Julia Pro V0.6.2.2 Package API Manual

User Manual: Pdf

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JuliaPro v0.6.2.2 API Manual
March 1, 2018
Julia Computing Inc.
info@juliacomputing.com
Contents
1 Distributions 1
2 StatsBase 46
3 PyPlot 81
4 IndexedTables 108
5 Images 139
6 Knet 155
7 DataFrames 171
8 DataStructures 185
9 JDBC 197
10 NNlib 208
11 ImageCore 214
12 Reactive 223
13 JuliaDB 230
14 Combinatorics 237
15 HypothesisTests 243
16 DataArrays 249
17 GLM 257
18 Documenter 263
19 ColorTypes 268
I
II CONTENTS
20 Primes 273
21 Roots 279
22 ImageTransformations 284
23 PyCall 288
24 Gadfly 293
25 IterTools 298
26 Iterators 306
27 Polynomials 314
28 Colors 318
29 FileIO 322
30 Interact 326
31 FFTW 330
32 AxisArrays 334
33 QuadGK 338
34 BusinessDays 341
35 NLSolversBase 344
36 Dagger 347
37 ShowItLikeYouBuildIt 350
38 CoordinateTransformations 353
39 Graphics 355
40 MacroTools 358
41 ImageMorphology 361
42 Contour 363
43 Compose 365
44 CoupledFields 368
CONTENTS III
45 AxisAlgorithms 370
46 Libz 372
47 NullableArrays 375
48 ImageMetadata 377
49 LegacyStrings 379
50 BufferedStreams 381
51 MbedTLS 383
52 DataValues 385
53 OnlineStats 387
54 NearestNeighbors 389
55 IJulia 390
56 WebSockets 392
57 AutoGrad 394
58 ComputationalResources 396
59 Clustering 398
60 JuliaWebAPI 400
61 DecisionTree 402
62 Blosc 404
63 Missings 406
64 Parameters 408
65 HDF5 410
66 HttpServer 412
67 MappedArrays 414
68 TextParse 415
69 LossFunctions 417
IV CONTENTS
70 LearnBase 418
71 Juno 420
72 HttpParser 422
73 ImageAxes 423
74 Flux 424
75 IntervalSets 425
76 Media 426
77 Rotations 427
78 Mustache 428
79 TiledIteration 429
80 Distances 430
81 HttpCommon 431
82 StaticArrays 432
83 SweepOperator 433
84 PaddedViews 434
85 SpecialFunctions 435
86 NamedTuples 436
87 Loess 437
88 Nulls 438
89 WoodburyMatrices 439
90 Requests 440
91 MemPool 441
92 SimpleTraits 442
93 BinDeps 443
Chapter 1
Distributions
1.1 Base.LinAlg.scale!
Base.LinAlg.scale! —Method.
scale!{D<:AbstractMvLogNormal}(::Type{D},s::Symbol,m::AbstractVector,S::AbstractMatrix,::AbstractMatrix)
Calculate the scale parameter, as defined for the location parameter above
and store the result in .
source
1.2 Base.mean
Base.mean —Method.
mean(d::Union{UnivariateMixture, MultivariateMixture})
Compute the overall mean (expectation).
source
1.3 Base.mean
Base.mean —Method.
mean(d::MatrixDistribution)
Return the mean matrix of d.
source
1
2CHAPTER 1. DISTRIBUTIONS
1.4 Base.mean
Base.mean —Method.
mean(d::MultivariateDistribution)
Compute the mean vector of distribution d.
source
1.5 Base.mean
Base.mean —Method.
mean(d::UnivariateDistribution)
Compute the expectation.
source
1.6 Base.median
Base.median —Method.
median(d::UnivariateDistribution)
Return the median value of distribution d.
source
1.7 Base.median
Base.median —Method.
median(d::MvLogNormal)
Return the median vector of the lognormal distribution. which is strictly
smaller than the mean.
source
1.8 Base.quantile
Base.quantile —Method.
quantile(d::UnivariateDistribution, q::Real)
Evaluate the inverse cumulative distribution function at q.
See also: cquantile,invlogcdf, and invlogccdf.
source
1.9. BASE.STD 3
1.9 Base.std
Base.std —Method.
std(d::UnivariateDistribution)
Return the standard deviation of distribution d, i.e. sqrt(var(d)).
source
1.10 Base.var
Base.var —Method.
var(d::UnivariateMixture)
Compute the overall variance (only for UnivariateMixture).
source
1.11 Base.var
Base.var —Method.
var(d::MultivariateDistribution)
Compute the vector of element-wise variances for distribution d.
source
1.12 Base.var
Base.var —Method.
var(d::UnivariateDistribution)
Compute the variance. (A generic std is provided as std(d) = sqrt(var(d)))
source
1.13 Distributions.TruncatedNormal
Distributions.TruncatedNormal —Method.
TruncatedNormal(mu, sigma, l, u)
The truncated normal distribution is a particularly important one in the fam-
ily of truncated distributions. We provide additional support for this type with
TruncatedNormal which calls Truncated(Normal(mu, sigma), l, u). Unlike
the general case, truncated normal distributions support mean,mode,modes,
var,std, and entropy.
source
4CHAPTER 1. DISTRIBUTIONS
1.14 Distributions.ccdf
Distributions.ccdf —Method.
ccdf(d::UnivariateDistribution, x::Real)
The complementary cumulative function evaluated at x, i.e. 1 - cdf(d,
x).
source
1.15 Distributions.cdf
Distributions.cdf —Method.
cdf(d::UnivariateDistribution, x::Real)
Evaluate the cumulative probability at x.
See also ccdf,logcdf, and logccdf.
source
1.16 Distributions.cf
Distributions.cf —Method.
cf(d::UnivariateDistribution, t)
Evaluate the characteristic function of distribution d.
source
1.17 Distributions.components
Distributions.components —Method.
components(d::AbstractMixtureModel)
Get a list of components of the mixture model d.
source
1.18 Distributions.cquantile
Distributions.cquantile —Method.
cquantile(d::UnivariateDistribution, q::Real)
The complementary quantile value, i.e. quantile(d, 1-q).
source
1.19. DISTRIBUTIONS.FAILPROB 5
1.19 Distributions.failprob
Distributions.failprob —Method.
failprob(d::UnivariateDistribution)
Get the probability of failure.
source
1.20 Distributions.fit mle
Distributions.fit mle —Method.
fit_mle(D, x, w)
Fit a distribution of type Dto a weighted data set x, with weights given by
w.
Here, wshould be an array with length n, where nis the number of samples
contained in x.
source
1.21 Distributions.fit mle
Distributions.fit mle —Method.
fit_mle(D, x)
Fit a distribution of type Dto a given data set x.
•For univariate distribution, x can be an array of arbitrary size.
•For multivariate distribution, x should be a matrix, where each column is
a sample.
source
1.22 Distributions.insupport
Distributions.insupport —Method.
insupport(d::MultivariateMixture, x)
Evaluate whether xis within the support of mixture distribution d.
source
6CHAPTER 1. DISTRIBUTIONS
1.23 Distributions.insupport
Distributions.insupport —Method.
insupport(d::MultivariateDistribution, x::AbstractArray)
If xis a vector, it returns whether x is within the support of d. If xis a
matrix, it returns whether every column in xis within the support of d.
source
1.24 Distributions.insupport
Distributions.insupport —Method.
insupport(d::UnivariateDistribution, x::Any)
When xis a scalar, it returns whether x is within the support of d(e.g.,
insupport(d, x) = minimum(d) <= x <= maximum(d)). When xis an array,
it returns whether every element in x is within the support of d.
Generic fallback methods are provided, but it is often the case that insupport
can be done more efficiently, and a specialized insupport is thus desirable. You
should also override this function if the support is composed of multiple disjoint
intervals.
source
1.25 Distributions.invcov
Distributions.invcov —Method.
invcov(d::AbstractMvNormal)
Return the inversed covariance matrix of d.
source
1.26 Distributions.invlogccdf
Distributions.invlogccdf —Method.
invlogcdf(d::UnivariateDistribution, lp::Real)
The inverse function of logcdf.
source
1.27. DISTRIBUTIONS.INVLOGCDF 7
1.27 Distributions.invlogcdf
Distributions.invlogcdf —Method.
invlogcdf(d::UnivariateDistribution, lp::Real)
The inverse function of logcdf.
source
1.28 Distributions.isleptokurtic
Distributions.isleptokurtic —Method.
isleptokurtic(d)
Return whether dis leptokurtic (i.e kurtosis(d) <0).
source
1.29 Distributions.ismesokurtic
Distributions.ismesokurtic —Method.
ismesokurtic(d)
Return whether dis mesokurtic (i.e kurtosis(d) == 0).
source
1.30 Distributions.isplatykurtic
Distributions.isplatykurtic —Method.
isplatykurtic(d)
Return whether dis platykurtic (i.e kurtosis(d) >0).
source
1.31 Distributions.location!
Distributions.location! —Method.
location!{D<:AbstractMvLogNormal}(::Type{D},s::Symbol,m::AbstractVector,S::AbstractMatrix,::AbstractVector)
Calculate the location vector (as above) and store the result in
source
8CHAPTER 1. DISTRIBUTIONS
1.32 Distributions.location
Distributions.location —Method.
location(d::UnivariateDistribution)
Get the location parameter.
source
1.33 Distributions.location
Distributions.location —Method.
location(d::MvLogNormal)
Return the location vector of the distribution (the mean of the underlying
normal distribution).
source
1.34 Distributions.location
Distributions.location —Method.
location{D<:AbstractMvLogNormal}(::Type{D},s::Symbol,m::AbstractVector,S::AbstractMatrix)
Calculate the location vector (the mean of the underlying normal distribu-
tion).
•If s == :meancov, then m is taken as the mean, and S the covariance
matrix of a lognormal distribution.
•If s == :mean |:median |:mode, then m is taken as the mean, median
or mode of the lognormal respectively, and S is interpreted as the scale
matrix (the covariance of the underlying normal distribution).
It is not possible to analytically calculate the location vector from e.g., me-
dian + covariance, or from mode + covariance.
source
1.35 Distributions.logccdf
Distributions.logccdf —Method.
logccdf(d::UnivariateDistribution, x::Real)
The logarithm of the complementary cumulative function values evaluated
at x, i.e. log(ccdf(x)).
source
1.36. DISTRIBUTIONS.LOGCDF 9
1.36 Distributions.logcdf
Distributions.logcdf —Method.
logcdf(d::UnivariateDistribution, x::Real)
The logarithm of the cumulative function value(s) evaluated at x, i.e. log(cdf(x)).
source
1.37 Distributions.logdetcov
Distributions.logdetcov —Method.
logdetcov(d::AbstractMvNormal)
Return the log-determinant value of the covariance matrix.
source
1.38 Distributions.logpdf
Distributions.logpdf —Method.
logpdf(d::Union{UnivariateMixture, MultivariateMixture}, x)
Evaluate the logarithm of the (mixed) probability density function over x.
Here, xcan be a single sample or an array of multiple samples.
source
1.39 Distributions.logpdf
Distributions.logpdf —Method.
logpdf(d::MultivariateDistribution, x::AbstractArray)
Return the logarithm of probability density evaluated at x.
•If xis a vector, it returns the result as a scalar.
•If xis a matrix with n columns, it returns a vector rof length n, where
r[i] corresponds to x[:,i].
logpdf!(r, d, x) will write the results to a pre-allocated array r.
source
10 CHAPTER 1. DISTRIBUTIONS
1.40 Distributions.logpdf
Distributions.logpdf —Method.
logpdf(d::UnivariateDistribution, x::Real)
Evaluate the logarithm of probability density (mass) at x. Whereas there
is a fallback implemented logpdf(d, x) = log(pdf(d, x)). Relying on this
fallback is not recommended in general, as it is prone to overflow or underflow.
source
1.41 Distributions.logpdf
Distributions.logpdf —Method.
logpdf(d::MatrixDistribution, AbstractMatrix)
Compute the logarithm of the probability density at the input matrix x.
source
1.42 Distributions.mgf
Distributions.mgf —Method.
mgf(d::UnivariateDistribution, t)
Evaluate the moment generating function of distribution d.
source
1.43 Distributions.ncategories
Distributions.ncategories —Method.
ncategories(d::UnivariateDistribution)
Get the number of categories.
source
1.44 Distributions.nsamples
Distributions.nsamples —Method.
nsamples(s::Sampleable)
The number of samples contained in A. Multiple samples are often organized
into an array, depending on the variate form.
source
1.45. DISTRIBUTIONS.NTRIALS 11
1.45 Distributions.ntrials
Distributions.ntrials —Method.
ntrials(d::UnivariateDistribution)
Get the number of trials.
source
1.46 Distributions.pdf
Distributions.pdf —Method.
pdf(d::Union{UnivariateMixture, MultivariateMixture}, x)
Evaluate the (mixed) probability density function over x. Here, xcan be a
single sample or an array of multiple samples.
source
1.47 Distributions.pdf
Distributions.pdf —Method.
pdf(d::MultivariateDistribution, x::AbstractArray)
Return the probability density of distribution devaluated at x.
•If xis a vector, it returns the result as a scalar.
•If xis a matrix with n columns, it returns a vector rof length n, where
r[i] corresponds
to x[:,i] (i.e. treating each column as a sample).
pdf!(r, d, x) will write the results to a pre-allocated array r.
source
1.48 Distributions.pdf
Distributions.pdf —Method.
pdf(d::UnivariateDistribution, x::Real)
Evaluate the probability density (mass) at x.
See also: logpdf.
source
12 CHAPTER 1. DISTRIBUTIONS
1.49 Distributions.pdf
Distributions.pdf —Method.
pdf(d::MatrixDistribution, x::AbstractArray)
Compute the probability density at the input matrix x.
source
1.50 Distributions.probs
Distributions.probs —Method.
probs(d::AbstractMixtureModel)
Get the vector of prior probabilities of all components of d.
source
1.51 Distributions.rate
Distributions.rate —Method.
rate(d::UnivariateDistribution)
Get the rate parameter.
source
1.52 Distributions.sampler
Distributions.sampler —Method.
sampler(d::Distribution) -> Sampleable
Samplers can often rely on pre-computed quantities (that are not parameters
themselves) to improve efficiency. If such a sampler exists, it can be provide
with this sampler method, which would be used for batch sampling. The general
fallback is sampler(d::Distribution) = d.
source
1.53 Distributions.scale
Distributions.scale —Method.
scale(d::UnivariateDistribution)
Get the scale parameter.
source
1.54. DISTRIBUTIONS.SCALE 13
1.54 Distributions.scale
Distributions.scale —Method.
scale(d::MvLogNormal)
Return the scale matrix of the distribution (the covariance matrix of the
underlying normal distribution).
source
1.55 Distributions.scale
Distributions.scale —Method.
scale{D<:AbstractMvLogNormal}(::Type{D},s::Symbol,m::AbstractVector,S::AbstractMatrix)
Calculate the scale parameter, as defined for the location parameter above.
source
1.56 Distributions.shape
Distributions.shape —Method.
shape(d::UnivariateDistribution)
Get the shape parameter.
source
1.57 Distributions.sqmahal
Distributions.sqmahal —Method.
sqmahal(d, x)
Return the squared Mahalanobis distance from x to the center of d, w.r.t.
the covariance. When x is a vector, it returns a scalar value. When x is a
matrix, it returns a vector of length size(x,2).
sqmahal!(r, d, x) with write the results to a pre-allocated array r.
source
1.58 Distributions.succprob
Distributions.succprob —Method.
succprob(d::UnivariateDistribution)
Get the probability of success.
source
14 CHAPTER 1. DISTRIBUTIONS
1.59 StatsBase.dof
StatsBase.dof —Method.
dof(d::UnivariateDistribution)
Get the degrees of freedom.
source
1.60 StatsBase.entropy
StatsBase.entropy —Method.
entropy(d::MultivariateDistribution, b::Real)
Compute the entropy value of distribution d, w.r.t. a given base.
source
1.61 StatsBase.entropy
StatsBase.entropy —Method.
entropy(d::MultivariateDistribution)
Compute the entropy value of distribution d.
source
1.62 StatsBase.entropy
StatsBase.entropy —Method.
entropy(d::UnivariateDistribution, b::Real)
Compute the entropy value of distribution d, w.r.t. a given base.
source
1.63 StatsBase.entropy
StatsBase.entropy —Method.
entropy(d::UnivariateDistribution)
Compute the entropy value of distribution d.
source
1.64. STATSBASE.KURTOSIS 15
1.64 StatsBase.kurtosis
StatsBase.kurtosis —Method.
kurtosis(d::Distribution, correction::Bool)
Computes excess kurtosis by default. Proper kurtosis can be returned with
correction=false
source
1.65 StatsBase.kurtosis
StatsBase.kurtosis —Method.
kurtosis(d::UnivariateDistribution)
Compute the excessive kurtosis.
source
1.66 StatsBase.loglikelihood
StatsBase.loglikelihood —Method.
loglikelihood(d::MultivariateDistribution, x::AbstractMatrix)
The log-likelihood of distribution dw.r.t. all columns contained in matrix
x.
source
1.67 StatsBase.loglikelihood
StatsBase.loglikelihood —Method.
loglikelihood(d::UnivariateDistribution, X::AbstractArray)
The log-likelihood of distribution dw.r.t. all samples contained in array x.
source
1.68 StatsBase.mode
StatsBase.mode —Method.
mode(d::UnivariateDistribution)
Returns the first mode.
source
16 CHAPTER 1. DISTRIBUTIONS
1.69 StatsBase.mode
StatsBase.mode —Method.
mode(d::MvLogNormal)
Return the mode vector of the lognormal distribution, which is strictly
smaller than the mean and median.
source
1.70 StatsBase.modes
StatsBase.modes —Method.
modes(d::UnivariateDistribution)
Get all modes (if this makes sense).
source
1.71 StatsBase.params!
StatsBase.params! —Method.
params!{D<:AbstractMvLogNormal}(::Type{D},m::AbstractVector,S::AbstractMatrix,::AbstractVector,::AbstractMatrix)
Calculate (scale,location) for a given mean and covariance, and store the
results in and
source
1.72 StatsBase.params
StatsBase.params —Method.
params(d::UnivariateDistribution)
Return a tuple of parameters. Let dbe a distribution of type D, then
D(params(d)...) will construct exactly the same distribution as d.
source
1.73 StatsBase.params
StatsBase.params —Method.
params{D<:AbstractMvLogNormal}(::Type{D},m::AbstractVector,S::AbstractMatrix)
Return (scale,location) for a given mean and covariance
source
1.74. STATSBASE.SKEWNESS 17
1.74 StatsBase.skewness
StatsBase.skewness —Method.
skewness(d::UnivariateDistribution)
Compute the skewness.
source
1.75 Distributions.AbstractMvNormal
Distributions.AbstractMvNormal —Type.
The Multivariate normal distribution is a multidimensional generalization
of the normal distribution. The probability density function of a d-dimensional
multivariate normal distribution with mean vector µand covariance matrix Σ
is:
f(x;µ,Σ) = 1
(2π)d/2|Σ|1/2exp −1
2(x−µ)TΣ−1(x−µ)
We realize that the mean vector and the covariance often have special forms
in practice, which can be exploited to simplify the computation. For example,
the mean vector is sometimes just a zero vector, while the covariance matrix
can be a diagonal matrix or even in the form of σI. To take advantage of such
special cases, we introduce a parametric type MvNormal, defined as below, which
allows users to specify the special structure of the mean and covariance.
[] immutable MvNormal{Cov¡:AbstractPDMat,Mean¡:Union{Vector,ZeroVector}}
¡: AbstractMvNormal ::Mean ::Cov end
Here, the mean vector can be an instance of either Vector or ZeroVector,
where the latter is simply an empty type indicating a vector filled with zeros.
The covariance can be of any subtype of AbstractPDMat. Particularly, one can
use PDMat for full covariance, PDiagMat for diagonal covariance, and ScalMat
for the isotropic covariance – those in the form of σI. (See the Julia package
PDMats for details).
We also define a set of alias for the types using different combinations of
mean vectors and covariance:
[] const IsoNormal = MvNormal{ScalMat, Vector{Float64}} const DiagNor-
mal = MvNormal{PDiagMat, Vector{Float64}} const FullNormal = MvNor-
mal{PDMat, Vector{Float64}}
const ZeroMeanIsoNormal = MvNormal{ScalMat, ZeroVector{Float64}} const
ZeroMeanDiagNormal = MvNormal{PDiagMat, ZeroVector{Float64}} const
ZeroMeanFullNormal = MvNormal{PDMat, ZeroVector{Float64}}
source
18 CHAPTER 1. DISTRIBUTIONS
1.76 Distributions.Arcsine
Distributions.Arcsine —Type.
Arcsine(a,b)
The Arcsine distribution has probability density function
f(x) = 1
πp(x−a)(b−x), x ∈[a, b]
[] Arcsine() Arcsine distribution with support [0, 1] Arcsine(b) Arcsine
distribution with support [0, b] Arcsine(a, b) Arcsine distribution with support
[a, b]
params(d) Get the parameters, i.e. (a, b) minimum(d) Get the lower
bound, i.e. a maximum(d) Get the upper bound, i.e. b location(d) Get the
left bound, i.e. a scale(d) Get the span of the support, i.e. b - a
External links
•Arcsine distribution on Wikipedia
source
1.77 Distributions.Bernoulli
Distributions.Bernoulli —Type.
Bernoulli(p)
ABernoulli distribution is parameterized by a success rate p, which takes
value 1 with probability pand 0 with probability 1-p.
P(X=k) = (1−pfor k= 0,
pfor k= 1.
[] Bernoulli() Bernoulli distribution with p = 0.5 Bernoulli(p) Bernoulli
distribution with success rate p
params(d) Get the parameters, i.e. (p,) succprob(d) Get the success rate,
i.e. p failprob(d) Get the failure rate, i.e. 1 - p
External links:
•Bernoulli distribution on Wikipedia
source
1.78. DISTRIBUTIONS.BETA 19
1.78 Distributions.Beta
Distributions.Beta —Type.
Beta(,)
The Beta distribution has probability density function
f(x;α, β) = 1
B(α, β)xα−1(1 −x)β−1, x ∈[0,1]
The Beta distribution is related to the Gamma distribution via the property
that if X∼Gamma(α) and Y∼Gamma(β) independently, then X/(X+Y)∼
Beta(α, β).
