object of class "splitppp", effectively a list of point patterns, usually created by split.ppp).

...

Ignored.

Details
This is a method for the generic function summary.
An object of the class "splitppp" is effectively a list of point patterns (objects of class "ppp")
representing different sub-patterns of an original point pattern.
This function extracts a useful summary of each of the sub-patterns.
Author(s)
Adrian Baddeley 
and Rolf Turner 

1412

sumouter

See Also
summary, split, split.ppp
Examples
data(amacrine)
# multitype point pattern
summary(split(amacrine))

sumouter

Compute Quadratic Forms

Description
Calculates certain quadratic forms of matrices.
Usage
sumouter(x, w=NULL, y=x)
quadform(x, v)
bilinearform(x, v, y)
Arguments
x,y

A matrix, whose rows are the vectors in the quadratic form.

w

Optional vector of weights

v

Matrix determining the quadratic form

Details
The matrices x and y will be interpreted as collections of row vectors. They must have the same
number of rows.
The command sumouter computes the sum of the outer products of corresponding row vectors,
weighted by the entries of w:
X
M=
wi xi yi>
i

where the sum is over all rows of x (after removing any rows containing NA or other non-finite values). If w is missing, the weights will be taken as 1. The result is a p × q matrix where p = ncol(x)
and q = ncol(y).
The command quadform evaluates the quadratic form, defined by the matrix v, for each of the row
vectors of x:
yi = xi V x>
i
The result y is a numeric vector of length n where n = nrow(x). If x[i,] contains NA or other
non-finite values, then y[i] = NA.
The command bilinearform evaluates the more general bilinear form defined by the matrix v.
Here x and y must be matrices of the same dimensions. For each row vector of x and corresponding
row vector of y, the bilinear form is
zi = xi V yi>
The result z is a numeric vector of length n where n = nrow(x). If x[i,] or y[i,] contains NA or
other non-finite values, then z[i] = NA.

superimpose

1413

Value
A vector or matrix.
Author(s)
Adrian Baddeley 
and Rolf Turner 
Examples
x <- matrix(1:12, 4, 3)
dimnames(x) <- list(c("Wilma", "Fred", "Barney", "Betty"), letters[1:3])
x
sumouter(x)
w <- 4:1
sumouter(x, w)
v <- matrix(1, 3, 3)
quadform(x, v)
# should be the same as quadform(x, v)
bilinearform(x, v, x)
# See what happens with NA's
x[3,2] <- NA
sumouter(x, w)
quadform(x, v)

superimpose

Superimpose Several Geometric Patterns

Description
Superimpose any number of point patterns or line segment patterns.
Usage
superimpose(...)
## S3 method for class 'ppp'
superimpose(..., W=NULL, check=TRUE)
## S3 method for class 'psp'
superimpose(..., W=NULL, check=TRUE)
## S3 method for class 'splitppp'
superimpose(..., W=NULL, check=TRUE)
## S3 method for class 'ppplist'
superimpose(..., W=NULL, check=TRUE)

1414

superimpose

## Default S3 method:
superimpose(...)
Arguments
...

Any number of arguments, each of which represents either a point pattern or a
line segment pattern or a list of point patterns.

W

Optional. Data determining the window for the resulting pattern. Either a window (object of class "owin", or something acceptable to as.owin), or a function
which returns a window, or one of the strings "convex", "rectangle", "bbox"
or "none".

check

Logical value (passed to ppp or psp as appropriate) determining whether to
check the geometrical validity of the resulting pattern.

Details
This function is used to superimpose several geometric patterns of the same kind, producing a single
pattern of the same kind.
The function superimpose is generic, with methods for the class ppp of point patterns, the class psp
of line segment patterns, and a default method. There is also a method for lpp, described separately
in superimpose.lpp.
The dispatch to a method is initially determined by the class of the first argument in ....
• default: If the first argument is not an object of class ppp or psp, then the default method
superimpose.default is executed. This checks the class of all arguments, and dispatches to
the appropriate method. Arguments of class ppplist can be handled.
• ppp: If the first ... argument is an object of class ppp then the method superimpose.ppp is
executed. All arguments in ... must be either ppp objects or lists with components x and y.
The result will be an object of class ppp.
• psp: If the first ... argument is an object of class psp then the psp method is dispatched and
all ... arguments must be psp objects. The result is a psp object.
The patterns are not required to have the same window of observation.
The window for the superimposed pattern is controlled by the argument W.
• If W is a window (object of class "W" or something acceptable to as.owin) then this determines
the window for the superimposed pattern.
• If W is NULL, or the character string "none", then windows are extracted from the geometric
patterns, as follows. For superimpose.psp, all arguments ... are line segment patterns
(objects of class "psp"); their observation windows are extracted; the union of these windows
is computed; and this union is taken to be the window for the superimposed pattern. For
superimpose.ppp and superimpose.default, the arguments ... are inspected, and any
arguments which are point patterns (objects of class "ppp") are selected; their observation
windows are extracted, and the union of these windows is taken to be the window for the
superimposed point pattern. For superimpose.default if none of the arguments is of class
"ppp" then no window is computed and the result of superimpose is a list(x,y).
• If W is one of the strings "convex", "rectangle" or "bbox" then a window for the superimposed pattern is computed from the coordinates of the points or the line segments as follows.
"bbox": the bounding box of the points or line segments (see bounding.box.xy);
"convex": the Ripley-Rasson estimator of a convex window (see ripras);

superimpose

1415

"rectangle": the Ripley-Rasson estimator of a rectangular window (using ripras with argument shape="rectangle").
• If W is a function, then this function is used to compute a window for the superimposed pattern
from the coordinates of the points or the line segments. The function should accept input of
the form list(x,y) and is expected to return an object of class "owin". Examples of such
functions are ripras and bounding.box.xy.
The arguments ... may be marked patterns. The marks of each component pattern must have
the same format. Numeric and character marks may be “mixed”. If there is such mixing then the
numeric marks are coerced to character in the combining process. If the mark structures are all data
frames, then these data frames must have the same number of columns and identical column names.
If the arguments ... are given in the form name=value, then the names will be used as an extra
column of marks attached to the elements of the corresponding patterns.
Value
For superimpose.ppp, a point pattern (object of class "ppp"). For superimpose.default, either a
point pattern (object of class "ppp") or a list(x,y). For superimpose.psp, a line segment pattern
(object of class "psp").
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
superimpose.lpp, concatxy, quadscheme.
Examples
# superimposing point patterns
p1 <- runifrect(30)
p2 <- runifrect(42)
s1 <- superimpose(p1,p2) # Unmarked pattern.
p3 <- list(x=rnorm(20),y=rnorm(20))
s2 <- superimpose(p3,p2,p1) # Default method gets called.
s2a <- superimpose(p1,p2,p3) # Same as s2 except for order of points.
s3 <- superimpose(clyde=p1,irving=p2) # Marked pattern; marks a factor
# with levels "clyde" and "irving";
# warning given.
marks(p1) <- factor(sample(LETTERS[1:3],30,TRUE))
marks(p2) <- factor(sample(LETTERS[1:3],42,TRUE))
s5 <- superimpose(clyde=p1,irving=p2) # Marked pattern with extra column
marks(p2) <- data.frame(a=marks(p2),b=runif(42))
s6 <- try(superimpose(p1,p2)) # Gives an error.
marks(p1) <- data.frame(a=marks(p1),b=1:30)
s7 <- superimpose(p1,p2) # O.K.
# how to make a 2-type point pattern with types "a" and "b"
u <- superimpose(a = rpoispp(10), b = rpoispp(20))
# how to make a 2-type point pattern with types 1 and 2
u <- superimpose("1" = rpoispp(10), "2" = rpoispp(20))

1416

superimpose.lpp
#
X
Y
Z

superimposing line segment patterns
<- rpoisline(10)
<- as.psp(matrix(runif(40), 10, 4), window=owin())
<- superimpose(X, Y)

# being unreasonable
## Not run:
if(FALSE) {
crud <- try(superimpose(p1,p2,X,Y)) # Gives an error, of course!
}
## End(Not run)

superimpose.lpp

Superimpose Several Point Patterns on Linear Network

Description
Superimpose any number of point patterns on the same linear network.
Usage
## S3 method for class 'lpp'
superimpose(..., L=NULL)
Arguments
...

Any number of arguments, each of which represents a point pattern on the same
linear network. Each argument can be either an object of class "lpp", giving
both the spatial coordinates of the points and the linear network, or a list(x,y)
or list(x,y,seg,tp) giving just the spatial coordinates of the points.

L

Optional. The linear network. An object of class "linnet". This argument is
required if none of the other arguments is of class "lpp".

Details
This function is used to superimpose several point patterns on the same linear network. It is a
method for the generic function superimpose.
Each of the arguments ... can be either a point pattern on a linear network (object of class
"lpp" giving both the spatial coordinates of the points and the linear network), or a list(x,y)
or list(x,y,seg,tp) giving just the spatial coordinates of the points. These arguments must represent point patterns on the same linear network.
The argument L is an alternative way to specify the linear network, and is required if none of the
arguments ... is an object of class "lpp".
The arguments ... may be marked patterns. The marks of each component pattern must have
the same format. Numeric and character marks may be “mixed”. If there is such mixing then the
numeric marks are coerced to character in the combining process. If the mark structures are all data
frames, then these data frames must have the same number of columns and identical column names.
If the arguments ... are given in the form name=value, then the names will be used as an extra
column of marks attached to the elements of the corresponding patterns.

symbolmap

1417

Value
An object of class "lpp" representing the combined point pattern on the linear network.
Author(s)
Adrian Baddeley 
Rolf Turner 
Ege Rubak 
and Greg McSwiggan.
See Also
superimpose
Examples
X <- rpoislpp(5, simplenet)
Y <- rpoislpp(10, simplenet)
superimpose(X,Y) # not marked
superimpose(A=X, B=Y) # multitype with types A and B

symbolmap

Graphics Symbol Map

Description
Create a graphics symbol map that associates data values with graphical symbols.
Usage
symbolmap(..., range = NULL, inputs = NULL)
Arguments
...

Named arguments specifying the graphical parameters. See Details.

range

Optional. Range of numbers that are mapped. A numeric vector of length 2
giving the minimum and maximum values that will be mapped. Incompatible
with inputs.

inputs

Optional. A vector containing all the data values that will be mapped to symbols.
Incompatible with range.

Details
A graphical symbol map is an association between data values and graphical symbols. The command symbolmap creates an object of class "symbolmap" that represents a graphical symbol map.
Once a symbol map has been created, it can be applied to any suitable data to generate a plot of
those data. This makes it easy to ensure that the same symbol map is used in two different plots.
The symbol map can be plotted as a legend to the plots, and can also be plotted in its own right.
The possible values of data that will be mapped are specified by range or inputs.

1418

symbolmap
• if range is given, it should be a numeric vector of length 2 giving the minimum and maximum
values of the range of numbers that will be mapped. These limits must be finite.
• if inputs is given, it should be a vector of any atomic type (e.g. numeric, character, logical,
factor). This vector contains all the possible data values that will be mapped.
• If neither range nor inputs is given, it is assumed that the possible values are real numbers.

The association of data values with graphical symbols is specified by the other arguments ... which
are given in name=value form. These arguments specify the kinds of symbols that will be used, the
sizes of the symbols, and graphics parameters for drawing the symbols.
Each graphics parameter can be either a single value, for example shape="circles", or a function(x)
which determines the value of the graphics parameter as a function of the data x, for example
shape=function(x) ifelse(x > 0, "circles", "squares"). Colourmaps (see colourmap)
are also acceptable because they are functions.
Currently recognised graphics parameters, and their allowed values, are:
shape The shape of the symbol: currently either "circles", "squares", "arrows" or NA. This
parameter takes precedence over pch.
size The size of the symbol: a positive number or zero.
pch Graphics character code: a positive integer, or a single character. See par.
cex Graphics character expansion factor.
cols Colour of plotting characters.
fg,bg Colour of foreground (or symbol border) and background (or symbol interior).
col,lwd,lty Colour, width and style of lines.
etch Logical. If TRUE, each symbol is surrounded by a border drawn in the opposite colour, which
improves its visibility against the background. Default is FALSE.
direction,headlength,headangle,arrowtype Numeric parameters of arrow symbols, applicable when
shape="arrows". Here direction is the direction of the arrow in degrees anticlockwise from
the x axis; headlength is the length of the head of the arrow in coordinate units; headangle
is the angle subtended by the point of the arrow; and arrowtype is an integer code specifying
which ends of the shaft have arrowheads attached (0 means no arrowheads, 1 is an arrowhead
at the start of the shaft, 2 is an arrowhead at the end of the shaft, and 3 is arrowheads at both
ends).
A vector of colour values is also acceptable for the arguments col,cols,fg,bg if range is specified.
Value
An object of class "symbolmap".
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
plot.symbolmap to plot the symbol map itself.
invoke.symbolmap to apply the symbol map to some data and plot the resulting symbols.
update.symbolmap to change the symbol map.

tess

1419

Examples
g <- symbolmap(inputs=letters[1:10], pch=11:20)
g1 <- symbolmap(range=c(0,100), size=function(x) x/50)
g2 <- symbolmap(shape=function(x) ifelse(x > 0, "circles", "squares"),
size=function(x) sqrt(ifelse(x > 0, x/pi, -x)),
bg = function(x) ifelse(abs(x) < 1, "red", "black"))
colmap <- colourmap(topo.colors(20), range=c(0,10))
g3 <- symbolmap(pch=21, bg=colmap, range=c(0,10))
plot(g3)

tess

Create a Tessellation

Description
Creates an object of class "tess" representing a tessellation of a spatial region.
Usage
tess(..., xgrid = NULL, ygrid = NULL, tiles = NULL, image = NULL,
window=NULL, marks=NULL, keepempty=FALSE, unitname=NULL, check=TRUE)
Arguments
...

Ignored.

xgrid,ygrid

Cartesian coordinates of vertical and horizontal lines determining a grid of rectangles. Incompatible with other arguments.

tiles

List of tiles in the tessellation. A list, each of whose elements is a window
(object of class "owin"). Incompatible with other arguments.

image

Pixel image which specifies the tessellation. Incompatible with other arguments.

window

Optional. The spatial region which is tessellated (i.e. the union of all the tiles).
An object of class "owin".

marks

Optional vector or data frame of marks associated with the tiles.

keepempty

Logical flag indicating whether empty tiles should be retained or deleted.

unitname

Optional. Name of unit of length. Either a single character string, or a vector of
two character strings giving the singular and plural forms, respectively. If this
argument is missing or NULL, information about the unitname will be extracted
from the other arguments. If this argument is given, it overrides any other information about the unitname.

check

Logical value indicating whether to check the validity of the input data. It is
strongly recommended to use the default value check=TRUE.

1420

tess

Details
A tessellation is a collection of disjoint spatial regions (called tiles) that fit together to form a larger
spatial region. This command creates an object of class "tess" that represents a tessellation.
Three types of tessellation are supported:
rectangular: tiles are rectangles, with sides parallel to the x and y axes. They may or may not have
equal size and shape. The arguments xgrid and ygrid determine the positions of the vertical
and horizontal grid lines, respectively. (See quadrats for another way to do this.)
tile list: tiles are arbitrary spatial regions. The argument tiles is a list of these tiles, which are
objects of class "owin".
pixel image: Tiles are subsets of a fine grid of pixels. The argument image is a pixel image (object
of class "im") with factor values. Each level of the factor represents a different tile of the
tessellation. The pixels that have a particular value of the factor constitute a tile.
The optional argument window specifies the spatial region formed by the union of all the tiles. In
other words it specifies the spatial region that is divided into tiles by the tessellation. If this argument
is missing or NULL, it will be determined by computing the set union of all the tiles. This is a timeconsuming computation. For efficiency it is advisable to specify the window. Note that the validity
of the window will not be checked.
Empty tiles may occur, either because one of the entries in the list tiles is an empty window, or
because one of the levels of the factor-valued pixel image image does not occur in the pixel data.
When keepempty=TRUE, empty tiles are permitted. When keepempty=FALSE (the default), tiles are
not allowed to be empty, and any empty tiles will be removed from the tessellation.
There are methods for print, plot, [ and [<- for tessellations. Use tiles to extract the list of
tiles in a tessellation, tilenames to extract the names of the tiles, and tile.areas to compute their
areas.
The tiles may have marks, which can be extracted by marks.tess and changed by marks<-.tess.
Tessellations can be used to classify the points of a point pattern, in split.ppp, cut.ppp and
by.ppp.
To construct particular tessellations, see quadrats, hextess, dirichlet, delaunay and rpoislinetess.
Value
An object of class "tess" representing the tessellation.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
marks.tess, plot.tess, [.tess, as.tess, tiles, intersect.tess, split.ppp, cut.ppp, by.ppp,
bdist.tiles, tile.areas.
To construct particular tessellations, see quadrats, hextess, dirichlet, delaunay and rpoislinetess.
To divide space into pieces containing equal amounts of stuff, use quantess.

test.crossing.psp

1421

Examples
A <- tess(xgrid=0:4,ygrid=0:4)
A
B <- A[c(1, 2, 5, 7, 9)]
B
v <- as.im(function(x,y){factor(round(5 * (x^2 + y^2)))}, W=owin())
levels(v) <- letters[seq(length(levels(v)))]
E <- tess(image=v)
E

test.crossing.psp

Check Whether Segments Cross

Description
Determine whether there is a crossing (intersection) between each pair of line segments.
Usage
test.crossing.psp(A, B)
test.selfcrossing.psp(A)
Arguments
A,B

Line segment patterns (objects of class "psp").

Details
These functions decide whether the given line segments intersect each other.
If A and B are two spatial patterns of line segments, test.crossing.psp(A, B) returns a logical
matrix in which the entry on row i, column j is equal to TRUE if segment A[i] has an intersection
with segment B[j].
If A is a pattern of line segments, test.selfcross.psp(A) returns a symmetric logical matrix in
which the entry on row i, column j is equal to TRUE if segment A[i] has an intersection with
segment A[j].
Value
A logical matrix.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
psp

1422

text.ppp

Examples
B <- edges(letterR)
A <- rpoisline(5, Frame(B))
MA <- test.selfcrossing.psp(A)
MAB <- test.crossing.psp(A, B)

text.ppp

Add Text Labels to Spatial Pattern

Description
Plots a text label at the location of each point in a spatial point pattern, or each object in a spatial
pattern of objects.
Usage
## S3 method for class 'ppp'
text(x, ...)
## S3 method for class 'lpp'
text(x, ...)
## S3 method for class 'psp'
text(x, ...)
Arguments
x

A spatial point pattern (object of class "ppp"), a point pattern on a linear network
(class "lpp") or a spatial pattern of line segments (class "psp").

...

Additional arguments passed to text.default.

Details
These functions are methods for the generic text. A text label is added to the existing plot, at the
location of each point in the point pattern x, or near the location of the midpoint of each segment in
the segment pattern x.
Additional arguments ... are passed to text.default and may be used to control the placement
of the labels relative to the point locations, and the size and colour of the labels.
By default, the labels are the serial numbers 1 to n, where n is the number of points or segments in
x. This can be changed by specifying the argument labels, which should be a vector of length n.
Value
Null.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .

texturemap

1423

See Also
text.default
Examples
plot(cells)
text(cells, pos=2)
plot(Frame(cells))
text(cells, cex=1.5)
S <- as.psp(simplenet)
plot(S)
text(S)

texturemap

Texture Map

Description
Create a map that associates data values with graphical textures.
Usage
texturemap(inputs, textures, ...)
Arguments
inputs

A vector containing all the data values that will be mapped to textures.

textures

Optional. A vector of integer codes specifying the textures to which the inputs
will be mapped.

...

Other graphics parameters such as col, lwd, lty.

Details
A texture map is an association between data values and graphical textures. The command texturemap
creates an object of class "texturemap" that represents a texture map.
Once a texture map has been created, it can be applied to any suitable data to generate a texture plot
of those data using textureplot. This makes it easy to ensure that the same texture map is used in
two different plots. The texture map can also be plotted in its own right.
The argument inputs should be a vector containing all the possible data values (such as the levels
of a factor) that are to be mapped.
The textures should be integer values between 1 and 8, representing the eight possible textures
described in the help for add.texture. The default is textures = 1:n where n is the length of
inputs.
Value
An object of class "texturemap" representing the texture map.

1424

textureplot

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
textureplot
Examples
texturemap(letters[1:4], 2:5, col=1:4, lwd=2)

textureplot

Plot Image or Tessellation Using Texture Fill

Description
For a factor-valued pixel image, this command plots each level of the factor using a different texture.
For a tessellation, each tile is plotted using a different texture.
Usage
textureplot(x, ...,
main, add=FALSE, clipwin=NULL, do.plot = TRUE,
border=NULL, col = NULL, lwd = NULL, lty = NULL, spacing = NULL,
textures=1:8,
legend=TRUE,
leg.side=c("right", "left", "bottom", "top"),
legsep=0.1, legwid=0.2)
Arguments
x

A tessellation (object of class "tess" or something acceptable to as.tess) with
at most 8 tiles, or a pixel image (object of class "im" or something acceptable to
as.im) whose pixel values are a factor with at most 8 levels.

...

Other arguments passed to add.texture.

main

Character string giving a main title for the plot.

add

Logical value indicating whether to draw on the current plot (add=TRUE) or to
initialise a new plot (add=FALSE).

clipwin

Optional. A window (object of class "owin"). Only this subset of the image will
be displayed.

do.plot

Logical. Whether to actually do the plot.

border

Colour for drawing the boundaries between the different regions. The default
(border=NULL) means to use par("fg"). Use border=NA to omit borders.

col

Numeric value or vector giving the colour or colours in which the textures should
be plotted.

lwd

Numeric value or vector giving the line width or widths to be used.

lty

Numeric value or vector giving the line type or types to be used.

thinNetwork

1425

spacing

Numeric value or vector giving the spacing parameter for the textures.

textures

Textures to be used for each level. Either a texture map (object of class "texturemap")
or a vector of integer codes (to be interpreted by add.texture).

legend

Logical. Whether to display an explanatory legend.

leg.side

Position of legend relative to main plot.

legsep

Separation between legend and main plot, as a fraction of the shortest side length
of the main plot.

legwid

Width (if vertical) or height (if horizontal) of the legend as a fraction of the
shortest side length of the main plot.

Details
If x is a tessellation, then each tile of the tessellation is plotted and filled with a texture using
add.texture.
If x is a factor-valued pixel image, then for each level of the factor, the algorithm finds the region
where the image takes this value, and fills the region with a texture using add.texture.
Value
(Invisible) A texture map (object of class "texturemap") associating a texture with each level of
the factor.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
im, plot.im, add.texture.
Examples
nd <- if(interactive()) 128 else 32
Z <- setcov(owin(), dimyx=nd)
Zcut <- cut(Z, 3, labels=c("Lo", "Med", "Hi"))
textureplot(Zcut)
textureplot(dirichlet(runifpoint(6)))

thinNetwork

Remove Vertices or Segments from a Linear Network

Description
Delete some vertices and/or segments from a linear network or related object.
Usage
thinNetwork(X, retainvertices, retainedges)

1426

thinNetwork

Arguments
X

A linear network (object of class "linnet"), or a point pattern on a linear network (object of class "lpp").

retainvertices Optional. Subset index specifying which vertices should be retained (not deleted).
retainedges

Optional. Subset index specifying which edges (segments) should be retained
(not deleted).

Details
This function deletes some of the vertices and edges (segments) in the linear network.
The arguments retainvertices and retainedges can be any kind of subset index: a vector of
positive integers specifying which vertices/edges should be retained; a vector of negative integers
specifying which vertices/edges should be deleted; or a logical vector specifying whether each
vertex/edge should be retained (TRUE) or deleted (FALSE).
Vertices are indexed in the same sequence as in vertices(as.linnet(X)). Segments are indexed
in the same sequence as in as.psp(as.linnet(X)).
The argument retainedges has higher precedence than retainvertices in the sense that:
• If retainedges is given, then any vertex which is an endpoint of a retained edge will also be
retained.
• If retainvertices is given and retainedges is missing, then any segment joining two retained vertices will also be retained.
• Thus, when both retainvertices and retainedges are given, it is possible that more vertices will be retained than those specified by retainvertices.
After the network has been altered, other consequential changes will occur, including renumbering
of the segments and vertices. If X is a point pattern on a linear network, then data points will be
deleted if they lie on a deleted edge.
Value
An object of the same kind as X.
Author(s)
Adrian Baddeley  and Suman Rakshit.
See Also
linnet to make a network;
connected.linnet to extract connected components.
Examples
L <- simplenet
plot(L, main="thinNetwork(L, retainedges=c(-3, -5))")
text(midpoints.psp(as.psp(L)), labels=1:nsegments(L), pos=3)
Lsub <- thinNetwork(L, retainedges=c(-3, -5))
plot(Lsub, add=TRUE, col="blue", lwd=2)

thomas.estK

thomas.estK

1427

Fit the Thomas Point Process by Minimum Contrast

Description
Fits the Thomas point process to a point pattern dataset by the Method of Minimum Contrast using
the K function.
Usage
thomas.estK(X, startpar=c(kappa=1,scale=1), lambda=NULL,
q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...)
Arguments
X

Data to which the Thomas model will be fitted. Either a point pattern or a
summary statistic. See Details.

startpar

Vector of starting values for the parameters of the Thomas process.

lambda

Optional. An estimate of the intensity of the point process.

q,p

Optional. Exponents for the contrast criterion.

rmin, rmax

Optional. The interval of r values for the contrast criterion.

...

Optional arguments passed to optim to control the optimisation algorithm. See
Details.

Details
This algorithm fits the Thomas point process model to a point pattern dataset by the Method of
Minimum Contrast, using the K function.
The argument X can be either
a point pattern: An object of class "ppp" representing a point pattern dataset. The K function of
the point pattern will be computed using Kest, and the method of minimum contrast will be
applied to this.
a summary statistic: An object of class "fv" containing the values of a summary statistic, computed for a point pattern dataset. The summary statistic should be the K function, and this
object should have been obtained by a call to Kest or one of its relatives.
The algorithm fits the Thomas point process to X, by finding the parameters of the Thomas model
which give the closest match between the theoretical K function of the Thomas process and the
observed K function. For a more detailed explanation of the Method of Minimum Contrast, see
mincontrast.
The Thomas point process is described in Møller and Waagepetersen (2003, pp. 61–62). It is
a cluster process formed by taking a pattern of parent points, generated according to a Poisson
process with intensity κ, and around each parent point, generating a random number of offspring
points, such that the number of offspring of each parent is a Poisson random variable with mean µ,
and the locations of the offspring points of one parent are independent and isotropically Normally
distributed around the parent point with standard deviation σ which is equal to the parameter scale.
The named vector of stating values can use either sigma2 (σ 2 ) or scale as the name of the second
component, but the latter is recommended for consistency with other cluster models.

1428

thomas.estK

The theoretical K-function of the Thomas process is
K(r) = πr2 +

r2
1
(1 − exp(− 2 )).
κ
4σ

The theoretical intensity of the Thomas process is λ = κµ.
In this algorithm, the Method of Minimum Contrast is first used to find optimal values of the parameters κ and σ 2 . Then the remaining parameter µ is inferred from the estimated intensity λ.
If the argument lambda is provided, then this is used as the value of λ. Otherwise, if X is a point
pattern, then λ will be estimated from X. If X is a summary statistic and lambda is missing, then the
intensity λ cannot be estimated, and the parameter µ will be returned as NA.
The remaining arguments rmin,rmax,q,p control the method of minimum contrast; see mincontrast.
The Thomas process can be simulated, using rThomas.
Homogeneous or inhomogeneous Thomas process models can also be fitted using the function kppm.
The optimisation algorithm can be controlled through the additional arguments "..." which are
passed to the optimisation function optim. For example, to constrain the parameter values to a
certain range, use the argument method="L-BFGS-B" to select an optimisation algorithm that respects box constraints, and use the arguments lower and upper to specify (vectors of) minimum
and maximum values for each parameter.
Value
An object of class "minconfit". There are methods for printing and plotting this object. It contains
the following main components:
par

Vector of fitted parameter values.

fit

Function value table (object of class "fv") containing the observed values of the
summary statistic (observed) and the theoretical values of the summary statistic
computed from the fitted model parameters.

Author(s)

Rasmus Waagepetersen  Adapted for spatstat by Adrian Baddeley 
References
Diggle, P. J., Besag, J. and Gleaves, J. T. (1976) Statistical analysis of spatial point patterns by
means of distance methods. Biometrics 32 659–667.
Møller, J. and Waagepetersen, R. (2003). Statistical Inference and Simulation for Spatial Point
Processes. Chapman and Hall/CRC, Boca Raton.
Thomas, M. (1949) A generalisation of Poisson’s binomial limit for use in ecology. Biometrika 36,
18–25.
Waagepetersen, R. (2007) An estimating function approach to inference for inhomogeneous NeymanScott processes. Biometrics 63, 252–258.

tile.areas

1431

See Also
thomas.estK mincontrast, pcf, rThomas to simulate the fitted model.
Examples
data(redwood)
u <- thomas.estpcf(redwood, c(kappa=10, scale=0.1))
u
plot(u, legendpos="topright")
u2 <- thomas.estpcf(redwood, c(kappa=10, scale=0.1),
pcfargs=list(stoyan=0.12))

tile.areas

Compute Areas of Tiles in a Tessellation

Description
Computes the area of each tile in a tessellation.
Usage
tile.areas(x)
Arguments
x

A tessellation (object of class "tess").

Details
A tessellation is a collection of disjoint spatial regions (called tiles) that fit together to form a larger
spatial region. See tess.
This command computes the area of each of the tiles that make up the tessellation x. The result is a
numeric vector in the same order as the tiles would be listed by tiles(x).
Value
A numeric vector.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
tess, tiles, tilenames, tiles.empty
Examples
A <- tess(xgrid=0:2,ygrid=0:2)
tile.areas(A)
v <- as.im(function(x,y){factor(round(x^2 + y^2))}, W=owin())
E <- tess(image=v)
tile.areas(E)

1432

tile.lengths

tile.lengths

Compute Lengths of Tiles in a Tessellation on a Network

Description
Computes the length of each tile in a tessellation on a linear network.
Usage
tile.lengths(x)
Arguments
x

A tessellation on a linear network (object of class "lintess").

Details
A tessellation on a linear network L is a partition of the network into non-overlapping pieces (tiles).
Each tile consists of one or more line segments which are subsets of the line segments making up
the network. A tile can consist of several disjoint pieces.
This command computes the length of each of the tiles that make up the tessellation x. The result
is a numeric vector.
Value
A numeric vector.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
lintess
Examples
X <- runiflpp(5, simplenet)
A <- lineardirichlet(X)
plot(A)
tile.lengths(A)

tileindex

1433

tileindex

Determine Which Tile Contains Each Given Point

Description
Given a tessellation and a list of spatial points, determine which tile of the tessellation contains each
of the given points.
Usage
tileindex(x, y, Z)
Arguments
x,y

Spatial coordinates. Numeric vectors of equal length.

Z

A tessellation (object of class "tess").

Details
This function determines which tile of the tessellation Z contains each of the spatial points with
coordinates (x[i],y[i]).
The result is a factor, of the same length as x and y, indicating which tile contains each point. The
levels of the factor are the names of the tiles of Z. Values are NA if the corresponding point lies
outside the tessellation.
Value
A factor, of the same length as x and y, whose levels are the names of the tiles of Z.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
cut.ppp and split.ppp to divide up the points of a point pattern according to a tessellation.
as.function.tess to create a function whose value is the tile index.
Examples
X <- runifpoint(7)
V <- dirichlet(X)
tileindex(0.1, 0.4, V)

1434

tilenames

tilenames

Names of Tiles in a Tessellation

Description
Extract or Change the Names of the Tiles in a Tessellation.

Usage
tilenames(x)
tilenames(x) <- value
Arguments
x

A tessellation (object of class "tess").

value

Character vector giving new names for the tiles.

Details
These functions extract or change the names of the tiles that make up the tessellation x.
If the tessellation is a regular grid, the tile names cannot be changed.

Value
tilenames returns a character vector.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
tess, tiles
Examples
D <- dirichlet(runifpoint(10))
tilenames(D)
tilenames(D) <- paste("Cell", 1:10)

tiles

1435

tiles

Extract List of Tiles in a Tessellation

Description
Extracts a list of the tiles that make up a tessellation.

Usage
tiles(x)
Arguments
x

A tessellation (object of class "tess").

Details
A tessellation is a collection of disjoint spatial regions (called tiles) that fit together to form a larger
spatial region. See tess.
The tiles that make up the tessellation x are returned in a list.
Value
A list of windows (objects of class "owin").
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
tess, tilenames, tile.areas, tiles.empty
Examples
A <- tess(xgrid=0:2,ygrid=0:2)
tiles(A)
v <- as.im(function(x,y){factor(round(x^2 + y^2))}, W=owin())
E <- tess(image=v)
tiles(E)

1436

tiles.empty

tiles.empty

Check For Empty Tiles in a Tessellation

Description
Checks whether each tile in a tessellation is empty or non-empty.
Usage
tiles.empty(x)
Arguments
x

A tessellation (object of class "tess").

Details
A tessellation is a collection of disjoint spatial regions (called tiles) that fit together to form a larger
spatial region. See tess.
It is possible for some tiles of a tessellation to be empty. For example, this can happen when the
tessellation x is obtained by restricting another tessellation y to a smaller spatial domain w.
The function tiles.empty checks whether each tile is empty or non-empty. The result is a logical
vector, with entries equal to TRUE when the corresponding tile is empty. Results are given in the
same order as the tiles would be listed by tiles(x).
Value
A logical vector.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
tess, tiles, tilenames, tile.areas
Examples
A <- tess(xgrid=0:2,ygrid=0:2)
tiles.empty(A)
v <- as.im(function(x,y){factor(round(x^2 + y^2))}, W=owin())
E <- tess(image=v)
tiles.empty(E)

timed

1437

timed

Record the Computation Time

Description
Saves the result of a calculation as an object of class "timed" which includes information about the
time taken to compute the result. The computation time is printed when the object is printed.
Usage
timed(x, ..., starttime = NULL, timetaken = NULL)
Arguments
x

An expression to be evaluated, or an object that has already been evaluated.

starttime

The time at which the computation is defined to have started. The default is the
current time. Ignored if timetaken is given.

timetaken

The length of time taken to perform the computation. The default is the time
taken to evaluate x.

...

Ignored.

Details
This is a simple mechanism for recording how long it takes to perform complicated calculations
(usually for the purposes of reporting in a publication).
If x is an expression to be evaluated, timed(x) evaluates the expression and measures the time taken
to evaluate it. The result is saved as an object of the class "timed". Printing this object displays the
computation time.
If x is an object which has already been computed, then the time taken to compute the object can be
specified either directly by the argument timetaken, or indirectly by the argument starttime.
• timetaken is the duration of time taken to perform the computation. It should be the difference of two clock times returned by proc.time. Typically the user sets begin <- proc.time()
before commencing the calculations, then end <- proc.time() after completing the calculations, and then sets timetaken <- end - begin.
• starttime is the clock time at which the computation started. It should be a value that was returned by proc.time at some earlier time when the calculations commenced. When timed is
called, the computation time will be taken as the difference between the current clock time and
starttime. Typically the user sets begin <- proc.time() before commencing the calculations, and when the calculations are completed, the user calls result <- timed(result, starttime=begin).
If the result of evaluating x belongs to other S3 classes, then the result of timed(x, ...) also
inherits these classes, and printing the object will display the appropriate information for these
classes as well.
Value
An object inheriting the class "timed".

1438

timeTaken

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
timeTaken to extract the time taken.
Examples
timed(clarkevans(cells))
timed(Kest(cells))
answer <- timed(42, timetaken=4.1e17)
answer

timeTaken

Extract the Total Computation Time

Description
Given an object or objects that contain timing information (reporting the amount of computer time
taken to compute each object), this function extracts the timing data and evaluates the total time
taken.
Usage
timeTaken(..., warn=TRUE)
Arguments
...

One or more objects of class "timed" containing timing data.

warn

Logical value indicating whether a warning should be issued if some of the
arguments do not contain timing information.

Details
An object of class "timed" contains information on the amount of computer time that was taken to
compute the object. See timed.
This function extracts the timing information from one or more such objects, and calculates the total
time.
Value
An object inheriting the class "timed".
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .

transect.im

1439

See Also
timed
Examples
A <- timed(Kest(cells))
B <- timed(Gest(cells))
A
B
timeTaken(A,B)

transect.im

Pixel Values Along a Transect

Description
Extract the pixel values of a pixel image at each point along a linear transect.
Usage
transect.im(X, ..., from="bottomleft", to="topright",
click=FALSE, add=FALSE)
Arguments
X

A pixel image (object of class "im").

...

Ignored.

from,to

Optional. Start point and end point of the transect. Pairs of (x, y) coordinates
in a format acceptable to xy.coords, or keywords "bottom", "left", "top",
"right", "bottomleft" etc.

click

Optional. Logical value. If TRUE, the linear transect is determined interactively
by the user, who clicks two points on the current plot.

add

Logical. If click=TRUE, this argument determines whether to perform interactive tasks on the current plot (add=TRUE) or to start by plotting X (add=FALSE).

Details
The pixel values of the image X along a line segment will be extracted. The result is a function table
("fv" object) which can be plotted directly.
If click=TRUE, then the user is prompted to click two points on the plot of X. These endpoints define
the transect.
Otherwise, the transect is defined by the endpoints from and to. The default is a diagonal transect
from bottom left to top right of the frame.
Value
An object of class "fv" which can be plotted.

1440

transmat

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
im
Examples
Z <- density(redwood)
plot(transect.im(Z))
## Not run:
if(FALSE) {
plot(transect.im(Z, click=TRUE))
}
## End(Not run)

transmat

Convert Pixel Array Between Different Conventions

Description
This function provides a simple way to convert arrays of pixel data between different display conventions.
Usage
transmat(m, from, to)
Arguments
m

A matrix.

from,to

Specifications of the spatial arrangement of the pixels. See Details.

Details
Pixel images are handled by many different software packages. In virtually all of these, the pixel
values are stored in a matrix, and are accessed using the row and column indices of the matrix.
However, different pieces of software use different conventions for mapping the matrix indices
[i, j] to the spatial coordinates (x, y).
• In the Cartesian convention, the first matrix index i is associated with the first Cartesian
coordinate x, and j is associated with y. This convention is used in image.default.
• In the European reading order convention, a matrix is displayed in the spatial coordinate
system as it would be printed in a page of text: i is effectively associated with the negative y
coordinate, and j is associated with x. This convention is used in some image file formats.
• In the spatstat convention, i is associated with the increasing y coordinate, and j is associated with x. This is also used in some image file formats.

treebranchlabels

1441

To convert between these conventions, use the function transmat. If a matrix m contains pixel image
data that is correctly displayed by software that uses the Cartesian convention, and we wish to convert it to the European reading convention, we can type mm <- transmat(m, from="Cartesian", to="European").
The transformed matrix mm will then be correctly displayed by software that uses the European convention.
Each of the arguments from and to can be one of the names "Cartesian", "European" or "spatstat"
(partially matched) or it can be a list specifying another convention. For example to=list(x="-i", y="-j")!
specifies that rows of the output matrix are expected to be displayed as vertical columns in the plot,
starting at the right side of the plot, as in the traditional Chinese, Japanese and Korean writing order.
Value
Another matrix obtained by rearranging the entries of m.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
Examples
opa <- par(mfrow=c(1,2))
# image in spatstat format
Z <- bei.extra$elev
plot(Z, main="plot.im", ribbon=FALSE)
m <- as.matrix(Z)
# convert matrix to format suitable for display by image.default
Y <- transmat(m, from="spatstat", to="Cartesian")
image(Y, asp=0.5, main="image.default", axes=FALSE)
par(opa)

treebranchlabels

Label Vertices of a Tree by Branch Membership

Description
Given a linear network which is a tree (acyclic graph), this function assigns a label to each vertex,
indicating its position in the tree.
Usage
treebranchlabels(L, root = 1)
Arguments
L

Linear network (object of class "linnet"). The network must have no loops.

root

Root of the tree. An integer index identifying which point in vertices(L) is
the root of the tree.

1442

treeprune

Details
The network L should be a tree, that is, it must have no loops.
This function computes a character string label for each vertex of the network L. The vertex identified by root (that is, vertices(L)[root]) is taken as the root of the tree and is given the empty
label "".
• If there are several line segments which meet at the root vertex, each of these segments is the
start of a new branch of the tree; the other endpoints of these segments are assigned the labels
"a", "b", "c" and so on.
• If only one segment issues from the root vertex, the other endpoint of this segment is assigned
the empty label "".
A similar rule is then applied to each of the newly-labelled vertices. If the vertex labelled "a" is
joined to two other unlabelled vertices, these will be labelled "aa" and "ab". The rule is applied
recursively until all vertices have been labelled.
If L is not a tree, the algorithm will terminate, but the results will be nonsense.
Value
A vector of character strings, with one entry for each point in vertices(L).
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
deletebranch, extractbranch, treeprune for manipulating a network using the branch labels.
linnet for creating a network.
Examples
# make a simple tree
m <- simplenet$m
m[8,10] <- m[10,8] <- FALSE
L <- linnet(vertices(simplenet), m)
plot(L, main="")
# compute branch labels
tb <- treebranchlabels(L, 1)
tbc <- paste0("[", tb, "]")
text(vertices(L), labels=tbc, cex=2)

treeprune

Prune Tree to Given Level

Description
Prune a tree by removing all the branches above a given level.

treeprune

1443

Usage
treeprune(X, root = 1, level = 0)
Arguments
X

Object of class "linnet" or "lpp".

root

Index of the root vertex amongst the vertices of as.linnet(X).

level

Integer specifying the level above which the tree should be pruned.

Details
The object X must be either a linear network, or a derived object such as a point pattern on a linear
network. The linear network must be an acyclic graph (i.e. must not contain any loops) so that it
can be interpreted as a tree.
This function removes all vertices for which treebranchlabels gives a string more than level
characters long.
Value
Object of the same kind as X.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
treebranchlabels for calculating the branch labels.
deletebranch for removing entire branches. extractbranch for extracting entire branches.
linnet for creating networks.
Examples
# make a simple tree
m <- simplenet$m
m[8,10] <- m[10,8] <- FALSE
L <- linnet(vertices(simplenet), m)
plot(L, main="")
# compute branch labels
tb <- treebranchlabels(L, 1)
tbc <- paste0("[", tb, "]")
text(vertices(L), labels=tbc, cex=2)
# prune tree
tp <- treeprune(L, root=1, 1)
plot(tp, add=TRUE, col="blue", lwd=3)

1444

triangulate.owin

triangulate.owin

Decompose Window into Triangles

Description
Given a spatial window, this function decomposes the window into disjoint triangles. The result is
a tessellation of the window in which each tile is a triangle.

Usage
triangulate.owin(W)
Arguments
W

Window (object of class "owin").

Details
The window W will be decomposed into disjoint triangles. The result is a tessellation of W in which
each tile is a triangle. All triangle vertices lie on the boundary of the original polygon.
The window is first converted to a polygonal window using as.polygonal. The vertices of the
polygonal window are extracted, and the Delaunay triangulation of these vertices is computed using
delaunay. Each Delaunay triangle is intersected with the window: if the result is not a triangle, the
triangulation procedure is applied recursively to this smaller polygon.

Value
Tessellation (object of class "tess").
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
tess, delaunay, as.polygonal
Examples
plot(triangulate.owin(letterR))

trim.rectangle

trim.rectangle

1445

Cut margins from rectangle

Description
Trims a margin from a rectangle.
Usage
trim.rectangle(W, xmargin=0, ymargin=xmargin)
Arguments
W

A window (object of class "owin"). Must be of type "rectangle".

xmargin

Width of horizontal margin to be trimmed. A single nonnegative number, or a
vector of length 2 indicating margins of unequal width at left and right.

ymargin

Height of vertical margin to be trimmed. A single nonnegative number, or a
vector of length 2 indicating margins of unequal width at bottom and top.

Details
This is a simple convenience function to trim off a margin of specified width and height from each
side of a rectangular window. Unequal margins can also be trimmed.
Value
Another object of class "owin" representing the window after margins are trimmed.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
grow.rectangle, erosion, owin.object
Examples
w <- square(10)
# trim a margin of width 1 from all four sides
square9 <- trim.rectangle(w, 1)
# trim margin of width 3 from the right side
# and margin of height 4 from top edge.
v <- trim.rectangle(w, c(0,3), c(0,4))

1446

Triplets

triplet.family

Triplet Interaction Family

Description
An object describing the family of all Gibbs point processes with interaction order equal to 3.
Details
Advanced Use Only!
This structure would not normally be touched by the user. It describes the interaction structure of
Gibbs point processes which have infinite order of interaction, such as the triplet interaction process
Triplets.
Anyway, triplet.family is an object of class "isf" containing a function triplet.family$eval
for evaluating the sufficient statistics of a Gibbs point process model taking an exponential family
form.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Baddeley, A. and Turner, R. (2000) Practical maximum pseudolikelihood for spatial point patterns.
Australian and New Zealand Journal of Statistics 42, 283–322.
See Also
Triplets to create the triplet interaction process structure.
Other families: pairwise.family, pairsat.family, inforder.family, ord.family.

Triplets

The Triplet Point Process Model

Description
Creates an instance of Geyer’s triplet interaction point process model which can then be fitted to
point pattern data.
Usage
Triplets(r)
Arguments
r

The interaction radius of the Triplets process

Triplets

1447

Details
The (stationary) Geyer triplet process (Geyer, 1999) with interaction radius r and parameters β and
γ is the point process in which each point contributes a factor β to the probability density of the
point pattern, and each triplet of close points contributes a factor γ to the density. A triplet of close
points is a group of 3 points, each pair of which is closer than r units apart.
Thus the probability density is
f (x1 , . . . , xn ) = αβ n(x) γ s(x)
where x1 , . . . , xn represent the points of the pattern, n(x) is the number of points in the pattern,
s(x) is the number of unordered triples of points that are closer than r units apart, and α is the
normalising constant.
The interaction parameter γ must be less than or equal to 1 so that this model describes an “ordered”
or “inhibitive” pattern.
The nonstationary Triplets process is similar except that the contribution of each individual point xi
is a function β(xi ) of location, rather than a constant beta.
The function ppm(), which fits point process models to point pattern data, requires an argument of
class "interact" describing the interpoint interaction structure of the model to be fitted. The appropriate description of the Triplets process pairwise interaction is yielded by the function Triplets().
See the examples below.
Note the only argument is the interaction radius r. When r is fixed, the model becomes an exponential family. The canonical parameters log(β) and log(γ) are estimated by ppm(), not fixed in
Triplets().
Value
An object of class "interact" describing the interpoint interaction structure of the Triplets process
with interaction radius r.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Geyer, C.J. (1999) Likelihood Inference for Spatial Point Processes. Chapter 3 in O.E. BarndorffNielsen, W.S. Kendall and M.N.M. Van Lieshout (eds) Stochastic Geometry: Likelihood and Computation, Chapman and Hall / CRC, Monographs on Statistics and Applied Probability, number 80.
Pages 79–140.
See Also
ppm, triplet.family, ppm.object
Examples
Triplets(r=0.1)
# prints a sensible description of itself
## Not run:
ppm(cells, ~1, Triplets(r=0.2))

1448

Tstat
# fit the stationary Triplets process to `cells'

## End(Not run)
ppm(cells, ~polynom(x,y,3), Triplets(r=0.2))
# fit a nonstationary Triplets process with log-cubic polynomial trend

Tstat

Third order summary statistic

Description
Computes the third order summary statistic T (r) of a spatial point pattern.
Usage
Tstat(X, ..., r = NULL, rmax = NULL,
correction = c("border", "translate"), ratio = FALSE, verbose=TRUE)
Arguments
X

The observed point pattern, from which an estimate of T (r) will be computed.
An object of class "ppp", or data in any format acceptable to as.ppp().

...

Ignored.

r

Optional. Vector of values for the argument r at which T (r) should be evaluated.
Users are advised not to specify this argument; there is a sensible default.

rmax

Optional. Numeric. The maximum value of r for which T (r) should be estimated.

correction

Optional. A character vector containing any selection of the options "none",
"border", "bord.modif", "translate", "translation", or "best". It specifies the edge correction(s) to be applied. Alternatively correction="all" selects all options.

ratio

Logical. If TRUE, the numerator and denominator of each edge-corrected estimate will also be saved, for use in analysing replicated point patterns.

verbose

Logical. If TRUE, an estimate of the computation time is printed.

Details
This command calculates the third-order summary statistic T (r) for a spatial point patterns, defined
by Schladitz and Baddeley (2000).
The definition of T (r) is similar to the definition of Ripley’s K function K(r), except that K(r)
counts pairs of points while T (r) counts triples of points. Essentially T (r) is a rescaled cumulative
distribution function of the diameters of triangles in the point pattern. The diameter of a triangle is
the length of its longest side.
Value
An object of class "fv", see fv.object, which can be plotted directly using plot.fv.

tweak.colourmap

1449

Computation time
If the number of points is large, the algorithm can take a very long time to inspect all possible
triangles. A rough estimate of the total computation time will be printed at the beginning of the
calculation. If this estimate seems very large, stop the calculation using the user interrupt signal,
and call Tstat again, using rmax to restrict the range of r values, thus reducing the number of
triangles to be inspected.
Author(s)
Adrian Baddeley 
References
Schladitz, K. and Baddeley, A. (2000) A third order point process characteristic. Scandinavian
Journal of Statistics 27 (2000) 657–671.
See Also
Kest
Examples
plot(Tstat(redwood))

tweak.colourmap

Change Colour Values in a Colour Map

Description
Assign new colour values to some of the entries in a colour map.
Usage
tweak.colourmap(m, col, ..., inputs=NULL, range=NULL)
Arguments
m

A colour map (object of class "colourmap").

inputs

Input values to the colour map, to be assigned new colours. Incompatible with
range.

range

Numeric vector of length 2 specifying a range of numerical values which should
be assigned a new colour. Incompatible with inputs.

col

Replacement colours for the specified inputs or the specified range of values.

...

Other arguments are ignored.

Details
This function changes the colour map m by assigning new colours to each of the input values specified by inputs, or by assigning a single new colour to the range of input values specified by range.
The modified colour map is returned.

1450

union.quad

Value
Another colour map (object of class "colourmap").
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
colourmap, interp.colourmap, colouroutputs, colourtools.
Examples
co <- colourmap(rainbow(32), range=c(0,1))
plot(tweak.colourmap(co, inputs=c(0.5, 0.6), "white"))
plot(tweak.colourmap(co, range=c(0.5,0.6), "white"))

union.quad

Union of Data and Dummy Points

Description
Combines the data and dummy points of a quadrature scheme into a single point pattern.
Usage
union.quad(Q)
Arguments
Q

A quadrature scheme (an object of class "quad").

Details
The argument Q should be a quadrature scheme (an object of class "quad", see quad.object for
details).
This function combines the data and dummy points of Q into a single point pattern. If either the data
or the dummy points are marked, the result is a marked point pattern.
The function as.ppp will perform the same task.
Value
A point pattern (of class "ppp").
Author(s)
Adrian Baddeley 
and Rolf Turner 

unique.ppp

1451

See Also
quad.object, as.ppp
Examples
data(simdat)
Q <- quadscheme(simdat, default.dummy(simdat))
U <- union.quad(Q)
## Not run: plot(U)
# equivalent:
U <- as.ppp(Q)

unique.ppp

Extract Unique Points from a Spatial Point Pattern

Description
Removes any points that are identical to other points in a spatial point pattern.
Usage
## S3 method for class 'ppp'
unique(x, ..., warn=FALSE)
## S3 method for class 'ppx'
unique(x, ..., warn=FALSE)
Arguments
x

A spatial point pattern (object of class "ppp" or "ppx").

...

Arguments passed to duplicated.ppp or duplicated.data.frame.

warn

Logical. If TRUE, issue a warning message if any duplicated points were found.

Details
These are methods for the generic function unique for point pattern datasets (of class "ppp", see
ppp.object, or class "ppx").
This function removes duplicate points in x, and returns a point pattern.
Two points in a point pattern are deemed to be identical if their x, y coordinates are the same, and
their marks are the same (if they carry marks). This is the default rule: see duplicated.ppp for
other options.
Value
Another point pattern object.
Author(s)
Adrian Baddeley 
and Rolf Turner 

1452

unitname

See Also
ppp.object, duplicated.ppp, multiplicity.ppp
Examples
X <- ppp(c(1,1,0.5), c(2,2,1), window=square(3))
unique(X)
unique(X, rule="deldir")

unitname

Name for Unit of Length

Description
Inspect or change the name of the unit of length in a spatial dataset.
Usage
unitname(x)
## S3 method for class 'dppm'
unitname(x)
## S3 method for class 'im'
unitname(x)
## S3 method for class 'kppm'
unitname(x)
## S3 method for class 'minconfit'
unitname(x)
## S3 method for class 'owin'
unitname(x)
## S3 method for class 'ppm'
unitname(x)
## S3 method for class 'ppp'
unitname(x)
## S3 method for class 'psp'
unitname(x)
## S3 method for class 'quad'
unitname(x)
## S3 method for class 'slrm'
unitname(x)
## S3 method for class 'tess'
unitname(x)
unitname(x) <- value
## S3 replacement method for class
unitname(x) <- value
## S3 replacement method for class
unitname(x) <- value
## S3 replacement method for class
unitname(x) <- value
## S3 replacement method for class
unitname(x) <- value
## S3 replacement method for class

'dppm'
'im'
'kppm'
'minconfit'
'owin'

unitname

1453

unitname(x) <- value
## S3 replacement method
unitname(x) <- value
## S3 replacement method
unitname(x) <- value
## S3 replacement method
unitname(x) <- value
## S3 replacement method
unitname(x) <- value
## S3 replacement method
unitname(x) <- value
## S3 replacement method
unitname(x) <- value

for class 'ppm'
for class 'ppp'
for class 'psp'
for class 'quad'
for class 'slrm'
for class 'tess'

Arguments
x

A spatial dataset. Either a point pattern (object of class "ppp"), a line segment
pattern (object of class "psp"), a window (object of class "owin"), a pixel image (object of class "im"), a tessellation (object of class "tess"), a quadrature
scheme (object of class "quad"), or a fitted point process model (object of class
"ppm" or "kppm" or "slrm" or "dppm" or "minconfit").

value

Name of the unit of length. See Details.

Details
Spatial datasets in the spatstat package may include the name of the unit of length. This name is
used when printing or plotting the dataset, and in some other applications.
unitname(x) extracts this name, and unitname(x) <- value sets the name to value.
A valid name is either
• a single character string
• a vector of two character strings giving the singular and plural forms of the unit name
• a list of length 3, containing two character strings giving the singular and plural forms of the
basic unit, and a number specifying the multiple of this unit.
Note that re-setting the name of the unit of length does not affect the numerical values in x. It
changes only the string containing the name of the unit of length. To rescale the numerical values,
use rescale.
Value
The return value of unitname is an object of class "unitname" containing the name of the unit of
length in x. There are methods for print, summary, as.character, rescale and compatible.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
rescale, owin, ppp

1454

unmark

Examples
X <- runifpoint(20)
# if the unit of length is 1 metre:
unitname(X) <- c("metre", "metres")
# if the unit of length is 6 inches:
unitname(X) <- list("inch", "inches", 6)

unmark

Remove Marks

Description
Remove the mark information from a spatial dataset.
Usage
unmark(X)
## S3 method
unmark(X)
## S3 method
unmark(X)
## S3 method
unmark(X)
## S3 method
unmark(X)

for class 'ppp'
for class 'splitppp'
for class 'psp'
for class 'ppx'

Arguments
X

A point pattern (object of class "ppp"), a split point pattern (object of class
"splitppp"), a line segment pattern (object of class "psp") or a multidimensional space-time point pattern (object of class "ppx").

Details
A ‘mark’ is a value attached to each point in a spatial point pattern, or attached to each line segment
in a line segment pattern, etc.
The function unmark is a simple way to remove the marks from such a dataset.
Value
An object of the same class as X with any mark information deleted.
Author(s)
Adrian Baddeley  and Rolf Turner 
See Also
ppp.object, psp.object

unnormdensity

1455

Examples
data(lansing)
hicks <- lansing[lansing$marks == "hickory", ]
## Not run:
plot(hicks) # still a marked point pattern, but only 1 value of marks
plot(unmark(hicks)) # unmarked
## End(Not run)

unnormdensity

Weighted kernel smoother

Description
An unnormalised version of kernel density estimation where the weights are not required to sum to
1. The weights may be positive, negative or zero.
Usage
unnormdensity(x, ..., weights = NULL)
Arguments
x

Numeric vector of data

...

Arguments passed to density.default. Arguments must be named. ‘

weights

Optional numeric vector of weights for the data.

Details
This is an alternative to the standard R kernel density estimation function density.default.
The standard density.default requires the weights to be nonnegative numbers that add up to 1,
and returns a probability density (a function that integrates to 1).
This function unnormdensity does not impose any requirement on the weights except that they be
finite. Individual weights may be positive, negative or zero. The result is a function that does not
necessarily integrate to 1 and may be negative. The result is the convolution of the kernel k with
the weighted data,
X
f (x) =
wi k(x − xi )
i

where xi are the data points and wi are the weights.
The algorithm first selects the kernel bandwidth by applying density.default to the data x with
normalised, positive weight vector w = abs(weights)/sum(abs(weights)) and extracting the
selected bandwidth. Then the result is computed by applying applying density.default to x
twice using the normalised positive and negative parts of the weights.
Note that the arguments ... must be passed by name, i.e. in the form (name=value). Arguments
that do not match an argument of density.default will be ignored silently.
Value
Object of class "density" as described in density.default.

1456

unstack.msr

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
density.default
Examples
d <- unnormdensity(1:3, weights=c(-1,0,1))
if(interactive()) plot(d)

unstack.msr

Separate a Vector Measure into its Scalar Components

Description
Converts a vector-valued measure into a list of scalar-valued measures.
Usage
## S3 method for class 'msr'
unstack(x, ...)
Arguments
x

A measure (object of class "msr").

...

Ignored.

Details
This is a method for the generic unstack for the class "msr" of measures.
If x is a vector-valued measure, then y <- unstack(x) is a list of scalar-valued measures defined
by the components of x. The jth entry of the list, y[[j]], is equivalent to the jth component of the
vector measure x.
If x is a scalar-valued measure, then the result is a list consisting of one entry, which is x.
Value
A list of measures, of class "solist".
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .

unstack.ppp

1457

See Also
unstack
unstack.ppp
split.msr.
Examples
fit <- ppm(cells ~ x)
m <- residuals(fit, type="score")
m
unstack(m)

unstack.ppp

Separate Multiple Columns of Marks

Description
Given a spatial pattern with several columns of marks, take one column at a time, and return a list
of spatial patterns each having only one column of marks.
Usage
## S3 method for class 'ppp'
unstack(x, ...)
## S3 method for class 'psp'
unstack(x, ...)
## S3 method for class 'lpp'
unstack(x, ...)
Arguments
x

A spatial point pattern (object of class "ppp" or "lpp") or a spatial pattern of
line segments (object of class "psp").

...

Ignored.

Details
The functions defined here are methods for the generic unstack. The functions expect a spatial
object x which has several columns of marks; they separate the columns, and return a list of spatial
objects, each having only one column of marks.
If x has several columns of marks (i.e. marks(x) is a matrix, data frame or hyperframe with several
columns), then y <- unstack(x) is a list of spatial objects, each of the same kind as x. The jth
entry y[[j]] is equivalent to x except that it only includes the jth column of marks(x).
If x has no marks, or has only a single column of marks, the result is a list consisting of one entry,
which is x.
Value
A list, of class "solist", whose entries are objects of the same type as x.

1458

update.detpointprocfamily

Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
unstack
unstack.msr
See also methods for the generic split such as split.ppp.
Examples
finpines
unstack(finpines)

update.detpointprocfamily
Set Parameter Values in a Determinantal Point Process Model

Description
Set parameter values in a determinantal point process model object.

Usage
## S3 method for class 'detpointprocfamily'
update(object, ...)
Arguments
object

object of class "detpointprocfamily".

...

arguments of the form tag=value specifying the parameters values to set.

Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 

update.interact

1459

update.interact

Update an Interpoint Interaction

Description
This command updates the object using the arguments given.
Usage
## S3 method for class 'interact'
update(object, ...)
Arguments
object

Interpoint interaction (object of class "interact").

...

Additional or replacement values of parameters of object.

Details
This is a method for the generic function update for the class "interact" of interpoint interactions.
It updates the object using the parameters given in the extra arguments ....
The extra arguments must be given in the form name=value and must be recognisable to the interaction object. They override any parameters of the same name in object.
Value
Another object of class "interact", equivalent to object except for changes in parameter values.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
update.ppm
Examples
Str <- Strauss(r=1)
Str
update(Str, r=2)
M <- MultiStrauss(radii=matrix(1,2,2))
update(M, types=c("on", "off"))

1460

update.kppm

update.kppm

Update a Fitted Cluster Point Process Model

Description
update method for class "kppm".
Usage
## S3 method for class 'kppm'
update(object, ..., evaluate=TRUE)
Arguments
object

Fitted cluster point process model. An object of class "kppm", obtained from
kppm.

...

Arguments passed to kppm.

evaluate

Logical value indicating whether to return the updated fitted model (evaluate=TRUE,
the default) or just the updated call to kppm (evaluate=FALSE).

Details
object should be a fitted cluster point process model, obtained from the model-fitting function
kppm. The model will be updated according to the new arguments provided.
If the argument trend is provided, it determines the intensity in the updated model. It should be an
R formula (with or without a left hand side). It may include the symbols + or - to specify addition
or deletion of terms in the current model formula, as shown in the Examples below. The symbol .
refers to the current contents of the formula.
The intensity in the updated model is determined by the argument trend if it is provided, or otherwise by any unnamed argument that is a formula, or otherwise by the formula of the original model,
formula(object).
The spatial point pattern data to which the new model is fitted is determined by the left hand side
of the updated model formula, if this is present. Otherwise it is determined by the argument X if it
is provided, or otherwise by any unnamed argument that is a point pattern or a quadrature scheme.
The model is refitted using kppm.
Value
Another fitted cluster point process model (object of class "kppm".
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
kppm, plot.kppm, predict.kppm, simulate.kppm, methods.kppm, vcov.kppm

update.ppm

1461

Examples
fit <- kppm(redwood ~1, "Thomas")
fitx <- update(fit, ~ . + x)
fitM <- update(fit, clusters="MatClust")
fitC <- update(fit, cells)
fitCx <- update(fit, cells ~ x)

update.ppm

Update a Fitted Point Process Model

Description
update method for class "ppm".
Usage
## S3 method for class 'ppm'
update(object, ..., fixdummy=TRUE, use.internal=NULL,
envir=environment(terms(object)))

Arguments
object

An existing fitted point process model, typically produced by ppm.

...

Arguments to be updated in the new call to ppm.

fixdummy

Logical flag indicating whether the quadrature scheme for the call to ppm should
use the same set of dummy points as that in the original call.

use.internal

Optional. Logical flag indicating whether the model should be refitted using the
internally saved data (use.internal=TRUE) or by re-evaluating these data in the
current frame (use.internal=FALSE).

envir

Environment in which to re-evaluate the call to ppm.

Details
This is a method for the generic function update for the class "ppm". An object of class "ppm"
describes a fitted point process model. See ppm.object) for details of this class.
update.ppm will modify the point process model specified by object according to the new arguments given, then re-fit it. The actual re-fitting is performed by the model-fitting function ppm.
If you are comparing several model fits to the same data, or fits of the same model to different data,
it is strongly advisable to use update.ppm rather than trying to fit them by hand. This is because
update.ppm re-fits the model in a way which is comparable to the original fit.
The arguments ... are matched to the formal arguments of ppm as follows.
First, all the named arguments in ... are matched with the formal arguments of ppm. Use name=NULL
to remove the argument name from the call.
Second, any unnamed arguments in ... are matched with formal arguments of ppm if the matching
is obvious from the class of the object. Thus ... may contain
• exactly one argument of class "ppp" or "quad", which will be interpreted as the named argument Q;

1462

update.ppm
• exactly one argument of class "formula", which will be interpreted as the named argument
trend (or as specifying a change to the trend formula);
• exactly one argument of class "interact", which will be interpreted as the named argument
interaction;
• exactly one argument of class "data.frame", which will be interpreted as the named argument covariates.