[] Beta() equivalent to Beta(1, 1) Beta(a) equivalent to Beta(a, a) Beta(a,
b) Beta distribution with shape parameters a and b
params(d) Get the parameters, i.e. (a, b)
External links
•Beta distribution on Wikipedia
source
1.79 Distributions.BetaBinomial
Distributions.BetaBinomial —Type.
BetaBinomial(n,,)
ABeta-binomial distribution is the compound distribution of the Binomial
distribution where the probability of success pis distributed according to the
Beta. It has three parameters: n, the number of trials and two shape parameters
,
P(X=k) = n
kB(k+α, n −k+β)/B(α, β),for k= 0,1,2, . . . , n.
[] BetaBinomial(n, a, b) BetaBinomial distribution with n trials and shape
parameters a, b
params(d) Get the parameters, i.e. (n, a, b) ntrials(d) Get the number of
trials, i.e. n
External links:
•Beta-binomial distribution on Wikipedia
source
20 CHAPTER 1. DISTRIBUTIONS
1.80 Distributions.BetaPrime
Distributions.BetaPrime —Type.
BetaPrime(,)
The Beta prime distribution has probability density function
f(x;α, β) = 1
B(α, β)xα−1(1 + x)−(α+β), x > 0
The Beta prime distribution is related to the Beta distribution via the rela-
tion ship that if X∼Beta(α, β) then X
1−X∼BetaPrime(α, β)
[] BetaPrime() equivalent to BetaPrime(1, 1) BetaPrime(a) equivalent to
BetaPrime(a, a) BetaPrime(a, b) Beta prime distribution with shape parame-
ters a and b
params(d) Get the parameters, i.e. (a, b)
External links
•Beta prime distribution on Wikipedia
source
1.81 Distributions.Binomial
Distributions.Binomial —Type.
Binomial(n,p)
ABinomial distribution characterizes the number of successes in a sequence
of independent trials. It has two parameters: n, the number of trials, and p, the
probability of success in an individual trial, with the distribution:
P(X=k) = n
kpk(1 −p)n−k,for k= 0,1,2, . . . , n.
[] Binomial() Binomial distribution with n = 1 and p = 0.5 Binomial(n)
Binomial distribution for n trials with success rate p = 0.5 Binomial(n, p)
Binomial distribution for n trials with success rate p
params(d) Get the parameters, i.e. (n, p) ntrials(d) Get the number of
trials, i.e. n succprob(d) Get the success rate, i.e. p failprob(d) Get the failure
rate, i.e. 1 - p
External links:
•Binomial distribution on Wikipedia
source
1.82. DISTRIBUTIONS.BIWEIGHT 21
1.82 Distributions.Biweight
Distributions.Biweight —Type.
Biweight(, )
source
1.83 Distributions.Categorical
Distributions.Categorical —Type.
Categorical(p)
ACategorical distribution is parameterized by a probability vector p(of
length K).
P(X=k) = p[k] for k= 1,2, . . . , K.
[] Categorical(p) Categorical distribution with probability vector p params(d)
Get the parameters, i.e. (p,) probs(d) Get the probability vector, i.e. p ncate-
gories(d) Get the number of categories, i.e. K
Here, pmust be a real vector, of which all components are nonnegative
and sum to one. Note: The input vector pis directly used as a field of the
constructed distribution, without being copied. External links:
•Categorical distribution on Wikipedia
source
1.84 Distributions.Cauchy
Distributions.Cauchy —Type.
Cauchy(, )
The Cauchy distribution with location and scale has probability density
function
f(x;µ, σ) = 1
πσ 1 + x−µ
σ2
[] Cauchy() Standard Cauchy distribution, i.e. Cauchy(0, 1) Cauchy(u)
Cauchy distribution with location u and unit scale, i.e. Cauchy(u, 1) Cauchy(u,
b) Cauchy distribution with location u and scale b
params(d) Get the parameters, i.e. (u, b) location(d) Get the location
parameter, i.e. u scale(d) Get the scale parameter, i.e. b
External links
•Cauchy distribution on Wikipedia
source
22 CHAPTER 1. DISTRIBUTIONS
1.85 Distributions.Chi
Distributions.Chi —Type.
Chi()
The Chi distribution degrees of freedom has probability density function
f(x;k) = 1
Γ(k/2)21−k/2xk−1e−x2/2, x > 0
It is the distribution of the square-root of a Chisq variate.
[] Chi(k) Chi distribution with k degrees of freedom
params(d) Get the parameters, i.e. (k,) dof(d) Get the degrees of freedom,
i.e. k
External links
•Chi distribution on Wikipedia
source
1.86 Distributions.Chisq
Distributions.Chisq —Type.
Chisq()
The Chi squared distribution (typically written ) with degrees of freedom
has the probability density function
f(x;k) = xk/2−1e−x/2
2k/2Γ(k/2) , x > 0.
If is an integer, then it is the distribution of the sum of squares of inde-
pendent standard Normal variates.
[] Chisq(k) Chi-squared distribution with k degrees of freedom
params(d) Get the parameters, i.e. (k,) dof(d) Get the degrees of freedom,
i.e. k
External links
•Chi-squared distribution on Wikipedia
source
1.87. DISTRIBUTIONS.COSINE 23
1.87 Distributions.Cosine
Distributions.Cosine —Type.
Cosine(, )
A raised Cosine distribution.
External link:
•Cosine distribution on wikipedia
source
1.88 Distributions.Dirichlet
Distributions.Dirichlet —Type.
Dirichlet
The Dirichlet distribution is often used the conjugate prior for Categorical
or Multinomial distributions. The probability density function of a Dirichlet
distribution with parameter α= (α1, . . . , αk) is:
f(x;α) = 1
B(α)
k
Y
i=1
xαi−1
i,with B(α) = Qk
i=1 Γ(αi)
ΓPk
i=1 αi, x1+··· +xk= 1
[] Let alpha be a vector Dirichlet(alpha) Dirichlet distribution with param-
eter vector alpha
Let a be a positive scalar Dirichlet(k, a) Dirichlet distribution with param-
eter a * ones(k)
source
1.89 Distributions.DiscreteUniform
Distributions.DiscreteUniform —Type.
DiscreteUniform(a,b)
ADiscrete uniform distribution is a uniform distribution over a consecutive
sequence of integers between aand b, inclusive.
P(X=k)=1/(b−a+ 1) for k=a, a + 1, . . . , b.
[] DiscreteUniform(a, b) a uniform distribution over {a, a+1, ..., b}
params(d) Get the parameters, i.e. (a, b) span(d) Get the span of the
support, i.e. (b - a + 1) probval(d) Get the probability value, i.e. 1 / (b - a +
1) minimum(d) Return a maximum(d) Return b
External links
24 CHAPTER 1. DISTRIBUTIONS
•Discrete uniform distribution on Wikipedia
source
1.90 Distributions.Epanechnikov
Distributions.Epanechnikov —Type.
Epanechnikov(, )
source
1.91 Distributions.Erlang
Distributions.Erlang —Type.
Erlang(,)
The Erlang distribution is a special case of a Gamma distribution with integer
shape parameter.
[] Erlang() Erlang distribution with unit shape and unit scale, i.e. Erlang(1,
1) Erlang(a) Erlang distribution with shape parameter a and unit scale, i.e.
Erlang(a, 1) Erlang(a, s) Erlang distribution with shape parameter a and scale
b
External links
•Erlang distribution on Wikipedia
source
1.92 Distributions.Exponential
Distributions.Exponential —Type.
Exponential()
The Exponential distribution with scale parameter has probability density
function
f(x;θ) = 1
θe−x
θ, x > 0
[] Exponential() Exponential distribution with unit scale, i.e. Exponen-
tial(1) Exponential(b) Exponential distribution with scale b
params(d) Get the parameters, i.e. (b,) scale(d) Get the scale parameter,
i.e. b rate(d) Get the rate parameter, i.e. 1 / b
External links
•Exponential distribution on Wikipedia
source
1.93. DISTRIBUTIONS.FDIST 25
1.93 Distributions.FDist
Distributions.FDist —Type.
FDist(1, 2)
The F distribution has probability density function
f(x;ν1, ν2) = 1
xB(ν1/2, ν2/2) s(ν1x)ν1·νν2
2
(ν1x+ν2)ν1+ν2, x > 0
It is related to the Chisq distribution via the property that if X1∼Chisq(ν1)
and X2∼Chisq(ν2), then (X1/ν1)/(X2/ν2)∼FDist(ν1, ν2).
[] FDist(1, 2) F-Distribution with parameters 1 and 2
params(d) Get the parameters, i.e. (1, 2)
External links
•F distribution on Wikipedia
source
1.94 Distributions.Frechet
Distributions.Frechet —Type.
Frechet(,)
The Frchet distribution with shape and scale has probability density func-
tion
f(x;α, θ) = α
θx
θ−α−1e−(x/θ)−α, x > 0
[] Frechet() Frchet distribution with unit shape and unit scale, i.e. Frechet(1,
1) Frechet(a) Frchet distribution with shape a and unit scale, i.e. Frechet(a, 1)
Frechet(a, b) Frchet distribution with shape a and scale b
params(d) Get the parameters, i.e. (a, b) shape(d) Get the shape param-
eter, i.e. a scale(d) Get the scale parameter, i.e. b
External links
•Frchet distribution on Wikipedia
source
26 CHAPTER 1. DISTRIBUTIONS
1.95 Distributions.Gamma
Distributions.Gamma —Type.
Gamma(,)
The Gamma distribution with shape parameter and scale has probability
density function
f(x;α, θ) = xα−1e−x/θ
Γ(α)θα, x > 0
[] Gamma() Gamma distribution with unit shape and unit scale, i.e. Gamma(1,
1) Gamma() Gamma distribution with shape and unit scale, i.e. Gamma(, 1)
Gamma(, ) Gamma distribution with shape and scale
params(d) Get the parameters, i.e. (, ) shape(d) Get the shape parameter,
i.e. scale(d) Get the scale parameter, i.e.
External links
•Gamma distribution on Wikipedia
source
1.96 Distributions.GeneralizedExtremeValue
Distributions.GeneralizedExtremeValue —Type.
GeneralizedExtremeValue(, , )
The Generalized extreme value distribution with shape parameter , scale and
location has probability density function
f(x;ξ, σ, µ) = (1
σ1 + x−µ
σξ−1/ξ−1exp n−1 + x−µ
σξ−1/ξofor ξ6= 0
1
σexp −x−µ
σexp −exp −x−µ
σ for ξ= 0
for
x∈
hµ−σ
ξ,+∞for ξ > 0
(−∞,+∞) for ξ= 0
−∞, µ −σ
ξifor ξ < 0
[] GeneralizedExtremeValue(m, s, k) Generalized Pareto distribution with
shape k, scale s and location m.
params(d) Get the parameters, i.e. (m, s, k) location(d) Get the location
parameter, i.e. m scale(d) Get the scale parameter, i.e. s shape(d) Get the
shape parameter, i.e. k (sometimes called c)
External links
1.97. DISTRIBUTIONS.GENERALIZEDPARETO 27
•Generalized extreme value distribution on Wikipedia
source
1.97 Distributions.GeneralizedPareto
Distributions.GeneralizedPareto —Type.
GeneralizedPareto(, , )
The Generalized Pareto distribution with shape parameter , scale and loca-
tion has probability density function
f(x;µ, σ, ξ) = (1
σ(1 + ξx−µ
σ)−1
ξ−1for ξ6= 0
1
σe−(x−µ)
σfor ξ= 0 , x ∈([µ, ∞] for ξ≥0
[µ, µ −σ/ξ] for ξ < 0
[] GeneralizedPareto() Generalized Pareto distribution with unit shape and
unit scale, i.e. GeneralizedPareto(0, 1, 1) GeneralizedPareto(k, s) Generalized
Pareto distribution with shape k and scale s, i.e. GeneralizedPareto(0, k, s)
GeneralizedPareto(m, k, s) Generalized Pareto distribution with shape k, scale
s and location m.
params(d) Get the parameters, i.e. (m, s, k) location(d) Get the location
parameter, i.e. m scale(d) Get the scale parameter, i.e. s shape(d) Get the
shape parameter, i.e. k
External links
•Generalized Pareto distribution on Wikipedia
source
1.98 Distributions.Geometric
Distributions.Geometric —Type.
Geometric(p)
AGeometric distribution characterizes the number of failures before the first
success in a sequence of independent Bernoulli trials with success rate p.
P(X=k) = p(1 −p)k,for k= 0,1,2,....
[] Geometric() Geometric distribution with success rate 0.5 Geometric(p)
Geometric distribution with success rate p
params(d) Get the parameters, i.e. (p,) succprob(d) Get the success rate,
i.e. p failprob(d) Get the failure rate, i.e. 1 - p
External links
•Geometric distribution on Wikipedia
source
28 CHAPTER 1. DISTRIBUTIONS
1.99 Distributions.Gumbel
Distributions.Gumbel —Type.
Gumbel(, )
The Gumbel distribution with location and scale has probability density
function
f(x;µ, θ) = 1
θe−(z+ez),with z=x−µ
θ
[] Gumbel() Gumbel distribution with zero location and unit scale, i.e.
Gumbel(0, 1) Gumbel(u) Gumbel distribution with location u and unit scale,
i.e. Gumbel(u, 1) Gumbel(u, b) Gumbel distribution with location u and scale
b
params(d) Get the parameters, i.e. (u, b) location(d) Get the location
parameter, i.e. u scale(d) Get the scale parameter, i.e. b
External links
•Gumbel distribution on Wikipedia
source
1.100 Distributions.Hypergeometric
Distributions.Hypergeometric —Type.
Hypergeometric(s, f, n)
AHypergeometric distribution describes the number of successes in ndraws
without replacement from a finite population containing ssuccesses and ffail-
ures.
P(X=k) = s
k f
n−k
s+f
n,for k= max(0, n −f),...,min(n, s).
[] Hypergeometric(s, f, n) Hypergeometric distribution for a population with
s successes and f failures, and a sequence of n trials.
params(d) Get the parameters, i.e. (s, f, n)
External links
•Hypergeometric distribution on Wikipedia
source
1.101. DISTRIBUTIONS.INVERSEGAMMA 29
1.101 Distributions.InverseGamma
Distributions.InverseGamma —Type.
InverseGamma(, )
The inverse gamma distribution with shape parameter and scale has prob-
ability density function
f(x;α, θ) = θαx−(α+1)
Γ(α)e−θ
x, x > 0
It is related to the Gamma distribution: if X∼Gamma(α, β), then ‘1 / X
∼InverseGamma(α, βˆ{−1})“.
[] InverseGamma() Inverse Gamma distribution with unit shape and unit
scale, i.e. InverseGamma(1, 1) InverseGamma(a) Inverse Gamma distribution
with shape a and unit scale, i.e. InverseGamma(a, 1) InverseGamma(a, b)
Inverse Gamma distribution with shape a and scale b
params(d) Get the parameters, i.e. (a, b) shape(d) Get the shape param-
eter, i.e. a scale(d) Get the scale parameter, i.e. b
External links
•Inverse gamma distribution on Wikipedia
source
1.102 Distributions.InverseGaussian
Distributions.InverseGaussian —Type.
InverseGaussian(,)
The inverse Gaussian distribution with mean and shape has probability
density function
f(x;µ, λ) = rλ
2πx3exp−λ(x−µ)2
2µ2x, x > 0
[] InverseGaussian() Inverse Gaussian distribution with unit mean and unit
shape, i.e. InverseGaussian(1, 1) InverseGaussian(mu), Inverse Gaussian distri-
bution with mean mu and unit shape, i.e. InverseGaussian(u, 1) InverseGaus-
sian(mu, lambda) Inverse Gaussian distribution with mean mu and shape
lambda
params(d) Get the parameters, i.e. (mu, lambda) mean(d) Get the mean
parameter, i.e. mu shape(d) Get the shape parameter, i.e. lambda
External links
•Inverse Gaussian distribution on Wikipedia
source
30 CHAPTER 1. DISTRIBUTIONS
1.103 Distributions.InverseWishart
Distributions.InverseWishart —Type.
InverseWishart(nu, P)
The [Inverse Wishart distribution](http://en.wikipedia.org/wiki/Inverse-Wishart distribution
is usually used a the conjugate prior for the covariance matrix of a multivariate
normal distribution, which is characterized by a degree of freedom , and a base
matrix .
source
1.104 Distributions.KSDist
Distributions.KSDist —Type.
KSDist(n)
Distribution of the (two-sided) Kolmogorov-Smirnoff statistic
Dn= sup
x|ˆ
Fn(x)−F(x)|p(n)
Dnconverges a.s. to the Kolmogorov distribution.
source
1.105 Distributions.KSOneSided
Distributions.KSOneSided —Type.
KSOneSided(n)
Distribution of the one-sided Kolmogorov-Smirnov test statistic:
D+
n= sup
x
(ˆ
Fn(x)−F(x))
source
1.106 Distributions.Kolmogorov
Distributions.Kolmogorov —Type.
Kolmogorov()
Kolmogorov distribution defined as
sup
t∈[0,1] |B(t)|
where B(t) is a Brownian bridge used in the Kolmogorov–Smirnov test for
large n.
source
1.107. DISTRIBUTIONS.LAPLACE 31
1.107 Distributions.Laplace
Distributions.Laplace —Type.
Laplace(,)
The Laplace distribution with location and scale has probability density
function
f(x;µ, β) = 1
2βexp −|x−µ|
β
[] Laplace() Laplace distribution with zero location and unit scale, i.e.
Laplace(0, 1) Laplace(u) Laplace distribution with location u and unit scale,
i.e. Laplace(u, 1) Laplace(u, b) Laplace distribution with location u ans scale
b
params(d) Get the parameters, i.e. (u, b) location(d) Get the location
parameter, i.e. u scale(d) Get the scale parameter, i.e. b
External links
•Laplace distribution on Wikipedia
source
1.108 Distributions.Levy
Distributions.Levy —Type.
Levy(, )
The Lvy distribution with location and scale has probability density func-
tion
f(x;µ, σ) = rσ
2π(x−µ)3exp −σ
2(x−µ), x > µ
[] Levy() Levy distribution with zero location and unit scale, i.e. Levy(0,
1) Levy(u) Levy distribution with location u and unit scale, i.e. Levy(u, 1)
Levy(u, c) Levy distribution with location u ans scale c
params(d) Get the parameters, i.e. (u, c) location(d) Get the location
parameter, i.e. u
External links
•Lvy distribution on Wikipedia
source
32 CHAPTER 1. DISTRIBUTIONS
1.109 Distributions.LocationScale
Distributions.LocationScale —Type.
LocationScale(,,)
A location-scale transformed distribution with location parameter , scale
parameter , and given distribution .
f(x) = 1x−
[] LocationScale(,,) location-scale transformed distribution params(d) Get
the parameters, i.e. (, , and the base distribution) location(d) Get the location
parameter scale(d) Get the scale parameter
External links Location-Scale family on Wikipedia
source
1.110 Distributions.LogNormal
Distributions.LogNormal —Type.
LogNormal(,)
The log normal distribution is the distribution of the exponential of a Normal
variate: if X∼Normal(µ, σ) then exp(X)∼LogNormal(µ, σ). The probability
density function is
f(x;µ, σ) = 1
x√2πσ2exp −(log(x)−µ)2
2σ2, x > 0
[] LogNormal() Log-normal distribution with zero log-mean and unit scale
LogNormal(mu) Log-normal distribution with log-mean mu and unit scale Log-
Normal(mu, sig) Log-normal distribution with log-mean mu and scale sig
params(d) Get the parameters, i.e. (mu, sig) meanlogx(d) Get the mean of
log(X), i.e. mu varlogx(d) Get the variance of log(X), i.e. sig2stdlogx(d)Getthestandarddeviationoflog(X), i.e.sig
External links
•Log normal distribution on Wikipedia
source
1.111. DISTRIBUTIONS.LOGISTIC 33
1.111 Distributions.Logistic
Distributions.Logistic —Type.
Logistic(,)
The Logistic distribution with location and scale has probability density
function
f(x;µ, θ) = 1
4θsech2x−µ
2θ
[] Logistic() Logistic distribution with zero location and unit scale, i.e.
Logistic(0, 1) Logistic(u) Logistic distribution with location u and unit scale,
i.e. Logistic(u, 1) Logistic(u, b) Logistic distribution with location u ans scale
b
params(d) Get the parameters, i.e. (u, b) location(d) Get the location
parameter, i.e. u scale(d) Get the scale parameter, i.e. b
External links
•Logistic distribution on Wikipedia
source
1.112 Distributions.MixtureModel
Distributions.MixtureModel —Method.
MixtureModel(components, [prior])
Construct a mixture model with a vector of components and a prior prob-
ability vector. If no prior is provided then all components will have the same
prior probabilities.
source
1.113 Distributions.MixtureModel
Distributions.MixtureModel —Method.
MixtureModel(C, params, [prior])
Construct a mixture model with component type C, a vector of parame-
ters for constructing the components given by params, and a prior probability
vector. If no prior is provided then all components will have the same prior
probabilities.
source
34 CHAPTER 1. DISTRIBUTIONS
1.114 Distributions.Multinomial
Distributions.Multinomial —Type.
The Multinomial distribution generalizes the binomial distribution. Consider
n independent draws from a Categorical distribution over a finite set of size k,
and let X= (X1, ..., Xk) where Xirepresents the number of times the element
ioccurs, then the distribution of Xis a multinomial distribution. Each sample
of a multinomial distribution is a k-dimensional integer vector that sums to n.
The probability mass function is given by
f(x;n, p) = n!
x1!···xk!
k
Y
i=1
pxi
i, x1+··· +xk=n
[] Multinomial(n, p) Multinomial distribution for n trials with probability
vector p Multinomial(n, k) Multinomial distribution for n trials with equal
probabilities over 1:k
source
1.115 Distributions.MvLogNormal
Distributions.MvLogNormal —Type.
MvLogNormal(d::MvNormal)
The Multivariate lognormal distribution is a multidimensional generalization
of the lognormal distribution.
If X∼ N(µ,Σ) has a multivariate normal distribution then Y= exp(X)
has a multivariate lognormal distribution.
Mean vector µand covariance matrix Σof the underlying normal distri-
bution are known as the location and scale parameters of the corresponding
lognormal distribution.
source
1.116 Distributions.MvNormal
Distributions.MvNormal —Type.
MvNormal
Generally, users don’t have to worry about these internal details. We pro-
vide a common constructor MvNormal, which will construct a distribution of
appropriate type depending on the input arguments.