The trend argument can be a formula that specifies a change to the current trend formula. For
example, the formula ~ . + Z specifies that the additional covariate Z will be added to the right
hand side of the trend formula in the existing object.
The argument fixdummy=TRUE ensures comparability of the objects before and after updating.
When fixdummy=FALSE, calling update.ppm is exactly the same as calling ppm with the updated
arguments. However, the original and updated models are not strictly comparable (for example,
their pseudolikelihoods are not strictly comparable) unless they used the same set of dummy points
for the quadrature scheme. Setting fixdummy=TRUE ensures that the re-fitting will be performed
using the same set of dummy points. This is highly recommended.
The value of use.internal determines where to find data to re-evaluate the model (data for the arguments mentioned in the original call to ppm that are not overwritten by arguments to update.ppm).
If use.internal=FALSE, then arguments to ppm are re-evaluated in the frame where you call
update.ppm. This is like the behaviour of the other methods for update. This means that if you
have changed any of the objects referred to in the call, these changes will be taken into account.
Also if the original call to ppm included any calls to random number generators, these calls will be
recomputed, so that you will get a different outcome of the random numbers.
If use.internal=TRUE, then arguments to ppm are extracted from internal data stored inside the
current fitted model object. This is useful if you don’t want to re-evaluate anything. It is also
necessary if if object has been restored from a dump file using load or source. In such cases, we
have lost the environment in which object was fitted, and data cannot be re-evaluated.
By default, if use.internal is missing, update.ppm will re-evaluate the arguments if this is possible, and use internal data if not.
Value
Another fitted point process model (object of class "ppm").
Author(s)
Adrian Baddeley 
and Rolf Turner 
Examples
data(nztrees)
data(cells)
# fit the stationary Poisson process
fit <- ppm(nztrees, ~ 1)
# fit a nonstationary Poisson process
fitP <- update(fit, trend=~x)
fitP <- update(fit, ~x)
# change the trend formula: add another term to the trend

update.rmhcontrol

1463

fitPxy <- update(fitP, ~ . + y)
# change the trend formula: remove the x variable
fitPy <- update(fitPxy, ~ . - x)
# fit a stationary Strauss process
fitS <- update(fit, interaction=Strauss(13))
fitS <- update(fit, Strauss(13))
# refit using a different edge correction
fitS <- update(fitS, correction="isotropic")
# re-fit the model to a subset
# of the original point pattern
nzw <- owin(c(0,148),c(0,95))
nzsub <- nztrees[,nzw]
fut <- update(fitS, Q=nzsub)
fut <- update(fitS, nzsub)
# WARNING: the point pattern argument is called 'Q'
ranfit <- ppm(rpoispp(42), ~1, Poisson())
ranfit
# different random data!
update(ranfit)
# the original data
update(ranfit, use.internal=TRUE)

update.rmhcontrol

Update Control Parameters of Metropolis-Hastings Algorithm

Description
update method for class "rmhcontrol".
Usage
## S3 method for class 'rmhcontrol'
update(object, ...)
Arguments
object

Object of class "rmhcontrol" containing control parameters for a MetropolisHastings algorithm.

...

Arguments to be updated in the new call to rmhcontrol.

Details
This is a method for the generic function update for the class "rmhcontrol". An object of class
"rmhcontrol" describes a set of control parameters for the Metropolis-Hastings simulation algorithm. See rmhcontrol).
update.rmhcontrol will modify the parameters specified by object according to the new arguments given.

1464

update.symbolmap

Value
Another object of class "rmhcontrol".
Author(s)
Adrian Baddeley 
and Rolf Turner 
Examples
a <- rmhcontrol(expand=1)
update(a, expand=2)

update.symbolmap

Update a Graphics Symbol Map.

Description
This command updates the object using the arguments given.
Usage
## S3 method for class 'symbolmap'
update(object, ...)
Arguments
object

Graphics symbol map (object of class "symbolmap").

...

Additional or replacement arguments to symbolmap.

Details
This is a method for the generic function update for the class "symbolmap" of graphics symbol
maps. It updates the object using the parameters given in the extra arguments ....
The extra arguments must be given in the form name=value and must be recognisable to symbolmap.
They override any parameters of the same name in object.
Value
Another object of class "symbolmap".
Author(s)
Adrian Baddeley 
, Rolf Turner 
and Ege Rubak .
See Also
symbolmap to create a graphics symbol map.

valid

1465

Examples
g <- symbolmap(size=function(x) x/50)
g
update(g, range=c(0,1))
update(g, size=42)
update(g, shape="squares", range=c(0,1))

valid

Check Whether Point Process Model is Valid

Description
Determines whether a point process model object corresponds to a valid point process.
Usage
valid(object, ...)
Arguments
object

Object of some class, describing a point process model.

...

Additional arguments passed to methods.

Details
The function valid is generic, with methods for the classes "ppm" and "dppmodel".
An object representing a point process is called valid if all its parameter values are known (for
example, no parameter takes the value NA or NaN) and the parameter values correspond to a welldefined point process (for example, the parameter values satisfy all the constraints that are imposed
by mathematical theory.)
See the methods for further details.
Value
A logical value, or NA.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
valid.ppm, valid.detpointprocfamily

1466

valid.detpointprocfamily

valid.detpointprocfamily
Check Validity of a Determinantal Point Process Model

Description
Checks the validity of a determinantal point process model.

Usage
## S3 method for class 'detpointprocfamily'
valid(object, ...)

Arguments
object

Model of class "detpointprocfamily".

...

Ignored.

Value
Logical

Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 

See Also
valid

Examples
model1 <- dppMatern(lambda=100, alpha=.01, nu=1, d=2)
valid(model1)
model2 <- dppMatern(lambda=100, alpha=1, nu=1, d=2)
valid(model2)

valid.ppm

1467

valid.ppm

Check Whether Point Process Model is Valid

Description
Determines whether a fitted point process model satisfies the integrability conditions for existence
of the point process.
Usage
## S3 method for class 'ppm'
valid(object, warn=TRUE, ...)
Arguments
object

Fitted point process model (object of class "ppm").

warn

Logical value indicating whether to issue a warning if the validity of the model
cannot be checked (due to unavailability of the required code).

...

Ignored.

Details
This is a method for the generic function valid for Poisson and Gibbs point process models (class
"ppm").
The model-fitting function ppm fits Gibbs point process models to point pattern data. By default,
ppm does not check whether the fitted model actually exists as a point process. This checking is
done by valid.ppm.
Unlike a regression model, which is well-defined for any values of the fitted regression coefficients,
a Gibbs point process model is only well-defined if the fitted interaction parameters satisfy some
constraints. A famous example is the Strauss process (see Strauss) which exists only when the
interaction parameter γ is less than or equal to 1. For values γ > 1, the probability density is not
integrable and the process does not exist (and cannot be simulated).
By default, ppm does not enforce the constraint that a fitted Strauss process (for example) must
satisfy γ ≤ 1. This is because a fitted parameter value of γ > 1 could be useful information for data
analysis, as it indicates that the Strauss model is not appropriate, and suggests a clustered model
should be fitted.
The function valid.ppm checks whether the fitted model object specifies a well-defined point
process. It returns TRUE if the model is well-defined.
Another possible reason for invalid models is that the data may not be adequate for estimation of
the model parameters. In this case, some of the fitted coefficients could be NA or infinite values. If
this happens then valid.ppm returns FALSE.
Use the function project.ppm to force the fitted model to be valid.
Value
A logical value, or NA.

1468

varblock

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
ppm, project.ppm
Examples
fit1 <- ppm(cells, ~1, Strauss(0.1))
valid(fit1)
fit2 <- ppm(redwood, ~1, Strauss(0.1))
valid(fit2)

varblock

Estimate Variance of Summary Statistic by Subdivision

Description
This command estimates the variance of any summary statistic (such as the K-function) by spatial
subdivision of a single point pattern dataset.
Usage
varblock(X, fun = Kest,
blocks = quadrats(X, nx = nx, ny = ny),
...,
nx = 3, ny = nx,
confidence=0.95)
Arguments
X

Point pattern dataset (object of class "ppp").

fun

Function that computes the summary statistic.

blocks

Optional. A tessellation that specifies the division of the space into blocks.

...

Arguments passed to fun.

nx,ny

Optional. Number of rectangular blocks in the x and y directions. Incompatible
with blocks.

confidence

Confidence level, as a fraction between 0 and 1.

Details
This command computes an estimate of the variance of the summary statistic fun(X) from a single
point pattern dataset X using a subdivision method. It can be used to plot confidence intervals for
the true value of a summary function such as the K-function.
The window containing X is divided into pieces by an nx * ny array of rectangles (or is divided
into pieces of more general shape, according to the argument blocks if it is present). The summary
statistic fun is applied to each of the corresponding sub-patterns of X as described below. Then the

varblock

1469

pointwise sample mean, sample variance and sample standard deviation of these summary statistics are computed. Then pointwise confidence intervals are computed, for the specified level of
confidence, defaulting to 95 percent.
The variance is estimated by equation (4.21) of Diggle (2003, page 52). This assumes that the point
pattern X is stationary. For further details see Diggle (2003, pp 52–53).
The estimate of the summary statistic from each block is computed as follows. For most functions
fun, the estimate from block B is computed by finding the subset of X consisting of points that fall
inside B, and applying fun to these points, by calling fun(X[B]).
However if fun is the K-function Kest, or any function which has an argument called domain, the
estimate for each block B is computed by calling fun(X, domain=B). In the case of the K-function
this means that the estimate from block B is computed by counting pairs of points in which the first
point lies in B, while the second point may lie anywhere.
Value
A function value table (object of class "fv") that contains the result of fun(X) as well as the sample
mean, sample variance and sample standard deviation of the block estimates, together with the upper
and lower two-standard-deviation confidence limits.
Errors
If the blocks are too small, there may be insufficient data in some blocks, and the function fun may
report an error. If this happens, you need to take larger blocks.
An error message about incompatibility may occur. The different function estimates may be incompatible in some cases, for example, because they use different default edge corrections (typically
because the tiles of the tessellation are not the same kind of geometric object as the window of X, or
because the default edge correction depends on the number of points). To prevent this, specify the
choice of edge correction, in the correction argument to fun, if it has one.
An alternative to varblock is Loh’s mark bootstrap lohboot.
Author(s)
Adrian Baddeley 
and Rolf Turner 
References
Diggle, P.J. (2003) Statistical analysis of spatial point patterns, Second edition. Arnold.
See Also
tess, quadrats for basic manipulation.
lohboot for an alternative bootstrap technique.
Examples
v <- varblock(amacrine, Kest, nx=4, ny=2)
v <- varblock(amacrine, Kcross, nx=4, ny=2)
if(interactive()) plot(v, iso ~ r, shade=c("hiiso", "loiso"))

1470

varcount

varcount

Predicted Variance of the Number of Points

Description
Given a fitted point process model, calculate the predicted variance of the number of points in a
nominated set B.
Usage
varcount(model, B, ..., dimyx = NULL)
Arguments
model

A fitted point process model (object of class "ppm", "kppm" or "dppm").

B

A window (object of class "owin" specifying the region in which the points
are counted. Alternatively a pixel image (object of class "im") or a function of
spatial coordinates specifying a numerical weight for each random point.

...

Additional arguments passed to B when it is a function.

dimyx

Spatial resolution for the calculations. Argument passed to as.mask.

Details
This command calculates the variance of the number of points falling in a specified window B
according to the model. It can also calculate the variance of a sum of weights attached to each
random point.
The model should be a fitted point process model (object of class "ppm", "kppm" or "dppm").
• If B is a window, this command calculates the variance of the number of points falling in B,
according to the fitted model.
If the model depends on spatial covariates other than the Cartesian coordinates, then B should
be a subset of the domain in which these covariates are defined.
P
• If B is a pixel image, this command calculates the variance of T = i B(xi ), the sum of the
values of B over all random points falling in the domain of the image.
If the model depends on spatial covariates other than the Cartesian coordinates, then the domain of the pixel image, as.owin(B), should be a subset of the domain in which these covariates are defined.
• If B isPa function(x,y) or function(x,y,...) this command calculates the variance of
T = i B(xi ), the sum of the values of B over all random points falling inside the window
W=as.owin(model), the window in which the original data were observed.
The variance calculation involves the intensity and the pair correlation function of the model. The
calculation is exact (up to discretisation error) for models of class "kppm" and "dppm", and for
Poisson point process models of class "ppm". For Gibbs point process models of class "ppm" the
calculation depends on the Poisson-saddlepoint approximations to the intensity and pair correlation
function, which are rough approximations. The approximation is not yet implemented for some
Gibbs models.

vargamma.estK

1471

Value
A single number.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak 
See Also
predict.ppm, predict.kppm, predict.dppm
Examples
fitT <- kppm(redwood ~ 1, "Thomas")
B <- owin(c(0, 0.5), c(-0.5, 0))
varcount(fitT, B)
fitS <- ppm(swedishpines ~ 1, Strauss(9))
BS <- square(50)
varcount(fitS, BS)

vargamma.estK

Fit the Neyman-Scott Cluster Point Process with Variance Gamma kernel

Description
Fits the Neyman-Scott cluster point process, with Variance Gamma kernel, to a point pattern dataset
by the Method of Minimum Contrast.
Usage
vargamma.estK(X, startpar=c(kappa=1,scale=1), nu = -1/4, lambda=NULL,
q = 1/4, p = 2, rmin = NULL, rmax = NULL, ...)
Arguments
X

Data to which the model will be fitted. Either a point pattern or a summary
statistic. See Details.

startpar

Vector of starting values for the parameters of the model.

nu

Numerical value controlling the shape of the tail of the clusters. A number
greater than -1/2.

lambda

Optional. An estimate of the intensity of the point process.

q,p

Optional. Exponents for the contrast criterion.

rmin, rmax

Optional. The interval of r values for the contrast criterion.

...

Optional arguments passed to optim to control the optimisation algorithm. See
Details.

1472

vargamma.estK

Details
This algorithm fits the Neyman-Scott Cluster point process model with Variance Gamma kernel
(Jalilian et al, 2013) to a point pattern dataset by the Method of Minimum Contrast, using the K
function.
The argument X can be either
a point pattern: An object of class "ppp" representing a point pattern dataset. The K function of
the point pattern will be computed using Kest, and the method of minimum contrast will be
applied to this.
a summary statistic: An object of class "fv" containing the values of a summary statistic, computed for a point pattern dataset. The summary statistic should be the K function, and this
object should have been obtained by a call to Kest or one of its relatives.
The algorithm fits the Neyman-Scott Cluster point process with Variance Gamma kernel to X, by
finding the parameters of the model which give the closest match between the theoretical K function
of the model and the observed K function. For a more detailed explanation of the Method of
Minimum Contrast, see mincontrast.
The Neyman-Scott cluster point process with Variance Gamma kernel is described in Jalilian et
al (2013). It is a cluster process formed by taking a pattern of parent points, generated according
to a Poisson process with intensity κ, and around each parent point, generating a random number
of offspring points, such that the number of offspring of each parent is a Poisson random variable
with mean µ, and the locations of the offspring points of one parent have a common distribution
described in Jalilian et al (2013).
The shape of the kernel is determined by the dimensionless index nu. This is the parameter ν 0 =
α/2 − 1 appearing in equation (12) on page 126 of Jalilian et al (2013). In previous versions of
spatstat instead of specifying nu (called nu.ker at that time) the user could specify nu.pcf which
is the parameter ν = α − 1 appearing in equation (13), page 127 of Jalilian et al (2013). These
are related by nu.pcf = 2 * nu.ker + 1 and nu.ker = (nu.pcf - 1)/2. This syntax is
still supported but not recommended for consistency across the package. In that case exactly one of
nu.ker or nu.pcf must be specified.
If the argument lambda is provided, then this is used as the value of the point process intensity λ.
Otherwise, if X is a point pattern, then λ will be estimated from X. If X is a summary statistic and
lambda is missing, then the intensity λ cannot be estimated, and the parameter µ will be returned
as NA.
The remaining arguments rmin,rmax,q,p control the method of minimum contrast; see mincontrast.
The corresponding model can be simulated using rVarGamma.
The parameter eta appearing in startpar is equivalent to the scale parameter omega used in
rVarGamma.
Homogeneous or inhomogeneous Neyman-Scott/VarGamma models can also be fitted using the
function kppm and the fitted models can be simulated using simulate.kppm.
The optimisation algorithm can be controlled through the additional arguments "..." which are
passed to the optimisation function optim. For example, to constrain the parameter values to a
certain range, use the argument method="L-BFGS-B" to select an optimisation algorithm that respects box constraints, and use the arguments lower and upper to specify (vectors of) minimum
and maximum values for each parameter.
Value
An object of class "minconfit". There are methods for printing and plotting this object. It contains
the following main components:

vargamma.estpcf

1473

par

Vector of fitted parameter values.

fit

Function value table (object of class "fv") containing the observed values of the
summary statistic (observed) and the theoretical values of the summary statistic
computed from the fitted model parameters.

Author(s)

Abdollah Jalilian and Rasmus Waagepetersen. Adapted for spatstat by Adrian Baddeley .
References
Waagepetersen, R. (2007) Estimating functions for inhomogeneous spatial point processes with
incomplete covariate data. Biometrika 95, 351–363.
See Also
kppm, vcov, vcov.ppm
Examples
data(redwood)
fit <- kppm(redwood ~ x + y)
vcov(fit)
vcov(fit, what="corr")
# confidence interval
confint(fit)
# cross-check the confidence interval by hand:
sd <- sqrt(diag(vcov(fit)))
t(coef(fit) + 1.96 * outer(sd, c(lower=-1, upper=1)))

vcov.mppm

Calculate Variance-Covariance Matrix for Fitted Multiple Point Process Model

Description
Given a fitted multiple point process model, calculate the variance-covariance matrix of the parameter estimates.
Usage
## S3 method for class 'mppm'
vcov(object, ..., what="vcov", err="fatal")
Arguments
object

A multiple point process model (object of class "mppm").

...

Arguments recognised by vcov.ppm.

what

Character string indicating which quantity should be calculated. Options include
"vcov" for the variance-covariance matrix, "corr" for the correlation matrix,
and "fisher" for the Fisher information matrix.

err

Character string indicating what action to take if an error occurs. Either "fatal",
"warn" or "null".

1478

vcov.mppm

Details
This is a method for the generic function vcov.
The argument object should be a fitted multiple point process model (object of class "mppm")
generated by mppm.
The variance-covariance matrix of the parameter estimates is computed using asymptotic theory for
maximum likelihood (for Poisson processes) or estimating equations (for other Gibbs models).
If what="vcov" (the default), the variance-covariance matrix is returned. If what="corr", the
variance-covariance matrix is normalised to yield a correlation matrix, and this is returned. If
what="fisher", the Fisher information matrix is returned instead.
In all three cases, the rows and columns of the matrix correspond to the parameters (coefficients) in
the same order as in coef{model}.
If errors or numerical problems occur, the argument err determines what will happen. If err="fatal"
an error will occur. If err="warn" a warning will be issued and NA will be returned. If err="null",
no warning is issued, but NULL is returned.
Value
A numeric matrix (or NA or NULL).
Error messages
An error message that reports system is computationally singular indicates that the determinant of
the Fisher information matrix of one of the models was either too large or too small for reliable
numerical calculation. See vcov.ppm for suggestions on how to handle this.
Author(s)

Adrian Baddeley, Ida-Maria Sintorn and Leanne Bischoff. Implemented by Adrian Baddeley  and Ege Rubak .
References
Baddeley, A., Rubak, E. and Turner, R. (2015) Spatial Point Patterns: Methodology and Applications with R. London: Chapman and Hall/CRC Press.
See Also
vcov, vcov.ppm, mppm
Examples
fit <- mppm(Wat ~x, data=hyperframe(Wat=waterstriders))
vcov(fit)

vcov.ppm

vcov.ppm

1479

Variance-Covariance Matrix for a Fitted Point Process Model

Description
Returns the variance-covariance matrix of the estimates of the parameters of a fitted point process
model.
Usage
## S3 method for class 'ppm'
vcov(object, ..., what = "vcov", verbose = TRUE,
fine=FALSE,
gam.action=c("warn", "fatal", "silent"),
matrix.action=c("warn", "fatal", "silent"),
logi.action=c("warn", "fatal", "silent"),
hessian=FALSE)
Arguments
object

A fitted point process model (an object of class "ppm".)

...

Ignored.

what

Character string (partially-matched) that specifies what matrix is returned. Options are "vcov" for the variance-covariance matrix, "corr" for the correlation
matrix, and "fisher" or "Fisher" for the Fisher information matrix.

fine

Logical value indicating whether to use a quick estimate (fine=FALSE, the default) or a slower, more accurate estimate (fine=TRUE).

verbose

Logical. If TRUE, a message will be printed if various minor problems are encountered.

gam.action

String indicating what to do if object was fitted by gam.

matrix.action

String indicating what to do if the matrix is ill-conditioned (so that its inverse
cannot be calculated).

logi.action

String indicating what to do if object was fitted via the logistic regression approximation using a non-standard dummy point process.

hessian

Logical. Use the negative Hessian matrix of the log pseudolikelihood instead of
the Fisher information.

Details
This function computes the asymptotic variance-covariance matrix of the estimates of the canonical
parameters in the point process model object. It is a method for the generic function vcov.
object should be an object of class "ppm", typically produced by ppm.
The canonical parameters of the fitted model object are the quantities returned by coef.ppm(object).
The function vcov calculates the variance-covariance matrix for these parameters.
The argument what provides three options:
what="vcov" return the variance-covariance matrix of the parameter estimates
what="corr" return the correlation matrix of the parameter estimates

1480

vcov.ppm

what="fisher" return the observed Fisher information matrix.
In all three cases, the result is a square matrix. The rows and columns of the matrix correspond to
the canonical parameters given by coef.ppm(object). The row and column names of the matrix
are also identical to the names in coef.ppm(object).
For models fitted by the Berman-Turner approximation (Berman and Turner, 1992; Baddeley and
Turner, 2000) to the maximum pseudolikelihood (using the default method="mpl" in the call to
ppm), the implementation works as follows.
• If the fitted model object is a Poisson process, the calculations are based on standard asymptotic theory for the maximum likelihood estimator (Kutoyants, 1998). The observed Fisher information matrix of the fitted model object is first computed, by summing over the BermanTurner quadrature points in the fitted model. The asymptotic variance-covariance matrix is
calculated as the inverse of the observed Fisher information. The correlation matrix is then
obtained by normalising.
• If the fitted model is not a Poisson process (i.e. it is some other Gibbs point process) then
the calculations are based on Coeurjolly and Rubak (2012). A consistent estimator of the
variance-covariance matrix is computed by summing terms over all pairs of data points. If
required, the Fisher information is calculated as the inverse of the variance-covariance matrix.
For models fitted by the Huang-Ogata method (method="ho" in the call to ppm), the implementation
uses the Monte Carlo estimate of the Fisher information matrix that was computed when the original
model was fitted.
For models fitted by the logistic regression approximation to the maximum pseudolikelihood (method="logi"
in the call to ppm), calculations are based on (Baddeley et al., 2013). A consistent estimator of the
variance-covariance matrix is computed by summing terms over all pairs of data points. If required,
the Fisher information is calculated as the inverse of the variance-covariance matrix. In this case the
calculations depend on the type of dummy pattern used, and currently only the types "stratrand",
"binomial" and "poisson" as generated by quadscheme.logi are implemented. For other types
the behavior depends on the argument logi.action. If logi.action="fatal" an error is produced. Otherwise, for types "grid" and "transgrid" the formulas for "stratrand" are used
which in many cases should be conservative. For an arbitrary user specified dummy pattern (type
"given") the formulas for "poisson" are used which in many cases should be conservative. If
logi.action="warn" a warning is issued otherwise the calculation proceeds without a warning.
The argument verbose makes it possible to suppress some diagnostic messages.
The asymptotic theory is not correct if the model was fitted using gam (by calling ppm with use.gam=TRUE).
The argument gam.action determines what to do in this case. If gam.action="fatal", an error
is generated. If gam.action="warn", a warning is issued and the calculation proceeds using the
incorrect theory for the parametric case, which is probably a reasonable approximation in many
applications. If gam.action="silent", the calculation proceeds without a warning.
If hessian=TRUE then the negative Hessian (second derivative) matrix of the log pseudolikelihood,
and its inverse, will be computed. For non-Poisson models, this is not a valid estimate of variance,
but is useful for other calculations.
Note that standard errors and 95% confidence intervals for the coefficients can also be obtained
using confint(object) or coef(summary(object)).
Value
A square matrix.

vcov.ppm

1481

Error messages
An error message that reports system is computationally singular indicates that the determinant of
the Fisher information matrix was either too large or too small for reliable numerical calculation.
If this message occurs, try repeating the calculation using fine=TRUE.
Singularity can occur because of numerical overflow or collinearity in the covariates. To check this,
rescale the coordinates of the data points and refit the model. See the Examples.
In a Gibbs model, a singular matrix may also occur if the fitted model is a hard core process: this is
a feature of the variance estimator.
Author(s)
Original code for Poisson point process was written by Adrian Baddeley 
and Rolf Turner  . New code for stationary Gibbs point processes
was generously contributed by Ege Rubak  and Jean-Francois Coeurjolly.
New code for generic Gibbs process written by Adrian Baddeley .
New code for logistic method contributed by Ege Rubak .
References
Baddeley, A., Coeurjolly, J.-F., Rubak, E. and Waagepetersen, R. (2014) Logistic regression for
spatial Gibbs point processes. Biometrika 101 (2) 377–392.
Coeurjolly, J.-F. and Rubak, E. (2013) Fast covariance estimation for innovations computed from a
spatial Gibbs point process. Scandinavian Journal of Statistics 40 669–684.
Kutoyants, Y.A. (1998) Statistical Inference for Spatial Poisson Processes, Lecture Notes in
Statistics 134. New York: Springer 1998.
See Also
vcov for the generic,
ppm for information about fitted models,
confint for confidence intervals.
Examples
X <- rpoispp(42)
fit <- ppm(X, ~ x + y)
vcov(fit)
vcov(fit, what="Fish")
# example of singular system
m <- ppm(demopat ~polynom(x,y,2))
## Not run:
try(v <- vcov(m))
## End(Not run)
# rescale x, y coordinates to range [0,1] x [0,1] approximately
demopatScale <- rescale(demopat, 10000)
m <- ppm(demopatScale ~ polynom(x,y,2))
v <- vcov(m)
# Gibbs example
fitS <- ppm(swedishpines ~1, Strauss(9))

1482

vcov.slrm
coef(fitS)
sqrt(diag(vcov(fitS)))

vcov.slrm

Variance-Covariance Matrix for a Fitted Spatial Logistic Regression

Description
Returns the variance-covariance matrix of the estimates of the parameters of a point process model
that was fitted by spatial logistic regression.
Usage
## S3 method for class 'slrm'
vcov(object, ...,
what=c("vcov", "corr", "fisher", "Fisher"))
Arguments
object

A fitted point process model of class "slrm".

...

Ignored.

what

Character string (partially-matched) that specifies what matrix is returned. Options are "vcov" for the variance-covariance matrix, "corr" for the correlation
matrix, and "fisher" or "Fisher" for the Fisher information matrix.

Details
This function computes the asymptotic variance-covariance matrix of the estimates of the canonical
parameters in the point process model object. It is a method for the generic function vcov.
object should be an object of class "slrm", typically produced by slrm. It represents a Poisson
point process model fitted by spatial logistic regression.
The canonical parameters of the fitted model object are the quantities returned by coef.slrm(object).
The function vcov calculates the variance-covariance matrix for these parameters.
The argument what provides three options:
what="vcov" return the variance-covariance matrix of the parameter estimates
what="corr" return the correlation matrix of the parameter estimates
what="fisher" return the observed Fisher information matrix.
In all three cases, the result is a square matrix. The rows and columns of the matrix correspond to
the canonical parameters given by coef.slrm(object). The row and column names of the matrix
are also identical to the names in coef.slrm(object).
Note that standard errors and 95% confidence intervals for the coefficients can also be obtained
using confint(object) or coef(summary(object)).
Standard errors for the fitted intensity can be obtained using predict.slrm.
Value
A square matrix.

vertices

1483

Error messages
An error message that reports system is computationally singular indicates that the determinant of
the Fisher information matrix was either too large or too small for reliable numerical calculation.
This can occur because of numerical overflow or collinearity in the covariates.
Author(s)
Adrian Baddeley 
and Rolf Turner  .
References
Baddeley, A., Berman, M., Fisher, N.I., Hardegen, A., Milne, R.K., Schuhmacher, D., Shah, R.
and Turner, R. (2010) Spatial logistic regression and change-of-support for spatial Poisson point
processes. Electronic Journal of Statistics 4, 1151–1201. doi: 10.1214/10-EJS581
See Also
vcov for the generic,
slrm for information about fitted models,
predict.slrm for other kinds of calculation about the model,
confint for confidence intervals.
Examples
X <- rpoispp(42)
fit <- slrm(X ~ x + y)
vcov(fit)
vcov(fit, what="corr")
vcov(fit, what="f")

vertices

Vertices of a Window

Description
Finds the vertices of a window, or similar object.
Usage
vertices(w)
## S3 method for class 'owin'
vertices(w)
Arguments
w

A window (object of class "owin") or similar object.

1484

volume

Details
This function computes the vertices (‘corners’) of a spatial window or other object.
For vertices.owin, the argument w should be a window (an object of class "owin", see owin.object
for details).
If w is a rectangle, the coordinates of the four corner points are returned.
If w is a polygonal window (consisting of one or more polygons), the coordinates of the vertices of
all polygons are returned.
If w is a binary mask, then a ‘boundary pixel’ is defined to be a pixel inside the window which has
at least one neighbour outside the window. The coordinates of the centres of all boundary pixels are
returned.
Value
A list with components x and y giving the coordinates of the vertices.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
owin.object.
Examples
data(letterR)
vert <- vertices(letterR)
plot(letterR, main="Polygonal vertices")
points(vert)
plot(letterR, main="Boundary pixels")
points(vertices(as.mask(letterR)))

volume

Volume of an Object

Description
Computes the volume of a spatial object such as a three-dimensional box.
Usage
volume(x)
Arguments
x

An object whose volume will be computed.

weighted.median

1485

Details
This function computes the volume of an object such as a three-dimensional box.
The function volume is generic, with methods for the classes "box3" (three-dimensional boxes) and
"boxx" (multi-dimensional boxes).
There is also a method for the class "owin" (two-dimensional windows), which is identical to
area.owin, and a method for the class "linnet" of linear networks, which returns the length of
the network.
Value
The numerical value of the volume of the object.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
area.owin, volume.box3, volume.boxx, volume.linnet

weighted.median

Weighted Median, Quantiles or Variance

Description
Compute the median, quantiles or variance of a set of numbers which have weights associated with
them.
Usage
weighted.median(x, w, na.rm = TRUE)
weighted.quantile(x, w, probs=seq(0,1,0.25), na.rm = TRUE)
weighted.var(x, w, na.rm = TRUE)
Arguments
x

Data values. A vector of numeric values, for which the median or quantiles are
required.

w

Weights. A vector of nonnegative numbers, of the same length as x.

probs

Probabilities for which the quantiles should be computed. A numeric vector of
values between 0 and 1.

na.rm

Logical. Whether to ignore NA values.

1486

where.max

Details
The ith observation x[i] is treated as having a weight proportional to w[i].
The weighted median is a value m such that the total weight of data to the left of m is equal to half
the total weight. If there is no such value, linear interpolation is performed.
Value
A numeric value or vector.
Author(s)
Adrian Baddeley .
See Also
quantile, median.
Examples
x <- 1:20
w <- runif(20)
weighted.median(x, w)
weighted.quantile(x, w)
weighted.var(x, w)

where.max

Find Location of Maximum in a Pixel Image

Description
Finds the spatial location(s) where a given pixel image attains its maximum or minimum value.
Usage
where.max(x, first = TRUE)
where.min(x, first = TRUE)
Arguments
x

A pixel image (object of class "im").

first

Logical value. If TRUE (the default), then only one location will be returned. If
FALSE, then all locations where the maximum is achieved will be returned.

Details
This function finds the spatial location or locations where the pixel image x attains its maximum or
minimum value. The result is a point pattern giving the locations.
If first=TRUE (the default), then only one location will be returned, namely the location with the
smallest y coordinate value which attains the maximum or minimum. This behaviour is analogous
to the functions which.min and which.max.
If first=FALSE, then the function returns the locations of all pixels where the maximum (or minimum) value is attained. This could be a large number of points.

whichhalfplane

1487

Value
A point pattern (object of class "ppp").
Author(s)
Adrian Baddeley 
See Also
Summary.im for computing the minimum and maximum of pixel values; eval.im and Math.im
for mathematical expressions involving images; solutionset for finding the set of pixels where a
statement is true.
Examples
D <- distmap(letterR, invert=TRUE)
plot(D)
plot(where.max(D), add=TRUE, pch=16, cols="green")

whichhalfplane

Test Which Side of Infinite Line a Point Falls On

Description
Given an infinite line and a spatial point location, determine which side of the line the point falls
on.
Usage
whichhalfplane(L, x, y = NULL)
Arguments
L

Object of class "infline" specifying one or more infinite straight lines in two
dimensions.

x,y

Arguments acceptable to xy.coords specifying the locations of the points.

Details
An infinite line L divides the two-dimensional plane into two half-planes. This function returns a
matrix M of logical values in which M[i,j] = TRUE if the jth spatial point lies below or to the left
of the ith line.
Value
A logical matrix.
Author(s)
Adrian Baddeley .

1488

whist

See Also
infline
Examples
L <- infline(p=runif(3), theta=runif(3, max=2*pi))
X <- runifpoint(4)
whichhalfplane(L, X)

whist

Weighted Histogram

Description
Computes the weighted histogram of a set of observations with a given set of weights.
Usage
whist(x, breaks, weights = NULL)
Arguments
x

Numeric vector of observed values.

breaks

Vector of breakpoints for the histogram.

weights

Numeric vector of weights for the observed values.

Details
This low-level function computes (but does not plot) the weighted histogram of a vector of observations x using a given vector of weights.
The arguments x and weights should be numeric vectors of equal length. They may include NA or
infinite values.
The argument breaks should be a numeric vector whose entries are strictly increasing. These values
define the boundaries between the successive histogram cells. The breaks do not have to span the
range of the observations.
There are N-1 histogram cells, where N = length(breaks). An observation x[i] falls in the jth
cell if breaks[j] <= x[i] < breaks[j+1] (for j < N-1) or breaks[j] <= x[i] <= breaks[j+1]
(for j = N-1). The weighted histogram value h[j] for the jth cell is the sum of weights[i] for
all observations x[i] that fall in the cell.
Note that, in contrast to the function hist, the function whist does not require the breakpoints to
span the range of the observations x. Values of x that fall outside the range of breaks are handled
separately; their total weight is returned as an attribute of the histogram.
Value
A numeric vector of length N-1 containing the histogram values, where N = length(breaks).
The return value also has attributes "low" and "high" giving the total weight of all observations
that are less than the lowest breakpoint, or greater than the highest breakpoint, respectively.

will.expand

1489

Author(s)
Adrian Baddeley 
and Rolf Turner 
with thanks to Peter Dalgaard.
Examples
x <- rnorm(100)
b <- seq(-1,1,length=21)
w <- runif(100)
whist(x,b,w)

will.expand

Test Expansion Rule

Description
Determines whether an expansion rule will actually expand the window or not.
Usage
will.expand(x)
Arguments
x

Expansion rule. An object of class "rmhexpand".

Details
An object of class "rmhexpand" describes a rule for expanding a simulation window. See rmhexpand
for details.
One possible expansion rule is to do nothing, i.e. not to expand the window.
This command inspects the expansion rule x and determines whether it will or will not actually
expand the window. It returns TRUE if the window will be expanded.
Value
Logical value.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
rmhexpand, expand.owin

1490

Window

Examples
x <- rmhexpand(distance=0.2)
y <- rmhexpand(area=1)
will.expand(x)
will.expand(y)

Window

Extract or Change the Window of a Spatial Object

Description
Given a spatial object (such as a point pattern or pixel image) in two dimensions, these functions
extract or change the window in which the object is defined.
Usage
Window(X, ...)
Window(X, ...) <- value
## S3 method for class 'ppp'
Window(X, ...)
## S3 replacement method for class 'ppp'
Window(X, ...) <- value
## S3 method for class 'psp'
Window(X, ...)
## S3 replacement method for class 'psp'
Window(X, ...) <- value
## S3 method for class 'im'
Window(X, ...)
## S3 replacement method for class 'im'
Window(X, ...) <- value
Arguments
X

A spatial object such as a point pattern, line segment pattern or pixel image.

...

Extra arguments. They are ignored by all the methods listed here.

value

Another window (object of class "owin") to be used as the window for X.

Details
The functions Window and Window<- are generic.
Window(X) extracts the spatial window in which X is defined.
Window(X) <- W changes the window in which X is defined to the new window W, and discards any
data outside W. In particular:

WindowOnly

1491

• If X is a point pattern (object of class "ppp") then Window(X) <- W discards any points of X
which fall outside W.
• If X is a line segment pattern (object of class "psp") then Window(X) <- W clips the segments
of X to the boundaries of W.
• If X is a pixel image (object of class "im") then Window(X) <- W has the effect that pixels
lying outside W are retained but their pixel values are set to NA.
Many other classes of spatial object have a method for Window, but not Window<-. See Window.ppm.
Value
The result of Window is a window (object of class "owin").
The result of Window<- is the updated object X, of the same class as X.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
Window.ppm
Examples
## point patterns
Window(cells)
X <- demopat
Window(X)
Window(X) <- as.rectangle(Window(X))
## line segment patterns
X <- psp(runif(10), runif(10), runif(10), runif(10), window=owin())
Window(X)
Window(X) <- square(0.5)
## images
Z <- setcov(owin())
Window(Z)
Window(Z) <- square(0.5)

WindowOnly

Extract Window of Spatial Object

Description
Given a spatial object (such as a point pattern or pixel image) in two dimensions, these functions
extract the window in which the object is defined.

1492
Usage
## S3 method for class 'ppm'
Window(X, ..., from=c("points", "covariates"))
## S3 method for class 'kppm'
Window(X, ..., from=c("points", "covariates"))
## S3 method for class 'dppm'
Window(X, ..., from=c("points", "covariates"))
## S3 method for class 'lpp'
Window(X, ...)
## S3 method for class 'lppm'
Window(X, ...)
## S3 method for class 'msr'
Window(X, ...)
## S3 method for class 'quad'
Window(X, ...)
## S3 method for class 'quadratcount'
Window(X, ...)
## S3 method for class 'quadrattest'
Window(X, ...)
## S3 method for class 'tess'
Window(X, ...)
## S3 method for class 'layered'
Window(X, ...)
## S3 method for class 'distfun'
Window(X, ...)
## S3 method for class 'nnfun'
Window(X, ...)
## S3 method for class 'funxy'
Window(X, ...)
## S3 method for class 'rmhmodel'
Window(X, ...)
## S3 method for class 'leverage.ppm'
Window(X, ...)
## S3 method for class 'influence.ppm'
Window(X, ...)

WindowOnly

with.fv

1493

Arguments
X

A spatial object.

...

Ignored.

from

Character string. See Details.

Details
These are methods for the generic function Window which extract the spatial window in which the
object X is defined.
The argument from applies when X is a fitted point process model (object of class "ppm", "kppm"
or "dppm"). If from="data" (the default), Window extracts the window of the original point pattern
data to which the model was fitted. If from="covariates" then Window returns the window in
which the spatial covariates of the model were provided.
Value
An object of class "owin" (see owin.object) specifying an observation window.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 
See Also
Window, Window.ppp, Window.psp.
owin.object
Examples
X <- quadratcount(cells, 4)
Window(X)

with.fv

Evaluate an Expression in a Function Table

Description
Evaluate an R expression in a function value table (object of class "fv").
Usage
## S3 method for class 'fv'
with(data, expr, ..., fun = NULL, enclos=NULL)

1494

with.fv

Arguments
data

A function value table (object of class "fv") in which the expression will be
evaluated.

expr

The expression to be evaluated. An R language expression, which may involve
the names of columns in data, the special abbreviations ., .x and .y, and global
constants or functions.

...

Ignored.

fun

Logical value, specifying whether the result should be interpreted as another
function (fun=TRUE) or simply returned as a numeric vector or array (fun=FALSE).
See Details.

enclos

An environment in which to search for variables that are not found in data.
Defaults to parent.frame().

Details
This is a method for the generic command with for an object of class "fv" (function value table).
An object of class "fv" is a convenient way of storing and plotting several different estimates of
the same function. It is effectively a data frame with extra attributes. See fv.object for further
explanation.
This command makes it possible to perform computations that involve different estimates of the
same function. For example we use it to compute the arithmetic difference between two different
edge-corrected estimates of the K function of a point pattern.
The argument expr should be an R language expression. The expression may involve
• the name of any column in data, referring to one of the estimates of the function;
• the symbol . which stands for all the available estimates of the function;
• the symbol .y which stands for the recommended estimate of the function (in an "fv" object,
one of the estimates is always identified as the recommended estimate);
• the symbol .x which stands for the argument of the function;
• global constants or functions.
See the Examples. The expression should be capable of handling vectors and matrices.
The interpretation of the argument fun is as follows:
• If fun=FALSE, the result of evaluating the expression expr will be returned as a numeric vector,
matrix or data frame.
• If fun=TRUE, then the result of evaluating expr will be interpreted as containing the values of
a new function. The return value will be an object of class "fv". (This can only happen if the
result has the right dimensions.)
• The default is fun=TRUE if the result of evaluating expr has more than one column, and
fun=FALSE otherwise.
To perform calculations involving several objects of class "fv", use eval.fv.
Value
A function value table (object of class "fv") or a numeric vector or data frame.

with.hyperframe

1495

Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
with, fv.object, eval.fv, Kest
Examples
# compute 4 estimates of the K function
X <- rpoispp(42)
K <- Kest(X)
plot(K)
# derive 4 estimates of the L function L(r) = sqrt(K(r)/pi)
L <- with(K, sqrt(./pi))
plot(L)
# compute 4 estimates of V(r) = L(r)/r
V <- with(L, ./.x)
plot(V)
# compute the maximum absolute difference between
# the isotropic and translation correction estimates of K(r)
D <- with(K, max(abs(iso - trans)))

with.hyperframe

Evaluate an Expression in Each Row of a Hyperframe

Description
An expression, involving the names of columns in a hyperframe, is evaluated separately for each
row of the hyperframe.
Usage
## S3 method for class 'hyperframe'
with(data, expr, ...,
simplify = TRUE,
ee = NULL, enclos=NULL)
Arguments
data
expr
...
simplify
ee
enclos

A hyperframe (object of class "hyperframe") containing data.
An R language expression to be evaluated.
Ignored.
Logical. If TRUE, the return value will be simplified to a vector whenever possible.
Alternative form of expr, as an object of class "expression".
An environment in which to search for objects that are not found in the hyperframe. Defaults to parent.frame().

1496

with.msr

Details
This function evaluates the expression expr in each row of the hyperframe data. It is a method for
the generic function with.
The argument expr should be an R language expression in which each variable name is either
the name of a column in the hyperframe data, or the name of an object in the parent frame (the
environment in which with was called.) The argument ee can be used as an alternative to expr and
should be an expression object (of class "expression").
For each row of data, the expression will be evaluated so that variables which are column names
of data are interpreted as the entries for those columns in the current row.
For example, if a hyperframe h has columns called A and B, then with(h, A != B) inspects each
row of data in turn, tests whether the entries in columns A and B are equal, and returns the n logical
values.
Value
Normally a list of length n (where n is the number of rows) containing the results of evaluating the
expression for each row. If simplify=TRUE and each result is a single atomic value, then the result
is a vector or factor containing the same values.
Author(s)
Adrian Baddeley 
and Rolf Turner 
See Also
hyperframe, plot.hyperframe
Examples
# generate Poisson point patterns with intensities 10 to 100
H <- hyperframe(L=seq(10,100, by=10))
X <- with(H, rpoispp(L))

with.msr

Evaluate Expression Involving Components of a Measure

Description
An expression involving the names of components of a measure is evaluated.
Usage
## S3 method for class 'msr'
with(data, expr, ...)
Arguments
data

A measure (object of class "msr").

expr

An expression to be evaluated.

...

Ignored.

with.msr

1497

Details
This is a method for the generic function with for the class "msr". The argument data should be
an object of class "msr" representing a measure (a function which assigns a value to each subset of
two-dimensional space).
This function can be used to extract the components of the measure, or to perform more complicated
manipulations of the components.
The argument expr should be an un-evaluated expression in the R language. The expression may
involve any of the variable names listed below with their corresponding meanings.
qlocations
qweights
density
discrete
continuous
increment
is.atom
atoms
atommass

(point pattern) all quadrature locations
(numeric) all quadrature weights
(numeric) density value at each quadrature point
(numeric) discrete mass at each quadrature point
(numeric) increment of continuous component
(numeric) increment of measure
(logical) whether quadrature point is an atom
(point pattern) locations of atoms
(numeric) massess of atoms

The measure is the sum of discrete and continuous components. The discrete component assigns
non-zero mass to several points called atoms. The continuous component has a density which
should be integrated over a region to determine the value for that region.
An object of class "msr" approximates the continuous component by a sum over quadrature points.
The quadrature points are chosen so that they include the atoms of the measure. In the list above,
we have increment = continuous + discrete, continuous = density * qweights,
is.atom = (discrete > 0), atoms = qlocations[is.atom] and atommass = discrete[is.atom].
Value
The result of evaluating the expression could be an object of any kind.
Author(s)
Adrian Baddeley , Rolf Turner 
and Ege Rubak .
See Also
msr, split.msr
Examples
X <- rpoispp(function(x,y) { exp(3+3*x) })
fit <- ppm(X, ~x+y)
rp <- residuals(fit, type="pearson")
with(rp, atoms)
with(rp, qlocations %mark% continuous)

1498

with.ssf

with.ssf

Evaluate Expression in a Spatially Sampled Function

Description
Given a spatially sampled function, evaluate an expression involving the function values.
Usage
apply.ssf(X, ...)
## S3 method for class 'ssf'
with(data, ...)
Arguments
X, data

A spatially sampled function (object of class "ssf").

...

Arguments passed to with.default or apply specifying what to compute.

Details
An object of class "ssf" represents a function (real- or vector-valued) that has been sampled at a
finite set of points. It contains a data frame which provides the function values at the sample points.
In with.ssf, the expression specified by ... will be evaluated in this dataframe. In apply.ssf,
the dataframe will be subjected to the apply operator using the additional arguments ....
If the result of evaluation is a data frame with one row for each data point, or a numeric vector
with one entry for each data point, then the result will be an object of class "ssf" containing this
information. Otherwise, the result will be a numeric vector.
Value
An object of class "ssf" or a numeric vector.
Author(s)
Adrian Baddeley .
See Also
ssf
Examples
a <- ssf(cells, data.frame(d=nndist(cells), i=1:npoints(cells)))
with(a, i/d)

yardstick

yardstick

1499

Text, Arrow or Scale Bar in a Diagram

Description
Create spatial objects that represent a text string, an arrow, or a yardstick (scale bar).
Usage
textstring(x, y, txt = NULL, ...)
onearrow(x0, y0, x1, y1, txt = NULL, ...)
yardstick(x0, y0, x1, y1, txt = NULL, ...)
Arguments
x,y

Coordinates where the text should be placed.

x0,y0,x1,y1

Spatial coordinates of both ends of the arrow or yardstick. Alternatively x0 can
be a point pattern (class "ppp") containing exactly two points, or a line segment
pattern (class "psp") consisting of exactly one line segment.

txt

The text to be displayed beside the line segment. Either a character string or an
expression.

...

Additional named arguments for plotting the object.

Details
These commands create objects that represent components of a diagram:
• textstring creates an object that represents a string of text at a particular spatial location.
• onearrow creates an object that represents an arrow between two locations.
• yardstick creates an object that represents a scale bar: a line segment indicating the scale of
the plot.
To display the relevant object, it should be plotted, using plot. See the help files for the plot
methods plot.textstring, plot.onearrow and plot.yardstick.
These objects are designed to be included as components in a layered object or a solist. This
makes it possible to build up a diagram consisting of many spatial objects, and to annotate the
diagram with arrows, text and so on, so that ultimately the entire diagram is plotted using plot.
Value
An object of class "diagramobj" which also belongs to one of the special classes "textstring",
"onearrow" or "yardstick". There are methods for plot, print, "[" and shift.
Author(s)
Adrian Baddeley 
Rolf Turner 
and Ege Rubak 

1500

zapsmall.im

See Also
plot.textstring, plot.onearrow, plot.yardstick.
Examples
X <- rescale(swedishpines)
plot(X, pch=16, main="")
ys <- yardstick(as.psp(list(xmid=4, ymid=0.5, length=1, angle=0),
window=Window(X)),
txt="1 m")
plot(ys, angle=90)

zapsmall.im

Rounding of Pixel Values

Description
Modifies a pixel image, identifying those pixels that have values very close to zero, and replacing
the value by zero.
Usage
zapsmall.im(x, digits)
Arguments
x

Pixel image (object of class "im").

digits

Argument passed to zapsmall indicating the precision to be used.

Details
The function zapsmall is applied to each pixel value of the image x.
Value
Another pixel image.
Author(s)
Ege Rubak  and Adrian Baddeley 
See Also
zapsmall
Examples
data(cells)
D <- density(cells)
zapsmall.im(D)

zclustermodel

1501

zclustermodel

Cluster Point Process Model

Description
Experimental code. Creates an object representing a cluster point process model. Typically used
for theoretical calculations about such a model.

Usage
zclustermodel(name = "Thomas", ..., mu, kappa, scale)
Arguments
name

Name of the cluster process. One of "Thomas", "MatClust", "VarGamma" or
"Cauchy".

...

Other arguments needed for the model.

mu

Mean cluster size. A single number, or a pixel image.

kappa

Parent intensity. A single number.

scale

Cluster scale parameter of the model.

Details
Experimental.

Value
Object of the experimental class "zclustermodel".
Author(s)
Adrian Baddeley 
See Also
methods.zclustermodel
Examples
m <- zclustermodel("Thomas", kappa=10, mu=5, scale=0.1)

1502

[.ssf

[.ssf

Subset of spatially sampled function

Description
Extract a subset of the data for a spatially sampled function.
Usage
## S3 method for class 'ssf'
x[i, j, ..., drop]
Arguments
x

Object of class "ssf".

i

Subset index applying to the locations where the function is sampled.

j

Subset index applying to the columns (variables) measured at each location.

..., drop

Ignored.

Details
This is the subset operator for the class "ssf".
Value
Another object of class "ssf".
Author(s)
Adrian Baddeley .
See Also
ssf, with.ssf
Examples
f <- ssf(cells, data.frame(d=nndist(cells), i=1:42))
f
f[1:10,]
f[ ,1]

Index
default.dummy, 276
default.expand, 277
default.rmhcontrol, 279
disc, 329
discs, 333
ellipse, 377
gridcentres, 503
gridweights, 504
hextess, 520
im, 540
infline, 549
linequad, 717
lintess, 723
owin, 890
pixelquad, 951
pp3, 1039
ppp, 1060
pppmatching, 1068
ppx, 1072
psp, 1100
quadratresample, 1125
quadrats, 1126
quadscheme, 1127
quadscheme.logi, 1129
quasirandom, 1136
rags, 1137
ragsAreaInter, 1138
ragsMultiHard, 1140
rCauchy, 1145
rcell, 1147
rcellnumber, 1149
rDGS, 1150
rDiggleGratton, 1151
rdpp, 1153
regularpolygon, 1159
rGaussPoisson, 1186
rgbim, 1187
rHardcore, 1188
rjitter, 1197
rlabel, 1199
rLGCP, 1200
rlinegrid, 1202
rlpp, 1203

∗Topic IO
scanpp, 1322
∗Topic algebra
dimhat, 324
matrixpower, 780
subspaceDistance, 1398
∗Topic arith
polynom, 1031
whist, 1488
∗Topic array
dimhat, 324
matrixpower, 780
sumouter, 1412
∗Topic attribute
bind.fv, 149
fasp.object, 446
fv.object, 476
im.object, 543
owin.object, 893
ppm.object, 1046
ppp.object, 1063
pppmatching.object, 1069
psp.object, 1101
quad.object, 1113
∗Topic character
begins, 146
∗Topic classes
fv, 473
∗Topic classif
clusterset, 215
nnclean, 843
∗Topic color
beachcolours, 144
colourmap, 222
colouroutputs, 224
colourtools, 225
interp.colourmap, 568
plot.colourmap, 958
tweak.colourmap, 1449
∗Topic datagen
box3, 160
boxx, 161
clickjoin, 198
1503

1504

INDEX
rMatClust, 1204
rMaternI, 1207
rMaternII, 1208
rmh, 1209
rmh.default, 1210
rmh.ppm, 1220
rmhcontrol, 1223
rmhexpand, 1227
rmhmodel, 1229
rmhmodel.default, 1230
rmhmodel.list, 1237
rmhmodel.ppm, 1239
rmhstart, 1241
rMosaicField, 1242
rMosaicSet, 1243
rmpoint, 1244
rmpoispp, 1248
rNeymanScott, 1251
rnoise, 1254
rPenttinen, 1268
rpoint, 1270
rpoisline, 1271
rpoislinetess, 1272
rpoislpp, 1273
rpoispp, 1274
rpoispp3, 1276
rpoisppOnLines, 1277
rpoisppx, 1279
rPoissonCluster, 1280
rQuasi, 1283
rshift, 1284
rshift.ppp, 1285
rshift.psp, 1287
rshift.splitppp, 1289
rSSI, 1290
rstrat, 1292
rStrauss, 1293
rStraussHard, 1295
rsyst, 1296
rtemper, 1297
rthin, 1299
rThomas, 1300
runifdisc, 1305
runiflpp, 1306
runifpoint, 1307
runifpoint3, 1308
runifpointOnLines, 1309
runifpointx, 1310
rVarGamma, 1311
simulate.dppm, 1343
spokes, 1382
square, 1383

ssf, 1384
stratrand, 1387
tess, 1419
∗Topic data
sessionLibs, 1330
∗Topic distribution
dmixpois, 345
rknn, 1198
∗Topic documentation
beginner, 145
bugfixes, 163
foo, 465
latest.news, 671
∗Topic environment
requireversion, 1171
∗Topic fit model
improve.kppm, 545
∗Topic hplot
add.texture, 50
contour.im, 244
contour.imlist, 246
dg.envelope, 300
diagnose.ppm, 309
envelope, 384
envelope.envelope, 393
envelope.pp3, 398
invoke.symbolmap, 573
iplot, 574
istat, 597
layered, 672
layerplotargs, 673
lurking, 745
lurking.mppm, 749
methods.objsurf, 803
pairs.im, 905
pairs.linim, 907
panel.contour, 911
persp.im, 942
perspPoints, 944
plot.anylist, 952
plot.bermantest, 955
plot.cdftest, 957
plot.colourmap, 958
plot.envelope, 961
plot.fasp, 962
plot.fv, 964
plot.hyperframe, 967
plot.im, 969
plot.imlist, 974
plot.laslett, 978
plot.layered, 979
plot.lintess, 985

INDEX
plot.listof, 986
plot.lpp, 989
plot.mppm, 991
plot.msr, 992
plot.onearrow, 994
plot.owin, 995
plot.plotppm, 998
plot.pp3, 999
plot.ppm, 1001
plot.ppp, 1003
plot.psp, 1007
plot.quad, 1009
plot.quadratcount, 1010
plot.quadrattest, 1012
plot.rppm, 1013
plot.slrm, 1015
plot.solist, 1016
plot.splitppp, 1019
plot.ssf, 1020
plot.symbolmap, 1021
plot.tess, 1023
plot.textstring, 1024
plot.texturemap, 1025
plot.yardstick, 1026
points.lpp, 1027
pool.envelope, 1033
pool.fasp, 1035
pool.fv, 1036
qqplot.ppm, 1109
rose, 1256
symbolmap, 1417
text.ppp, 1422
texturemap, 1423
textureplot, 1424
transmat, 1440
update.symbolmap, 1464
yardstick, 1499
∗Topic htest
berman.test, 147
bits.test, 151
cdf.test, 183
cdf.test.mppm, 187
clarkevans.test, 195
dclf.progress, 269
dclf.sigtrace, 271
dclf.test, 273
dg.envelope, 300
dg.progress, 302
dg.sigtrace, 304
dg.test, 307
envelope, 384
envelope.envelope, 393

1505
envelope.pp3, 398
hopskel, 529
plot.quadrattest, 1012
plot.scan.test, 1014
pool.envelope, 1033
pool.fasp, 1035
pool.fv, 1036
pool.quadrattest, 1037
quadrat.test, 1116
quadrat.test.mppm, 1119
quadrat.test.splitppp, 1121
scan.test, 1318
scanLRTS, 1320
segregation.test, 1327
studpermu.test, 1392
∗Topic iplot
clickbox, 196
clickdist, 197
clicklpp, 199
clickpoly, 200
clickppp, 201
identify.ppp, 534
identify.psp, 535
run.simplepanel, 1302
simplepanel, 1339
transect.im, 1439
∗Topic iteration
applynbd, 78
dg.envelope, 300
envelope, 384
envelope.envelope, 393
envelope.pp3, 398
envelopeArray, 401
pool.envelope, 1033
pool.fasp, 1035
pool.fv, 1036
∗Topic list
anylist, 75
as.solist, 134
Extract.anylist, 415
Extract.solist, 440
solapply, 1364
solist, 1365
∗Topic manip
[.ssf, 1502
anylist, 75
append.psp, 77
as.box3, 87
as.data.frame.envelope, 89
as.data.frame.hyperframe, 90
as.data.frame.ppp, 93
as.data.frame.psp, 94

1506

INDEX
as.function.im, 97
as.function.leverage.ppm, 98
as.function.owin, 99
as.function.tess, 100
as.fv, 101
as.hyperframe, 102
as.hyperframe.ppx, 104
as.im, 105
as.layered, 110
as.linfun, 112
as.linim, 113
as.linnet.linim, 114
as.linnet.psp, 116
as.mask, 118
as.mask.psp, 120
as.owin, 123
as.polygonal, 126
as.ppp, 129
as.psp, 131
as.rectangle, 133
as.solist, 134
as.tess, 135
by.im, 175
by.ppp, 176
cbind.hyperframe, 181
collapse.fv, 221
commonGrid, 227
compatible, 230
compatible.fasp, 231
compatible.fv, 231
compatible.im, 232
concatxy, 236
connected.linnet, 241
connected.lpp, 242
coords, 251
crossing.linnet, 261
crossing.psp, 262
data.ppm, 268
delaunay, 280
delaunayDistance, 281
delaunayNetwork, 282
deletebranch, 283
dirichlet, 325
dirichletAreas, 326
dirichletVertices, 327
discretise, 331
divide.linnet, 343
domain, 346
edges, 369
edges2triangles, 370
edges2vees, 371
edit.hyperframe, 372

edit.ppp, 373
endpoints.psp, 382
eval.fasp, 405
eval.fv, 407
eval.im, 409
eval.linim, 410
expand.owin, 414
Extract.anylist, 415
Extract.fasp, 416
Extract.fv, 417
Extract.hyperframe, 419
Extract.im, 420
Extract.influence.ppm, 423
Extract.layered, 424
Extract.leverage.ppm, 426
Extract.linim, 427
Extract.linnet, 428
Extract.listof, 429
Extract.lpp, 430
Extract.msr, 431
Extract.owin, 432
Extract.ppp, 433
Extract.ppx, 436
Extract.psp, 437
Extract.quad, 439
Extract.solist, 440
Extract.splitppp, 441
Extract.tess, 442
Frame, 469
fvnames, 477
grow.boxx, 505
grow.rectangle, 506
harmonise, 510
harmonise.fv, 511
harmonise.im, 512
harmonise.msr, 513
harmonise.owin, 514
headtail, 517
hyperframe, 533
im, 540
im.apply, 542
insertVertices, 552
interp.im, 569
is.convex, 580
is.dppm, 581
is.empty, 581
is.im, 584
is.lpp, 584
is.marked, 585
is.marked.ppm, 586
is.marked.ppp, 587
is.multitype, 588

INDEX
is.multitype.ppm, 589
is.multitype.ppp, 590
is.owin, 591
is.ppm, 592
is.ppp, 593
is.rectangle, 593
laslett, 669
levelset, 685
lineardirichlet, 694
lixellate, 724
lut, 751
marks, 760
marks.psp, 762
marks.tess, 763
mergeLevels, 785
nearest.raster.point, 840
nestsplit, 842
nobjects, 878
npoints, 880
nsegments, 881
nvertices, 882
padimage, 894
periodify, 940
pixelcentres, 945
pixellate, 946
pixellate.owin, 947
pixellate.ppp, 948
pixellate.psp, 950
pointsOnLines, 1028
PPversion, 1071
quad.ppm, 1114
quantess, 1131
raster.x, 1143
rat, 1144
relevel.im, 1160
Replace.im, 1168
Replace.linim, 1170
rescue.rectangle, 1178
rgbim, 1187
rotate.infline, 1260
round.ppp, 1266
selfcrossing.psp, 1328
selfcut.psp, 1329
shift, 1333
shift.im, 1334
shift.owin, 1335
shift.ppp, 1336
shift.psp, 1337
solapply, 1364
solist, 1365
solutionset, 1366
split.hyperframe, 1374

1507
split.im, 1375
split.msr, 1376
split.ppp, 1378
split.ppx, 1380
stienen, 1386
subset.hyperframe, 1395
subset.ppp, 1396
superimpose, 1413
superimpose.lpp, 1416
thinNetwork, 1425
tile.areas, 1431
tile.lengths, 1432
tileindex, 1433
tilenames, 1434
tiles, 1435
tiles.empty, 1436
transect.im, 1439
transmat, 1440
treeprune, 1442
triangulate.owin, 1444
trim.rectangle, 1445
union.quad, 1450
unitname, 1452
unmark, 1454
unstack.msr, 1456
unstack.ppp, 1457
whichhalfplane, 1487
will.expand, 1489
Window, 1490
WindowOnly, 1491
with.fv, 1493
with.hyperframe, 1495
with.msr, 1496
with.ssf, 1498
∗Topic math
affine, 54
affine.im, 54
affine.linnet, 55
affine.lpp, 57
affine.owin, 58
affine.ppp, 59
affine.psp, 61
affine.tess, 62
angles.psp, 67
area.owin, 80
areaGain, 82
areaLoss, 86
as.lpp, 117
bc.ppm, 140
bdist.pixels, 141
bdist.points, 142
bdist.tiles, 143

1508

INDEX
border, 154
boundingcircle, 158
branchlabelfun, 162
centroid.owin, 189
chop.tess, 191
clip.infline, 202
closepairs, 203
closepairs.pp3, 205
closetriples, 207
closing, 208
complement.owin, 235
connected, 239
connected.ppp, 243
convolve.im, 250
covering, 253
crossdist, 254
crossdist.default, 255
crossdist.lpp, 256
crossdist.pp3, 257
crossdist.ppp, 258
crossdist.ppx, 259
crossdist.psp, 260
deltametric, 284
deriv.fv, 295
diameter, 313
diameter.box3, 314
diameter.boxx, 316
diameter.linnet, 317
diameter.owin, 318
dilated.areas, 321
dilation, 322
dirichletAreas, 326
dirichletVertices, 327
discpartarea, 330
distcdf, 334
distfun, 335
distfun.lpp, 337
distmap, 338
distmap.owin, 339
distmap.ppp, 341
distmap.psp, 342
dummify, 362
eroded.areas, 402
erosion, 403
erosionAny, 404
fardist, 445
flipxy, 463
funxy, 472
gauss.hermite, 480
has.close, 515
imcov, 544
incircle, 547

increment.fv, 548
inside.boxx, 554
inside.owin, 555
integral.im, 556
integral.linim, 557
integral.msr, 558
intersect.owin, 570
intersect.tess, 572
is.connected, 578
is.connected.ppp, 579
is.subset.owin, 596
LambertW, 668
lengths.psp, 681
linfun, 718
matchingdist, 769
maxnndist, 781
measureVariation, 784
methods.linfun, 795
methods.linim, 796
midpoints.psp, 814
MinkowskiSum, 817
nearestsegment, 841
nncross, 848
nncross.lpp, 850
nncross.pp3, 852
nndist, 856
nndist.pp3, 860
nndist.ppx, 861
nndist.psp, 863
nnfromvertex, 864
nnfun, 865
nnfun.lpp, 866
nnmap, 867
nnwhich, 872
nnwhich.pp3, 876
nnwhich.ppx, 877
opening, 884
overlap.owin, 889
pairdist, 895
pairdist.default, 896
pairdist.pp3, 898
pairdist.ppp, 899
pairdist.ppx, 900
pairdist.psp, 901
perimeter, 939
pppdist, 1065
project2segment, 1095
project2set, 1096
quadratcount, 1122
range.fv, 1142
reflect, 1158
rescale, 1172

INDEX
rescale.im, 1173
rescale.owin, 1174
rescale.ppp, 1176
rescale.psp, 1177
rex, 1185
rotate, 1258
rotate.im, 1259
rotate.owin, 1261
rotate.ppp, 1262
rotate.psp, 1263
rotmean, 1264
rounding, 1267
scalardilate, 1316
setcov, 1331
sidelengths.owin, 1338
simplify.owin, 1342
stieltjes, 1385
stienen, 1386
test.crossing.psp, 1421
treebranchlabels, 1441
vertices, 1483
volume, 1484
weighted.median, 1485
where.max, 1486
∗Topic methods
adaptive.density, 49
anova.lppm, 68
anova.mppm, 70
anova.ppm, 72
anova.slrm, 74
anyNA.im, 76
as.data.frame.im, 91
as.data.frame.owin, 92
as.data.frame.tess, 95
as.function.fv, 96
as.matrix.im, 121
as.matrix.owin, 122
bw.diggle, 164
bw.frac, 165
bw.pcf, 166
bw.ppl, 168
bw.relrisk, 170
bw.scott, 171
bw.smoothppp, 172
bw.stoyan, 174
by.im, 175
by.ppp, 176
coef.mppm, 217
coef.ppm, 218
coef.slrm, 220
cut.im, 263
cut.lpp, 264