MvNormal(sig)
Construct a multivariate normal distribution with zero mean and covariance
represented by sig.
1.117. DISTRIBUTIONS.MVNORMALCANON 35
MvNormal(mu, sig)
Construct a multivariate normal distribution with mean mu and covariance
represented by sig.
MvNormal(d, sig)
Construct a multivariate normal distribution of dimension d, with zero mean,
and an isotropic covariance as abs2(sig) * eye(d).
Arguments
•mu::Vector{T<:Real}: The mean vector.
•d::Real: dimension of distribution.
•sig: The covariance, which can in of either of the following forms (with
T<:Real):
1. subtype of AbstractPDMat
2. symmetric matrix of type Matrix{T}
3. vector of type Vector{T}: indicating a diagonal covariance as diagm(abs2(sig)).
4. real-valued number: indicating an isotropic covariance as abs2(sig)
* eye(d).
Note: The constructor will choose an appropriate covariance form inter-
nally, so that special structure of the covariance can be exploited.
source
1.117 Distributions.MvNormalCanon
Distributions.MvNormalCanon —Type.
MvNormalCanon
Multivariate normal distribution is an exponential family distribution, with
two canonical parameters: the potential vector hand the precision matrix J.
The relation between these parameters and the conventional representation (i.e.
the one using mean µand covariance Σ) is:
h=Σ−1µ,and J=Σ−1
The canonical parameterization is widely used in Bayesian analysis. We pro-
vide a type MvNormalCanon, which is also a subtype of AbstractMvNormal to
represent a multivariate normal distribution using canonical parameters. Par-
ticularly, MvNormalCanon is defined as:
[] immutable MvNormalCanon{P¡:AbstractPDMat,V¡:Union{Vector,ZeroVector}}
¡: AbstractMvNormal ::V the mean vector h::V potential vector, i.e. inv() *
J::P precision matrix, i.e. inv() end
36 CHAPTER 1. DISTRIBUTIONS
We also define aliases for common specializations of this parametric type:
[] const FullNormalCanon = MvNormalCanon{PDMat, Vector{Float64}}
const DiagNormalCanon = MvNormalCanon{PDiagMat, Vector{Float64}} const
IsoNormalCanon = MvNormalCanon{ScalMat, Vector{Float64}}
const ZeroMeanFullNormalCanon = MvNormalCanon{PDMat, ZeroVector{Float64}}
const ZeroMeanDiagNormalCanon = MvNormalCanon{PDiagMat, ZeroVector{Float64}}
const ZeroMeanIsoNormalCanon = MvNormalCanon{ScalMat, ZeroVector{Float64}}
A multivariate distribution with canonical parameterization can be con-
structed using a common constructor MvNormalCanon as:
MvNormalCanon(h, J)
Construct a multivariate normal distribution with potential vector hand
precision matrix represented by J.
MvNormalCanon(J)
Construct a multivariate normal distribution with zero mean (thus zero po-
tential vector) and precision matrix represented by J.
MvNormalCanon(d, J)
Construct a multivariate normal distribution of dimension d, with zero mean
and a precision matrix as J * eye(d).
Arguments
•d::Int: dimension of distribution
•h::Vector{T<:Real}: the potential vector, of type Vector{T}with T<:Real.
•J: the representation of the precision matrix, which can be in either of the
following forms (T<:Real):
1. an instance of a subtype of AbstractPDMat
2. a square matrix of type Matrix{T}
3. a vector of type Vector{T}: indicating a diagonal precision matrix
as diagm(J).
4. a real number: indicating an isotropic precision matrix as J * eye(d).
Note: MvNormalCanon share the same set of methods as MvNormal.
source
1.118. DISTRIBUTIONS.NEGATIVEBINOMIAL 37
1.118 Distributions.NegativeBinomial
Distributions.NegativeBinomial —Type.
NegativeBinomial(r,p)
ANegative binomial distribution describes the number of failures before the
rth success in a sequence of independent Bernoulli trials. It is parameterized
by r, the number of successes, and p, the probability of success in an individual
trial.
P(X=k) = k+r−1
kpr(1 −p)k,for k= 0,1,2,....
The distribution remains well-defined for any positive r, in which case
P(X=k) = Γ(k+r)
k!Γ(r)pr(1 −p)k,for k= 0,1,2,....
[] NegativeBinomial() Negative binomial distribution with r = 1 and p =
0.5 NegativeBinomial(r, p) Negative binomial distribution with r successes and
success rate p
params(d) Get the parameters, i.e. (r, p) succprob(d) Get the success rate,
i.e. p failprob(d) Get the failure rate, i.e. 1 - p
External links:
•Negative binomial distribution on Wikipedia
source
1.119 Distributions.NoncentralBeta
Distributions.NoncentralBeta —Type.
NoncentralBeta(, , )
source
1.120 Distributions.NoncentralChisq
Distributions.NoncentralChisq —Type.
NoncentralChisq(, )
The noncentral chi-squared distribution with degrees of freedom and non-
centrality parameter has the probability density function
f(x;ν, λ) = 1
2e−(x+λ)/2x
λν/4−1/2Iν/2−1(√λx), x > 0
38 CHAPTER 1. DISTRIBUTIONS
It is the distribution of the sum of squares of independent Normal variates
with individual means µiand
λ=
ν
X
i=1
µ2
i
[] NoncentralChisq(, ) Noncentral chi-squared distribution with degrees of
freedom and noncentrality parameter
params(d) Get the parameters, i.e. (, )
External links
•Noncentral chi-squared distribution on Wikipedia
source
1.121 Distributions.NoncentralF
Distributions.NoncentralF —Type.
NoncentralF(1, 2, )
source
1.122 Distributions.NoncentralT
Distributions.NoncentralT —Type.
NoncentralT(, )
source
1.123 Distributions.Normal
Distributions.Normal —Type.
Normal(,)
The Normal distribution with mean and standard deviation has probability
density function
f(x;µ, σ) = 1
√2πσ2exp −(x−µ)2
2σ2
[] Normal() standard Normal distribution with zero mean and unit variance
Normal(mu) Normal distribution with mean mu and unit variance Normal(mu,
sig) Normal distribution with mean mu and variance sig2
params(d) Get the parameters, i.e. (mu, sig) mean(d) Get the mean, i.e.
mu std(d) Get the standard deviation, i.e. sig
External links
1.124. DISTRIBUTIONS.NORMALCANON 39
•Normal distribution on Wikipedia
source
1.124 Distributions.NormalCanon
Distributions.NormalCanon —Type.
NormalCanon(, )
Canonical Form of Normal distribution
source
1.125 Distributions.NormalInverseGaussian
Distributions.NormalInverseGaussian —Type.
NormalInverseGaussian(,,,)
The Normal-inverse Gaussian distribution with location , tail heaviness ,
asymmetry parameter and scale has probability density function
f(x;µ, α, β, δ) =
αδK1αpδ2+ (x−µ)2
πpδ2+ (x−µ)2eδγ+β(x−µ)
where Kjdenotes a modified Bessel function of the third kind.
External links
•Normal-inverse Gaussian distribution on Wikipedia
source
1.126 Distributions.Pareto
Distributions.Pareto —Type.
Pareto(,)
The Pareto distribution with shape and scale has probability density func-
tion
f(x;α, θ) = αθα
xα+1 , x ≥θ
[] Pareto() Pareto distribution with unit shape and unit scale, i.e. Pareto(1,
1) Pareto(a) Pareto distribution with shape a and unit scale, i.e. Pareto(a, 1)
Pareto(a, b) Pareto distribution with shape a and scale b
params(d) Get the parameters, i.e. (a, b) shape(d) Get the shape param-
eter, i.e. a scale(d) Get the scale parameter, i.e. b
External links
40 CHAPTER 1. DISTRIBUTIONS
•Pareto distribution on Wikipedia
source
1.127 Distributions.Poisson
Distributions.Poisson —Type.
Poisson()
APoisson distribution descibes the number of independent events occurring
within a unit time interval, given the average rate of occurrence .
P(X=k) = λk
k!e−λ,for k= 0,1,2,....
[] Poisson() Poisson distribution with rate parameter 1 Poisson(lambda)
Poisson distribution with rate parameter lambda
params(d) Get the parameters, i.e. (,) mean(d) Get the mean arrival rate,
i.e.
External links:
•Poisson distribution on Wikipedia
source
1.128 Distributions.PoissonBinomial
Distributions.PoissonBinomial —Type.
PoissonBinomial(p)
APoisson-binomial distribution describes the number of successes in a se-
quence of independent trials, wherein each trial has a different success rate. It
is parameterized by a vector p(of length K), where Kis the total number of
trials and p[i] corresponds to the probability of success of the ith trial.
P(X=k) = X
A∈FkY
i∈A
p[i]Y
j∈Ac
(1 −p[j]),for k= 0,1,2, . . . , K,
where Fkis the set of all subsets of kintegers that can be selected from
{1,2,3, ..., K}.
[] PoissonBinomial(p) Poisson Binomial distribution with success rate vector
p
params(d) Get the parameters, i.e. (p,) succprob(d) Get the vector of
success rates, i.e. p failprob(d) Get the vector of failure rates, i.e. 1-p
External links:
•Poisson-binomial distribution on Wikipedia
source
1.129. DISTRIBUTIONS.RAYLEIGH 41
1.129 Distributions.Rayleigh
Distributions.Rayleigh —Type.
Rayleigh()
The Rayleigh distribution with scale has probability density function
f(x;σ) = x
σ2e−x2
2σ2, x > 0
It is related to the Normal distribution via the property that if X, Y ∼
Normal(0, σ), independently, then √X2+Y2∼Rayleigh(σ).
[] Rayleigh() Rayleigh distribution with unit scale, i.e. Rayleigh(1) Rayleigh(s)
Rayleigh distribution with scale s
params(d) Get the parameters, i.e. (s,) scale(d) Get the scale parameter,
i.e. s
External links
•Rayleigh distribution on Wikipedia
source
1.130 Distributions.Semicircle
Distributions.Semicircle —Type.
Semicircle(r)
The Wigner semicircle distribution with radius parameter rhas probability
density function
f(x;r) = 2
πr2pr2−x2, x ∈[−r, r].
[] Semicircle(r) Wigner semicircle distribution with radius r
params(d) Get the radius parameter, i.e. (r,)
External links
•Wigner semicircle distribution on Wikipedia
source
42 CHAPTER 1. DISTRIBUTIONS
1.131 Distributions.Skellam
Distributions.Skellam —Type.
Skellam(1, 2)
ASkellam distribution describes the difference between two independent
Poisson variables, respectively with rate 1and 2.
P(X=k) = e−(µ1+µ2)µ1
µ2k/2
Ik(2√µ1µ2) for integer k
where Ikis the modified Bessel function of the first kind.
[] Skellam(mu1, mu2) Skellam distribution for the difference between two
Poisson variables, respectively with expected values mu1 and mu2.
params(d) Get the parameters, i.e. (mu1, mu2)
External links:
•Skellam distribution on Wikipedia
source
1.132 Distributions.SymTriangularDist
Distributions.SymTriangularDist —Type.
SymTriangularDist(,)
The Symmetric triangular distribution with location and scale has proba-
bility density function
f(x;µ, σ) = 1
σ1−
x−µ
σ, µ −σ≤x≤µ+σ
[] SymTriangularDist() Symmetric triangular distribution with zero loca-
tion and unit scale SymTriangularDist(u) Symmetric triangular distribution
with location u and unit scale SymTriangularDist(u, s) Symmetric triangular
distribution with location u and scale s
params(d) Get the parameters, i.e. (u, s) location(d) Get the location
parameter, i.e. u scale(d) Get the scale parameter, i.e. s
source
1.133 Distributions.TDist
Distributions.TDist —Type.
TDist()
1.134. DISTRIBUTIONS.TRIANGULARDIST 43
The Students T distribution with degrees of freedom has probability density
function
f(x;d) = 1
√dB(1/2, d/2) 1 + x2
d−d+1
2
[] TDist(d) t-distribution with d degrees of freedom
params(d) Get the parameters, i.e. (d,) dof(d) Get the degrees of freedom,
i.e. d
External links
Student’s T distribution on Wikipedia
source
1.134 Distributions.TriangularDist
Distributions.TriangularDist —Type.
TriangularDist(a,b,c)
The triangular distribution with lower limit a, upper limit band mode chas
probability density function
f(x;a, b, c) =
0 for x < a,
2(x−a)
(b−a)(c−a)for a≤x≤c,
2(b−x)
(b−a)(b−c)for c < x ≤b,
0 for b < x,
[] TriangularDist(a, b) Triangular distribution with lower limit a, upper
limit b, and mode (a+b)/2 TriangularDist(a, b, c) Triangular distribution with
lower limit a, upper limit b, and mode c
params(d) Get the parameters, i.e. (a, b, c) minimum(d) Get the lower
bound, i.e. a maximum(d) Get the upper bound, i.e. b mode(d) Get the mode,
i.e. c
External links
•Triangular distribution on Wikipedia
source
1.135 Distributions.Triweight
Distributions.Triweight —Type.
Triweight(, )
source
44 CHAPTER 1. DISTRIBUTIONS
1.136 Distributions.Truncated
Distributions.Truncated —Type.
Truncated(d, l, u):
Construct a truncated distribution.
Arguments
•d::UnivariateDistribution: The original distribution.
•l::Real: The lower bound of the truncation, which can be a finite value
or -Inf.
•u::Real: The upper bound of the truncation, which can be a finite value
of Inf.
source
1.137 Distributions.Uniform
Distributions.Uniform —Type.
Uniform(a,b)
The continuous uniform distribution over an interval [a, b] has probability
density function
f(x;a, b) = 1
b−a, a ≤x≤b
[] Uniform() Uniform distribution over [0, 1] Uniform(a, b) Uniform distri-
bution over [a, b]
params(d) Get the parameters, i.e. (a, b) minimum(d) Get the lower
bound, i.e. a maximum(d) Get the upper bound, i.e. b location(d) Get the
location parameter, i.e. a scale(d) Get the scale parameter, i.e. b - a
External links
•Uniform distribution (continuous) on Wikipedia
source
1.138 Distributions.VonMises
Distributions.VonMises —Type.
VonMises(, )
1.139. DISTRIBUTIONS.WEIBULL 45
The von Mises distribution with mean and concentration has probability
density function
f(x;µ, κ) = 1
2πI0(κ)exp (κcos(x−µ))
[] VonMises() von Mises distribution with zero mean and unit concentration
VonMises() von Mises distribution with zero mean and concentration Von-
Mises(, ) von Mises distribution with mean and concentration
External links
•von Mises distribution on Wikipedia
source
1.139 Distributions.Weibull
Distributions.Weibull —Type.
Weibull(,)
The Weibull distribution with shape and scale has probability density
function
f(x;α, θ) = α
θx
θα−1e−(x/θ)α, x ≥0
[] Weibull() Weibull distribution with unit shape and unit scale, i.e. Weibull(1,
1) Weibull(a) Weibull distribution with shape a and unit scale, i.e. Weibull(a,
1) Weibull(a, b) Weibull distribution with shape a and scale b
params(d) Get the parameters, i.e. (a, b) shape(d) Get the shape param-
eter, i.e. a scale(d) Get the scale parameter, i.e. b
External links
•Weibull distribution on Wikipedia
source
1.140 Distributions.Wishart
Distributions.Wishart —Type.
Wishart(nu, S)
The Wishart distribution is a multidimensional generalization of the Chi-
square distribution, which is characterized by a degree of freedom , and a base
matrix S.
source
Chapter 2
StatsBase
2.1 Base.stdm
Base.stdm —Method.
stdm(v, w::AbstractWeights, m, [dim]; corrected=false)
Compute the standard deviation of a real-valued array xwith a known mean
m, optionally over a dimension dim. Observations in xare weighted using weight
vector w. The uncorrected (when corrected=false) sample standard deviation
is defined as:
v
u
u
t
1
Pw
n
X
i=1
wi(xi−m)2
where nis the length of the input. The unbiased estimate (when corrected=true)
of the population standard deviation is computed by replacing 1
Pwwith a factor
dependent on the type of weights used:
•AnalyticWeights:1
Pw−Pw2/Pw
•FrequencyWeights:1
Pw−1
•ProbabilityWeights:n
(n−1) Pwwhere nequals count(!iszero, w)
•Weights:ArgumentError (bias correction not supported)
source
46
2.2. BASE.VARM 47
2.2 Base.varm
Base.varm —Method.
varm(x, w::AbstractWeights, m, [dim]; corrected=false)
Compute the variance of a real-valued array xwith a known mean m, option-
ally over a dimension dim. Observations in xare weighted using weight vector
w. The uncorrected (when corrected=false) sample variance is defined as:
1
Pw
n
X
i=1
wi(xi−m)2
where nis the length of the input. The unbiased estimate (when corrected=true)
of the population variance is computed by replacing 1
Pwwith a factor dependent
on the type of weights used:
•AnalyticWeights:1
Pw−Pw2/Pw
•FrequencyWeights:1
Pw−1
•ProbabilityWeights:n
(n−1) Pwwhere nequals count(!iszero, w)
•Weights:ArgumentError (bias correction not supported)
source
2.3 StatsBase.L1dist
StatsBase.L1dist —Method.
L1dist(a, b)
Compute the L1 distance between two arrays: Pn
i=1 |ai−bi|. Efficient equiv-
alent of sum(abs, a - b).
source
2.4 StatsBase.L2dist
StatsBase.L2dist —Method.
L2dist(a, b)
Compute the L2 distance between two arrays: pPn
i=1 |ai−bi|2. Efficient
equivalent of sqrt(sumabs2(a - b)).
source
48 CHAPTER 2. STATSBASE
2.5 StatsBase.Linfdist
StatsBase.Linfdist —Method.
Linfdist(a, b)
Compute the L distance, also called the Chebyshev distance, between two
arrays: maxi∈1:n|ai−bi|. Efficient equivalent of maxabs(a - b).
source
2.6 StatsBase.addcounts!
StatsBase.addcounts! —Method.
addcounts!(r, x, levels::UnitRange{<:Int}, [wv::AbstractWeights])
Add the number of occurrences in xof each value in levels to an existing
array r. If a weighting vector wv is specified, the sum of weights is used rather
than the raw counts.
source
2.7 StatsBase.addcounts!
StatsBase.addcounts! —Method.
addcounts!(dict, x[, wv]; alg = :auto)
Add counts based on xto a count map. New entries will be added if new
values come up. If a weighting vector wv is specified, the sum of the weights is
used rather than the raw counts.
alg can be one of:
•:auto (default): if StatsBase.radixsort safe(eltype(x)) == true then
use :radixsort, otherwise use :dict.
•:radixsort: if radixsort safe(eltype(x)) == true then use the radix
sort algorithm to sort the input vector which will generally lead to shorter
running time. However the radix sort algorithm creates a copy of the
input vector and hence uses more RAM. Choose :dict if the amount of
available RAM is a limitation.
•:dict: use Dict-based method which is generally slower but uses less
RAM and is safe for any data type.
source
2.8. STATSBASE.ADJR2 49
2.8 StatsBase.adjr2
StatsBase.adjr2 —Method.
adjr2(obj::StatisticalModel, variant::Symbol)
adjr(obj::StatisticalModel, variant::Symbol)
Adjusted coefficient of determination (adjusted R-squared).
For linear models, the adjusted R is defined as 1−(1−(1−R2)(n−1)/(n−p)),
with R2the coefficient of determination, nthe number of observations, and p
the number of coefficients (including the intercept). This definition is generally
known as the Wherry Formula I.
For other models, one of the several pseudo R definitions must be chosen
via variant. The only currently supported variant is :MacFadden, defined as
1−(log L−k)/log L0. In this formula, Lis the likelihood of the model, L0
that of the null model (the model including only the intercept). These two
quantities are taken to be minus half deviance of the corresponding models.
kis the number of consumed degrees of freedom of the model (as returned by
dof).
source
2.9 StatsBase.aic
StatsBase.aic —Method.
aic(obj::StatisticalModel)
Akaike’s Information Criterion, defined as −2 log L+2k, with Lthe likelihood
of the model, and kits number of consumed degrees of freedom (as returned by
dof).
source
2.10 StatsBase.aicc
StatsBase.aicc —Method.
aicc(obj::StatisticalModel)
Corrected Akaike’s Information Criterion for small sample sizes (Hurvich and
Tsai 1989), defined as −2 log L+2k+2k(k−1)/(n−k−1), with Lthe likelihood
of the model, kits number of consumed degrees of freedom (as returned by dof),
and nthe number of observations (as returned by nobs).
source
50 CHAPTER 2. STATSBASE
2.11 StatsBase.autocor!
StatsBase.autocor! —Method.
autocor!(r, x, lags; demean=true)
Compute the autocorrelation function (ACF) of a vector or matrix xat
lags and store the result in r.demean denotes whether the mean of xshould
be subtracted from xbefore computing the ACF.
If xis a vector, rmust be a vector of the same length as x. If xis a matrix, r
must be a matrix of size (length(lags), size(x,2)), and where each column
in the result will correspond to a column in x.
The output is normalized by the variance of x, i.e. so that the lag 0 auto-
correlation is 1. See autocov! for the unnormalized form.
source
2.12 StatsBase.autocor
StatsBase.autocor —Method.
autocor(x, [lags]; demean=true)
Compute the autocorrelation function (ACF) of a vector or matrix x, op-
tionally specifying the lags.demean denotes whether the mean of xshould be
subtracted from xbefore computing the ACF.
If xis a vector, return a vector of the same length as x. If xis a matrix,
return a matrix of size (length(lags), size(x,2)), where each column in the
result corresponds to a column in x.
When left unspecified, the lags used are the integers from 0 to min(size(x,1)-1,
10*log10(size(x,1))).
The output is normalized by the variance of x, i.e. so that the lag 0 auto-
correlation is 1. See autocov for the unnormalized form.
source
2.13 StatsBase.autocov!
StatsBase.autocov! —Method.
autocov!(r, x, lags; demean=true)
Compute the autocovariance of a vector or matrix xat lags and store the
result in r.demean denotes whether the mean of xshould be subtracted from
xbefore computing the autocovariance.
If xis a vector, rmust be a vector of the same length as x. If xis a matrix, r
must be a matrix of size (length(lags), size(x,2)), and where each column
in the result will correspond to a column in x.