1509
cut.ppp, 266
density.lpp, 286
density.ppp, 288
density.psp, 292
density.splitppp, 294
dkernel, 344
duplicated.ppp, 364
fitted.lppm, 456
fitted.ppm, 459
fitted.slrm, 461
fixef.mppm, 462
formula.fv, 466
formula.ppm, 467
hist.funxy, 527
hist.im, 528
idw, 536
kernel.factor, 628
kernel.moment, 629
kernel.squint, 630
logLik.slrm, 739
Math.im, 775
Math.imlist, 776
Math.linim, 778
mean.im, 782
mean.linim, 783
methods.box3, 786
methods.boxx, 788
methods.dppm, 789
methods.fii, 790
methods.funxy, 791
methods.kppm, 792
methods.layered, 793
methods.linnet, 798
methods.lpp, 800
methods.pp3, 804
methods.rho2hat, 806
methods.rhohat, 807
methods.slrm, 809
methods.ssf, 810
methods.unitname, 812
nndensity.ppp, 855
nnmark, 869
Ops.msr, 885
predict.slrm, 1084
quantile.density, 1133
quantile.im, 1135
ranef.mppm, 1141
relrisk.ppp, 1165
residuals.dppm, 1179
residuals.kppm, 1180
residuals.ppm, 1182
scaletointerval, 1317

1510
Smooth, 1354
Smooth.ppp, 1358
Smooth.ssf, 1360
Smoothfun.ppp, 1361
split.im, 1375
split.ppp, 1378
split.ppx, 1380
summary.anylist, 1401
summary.im, 1402
summary.kppm, 1403
summary.listof, 1404
summary.owin, 1405
summary.ppm, 1406
summary.ppp, 1407
summary.psp, 1408
summary.quad, 1409
summary.solist, 1410
summary.splitppp, 1411
unique.ppp, 1451
update.ppm, 1461
update.rmhcontrol, 1463
vcov.kppm, 1476
vcov.mppm, 1477
vcov.ppm, 1479
vcov.slrm, 1482
zapsmall.im, 1500
∗Topic models
addvar, 51
anova.lppm, 68
anova.mppm, 70
anova.ppm, 72
anova.slrm, 74
AreaInter, 83
as.interact, 109
as.ppm, 127
BadGey, 138
bc.ppm, 140
cauchy.estK, 177
cauchy.estpcf, 179
clusterfit, 211
coef.mppm, 217
coef.ppm, 218
coef.slrm, 220
compareFit, 228
Concom, 237
data.ppm, 268
detpointprocfamilyfun, 296
dfbetas.ppm, 299
diagnose.ppm, 309
DiggleGatesStibbard, 319
DiggleGratton, 320
dim.detpointprocfamily, 324

INDEX
dppeigen, 352
dppkernel, 354
dppm, 354
dppparbounds, 359
dppspecden, 360
dppspecdenrange, 361
dummy.ppm, 363
eem, 374
effectfun, 375
emend, 380
emend.ppm, 381
exactMPLEstrauss, 413
Fiksel, 451
fitin.ppm, 455
fitted.lppm, 456
fitted.mppm, 457
fitted.ppm, 459
fitted.slrm, 461
fixef.mppm, 462
Gcom, 481
Geyer, 493
Gres, 502
Hardcore, 507
harmonic, 509
HierHard, 522
hierpair.family, 523
HierStrauss, 524
HierStraussHard, 525
Hybrid, 531
hybrid.family, 532
influence.ppm, 550
inforder.family, 552
intensity, 560
intensity.dppm, 561
intensity.ppm, 562
ippm, 576
is.dppm, 581
is.hybrid, 582
is.marked.ppm, 586
is.multitype.ppm, 589
is.ppm, 592
is.stationary, 594
Kcom, 613
Kmodel, 647
Kmodel.kppm, 649
Kmodel.ppm, 650
kppm, 657
Kres, 662
LennardJones, 682
leverage.ppm, 686
lgcp.estK, 688
lgcp.estpcf, 691

INDEX
logLik.dppm, 732
logLik.kppm, 733
logLik.mppm, 735
logLik.ppm, 737
logLik.slrm, 739
lppm, 744
lurking, 745
lurking.mppm, 749
matclust.estK, 770
matclust.estpcf, 772
methods.lppm, 801
methods.zclustermodel, 813
mincontrast, 815
model.depends, 820
model.frame.ppm, 821
model.images, 823
model.matrix.mppm, 825
model.matrix.ppm, 826
model.matrix.slrm, 828
mppm, 829
msr, 832
MultiHard, 834
MultiStrauss, 837
MultiStraussHard, 838
objsurf, 883
Ord, 886
ord.family, 888
OrdThresh, 888
PairPiece, 904
pairsat.family, 908
Pairwise, 909
pairwise.family, 910
parameters, 912
parres, 913
Penttinen, 938
plot.dppm, 960
plot.influence.ppm, 975
plot.kppm, 976
plot.leverage.ppm, 980
plot.lppm, 990
plot.mppm, 991
plot.plotppm, 998
plot.ppm, 1001
plot.rppm, 1013
plot.slrm, 1015
Poisson, 1029
ppm, 1040
ppm.ppp, 1048
ppmInfluence, 1058
predict.dppm, 1073
predict.kppm, 1074
predict.lppm, 1075

1511
predict.mppm, 1076
predict.ppm, 1078
predict.rppm, 1083
predict.slrm, 1084
print.ppm, 1087
profilepl, 1091
prune.rppm, 1097
pseudoR2, 1098
psib, 1099
psst, 1102
psstA, 1104
psstG, 1107
qqplot.ppm, 1109
quad.ppm, 1114
ranef.mppm, 1141
rdpp, 1153
reach, 1154
reach.dppm, 1156
relrisk.ppm, 1163
residuals.dppm, 1179
residuals.kppm, 1180
residuals.mppm, 1181
residuals.ppm, 1182
rho2hat, 1190
rhohat, 1191
rmh.ppm, 1220
rppm, 1282
SatPiece, 1313
Saturated, 1315
simulate.dppm, 1343
simulate.kppm, 1345
simulate.lppm, 1347
simulate.mppm, 1348
simulate.ppm, 1349
simulate.slrm, 1351
slrm, 1352
Smooth.msr, 1356
Softcore, 1362
Strauss, 1389
StraussHard, 1390
subfits, 1394
suffstat, 1399
summary.kppm, 1403
summary.ppm, 1406
thomas.estK, 1427
thomas.estpcf, 1429
triplet.family, 1446
Triplets, 1446
update.detpointprocfamily, 1458
update.interact, 1459
update.kppm, 1460
update.ppm, 1461

1512
update.rmhcontrol, 1463
valid, 1465
valid.detpointprocfamily, 1466
valid.ppm, 1467
varcount, 1470
vargamma.estK, 1471
vargamma.estpcf, 1473
vcov.kppm, 1476
vcov.mppm, 1477
vcov.ppm, 1479
vcov.slrm, 1482
zclustermodel, 1501
∗Topic multivariate
dimhat, 324
sdr, 1323
subspaceDistance, 1398
∗Topic nonparametric
allstats, 63
alltypes, 64
blur, 153
CDF, 182
circdensity, 192
clarkevans, 193
clarkevans.test, 195
compileK, 233
deriv.fv, 295
dkernel, 344
edge.Ripley, 365
edge.Trans, 367
Emark, 378
envelopeArray, 401
ewcdf, 412
F3est, 443
Fest, 447
Finhom, 453
FmultiInhom, 464
fryplot, 470
G3est, 478
Gcross, 484
Gdot, 487
Gest, 490
Gfox, 494
Ginhom, 496
Gmulti, 498
GmultiInhom, 500
Hest, 518
hopskel, 529
Iest, 538
increment.fv, 548
intensity.lpp, 561
intensity.ppp, 564
intensity.psp, 566

INDEX
intensity.quadratcount, 567
Jcross, 598
Jdot, 600
Jest, 603
Jinhom, 606
Jmulti, 608
K3est, 610
kaplan.meier, 611
Kcross, 616
Kcross.inhom, 618
Kdot, 622
Kdot.inhom, 625
kernel.factor, 628
kernel.moment, 629
kernel.squint, 630
Kest, 631
Kest.fft, 635
Kinhom, 636
km.rs, 641
Kmark, 642
Kmeasure, 644
Kmulti, 651
Kmulti.inhom, 654
Kscaled, 664
Ksector, 667
Lcross, 675
Lcross.inhom, 677
Ldot, 678
Ldot.inhom, 680
Lest, 684
linearK, 696
linearKcross, 697
linearKcross.inhom, 699
linearKdot, 700
linearKdot.inhom, 702
linearKinhom, 703
linearmarkconnect, 705
linearmarkequal, 707
linearpcf, 708
linearpcfcross, 709
linearpcfcross.inhom, 711
linearpcfdot, 712
linearpcfdot.inhom, 714
linearpcfinhom, 715
Linhom, 719
localK, 726
localKinhom, 728
localpcf, 730
lohboot, 740
markconnect, 752
markcorr, 754
markcrosscorr, 758

INDEX
markvario, 767
miplot, 818
nncorr, 845
nnorient, 871
npfun, 879
pairorient, 902
pcf, 916
pcf.fasp, 917
pcf.fv, 919
pcf.ppp, 921
pcf3est, 924
pcfcross, 926
pcfcross.inhom, 928
pcfdot, 930
pcfdot.inhom, 932
pcfinhom, 934
pcfmulti, 936
pool.anylist, 1032
pool.rat, 1038
PPversion, 1071
quantile.density, 1133
quantile.ewcdf, 1134
reduced.sample, 1157
sdrPredict, 1326
sharpen, 1332
Smooth.fv, 1355
spatialcdf, 1369
Tstat, 1448
varblock, 1468
∗Topic optimize
bc.ppm, 140
rex, 1185
∗Topic package
spatstat-package, 25
∗Topic print
print.im, 1085
print.owin, 1086
print.ppm, 1087
print.ppp, 1088
print.psp, 1089
print.quad, 1090
progressreport, 1093
∗Topic programming
applynbd, 78
eval.fasp, 405
eval.fv, 407
eval.im, 409
eval.linim, 410
im.apply, 542
levelset, 685
markstat, 764
marktable, 766

1513
solutionset, 1366
with.fv, 1493
with.hyperframe, 1495
with.ssf, 1498
∗Topic smooth
adaptive.density, 49
bw.diggle, 164
bw.frac, 165
bw.pcf, 166
bw.ppl, 168
bw.relrisk, 170
bw.scott, 171
bw.smoothppp, 172
bw.stoyan, 174
circdensity, 192
density.lpp, 286
density.ppp, 288
density.psp, 292
density.splitppp, 294
dkernel, 344
idw, 536
kernel.factor, 628
kernel.moment, 629
kernel.squint, 630
nndensity.ppp, 855
nnmark, 869
relrisk.ppp, 1165
Smooth, 1354
Smooth.ppp, 1358
Smooth.ssf, 1360
Smoothfun.ppp, 1361
unnormdensity, 1455
∗Topic spatial
[.ssf, 1502
adaptive.density, 49
add.texture, 50
addvar, 51
affine, 54
affine.im, 54
affine.linnet, 55
affine.lpp, 57
affine.owin, 58
affine.ppp, 59
affine.psp, 61
affine.tess, 62
allstats, 63
alltypes, 64
angles.psp, 67
anova.lppm, 68
anova.mppm, 70
anova.ppm, 72
anova.slrm, 74

1514

INDEX
anyNA.im, 76
append.psp, 77
applynbd, 78
area.owin, 80
areaGain, 82
AreaInter, 83
areaLoss, 86
as.box3, 87
as.data.frame.envelope, 89
as.data.frame.hyperframe, 90
as.data.frame.im, 91
as.data.frame.owin, 92
as.data.frame.ppp, 93
as.data.frame.psp, 94
as.data.frame.tess, 95
as.function.fv, 96
as.function.im, 97
as.function.leverage.ppm, 98
as.function.owin, 99
as.function.tess, 100
as.fv, 101
as.hyperframe, 102
as.hyperframe.ppx, 104
as.im, 105
as.interact, 109
as.layered, 110
as.linfun, 112
as.linim, 113
as.linnet.linim, 114
as.linnet.psp, 116
as.lpp, 117
as.mask, 118
as.mask.psp, 120
as.matrix.im, 121
as.matrix.owin, 122
as.owin, 123
as.polygonal, 126
as.ppm, 127
as.ppp, 129
as.psp, 131
as.rectangle, 133
as.solist, 134
as.tess, 135
auc, 137
BadGey, 138
bc.ppm, 140
bdist.pixels, 141
bdist.points, 142
bdist.tiles, 143
beachcolours, 144
berman.test, 147
bind.fv, 149

bits.test, 151
blur, 153
border, 154
bounding.box.xy, 156
boundingbox, 157
boundingcircle, 158
box3, 160
boxx, 161
branchlabelfun, 162
bw.diggle, 164
bw.frac, 165
bw.pcf, 166
bw.ppl, 168
bw.relrisk, 170
bw.scott, 171
bw.smoothppp, 172
bw.stoyan, 174
by.im, 175
by.ppp, 176
cauchy.estK, 177
cauchy.estpcf, 179
cbind.hyperframe, 181
cdf.test, 183
cdf.test.mppm, 187
centroid.owin, 189
chop.tess, 191
clarkevans, 193
clarkevans.test, 195
clickbox, 196
clickdist, 197
clickjoin, 198
clicklpp, 199
clickpoly, 200
clickppp, 201
clip.infline, 202
closepairs, 203
closepairs.pp3, 205
closetriples, 207
closing, 208
clusterfield, 209
clusterfit, 211
clusterkernel, 213
clusterradius, 214
clusterset, 215
coef.mppm, 217
coef.ppm, 218
coef.slrm, 220
collapse.fv, 221
colourmap, 222
colouroutputs, 224
commonGrid, 227
compareFit, 228

INDEX
compatible, 230
compatible.fasp, 231
compatible.fv, 231
compatible.im, 232
compileK, 233
complement.owin, 235
concatxy, 236
Concom, 237
connected, 239
connected.linnet, 241
connected.lpp, 242
connected.ppp, 243
contour.im, 244
contour.imlist, 246
convexhull, 247
convexhull.xy, 248
convexify, 249
convolve.im, 250
coords, 251
corners, 252
covering, 253
crossdist, 254
crossdist.default, 255
crossdist.lpp, 256
crossdist.pp3, 257
crossdist.ppp, 258
crossdist.ppx, 259
crossdist.psp, 260
crossing.linnet, 261
crossing.psp, 262
cut.im, 263
cut.lpp, 264
cut.ppp, 266
data.ppm, 268
dclf.progress, 269
dclf.sigtrace, 271
dclf.test, 273
default.dummy, 276
default.expand, 277
default.rmhcontrol, 279
delaunay, 280
delaunayDistance, 281
delaunayNetwork, 282
deletebranch, 283
deltametric, 284
density.lpp, 286
density.ppp, 288
density.psp, 292
density.splitppp, 294
deriv.fv, 295
detpointprocfamilyfun, 296
dfbetas.ppm, 299

1515
dg.envelope, 300
dg.progress, 302
dg.sigtrace, 304
dg.test, 307
diagnose.ppm, 309
diameter, 313
diameter.box3, 314
diameter.boxx, 316
diameter.linnet, 317
diameter.owin, 318
DiggleGatesStibbard, 319
DiggleGratton, 320
dilated.areas, 321
dilation, 322
dim.detpointprocfamily, 324
dirichlet, 325
dirichletAreas, 326
dirichletVertices, 327
dirichletWeights, 328
disc, 329
discpartarea, 330
discretise, 331
discs, 333
distcdf, 334
distfun, 335
distfun.lpp, 337
distmap, 338
distmap.owin, 339
distmap.ppp, 341
distmap.psp, 342
divide.linnet, 343
domain, 346
dppeigen, 352
dppkernel, 354
dppm, 354
dppparbounds, 359
dppspecden, 360
dppspecdenrange, 361
dummy.ppm, 363
duplicated.ppp, 364
edge.Ripley, 365
edge.Trans, 367
edges, 369
edges2triangles, 370
edges2vees, 371
edit.hyperframe, 372
edit.ppp, 373
eem, 374
effectfun, 375
ellipse, 377
Emark, 378
emend, 380

1516

INDEX
emend.ppm, 381
endpoints.psp, 382
envelope, 384
envelope.envelope, 393
envelope.lpp, 395
envelope.pp3, 398
envelopeArray, 401
eroded.areas, 402
erosion, 403
erosionAny, 404
eval.fasp, 405
eval.fv, 407
eval.im, 409
eval.linim, 410
exactMPLEstrauss, 413
expand.owin, 414
Extract.anylist, 415
Extract.fasp, 416
Extract.fv, 417
Extract.hyperframe, 419
Extract.im, 420
Extract.influence.ppm, 423
Extract.layered, 424
Extract.leverage.ppm, 426
Extract.linim, 427
Extract.linnet, 428
Extract.listof, 429
Extract.lpp, 430
Extract.msr, 431
Extract.owin, 432
Extract.ppp, 433
Extract.ppx, 436
Extract.psp, 437
Extract.quad, 439
Extract.solist, 440
Extract.splitppp, 441
Extract.tess, 442
F3est, 443
fardist, 445
fasp.object, 446
Fest, 447
Fiksel, 451
Finhom, 453
fitin.ppm, 455
fitted.lppm, 456
fitted.mppm, 457
fitted.ppm, 459
fitted.slrm, 461
fixef.mppm, 462
flipxy, 463
FmultiInhom, 464
formula.fv, 466

formula.ppm, 467
Frame, 469
fryplot, 470
funxy, 472
fv, 473
fv.object, 476
fvnames, 477
G3est, 478
Gcom, 481
Gcross, 484
Gdot, 487
Gest, 490
Geyer, 493
Gfox, 494
Ginhom, 496
Gmulti, 498
GmultiInhom, 500
Gres, 502
gridcentres, 503
gridweights, 504
grow.boxx, 505
grow.rectangle, 506
Hardcore, 507
harmonic, 509
harmonise, 510
harmonise.fv, 511
harmonise.im, 512
harmonise.msr, 513
harmonise.owin, 514
has.close, 515
headtail, 517
Hest, 518
hextess, 520
HierHard, 522
hierpair.family, 523
HierStrauss, 524
HierStraussHard, 525
hist.funxy, 527
hist.im, 528
hopskel, 529
Hybrid, 531
hybrid.family, 532
hyperframe, 533
identify.ppp, 534
identify.psp, 535
idw, 536
Iest, 538
im, 540
im.apply, 542
im.object, 543
imcov, 544
improve.kppm, 545

INDEX
incircle, 547
increment.fv, 548
infline, 549
influence.ppm, 550
inforder.family, 552
insertVertices, 552
inside.boxx, 554
inside.owin, 555
integral.im, 556
integral.linim, 557
integral.msr, 558
intensity, 560
intensity.dppm, 561
intensity.lpp, 561
intensity.ppm, 562
intensity.ppp, 564
intensity.psp, 566
intensity.quadratcount, 567
interp.colourmap, 568
interp.im, 569
intersect.owin, 570
intersect.tess, 572
invoke.symbolmap, 573
iplot, 574
ippm, 576
is.connected, 578
is.connected.ppp, 579
is.convex, 580
is.dppm, 581
is.empty, 581
is.hybrid, 582
is.im, 584
is.lpp, 584
is.marked, 585
is.marked.ppm, 586
is.marked.ppp, 587
is.multitype, 588
is.multitype.ppm, 589
is.multitype.ppp, 590
is.owin, 591
is.ppm, 592
is.ppp, 593
is.rectangle, 593
is.stationary, 594
is.subset.owin, 596
istat, 597
Jcross, 598
Jdot, 600
Jest, 603
Jinhom, 606
Jmulti, 608
K3est, 610

1517
kaplan.meier, 611
Kcom, 613
Kcross, 616
Kcross.inhom, 618
Kdot, 622
Kdot.inhom, 625
Kest, 631
Kest.fft, 635
Kinhom, 636
km.rs, 641
Kmark, 642
Kmeasure, 644
Kmodel, 647
Kmodel.kppm, 649
Kmodel.ppm, 650
Kmulti, 651
Kmulti.inhom, 654
kppm, 657
Kres, 662
Kscaled, 664
Ksector, 667
laslett, 669
layered, 672
layerplotargs, 673
Lcross, 675
Lcross.inhom, 677
Ldot, 678
Ldot.inhom, 680
lengths.psp, 681
LennardJones, 682
Lest, 684
levelset, 685
leverage.ppm, 686
lgcp.estK, 688
lgcp.estpcf, 691
lineardirichlet, 694
lineardisc, 695
linearK, 696
linearKcross, 697
linearKcross.inhom, 699
linearKdot, 700
linearKdot.inhom, 702
linearKinhom, 703
linearmarkconnect, 705
linearmarkequal, 707
linearpcf, 708
linearpcfcross, 709
linearpcfcross.inhom, 711
linearpcfdot, 712
linearpcfdot.inhom, 714
linearpcfinhom, 715
linequad, 717

1518

INDEX
linfun, 718
Linhom, 719
linim, 720
linnet, 722
lintess, 723
lixellate, 724
localK, 726
localKinhom, 728
localpcf, 730
logLik.dppm, 732
logLik.kppm, 733
logLik.mppm, 735
logLik.ppm, 737
logLik.slrm, 739
lohboot, 740
lpp, 742
lppm, 744
lurking, 745
lurking.mppm, 749
lut, 751
markconnect, 752
markcorr, 754
markcrosscorr, 758
marks, 760
marks.psp, 762
marks.tess, 763
markstat, 764
marktable, 766
markvario, 767
matchingdist, 769
matclust.estK, 770
matclust.estpcf, 772
Math.im, 775
Math.imlist, 776
Math.linim, 778
maxnndist, 781
mean.im, 782
mean.linim, 783
measureVariation, 784
mergeLevels, 785
methods.box3, 786
methods.boxx, 788
methods.dppm, 789
methods.fii, 790
methods.funxy, 791
methods.kppm, 792
methods.layered, 793
methods.linfun, 795
methods.linim, 796
methods.linnet, 798
methods.lpp, 800
methods.lppm, 801

methods.objsurf, 803
methods.pp3, 804
methods.ppx, 805
methods.rho2hat, 806
methods.rhohat, 807
methods.slrm, 809
methods.ssf, 810
methods.unitname, 812
methods.zclustermodel, 813
midpoints.psp, 814
mincontrast, 815
MinkowskiSum, 817
miplot, 818
model.depends, 820
model.frame.ppm, 821
model.images, 823
model.matrix.mppm, 825
model.matrix.ppm, 826
model.matrix.slrm, 828
mppm, 829
msr, 832
MultiHard, 834
multiplicity.ppp, 835
MultiStrauss, 837
MultiStraussHard, 838
nearest.raster.point, 840
nearestsegment, 841
nestsplit, 842
nnclean, 843
nncorr, 845
nncross, 848
nncross.lpp, 850
nncross.pp3, 852
nndensity.ppp, 855
nndist, 856
nndist.lpp, 858
nndist.pp3, 860
nndist.ppx, 861
nndist.psp, 863
nnfromvertex, 864
nnfun, 865
nnfun.lpp, 866
nnmap, 867
nnmark, 869
nnorient, 871
nnwhich, 872
nnwhich.lpp, 875
nnwhich.pp3, 876
nnwhich.ppx, 877
nobjects, 878
npfun, 879
npoints, 880

INDEX
nsegments, 881
nvertices, 882
objsurf, 883
opening, 884
Ops.msr, 885
Ord, 886
ord.family, 888
OrdThresh, 888
overlap.owin, 889
owin, 890
owin.object, 893
padimage, 894
pairdist, 895
pairdist.default, 896
pairdist.lpp, 897
pairdist.pp3, 898
pairdist.ppp, 899
pairdist.ppx, 900
pairdist.psp, 901
pairorient, 902
PairPiece, 904
pairs.im, 905
pairs.linim, 907
pairsat.family, 908
Pairwise, 909
pairwise.family, 910
panel.contour, 911
parameters, 912
parres, 913
pcf, 916
pcf.fasp, 917
pcf.fv, 919
pcf.ppp, 921
pcf3est, 924
pcfcross, 926
pcfcross.inhom, 928
pcfdot, 930
pcfdot.inhom, 932
pcfinhom, 934
pcfmulti, 936
Penttinen, 938
perimeter, 939
periodify, 940
persp.im, 942
perspPoints, 944
pixelcentres, 945
pixellate, 946
pixellate.owin, 947
pixellate.ppp, 948
pixellate.psp, 950
pixelquad, 951
plot.anylist, 952

1519
plot.bermantest, 955
plot.cdftest, 957
plot.colourmap, 958
plot.dppm, 960
plot.envelope, 961
plot.fasp, 962
plot.fv, 964
plot.hyperframe, 967
plot.im, 969
plot.imlist, 974
plot.influence.ppm, 975
plot.kppm, 976
plot.laslett, 978
plot.layered, 979
plot.leverage.ppm, 980
plot.linim, 982
plot.linnet, 984
plot.lintess, 985
plot.listof, 986
plot.lpp, 989
plot.lppm, 990
plot.mppm, 991
plot.msr, 992
plot.onearrow, 994
plot.owin, 995
plot.plotppm, 998
plot.pp3, 999
plot.ppm, 1001
plot.ppp, 1003
plot.psp, 1007
plot.quad, 1009
plot.quadratcount, 1010
plot.quadrattest, 1012
plot.rppm, 1013
plot.scan.test, 1014
plot.slrm, 1015
plot.solist, 1016
plot.splitppp, 1019
plot.ssf, 1020
plot.symbolmap, 1021
plot.tess, 1023
plot.textstring, 1024
plot.texturemap, 1025
plot.yardstick, 1026
points.lpp, 1027
pointsOnLines, 1028
Poisson, 1029
pool, 1032
pool.anylist, 1032
pool.envelope, 1033
pool.fasp, 1035
pool.fv, 1036

1520

INDEX
pool.quadrattest, 1037
pool.rat, 1038
pp3, 1039
ppm, 1040
ppm.object, 1046
ppm.ppp, 1048
ppmInfluence, 1058
ppp, 1060
ppp.object, 1063
pppdist, 1065
pppmatching, 1068
pppmatching.object, 1069
PPversion, 1071
ppx, 1072
predict.dppm, 1073
predict.kppm, 1074
predict.lppm, 1075
predict.mppm, 1076
predict.ppm, 1078
predict.rppm, 1083
predict.slrm, 1084
print.im, 1085
print.owin, 1086
print.ppm, 1087
print.ppp, 1088
print.psp, 1089
print.quad, 1090
profilepl, 1091
project2segment, 1095
project2set, 1096
prune.rppm, 1097
pseudoR2, 1098
psib, 1099
psp, 1100
psp.object, 1101
psst, 1102
psstA, 1104
psstG, 1107
qqplot.ppm, 1109
quad.object, 1113
quad.ppm, 1114
quadrat.test, 1116
quadrat.test.mppm, 1119
quadrat.test.splitppp, 1121
quadratcount, 1122
quadratresample, 1125
quadscheme, 1127
quadscheme.logi, 1129
quantess, 1131
quantile.ewcdf, 1134
quantile.im, 1135
quasirandom, 1136

rags, 1137
ragsAreaInter, 1138
ragsMultiHard, 1140
ranef.mppm, 1141
range.fv, 1142
raster.x, 1143
rat, 1144
rCauchy, 1145
rcell, 1147
rDGS, 1150
rDiggleGratton, 1151
rdpp, 1153
reach, 1154
reach.dppm, 1156
reduced.sample, 1157
reflect, 1158
regularpolygon, 1159
relevel.im, 1160
relrisk, 1162
relrisk.ppm, 1163
relrisk.ppp, 1165
Replace.im, 1168
Replace.linim, 1170
rescale, 1172
rescale.im, 1173
rescale.owin, 1174
rescale.ppp, 1176
rescale.psp, 1177
rescue.rectangle, 1178
residuals.dppm, 1179
residuals.kppm, 1180
residuals.mppm, 1181
residuals.ppm, 1182
rGaussPoisson, 1186
rgbim, 1187
rHardcore, 1188
rho2hat, 1190
rhohat, 1191
ripras, 1195
rjitter, 1197
rknn, 1198
rlabel, 1199
rLGCP, 1200
rlinegrid, 1202
rlpp, 1203
rMatClust, 1204
rMaternI, 1207
rMaternII, 1208
rmh, 1209
rmh.default, 1210
rmh.ppm, 1220
rmhcontrol, 1223

INDEX
rmhexpand, 1227
rmhmodel, 1229
rmhmodel.default, 1230
rmhmodel.list, 1237
rmhmodel.ppm, 1239
rmhstart, 1241
rMosaicField, 1242
rMosaicSet, 1243
rmpoint, 1244
rmpoispp, 1248
rNeymanScott, 1251
rnoise, 1254
roc, 1255
rose, 1256
rotate, 1258
rotate.im, 1259
rotate.infline, 1260
rotate.owin, 1261
rotate.ppp, 1262
rotate.psp, 1263
rotmean, 1264
round.ppp, 1266
rounding, 1267
rPenttinen, 1268
rpoint, 1270
rpoisline, 1271
rpoislinetess, 1272
rpoislpp, 1273
rpoispp, 1274
rpoispp3, 1276
rpoisppOnLines, 1277
rpoisppx, 1279
rPoissonCluster, 1280
rppm, 1282
rQuasi, 1283
rshift, 1284
rshift.ppp, 1285
rshift.psp, 1287
rshift.splitppp, 1289
rSSI, 1290
rstrat, 1292
rStrauss, 1293
rStraussHard, 1295
rsyst, 1296
rtemper, 1297
rthin, 1299
rThomas, 1300
runifdisc, 1305
runiflpp, 1306
runifpoint, 1307
runifpoint3, 1308
runifpointOnLines, 1309

1521
runifpointx, 1310
rVarGamma, 1311
SatPiece, 1313
Saturated, 1315
scalardilate, 1316
scaletointerval, 1317
scan.test, 1318
scanLRTS, 1320
scanpp, 1322
sdr, 1323
sdrPredict, 1326
segregation.test, 1327
selfcrossing.psp, 1328
selfcut.psp, 1329
setcov, 1331
sharpen, 1332
shift, 1333
shift.im, 1334
shift.owin, 1335
shift.ppp, 1336
shift.psp, 1337
sidelengths.owin, 1338
simplify.owin, 1342
simulate.dppm, 1343
simulate.kppm, 1345
simulate.lppm, 1347
simulate.mppm, 1348
simulate.ppm, 1349
simulate.slrm, 1351
slrm, 1352
Smooth, 1354
Smooth.fv, 1355
Smooth.msr, 1356
Smooth.ppp, 1358
Smooth.ssf, 1360
Smoothfun.ppp, 1361
Softcore, 1362
solapply, 1364
solist, 1365
solutionset, 1366
spatialcdf, 1369
spatstat-package, 25
spatstat.options, 1370
split.hyperframe, 1374
split.im, 1375
split.msr, 1376
split.ppp, 1378
split.ppx, 1380
spokes, 1382
square, 1383
ssf, 1384
stieltjes, 1385

1522

INDEX
stienen, 1386
stratrand, 1387
Strauss, 1389
StraussHard, 1390
studpermu.test, 1392
subfits, 1394
subset.hyperframe, 1395
subset.ppp, 1396
suffstat, 1399
summary.anylist, 1401
summary.im, 1402
summary.kppm, 1403
summary.listof, 1404
summary.owin, 1405
summary.ppm, 1406
summary.ppp, 1407
summary.psp, 1408
summary.quad, 1409
summary.solist, 1410
summary.splitppp, 1411
superimpose, 1413
superimpose.lpp, 1416
symbolmap, 1417
tess, 1419
test.crossing.psp, 1421
text.ppp, 1422
texturemap, 1423
textureplot, 1424
thinNetwork, 1425
thomas.estK, 1427
thomas.estpcf, 1429
tile.areas, 1431
tile.lengths, 1432
tileindex, 1433
tilenames, 1434
tiles, 1435
tiles.empty, 1436
transect.im, 1439
transmat, 1440
treebranchlabels, 1441
treeprune, 1442
triangulate.owin, 1444
trim.rectangle, 1445
triplet.family, 1446
Triplets, 1446
Tstat, 1448
tweak.colourmap, 1449
union.quad, 1450
unique.ppp, 1451
unitname, 1452
unmark, 1454
unstack.msr, 1456

unstack.ppp, 1457
update.detpointprocfamily, 1458
update.interact, 1459
update.kppm, 1460
update.ppm, 1461
update.rmhcontrol, 1463
update.symbolmap, 1464
valid, 1465
valid.detpointprocfamily, 1466
valid.ppm, 1467
varblock, 1468
varcount, 1470
vargamma.estK, 1471
vargamma.estpcf, 1473
vcov.kppm, 1476
vcov.mppm, 1477
vcov.ppm, 1479
vcov.slrm, 1482
vertices, 1483
volume, 1484
where.max, 1486
whichhalfplane, 1487
will.expand, 1489
Window, 1490
WindowOnly, 1491
with.fv, 1493
with.hyperframe, 1495
with.msr, 1496
with.ssf, 1498
yardstick, 1499
zapsmall.im, 1500
zclustermodel, 1501
∗Topic univar
CDF, 182
ewcdf, 412
mean.im, 782
mean.linim, 783
quantile.density, 1133
quantile.im, 1135
scaletointerval, 1317
zapsmall.im, 1500
∗Topic utilities
bounding.box.xy, 156
boundingbox, 157
convexhull, 247
convexhull.xy, 248
convexify, 249
corners, 252
dirichletWeights, 328
dummy.ppm, 363
layout.boxes, 674
multiplicity.ppp, 835

INDEX
quadrats, 1126
reload.or.compute, 1161
ripras, 1195
run.simplepanel, 1302
simplepanel, 1339
timed, 1437
timeTaken, 1438
.Random.seed, 1212, 1214
[, 418, 419, 424–426, 430, 434, 437–439, 1169
[.anylist (Extract.anylist), 415
[.fasp, 447
[.fasp (Extract.fasp), 416
[.fv (Extract.fv), 417
[.hyperframe, 534, 1375, 1395, 1396
[.hyperframe (Extract.hyperframe), 419
[.im, 32, 98, 263, 426, 541, 543, 544, 557,
1169
[.im (Extract.im), 420
[.influence.ppm
(Extract.influence.ppm), 423
[.layered, 35, 673, 674, 979, 980
[.layered (Extract.layered), 424
[.leverage.ppm, 426
[.leverage.ppm (Extract.leverage.ppm),
426
[.linim (Extract.linim), 427
[.linnet (Extract.linnet), 428
[.lpp, 265, 1398
[.lpp (Extract.lpp), 430
[.msr, 833, 1377
[.msr (Extract.msr), 431
[.owin (Extract.owin), 432
[.pp3, 1397, 1398
[.ppp, 29, 266, 423, 439, 464, 501, 1063,
1064, 1397, 1398
[.ppp (Extract.ppp), 433
[.ppx, 1397, 1398
[.ppx (Extract.ppx), 436
[.psp, 32, 1102
[.psp (Extract.psp), 437
[.quad (Extract.quad), 439
[.solist (Extract.solist), 440
[.splitppp (Extract.splitppp), 441
[.ssf, 1385, 1502
[.tess, 33, 1420
[.tess (Extract.tess), 442
[<-.im, 32
[<-.tess, 33
[<-.anylist (Extract.anylist), 415
[<-.fv (Extract.fv), 417
[<-.hyperframe (Extract.hyperframe), 419
[<-.im (Replace.im), 1168

1523
[<-.layered (Extract.layered), 424
[<-.linim (Replace.linim), 1170
[<-.listof (Extract.listof), 429
[<-.ppp (Extract.ppp), 433
[<-.solist (Extract.solist), 440
[<-.splitppp (Extract.splitppp), 441
[<-.tess (Extract.tess), 442
[[<-.layered (Extract.layered), 424
$, 420
$.hyperframe (Extract.hyperframe), 419
$<-.fv (Extract.fv), 417
$<-.hyperframe (Extract.hyperframe), 419
%mark% (marks), 760
abline, 549
ad.test, 184–186, 188
adaptive.density, 49, 290, 292
add.texture, 50, 996, 1025, 1026,
1423–1425
add1, 1046
addvar, 46, 51, 915
affine, 29, 30, 54, 55–63, 415, 463, 794, 797,
893, 1159, 1173, 1175–1178, 1305,
1316, 1317, 1334, 1336–1338
affine.im, 32, 54, 54, 59–61, 63, 1260
affine.layered (methods.layered), 793
affine.linim (methods.linim), 796
affine.linnet, 55
affine.lpp, 57
affine.owin, 54, 55, 58, 60, 61, 63, 893
affine.ppp, 54, 55, 59, 59, 61
affine.psp, 33, 54, 55, 59, 60, 61
affine.tess, 62
aggregate, 857
AIC, 40, 42, 736
AIC.dppm (logLik.dppm), 732
AIC.kppm (logLik.kppm), 733
AIC.mppm (logLik.mppm), 735
AIC.ppm, 1091
AIC.ppm (logLik.ppm), 737
allstats, 36, 63
alltypes, 37, 64, 231, 402, 406, 446, 447,
754, 917–920, 963, 1035
amacrine, 28, 585–589, 591, 1063
anemones, 28
angles.psp, 33, 67, 682, 814, 1102
anova, 69, 70, 72, 74
anova.glm, 69–72, 74
anova.lppm, 43, 68
anova.mppm, 70
anova.ppm, 41, 46, 72, 1046, 1118, 1122
anova.slrm, 44, 74, 1354
ants, 28

1524
anyDuplicated, 364
anyDuplicated.ppp (duplicated.ppp), 364
anyDuplicated.ppx (duplicated.ppp), 364
anylapply, 76
anylapply (solapply), 1364
anylist, 75, 416, 1033, 1365, 1366, 1401
anyNA, 76
anyNA.im, 76, 783, 1402
append.psp, 77
apply, 78, 79, 765, 1498
apply.ssf (with.ssf), 1498
applynbd, 38, 78, 205, 765–767
approxfun, 511
area (area.owin), 80
area.owin, 31, 80, 318, 890, 893, 894, 940,
1339, 1485
areaGain, 82, 86, 1372
AreaInter, 42, 82, 83, 83, 86, 552, 911, 938,
939, 1041, 1043, 1051, 1056, 1092,
1106, 1139, 1215, 1221, 1222, 1235,
1240
areaLoss, 83, 86, 1372
array, 121
as.anylist, 135
as.anylist (anylist), 75
as.array.im (as.matrix.im), 121
as.box3, 34, 87, 160, 315, 1040
as.boxx, 88, 554, 1344
as.data.frame, 89, 90, 93, 94, 103, 105, 795,
797
as.data.frame.default, 91, 92, 95
as.data.frame.envelope, 89
as.data.frame.hyperframe, 35, 90, 534
as.data.frame.im, 31, 91, 92, 95
as.data.frame.linfun (methods.linfun),
795
as.data.frame.linim (methods.linim), 796
as.data.frame.owin, 30, 92, 95
as.data.frame.ppp, 93, 518
as.data.frame.ppx, 518
as.data.frame.ppx (as.hyperframe.ppx),
104
as.data.frame.psp, 32, 94, 518
as.data.frame.tess, 95
as.function, 97, 795, 811
as.function.fv, 96, 475, 476
as.function.im, 31, 97
as.function.leverage.ppm, 98, 687
as.function.linfun (methods.linfun), 795
as.function.owin, 99
as.function.rhohat (as.function.fv), 96
as.function.ssf (methods.ssf), 810

INDEX
as.function.tess, 100, 1433
as.fv, 101
as.hyperframe, 34, 35, 102, 104, 105, 181,
182, 534
as.hyperframe.ppx, 103, 104, 104, 534
as.im, 31, 49, 105, 114, 122, 228, 334, 336,
410, 512, 513, 527, 541, 543, 544,
792, 797, 811, 865, 946–949, 1254,
1362, 1424
as.im.function, 792
as.im.leverage.ppm, 99, 687
as.im.linim (methods.linim), 796
as.im.owin, 30, 99
as.im.ppp, 30, 107, 108
as.im.ppp (pixellate.ppp), 948
as.im.scan.test, 1319–1322
as.im.scan.test (plot.scan.test), 1014
as.im.ssf (methods.ssf), 810
as.interact, 41, 109, 791, 1046
as.interact.fii, 456, 791
as.interact.ppm, 1046
as.layered, 110, 673
as.linfun, 112
as.linfun.lintess, 724
as.linim, 113, 338, 719, 724, 795, 867
as.linim.linfun, 795
as.linnet, 57, 115
as.linnet (methods.linnet), 798
as.linnet.linfun (as.linnet.linim), 114
as.linnet.linim, 114
as.linnet.lintess, 724
as.linnet.lintess (as.linnet.linim), 114
as.linnet.lpp (as.linnet.linim), 114
as.linnet.lppm (methods.lppm), 801
as.linnet.psp, 116, 282
as.lpp, 117, 427, 552, 553, 743
as.mask, 30, 31, 55, 59, 106, 113, 118, 120,
122, 127, 137, 141, 147, 159, 208,
210, 215, 227, 228, 239, 253,
284–286, 288, 290, 293, 299, 323,
329, 330, 332–335, 338–342, 377,
403, 404, 445, 453, 495, 496, 505,
518, 519, 536, 537, 546, 567, 570,
572, 596, 606, 635, 645, 669, 687,
841, 855, 868, 870, 884, 893, 945,
947–951, 1002, 1075, 1096, 1111,
1144, 1194, 1201, 1242, 1254, 1255,
1262, 1318, 1320, 1331, 1369, 1470
as.mask.psp, 33, 120, 951
as.matrix, 1153
as.matrix.im, 31, 121, 122, 541, 543, 544
as.matrix.owin, 121, 122

INDEX
as.matrix.ppx (as.hyperframe.ppx), 104
as.owin, 30, 81, 118, 123, 127, 130, 132–134,
155, 156, 158, 166, 190, 209, 247,
248, 252, 318, 322, 323, 369,
402–404, 449, 468, 491, 503, 514,
515, 545, 555, 556, 570, 596, 633,
686, 738, 795, 799, 885, 891–893,
923, 939, 940, 951, 996, 1046, 1061,
1062, 1096, 1100, 1101, 1126, 1146,
1148, 1179, 1186, 1196, 1203, 1205,
1207, 1208, 1211, 1234, 1239, 1248,
1251, 1272, 1274, 1280, 1290, 1292,
1296, 1301, 1307, 1311, 1331, 1339,
1370, 1371, 1388, 1414
as.owin.linfun (methods.linfun), 795
as.owin.linnet (methods.linnet), 798
as.owin.ppm, 1046
as.polygonal, 31, 119, 126, 939, 1444
as.ppm, 127, 1092, 1179, 1180
as.ppm.dppm, 357, 789
as.ppm.kppm, 661, 793
as.ppp, 27, 78, 129, 276, 330, 332, 378, 379,
421, 422, 448, 453, 471, 484, 487,
490, 491, 496, 498, 538, 539, 598,
601, 603, 606, 609, 617, 620, 623,
626, 631, 632, 635, 637, 642, 645,
652, 655, 664, 667, 684, 742, 752,
753, 755, 756, 758, 759, 767, 801,
811, 818, 1003, 1061–1064,
1127–1131, 1168–1170, 1211, 1241,
1323, 1448, 1450, 1451
as.ppp.influence.ppm, 551
as.ppp.lpp (methods.lpp), 800
as.ppp.ssf (methods.ssf), 810
as.psp, 32, 77, 131, 799, 801, 1008, 1101,
1102, 1370
as.psp.linnet, 799
as.psp.linnet (methods.linnet), 798
as.psp.lpp (methods.lpp), 800
as.rectangle, 119, 133, 157, 158, 190, 893,
1252, 1281, 1339
as.solist, 76, 134, 1364, 1366
as.tess, 33, 135, 572, 573, 1117, 1123, 1420,
1424
atan2, 68
auc, 137, 1255, 1256
axis, 959, 970, 983, 1022
axis.default, 1025
axisTicks, 970
BadGey, 42, 138, 139, 1041, 1043, 1051, 1056,
1092, 1215, 1235, 1240, 1314
barplot, 528, 529

1525
bc, 1186
bc (bc.ppm), 140
bc.ppm, 140
bdist.pixels, 31, 141, 143, 893, 894
bdist.points, 31, 142, 142, 143, 893, 894
bdist.tiles, 31, 33, 142, 143, 143, 1420
bdspots, 28
beachcolourmap, 35, 942, 971
beachcolourmap (beachcolours), 144
beachcolours, 144, 942, 971
beginner, 145, 466
begins, 146
bei, 28
berman.test, 46, 147, 186, 955, 956, 1047
berman.test.ppm, 1047
betacells, 28, 1063
bilinearform (sumouter), 1412
bind.fv, 149, 234, 475, 476
bits.test, 46, 151, 308
blur, 32, 153, 454, 497, 607, 638, 665, 1371
border, 30, 154
bounding.box.xy, 156, 248, 1196, 1414,
1415
boundingbox, 30, 134, 157, 571, 893, 894
boundingcentre (boundingcircle), 158
boundingcircle, 158
boundingradius, 317, 318
boundingradius (boundingcircle), 158
boundingradius.linnet, 159
boundingradius.linnet
(diameter.linnet), 317
box3, 34, 87, 160, 506, 787, 813, 1040, 1277,
1309
boxx, 34, 88, 161, 317, 506, 554, 788, 1280,
1311
bramblecanes, 28, 1063
branchlabelfun, 162, 284
bronzefilter, 28
bugfixes, 163, 671
bw.diggle, 36, 102, 164, 166, 169, 171, 172,
289, 290, 292, 1167
bw.frac, 36, 165, 172
bw.pcf, 166, 923, 936
bw.ppl, 36, 165, 166, 168, 172, 289, 290, 292
bw.relrisk, 36, 166, 170, 174, 1166–1168,
1371
bw.scott, 36, 165, 166, 169, 171
bw.smoothppp, 36, 166, 172, 1358–1360
bw.stoyan, 36, 166, 170, 171, 174, 923, 934,
936
by, 175, 176
by.im, 175, 1376

1526
by.ppp, 29, 176, 1420
cauchy.estK, 40, 177, 181, 661, 1473
cauchy.estpcf, 40, 179, 179, 661, 1475
cbind, 150, 181
cbind.fv, 221, 222, 475, 476, 511, 512
cbind.fv (bind.fv), 149
cbind.hyperframe, 35, 181, 534
CDF, 182, 1134
cdf.test, 46, 149, 183, 187–189, 957, 958,
1047, 1118, 1122, 1370
cdf.test.mppm, 187
cdf.test.ppm, 1047
cells, 28, 1063
centroid.owin, 31, 189, 547, 1336
chicago, 28, 35, 743
chisq.test, 1118, 1122
chop.tess, 33, 191, 202, 550
chorley, 28
chull, 580
circdensity, 192, 871, 902, 903, 1258
clarkevans, 36, 193, 195, 196, 530
clarkevans.test, 46, 193, 194, 195, 530
clear.simplepanel (run.simplepanel),
1302
clickbox, 30, 196, 197, 200–202
clickdist, 31, 197, 197, 200–202
clickjoin, 34, 198
clicklpp, 197, 199
clickpoly, 30, 197, 200, 200, 202
clickppp, 27, 197, 198, 200, 201, 201, 535
clip.infline, 191, 202, 550
clip.psp, 1102
clmfires, 28
closepaircounts (closepairs), 203
closepairs, 203, 206, 207, 923, 936, 1373
closepairs.pp3, 205, 205
closetriples, 207
closing, 30, 208, 885, 893
clusterfield, 40, 209, 213, 977, 1146, 1205,
1301, 1312
clusterfit, 211, 1147, 1206, 1302
clusterkernel, 213, 215
clusterradius, 40, 214, 1146, 1301, 1312
clusterset, 36, 215
cm.colors, 942, 971
coef, 217, 219, 220, 455, 789, 790, 793, 802,
913
coef.dppm (methods.dppm), 789
coef.fii (methods.fii), 790
coef.kppm, 40
coef.kppm (methods.kppm), 792
coef.lppm (methods.lppm), 801

INDEX
coef.mppm, 217, 462, 831, 1142
coef.ppm, 41, 218, 468, 738, 1046, 1047, 1480
coef.slrm, 44, 220, 810, 1354, 1482
coef.summary.fii (methods.fii), 790
col2hex (colourtools), 225
col2rgb, 225, 227
collapse, 221
collapse.anylist (collapse.fv), 221
collapse.fv, 221, 229, 511
colourmap, 35, 145, 222, 225, 227, 245, 569,
752, 942, 959, 969, 971, 973, 1418,
1450
colouroutputs, 223, 224, 1450
colouroutputs<- (colouroutputs), 224
colours, 223
colourtools, 145, 223, 225, 225, 569, 1188,
1450
commonGrid, 31, 32, 227, 233, 513, 515
compareFit, 47, 228
compatible, 230, 231, 232, 510, 511, 812,
1144, 1145, 1453
compatible.fasp, 230, 231
compatible.fv, 230, 231, 231, 511, 512
compatible.im, 32, 228, 230, 232, 410, 513
compatible.unitname, 230
compatible.unitname (methods.unitname),
812
compileK, 233, 697
compilepcf (compileK), 233
complement.owin, 30, 235, 582, 892, 893
complementarycolour (colourtools), 225
Complex.im, 782
Complex.im (Math.im), 775
Complex.imlist (Math.imlist), 776
Complex.linim (Math.linim), 778
concatxy, 236, 1415
Concom, 42, 237, 1041, 1043, 1051, 1056
confint, 1046, 1353, 1476, 1481, 1483
connected, 239, 579
connected.im, 32, 244
connected.linnet, 241, 242, 723, 799, 1426
connected.lpp, 242
connected.owin, 31
connected.pp3 (connected.ppp), 243
connected.ppp, 29, 239, 240, 243, 580
contour, 792, 803, 998, 1001, 1002, 1021
contour.default, 107, 244, 245, 804, 981,
1020
contour.funxy (methods.funxy), 791
contour.im, 32, 244, 246, 792, 911, 943, 973,
981, 1372
contour.imlist, 246

INDEX
contour.leverage.ppm
(plot.leverage.ppm), 980
contour.listof, 955, 988, 1019
contour.listof (contour.imlist), 246
contour.objsurf (methods.objsurf), 803
contour.ssf (plot.ssf), 1020
convexhull, 30, 31, 247, 248, 249
convexhull.xy, 156, 247, 248, 280, 581, 1196
convexify, 249
convolve.im, 32, 250, 545
coords, 29, 34, 251
coords.ppx, 259, 861, 877, 900
coords<- (coords), 251
coplot, 911
copper, 28
cor, 845, 847
corners, 43, 252, 277, 1128, 1129
countends (lineardisc), 695
covering, 253
crossdist, 37, 254, 255–260, 896, 897,
899–902
crossdist.default, 254, 255, 259
crossdist.lpp, 39, 256
crossdist.pp3, 39, 257
crossdist.ppp, 254, 256, 257, 258
crossdist.ppx, 39, 259
crossdist.psp, 254, 256, 259, 260
crossing.linnet, 261
crossing.psp, 33, 262, 262, 1329, 1373
crosspaircounts (closepairs), 203
crosspairs, 1373
crosspairs (closepairs), 203
crosspairs.pp3 (closepairs.pp3), 205
cut, 263–265, 267
cut.default, 263–266
cut.im, 32, 263, 541, 543, 1136
cut.lpp, 264
cut.ppp, 29, 38, 100, 177, 266, 435, 845,
1379, 1420, 1433
cvm.test, 184–186, 188
data, 28
data.frame, 533
data.ppm, 268, 374, 375, 1047
dclf.progress, 46, 269, 272, 276, 304
dclf.sigtrace, 271, 306
dclf.test, 46, 151, 152, 269–272, 273, 307,
308, 390
default.dummy, 43, 276, 1052, 1128, 1372
default.expand, 277, 279, 390
default.rmhcontrol, 279, 1221, 1222
delaunay, 30, 33, 280, 281, 282, 326, 1420,
1444

1527
delaunayDistance, 30, 280, 281, 282
delaunayNetwork, 34, 280, 281, 282, 723
deldir, 325
deletebranch, 283, 1442, 1443
deltametric, 284
demohyper, 28
demopat, 28, 1063
dendrite, 28, 35, 743
density, 182, 183, 293, 378, 379, 753, 755,
756, 759, 767, 927, 931, 937, 1133
density.default, 52, 168, 192, 233, 344,
345, 628–631, 708, 710, 711, 713,
714, 716, 914, 915, 921–923, 926,
927, 929–931, 933, 934, 936, 937,
1193, 1194, 1258, 1455, 1456
density.linnet (density.psp), 292
density.lpp, 39, 286
density.ppp, 29, 31, 36, 37, 49, 50, 154,
164–166, 168, 169, 171, 172, 209,
210, 288, 294, 310, 453, 454, 496,
497, 537, 606, 607, 620, 626, 627,
637, 638, 655, 656, 664, 665, 728,
730, 856, 911, 929, 933, 935, 949,
1166, 1168, 1190, 1332, 1333,
1357–1360, 1371
density.ppplist (density.splitppp), 294
density.psp, 33, 292, 1371
density.splitppp, 294, 955, 988, 1019
density.splitppx (density.lpp), 286
deriv, 296
deriv.fv, 37, 295, 548, 872, 903
detpointprocfamilyfun, 296
dev.capabilities, 972
dev.new, 1303
dev.off, 1303
dev.set, 1303
deviance, 737, 802
deviance.lppm, 1098, 1099
deviance.lppm (methods.lppm), 801
deviance.ppm, 1098, 1099
deviance.ppm (logLik.ppm), 737
dfbetas, 299
dfbetas.ppm, 46, 299, 457, 460, 551, 687,
1058, 1059
dg.envelope, 300
dg.progress, 302
dg.sigtrace, 272, 304
dg.test, 46, 152, 302–306, 307
diagnose.ppm, 46, 309, 374, 375, 746–749,
1109–1112, 1183, 1184
diameter, 313, 317, 318, 894
diameter.box3, 34, 160, 314, 314

1528
diameter.boxx, 34, 161, 314, 316
diameter.linnet, 314, 317
diameter.owin, 31, 81, 314, 318, 893, 940,
1339
DiggleGatesStibbard, 42, 319, 1041, 1043,
1051, 1056, 1092, 1151, 1215, 1221,
1222, 1235, 1240
DiggleGratton, 42, 320, 320, 1041, 1043,
1051, 1056, 1092, 1153, 1155, 1215,
1221, 1222, 1230, 1235, 1238, 1240
dilated.areas, 31, 86, 321
dilation, 30, 155, 208, 209, 278, 322, 322,
404, 507, 817, 818, 885, 893
dilation.owin, 1228
dilationAny, 323
dilationAny (MinkowskiSum), 817
dim.detpointprocfamily, 324
dimhat, 324, 1325
dirichlet, 30, 33, 49, 50, 280–282, 325, 327,
328, 1420
dirichletAreas, 326, 328
dirichletEdges (dirichletVertices), 327
dirichletNetwork, 34
dirichletNetwork (delaunayNetwork), 282
dirichletVertices, 326, 327, 327
dirichletWeights, 43, 328, 505, 1129
disc, 30, 323, 329, 331, 333, 334, 377, 891,
892, 1061, 1160, 1305
discpartarea, 330
discretise, 30, 331
discs, 330, 333
distcdf, 166, 334
distfun, 37, 107, 335, 337, 339, 341–343,
792, 866
distfun.lpp, 39, 337, 719, 743, 867
distfun.owin, 31
distfun.psp, 33
distmap, 37, 82, 107, 284, 285, 321, 322, 336,
337, 338, 340–343, 519, 543, 547,
869
distmap.owin, 31, 141, 142, 339, 339, 341,
343
distmap.ppp, 339, 340, 341, 343
distmap.psp, 33, 339–341, 342, 841, 1102
Distributions, 1254
divide.linnet, 343, 724
dkernel, 286, 344, 629–631
dknn (rknn), 1198
dmixpois, 345
domain, 346
dpois, 346
dppapproxkernel, 349, 354

INDEX
dppapproxpcf, 350
dppBessel, 350, 352, 353, 355, 357, 358, 360
dppCauchy, 351, 351, 353, 355, 357, 358, 360
dppeigen, 352
dppGauss, 351, 352, 353, 355, 357, 358, 360
dppkernel, 354
dppm, 44, 102, 354, 376, 595, 732, 733, 789,
822–824, 827, 883, 960, 961, 1044,
1073, 1074, 1180
dppMatern, 351–353, 355, 357, 358, 360
dppparbounds, 359
dppPowerExp, 351–353, 355, 357, 358, 359
dppspecden, 360, 361
dppspecdenrange, 361, 361
drop1, 40, 42, 1046
dummify, 362
dummy.ppm, 363, 1047
duplicated, 364
duplicated.data.frame, 365, 1451
duplicated.ppp, 29, 364, 836, 1408, 1451,
1452
duplicated.ppx (duplicated.ppp), 364
duplicatedxy, 365
ecdf, 412, 1385
edge.Ripley, 365, 368, 1371
edge.Trans, 366, 367, 1371
edges, 31, 32, 133, 318, 369
edges2triangles, 370, 371
edges2vees, 370, 371
edit, 372, 373, 518
edit.data.frame, 372, 373
edit.hyperframe, 372, 373
edit.im (edit.ppp), 373
edit.ppp, 29, 372, 373
edit.psp (edit.ppp), 373
eem, 311–313, 374, 748, 749, 1112, 1183
effectfun, 41, 375
eigen, 780
ellipse, 30, 330, 377, 892, 1061, 1160
Emark, 38, 378
emend, 380, 381, 382, 802
emend.lppm, 381
emend.lppm (methods.lppm), 801
emend.ppm, 381, 381, 1050, 1055, 1056
endpoints.psp, 33, 382
envelope, 37, 43, 45, 46, 65, 66, 89, 151, 152,
269, 271, 273–276, 278, 279, 301,
302, 305, 307, 308, 384, 394, 395,
397, 398, 400–402, 474, 499, 609,
634, 652, 655, 937, 961, 1034, 1047,
1110

INDEX
envelope.envelope, 386, 389, 390, 393,
1034
envelope.lpp, 39, 386, 395
envelope.lppm, 43
envelope.lppm (envelope.lpp), 395
envelope.pp3, 34, 39, 386, 398
envelope.ppm, 1047
envelopeArray, 401
eroded.areas, 31, 322, 402, 404, 893, 894
eroded.volumes, 34, 160, 161
eroded.volumes (diameter.box3), 314
eroded.volumes.boxx, 34, 161
eroded.volumes.boxx (diameter.boxx), 316
erosion, 30, 142, 143, 155, 208, 209, 323,
402, 403, 403, 405, 507, 545, 582,
885, 893, 1286, 1331, 1445
erosionAny, 404, 404, 818
eval.fasp, 37, 231, 405, 447
eval.fv, 37, 232, 406, 407, 511, 512, 1494,
1495
eval.im, 32, 233, 409, 411, 541–544, 557,
775–778, 1168, 1174, 1367, 1487
eval.linim, 43, 410, 721, 779
ewcdf, 185, 188, 412, 1134, 1135, 1369, 1370,
1385
exactdt, 37
exactMPLEstrauss, 413
expand.owin, 414, 1229, 1489
expression, 968
Extract.anylist, 415
Extract.fasp, 416
Extract.fv, 417
Extract.hyperframe, 419
Extract.im, 420
Extract.influence.ppm, 423
Extract.layered, 424
Extract.leverage.ppm, 426
Extract.linim, 427
Extract.linnet, 428
Extract.listof, 429
Extract.lpp, 430
Extract.msr, 431
Extract.owin, 432
Extract.ppp, 433
Extract.ppx, 436
Extract.psp, 437
Extract.quad, 439
Extract.solist, 440
Extract.splitppp, 441
Extract.tess, 442
extractAIC, 732, 734, 736, 737, 802
extractAIC.dppm (logLik.dppm), 732

1529
extractAIC.kppm (logLik.kppm), 733
extractAIC.lppm (methods.lppm), 801
extractAIC.mppm (logLik.mppm), 735
extractAIC.ppm, 468, 1046
extractAIC.ppm (logLik.ppm), 737
extractbranch, 1442, 1443
extractbranch (deletebranch), 283
F3est, 39, 401, 443, 479, 611, 1372
factor, 786
fardist, 445
fasp, 1035
fasp.object, 63, 66, 406, 416, 417, 446, 917,
918, 963
Fest, 36, 63–66, 85, 390, 408, 447, 453, 454,
476, 492, 495, 519, 520, 597, 599,
601–605, 609, 634, 1071
fft, 645
Fhazard (Fest), 447
Fiksel, 42, 451, 911, 1041, 1043, 1051, 1056,
1092, 1215, 1235, 1240
Finhom, 36, 453, 465, 497, 607
finpines, 28
fitin, 41, 110, 790, 791, 1047, 1092, 1155
fitin (fitin.ppm), 455
fitin.ppm, 455, 1047
fitted, 457, 461, 1073, 1074, 1083
fitted.dppm, 357
fitted.dppm (predict.dppm), 1073
fitted.kppm, 40, 661
fitted.kppm (predict.kppm), 1074
fitted.lppm, 43, 456, 704, 716
fitted.mppm, 457, 1078
fitted.ppm, 41, 459, 468, 620, 626, 637, 655,
704, 716, 738, 935, 1046, 1047,
1073–1075, 1082
fitted.rppm (predict.rppm), 1083
fitted.slrm, 44, 461, 1354
fixef, 462
fixef.mppm, 217, 218, 462, 1142
flipxy, 29, 30, 33, 54, 60, 61, 463, 463, 794,
1159, 1260
flipxy.infline (rotate.infline), 1260
flipxy.layered (methods.layered), 793
flu, 28
FmultiInhom, 464
foo, 465
formula, 466, 467, 789, 793, 802, 809, 820,
831
formula.dppm (methods.dppm), 789
formula.fv, 466
formula.kppm, 40
formula.kppm (methods.kppm), 792

1530
formula.lppm (methods.lppm), 801
formula.ppm, 41, 467, 738, 1046
formula.slrm (methods.slrm), 809
formula<- (formula.fv), 466
fourierbasis, 298, 468
fourierbasisraw (fourierbasis), 468
Frame, 30, 349, 469
Frame<- (Frame), 469
fryplot, 36, 470, 645, 646
frypoints (fryplot), 470
funxy, 472, 792
fv, 97, 149, 150, 467, 473, 1258
fv.object, 64, 97, 101, 102, 221, 222, 295,
296, 335, 376, 379, 389, 390, 408,
417, 418, 449, 454, 466, 474, 475,
476, 477, 478, 483, 485, 488, 491,
497, 499, 502, 512, 519, 539, 548,
599, 601, 604, 607, 609, 615, 617,
621, 623, 627, 633, 635, 639, 643,
652, 656, 662, 666, 676, 677, 679,
680, 684, 698, 700, 701, 703, 706,
707, 710, 712, 713, 715, 720, 727,
729, 731, 753, 757, 768, 816, 917,
920, 927, 931, 966, 1104, 1106,
1108, 1355, 1356, 1385, 1386, 1448,
1494, 1495
fvnames, 96, 97, 477, 966, 1071
fvnames<- (fvnames), 477
G3est, 39, 258, 401, 445, 478, 611, 860, 861
gam, 509, 829
gam.control, 829, 1050
ganglia, 1063
gauss.hermite, 346, 480
Gcom, 47, 229, 481, 502, 615, 1371
Gcross, 37, 65, 66, 484, 487, 489, 498, 500,
599
Gdot, 37, 66, 484, 486, 487, 498, 500, 599,
602, 609
Gest, 36, 63, 64, 66, 193, 194, 256, 259, 261,
390, 451, 476, 483–489, 490,
495–500, 502, 597, 603–605, 634,
857, 858, 1071, 1385, 1386
getCall, 736
getCall.mppm (logLik.mppm), 735
Geyer, 42, 139, 493, 493, 888, 908, 911, 1041,
1043, 1051, 1056, 1092, 1108, 1155,
1215, 1221, 1222, 1235, 1240,
1313–1315
Gfox, 40, 494
Ginhom, 36, 454, 496, 501, 607
glm, 25, 509, 820, 829, 830, 1041, 1049–1051
glm.control, 829, 1050

INDEX
Gmulti, 37, 38, 484, 486, 487, 489, 498, 501,
599, 601
GmultiInhom, 500
gordon, 28
gorillas, 28
Gres, 47, 229, 483, 502, 663, 1104, 1106,
1108, 1371
gridcenters (gridcentres), 503
gridcentres, 43, 277, 503, 1128, 1129, 1383,
1388
gridweights, 43, 329, 504, 1129
grow.box3 (grow.boxx), 505
grow.boxx, 505
grow.rectangle, 506, 506, 1445
grow.simplepanel, 1303
grow.simplepanel (simplepanel), 1339
Halton, 1283, 1284
Halton (quasirandom), 1136
Hammersley, 1283
Hammersley (quasirandom), 1136
hamster, 28
Hardcore, 42, 507, 911, 1041, 1043, 1051,
1054, 1056, 1092, 1190, 1215, 1221,
1222, 1235, 1240
harmonic, 509, 1031
harmonise, 510, 511, 513–515, 812
harmonise.fv, 37, 408, 510, 511, 511
harmonise.im, 32, 228, 233, 410, 510, 511,
512, 515, 542, 775, 777
harmonise.msr, 513
harmonise.owin, 514
harmonise.unitname (methods.unitname),
812
harmonize (harmonise), 510
harmonize.fv (harmonise.fv), 511
harmonize.im (harmonise.im), 512
harmonize.owin (harmonise.owin), 514
harmonize.unitname (methods.unitname),
812
has.close, 515
has.offset (model.depends), 820
head, 517
head.hyperframe, 35
head.ppp (headtail), 517
head.ppx (headtail), 517
head.psp (headtail), 517
head.tess (headtail), 517
headtail, 517
heat.colors, 942, 971
Hest, 39, 494, 495, 518
hexagon, 521, 892, 1061
hexagon (regularpolygon), 1159