2.14. STATSBASE.AUTOCOV 51
The output is not normalized. See autocor! for a method with normaliza-
tion.
source
2.14 StatsBase.autocov
StatsBase.autocov —Method.
autocov(x, [lags]; demean=true)
Compute the autocovariance of a vector or matrix x, optionally specifying
the lags at which to compute the autocovariance. demean denotes whether the
mean of xshould be subtracted from xbefore computing the autocovariance.
If xis a vector, return a vector of the same length as x. If xis a matrix,
return a matrix of size (length(lags), size(x,2)), where each column in the
result corresponds to a column in x.
When left unspecified, the lags used are the integers from 0 to min(size(x,1)-1,
10*log10(size(x,1))).
The output is not normalized. See autocor for a function with normaliza-
tion.
source
2.15 StatsBase.aweights
StatsBase.aweights —Method.
aweights(vs)
Construct an AnalyticWeights vector from array vs. See the documenta-
tion for AnalyticWeights for more details.
source
2.16 StatsBase.bic
StatsBase.bic —Method.
bic(obj::StatisticalModel)
Bayesian Information Criterion, defined as −2 log L+klog n, with Lthe like-
lihood of the model, kits number of consumed degrees of freedom (as returned
by dof), and nthe number of observations (as returned by nobs).
source
52 CHAPTER 2. STATSBASE
2.17 StatsBase.coef
StatsBase.coef —Method.
coef(obj::StatisticalModel)
Return the coefficients of the model.
source
2.18 StatsBase.coefnames
StatsBase.coefnames —Method.
coefnames(obj::StatisticalModel)
Return the names of the coefficients.
source
2.19 StatsBase.coeftable
StatsBase.coeftable —Method.
coeftable(obj::StatisticalModel)
Return a table of class CoefTable with coefficients and related statistics.
source
2.20 StatsBase.competerank
StatsBase.competerank —Method.
competerank(x; lt = isless, rev::Bool = false)
Return the standard competition ranking (“1224” ranking) of an array. The
lt keyword allows providing a custom “less than” function; use rev=true to
reverse the sorting order. Items that compare equal are given the same rank,
then a gap is left in the rankings the size of the number of tied items - 1. Missing
values are assigned rank missing.
source
2.21 StatsBase.confint
StatsBase.confint —Method.
confint(obj::StatisticalModel)
Compute confidence intervals for coefficients.
source
2.22. STATSBASE.COR2COV 53
2.22 StatsBase.cor2cov
StatsBase.cor2cov —Method.
cor2cov(C, s)
Compute the covariance matrix from the correlation matrix Cand a vector
of standard deviations s. Use StatsBase.cor2cov! for an in-place version.
source
2.23 StatsBase.corkendall
StatsBase.corkendall —Method.
corkendall(x, y=x)
Compute Kendall’s rank correlation coefficient, . xand ymust both be either
matrices or vectors.
source
2.24 StatsBase.corspearman
StatsBase.corspearman —Method.
corspearman(x, y=x)
Compute Spearman’s rank correlation coefficient. If xand yare vectors,
the output is a float, otherwise it’s a matrix corresponding to the pairwise
correlations of the columns of xand y.
source
2.25 StatsBase.counteq
StatsBase.counteq —Method.
counteq(a, b)
Count the number of indices at which the elements of the arrays aand bare
equal.
source
54 CHAPTER 2. STATSBASE
2.26 StatsBase.countmap
StatsBase.countmap —Method.
countmap(x; alg = :auto)
Return a dictionary mapping each unique value in xto its number of occur-
rences.
•:auto (default): if StatsBase.radixsort safe(eltype(x)) == true then
use :radixsort, otherwise use :dict.
•:radixsort: if radixsort safe(eltype(x)) == true then use the radix
sort algorithm to sort the input vector which will generally lead to shorter
running time. However the radix sort algorithm creates a copy of the
input vector and hence uses more RAM. Choose :dict if the amount of
available RAM is a limitation.
•:dict: use Dict-based method which is generally slower but uses less
RAM and is safe for any data type.
source
2.27 StatsBase.countne
StatsBase.countne —Method.
countne(a, b)
Count the number of indices at which the elements of the arrays aand bare
not equal.
source
2.28 StatsBase.counts
StatsBase.counts —Function.
counts(x, [wv::AbstractWeights])
counts(x, levels::UnitRange{<:Integer}, [wv::AbstractWeights])
counts(x, k::Integer, [wv::AbstractWeights])
Count the number of times each value in xoccurs. If levels is provided,
only values falling in that range will be considered (the others will be ignored
without raising an error or a warning). If an integer kis provided, only values
in the range 1:k will be considered.
If a weighting vector wv is specified, the sum of the weights is used rather
than the raw counts.
The output is a vector of length length(levels).
source
2.29. STATSBASE.COV2COR 55
2.29 StatsBase.cov2cor
StatsBase.cov2cor —Method.
cov2cor(C, s)
Compute the correlation matrix from the covariance matrix Cand a vector
of standard deviations s. Use Base.cov2cor! for an in-place version.
source
2.30 StatsBase.crosscor!
StatsBase.crosscor! —Method.
crosscor!(r, x, y, lags; demean=true)
Compute the cross correlation between real-valued vectors or matrices x
and yat lags and store the result in r.demean specifies whether the respective
means of xand yshould be subtracted from them before computing their cross
correlation.
If both xand yare vectors, rmust be a vector of the same length as lags. If
either xis a matrix and yis a vector, rmust be a matrix of size (length(lags),
size(x, 2)); if xis a vector and yis a matrix, rmust be a matrix of size
(length(lags), size(y, 2)). If both xand yare matrices, rmust be a
three-dimensional array of size (length(lags), size(x, 2), size(y, 2)).
The output is normalized by sqrt(var(x)*var(y)). See crosscov! for the
unnormalized form.
source
2.31 StatsBase.crosscor
StatsBase.crosscor —Method.
crosscor(x, y, [lags]; demean=true)
Compute the cross correlation between real-valued vectors or matrices x
and y, optionally specifying the lags.demean specifies whether the respective
means of xand yshould be subtracted from them before computing their cross
correlation.
If both xand yare vectors, return a vector of the same length as lags.
Otherwise, compute cross covariances between each pairs of columns in xand
y.
When left unspecified, the lags used are the integers from -min(size(x,1)-1,
10*log10(size(x,1))) to min(size(x,1), 10*log10(size(x,1))).
The output is normalized by sqrt(var(x)*var(y)). See crosscov for the
unnormalized form.
source
56 CHAPTER 2. STATSBASE
2.32 StatsBase.crosscov!
StatsBase.crosscov! —Method.
crosscov!(r, x, y, lags; demean=true)
Compute the cross covariance function (CCF) between real-valued vectors or
matrices xand yat lags and store the result in r.demean specifies whether the
respective means of xand yshould be subtracted from them before computing
their CCF.
If both xand yare vectors, rmust be a vector of the same length as lags. If
either xis a matrix and yis a vector, rmust be a matrix of size (length(lags),
size(x, 2)); if xis a vector and yis a matrix, rmust be a matrix of size
(length(lags), size(y, 2)). If both xand yare matrices, rmust be a
three-dimensional array of size (length(lags), size(x, 2), size(y, 2)).
The output is not normalized. See crosscor! for a function with normal-
ization.
source
2.33 StatsBase.crosscov
StatsBase.crosscov —Method.
crosscov(x, y, [lags]; demean=true)
Compute the cross covariance function (CCF) between real-valued vectors or
matrices xand y, optionally specifying the lags.demean specifies whether the
respective means of xand yshould be subtracted from them before computing
their CCF.
If both xand yare vectors, return a vector of the same length as lags.
Otherwise, compute cross covariances between each pairs of columns in xand
y.
When left unspecified, the lags used are the integers from -min(size(x,1)-1,
10*log10(size(x,1))) to min(size(x,1), 10*log10(size(x,1))).
The output is not normalized. See crosscor for a function with normaliza-
tion.
source
2.34 StatsBase.crossentropy
StatsBase.crossentropy —Method.
crossentropy(p, q, [b])
Compute the cross entropy between pand q, optionally specifying a real
number bsuch that the result is scaled by 1/log(b).
source
2.35. STATSBASE.DENSERANK 57
2.35 StatsBase.denserank
StatsBase.denserank —Method.
denserank(x)
Return the dense ranking (“1223” ranking) of an array. The lt keyword
allows providing a custom “less than” function; use rev=true to reverse the
sorting order. Items that compare equal receive the same ranking, and the
next subsequent rank is assigned with no gap. Missing values are assigned rank
missing.
source
2.36 StatsBase.describe
StatsBase.describe —Method.
describe(a)
Pretty-print the summary statistics provided by summarystats: the mean,
minimum, 25th percentile, median, 75th percentile, and maximum.
source
2.37 StatsBase.deviance
StatsBase.deviance —Method.
deviance(obj::StatisticalModel)
Return the deviance of the model relative to a reference, which is usually
when applicable the saturated model. It is equal, up to a constant, to −2 log L,
with Lthe likelihood of the model.
source
2.38 StatsBase.dof
StatsBase.dof —Method.
dof(obj::StatisticalModel)
Return the number of degrees of freedom consumed in the model, including
when applicable the intercept and the distribution’s dispersion parameter.
source
58 CHAPTER 2. STATSBASE
2.39 StatsBase.dof residual
StatsBase.dof residual —Method.
dof_residual(obj::RegressionModel)
Return the residual degrees of freedom of the model.
source
2.40 StatsBase.ecdf
StatsBase.ecdf —Method.
ecdf(X)
Return an empirical cumulative distribution function (ECDF) based on a
vector of samples given in X.
Note: this is a higher-level function that returns a function, which can then
be applied to evaluate CDF values on other samples.
source
2.41 StatsBase.entropy
StatsBase.entropy —Method.
entropy(p, [b])
Compute the entropy of an array p, optionally specifying a real number b
such that the entropy is scaled by 1/log(b).
source
2.42 StatsBase.findat
StatsBase.findat —Method.
findat(a, b)
For each element in b, find its first index in a. If the value does not occur
in a, the corresponding index is 0.
source
2.43 StatsBase.fit!
StatsBase.fit! —Method.
Fit a statistical model in-place.
source
2.44. STATSBASE.FIT 59
2.44 StatsBase.fit
StatsBase.fit —Method.
Fit a statistical model.
source
2.45 StatsBase.fit
StatsBase.fit —Method.
fit(Histogram, data[, weight][, edges]; closed=:right, nbins)
Fit a histogram to data.
Arguments
•data: either a vector (for a 1-dimensional histogram), or a tuple of vectors
of equal length (for an n-dimensional histogram).
•weight: an optional AbstractWeights (of the same length as the data
vectors), denoting the weight each observation contributes to the bin. If
no weight vector is supplied, each observation has weight 1.
•edges: a vector (typically an AbstractRange object), or tuple of vectors,
that gives the edges of the bins along each dimension. If no edges are
provided, these are determined from the data.
Keyword arguments
•closed=:right: if :left, the bin intervals are left-closed [a,b); if :right
(the default), intervals are right-closed (a,b].
•nbins: if no edges argument is supplied, the approximate number of bins
to use along each dimension (can be either a single integer, or a tuple of
integers).
Examples
[] Univariate h = fit(Histogram, rand(100)) h = fit(Histogram, rand(100),
0:0.1:1.0) h = fit(Histogram, rand(100), nbins=10) h = fit(Histogram, rand(100),
weights(rand(100)), 0:0.1:1.0) h = fit(Histogram, [20], 0:20:100) h = fit(Histogram,
[20], 0:20:100, closed=:left)
Multivariate h = fit(Histogram, (rand(100),rand(100))) h = fit(Histogram,
(rand(100),rand(100)),nbins=10)
source
60 CHAPTER 2. STATSBASE
2.46 StatsBase.fitted
StatsBase.fitted —Method.
fitted(obj::RegressionModel)
Return the fitted values of the model.
source
2.47 StatsBase.fweights
StatsBase.fweights —Method.
fweights(vs)
Construct a FrequencyWeights vector from a given array. See the docu-
mentation for FrequencyWeights for more details.
source
2.48 StatsBase.genmean
StatsBase.genmean —Method.
genmean(a, p)
Return the generalized/power mean with exponent pof a real-valued array,
i.e. 1
nPn
i=1 ap
i1
p, where n = length(a). It is taken to be the geometric mean
when p == 0.
source
2.49 StatsBase.geomean
StatsBase.geomean —Method.
geomean(a)
Return the geometric mean of a real-valued array.
source
2.50 StatsBase.gkldiv
StatsBase.gkldiv —Method.
gkldiv(a, b)
Compute the generalized Kullback-Leibler divergence between two arrays:
Pn
i=1(ailog(ai/bi)−ai+bi). Efficient equivalent of sum(a*log(a/b)-a+b).
source
2.51. STATSBASE.HARMMEAN 61
2.51 StatsBase.harmmean
StatsBase.harmmean —Method.
harmmean(a)
Return the harmonic mean of a real-valued array.
source
2.52 StatsBase.indexmap
StatsBase.indexmap —Method.
indexmap(a)
Construct a dictionary that maps each unique value in ato the index of its
first occurrence in a.
source
2.53 StatsBase.indicatormat
StatsBase.indicatormat —Method.
indicatormat(x, c=sort(unique(x)); sparse=false)
Construct a boolean matrix Iof size (length(c), length(x)). Let ci be
the index of x[i] in c. Then I[ci, i] = true and all other elements are
false.
source
2.54 StatsBase.indicatormat
StatsBase.indicatormat —Method.
indicatormat(x, k::Integer; sparse=false)
Construct a boolean matrix Iof size (k, length(x)) such that I[x[i], i]
= true and all other elements are set to false. If sparse is true, the output
will be a sparse matrix, otherwise it will be dense (default).
Examples
julia> using StatsBase
julia> indicatormat([1 2 2], 2)
23 Array{Bool,2}:
true false false
false true true
source
62 CHAPTER 2. STATSBASE
2.55 StatsBase.inverse rle
StatsBase.inverse rle —Method.
inverse_rle(vals, lens)
Reconstruct a vector from its run-length encoding (see rle). vals is a vector
of the values and lens is a vector of the corresponding run lengths.
source
2.56 StatsBase.iqr
StatsBase.iqr —Method.
iqr(v)
Compute the interquartile range (IQR) of an array, i.e. the 75th percentile
minus the 25th percentile.
source
2.57 StatsBase.kldivergence
StatsBase.kldivergence —Method.
kldivergence(p, q, [b])
Compute the Kullback-Leibler divergence of qfrom p, optionally specifying
a real number bsuch that the divergence is scaled by 1/log(b).
source
2.58 StatsBase.kurtosis
StatsBase.kurtosis —Method.
kurtosis(v, [wv::AbstractWeights], m=mean(v))
Compute the excess kurtosis of a real-valued array v, optionally specifying
a weighting vector wv and a center m.
source
2.59 StatsBase.levelsmap
StatsBase.levelsmap —Method.
levelsmap(a)
Construct a dictionary that maps each of the nunique values in ato a
number between 1 and n.
source
2.60. STATSBASE.LOGLIKELIHOOD 63
2.60 StatsBase.loglikelihood
StatsBase.loglikelihood —Method.
loglikelihood(obj::StatisticalModel)
Return the log-likelihood of the model.
source
2.61 StatsBase.mad!
StatsBase.mad! —Method.
StatsBase.mad!(v; center=median!(v), normalize=true)
Compute the median absolute deviation (MAD) of varound center (by
default, around the median), overwriting vin the process.
If normalize is set to true, the MAD is multiplied by 1 / quantile(Normal(),
3/4) 1.4826, in order to obtain a consistent estimator of the standard devia-
tion under the assumption that the data is normally distributed.
source
2.62 StatsBase.mad
StatsBase.mad —Method.
mad(v; center=median(v), normalize=true)
Compute the median absolute deviation (MAD) of varound center (by
default, around the median).
If normalize is set to true, the MAD is multiplied by 1 / quantile(Normal(),
3/4) 1.4826, in order to obtain a consistent estimator of the standard devia-
tion under the assumption that the data is normally distributed.
source
2.63 StatsBase.maxad
StatsBase.maxad —Method.
maxad(a, b)
Return the maximum absolute deviation between two arrays: maxabs(a -
b).
source
64 CHAPTER 2. STATSBASE
2.64 StatsBase.mean and cov
StatsBase.mean and cov —Function.
mean_and_cov(x, [wv::AbstractWeights]; vardim=1, corrected=false) -> (mean, cov)
Return the mean and covariance matrix as a tuple. A weighting vector wv
can be specified. vardim that designates whether the variables are columns in
the matrix (1) or rows (2). Finally, bias correction is applied to the covariance
calculation if corrected=true. See cov documentation for more details.
source
2.65 StatsBase.mean and std
StatsBase.mean and std —Method.
mean_and_std(x, [w::AbstractWeights], [dim]; corrected=false) -> (mean, std)
Return the mean and standard deviation of a real-valued array x, optionally
over a dimension dim, as a tuple. A weighting vector wcan be specified to weight
the estimates. Finally, bias correction is applied to the standard deviation
calculation if corrected=true. See std documentation for more details.
source
2.66 StatsBase.mean and var
StatsBase.mean and var —Method.
mean_and_var(x, [w::AbstractWeights], [dim]; corrected=false) -> (mean, var)
Return the mean and variance of a real-valued array x, optionally over a
dimension dim, as a tuple. Observations in xcan be weighted using weight
vector w. Finally, bias correction is be applied to the variance calculation if
corrected=true. See var documentation for more details.
source
2.67 StatsBase.meanad
StatsBase.meanad —Method.
meanad(a, b)
Return the mean absolute deviation between two arrays: mean(abs(a -
b)).
source
2.68. STATSBASE.MODE 65
2.68 StatsBase.mode
StatsBase.mode —Method.
mode(a, [r])
Return the mode (most common number) of an array, optionally over a
specified range r. If several modes exist, the first one (in order of appearance)
is returned.
source
2.69 StatsBase.model response
StatsBase.model response —Method.
model_response(obj::RegressionModel)
Return the model response (a.k.a. the dependent variable).
source
2.70 StatsBase.modelmatrix
StatsBase.modelmatrix —Method.
modelmatrix(obj::RegressionModel)
Return the model matrix (a.k.a. the design matrix).
source
2.71 StatsBase.modes
StatsBase.modes —Method.
modes(a, [r])::Vector
Return all modes (most common numbers) of an array, optionally over a
specified range r.
source
2.72 StatsBase.moment
StatsBase.moment —Method.
moment(v, k, [wv::AbstractWeights], m=mean(v))
Return the kth order central moment of a real-valued array v, optionally
specifying a weighting vector wv and a center m.
source
66 CHAPTER 2. STATSBASE
2.73 StatsBase.msd
StatsBase.msd —Method.
msd(a, b)
Return the mean squared deviation between two arrays: mean(abs2(a -
b)).
source
2.74 StatsBase.nobs
StatsBase.nobs —Method.
nobs(obj::StatisticalModel)
Return the number of independent observations on which the model was fit-
ted. Be careful when using this information, as the definition of an independent
observation may vary depending on the model, on the format used to pass the
data, on the sampling plan (if specified), etc.
source
2.75 StatsBase.nquantile
StatsBase.nquantile —Method.
nquantile(v, n)
Return the n-quantiles of a real-valued array, i.e. the values which partition
vinto nsubsets of nearly equal size.
Equivalent to quantile(v, [0:n]/n). For example, nquantiles(x, 5) re-
turns a vector of quantiles, respectively at [0.0, 0.2, 0.4, 0.6, 0.8, 1.0].
source
2.76 StatsBase.nulldeviance
StatsBase.nulldeviance —Method.
nulldeviance(obj::StatisticalModel)
Return the deviance of the null model, that is the one including only the
intercept.
source
2.77. STATSBASE.NULLLOGLIKELIHOOD 67
2.77 StatsBase.nullloglikelihood
StatsBase.nullloglikelihood —Method.
loglikelihood(obj::StatisticalModel)
Return the log-likelihood of the null model corresponding to model obj. This
is usually the model containing only the intercept.
source
2.78 StatsBase.ordinalrank
StatsBase.ordinalrank —Method.
ordinalrank(x; lt = isless, rev::Bool = false)
Return the ordinal ranking (“1234” ranking) of an array. The lt keyword
allows providing a custom “less than” function; use rev=true to reverse the
sorting order. All items in xare given distinct, successive ranks based on their
position in sort(x; lt = lt, rev = rev). Missing values are assigned rank
missing.
source
2.79 StatsBase.pacf!
StatsBase.pacf! —Method.
pacf!(r, X, lags; method=:regression)
Compute the partial autocorrelation function (PACF) of a matrix Xat lags
and store the result in r.method designates the estimation method. Rec-
ognized values are :regression, which computes the partial autocorrelations
via successive regression models, and :yulewalker, which computes the partial
autocorrelations using the Yule-Walker equations.
rmust be a matrix of size (length(lags), size(x, 2)).
source
2.80 StatsBase.pacf
StatsBase.pacf —Method.
pacf(X, lags; method=:regression)
68 CHAPTER 2. STATSBASE
Compute the partial autocorrelation function (PACF) of a real-valued vector
or matrix Xat lags.method designates the estimation method. Recognized val-
ues are :regression, which computes the partial autocorrelations via successive
regression models, and :yulewalker, which computes the partial autocorrela-
tions using the Yule-Walker equations.
If xis a vector, return a vector of the same length as lags. If xis a matrix,
return a matrix of size (length(lags), size(x, 2)), where each column in
the result corresponds to a column in x.
source
2.81 StatsBase.percentile
StatsBase.percentile —Method.
percentile(v, p)
Return the pth percentile of a real-valued array v, i.e. quantile(x, p /
100).
source
2.82 StatsBase.predict
StatsBase.predict —Function.
predict(obj::RegressionModel, [newX])
Form the predicted response of model obj. An object with new covariate
values newX can be supplied, which should have the same type and structure as
that used to fit obj; e.g. for a GLM it would generally be a DataFrame with the
same variable names as the original predictors.
source
2.83 StatsBase.predict!
StatsBase.predict! —Function.
predict!