INDEX
hexgrid (hextess), 520
hextess, 33, 520, 1124, 1160, 1420
HierHard, 42, 522, 525, 527
hierpair.family, 523
HierStrauss, 42, 523, 524, 524, 527
HierStraussHard, 42, 523, 525, 525
hist, 449, 485, 488, 491, 499, 527–529, 609,
652, 656, 1257, 1258, 1488
hist.default, 527–529, 844
hist.funxy, 527
hist.im, 32, 527, 528, 543, 973
hopskel, 194, 529
hopskel.test, 196
hsv, 1188
hsvim, 32
hsvim (rgbim), 1187
humberside, 28
Hybrid, 42, 531, 532, 533, 583, 1041, 1043,
1051, 1056, 1221, 1222, 1235, 1240
hybrid.family, 532
hyperframe, 35, 90, 103–105, 181, 182, 188,
420, 533, 554, 761, 824, 829, 830,
967, 968, 1072, 1077, 1078, 1375,
1496
hyytiala, 28
identify, 535, 536
identify.default, 535
identify.lpp, 200
identify.lpp (identify.ppp), 534
identify.ppp, 29, 198, 201, 202, 534, 536
identify.psp, 535
idw, 536, 1359, 1360
Iest, 38, 538
im, 27, 31, 175, 177, 540, 543, 544, 721, 824,
895, 906, 949, 1174, 1376, 1379,
1425, 1440
im.apply, 542
im.object, 50, 76, 107, 240, 244, 245, 263,
264, 292, 293, 295, 409, 410, 421,
422, 528, 529, 537, 541, 543, 584,
646, 686, 782, 783, 824, 942, 943,
969, 973, 1042, 1051, 1081, 1086,
1136, 1169, 1188, 1234, 1246, 1249,
1251, 1270, 1275, 1281, 1360, 1367,
1402
image, 803, 998, 1001, 1002, 1021, 1372
image.default, 107, 959, 969–973, 983, 996,
997, 1020, 1440
image.im (plot.im), 969
image.imlist (plot.imlist), 974
image.listof, 954, 955, 987, 988, 1018, 1019
image.listof (plot.imlist), 974

1531
image.objsurf (methods.objsurf), 803
image.ssf (plot.ssf), 1020
imcov, 32, 251, 335, 544, 1331
improve.kppm, 40, 545, 546, 658, 659
incircle, 31, 547
increment.fv, 548
infline, 191, 202, 549, 1261, 1488
influence, 551
influence.ppm, 46, 300, 423, 424, 550, 687,
975, 976, 1058, 1059
inforder.family, 524, 533, 552, 908, 911,
1446
inradius, 31
inradius (incircle), 547
insertVertices, 552, 723, 799
inside.boxx, 554
inside.owin, 31, 503, 504, 555, 1383, 1388
integral, 559, 811
integral (integral.im), 556
integral.im, 32, 556, 558, 646, 1402
integral.linfun (integral.linim), 557
integral.linim, 114, 557
integral.msr, 558, 1183
integral.ssf (methods.ssf), 810
intensity, 36, 560, 562–568, 856
intensity.detpointprocfamily
(intensity.dppm), 561
intensity.dppm, 561
intensity.lpp, 561, 743, 801
intensity.ppm, 41, 560, 562, 565
intensity.ppp, 560, 562, 563, 564
intensity.ppx, 565
intensity.psp, 566
intensity.quadratcount, 36, 567, 1123,
1124, 1195
intensity.splitppp (intensity.ppp), 564
interp.colourmap, 35, 223, 225, 227, 568,
1450
interp.colours (colourtools), 225
interp.im, 32, 154, 569
intersect.owin, 31, 433, 570, 573, 582, 890
intersect.tess, 33, 443, 572, 1420
invoke.symbolmap, 573, 1022, 1418
iplot, 29, 574, 597, 1006
iplot.layered, 575
iplot.linnet, 34
iplot.ppp, 575
ippm, 300, 551, 576, 687, 827, 1043, 1044,
1054, 1056, 1059
is.colour (colourtools), 225
is.connected, 578, 580
is.connected.ppp, 579, 579

1532
is.convex, 31, 247, 580
is.dppm, 581
is.empty, 581
is.grey (colourtools), 225
is.hybrid, 41, 582
is.im, 32, 584
is.kppm (is.ppm), 592
is.lpp, 584
is.lppm (is.ppm), 592
is.marked, 585, 586–588, 595, 1047
is.marked.lppm (is.marked.ppm), 586
is.marked.ppm, 585, 586, 588, 1047
is.marked.ppp, 585, 586, 587
is.mask, 31
is.mask (is.rectangle), 593
is.multitype, 588, 590, 591, 1047
is.multitype.lpp (is.multitype.ppp), 590
is.multitype.lppm (is.multitype.ppm),
589
is.multitype.ppm, 589, 589, 591, 1047
is.multitype.ppp, 589, 590, 590
is.owin, 591
is.poisson, 1047
is.poisson (is.stationary), 594
is.poisson.ppm, 1047
is.polygonal, 31
is.polygonal (is.rectangle), 593
is.ppm, 592
is.ppp, 593
is.psp, 32
is.rectangle, 31, 593
is.slrm (is.ppm), 592
is.stationary, 594, 1047
is.stationary.ppm, 1047
is.subset.owin, 31, 571, 596
istat, 35, 575, 597
japanesepines, 28
Jcross, 37, 65, 66, 598, 601, 602, 608, 609
Jdot, 37, 66, 598, 600, 600, 608, 609
Jest, 36, 63, 64, 66, 390, 451, 476, 492, 495,
538, 539, 597–602, 603, 606–609,
634
Jfox, 40
Jfox (Gfox), 494
Jinhom, 36, 454, 497, 605, 606
Jmulti, 37, 38, 598, 600–602, 608
K3est, 39, 401, 445, 479, 610, 899, 925
kaplan.meier, 451, 492, 605, 611, 641, 642,
1158
Kcom, 47, 229, 483, 613, 662, 663, 1371

INDEX
Kcross, 37, 65, 66, 205, 380, 616, 620, 621,
623, 633, 634, 652, 653, 675, 676,
698, 754, 757, 767, 768, 916–920,
928, 931
Kcross.inhom, 38, 618, 627, 657, 677, 678,
699
Kdot, 37, 66, 380, 617, 618, 622, 624, 626,
627, 633, 634, 652, 653, 679, 701,
754, 757, 768, 916–920, 927, 931,
932
Kdot.inhom, 38, 621, 625, 657, 680, 681, 702
kernel.factor, 345, 628, 630, 631
kernel.moment, 629, 629, 631
kernel.squint, 629, 630
Kest, 36, 63, 64, 66, 96, 97, 164, 178, 179,
205, 212, 232, 234, 274, 356, 357,
366, 368, 379, 390, 394, 406–408,
451, 471, 472, 476, 492, 597, 605,
613–615, 617, 618, 621, 623, 624,
627, 631, 635, 636, 638, 640, 646,
647, 649, 651–653, 659, 661, 663,
665–668, 684, 685, 688, 690, 719,
720, 726–728, 741, 742, 753, 756,
771, 772, 816, 896, 897, 900, 903,
916–920, 922, 923, 927, 966, 1039,
1148, 1371, 1385, 1427, 1428, 1449,
1469, 1472, 1473, 1495
Kest.fft, 37, 635
Kinhom, 36, 212, 356, 357, 620, 621, 626, 627,
633, 634, 636, 656, 659, 661, 665,
719, 720, 729, 741, 742, 916–920,
935, 936
km.rs, 451, 492, 605, 612, 641, 1158
Kmark, 38, 642, 757
Kmeasure, 31, 37, 471, 472, 543, 635, 636, 644
Kmodel, 179, 647, 649–651, 1371, 1473
Kmodel.detpointprocfamily
(Kmodel.dppm), 648
Kmodel.dppm, 357, 648
Kmodel.kppm, 40, 647, 649, 651, 661
Kmodel.ppm, 41, 647, 649, 650
Kmulti, 37, 38, 617, 618, 623, 624, 633, 634,
651, 655, 657, 917, 919, 920
Kmulti.inhom, 621, 627, 654
kppm, 40, 45, 102, 128, 178–181, 210, 212,
213, 215, 546, 595, 649, 657, 734,
735, 772, 774, 793, 817, 822–824,
827, 883, 884, 977, 1041, 1044,
1074, 1075, 1100, 1146, 1147, 1180,
1201, 1202, 1206, 1252, 1302, 1312,
1313, 1346, 1403, 1404, 1428, 1430,
1460, 1472, 1473, 1475, 1477

INDEX
Kres, 47, 229, 502, 615, 662, 1104, 1106,
1108, 1371
ks.test, 184–186, 188
Kscaled, 37, 664
Ksector, 37, 667, 903
labels, 789, 793, 809
labels.dppm (methods.dppm), 789
labels.kppm (methods.kppm), 792
labels.slrm (methods.slrm), 809
LambertW, 668
lansing, 28, 1063
lapply, 1364
laslett, 669, 978
latest.news, 146, 163, 671
layered, 35, 111, 125, 425, 672, 673, 674,
794, 979, 980, 1499
layerplotargs, 673, 673, 979, 980
layerplotargs<- (layerplotargs), 673
layout.boxes, 674, 1340, 1341
Lcross, 37, 65, 66, 675, 677–679
Lcross.inhom, 38, 677, 681
Ldot, 37, 66, 676, 678, 680, 681
Ldot.inhom, 38, 680
legend, 965
lengths.psp, 33, 68, 681, 814, 1102
LennardJones, 42, 682, 911, 1041, 1043,
1051, 1056, 1092, 1155, 1215, 1232,
1235, 1240
Lest, 36, 66, 270, 274, 303, 597, 676, 679,
684, 720, 727, 741
letterR, 30
levelset, 32, 685, 1367
leverage (leverage.ppm), 686
leverage.ppm, 46, 98, 300, 426, 551, 686,
980–982, 1058, 1059
lgcp.estK, 40, 179, 661, 688, 693, 772, 774,
815, 817, 1202, 1371, 1428, 1473
lgcp.estpcf, 40, 181, 661, 690, 691, 1475
lineardirichlet, 694, 724
lineardisc, 34, 695
linearK, 38, 398, 696, 698, 700, 701, 703,
704, 709, 743
linearKcross, 38, 697, 701, 743
linearKcross.inhom, 38, 699
linearKdot, 38, 698, 700, 700, 703, 743
linearKdot.inhom, 38, 702
linearKinhom, 38, 703, 716, 717, 743
linearmarkconnect, 38, 705, 707, 743
linearmarkequal, 39, 706, 707
linearpcf, 38, 706, 707, 708, 710, 712, 713,
716, 717, 743
linearpcfcross, 39, 706, 707, 709, 713, 715

1533
linearpcfcross.inhom, 39, 711, 715
linearpcfdot, 39, 710, 712, 712
linearpcfdot.inhom, 39, 714
linearpcfinhom, 38, 709, 715
linequad, 717
lines, 746, 944
linfun, 43, 112, 162, 265, 338, 718, 795, 867
Linhom, 36, 677, 678, 680, 681, 719, 728, 729,
741
linim, 43, 114, 265, 287, 411, 558, 720, 779,
783, 983, 1076
linnet, 34, 57, 115–117, 198, 284, 318, 343,
696, 721, 722, 724, 725, 743, 799,
984, 1273, 1274, 1306, 1426, 1442,
1443
lintess, 265, 343, 694, 723, 985, 1432
lixellate, 723, 724, 799
lm, 25, 509, 820, 830
lme, 829, 831
lmeControl, 829
load, 1462
localK, 36, 634, 726, 728, 729, 731, 741, 742
localKinhom, 37, 727, 728, 731, 741, 742
localL, 36, 728, 729
localL (localK), 726
localLinhom, 37, 727
localLinhom (localKinhom), 728
localpcf, 36, 730, 740–742
localpcfinhom, 37, 741, 742
localpcfinhom (localpcf), 730
locator, 196, 197, 199–202
locfit, 1193, 1194
loess, 378, 753, 755, 759, 767, 1355, 1356
logLik, 732, 734, 736, 737, 739, 802
logLik.dppm, 732
logLik.kppm, 733
logLik.lppm (methods.lppm), 801
logLik.mppm, 735
logLik.ppm, 41, 468, 733, 735, 737, 1046
logLik.slrm, 44, 739, 1354
lohboot, 37, 45, 389, 634, 740, 923, 1469
longleaf, 29, 585–589, 591, 1063
lpp, 27, 35, 58, 117, 118, 265, 287, 431, 697,
705, 709, 717, 725, 742, 744, 745,
801, 859, 875, 898, 990, 1076, 1274,
1306
lppm, 43, 69, 457, 595, 718, 744, 803,
822–824, 827, 991, 1075, 1347
Lscaled (Kscaled), 664
lurking, 310, 312, 313, 745, 750, 1112
lurking.mppm, 749
lurking.ppm, 750

1534
lut, 223, 751
mad.progress, 46
mad.progress (dclf.progress), 269
mad.sigtrace (dclf.sigtrace), 271
mad.test, 46, 151, 152, 270, 272, 301, 302,
307, 308, 388, 390
mad.test (dclf.test), 273
markconnect, 37, 380, 706, 707, 752, 757,
768, 928, 932
markcorr, 38, 380, 643, 644, 754, 754, 759,
768
markcorrint (Kmark), 642
markcrosscorr, 38, 757, 758
markmean, 38
markmean (Smooth.ppp), 1358
marks, 29, 760, 762–764, 811
marks.psp, 32, 762, 1101
marks.ssf, 1384
marks.ssf (methods.ssf), 810
marks.tess, 763, 1420
marks<-, 27
marks<-.psp, 32
marks<- (marks), 760
marks<-.lpp (methods.lpp), 800
marks<-.psp (marks.psp), 762
marks<-.ssf (methods.ssf), 810
marks<-.tess (marks.tess), 763
markstat, 38, 79, 205, 764, 766, 767
marktable, 38, 79, 765, 766, 870
markvar, 38
markvar (Smooth.ppp), 1358
markvario, 38, 380, 754, 757, 767
matchingdist, 769, 1067, 1068, 1070
matclust.estK, 40, 661, 690, 770, 774, 815,
817, 1428
matclust.estpcf, 40, 661, 693, 772
Math.im, 775, 778, 782, 783, 1367, 1487
Math.imlist, 776
Math.linim, 721, 778
matrix, 541
matrixinvsqrt (matrixpower), 780
matrixpower, 780
matrixsqrt (matrixpower), 780
max, 811, 1142
max.fv (range.fv), 1142
max.ssf (methods.ssf), 810
maxnndist, 781
mctest.progress, 271
mctest.progress (dclf.progress), 269
mctest.sigtrace (dclf.sigtrace), 271
mean, 782–784
mean.im, 32, 543, 782, 784, 1402

INDEX
mean.linim, 783
measureNegative (measureVariation), 784
measurePositive (measureVariation), 784
measureVariation, 784, 833
median, 782–784, 1486
median.im (mean.im), 782
median.linim (mean.linim), 783
mergeLevels, 785, 1161
methods.box3, 786
methods.boxx, 788
methods.dppm, 357, 789
methods.fii, 456, 790
methods.funxy, 791
methods.kppm, 661, 792, 1460
methods.layered, 673, 674, 793
methods.linfun, 43, 338, 719, 795, 867
methods.linim, 796
methods.linnet, 34, 115, 116, 723, 798
methods.lpp, 35, 743, 800
methods.lppm, 745, 801, 991
methods.objsurf, 803, 883, 884
methods.pp3, 804
methods.ppm (ppm.object), 1046
methods.ppx, 743, 801, 805
methods.rho2hat, 806, 1191
methods.rhohat, 807, 1195
methods.slrm, 809
methods.ssf, 810, 1384, 1385
methods.unitname, 812
methods.zclustermodel, 813, 1501
midpoints.psp, 33, 68, 383, 682, 814, 1102
min, 811, 1142
min.fv (range.fv), 1142
min.ssf (methods.ssf), 810
mincontrast, 40, 102, 178–181, 211, 212,
356, 357, 659–661, 689, 690, 692,
693, 771–774, 815, 883, 884,
1427–1431, 1472–1475
MinkowskiSum, 405, 817
minnndist (maxnndist), 781
miplot, 36, 818, 1124
model.covariates (model.depends), 820
model.depends, 41, 820
model.frame, 822
model.frame.dppm (model.frame.ppm), 821
model.frame.glm, 822
model.frame.kppm (model.frame.ppm), 821
model.frame.lppm (model.frame.ppm), 821
model.frame.ppm, 41, 468, 738, 821, 1046
model.images, 41, 823, 827, 828
model.is.additive (model.depends), 820
model.matrix, 362, 820, 821, 824, 825, 827,

INDEX
828
model.matrix.dppm (model.matrix.ppm),
826
model.matrix.ippm (model.matrix.ppm),
826
model.matrix.kppm (model.matrix.ppm),
826
model.matrix.lm, 823–825, 827, 828
model.matrix.lppm (model.matrix.ppm),
826
model.matrix.mppm, 825
model.matrix.ppm, 468, 738, 822–824, 826,
1046
model.matrix.slrm, 828
mppm, 70, 71, 189, 217, 218, 313, 458, 462,
736, 750, 825, 829, 991, 992,
1076–1078, 1120, 1121, 1141, 1181,
1182, 1348, 1394, 1395, 1478
msr, 312, 431, 432, 514, 559, 748, 785, 832,
993, 1179, 1180, 1184, 1357, 1377,
1497
mucosa, 29
MultiHard, 42, 508, 523, 834, 838, 839, 911,
1043, 1054, 1138, 1140, 1215
multiplicity (multiplicity.ppp), 835
multiplicity.ppp, 365, 835, 1062, 1452
MultiStrauss, 42, 525, 835, 837, 839, 911,
1041, 1043, 1051, 1056, 1155, 1215,
1221, 1222, 1230, 1235, 1238, 1240
MultiStraussHard, 42, 527, 835, 838, 911,
1041, 1043, 1051, 1056, 1155, 1215,
1221, 1222, 1230, 1235, 1238, 1240
murchison, 29
nbfires, 29
nearest.neighbour (Gest), 490
nearest.raster.point, 31, 840, 893, 894
nearestsegment, 33, 343, 841, 1096
nestsplit, 842
news, 163, 671
nlm, 576, 577
nnclean, 36, 216, 843
nncorr, 845
nncross, 33, 37, 205, 341–343, 530, 848, 857,
858, 860, 862, 873, 874, 876–878,
1097
nncross.lpp, 39, 743, 850
nncross.pp3, 39, 852
nndensity (nndensity.ppp), 855
nndensity.ppp, 855
nndist, 37, 194, 205, 254, 256–261, 491, 492,
516, 530, 781, 845, 850, 854, 856,

1535
861–864, 873, 874, 876–878, 896,
897, 899–902, 1387
nndist.lpp, 39, 743, 852, 858, 865
nndist.pp3, 39, 860
nndist.ppp, 864
nndist.ppx, 39, 861
nndist.psp, 857, 858, 863
nnfromvertex, 864
nnfun, 37, 107, 865, 867
nnfun.lpp, 39, 338, 719, 743, 866
nnmap, 37, 855, 867
nnmark, 29, 537, 869, 1021, 1359, 1360
nnmean, 38
nnmean (nncorr), 845
nnorient, 871, 903
nnvario, 38
nnvario (nncorr), 845
nnwhich, 37, 491, 492, 857, 858, 860–862,
870, 872, 877, 878
nnwhich.lpp, 39, 743, 852, 875
nnwhich.pp3, 39, 873, 876
nnwhich.ppx, 39, 877
nobjects, 878
nobs, 732, 734, 737, 802
nobs.dppm (logLik.dppm), 732
nobs.kppm (logLik.kppm), 733
nobs.lppm (methods.lppm), 801
nobs.mppm (logLik.mppm), 735
nobs.ppm, 1046
nobs.ppm (logLik.ppm), 737
npfun, 879
npoints, 29, 34, 879, 880, 881
nsegments, 799, 801, 881
nsegments.linnet (methods.linnet), 798
nsegments.lpp (methods.lpp), 800
nvertices, 799, 882
nvertices.linnet (methods.linnet), 798
nztrees, 29, 1063
objsurf, 804, 883
offset, 820
onearrow, 994, 995, 1024
onearrow (yardstick), 1499
opening, 30, 209, 582, 884, 893
Ops.im (Math.im), 775
Ops.imlist (Math.imlist), 776
Ops.linim (Math.linim), 778
Ops.msr, 833, 885
optim, 177–180, 211, 355, 357, 413, 659, 660,
688, 689, 691, 692, 771–774, 815,
816, 1427–1430, 1471, 1472, 1474,
1475
options, 1370, 1373

1536
Ord, 42, 886, 888, 911, 1041, 1043, 1051,
1056, 1155
ord.family, 524, 533, 552, 888, 908, 911,
1446
OrdThresh, 42, 887, 888, 888, 911, 1041,
1043, 1051, 1056, 1092, 1154, 1155,
1240
osteo, 29
overlap.owin, 368, 571, 889
owin, 27, 30, 51, 81, 126, 127, 134, 156, 158,
190, 209, 235, 247, 248, 252, 318,
322, 323, 330, 331, 377, 403, 404,
438, 503, 545, 555, 581, 594, 885,
890, 893, 996, 1060–1064,
1100–1102, 1144, 1148, 1160, 1196,
1292, 1296, 1331, 1339, 1343, 1370,
1371, 1384, 1388, 1453
owin.object, 81, 107, 119, 124, 126, 130,
133, 142, 190, 235, 318, 330, 377,
402, 422, 435, 438, 507, 556, 571,
592, 686, 840, 841, 891, 892, 893,
940, 997, 1062, 1101, 1169, 1179,
1246, 1249, 1262, 1271, 1276, 1308,
1331, 1367, 1384, 1445, 1484, 1493
padimage, 894
pairdist, 37, 254, 256–261, 858, 861, 862,
895, 899–901
pairdist.default, 895, 896, 896, 900
pairdist.lpp, 39, 897
pairdist.pp3, 39, 898
pairdist.ppp, 895, 896, 899, 902
pairdist.ppx, 39, 900
pairdist.psp, 895, 896, 900, 901
pairorient, 872, 902
PairPiece, 42, 139, 888, 904, 911, 1041,
1043, 1051, 1056, 1155, 1215, 1221,
1222, 1230, 1235, 1238, 1240, 1313,
1314
pairs, 906, 907, 911
pairs.default, 905–907, 912
pairs.im, 905, 907, 911, 912
pairs.linim, 907
pairsat.family, 139, 524, 533, 552, 888,
908, 911, 1314, 1315, 1446
Pairwise, 42, 321, 888, 909, 911, 939, 1041,
1043, 1051, 1056, 1155
pairwise.family, 85, 238, 320, 452, 494,
508, 524, 533, 552, 683, 835, 838,
839, 888, 905, 908–910, 910, 1364,
1390, 1391, 1446
palette, 226, 227
paletteindex (colourtools), 225

INDEX
panel.contour, 906, 911
panel.histogram (panel.contour), 911
panel.image, 906
panel.image (panel.contour), 911
panel.smooth, 912
par, 199–201, 906, 954, 962, 968, 972, 988,
1006, 1009, 1010, 1018, 1418
paracou, 29
parameters, 40, 41, 912, 1092
parent.frame, 1494, 1495
parres, 46, 53, 913, 1195
pcf, 36, 167, 174, 180, 181, 212, 234, 356,
357, 390, 597, 617, 618, 621, 623,
624, 627, 633, 634, 639, 640, 647,
649, 651–653, 656, 657, 660, 661,
666, 685, 691, 693, 720, 731,
740–742, 773, 774, 916, 918, 920,
923, 925, 927, 928, 931, 932, 936,
1429, 1431, 1474, 1475
pcf.fasp, 917, 917
pcf.fv, 666, 916, 917, 919
pcf.ppp, 167, 168, 179, 691, 773, 916, 917,
920, 921, 928–930, 932–937, 1429,
1474
pcf3est, 39, 401, 445, 479, 611, 924
pcfcross, 37, 65, 710, 713, 754, 926,
930–932, 937
pcfcross.inhom, 38, 711, 712, 715, 928, 933
pcfdot, 37, 713, 927, 928, 930, 933, 937
pcfdot.inhom, 38, 714, 930, 932
pcfinhom, 36, 167, 168, 212, 356, 357, 660,
661, 731, 741, 742, 930, 933, 934
pcfmodel, 181, 649, 650, 1475
pcfmodel (Kmodel), 647
pcfmodel.detpointprocfamily
(Kmodel.dppm), 648
pcfmodel.dppm, 357
pcfmodel.dppm (Kmodel.dppm), 648
pcfmodel.kppm, 40, 661
pcfmodel.kppm (Kmodel.kppm), 649
pcfmodel.ppm, 41
pcfmodel.ppm (Kmodel.ppm), 650
pcfmodel.zclustermodel
(methods.zclustermodel), 813
pcfmulti, 37, 928, 932, 936
pdf.options, 972
Penttinen, 42, 938, 1041, 1043, 1051, 1056,
1233, 1235, 1240, 1269
perimeter, 31, 81, 318, 369, 893, 939, 1339
periodify, 29, 30, 33, 940, 1334, 1336–1338
persp, 792, 803, 998, 1001, 1002, 1372
persp.default, 804, 942, 943

INDEX
persp.funxy (methods.funxy), 791
persp.im, 32, 245, 543, 544, 792, 942, 944,
945, 973, 981
persp.leverage.ppm (plot.leverage.ppm),
980
persp.objsurf (methods.objsurf), 803
perspContour (perspPoints), 944
perspLines, 943
perspLines (perspPoints), 944
perspPoints, 943, 944
perspSegments (perspPoints), 944
pictex, 997
pixelcentres, 31, 32, 945, 1144
pixellate, 31, 799, 946, 948–951, 1084, 1352
pixellate.linnet, 34, 946
pixellate.linnet (methods.linnet), 798
pixellate.owin, 31, 946, 947
pixellate.ppp, 30, 210, 288, 290, 946, 948,
948
pixellate.psp, 33, 120, 293, 799, 946, 950
pixelquad, 43, 951, 1052
pkernel (dkernel), 344
pknn (rknn), 1198
plot, 790, 792, 795, 803, 806–808, 952, 960,
968, 977, 980, 986, 1002, 1006,
1009, 1015, 1016, 1021, 1023, 1025,
1092, 1114
plot.anylist, 75, 416, 952, 991, 1019, 1382,
1401
plot.bermantest, 148, 955
plot.cdftest, 185, 186, 957
plot.colourmap, 35, 223, 958
plot.default, 310, 746, 957, 1003
plot.diagppm (diagnose.ppm), 309
plot.dppm, 357, 789, 960, 1074
plot.ecdf, 955
plot.envelope, 388–390, 961
plot.fasp, 64–66, 447, 962
plot.fii (methods.fii), 790
plot.foo (foo), 465
plot.funxy, 337, 473, 866
plot.funxy (methods.funxy), 791
plot.fv, 37, 64, 97, 101, 270, 272, 335, 389,
390, 408, 449, 454, 466, 467,
474–478, 491, 497, 519, 539, 604,
607, 633, 676, 679, 684, 720, 727,
729, 731, 807, 808, 871, 903, 927,
931, 960–963, 964, 976, 977, 1072,
1372, 1373, 1448
plot.hyperframe, 35, 534, 967, 1496
plot.im, 32, 245, 422, 543, 544, 792, 795,
906, 911, 943, 969, 974, 975, 977,

1537
978, 981–983, 985, 993, 1013–1016,
1023, 1371, 1425
plot.imlist, 972, 974
plot.infline (infline), 549
plot.influence.ppm, 551, 975
plot.kppm, 40, 661, 793, 976, 1075, 1460
plot.laslett, 669, 671, 978
plot.layered, 35, 672, 673, 979
plot.leverage.ppm, 107, 687, 980
plot.linfun (methods.linfun), 795
plot.linim, 43, 721, 795, 796, 982, 990, 991
plot.linnet, 984, 989
plot.lintess, 724, 985
plot.listof, 429, 986, 992, 1005, 1019,
1020, 1405
plot.lpp, 535, 989, 1028
plot.lppm, 803, 990
plot.mppm, 991
plot.msr, 833, 992, 1179, 1180, 1183, 1357
plot.objsurf (methods.objsurf), 803
plot.onearrow, 994, 1499, 1500
plot.owin, 30, 51, 893, 894, 978, 995, 1005,
1006, 1008, 1009, 1023, 1372
plot.plotppm, 998, 1002
plot.pp3, 34, 999, 1372
plot.ppm, 41, 468, 738, 960, 961, 976, 977,
991, 992, 998, 999, 1001, 1046,
1047, 1081, 1082, 1088, 1372
plot.ppp, 29, 535, 975, 978, 989, 990, 993,
997, 998, 1003, 1010, 1019, 1020,
1028, 1062, 1064, 1372, 1373, 1386,
1387
plot.ppx (methods.ppx), 805
plot.psp, 32, 536, 984, 1007, 1102
plot.qqppm, 1111
plot.quad, 1009, 1090, 1114
plot.quadratcount, 1010, 1012, 1124
plot.quadrattest, 1011, 1012
plot.rho2hat (methods.rho2hat), 806
plot.rhohat (methods.rhohat), 807
plot.rpart, 1013
plot.rppm, 1013, 1084, 1097, 1283
plot.scan.test, 1014, 1320
plot.slrm, 44, 810, 1015, 1354
plot.solist, 64, 246, 441, 824, 974, 975,
978, 1016, 1365, 1411
plot.splitppp, 441, 1019, 1379
plot.ssf, 1020, 1384, 1385
plot.symbolmap, 574, 1004, 1021, 1418
plot.tess, 33, 1011, 1012, 1023, 1420
plot.textstring, 1024, 1499, 1500
plot.texturemap, 1025

1538
plot.yardstick, 1026, 1499, 1500
pmixpois (dmixpois), 345
points, 906, 944, 989, 1000, 1003, 1005,
1006, 1022, 1028
points.default, 1028
points.lpp, 990, 1027
pointsOnLines, 33, 1028, 1310
Poisson, 42, 110, 888, 911, 1029, 1041, 1043,
1051, 1056, 1092, 1155, 1215, 1221,
1222, 1235, 1240
poly, 1031, 1055
polyclip, 570
polygon, 196, 200, 982, 983, 996, 997, 1257
polynom, 509, 1031, 1371
polypath, 997
ponderosa, 29
pool, 234, 1032, 1033–1039, 1145
pool.anylist, 1032, 1036
pool.envelope, 389, 390, 1032, 1033, 1035
pool.fasp, 1032, 1034, 1035
pool.fv, 37, 1032, 1036, 1039
pool.quadrattest, 1037, 1121, 1122
pool.rat, 1032, 1036, 1038
pp3, 27, 33, 87, 160, 252, 401, 805, 1000,
1039, 1073, 1277, 1309
ppm, 41, 45, 52, 70, 72, 73, 83–85, 110, 128,
139, 149, 186, 219, 229, 237, 238,
268, 279, 309, 311, 313, 319–321,
355–357, 363, 374–376, 381, 382,
390, 413, 414, 452, 456, 457, 460,
467, 468, 481–483, 493, 494, 502,
508, 509, 522, 524, 526, 531, 532,
546, 576, 577, 583, 586, 589, 595,
613–615, 650, 651, 659, 661, 663,
683, 737, 738, 744–749, 820–824,
827, 829–831, 834, 835, 837–839,
887, 889, 904, 905, 910, 914, 939,
951, 952, 975, 998, 1001, 1002,
1030, 1040, 1046, 1047, 1050,
1078–1082, 1087, 1088, 1091, 1092,
1103, 1105–1107, 1110, 1112, 1114,
1115, 1128–1131, 1155, 1165, 1183,
1184, 1195, 1215, 1220–1222, 1230,
1235, 1238–1240, 1282, 1314, 1315,
1350, 1363, 1364, 1371, 1389–1391,
1395, 1399, 1400, 1447, 1461, 1467,
1468, 1479–1481
ppm.object, 85, 219, 238, 268, 313, 320, 321,
363, 375, 452, 455, 456, 460, 494,
508, 593, 683, 824, 827, 835, 838,
839, 887, 889, 905, 910, 939, 1001,
1002, 1042, 1043, 1046, 1054, 1056,

INDEX
1079, 1080, 1082, 1088, 1112, 1115,
1184, 1220, 1221, 1315, 1364, 1390,
1391, 1399, 1406, 1447, 1461
ppm.ppp, 661, 1040, 1043, 1048
ppm.quad, 1040, 1042, 1043
ppm.quad (ppm.ppp), 1048
ppmInfluence, 300, 551, 687, 1058
ppois, 346
ppp, 27, 129, 130, 177, 252, 280, 326, 332,
390, 723, 761, 892, 1029, 1043,
1056, 1060, 1063, 1064, 1215, 1278,
1310, 1323, 1414, 1453
ppp.object, 79, 129, 130, 143, 263, 267, 268,
277, 292, 295, 363–365, 383, 422,
435, 448, 491, 537, 539, 603, 632,
761, 765, 767, 836, 870, 881, 892,
1004, 1006, 1019, 1020, 1052, 1061,
1062, 1063, 1169, 1212, 1222, 1246,
1249, 1263, 1271, 1276, 1308, 1323,
1329, 1359, 1360, 1379, 1451, 1452,
1454
pppdist, 38, 770, 1065, 1068, 1070
pppmatching, 1068, 1070
pppmatching.object, 770, 1066–1068, 1069
PPversion, 1071
ppx, 27, 34, 104, 105, 161, 252, 437, 554, 761,
806, 1040, 1072, 1136, 1280, 1311,
1382
predict, 807, 808, 1073–1075, 1083, 1085
predict.dppm, 357, 789, 1073, 1471
predict.glm, 1055, 1082
predict.kppm, 40, 661, 793, 1074, 1460, 1471
predict.lppm, 43, 457, 745, 991, 1075, 1347
predict.mppm, 458, 1076
predict.ppm, 41, 376, 460, 468, 562, 563,
738, 1001, 1002, 1041, 1046, 1047,
1051, 1055, 1073–1075, 1078, 1083,
1088, 1164, 1371, 1471
predict.rho2hat (methods.rho2hat), 806
predict.rhohat (methods.rhohat), 807
predict.rppm, 1013, 1083, 1097, 1283
predict.slrm, 44, 810, 1015, 1016, 1084,
1351, 1354, 1482, 1483
predict.smooth.spline, 918–920
predict.zclustermodel
(methods.zclustermodel), 813
print, 549, 787–790, 793, 795, 797, 799,
801–803, 805–809, 811–813,
1086–1089, 1092
print.box3 (methods.box3), 786
print.boxx (methods.boxx), 788
print.default, 795, 812

INDEX
print.dppm (methods.dppm), 789
print.fii (methods.fii), 790
print.im, 543, 1085
print.infline (infline), 549
print.kppm (methods.kppm), 792
print.linfun (methods.linfun), 795
print.linim (methods.linim), 796
print.linnet (methods.linnet), 798
print.listof, 987, 988
print.lpp (methods.lpp), 800
print.lppm (methods.lppm), 801
print.mppm, 217, 218, 831
print.objsurf (methods.objsurf), 803
print.owin, 893, 1086, 1088, 1089, 1405
print.pp3, 881, 1040
print.pp3 (methods.pp3), 804
print.ppm, 41, 219, 531, 1002, 1041, 1046,
1047, 1051, 1082, 1087, 1372
print.ppp, 1087, 1088, 1373, 1408
print.ppx, 881, 1073
print.ppx (methods.ppx), 805
print.psp, 32, 1089, 1409
print.qqppm, 1111
print.quad, 1090
print.rho2hat (methods.rho2hat), 806
print.rhohat (methods.rhohat), 807
print.slrm (methods.slrm), 809
print.ssf (methods.ssf), 810
print.summary.fii (methods.fii), 790
print.summary.im (summary.im), 1402
print.summary.kppm (summary.kppm), 1403
print.summary.linim, 797
print.summary.lpp (methods.lpp), 800
print.summary.pp3 (methods.pp3), 804
print.summary.ppm, 1403
print.summary.ppm (summary.ppm), 1406
print.summary.quad (summary.quad), 1409
print.unitname (methods.unitname), 812
print.zclustermodel
(methods.zclustermodel), 813
proc.time, 1437
profilepl, 128, 452, 577, 1043, 1044, 1054,
1056, 1091
progressreport, 1093, 1372
project.ppm, 41, 1044, 1372, 1467, 1468
project.ppm (emend.ppm), 381
project2segment, 33, 342, 343, 841, 1095,
1097
project2set, 1096
prune, 1097
prune.rpart, 1097
prune.rppm, 1097, 1283

1539
ps.options, 972
pseudoR2, 1098
psib, 1099
psp, 27, 32, 77, 132, 133, 202, 723, 1029,
1100, 1102, 1203, 1272, 1278, 1310,
1370, 1414, 1421
psp.object, 132, 133, 263, 293, 383, 438,
763, 881, 1008, 1009, 1101, 1101,
1264, 1329, 1454
psst, 47, 229, 483, 502, 615, 663, 880, 1102,
1106, 1108, 1371
psstA, 47, 229, 483, 502, 615, 663, 1104,
1104, 1108, 1371, 1372
psstG, 47, 229, 483, 502, 615, 663, 1104,
1106, 1107, 1371, 1372
pyramidal, 29
qkernel (dkernel), 344
qknn (rknn), 1198
qmixpois (dmixpois), 345
qpois, 346
qqplot.ppm, 46, 278, 279, 310, 312, 313, 749,
1109
QQversion (PPversion), 1071
quad, 43
quad.mppm, 458
quad.object, 130, 253, 277, 328, 329, 439,
504, 505, 951, 952, 1010, 1052,
1090, 1113, 1115, 1128, 1129, 1383,
1388, 1410, 1450, 1451
quad.ppm, 460, 748, 822, 827, 833, 1047,
1114, 1182
quadform (sumouter), 1412
quadrat.test, 46, 136, 149, 186, 189, 1012,
1037, 1038, 1047, 1116, 1121–1124,
1127
quadrat.test.mppm, 1119
quadrat.test.ppm, 1047, 1120
quadrat.test.ppp, 1121
quadrat.test.splitppp, 1117, 1118, 1121
quadratcount, 36, 136, 567, 568, 819, 1011,
1116–1118, 1120, 1122, 1122, 1126,
1127, 1195
quadratresample, 28, 45, 47, 1118, 1122,
1124, 1125, 1127
quadrats, 33, 842, 1118, 1122, 1124, 1126,
1126, 1132, 1420, 1469
quadscheme, 42, 43, 73, 236, 253, 277, 482,
503, 504, 614, 833, 951, 952, 1043,
1052, 1056, 1090, 1103, 1105, 1107,
1114, 1127, 1182, 1184, 1292, 1293,
1297, 1382, 1383, 1388, 1415
quadscheme.logi, 1129, 1480

1540
quantess, 33, 843, 1131, 1420
quantile, 783, 784, 1132, 1134–1136, 1486
quantile.default, 740, 1132, 1134, 1135
quantile.density, 183, 1133
quantile.ewcdf, 412, 1134, 1134
quantile.im, 32, 543, 782, 783, 1134, 1135
quantile.linim (mean.linim), 783
quasirandom, 1136
quote, 475, 967, 968
rags, 1137, 1139, 1141
ragsAreaInter, 85, 1138, 1138, 1140, 1141
ragsMultiHard, 835, 1138, 1139, 1140
ranef, 1141
ranef.lme, 1141
ranef.mppm, 217, 218, 1141
range, 811, 1142
range.fv, 475, 1142
range.ssf (methods.ssf), 810
raster.x, 31, 891, 893, 1143
raster.xy, 31, 946
raster.xy (raster.x), 1143
raster.y, 31, 891, 893
raster.y (raster.x), 1143
rasterImage, 972
rat, 1038, 1039, 1144
rbind, 181
rbind.hyperframe, 35, 534
rbind.hyperframe (cbind.hyperframe), 181
rCauchy, 28, 40, 45, 178–181, 210, 214, 215,
1145, 1206, 1253, 1281, 1302
rcell, 28, 45, 1147, 1149, 1276
rcellnumber, 1148, 1149
rDGS, 27, 45, 320, 1150, 1153, 1190, 1269,
1294, 1296
rDiggleGratton, 27, 45, 1151, 1151, 1190,
1269, 1294, 1296
rdpp, 1153, 1344, 1345
reach, 73, 278, 791, 1047, 1154
reach.detpointprocfamily (reach.dppm),
1156
reach.dppm, 1156
reach.fii, 456, 791
reach.ppm, 1047
read.table, 1063, 1322
rect, 911
redraw.simplepanel (run.simplepanel),
1302
reduced.sample, 451, 492, 605, 612, 634,
641, 642, 1157
redwood, 29, 1063
redwoodfull, 29

INDEX
reflect, 29, 54, 62, 63, 250, 251, 463, 794,
1158, 1260
reflect.default, 63
reflect.im, 63
reflect.infline (rotate.infline), 1260
reflect.layered (methods.layered), 793
reflect.tess (affine.tess), 62
regularpolygon, 892, 1061, 1159
relevel, 786, 1160
relevel.im, 1160
relevel.ppp (relevel.im), 1160
relevel.ppx (relevel.im), 1160
reload.or.compute, 1161
relrisk, 36, 37, 170, 171, 290–292, 1162,
1162, 1164–1166, 1320, 1328
relrisk.ppm, 1163, 1163, 1168
relrisk.ppp, 1162, 1163, 1165, 1165, 1327
Replace.im, 1168
Replace.linim, 1170
requireversion, 1171
rescale, 56–58, 564, 566, 794, 812, 1054,
1172, 1174–1177, 1453
rescale.im, 1173, 1173
rescale.layered, 1173
rescale.layered (methods.layered), 793
rescale.linnet, 1173
rescale.linnet (affine.linnet), 55
rescale.lpp, 1173
rescale.lpp (affine.lpp), 57
rescale.owin, 1173, 1174, 1175, 1176
rescale.ppp, 1173, 1176
rescale.psp, 1173, 1177
rescale.unitname, 1173
rescale.unitname (methods.unitname), 812
rescue.rectangle, 59, 1178, 1262, 1275
reset.spatstat.options
(spatstat.options), 1370
residuals.dppm, 1179
residuals.kppm, 1180
residuals.mppm, 1181, 1182
residuals.ppm, 41, 311–313, 375, 468, 738,
748, 749, 827, 833, 1046, 1110,
1112, 1179–1181, 1182, 1357
residualspaper, 29, 47
rex, 141, 1185
RFsimulate, 1201
rGaussPoisson, 28, 45, 1147, 1186, 1202,
1206, 1253, 1276, 1281, 1302
rgb, 225, 226, 1188
rgb2hex (colourtools), 225
rgb2hsv, 226, 227
rgb2hsva (colourtools), 225

INDEX
rgbim, 32, 1187
rHardcore, 27, 45, 1151, 1153, 1188, 1269,
1294, 1296
rho2hat, 36, 46, 53, 807, 915, 1190, 1195
rhohat, 36, 46, 53, 808, 915, 1190, 1191, 1191
rhohat.lpp, 718
ripras, 30, 156, 248, 1195, 1323, 1414, 1415
rjitter, 27, 28, 45, 47, 1197
rkernel (dkernel), 344
rknn, 37, 1198
rlabel, 28, 1199, 1327
rLGCP, 40, 45, 659, 1200, 1346
rlinegrid, 33, 45, 1202
rlpp, 1203, 1274, 1306
rMatClust, 27, 40, 45, 210, 214, 215, 772,
774, 1064, 1147, 1187, 1202, 1204,
1208, 1209, 1253, 1276, 1281, 1302,
1346
rMaternI, 27, 45, 1064, 1207, 1209, 1276,
1291
rMaternII, 27, 45, 1064, 1208, 1276, 1291
rmax.Ripley (edge.Ripley), 365
rmax.Trans, 366, 368
rmax.Trans (edge.Trans), 367
rmh, 28, 45, 85, 125, 278, 279, 388, 835, 1047,
1050, 1053, 1110–1112, 1150–1153,
1189, 1190, 1209, 1211, 1215, 1221,
1222, 1224, 1225, 1227–1231, 1234,
1235, 1237–1242, 1269, 1294–1296,
1298, 1371, 1372, 1408
rmh.default, 1110, 1209, 1210, 1210,
1220–1222, 1276, 1298, 1349
rmh.ppm, 41, 43, 905, 1047, 1209, 1215, 1220,
1349, 1350
rmhcontrol, 278, 279, 1111, 1112,
1211–1213, 1220–1222, 1223, 1229,
1230, 1235, 1238–1242, 1349, 1350,
1371, 1463
rmhcontrol.default, 279, 1372
rmhexpand, 279, 414, 415, 1150, 1152, 1189,
1224, 1225, 1227, 1227, 1268, 1293,
1295, 1489
rmhmodel, 1047, 1155, 1222, 1224, 1227,
1229, 1229, 1231, 1237–1240, 1242,
1298
rmhmodel.default, 1211, 1230, 1230, 1237,
1238, 1240
rmhmodel.list, 1230, 1237, 1240
rmhmodel.ppm, 1047, 1230, 1238, 1239
rmhstart, 1211, 1220–1222, 1224–1227,
1230, 1235, 1238, 1240, 1241, 1298,
1349

1541
rmixpois (dmixpois), 345
RMmodel, 689, 690, 692, 693, 1201
rMosaicField, 45, 1242, 1244
rMosaicSet, 45, 1243, 1243
rmpoint, 27, 44, 1221, 1244, 1249, 1276
rmpoispp, 27, 45, 1140, 1209, 1221, 1246,
1248, 1275, 1276
rNeymanScott, 27, 45, 215, 1064, 1147, 1187,
1202, 1206, 1251, 1281, 1302, 1312,
1313
rnoise, 32, 1254
rnorm, 1254
roc, 36, 137, 138, 1255
rose, 192, 871, 903, 1256
rotate, 29, 30, 54–63, 415, 463, 794, 893,
1173, 1175, 1176, 1178, 1258, 1260,
1334, 1336–1338
rotate.im, 32, 63, 1259
rotate.infline, 550, 1260
rotate.layered (methods.layered), 793
rotate.linnet (affine.linnet), 55
rotate.lpp (affine.lpp), 57
rotate.owin, 63, 893, 1259, 1261, 1263, 1264
rotate.ppp, 1259, 1262, 1264
rotate.psp, 33, 1263
rotate.tess (affine.tess), 62
rotmean, 1264
round, 1266, 1267
round.pp3 (round.ppp), 1266
round.ppp, 1266, 1268
round.ppx (round.ppp), 1266
rounding, 1266, 1267
rpart, 1282
rPenttinen, 27, 45, 1151, 1153, 1190, 1233,
1268, 1294, 1296
rpoint, 27, 44, 1221, 1246, 1270, 1276, 1291,
1305, 1307, 1308
rpois, 346
rpoisline, 33, 45, 1203, 1271, 1273
rpoislinetess, 33, 45, 1243, 1244, 1272,
1420
rpoislpp, 35, 39, 743, 1273, 1306, 1347
rpoispp, 27, 44, 1064, 1147, 1187, 1201,
1202, 1206, 1208, 1209, 1221, 1249,
1251, 1253, 1274, 1278, 1281, 1291,
1302, 1308, 1312, 1313, 1351, 1352,
1371
rpoispp3, 34, 401, 1276, 1309
rpoisppOnLines, 28, 45, 1273, 1274, 1277
rpoisppx, 34, 1279, 1311
rPoissonCluster, 27, 1280
rppm, 1013, 1083, 1084, 1097, 1282

1542
rQuasi, 1137, 1283
rshift, 28, 45, 47, 1284, 1287, 1289, 1290
rshift.ppp, 1284, 1285, 1285, 1288–1290
rshift.psp, 1284, 1285, 1287, 1287
rshift.splitppp, 1284, 1285, 1289
rSSI, 27, 45, 1064, 1276, 1290
rstrat, 27, 43, 45, 1148, 1276, 1292, 1297
rStrauss, 27, 45, 1151, 1153, 1189, 1190,
1215, 1233, 1269, 1276, 1293, 1296
rStraussHard, 27, 45, 1151, 1153, 1190,
1269, 1294, 1295
rsyst, 27, 45, 1148, 1293, 1296
rtemper, 1297
rthin, 28, 45, 47, 1252, 1299, 1371
rThomas, 28, 40, 45, 210, 214, 215, 1064,
1147, 1187, 1206, 1253, 1276, 1281,
1300, 1346, 1428, 1430, 1431
rug, 807, 808
run.simplepanel, 1302, 1340, 1341
runif, 1254
runifdisc, 27, 45, 1305
runiflpp, 35, 39, 743, 1204, 1274, 1306
runifpoint, 27, 44, 1064, 1148, 1270, 1271,
1276, 1293, 1297, 1305, 1307, 1310,
1371
runifpoint3, 34, 1277, 1308
runifpointOnLines, 28, 45, 1029, 1278,
1306, 1309
runifpointx, 34, 1280, 1310
rVarGamma, 28, 40, 45, 210, 214, 215, 659,
1147, 1206, 1253, 1281, 1302, 1311,
1472, 1473, 1475
samecolour (colourtools), 225
sample, 1271, 1307
SatPiece, 42, 139, 494, 888, 908, 1041, 1043,
1051, 1056, 1313, 1314, 1315
Saturated, 42, 888, 908, 911, 1041, 1043,
1051, 1056, 1155, 1315
scalardilate, 29, 56–58, 62, 63, 794, 797,
1316
scalardilate.im, 63
scalardilate.layered (methods.layered),
793
scalardilate.linim (methods.linim), 796
scalardilate.linnet (affine.linnet), 55
scalardilate.lpp (affine.lpp), 57
scalardilate.owin, 63
scalardilate.tess (affine.tess), 62
scale, 1318
scaletointerval, 32, 1317
scan.test, 37, 46, 1014, 1015, 1318, 1321,
1322

INDEX
scanLRTS, 1015, 1319, 1320, 1320
scanpp, 1063, 1322
sdr, 324, 325, 1323, 1326
sdrPredict, 1325, 1326
segments, 198, 944, 985, 994, 1000, 1008,
1023, 1026
segregation.test, 46, 1327
selfcrossing.psp, 33, 263, 1328, 1329,
1373
selfcut.psp, 33, 116, 1329
sessionInfo, 1330
sessionLibs, 1330
set.seed, 1214, 1344, 1346, 1351
setcov, 31, 335, 368, 543, 545, 890, 1331,
1371
setmarks (marks), 760
setminus.owin, 31
setminus.owin (intersect.owin), 570
shapley, 29
sharpen, 216, 1332
sharpen.ppp, 29, 36, 37, 290, 292
shift, 29, 30, 54–63, 415, 463, 794, 797, 893,
941, 954, 988, 1018, 1173, 1175,
1176, 1178, 1260, 1317, 1333,
1334–1338, 1499
shift.im, 32, 63, 1260, 1334
shift.infline (rotate.infline), 1260
shift.layered (methods.layered), 793
shift.linim (methods.linim), 796
shift.linnet (affine.linnet), 55
shift.lpp (affine.lpp), 57
shift.owin, 63, 893, 1265, 1333, 1334, 1335,
1337, 1338
shift.ppp, 1333, 1334, 1336, 1336, 1338
shift.psp, 33, 1337
shift.tess (affine.tess), 62
shortside, 1339
shortside (diameter.box3), 314
shortside.box3, 34
shortside.boxx, 34
shortside.boxx (diameter.boxx), 316
shortside.owin (sidelengths.owin), 1338
sidelengths, 1339
sidelengths (diameter.box3), 314
sidelengths.boxx (diameter.boxx), 316
sidelengths.owin, 1338
simdat, 29, 1063
simplenet, 34, 723
simplepanel, 674, 675, 1303, 1304, 1339
simplify.owin, 30, 127, 249, 939, 1342
simulate, 808, 1092, 1344–1348, 1350–1352
simulate.detpointprocfamily, 349, 350

INDEX
simulate.detpointprocfamily
(simulate.dppm), 1343
simulate.dppm, 357, 789, 1343
simulate.kppm, 40, 45, 178, 180, 388, 661,
793, 1345, 1350, 1352, 1460, 1472,
1475
simulate.lppm, 1347
simulate.mppm, 1348
simulate.ppm, 28, 41, 43, 45, 278, 468, 738,
1046, 1222, 1346, 1348, 1349, 1352
simulate.rhohat (methods.rhohat), 807
simulate.slrm, 44, 810, 1351
slrm, 44, 74, 75, 220, 461, 595, 739, 809, 810,
823, 824, 828, 1015, 1016, 1085,
1352, 1352, 1482, 1483
Smooth, 153, 1354, 1356–1358, 1360, 1362
Smooth.fv, 37, 450, 1355, 1355
Smooth.im, 32, 1355
Smooth.im (blur), 153
Smooth.msr, 833, 993, 1355, 1356
Smooth.ppp, 29, 36, 37, 154, 173, 290–292,
537, 870, 949, 993, 1021, 1168,
1333, 1355, 1358, 1360, 1361, 1371
smooth.spline, 295, 296, 918–920, 1355,
1356
Smooth.ssf, 1360, 1385
Smoothfun (Smoothfun.ppp), 1361
Smoothfun.ppp, 1361
Softcore, 42, 888, 911, 1041, 1043, 1051,
1056, 1092, 1155, 1215, 1221, 1222,
1230, 1235, 1238, 1240, 1362
solapply, 135, 1364, 1366
solist, 76, 134, 135, 440, 441, 1364, 1365,
1365, 1411, 1499
solutionset, 32, 543, 686, 1366, 1487
source, 1462
spatdim, 1368
spatialcdf, 36, 528, 529, 1369, 1385
spatstat (spatstat-package), 25
spatstat-package, 25
spatstat.options, 30, 31, 42, 108, 119, 245,
382, 444, 483, 635, 636, 646, 792,
824, 966, 973, 996, 997, 999, 1000,
1002, 1005, 1080, 1082, 1106, 1225,
1227, 1308, 1370
spiders, 29, 35, 743
split, 1374, 1376, 1377, 1381, 1412, 1458
split.hyperframe, 534, 1374
split.im, 175, 1365, 1375
split.msr, 559, 785, 833, 1376, 1457, 1497
split.ppp, 29, 100, 111, 177, 294, 434, 435,
441, 842, 843, 845, 1019, 1020,

1543
1123, 1284, 1289, 1365, 1377, 1378,
1411, 1412, 1420, 1433, 1458
split.ppx, 1380
split<-.hyperframe (split.hyperframe),
1374
split<-.ppp (split.ppp), 1378
spokes, 43, 277, 1128, 1129, 1382
sporophores, 29
spruces, 29
square, 30, 891, 892, 1061, 1383
ssf, 811, 1021, 1361, 1384, 1498, 1502
step, 40, 42, 732, 734, 736, 738, 1046, 1353
stepfun, 1234
stieltjes, 1385
stienen, 1386
stienenSet (stienen), 1386
stratrand, 277, 504, 1128, 1129, 1383, 1387
Strauss, 42, 110, 381, 493, 494, 508, 835,
838, 839, 888, 904, 905, 911, 1030,
1041, 1043, 1051, 1056, 1091, 1092,
1154, 1155, 1215, 1221, 1222, 1230,
1235, 1238, 1240, 1294, 1389, 1467
StraussHard, 42, 452, 508, 888, 911, 1041,
1043, 1051, 1056, 1092, 1155, 1215,
1221, 1222, 1230, 1235, 1238, 1240,
1295, 1296, 1390
studpermu.test, 46, 1392
subfits, 313, 991, 1348, 1394
subset, 1395–1398
subset.hyperframe, 35, 1395
subset.lpp, 35, 265, 431
subset.lpp (subset.ppp), 1396
subset.pp3, 34
subset.pp3 (subset.ppp), 1396
subset.ppp, 29, 266, 435, 1396
subset.ppx, 34
subset.ppx (subset.ppp), 1396
subspaceDistance, 325, 1325, 1398
substitute, 475
suffstat, 1399
summary, 32, 35, 43, 790, 795, 797, 799, 801,
802, 805, 812, 1092, 1401–1403,
1405, 1406, 1408–1412
summary.anylist, 416, 1401
summary.fii (methods.fii), 790
Summary.im, 782, 1487
Summary.im (Math.im), 775
summary.im, 529, 543, 782, 783, 1086, 1402
Summary.imlist (Math.imlist), 776
summary.kppm, 40, 1403
summary.linfun (methods.linfun), 795
Summary.linim (Math.linim), 778

1544
summary.linim (methods.linim), 796
summary.linnet (methods.linnet), 798
summary.listof, 429, 1404
summary.lpp (methods.lpp), 800
summary.lppm (methods.lppm), 801
summary.mppm, 831
summary.owin, 893, 1087, 1405, 1408, 1409
summary.pp3 (methods.pp3), 804
summary.ppm, 41, 468, 595, 738, 1403, 1406
summary.ppp, 1088, 1212, 1405, 1407
summary.psp, 32, 68, 682, 814, 1089, 1408
summary.quad, 1090, 1409
summary.solist, 441, 1410
summary.splitppp, 441, 1411
summary.unitname (methods.unitname), 812
sumouter, 1412
superimpose, 29, 33, 77, 236, 1063, 1379,
1413, 1416, 1417
superimpose.lpp, 1414, 1415, 1416
svd, 780
Sweave, 1161, 1162, 1330
swedishpines, 29, 1063
symbolmap, 574, 1004, 1005, 1022, 1417, 1464
symbols, 1003, 1005, 1006, 1009, 1022
table, 767
tail, 517
tail.hyperframe, 35
tail.ppp (headtail), 517
tail.ppx (headtail), 517
tail.psp (headtail), 517
tail.tess (headtail), 517
terms, 467, 736, 789, 793, 802, 809, 820
terms.dppm (methods.dppm), 789
terms.kppm (methods.kppm), 792
terms.lppm (methods.lppm), 801
terms.mppm (logLik.mppm), 735
terms.ppm, 738, 1046
terms.ppm (formula.ppm), 467
terms.slrm (methods.slrm), 809
terrain.colors, 942, 971
tess, 27, 33, 136, 143, 175, 177, 240, 244,
266, 267, 280, 326, 442, 443, 521,
572, 573, 842, 1023, 1124, 1127,
1132, 1376, 1379, 1419, 1431,
1434–1436, 1444, 1469
test.crossing.psp, 1421
test.selfcrossing.psp
(test.crossing.psp), 1421
text, 1024, 1026, 1027, 1422
text.default, 535, 983, 1011, 1012, 1023,
1422, 1423
text.lpp, 990

INDEX
text.lpp (text.ppp), 1422
text.ppp, 1006, 1422
text.psp, 1009
text.psp (text.ppp), 1422
text.rpart, 1013
textstring, 1024
textstring (yardstick), 1499
texturemap, 51, 1025, 1026, 1423
textureplot, 51, 1025, 1026, 1423, 1424,
1424
thinNetwork, 241, 242, 723, 799, 1425
thomas.estK, 40, 179, 661, 690, 772, 774,
815, 817, 1427, 1431, 1473
thomas.estpcf, 40, 181, 661, 693, 774, 1429,
1475
tile.areas, 33, 1420, 1431, 1435, 1436
tile.lengths, 724, 1432
tileindex, 100, 1433
tilenames, 1132, 1420, 1431, 1434, 1435,
1436
tilenames<- (tilenames), 1434
tiles, 33, 443, 1420, 1431, 1434, 1435, 1436
tiles.empty, 1431, 1435, 1436
timed, 1437, 1438, 1439
timeTaken, 1438, 1438
to.grey, 972
to.grey (colourtools), 225
to.opaque (colourtools), 225
to.saturated (colourtools), 225
to.transparent (colourtools), 225
topo.colors, 942, 971
totalVariation (measureVariation), 784
transect.im, 32, 1439
transmat, 32, 1440
treebranchlabels, 162, 283, 284, 1441,
1443
treeprune, 1442, 1442
triangulate.owin, 31, 1444
trim.rectangle, 507, 1445
triplet.family, 1446, 1447
Triplets, 42, 1041, 1043, 1051, 1056, 1215,
1230, 1235, 1240, 1446, 1446
Tstat, 36, 207, 1448
tweak.colourmap, 35, 223–225, 227, 569,
1449
txtProgressBar, 1094
union.owin, 31, 334
union.owin (intersect.owin), 570
union.quad, 1114, 1450
unique.ppp, 29, 365, 836, 1062, 1451
unique.ppx (unique.ppp), 1451
uniroot, 668

INDEX
unit.square (square), 1383
unitname, 160, 161, 787, 788, 799, 801,
804–806, 813, 1047, 1172–1177,
1452
unitname.box3, 34
unitname.box3 (methods.box3), 786
unitname.boxx (methods.boxx), 788
unitname.linnet (methods.linnet), 798
unitname.lpp (methods.lpp), 800
unitname.pp3, 34
unitname.pp3 (methods.pp3), 804
unitname.ppm, 1047
unitname.ppx, 34
unitname.ppx (methods.ppx), 805
unitname<- (unitname), 1452
unitname<-.box3 (methods.box3), 786
unitname<-.boxx (methods.boxx), 788
unitname<-.linnet (methods.linnet), 798
unitname<-.lpp (methods.lpp), 800
unitname<-.pp3 (methods.pp3), 804
unitname<-.ppx (methods.ppx), 805
units, 1178
unmark, 29, 431, 434, 435, 437, 761, 801, 811,
1384, 1454
unmark.lpp (methods.lpp), 800
unmark.psp, 33
unmark.ssf (methods.ssf), 810
unmark.tess (marks.tess), 763
unnormdensity, 1455
unstack, 1456–1458
unstack.lpp (unstack.ppp), 1457
unstack.msr, 1456, 1458
unstack.ppp, 1457, 1457
unstack.psp (unstack.ppp), 1457
update, 802, 809, 831, 1459, 1461–1464
update.detpointprocfamily, 1458
update.interact, 1459
update.kppm, 40, 453, 496, 606, 620, 626,
637, 655, 661, 793, 934, 1460
update.lppm, 704, 716
update.lppm (methods.lppm), 801
update.ppm, 41, 453, 468, 481, 496, 606, 613,
620, 626, 637, 655, 704, 716, 738,
934, 1046, 1103, 1105, 1107, 1110,
1459, 1461
update.rmhcontrol, 279, 1221, 1222, 1463
update.slrm (methods.slrm), 809
update.symbolmap, 1418, 1464
urkiola, 29
valid, 381, 802, 1465, 1466, 1467
valid.detpointprocfamily, 1465, 1466
valid.lppm (methods.lppm), 801

1545
valid.ppm, 41, 382, 1044, 1055, 1056, 1465,
1467
varblock, 37, 45, 389, 634, 741, 742, 1126,
1468
varcount, 1470
vargamma.estK, 40, 179, 661, 1312, 1313,
1471, 1475
vargamma.estpcf, 40, 181, 661, 1312, 1313,
1473, 1473
vcov, 802, 1476–1479, 1481–1483
vcov.kppm, 40, 661, 793, 1075, 1460, 1476
vcov.lppm (methods.lppm), 801
vcov.mppm, 1477
vcov.ppm, 41, 70–73, 468, 738, 1046, 1164,
1406, 1477, 1478, 1479
vcov.slrm, 44, 810, 1354, 1482
vdCorput (quasirandom), 1136
vertexdegree (methods.linnet), 798
vertices, 580, 581, 799, 882, 1483
vertices.linnet, 34
vertices.linnet (methods.linnet), 798
View, 35, 517, 518
Vmark, 38
Vmark (Emark), 378
volume, 799, 1484
volume.box3, 34, 160, 1485
volume.box3 (diameter.box3), 314
volume.boxx, 34, 161, 1485
volume.boxx (diameter.boxx), 316
volume.linnet, 1485
volume.linnet (methods.linnet), 798
volume.owin (area.owin), 80
waka, 29
waterstriders, 29
weighted.median, 1485
weighted.quantile (weighted.median),
1485
weighted.var (weighted.median), 1485
where.max, 1486
where.min (where.max), 1486
which.max, 1486
which.min, 1486
whichhalfplane, 550, 1487
whist, 1488
will.expand, 1229, 1489
Window, 30, 349, 470, 799, 1490, 1493
Window.distfun (WindowOnly), 1491
Window.dppm (WindowOnly), 1491
Window.funxy (WindowOnly), 1491
Window.influence.ppm (WindowOnly), 1491
Window.kppm (WindowOnly), 1491
Window.layered (WindowOnly), 1491

1546
Window.leverage.ppm (WindowOnly), 1491
Window.linnet (methods.linnet), 798
Window.lpp (WindowOnly), 1491
Window.lppm (WindowOnly), 1491
Window.msr (WindowOnly), 1491
Window.nnfun (WindowOnly), 1491
Window.ppm, 1491
Window.ppm (WindowOnly), 1491
Window.ppp, 1493
Window.psp, 1493
Window.quad (WindowOnly), 1491
Window.quadratcount (WindowOnly), 1491
Window.quadrattest (WindowOnly), 1491
Window.rmhmodel (WindowOnly), 1491
Window.tess (WindowOnly), 1491
Window<- (Window), 1490
WindowOnly, 1491
with, 1494–1497
with.default, 1498
with.fv, 37, 150, 296, 388, 475, 1142, 1143,
1356, 1493
with.hyperframe, 35, 534, 968, 1395, 1495
with.msr, 785, 833, 886, 1377, 1496
with.ssf, 1385, 1498, 1502
X11, 997
X11.options, 997
xfig, 997
xy.coords, 117, 118, 569, 574, 944, 1252,
1439, 1487
yardstick, 995, 1024, 1026, 1027, 1499
zapsmall, 1500
zapsmall.im, 32, 1500
zclustermodel, 814, 1501

INDEX
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