In-place version of predict.
source
2.84. STATSBASE.PROPORTIONMAP 69
2.84 StatsBase.proportionmap
StatsBase.proportionmap —Method.
proportionmap(x)
Return a dictionary mapping each unique value in xto its proportion in x.
source
2.85 StatsBase.proportions
StatsBase.proportions —Method.
proportions(x, k::Integer, [wv::AbstractWeights])
Return the proportion of integers in 1 to kthat occur in x.
source
2.86 StatsBase.proportions
StatsBase.proportions —Method.
proportions(x, levels=span(x), [wv::AbstractWeights])
Return the proportion of values in the range levels that occur in x. Equiva-
lent to counts(x, levels) / length(x). If a weighting vector wv is specified,
the sum of the weights is used rather than the raw counts.
source
2.87 StatsBase.psnr
StatsBase.psnr —Method.
psnr(a, b, maxv)
Compute the peak signal-to-noise ratio between two arrays aand b.maxv is
the maximum possible value either array can take. The PSNR is computed as
10 * log10(maxv^2 / msd(a, b)).
source
2.88 StatsBase.pweights
StatsBase.pweights —Method.
pweights(vs)
Construct a ProbabilityWeights vector from a given array. See the docu-
mentation for ProbabilityWeights for more details.
source
70 CHAPTER 2. STATSBASE
2.89 StatsBase.r2
StatsBase.r2 —Method.
r2(obj::StatisticalModel, variant::Symbol)
r(obj::StatisticalModel, variant::Symbol)
Coefficient of determination (R-squared).
For a linear model, the R is defined as ESS/T SS, with ESS the explained
sum of squares and T SS the total sum of squares, and variant can be omitted.
For other models, one of several pseudo R definitions must be chosen via
variant. Supported variants are:
•:MacFadden (a.k.a. likelihood ratio index), defined as 1 −log L/ log L0.
•:CoxSnell, defined as 1 −(L0/L)2/n
•:Nagelkerke, defined as (1 −(L0/L)2/n)/(1 −L02/n), with nthe number
of observations (as returned by nobs).
In the above formulas, Lis the likelihood of the model, L0 that of the null
model (the model including only the intercept). These two quantities are taken
to be minus half deviance of the corresponding models.
source
2.90 StatsBase.renyientropy
StatsBase.renyientropy —Method.
renyientropy(p, )
Compute the Rnyi (generalized) entropy of order of an array p.
source
2.91 StatsBase.residuals
StatsBase.residuals —Method.
residuals(obj::RegressionModel)
Return the residuals of the model.
source
2.92. STATSBASE.RLE 71
2.92 StatsBase.rle
StatsBase.rle —Method.
rle(v) -> (vals, lens)
Return the run-length encoding of a vector as a tuple. The first element
of the tuple is a vector of values of the input and the second is the number of
consecutive occurrences of each element.
Examples
julia> using StatsBase
julia> rle([1,1,1,2,2,3,3,3,3,2,2,2])
([1, 2, 3, 2], [3, 2, 4, 3])
source
2.93 StatsBase.rmsd
StatsBase.rmsd —Method.
rmsd(a, b; normalize=false)
Return the root mean squared deviation between two optionally normalized
arrays. The root mean squared deviation is computed as sqrt(msd(a, b)).
source
2.94 StatsBase.sample!
StatsBase.sample! —Method.
sample!([rng], a, [wv::AbstractWeights], x; replace=true, ordered=false)
Draw a random sample of length(x) elements from an array aand store the
result in x. A polyalgorithm is used for sampling. Sampling probabilities are
proportional to the weights given in wv, if provided. replace dictates whether
sampling is performed with replacement and order dictates whether an ordered
sample, also called a sequential sample, should be taken.
Optionally specify a random number generator rng as the first argument
(defaults to Base.GLOBAL RNG).
source
72 CHAPTER 2. STATSBASE
2.95 StatsBase.sample
StatsBase.sample —Method.
sample([rng], a, [wv::AbstractWeights])
Select a single random element of a. Sampling probabilities are proportional
to the weights given in wv, if provided.
Optionally specify a random number generator rng as the first argument
(defaults to Base.GLOBAL RNG).
source
2.96 StatsBase.sample
StatsBase.sample —Method.
sample([rng], wv::AbstractWeights)
Select a single random integer in 1:length(wv) with probabilities propor-
tional to the weights given in wv.
Optionally specify a random number generator rng as the first argument
(defaults to Base.GLOBAL RNG).
source
2.97 StatsBase.sample
StatsBase.sample —Method.
sample([rng], a, [wv::AbstractWeights], n::Integer; replace=true, ordered=false)
Select a random, optionally weighted sample of size nfrom an array ausing
a polyalgorithm. Sampling probabilities are proportional to the weights given in
wv, if provided. replace dictates whether sampling is performed with replace-
ment and order dictates whether an ordered sample, also called a sequential
sample, should be taken.
Optionally specify a random number generator rng as the first argument
(defaults to Base.GLOBAL RNG).
source
2.98 StatsBase.sample
StatsBase.sample —Method.
sample([rng], a, [wv::AbstractWeights], dims::Dims; replace=true, ordered=false)
2.99. STATSBASE.SAMPLEPAIR 73
Select a random, optionally weighted sample from an array aspecifying the
dimensions dims of the output array. Sampling probabilities are proportional
to the weights given in wv, if provided. replace dictates whether sampling is
performed with replacement and order dictates whether an ordered sample,
also called a sequential sample, should be taken.
Optionally specify a random number generator rng as the first argument
(defaults to Base.GLOBAL RNG).
source
2.99 StatsBase.samplepair
StatsBase.samplepair —Method.
samplepair([rng], a)
Draw a pair of distinct elements from the array awithout replacement.
Optionally specify a random number generator rng as the first argument
(defaults to Base.GLOBAL RNG).
source
2.100 StatsBase.samplepair
StatsBase.samplepair —Method.
samplepair([rng], n)
Draw a pair of distinct integers between 1 and nwithout replacement.
Optionally specify a random number generator rng as the first argument
(defaults to Base.GLOBAL RNG).
source
2.101 StatsBase.scattermat
StatsBase.scattermat —Function.
scattermat(X, [wv::AbstractWeights]; mean=nothing, vardim=1)
Compute the scatter matrix, which is an unnormalized covariance matrix.
A weighting vector wv can be specified to weight the estimate.
Arguments
•mean=nothing: a known mean value. nothing indicates that the mean
is unknown, and the function will compute the mean. Specifying mean=0
indicates that the data are centered and hence there’s no need to subtract
the mean.
74 CHAPTER 2. STATSBASE
•vardim=1: the dimension along which the variables are organized. When
vardim = 1, the variables are considered columns with observations in
rows; when vardim = 2, variables are in rows with observations in columns.
source
2.102 StatsBase.sem
StatsBase.sem —Method.
sem(a)
Return the standard error of the mean of a, i.e. sqrt(var(a) / length(a)).
source
2.103 StatsBase.skewness
StatsBase.skewness —Method.
skewness(v, [wv::AbstractWeights], m=mean(v))
Compute the standardized skewness of a real-valued array v, optionally spec-
ifying a weighting vector wv and a center m.
source
2.104 StatsBase.span
StatsBase.span —Method.
span(x)
Return the span of an integer array, i.e. the range minimum(x):maximum(x).
The minimum and maximum of xare computed in one-pass using extrema.
source
2.105 StatsBase.sqL2dist
StatsBase.sqL2dist —Method.
sqL2dist(a, b)
Compute the squared L2 distance between two arrays: Pn
i=1 |ai−bi|2. Ef-
ficient equivalent of sumabs2(a - b).
source
2.106. STATSBASE.STDERR 75
2.106 StatsBase.stderr
StatsBase.stderr —Method.
stderr(obj::StatisticalModel)
Return the standard errors for the coefficients of the model.
source
2.107 StatsBase.summarystats
StatsBase.summarystats —Method.
summarystats(a)
Compute summary statistics for a real-valued array a. Returns a SummaryStats
object containing the mean, minimum, 25th percentile, median, 75th percentile,
and maxmimum.
source
2.108 StatsBase.tiedrank
StatsBase.tiedrank —Method.
tiedrank(x)
Return the tied ranking, also called fractional or “1 2.5 2.5 4” ranking, of
an array. The lt keyword allows providing a custom “less than” function; use
rev=true to reverse the sorting order. Items that compare equal receive the
mean of the rankings they would have been assigned under ordinal ranking.
Missing values are assigned rank missing.
source
2.109 StatsBase.trim!
StatsBase.trim! —Method.
trim!(x; prop=0.0, count=0)
A variant of trim that modifies xin place.
source
76 CHAPTER 2. STATSBASE
2.110 StatsBase.trim
StatsBase.trim —Method.
trim(x; prop=0.0, count=0)
Return a copy of xwith either count or proportion prop of the high-
est and lowest elements removed. To compute the trimmed mean of xuse
mean(trim(x)); to compute the variance use trimvar(x) (see trimvar).
Example
[] julia¿ trim([1,2,3,4,5], prop=0.2) 3-element Array{Int64,1}: 2 3 4
source
2.111 StatsBase.trimvar
StatsBase.trimvar —Method.
trimvar(x; prop=0.0, count=0)
Compute the variance of the trimmed mean of x. This function uses the
Winsorized variance, as described in Wilcox (2010).
source
2.112 StatsBase.variation
StatsBase.variation —Method.
variation(x, m=mean(x))
Return the coefficient of variation of an array x, optionally specifying a
precomputed mean m. The coefficient of variation is the ratio of the standard
deviation to the mean.
source
2.113 StatsBase.vcov
StatsBase.vcov —Method.
vcov(obj::StatisticalModel)
Return the variance-covariance matrix for the coefficients of the model.
source
2.114. STATSBASE.WEIGHTS 77
2.114 StatsBase.weights
StatsBase.weights —Method.
weights(vs)
Construct a Weights vector from array vs. See the documentation for
Weights for more details.
source
2.115 StatsBase.winsor!
StatsBase.winsor! —Method.
winsor!(x; prop=0.0, count=0)
A variant of winsor that modifies vector xin place.
source
2.116 StatsBase.winsor
StatsBase.winsor —Method.
winsor(x; prop=0.0, count=0)
Return a copy of xwith either count or proportion prop of the lowest
elements of xreplaced with the next-lowest, and an equal number of the highest
elements replaced with the previous-highest. To compute the Winsorized mean
of xuse mean(winsor(x)).
Example
[] julia¿ winsor([1,2,3,4,5], prop=0.2) 5-element Array{Int64,1}: 2 2 3 4 4
source
2.117 StatsBase.wmean
StatsBase.wmean —Method.
wmean(v, w::AbstractVector)
Compute the weighted mean of an array vwith weights w.
source
78 CHAPTER 2. STATSBASE
2.118 StatsBase.wmedian
StatsBase.wmedian —Method.
wmedian(v, w)
Compute the weighted median of an array vwith weights w, given as either
a vector or an AbstractWeights vector.
source
2.119 StatsBase.wquantile
StatsBase.wquantile —Method.
wquantile(v, w, p)
Compute the pth quantile(s) of vwith weights w, given as either a vector or
an AbstractWeights vector.
source
2.120 StatsBase.wsample!
StatsBase.wsample! —Method.
wsample!([rng], a, w, x; replace=true, ordered=false)
Select a weighted sample from an array aand store the result in x. Sam-
pling probabilities are proportional to the weights given in w.replace dictates
whether sampling is performed with replacement and order dictates whether
an ordered sample, also called a sequential sample, should be taken.
Optionally specify a random number generator rng as the first argument
(defaults to Base.GLOBAL RNG).
source
2.121 StatsBase.wsample
StatsBase.wsample —Method.
wsample([rng], [a], w)
Select a weighted random sample of size 1 from awith probabilities propor-
tional to the weights given in w. If ais not present, select a random weight from
w.
Optionally specify a random number generator rng as the first argument
(defaults to Base.GLOBAL RNG).
source
2.122. STATSBASE.WSAMPLE 79
2.122 StatsBase.wsample
StatsBase.wsample —Method.
wsample([rng], [a], w, n::Integer; replace=true, ordered=false)
Select a weighted random sample of size nfrom awith probabilities pro-
portional to the weights given in wif ais present, otherwise select a random
sample of size nof the weights given in w.replace dictates whether sampling
is performed with replacement and order dictates whether an ordered sample,
also called a sequential sample, should be taken.
Optionally specify a random number generator rng as the first argument
(defaults to Base.GLOBAL RNG).
source
2.123 StatsBase.wsample
StatsBase.wsample —Method.
wsample([rng], [a], w, dims::Dims; replace=true, ordered=false)
Select a weighted random sample from awith probabilities proportional to
the weights given in wif ais present, otherwise select a random sample of size
nof the weights given in w. The dimensions of the output are given by dims.
Optionally specify a random number generator rng as the first argument
(defaults to Base.GLOBAL RNG).
source
2.124 StatsBase.wsum!
StatsBase.wsum! —Method.
wsum!(R, A, w, dim; init=true)
Compute the weighted sum of Awith weights wover the dimension dim and
store the result in R. If init=false, the sum is added to Rrather than starting
from zero.
source
2.125 StatsBase.wsum
StatsBase.wsum —Method.
wsum(v, w::AbstractVector, [dim])
Compute the weighted sum of an array vwith weights w, optionally over the
dimension dim.
source
80 CHAPTER 2. STATSBASE
2.126 StatsBase.zscore!
StatsBase.zscore! —Method.
zscore!([Z], X, , )
Compute the z-scores of an array Xwith mean and standard deviation .
z-scores are the signed number of standard deviations above the mean that an
observation lies, i.e. (x−)/.
If a destination array Zis provided, the scores are stored in Zand it must
have the same shape as X. Otherwise Xis overwritten.
source
2.127 StatsBase.zscore
StatsBase.zscore —Method.
zscore(X, [, ])
Compute the z-scores of X, optionally specifying a precomputed mean and
standard deviation . z-scores are the signed number of standard deviations above
the mean that an observation lies, i.e. (x−)/.
and should be both scalars or both arrays. The computation is broadcast-
ing. In particular, when and are arrays, they should have the same size, and
size(, i) == 1 || size(, i) == size(X, i) for each dimension.
source
Chapter 3
PyPlot
3.1 Base.step
Base.step —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“step”,))
source
3.2 PyPlot.Axes3D
PyPlot.Axes3D —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x0000000031fb9228),
(“Axes3D”,))
source
3.3 PyPlot.acorr
PyPlot.acorr —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“acorr”,))
source
3.4 PyPlot.annotate
PyPlot.annotate —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“annotate”,))
source
81
82 CHAPTER 3. PYPLOT
3.5 PyPlot.arrow
PyPlot.arrow —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“arrow”,))
source
3.6 PyPlot.autoscale
PyPlot.autoscale —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“autoscale”,))
source
3.7 PyPlot.autumn
PyPlot.autumn —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“autumn”,))
source
3.8 PyPlot.axes
PyPlot.axes —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“axes”,))
source
3.9 PyPlot.axhline
PyPlot.axhline —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“axhline”,))
source
3.10 PyPlot.axhspan
PyPlot.axhspan —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“axhspan”,))
source
3.11. PYPLOT.AXIS 83
3.11 PyPlot.axis
PyPlot.axis —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“axis”,))
source
3.12 PyPlot.axvline
PyPlot.axvline —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“axvline”,))
source
3.13 PyPlot.axvspan
PyPlot.axvspan —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“axvspan”,))
source
3.14 PyPlot.bar
PyPlot.bar —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“bar”,))
source
3.15 PyPlot.bar3D
PyPlot.bar3D —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x0000000031fb9228),
(“Axes3D”, “bar3d”))
source
3.16 PyPlot.barbs
PyPlot.barbs —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“barbs”,))
source
84 CHAPTER 3. PYPLOT
3.17 PyPlot.barh
PyPlot.barh —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“barh”,))
source
3.18 PyPlot.bone
PyPlot.bone —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“bone”,))
source
3.19 PyPlot.box
PyPlot.box —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“box”,))
source
3.20 PyPlot.boxplot
PyPlot.boxplot —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“boxplot”,))
source
3.21 PyPlot.broken barh
PyPlot.broken barh —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“broken barh”,))
source
3.22 PyPlot.cla
PyPlot.cla —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“cla”,))
source
3.23. PYPLOT.CLABEL 85
3.23 PyPlot.clabel
PyPlot.clabel —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“clabel”,))
source
3.24 PyPlot.clf
PyPlot.clf —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“clf”,))
source
3.25 PyPlot.clim
PyPlot.clim —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“clim”,))
source
3.26 PyPlot.cohere
PyPlot.cohere —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“cohere”,))
source
3.27 PyPlot.colorbar
PyPlot.colorbar —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“colorbar”,))
source
3.28 PyPlot.colors
PyPlot.colors —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“colors”,))
source
86 CHAPTER 3. PYPLOT
3.29 PyPlot.contour
PyPlot.contour —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“contour”,))
source
3.30 PyPlot.contour3D
PyPlot.contour3D —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x0000000031fb9228),
(“Axes3D”, “contour3D”))
source
3.31 PyPlot.contourf
PyPlot.contourf —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“contourf”,))
source
3.32 PyPlot.contourf3D
PyPlot.contourf3D —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x0000000031fb9228),
(“Axes3D”, “contourf3D”))
source
3.33 PyPlot.cool
PyPlot.cool —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“cool”,))
source
3.34 PyPlot.copper
PyPlot.copper —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“copper”,))
source
3.35. PYPLOT.CSD 87
3.35 PyPlot.csd
PyPlot.csd —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“csd”,))
source
3.36 PyPlot.delaxes
PyPlot.delaxes —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“delaxes”,))
source
3.37 PyPlot.disconnect
PyPlot.disconnect —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“disconnect”,))
source
3.38 PyPlot.draw
PyPlot.draw —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“draw”,))
source
3.39 PyPlot.errorbar
PyPlot.errorbar —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“errorbar”,))
source
3.40 PyPlot.eventplot
PyPlot.eventplot —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“eventplot”,))
source
88 CHAPTER 3. PYPLOT
3.41 PyPlot.figaspect
PyPlot.figaspect —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“figaspect”,))
source
3.42 PyPlot.figimage
PyPlot.figimage —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“figimage”,))
source
3.43 PyPlot.figlegend
PyPlot.figlegend —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“figlegend”,))
source
3.44 PyPlot.figtext
PyPlot.figtext —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“figtext”,))
source
3.45 PyPlot.figure
PyPlot.figure —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x0000000031fc52f0),
())
source
3.46 PyPlot.fill between
PyPlot.fill between —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“fill between”,))
source
3.47. PYPLOT.FILL BETWEENX 89
3.47 PyPlot.fill betweenx
PyPlot.fill betweenx —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“fill betweenx”,))
source
3.48 PyPlot.findobj
PyPlot.findobj —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“findobj”,))
source
3.49 PyPlot.flag
PyPlot.flag —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“flag”,))
source
3.50 PyPlot.gca
PyPlot.gca —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“gca”,))
source
3.51 PyPlot.gcf
PyPlot.gcf —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x0000000031fc5400),
())
source
3.52 PyPlot.gci
PyPlot.gci —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“gci”,))
source
90 CHAPTER 3. PYPLOT
3.53 PyPlot.get cmap
PyPlot.get cmap —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002586f2f0),
())
source
3.54 PyPlot.get current fig manager
PyPlot.get current fig manager —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“get current fig manager”,))
source
3.55 PyPlot.get figlabels
PyPlot.get figlabels —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“get figlabels”,))
source
3.56 PyPlot.get fignums
PyPlot.get fignums —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“get fignums”,))
source
3.57 PyPlot.get plot commands
PyPlot.get plot commands —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“get plot commands”,))
source
3.58 PyPlot.ginput
PyPlot.ginput —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“ginput”,))
source
3.59. PYPLOT.GRAY 91
3.59 PyPlot.gray
PyPlot.gray —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“gray”,))
source
3.60 PyPlot.grid
PyPlot.grid —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“grid”,))
source
3.61 PyPlot.hexbin
PyPlot.hexbin —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“hexbin”,))
source
3.62 PyPlot.hist2D
PyPlot.hist2D —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“hist2d”,))
source
3.63 PyPlot.hlines
PyPlot.hlines —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“hlines”,))
source
3.64 PyPlot.hold
PyPlot.hold —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“hold”,))
source
92 CHAPTER 3. PYPLOT
3.65 PyPlot.hot
PyPlot.hot —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“hot”,))
source
3.66 PyPlot.hsv
PyPlot.hsv —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“hsv”,))
source
3.67 PyPlot.imread
PyPlot.imread —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“imread”,))
source
3.68 PyPlot.imsave
PyPlot.imsave —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“imsave”,))
source
3.69 PyPlot.imshow
PyPlot.imshow —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“imshow”,))
source
3.70 PyPlot.ioff
PyPlot.ioff —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“ioff”,))
source
3.71. PYPLOT.ION 93
3.71 PyPlot.ion
PyPlot.ion —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“ion”,))
source
3.72 PyPlot.ishold
PyPlot.ishold —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“ishold”,))
source
3.73 PyPlot.jet
PyPlot.jet —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“jet”,))
source
3.74 PyPlot.legend
PyPlot.legend —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“legend”,))
source
3.75 PyPlot.locator params
PyPlot.locator params —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“locator params”,))
source
3.76 PyPlot.loglog
PyPlot.loglog —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“loglog”,))
source
94 CHAPTER 3. PYPLOT
3.77 PyPlot.margins
PyPlot.margins —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“margins”,))
source
3.78 PyPlot.matshow
PyPlot.matshow —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“matshow”,))
source
3.79 PyPlot.mesh
PyPlot.mesh —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x0000000031fb9228),
(“Axes3D”, “plot wireframe”))
source
3.80 PyPlot.minorticks off
PyPlot.minorticks off —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“minorticks off”,))
source
3.81 PyPlot.minorticks on
PyPlot.minorticks on —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“minorticks on”,))
source
3.82 PyPlot.over
PyPlot.over —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“over”,))
source
3.83. PYPLOT.PAUSE 95
3.83 PyPlot.pause
PyPlot.pause —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“pause”,))
source
3.84 PyPlot.pcolor
PyPlot.pcolor —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“pcolor”,))
source
3.85 PyPlot.pcolormesh
PyPlot.pcolormesh —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“pcolormesh”,))
source
3.86 PyPlot.pie
PyPlot.pie —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“pie”,))
source
3.87 PyPlot.pink
PyPlot.pink —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“pink”,))
source
3.88 PyPlot.plot
PyPlot.plot —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“plot”,))
source
96 CHAPTER 3. PYPLOT
3.89 PyPlot.plot3D
PyPlot.plot3D —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x0000000031fb9228),
(“Axes3D”, “plot3D”))
source
3.90 PyPlot.plot date
PyPlot.plot date —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“plot date”,))
source
3.91 PyPlot.plot surface
PyPlot.plot surface —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x0000000031fb9228),
(“Axes3D”, “plot surface”))
source
3.92 PyPlot.plot trisurf
PyPlot.plot trisurf —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x0000000031fb9228),
(“Axes3D”, “plot trisurf”))
source
3.93 PyPlot.plot wireframe
PyPlot.plot wireframe —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x0000000031fb9228),
(“Axes3D”, “plot wireframe”))
source
3.94 PyPlot.plotfile
PyPlot.plotfile —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“plotfile”,))
source
3.95. PYPLOT.POLAR 97
3.95 PyPlot.polar
PyPlot.polar —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“polar”,))
source
3.96 PyPlot.prism
PyPlot.prism —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“prism”,))
source
3.97 PyPlot.psd
PyPlot.psd —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“psd”,))
source
3.98 PyPlot.quiver
PyPlot.quiver —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“quiver”,))
source
3.99 PyPlot.quiverkey
PyPlot.quiverkey —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“quiverkey”,))
source
3.100 PyPlot.rc
PyPlot.rc —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“rc”,))
source
98 CHAPTER 3. PYPLOT
3.101 PyPlot.rc context
PyPlot.rc context —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“rc context”,))
source
3.102 PyPlot.rcdefaults
PyPlot.rcdefaults —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“rcdefaults”,))
source
3.103 PyPlot.register cmap
PyPlot.register cmap —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002586f268),
())
source
3.104 PyPlot.rgrids
PyPlot.rgrids —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“rgrids”,))
source
3.105 PyPlot.savefig
PyPlot.savefig —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“savefig”,))
source
3.106 PyPlot.sca
PyPlot.sca —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“sca”,))
source
3.107. PYPLOT.SCATTER 99
3.107 PyPlot.scatter
PyPlot.scatter —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“scatter”,))
source
3.108 PyPlot.scatter3D
PyPlot.scatter3D —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x0000000031fb9228),
(“Axes3D”, “scatter3D”))
source
3.109 PyPlot.sci
PyPlot.sci —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“sci”,))
source
3.110 PyPlot.semilogx
PyPlot.semilogx —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“semilogx”,))
source
3.111 PyPlot.semilogy
PyPlot.semilogy —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“semilogy”,))
source
3.112 PyPlot.set cmap
PyPlot.set cmap —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“set cmap”,))
source
100 CHAPTER 3. PYPLOT
3.113 PyPlot.setp
PyPlot.setp —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“setp”,))
source
3.114 PyPlot.specgram
PyPlot.specgram —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“specgram”,))
source
3.115 PyPlot.spectral
PyPlot.spectral —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“spectral”,))
source
3.116 PyPlot.spring
PyPlot.spring —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“spring”,))
source
3.117 PyPlot.spy
PyPlot.spy —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“spy”,))
source
3.118 PyPlot.stackplot
PyPlot.stackplot —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“stackplot”,))
source
3.119. PYPLOT.STEM 101
3.119 PyPlot.stem
PyPlot.stem —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“stem”,))
source
3.120 PyPlot.streamplot
PyPlot.streamplot —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“streamplot”,))
source
3.121 PyPlot.subplot
PyPlot.subplot —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“subplot”,))
source
3.122 PyPlot.subplot2grid
PyPlot.subplot2grid —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“subplot2grid”,))
source
3.123 PyPlot.subplot tool
PyPlot.subplot tool —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“subplot tool”,))
source
3.124 PyPlot.subplots
PyPlot.subplots —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“subplots”,))
source
102 CHAPTER 3. PYPLOT
3.125 PyPlot.subplots adjust
PyPlot.subplots adjust —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“subplots adjust”,))
source
3.126 PyPlot.summer
PyPlot.summer —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“summer”,))
source
3.127 PyPlot.suptitle
PyPlot.suptitle —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“suptitle”,))
source
3.128 PyPlot.surf
PyPlot.surf —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x0000000031fb9228),
(“Axes3D”, “plot surface”))
source
3.129 PyPlot.table
PyPlot.table —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“table”,))
source
3.130 PyPlot.text
PyPlot.text —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“text”,))
source
3.131. PYPLOT.TEXT2D 103
3.131 PyPlot.text2D
PyPlot.text2D —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x0000000031fb9228),
(“Axes3D”, “text2D”))
source
3.132 PyPlot.text3D
PyPlot.text3D —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x0000000031fb9228),
(“Axes3D”, “text3D”))
source
3.133 PyPlot.thetagrids
PyPlot.thetagrids —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“thetagrids”,))
source
3.134 PyPlot.tick params
PyPlot.tick params —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“tick params”,))
source
3.135 PyPlot.ticklabel format
PyPlot.ticklabel format —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“ticklabel format”,))
source
3.136 PyPlot.tight layout
PyPlot.tight layout —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“tight layout”,))
source
104 CHAPTER 3. PYPLOT
3.137 PyPlot.title
PyPlot.title —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“title”,))
source
3.138 PyPlot.tricontour
PyPlot.tricontour —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“tricontour”,))
source
3.139 PyPlot.tricontourf
PyPlot.tricontourf —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“tricontourf”,))
source
3.140 PyPlot.tripcolor
PyPlot.tripcolor —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“tripcolor”,))
source
3.141 PyPlot.triplot
PyPlot.triplot —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“triplot”,))
source
3.142 PyPlot.twinx
PyPlot.twinx —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“twinx”,))
source
3.143. PYPLOT.TWINY 105
3.143 PyPlot.twiny
PyPlot.twiny —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“twiny”,))
source
3.144 PyPlot.vlines
PyPlot.vlines —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“vlines”,))
source
3.145 PyPlot.waitforbuttonpress
PyPlot.waitforbuttonpress —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“waitforbuttonpress”,))
source
3.146 PyPlot.winter
PyPlot.winter —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“winter”,))
source
3.147 PyPlot.xkcd
PyPlot.xkcd —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“xkcd”,))
source
3.148 PyPlot.xlabel
PyPlot.xlabel —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“xlabel”,))
source
106 CHAPTER 3. PYPLOT
3.149 PyPlot.xlim
PyPlot.xlim —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“xlim”,))
source
3.150 PyPlot.xscale
PyPlot.xscale —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“xscale”,))
source
3.151 PyPlot.xticks
PyPlot.xticks —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“xticks”,))
source
3.152 PyPlot.ylabel
PyPlot.ylabel —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“ylabel”,))
source
3.153 PyPlot.ylim
PyPlot.ylim —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“ylim”,))
source
3.154 PyPlot.yscale
PyPlot.yscale —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“yscale”,))
source
3.155. PYPLOT.YTICKS 107
3.155 PyPlot.yticks
PyPlot.yticks —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x000000002573e818),
(“yticks”,))
source
3.156 PyPlot.zlabel
PyPlot.zlabel —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x0000000031fb9228),
(“Axes3D”, “set zlabel”))
source
3.157 PyPlot.zlim
PyPlot.zlim —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x0000000031fb9228),
(“Axes3D”, “set zlim”))
source
3.158 PyPlot.zscale
PyPlot.zscale —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x0000000031fb9228),
(“Axes3D”, “set zscale”))
source
3.159 PyPlot.zticks
PyPlot.zticks —Method.
PyPlot.LazyHelp(PyCall.PyObject(Ptr{PyCall.PyObject struct}@0x0000000031fb9228),
(“Axes3D”, “set zticks”))
source
Chapter 4
IndexedTables
4.1 Base.Sort.select
Base.Sort.select —Method.
select(t::Table, which::Selection)
Select all or a subset of columns, or a single column from the table.
Selection is a type union of many types that can select from a table. It
can be:
1. Integer – returns the column at this position.
2. Symbol – returns the column with this name.
3. Pair{Selection =>Function}– selects and maps a function over the
selection, returns the result.
4. AbstractArray – returns the array itself. This must be the same length
as the table.
5. Tuple of Selection – returns a table containing a column for every se-
lector in the tuple. The tuple may also contain the type Pair{Symbol,
Selection}, which the selection a name. The most useful form of this
when introducing a new column.
Examples:
Selection with Integer – returns the column at this position.
julia> tbl = table([0.01, 0.05], [2,1], [3,4], names=[:t, :x, :y], pkey=:t)
Table with 2 rows, 3 columns:
t x y
0.01 2 3
0.05 1 4
108
4.1. BASE.SORT.SELECT 109
julia> select(tbl, 2)
2-element Array{Int64,1}:
2
1
Selection with Symbol – returns the column with this name.
julia> select(tbl, :t)
2-element Array{Float64,1}:
0.01
0.05
Selection with Pair{Selection =>Function}– selects some columns and
maps a function over it, then returns the mapped column.
julia> select(tbl, :t=>t->1/t)
2-element Array{Float64,1}:
100.0
20.0
Selection with AbstractArray – returns the array itself.
julia> select(tbl, [3,4])
2-element Array{Int64,1}:
3
4
Selection with Tuple– returns a table containing a column for every selector
in the tuple.
julia> select(tbl, (2,1))
Table with 2 rows, 2 columns:
x t
2 0.01
1 0.05
julia> vx = select(tbl, (:x, :t)=>p->p.x/p.t)
2-element Array{Float64,1}:
200.0
20.0
julia> select(tbl, (:x,:t=>-))
Table with 2 rows, 2 columns:
x t
1 -0.05
2 -0.01
110 CHAPTER 4. INDEXEDTABLES
Note that since tbl was initialized with tas the primary key column, se-
lections that retain the key column will retain its status as a key. The same
applies when multiple key columns are selected.
Selection with a custom array in the tuple will cause the name of the columns
to be removed and replaced with integers.
julia> select(tbl, (:x, :t, [3,4]))
Table with 2 rows, 3 columns:
1 2 3
2 0.01 3
1 0.05 4
This is because the third column’s name is unknown. In general if a column’s
name cannot be determined, then selection returns an iterable of tuples rather
than named tuples. In other words, it strips column names.
To specify a new name to a custom column, you can use Symbol =>Selection
selector.
julia> select(tbl, (:x,:t,:z=>[3,4]))
Table with 2 rows, 3 columns:
x t z
2 0.01 3
1 0.05 4
julia> select(tbl, (:x, :t, :minust=>:t=>-))
Table with 2 rows, 3 columns:
x t minust
2 0.01 -0.01
1 0.05 -0.05
julia> select(tbl, (:x, :t, :vx=>(:x,:t)=>p->p.x/p.t))
Table with 2 rows, 3 columns:
x t vx
2 0.01 200.0
1 0.05 20.0
source
4.2 DataValues.dropna
DataValues.dropna —Function.
dropna(t[, select])
Drop rows which contain NA values.
4.3. INDEXEDTABLES.AGGREGATE! 111
julia> t = table([0.1, 0.5, NA,0.7], [2,NA,4,5], [NA,6,NA,7],
names=[:t,:x,:y])
Table with 4 rows, 3 columns:
txy
0.1 2 #NA
0.5 #NA 6
#NA 4 #NA
0.7 5 7
julia> dropna(t)
Table with 1 rows, 3 columns:
t x y
0.7 5 7
Optionally select can be speicified to limit columns to look for NAs in.
julia> dropna(t, :y)
Table with 2 rows, 3 columns:
txy
0.5 #NA 6
0.7 5 7
julia> t1 = dropna(t, (:t, :x))
Table with 2 rows, 3 columns:
t x y
0.1 2 #NA
0.7 5 7
Any columns whose NA rows have been dropped will be converted to non-
na array type. In our last example, columns tand xgot converted from
Array{DataValue{Int}} to Array{Int}. Similarly if the vectors are of type
DataValueArray{T}(default for loadtable) they will be converted to Array{T}.
[] julia¿ typeof(column(dropna(t,:x), :x)) Array{Int64,1}
source
4.3 IndexedTables.aggregate!
IndexedTables.aggregate! —Method.
aggregate!(f::Function, arr::NDSparse)
Combine adjacent rows with equal indices using the given 2-argument re-
duction function, in place.
source
112 CHAPTER 4. INDEXEDTABLES
4.4 IndexedTables.asofjoin
IndexedTables.asofjoin —Method.
asofjoin(left::NDSparse, right::NDSparse)
asofjoin is most useful on two time-series. It joins rows from left with the
“most recent” value from right.
julia> x = ndsparse((["ko","ko", "xrx","xrx"],
Date.(["2017-11-11", "2017-11-12",
"2017-11-11", "2017-11-12"])), [1,2,3,4]);
julia> y = ndsparse((["ko","ko", "xrx","xrx"],
Date.(["2017-11-12", "2017-11-13",
"2017-11-10", "2017-11-13"])), [5,6,7,8])
julia> asofjoin(x,y)
2-d NDSparse with 4 values (Int64):
1 2
"ko" 2017-11-11 1
"ko" 2017-11-12 5
"xrx" 2017-11-11 7
"xrx" 2017-11-12 7
source
4.5 IndexedTables.colnames
IndexedTables.colnames —Function.
colnames(itr)
Returns the names of the “columns” in itr.
Examples:
julia> colnames([1,2,3])
1-element Array{Int64,1}:
1
julia> colnames(Columns([1,2,3], [3,4,5]))
2-element Array{Int64,1}:
1
2
julia> colnames(table([1,2,3], [3,4,5]))
2-element Array{Int64,1}:
4.6. INDEXEDTABLES.COLUMNS 113
1
2
julia> colnames(Columns(x=[1,2,3], y=[3,4,5]))
2-element Array{Symbol,1}:
:x
:y
julia> colnames(table([1,2,3], [3,4,5], names=[:x,:y]))
2-element Array{Symbol,1}:
:x
:y
julia> colnames(ndsparse(Columns(x=[1,2,3]), Columns(y=[3,4,5])))
2-element Array{Symbol,1}:
:x
:y
julia> colnames(ndsparse(Columns(x=[1,2,3]), [3,4,5]))
2-element Array{Any,1}:
:x
1
2
julia> colnames(ndsparse(Columns(x=[1,2,3]), [3,4,5]))
2-element Array{Any,1}:
:x
2
julia> colnames(ndsparse(Columns([1,2,3], [4,5,6]), Columns(x=[6,7,8])))
3-element Array{Any,1}:
1
2
:x
julia> colnames(ndsparse(Columns(x=[1,2,3]), Columns([3,4,5],[6,7,8])))
3-element Array{Any,1}:
:x
2
3
source
4.6 IndexedTables.columns
IndexedTables.columns —Function.
114 CHAPTER 4. INDEXEDTABLES
columns(itr[, select::Selection])
Select one or more columns from an iterable of rows as a tuple of vectors.
select specifies which columns to select. See Selection convention for
possible values. If unspecified, returns all columns.
itr can be NDSparse,Columns and AbstractVector, and their distributed
counterparts.
Examples
julia> t = table([1,2],[3,4], names=[:x,:y])
Table with 2 rows, 2 columns:
x y
1 3
2 4
julia> columns(t)
(x = [1, 2], y = [3, 4])
julia> columns(t, :x)
2-element Array{Int64,1}:
1
2
julia> columns(t, (:x,))
(x = [1, 2])
julia> columns(t, (:y,:x=>-))
(y = [3, 4], x = [-1, -2])
source
4.7 IndexedTables.columns
IndexedTables.columns —Method.
columns(itr, which)
Returns a vector or a tuple of vectors from the iterator.
source
4.8 IndexedTables.convertdim
IndexedTables.convertdim —Method.
convertdim(x::NDSparse, d::DimName, xlate; agg::Function, vecagg::Function,
name)
Apply function or dictionary xlate to each index in the specified dimension.
If the mapping is many-to-one, agg or vecagg is used to aggregate the results.
4.9. INDEXEDTABLES.DIMLABELS 115
If agg is passed, it is used as a 2-argument reduction function over the data. If
vecagg is passed, it is used as a vector-to-scalar function to aggregate the data.
name optionally specifies a new name for the translated dimension.
source
4.9 IndexedTables.dimlabels
IndexedTables.dimlabels —Method.
dimlabels(t::NDSparse)
Returns an array of integers or symbols giving the labels for the dimensions
of t.ndims(t) == length(dimlabels(t)).
source
4.10 IndexedTables.flatten
IndexedTables.flatten —Method.
flatten(t::Table, col)
Flatten col column which may contain a vector of vectors while repeating
the other fields.
Examples:
julia> x = table([1,2], [[3,4], [5,6]], names=[:x, :y])
Table with 2 rows, 2 columns:
x y
1 [3, 4]
2 [5, 6]
julia> flatten(x, 2)
Table with 4 rows, 2 columns:
x y
1 3
1 4
2 5
2 6
julia> x = table([1,2], [table([3,4],[5,6], names=[:a,:b]),
table([7,8], [9,10], names=[:a,:b])], names=[:x, :y]);
julia> flatten(x, :y)
Table with 4 rows, 3 columns:
xab
135
116 CHAPTER 4. INDEXEDTABLES
146
279
2 8 10
source
4.11 IndexedTables.flush!
IndexedTables.flush! —Method.
flush!(arr::NDSparse)
Commit queued assignment operations, by sorting and merging the internal
temporary buffer.
source
4.12 IndexedTables.groupby
IndexedTables.groupby —Function.
groupby(f, t[, by::Selection]; select::Selection, flatten)
Group rows by by, and apply fto each group. fcan be a function or a tuple
of functions. The result of fon each group is put in a table keyed by unique by
values. flatten will flatten the result and can be used when freturns a vector
instead of a single scalar value.
Examples
julia> t=table([1,1,1,2,2,2], [1,1,2,2,1,1], [1,2,3,4,5,6],
names=[:x,:y,:z]);
julia> groupby(mean, t, :x, select=:z)
Table with 2 rows, 2 columns:
x mean
1 2.0
2 5.0
julia> groupby(identity, t, (:x, :y), select=:z)
Table with 4 rows, 3 columns:
x y identity
1 1 [1, 2]
1 2 [3]
2 1 [5, 6]
2 2 [4]
julia> groupby(mean, t, (:x, :y), select=:z)
Table with 4 rows, 3 columns:
4.12. INDEXEDTABLES.GROUPBY 117
x y mean
1 1 1.5
1 2 3.0
2 1 5.5
2 2 4.0
multiple aggregates can be computed by passing a tuple of functions:
julia> groupby((mean, std, var), t, :y, select=:z)
Table with 2 rows, 4 columns:
y mean std var
1 3.5 2.38048 5.66667
2 3.5 0.707107 0.5
julia> groupby(@NT(q25=z->quantile(z, 0.25), q50=median,
q75=z->quantile(z, 0.75)), t, :y, select=:z)
Table with 2 rows, 4 columns:
y q25 q50 q75
1 1.75 3.5 5.25
2 3.25 3.5 3.75
Finally, it’s possible to select different inputs for different functions by using
a named tuple of slector =>function pairs:
julia> groupby(@NT(xmean=:z=>mean, ystd=(:y=>-)=>std), t, :x)
Table with 2 rows, 3 columns:
x xmean ystd
1 2.0 0.57735
2 5.0 0.57735
By default, the result of groupby when freturns a vector or iterator of values
will not be expanded. Pass the flatten option as true to flatten the grouped
column:
julia> t = table([1,1,2,2], [3,4,5,6], names=[:x,:y])
julia> groupby((:normy => x->Iterators.repeated(mean(x), length(x)),),
t, :x, select=:y, flatten=true)
Table with 4 rows, 2 columns:
x normy
1 3.5
1 3.5
2 5.5
2 5.5
118 CHAPTER 4. INDEXEDTABLES
The keyword option usekey = true allows to use information from the in-
dexing column. fwill need to accept two arguments, the first being the key (as
aTuple or NamedTuple) the second the data (as Columns).
julia> t = table([1,1,2,2], [3,4,5,6], names=[:x,:y])
julia> groupby((:x_plus_mean_y => (key, d) -> key.x + mean(d),),
t, :x, select=:y, usekey = true)
Table with 2 rows, 2 columns:
x x_plus_mean_y
1 4.5
2 7.5
source
4.13 IndexedTables.groupjoin
IndexedTables.groupjoin —Method.
groupjoin([f, ] left, right; how, <options>)
Join left and right creating groups of values with matching keys.
Inner join
Inner join is the default join (when how is unspecified). It looks up keys from
left in right and only joins them when there is a match. This generates the
“intersection” of keys from left and right.
One-to-many and many-to-many matches
If the same key appears multiple times in either table (say, mand ntimes
respectively), each row with a key from left is matched with each row from
right with that key. The resulting group has mn output elements.
julia> l = table([1,1,1,2], [1,2,2,1], [1,2,3,4],
names=[:a,:b,:c], pkey=(:a, :b))
Table with 4 rows, 3 columns:
abc
111
122
123
214
julia> r = table([0,1,1,2], [1,2,2,1], [1,2,3,4],
names=[:a,:b,:d], pkey=(:a, :b))
Table with 4 rows, 3 columns:
abd
011
4.13. INDEXEDTABLES.GROUPJOIN 119
122
123
214
julia> groupjoin(l,r)
Table with 2 rows, 3 columns:
a b groups
1 2 NamedTuples._NT_c_d{Int64,Int64}[(c = 2, d = 2), (c = 2, d = 3), (c = 3, d = 2), (c = 3, d = 3)]
2 1 NamedTuples._NT_c_d{Int64,Int64}[(c = 4, d = 4)]
Left join
Left join looks up rows from right where keys match that in left. If there
are no such rows in right, an NA value is used for every selected field from
right.
julia> groupjoin(l,r, how=:left)
Table with 3 rows, 3 columns:
a b groups
1 1 NamedTuples._NT_c_d{Int64,Int64}[]
1 2 NamedTuples._NT_c_d{Int64,Int64}[(c = 2, d = 2), (c = 2, d = 3), (c = 3, d = 2), (c = 3, d = 3)]
2 1 NamedTuples._NT_c_d{Int64,Int64}[(c = 4, d = 4)]
Outer join
Outer (aka Union) join looks up rows from right where keys match that in
left, and also rows from left where keys match those in left, if there are no
matches on either side, a tuple of NA values is used. The output is guarranteed
to contain
julia> groupjoin(l,r, how=:outer)
Table with 4 rows, 3 columns:
a b groups
0 1 NamedTuples._NT_c_d{Int64,Int64}[]
1 1 NamedTuples._NT_c_d{Int64,Int64}[]
1 2 NamedTuples._NT_c_d{Int64,Int64}[(c = 2, d = 2), (c = 2, d = 3), (c = 3, d = 2), (c = 3, d = 3)]
2 1 NamedTuples._NT_c_d{Int64,Int64}[(c = 4, d = 4)]
Options
•how::Symbol – join method to use. Described above.
•lkey::Selection – fields from left to match on
•rkey::Selection – fields from right to match on
120 CHAPTER 4. INDEXEDTABLES
•lselect::Selection – fields from left to use as input to use as output
columns, or input to fif it is specified. By default, this is all fields not
selected in lkey.
•rselect::Selection – fields from left to use as input to use as output
columns, or input to fif it is specified. By default, this is all fields not
selected in rkey.
julia> groupjoin(l,r, lkey=:a, rkey=:a, lselect=:c, rselect=:d, how=:outer)
Table with 3 rows, 2 columns:
a groups
0 NamedTuples._NT_c_d{Int64,Int64}[]
1 NamedTuples._NT_c_d{Int64,Int64}[(c = 1, d = 2), (c = 1, d = 3), (c = 2, d = 2), (c = 2, d = 3), (c = 3, d = 2), (c = 3, d = 3)]
2 NamedTuples._NT_c_d{Int64,Int64}[(c = 4, d = 4)]
source
4.14 IndexedTables.groupreduce
IndexedTables.groupreduce —Function.
groupreduce(f, t[, by::Selection]; select::Selection)
Group rows by by, and apply fto reduce each group. fcan be a function,
OnlineStat or a struct of these as described in reduce. Recommended: see
documentation for reduce first. The result of reducing each group is put in a
table keyed by unique by values, the names of the output columns are the same
as the names of the fields of the reduced tuples.
Examples
julia> t=table([1,1,1,2,2,2], [1,1,2,2,1,1], [1,2,3,4,5,6],
names=[:x,:y,:z]);
julia> groupreduce(+, t, :x, select=:z)
Table with 2 rows, 2 columns:
x +
1 6
2 15
julia> groupreduce(+, t, (:x, :y), select=:z)
Table with 4 rows, 3 columns:
xy+
113
123
2 1 11
4.15. INDEXEDTABLES.INSERTCOL 121
224
julia> groupreduce((+, min, max), t, (:x, :y), select=:z)
Table with 4 rows, 5 columns:
x y + min max
1 1 3 1 2
1 2 3 3 3
2 1 11 5 6
2 2 4 4 4
If fis a single function or a tuple of functions, the output columns will be
named the same as the functions themselves. To change the name, pass a named
tuple:
julia> groupreduce(@NT(zsum=+, zmin=min, zmax=max), t, (:x, :y), select=:z)
Table with 4 rows, 5 columns:
x y zsum zmin zmax
1 1 3 1 2
1 2 3 3 3
2 1 11 5 6
2 2 4 4 4
Finally, it’s possible to select different inputs for different reducers by using
a named tuple of slector =>function pairs:
julia> groupreduce(@NT(xsum=:x=>+, negysum=(:y=>-)=>+), t, :x)
Table with 2 rows, 3 columns:
x xsum negysum
1 3 -4
2 6 -4
source
4.15 IndexedTables.insertcol
IndexedTables.insertcol —Method.
insertcol(t, position::Integer, name, x)
Insert a column xnamed name at position. Returns a new table.
julia> t = table([0.01, 0.05], [2,1], [3,4], names=[:t, :x, :y], pkey=:t)
Table with 2 rows, 3 columns:
t x y
122 CHAPTER 4. INDEXEDTABLES
0.01 2 3
0.05 1 4
julia> insertcol(t, 2, :w, [0,1])
Table with 2 rows, 4 columns:
t w x y
0.01 0 2 3
0.05 1 1 4
source
4.16 IndexedTables.insertcolafter
IndexedTables.insertcolafter —Method.
insertcolafter(t, after, name, col)
Insert a column col named name after after. Returns a new table.
julia> t = table([0.01, 0.05], [2,1], [3,4], names=[:t, :x, :y], pkey=:t)
Table with 2 rows, 3 columns:
t x y
0.01 2 3
0.05 1 4
julia> insertcolafter(t, :t, :w, [0,1])
Table with 2 rows, 4 columns:
t w x y
0.01 0 2 3
0.05 1 1 4
source
4.17 IndexedTables.insertcolbefore
IndexedTables.insertcolbefore —Method.
insertcolbefore(t, before, name, col)
Insert a column col named name before before. Returns a new table.
julia> t = table([0.01, 0.05], [2,1], [3,4], names=[:t, :x, :y], pkey=:t)
Table with 2 rows, 3 columns:
t x y
0.01 2 3
4.18. INDEXEDTABLES.NCOLS 123
0.05 1 4
julia> insertcolbefore(t, :x, :w, [0,1])
Table with 2 rows, 4 columns:
t w x y
0.01 0 2 3
0.05 1 1 4
source
4.18 IndexedTables.ncols
IndexedTables.ncols —Function.
ncols(itr)
Returns the number of columns in itr.
julia> ncols([1,2,3])
1
julia> d = ncols(rows(([1,2,3],[4,5,6])))
2
julia> ncols(table(([1,2,3],[4,5,6])))
2
julia> ncols(table(@NT(x=[1,2,3],y=[4,5,6])))
2
julia> ncols(ndsparse(d, [7,8,9]))
3
source
4.19 IndexedTables.ndsparse
IndexedTables.ndsparse —Function.
ndsparse(indices, data; agg, presorted, copy, chunks)
Construct an NDSparse array with the given indices and data. Each vector
in indices represents the index values for one dimension. On construction, the
indices and data are sorted in lexicographic order of the indices.
Arguments:
•agg::Function: If indices contains duplicate entries, the corresponding
data items are reduced using this 2-argument function.
124 CHAPTER 4. INDEXEDTABLES
•presorted::Bool: If true, the indices are assumed to already be sorted
and no sorting is done.
•copy::Bool: If true, the storage for the new array will not be shared with
the passed indices and data. If false (the default), the passed arrays will
be copied only if necessary for sorting. The only way to guarantee sharing
of data is to pass presorted=true.
•chunks::Integer: distribute the table into chunks (Integer) chunks (a
safe bet is nworkers()). Not distributed by default. See Distributed docs.
Examples:
1-dimensional NDSparse can be constructed with a single array as index.
julia> x = ndsparse(["a","b"],[3,4])
1-d NDSparse with 2 values (Int64):
1
"a" 3
"b" 4
julia> keytype(x), eltype(x)
(Tuple{String}, Int64)
A dimension will be named if constructed with a named tuple of columns as
index.
julia> x = ndsparse(@NT(date=Date.(2014:2017)), [4:7;])
1-d NDSparse with 4 values (Int64):
date
2014-01-01 4
2015-01-01 5
2016-01-01 6
2017-01-01 7
julia> x[Date("2015-01-01")]
5
julia> keytype(x), eltype(x)
(Tuple{Date}, Int64)
Multi-dimensional NDSparse can be constructed by passing a tuple of index
columns:
julia> x = ndsparse((["a","b"],[3,4]), [5,6])
2-d NDSparse with 2 values (Int64):
1 2
4.19. INDEXEDTABLES.NDSPARSE 125
"a" 3 5
"b" 4 6
julia> keytype(x), eltype(x)
(Tuple{String,Int64}, Int64)
julia> x["a", 3]
5
The data itself can also contain tuples (these are stored in columnar format,
just like in table.)
julia> x = ndsparse((["a","b"],[3,4]), ([5,6], [7.,8.]))
2-d NDSparse with 2 values (2-tuples):
1 2 3 4
"a" 3 5 7.0
"b" 4 6 8.0
julia> x = ndsparse(@NT(x=["a","a","b"],y=[3,4,4]),
@NT(p=[5,6,7], q=[8.,9.,10.]))
2-d NDSparse with 3 values (2 field named tuples):
x y p q
"a" 3 5 8.0
"a" 4 6 9.0
"b" 4 7 10.0
julia> keytype(x), eltype(x)
(Tuple{String,Int64}, NamedTuples._NT_p_q{Int64,Float64})
julia> x["a", :]
2-d NDSparse with 2 values (2 field named tuples):
x y p q
"a" 3 5 8.0
"a" 4 6 9.0
Passing a chunks option to ndsparse, or constructing with a distributed
array will cause the result to be distributed. Use distribute function to dis-
tribute an array.
julia> x = ndsparse(@NT(date=Date.(2014:2017)), [4:7.;], chunks=2)
1-d Distributed NDSparse with 4 values (Float64) in 2 chunks:
date
126 CHAPTER 4. INDEXEDTABLES
2014-01-01 4.0
2015-01-01 5.0
2016-01-01 6.0
2017-01-01 7.0
julia> x = ndsparse(@NT(date=Date.(2014:2017)), distribute([4:7.0;], 2))
1-d Distributed NDSparse with 4 values (Float64) in 2 chunks:
date
2014-01-01 4.0
2015-01-01 5.0
2016-01-01 6.0
2017-01-01 7.0
Distribution is done to match the first distributed column from left to right.
Specify chunks to override this.
source
4.20 IndexedTables.pairs
IndexedTables.pairs —Method.
pairs(arr::NDSparse, indices...)
Similar to where, but returns an iterator giving index=>value pairs. index
will be a tuple.
source
4.21 IndexedTables.pkeynames
IndexedTables.pkeynames —Method.
pkeynames(t::Table)
Names of the primary key columns in t.
Example
julia> t = table([1,2], [3,4]);
julia> pkeynames(t)
()
julia> t = table([1,2], [3,4], pkey=1);
julia> pkeynames(t)
(1,)
julia> t = table([2,1],[1,3],[4,5], names=[:x,:y,:z], pkey=(1,2));
4.22. INDEXEDTABLES.PKEYNAMES 127
julia> pkeys(t)
2-element IndexedTables.Columns{NamedTuples._NT_x_y{Int64,Int64},NamedTuples._NT_x_y{Array{Int64,1},Array{Int64,1}}}:
(x=1,y=3)
(x=2,y=1)
source
4.22 IndexedTables.pkeynames
IndexedTables.pkeynames —Method.
pkeynames(t::NDSparse)
Names of the primary key columns in t.
Example
julia> x = ndsparse([1,2],[3,4])
1-d NDSparse with 2 values (Int64):
1
1 3
2 4
julia> pkeynames(x)
(1,)
source
4.23 IndexedTables.pkeys
IndexedTables.pkeys —Method.
pkeys(itr::Table)
Primary keys of the table. If Table doesn’t have any designated primary
key columns (constructed without pkey argument) then a default key of tuples
(1,):(n,) is generated.
Example
julia> a = table(["a","b"], [3,4]) # no pkey
Table with 2 rows, 2 columns:
1 2
128 CHAPTER 4. INDEXEDTABLES
"a" 3
"b" 4
julia> pkeys(a)
2-element Columns{Tuple{Int64}}:
(1,)
(2,)
julia> a = table(["a","b"], [3,4], pkey=1)
Table with 2 rows, 2 columns:
1 2
"a" 3
"b" 4
julia> pkeys(a)
2-element Columns{Tuple{String}}:
("a",)
("b",)
source
4.24 IndexedTables.popcol
IndexedTables.popcol —Method.
popcol(t, col)
Remove the column col from the table. Returns a new table.
julia> t = table([0.01, 0.05], [2,1], [3,4], names=[:t, :x, :y], pkey=:t)
Table with 2 rows, 3 columns:
t x y
0.01 2 3
0.05 1 4
julia> popcol(t, :x)
Table with 2 rows, 2 columns:
t y
0.01 3
0.05 4
source
4.25. INDEXEDTABLES.PUSHCOL 129
4.25 IndexedTables.pushcol
IndexedTables.pushcol —Method.
pushcol(t, name, x)
Push a column xto the end of the table. name is the name for the new
column. Returns a new table.
Example:
julia> t = table([0.01, 0.05], [2,1], [3,4], names=[:t, :x, :y], pkey=:t)
Table with 2 rows, 3 columns:
t x y
0.01 2 3
0.05 1 4
julia> pushcol(t, :z, [1//2, 3//4])
Table with 2 rows, 4 columns:
t x y z
0.01 2 3 1//2
0.05 1 4 3//4
source
4.26 IndexedTables.reducedim vec
IndexedTables.reducedim vec —Method.
reducedim vec(f::Function, arr::NDSparse, dims)
Like reducedim, except uses a function mapping a vector of values to a scalar
instead of a 2-argument scalar function.
source
4.27 IndexedTables.reindex
IndexedTables.reindex —Function.
reindex(t::Table, by[, select])
Reindex tby columns selected in by. Keeps columns selected by select as
non-indexed columns. By default all columns not mentioned in by are kept.
Use selectkeys to reindex and NDSparse object.
julia> t = table([2,1],[1,3],[4,5], names=[:x,:y,:z], pkey=(1,2))
julia> reindex(t, (:y, :z))
Table with 2 rows, 3 columns:
yzx
130 CHAPTER 4. INDEXEDTABLES
142
351
julia> pkeynames(t)
(:y, :z)
julia> reindex(t, (:w=>[4,5], :z))
Table with 2 rows, 4 columns:
wzxy
4513
5421
julia> pkeynames(t)
(:w, :z)
source
4.28 IndexedTables.renamecol
IndexedTables.renamecol —Method.
renamecol(t, col, newname)
Set newname as the new name for column col in t. Returns a new table.
julia> t = table([0.01, 0.05], [2,1], names=[:t, :x])
Table with 2 rows, 2 columns:
t x
0.01 2
0.05 1
julia> renamecol(t, :t, :time)
Table with 2 rows, 2 columns:
time x
0.01 2
0.05 1
source
4.29 IndexedTables.rows
IndexedTables.rows —Function.
rows(itr[, select::Selection])
4.30. INDEXEDTABLES.SETCOL 131
Select one or more fields from an iterable of rows as a vector of their values.
select specifies which fields to select. See Selection convention for pos-
sible values. If unspecified, returns all columns.
itr can be NDSparse,Columns and AbstractVector, and their distributed
counterparts.
Examples
julia> t = table([1,2],[3,4], names=[:x,:y])
Table with 2 rows, 2 columns:
x y
1 3
2 4
julia> rows(t)
2-element IndexedTables.Columns{NamedTuples._NT_x_y{Int64,Int64},NamedTuples._NT_x_y{Array{Int64,1},Array{Int64,1}}}:
(x=1,y=3)
(x=2,y=4)
julia> rows(t, :x)
2-element Array{Int64,1}:
1
2
julia> rows(t, (:x,))
2-element IndexedTables.Columns{NamedTuples._NT_x{Int64},NamedTuples._NT_x{Array{Int64,1}}}:
(x = 1)
(x = 2)
julia> rows(t, (:y,:x=>-))
2-element IndexedTables.Columns{NamedTuples._NT_y_x{Int64,Int64},NamedTuples._NT_y_x{Array{Int64,1},Array{Int64,1}}}:
(y = 3, x = -1)
(y = 4, x = -2)
Note that vectors of tuples returned are Columns object and have columnar
internal storage.
source
4.30 IndexedTables.setcol
IndexedTables.setcol —Method.
setcol(t::Table, col::Union{Symbol, Int}, x::Selection)
Sets a xas the column identified by col. Returns a new table.
setcol(t::Table, map::Pair...)
Set many columns at a time.
Examples:
132 CHAPTER 4. INDEXEDTABLES
julia> t = table([1,2], [3,4], names=[:x, :y])
Table with 2 rows, 2 columns:
x y
1 3
2 4
julia> setcol(t, 2, [5,6])
Table with 2 rows, 2 columns:
x y
1 5
2 6
xcan be any selection that transforms existing columns.
julia> setcol(t, :x, :x => x->1/x)
Table with 2 rows, 2 columns:
x y
1.0 5
0.5 6
setcol will result in a re-sorted copy if a primary key column is replaced.
julia> t = table([0.01, 0.05], [1,2], [3,4], names=[:t, :x, :y], pkey=:t)
Table with 2 rows, 3 columns:
t x y
0.01 1 3
0.05 2 4
julia> t2 = setcol(t, :t, [0.1,0.05])
Table with 2 rows, 3 columns:
t x y
0.05 2 4
0.1 1 3
julia> t == t2
false
source
4.31 IndexedTables.stack
IndexedTables.stack —Method.
4.32. INDEXEDTABLES.SUMMARIZE 133
stack(t, by = pkeynames(t); select = excludecols(t, by), variable
= :variable, value = :value)
Reshape a table from the wide to the long format. Columns in by are
kept as indexing columns. Columns in select are stacked. In addition to the
id columns, two additional columns labeled variable and value are added,
containg the column identifier and the stacked columns.
Examples
julia> t = table(1:4, [1, 4, 9, 16], [1, 8, 27, 64], names = [:x, :xsquare, :xcube], pkey = :x);
julia> stack(t)
Table with 8 rows, 3 columns:
x variable value
1 :xsquare 1
1 :xcube 1
2 :xsquare 4
2 :xcube 8
3 :xsquare 9
3 :xcube 27
4 :xsquare 16
4 :xcube 64
source
4.32 IndexedTables.summarize
IndexedTables.summarize —Function.
summarize(f, t, by = pkeynames(t); select = excludecols(t, by))
Apply summary functions column-wise to a table. Return a NamedTuple in
the non-grouped case and a table in the grouped case.
Examples
julia> t = table([1, 2, 3], [1, 1, 1], names = [:x, :y]);
julia> summarize((mean, std), t)
(x_mean = 2.0, y_mean = 1.0, x_std = 1.0, y_std = 0.0)
julia> s = table(["a","a","b","b"], [1,3,5,7], [2,2,2,2], names = [:x, :y, :z], pkey = :x);
julia> summarize(mean, s)
Table with 2 rows, 3 columns:
xyz
"a" 2.0 2.0
"b" 6.0 2.0
134 CHAPTER 4. INDEXEDTABLES
Use a NamedTuple to have different names for the summary functions:
julia> summarize(@NT(m = mean, s = std), t)
(x_m = 2.0, y_m = 1.0, x_s = 1.0, y_s = 0.0)
Use select to only summarize some columns:
julia> summarize(@NT(m = mean, s = std), t, select = :x)
(m = 2.0, s = 1.0)
source
4.33 IndexedTables.table
IndexedTables.table —Function.
table(cols::AbstractVector...; names, <options>)
Create a table with columns given by cols.
julia> a = table([1,2,3], [4,5,6])
Table with 3 rows, 2 columns:
1 2
1 4
2 5
3 6
names specify names for columns. If specified, the table will be an iterator
of named tuples.
julia> b = table([1,2,3], [4,5,6], names=[:x, :y])
Table with 3 rows, 2 columns:
x y
1 4
2 5
3 6
table(cols::Union{Tuple, NamedTuple};<options>)
Convert a struct of columns to a table of structs.
julia> table(([1,2,3], [4,5,6])) == a
true
julia> table(@NT(x=[1,2,3], y=[4,5,6])) == b
true
table(cols::Columns; <options>)
Construct a table from a vector of tuples. See rows.
4.33. INDEXEDTABLES.TABLE 135
julia> table(Columns([1,2,3], [4,5,6])) == a
true
julia> table(Columns(x=[1,2,3], y=[4,5,6])) == b
true
table(t::Union{Table, NDSparse};<options>)
Copy a Table or NDSparse to create a new table. The same primary keys
as the input are used.
julia> b == table(b)
true
table(iter; <options>)
Construct a table from an iterable table.
Options:
•pkey: select columns to act as the primary key. By default, no columns
are used as primary key.
•presorted: is the data pre-sorted by primary key columns? If so, skip
sorting. false by default. Irrelevant if chunks is specified.
•copy: creates a copy of the input vectors if true.true by default. Irrela-
vant if chunks is specified.
•chunks: distribute the table into chunks (Integer) chunks (a safe bet is
nworkers()). Table is not distributed by default. See Distributed docs.
Examples:
Specifying pkey will cause the table to be sorted by the columns named in
pkey:
julia> b = table([2,3,1], [4,5,6], names=[:x, :y], pkey=:x)
Table with 3 rows, 2 columns:
x y
1 6
2 4
3 5
julia> b = table([2,1,2,1],[2,3,1,3],[4,5,6,7],
names=[:x, :y, :z], pkey=(:x,:y))
Table with 4 rows, 3 columns:
xyz
135
137
216
224
136 CHAPTER 4. INDEXEDTABLES
Note that the keys do not have to be unique.
chunks option creates a distributed table.
chunks can be:
1. An integer – number of chunks to create
2. An vector of kintegers – number of elements in each of the kchunks.
3. The distribution of another array. i.e. vec.subdomains where vec is a
distributed array.
julia> t = table([2,3,1,4], [4,5,6,7],
names=[:x, :y], pkey=:x, chunks=2)
Distributed Table with 4 rows in 2 chunks:
x y
1 6
2 4
3 5
4 7
A distributed table will be constructed if one of the arrays passed into table
constructor is a distributed array. A distributed Array can be constructed using
distribute:
julia> x = distribute([1,2,3,4], 2);
julia> t = table(x, [5,6,7,8], names=[:x,:y])
Distributed Table with 4 rows in 2 chunks:
x y
1 5
2 6
3 7
4 8
julia> table(columns(t)..., [9,10,11,12],
names=[:x,:y,:z])
Distributed Table with 4 rows in 2 chunks:
xyz
159
2 6 10
3 7 11
4 8 12
4.34. INDEXEDTABLES.UNSTACK 137
Distribution is done to match the first distributed column from left to right.
Specify chunks to override this.
source
4.34 IndexedTables.unstack
IndexedTables.unstack —Method.
unstack(t, by = pkeynames(t); variable = :variable, value = :value)
Reshape a table from the long to the wide format. Columns in by are kept
as indexing columns. Keyword arguments variable and value denote which
column contains the column identifier and which the corresponding values.
Examples
julia> t = table(1:4, [1, 4, 9, 16], [1, 8, 27, 64], names = [:x, :xsquare, :xcube], pkey = :x);
julia> long = stack(t)
Table with 8 rows, 3 columns:
x variable value
1 :xsquare 1
1 :xcube 1
2 :xsquare 4
2 :xcube 8
3 :xsquare 9
3 :xcube 27
4 :xsquare 16
4 :xcube 64
julia> unstack(long)
Table with 4 rows, 3 columns:
x xsquare xcube
1 1 1
2 4 8
3 9 27
4 16 64
source
4.35 IndexedTables.update!
IndexedTables.update! —Method.
update!(f::Function, arr::NDSparse, indices...)
Replace data values xwith f(x) at each location that matches the given
indices.
source
138 CHAPTER 4. INDEXEDTABLES
4.36 IndexedTables.where
IndexedTables.where —Method.
where(arr::NDSparse, indices...)
Returns an iterator over data items where the given indices match. Accepts
the same index arguments as getindex.
source
Chapter 5
Images
5.1 ColorTypes.blue
ColorTypes.blue —Function.
b = blue(img) extracts the blue channel from an RGB image img
source
5.2 ColorTypes.green
ColorTypes.green —Function.
g = green(img) extracts the green channel from an RGB image img
source
5.3 ColorTypes.red
ColorTypes.red —Function.
r = red(img) extracts the red channel from an RGB image img
source
5.4 Images.adjust gamma
Images.adjust gamma —Method.
gamma_corrected_img = adjust_gamma(img, gamma)
Returns a gamma corrected image.
The adjust gamma function can handle a variety of input types. The re-
turned image depends on the input type. If the input is an Image then the
resulting image is of the same type and has the same properties.
139
140 CHAPTER 5. IMAGES
For coloured images, the input is converted to YIQ type and the Y channel
is gamma corrected. This is the combined with the I and Q channels and the
resulting image converted to the same type as the input.
source
5.5 Images.bilinear interpolation
Images.bilinear interpolation —Method.
P = bilinear_interpolation(img, r, c)
Bilinear Interpolation is used to interpolate functions of two variables on a
rectilinear 2D grid.
The interpolation is done in one direction first and then the values obtained
are used to do the interpolation in the second direction.
source
5.6 Images.blob LoG
Images.blob LoG —Method.
blob_LoG(img, scales, [edges], [shape]) -> Vector{BlobLoG}
Find “blobs” in an N-D image using the negative Lapacian of Gaussians with
the specifed vector or tuple of values. The algorithm searches for places where
the filtered image (for a particular ) is at a peak compared to all spatially- and
-adjacent voxels, where is scales[i] * shape for some i. By default, shape
is an ntuple of 1s.
The optional edges argument controls whether peaks on the edges are in-
cluded. edges can be true or false, or a N+1-tuple in which the first entry
controls whether edge- values are eligible to serve as peaks, and the remaining
N entries control each of the N dimensions of img.
Citation:
Lindeberg T (1998), “Feature Detection with Automatic Scale Selection”,
International Journal of Computer Vision, 30(2), 79–116.
See also: BlobLoG.
source
5.7 Images.boxdiff
Images.boxdiff —Method.
sum = boxdiff(integral_image, ytop:ybot, xtop:xbot)
sum = boxdiff(integral_image, CartesianIndex(tl_y, tl_x), CartesianIndex(br_y, br_x))
sum = boxdiff(integral_image, tl_y, tl_x, br_y, br_x)
5.8. IMAGES.CANNY 141
An integral image is a data structure which helps in efficient calculation of
sum of pixels in a rectangular subset of an image. It stores at each pixel the sum
of all pixels above it and to its left. The sum of a window in an image can be
directly calculated using four array references of the integral image, irrespective
of the size of the window, given the yrange and xrange of the window. Given
an integral image -
A------B-
-*******-
-*******-
-*******-
-*******-
-*******-
C******D-
---------
The sum of pixels in the area denoted by * is given by S = D + A - B - C.
source
5.8 Images.canny
Images.canny —Method.
canny_edges = canny(img, (upper, lower), sigma=1.4)
Performs Canny Edge Detection on the input image.
Parameters :
(upper, lower) : Bounds for hysteresis thresholding sigma : Specifies the
standard deviation of the gaussian filter
Example
[] imgedg = canny(img, (Percentile(80), Percentile(20)))
source
5.9 Images.clahe
Images.clahe —Method.
hist_equalised_img = clahe(img, nbins, xblocks = 8, yblocks = 8, clip = 3)
Performs Contrast Limited Adaptive Histogram Equalisation (CLAHE) on
the input image. It differs from ordinary histogram equalization in the respect
that the adaptive method computes several histograms, each corresponding to
a distinct section of the image, and uses them to redistribute the lightness
values of the image. It is therefore suitable for improving the local contrast and
enhancing the definitions of edges in each region of an image.
142 CHAPTER 5. IMAGES
In the straightforward form, CLAHE is done by calculation a histogram
of a window around each pixel and using the transformation function of the
equalised histogram to rescale the pixel. Since this is computationally expen-
sive, we use interpolation which gives a significant rise in efficiency without
compromising the result. The image is divided into a grid and equalised his-
tograms are calculated for each block. Then, each pixel is interpolated using
the closest histograms.
The xblocks and yblocks specify the number of blocks to divide the input
image into in each direction. nbins specifies the granularity of histogram cal-
culation of each local region. clip specifies the value at which the histogram is
clipped. The excess in the histogram bins with value exceeding clip is redis-
tributed among the other bins.
source
5.10 Images.cliphist
Images.cliphist —Method.
clipped_hist = cliphist(hist, clip)
Clips the histogram above a certain value clip. The excess left in the bins
exceeding clip is redistributed among the remaining bins.
source
5.11 Images.complement
Images.complement —Method.
y = complement(x)
Take the complement 1-x of x. If xis a color with an alpha channel, the
alpha channel is left untouched.
source
5.12 Images.component boxes
Images.component boxes —Method.
component boxes(labeled array) ->an array of bounding boxes for each
label, including the background label 0
source
5.13. IMAGES.COMPONENT CENTROIDS 143
5.13 Images.component centroids
Images.component centroids —Method.
component centroids(labeled array) ->an array of centroids for each
label, including the background label 0
source
5.14 Images.component indices
Images.component indices —Method.
component indices(labeled array) ->an array of pixels for each label,
including the background label 0
source
5.15 Images.component lengths
Images.component lengths —Method.
component lengths(labeled array) ->an array of areas (2D), volumes
(3D), etc. for each label, including the background label 0
source
5.16 Images.component subscripts
Images.component subscripts —Method.
component subscripts(labeled array) ->an array of pixels for each la-
bel, including the background label 0
source
5.17 Images.convexhull
Images.convexhull —Method.
chull = convexhull(img)
Computes the convex hull of a binary image and returns the vertices of
convex hull as a CartesianIndex array.
source
5.18 Images.corner2subpixel
Images.corner2subpixel —Method.
corners = corner2subpixel(responses::AbstractMatrix,corner_indicator::AbstractMatrix{Bool})
-> Vector{HomogeneousPoint{Float64,3}}
144 CHAPTER 5. IMAGES
Refines integer corner coordinates to sub-pixel precision.
The function takes as input a matrix representing corner responses and a
boolean indicator matrix denoting the integer coordinates of a corner in the
image. The output is a vector of type HomogeneousPoint storing the sub-pixel
coordinates of the corners.
The algorithm computes a correction factor which is added to the original
integer coordinates. In particular, a univariate quadratic polynomial is fit sep-
arately to the x-coordinates and y-coordinates of a corner and its immediate
east/west, and north/south neighbours. The fit is achieved using a local coor-
dinate system for each corner, where the origin of the coordinate system is a
given corner, and its immediate neighbours are assigned coordinates of minus
one and plus one.
The corner and its two neighbours form a system of three equations. For
example, let x1=−1, x2= 0 and x3= 1 denote the local xcoordinates of
the west, center and east pixels and let the vector b= [r1, r2, r3] denote the
corresponding corner response values. With
A=
x2
1x11
x2
2x21
x2
3x31
,
the coefficients of the quadratic polynomial can be found by solving the
system of equations b=Ax. The result is given by x=A−1b.
The vertex of the quadratic polynomial yields a sub-pixel estimate of the true
corner position. For example, for a univariate quadratic polynomial px2+qx+r,
the x-coordinate of the vertex is −q
2p. Hence, the refined sub-pixel coordinate is
equal to: c+−q
2p, where cis the integer coordinate.
!!! note Corners on the boundary of the image are not refined to sub-pixel
precision.
source
5.19 Images.entropy
Images.entropy —Method.
entropy(log, img)
entropy(img; [kind=:shannon])
Compute the entropy of a grayscale image defined as -sum(p.*log(p)). The
base of the logarithm (a.k.a. entropy unit) is one of the following:
•:shannon (log base 2, default), or use log = log2
•:nat (log base e), or use log = log
•:hartley (log base 10), or use log = log10
source
5.20. IMAGES.FASTCORNERS 145
5.20 Images.fastcorners
Images.fastcorners —Method.
fastcorners(img, n, threshold) -> corners
Performs FAST Corner Detection. nis the number of contiguous pixels which
need to be greater (lesser) than intensity + threshold (intensity - threshold) for
a pixel to be marked as a corner. The default value for n is 12.
source
5.21 Images.findlocalmaxima
Images.findlocalmaxima —Function.
findlocalmaxima(img, [region, edges]) ->Vector{CartesianIndex}
Returns the coordinates of elements whose value is larger than all of their
immediate neighbors. region is a list of dimensions to consider. edges is a
boolean specifying whether to include the first and last elements of each dimen-
sion, or a tuple-of-Bool specifying edge behavior for each dimension separately.
source
5.22 Images.findlocalminima
Images.findlocalminima —Function.
Like findlocalmaxima, but returns the coordinates of the smallest elements.
source
5.23 Images.gaussian pyramid
Images.gaussian pyramid —Method.
pyramid = gaussian_pyramid(img, n_scales, downsample, sigma)
Returns a gaussian pyramid of scales n scales, each downsampled by a
factor downsample and sigma for the gaussian kernel.
source
5.24 Images.harris
Images.harris —Method.
harris_response = harris(img; [k], [border], [weights])
Performs Harris corner detection. The covariances can be taken using either
a mean weighted filter or a gamma kernel.
source
146 CHAPTER 5. IMAGES
5.25 Images.hausdorff distance
Images.hausdorff distance —Method.
hausdorff distance(imgA, imgB) is the modified Hausdorff distance be-
tween binary images (or point sets).
References
Dubuisson, M-P; Jain, A. K., 1994. A Modified Hausdorff Distance for
Object-Matching.
source
5.26 Images.histeq
Images.histeq —Method.
hist_equalised_img = histeq(img, nbins)
hist_equalised_img = histeq(img, nbins, minval, maxval)
Returns a histogram equalised image with a granularity of approximately
nbins number of bins.
The histeq function can handle a variety of input types. The returned
image depends on the input type. If the input is an Image then the resulting
image is of the same type and has the same properties.
For coloured images, the input is converted to YIQ type and the Y channel
is equalised. This is the combined with the I and Q channels and the resulting
image converted to the same type as the input.
If minval and maxval are specified then intensities are equalized to the range
(minval, maxval). The default values are 0 and 1.
source
5.27 Images.histmatch
Images.histmatch —Function.
hist_matched_img = histmatch(img, oimg, nbins)
Returns a grayscale histogram matched image with a granularity of nbins
number of bins. img is the image to be matched and oimg is the image having
the desired histogram to be matched to.
source
5.28 Images.imROF
Images.imROF —Method.
imgr = imROF(img, , iterations)
5.29. IMAGES.IMADJUSTINTENSITY 147
Perform Rudin-Osher-Fatemi (ROF) filtering, more commonly known as To-
tal Variation (TV) denoising or TV regularization. is the regularization coeffi-
cient for the derivative, and iterations is the number of relaxation iterations
taken. 2d only.
See https://en.wikipedia.org/wiki/Total variation denoising and Chambolle,
A. (2004). “An algorithm for total variation minimization and applications”.
Journal of Mathematical Imaging and Vision. 20: 89–97
source
5.29 Images.imadjustintensity
Images.imadjustintensity —Method.
imadjustintensity(img [, (minval,maxval)]) ->Image
Map intensities over the interval (minval,maxval) to the interval [0,1].
This is equivalent to map(ScaleMinMax(eltype(img), minval, maxval),
img). (minval,maxval) defaults to extrema(img).
source
5.30 Images.imcorner
Images.imcorner —Method.
corners = imcorner(img; [method])
corners = imcorner(img, threshold, percentile; [method])
Performs corner detection using one of the following methods -
1. harris
2. shi_tomasi
3. kitchen_rosenfeld
The parameters of the individual methods are described in their documen-
tation. The maxima values of the resultant responses are taken as corners. If a
threshold is specified, the values of the responses are thresholded to give the cor-
ner pixels. The threshold is assumed to be a percentile value unless percentile
is set to false.
source
5.31 Images.imcorner subpixel
Images.imcorner subpixel —Method.
corners = imcorner_subpixel(img; [method])
-> Vector{HomogeneousPoint{Float64,3}}
corners = imcorner_subpixel(img, threshold, percentile; [method])
-> Vector{HomogeneousPoint{Float64,3}}
148 CHAPTER 5. IMAGES
Same as imcorner, but estimates corners to sub-pixel precision.
Sub-pixel precision is achieved by interpolating the corner response val-
ues using the 4-connected neighbourhood of a maximum response value. See
corner2subpixel for more details of the interpolation scheme.
source
5.32 Images.imedge
Images.imedge —Function.
grad_y, grad_x, mag, orient = imedge(img, kernelfun=KernelFactors.ando3, border="replicate")
Edge-detection filtering. kernelfun is a valid kernel function for imgradients,
defaulting to KernelFactors.ando3.border is any of the boundary conditions
specified in padarray.
Returns a tuple (grad y, grad x, mag, orient), which are the horizontal
gradient, vertical gradient, and the magnitude and orientation of the strongest
edge, respectively.
source
5.33 Images.imhist
Images.imhist —Method.
edges, count = imhist(img, nbins)
edges, count = imhist(img, nbins, minval, maxval)
Generates a histogram for the image over nbins spread between (minval,
maxval]. If minval and maxval are not given, then the minimum and maximum
values present in the image are taken.
edges is a vector that specifies how the range is divided; count[i+1] is the
number of values xthat satisfy edges[i] <= x <edges[i+1].count[1] is
the number satisfying x<edges[1], and count[end] is the number satisfying
x>= edges[end]. Consequently, length(count) == length(edges)+1.
source
5.34 Images.imstretch
Images.imstretch —Method.
imgs = imstretch(img, m, slope) enhances or reduces (for slope >1 or
<1, respectively) the contrast near saturation (0 and 1). This is essentially a
symmetric gamma-correction. For a pixel of brightness p, the new intensity is
1/(1+(m/(p+eps))^slope).
This assumes the input img has intensities between 0 and 1.
source
5.35. IMAGES.INTEGRAL IMAGE 149
5.35 Images.integral image
Images.integral image —Method.
integral_img = integral_image(img)
Returns the integral image of an image. The integral image is calculated
by assigning to each pixel the sum of all pixels above it and to its left, i.e. the
rectangle from (1, 1) to the pixel. An integral image is a data structure which
helps in efficient calculation of sum of pixels in a rectangular subset of an image.
See boxdiff for more information.
source
5.36 Images.kitchen rosenfeld
Images.kitchen rosenfeld —Method.
kitchen_rosenfeld_response = kitchen_rosenfeld(img; [border])
Performs Kitchen Rosenfeld corner detection. The covariances can be taken
using either a mean weighted filter or a gamma kernel.
source
5.37 Images.label components
Images.label components —Function.
label = label_components(tf, [connectivity])
label = label_components(tf, [region])
Find the connected components in a binary array tf. There are two forms
that connectivity can take:
•It can be a boolean array of the same dimensionality as tf, of size 1 or 3
along each dimension. Each entry in the array determines whether a given
neighbor is used for connectivity analyses. For example, connectivity = trues(3,3)
would use 8-connectivity and test all pixels that touch the current one, even the
corners.
•You can provide a list indicating which dimensions are used to
determine connectivity. For example, region = [1,3] would not test neigh-
bors along dimension 2 for connectivity. This corresponds to just the nearest
neighbors, i.e., 4-connectivity in 2d and 6-connectivity in 3d.
The default is region = 1:ndims(A).
The output label is an integer array, where 0 is used for background pixels,
and each connected region gets a different integer index.
source
150 CHAPTER 5. IMAGES
5.38 Images.magnitude
Images.magnitude —Method.
m = magnitude(grad_x, grad_y)
Calculates the magnitude of the gradient images given by grad x and grad y.
Equivalent to sqrt(gradx.2+grady.2).
Returns a magnitude image the same size as grad x and grad y.
source
5.39 Images.magnitude phase
Images.magnitude phase —Method.
magnitude_phase(grad_x, grad_y) -> m, p
Convenience function for calculating the magnitude and phase of the gradient
images given in grad x and grad y. Returns a tuple containing the magnitude
and phase images. See magnitude and phase for details.
source
5.40 Images.maxabsfinite
Images.maxabsfinite —Method.
m = maxabsfinite(A) calculates the maximum absolute value in A, ignoring
any values that are not finite (Inf or NaN).
source
5.41 Images.maxfinite
Images.maxfinite —Method.
m = maxfinite(A) calculates the maximum value in A, ignoring any values
that are not finite (Inf or NaN).
source
5.42 Images.meanfinite
Images.meanfinite —Method.
M = meanfinite(img, region) calculates the mean value along the dimen-
sions listed in region, ignoring any non-finite values.
source
5.43. IMAGES.MINFINITE 151
5.43 Images.minfinite
Images.minfinite —Method.
m = minfinite(A) calculates the minimum value in A, ignoring any values
that are not finite (Inf or NaN).
source
5.44 Images.ncc
Images.ncc —Method.
C = ncc(A, B) computes the normalized cross-correlation of Aand B.
source
5.45 Images.orientation
Images.orientation —Method.
orientation(grad_x, grad_y) -> orient
Calculate the orientation angle of the strongest edge from gradient images
given by grad x and grad y. Equivalent to atan2(grad x, grad y). When
both grad x and grad y are effectively zero, the corresponding angle is set to
zero.
source
5.46 Images.otsu threshold
Images.otsu threshold —Method.
thres = otsu_threshold(img)
thres = otsu_threshold(img, bins)
Computes threshold for grayscale image using Otsu’s method.
Parameters:
•img = Grayscale input image
•bins = Number of bins used to compute the histogram. Needed for
floating-point images.
source
152 CHAPTER 5. IMAGES
5.47 Images.phase
Images.phase —Method.
phase(grad_x, grad_y) -> p
Calculate the rotation angle of the gradient given by grad x and grad y.
Equivalent to atan2(-grad y, grad x), except that when both grad x and
grad y are effectively zero, the corresponding angle is set to zero.
source
5.48 Images.sad
Images.sad —Method.
s = sad(A, B) computes the sum-of-absolute differences over arrays/images
A and B
source
5.49 Images.sadn
Images.sadn —Method.
s = sadn(A, B) computes the sum-of-absolute differences over arrays/images
A and B, normalized by array size
source
5.50 Images.shepp logan
Images.shepp logan —Method.
phantom = shepp_logan(N,[M]; highContrast=true)
output the NxM Shepp-Logan phantom, which is a standard test image usu-
ally used for comparing image reconstruction algorithms in the field of computed
tomography (CT) and magnetic resonance imaging (MRI). If the argument M
is omitted, the phantom is of size NxN. When setting the keyword argument
highConstrast to false, the CT version of the phantom is created. Otherwise,
the high contrast MRI version is calculated.
source
5.51 Images.shi tomasi
Images.shi tomasi —Method.
shi_tomasi_response = shi_tomasi(img; [border], [weights])
5.52. IMAGES.SSD 153
Performs Shi Tomasi corner detection. The covariances can be taken using
either a mean weighted filter or a gamma kernel.
source
5.52 Images.ssd
Images.ssd —Method.
s = ssd(A, B) computes the sum-of-squared differences over arrays/images
A and B
source
5.53 Images.ssdn
Images.ssdn —Method.
s = ssdn(A, B) computes the sum-of-squared differences over arrays/images
A and B, normalized by array size
source
5.54 Images.thin edges
Images.thin edges —Method.
thinned = thin_edges(img, gradientangle, [border])
thinned, subpix = thin_edges_subpix(img, gradientangle, [border])
thinned, subpix = thin_edges_nonmaxsup(img, gradientangle, [border]; [radius::Float64=1.35], [theta=pi/180])
thinned, subpix = thin_edges_nonmaxsup_subpix(img, gradientangle, [border]; [radius::Float64=1.35], [theta=pi/180])
Edge thinning for 2D edge images. Currently the only algorithm available
is non-maximal suppression, which takes an edge image and its gradient angle,
and checks each edge point for local maximality in the direction of the gradient.
The returned image is non-zero only at maximal edge locations.
border is any of the boundary conditions specified in padarray.
In addition to the maximal edge image, the subpix versions of these func-
tions also return an estimate of the subpixel location of each local maxima, as a
2D array or image of Graphics.Point objects. Additionally, each local maxima
is adjusted to the estimated value at the subpixel location.
Currently, the nonmaxsup functions are identical to the first two function
calls, except that they also accept additional keyword arguments. radius indi-
cates the step size to use when searching in the direction of the gradient; values
between 1.2 and 1.5 are suggested (default 1.35). theta indicates the step size
to use when discretizing angles in the gradientangle image, in radians (default:
1 degree in radians = pi/180).
Example:
154 CHAPTER 5. IMAGES
g = rgb2gray(rgb_image)
gx, gy = imgradients(g)
mag, grad_angle = magnitude_phase(gx,gy)
mag[mag .< 0.5] = 0.0 # Threshold magnitude image
thinned, subpix = thin_edges_subpix(mag, grad_angle)
source
Chapter 6
Knet
6.1 Knet.accuracy
Knet.accuracy —Method.
accuracy(model, data, predict; average=true)
Compute accuracy(predict(model,x), y) for (x,y) in data and return
the ratio (if average=true) or the count (if average=false) of correct answers.
source
6.2 Knet.accuracy
Knet.accuracy —Method.
accuracy(scores, answers, d=1; average=true)
Given an unnormalized scores matrix and an Integer array of correct
answers, return the ratio of instances where the correct answer has the max-
imum score. d=1 means instances are in columns, d=2 means instances are in
rows. Use average=false to return the number of correct answers instead of
the ratio.
source
6.3 Knet.batchnorm
Knet.batchnorm —Function.
batchnorm(x[, moments, params]; kwargs...) performs batch normal-
ization to xwith optional scaling factor and bias stored in params.
2d, 4d and 5d inputs are supported. Mean and variance are computed over
dimensions (2,), (1,2,4) and (1,2,3,5) for 2d, 4d and 5d arrays, respectively.
155
156 CHAPTER 6. KNET
moments stores running mean and variance to be used in testing. It is op-
tional in the training mode, but mandatory in the test mode. Training and test
modes are controlled by the training keyword argument.
params stores the optional affine parameters gamma and beta. bnparams
function can be used to initialize params.
Example
# Inilization, C is an integer
moments = bnmoments()
params = bnparams(C)
...
# size(x) -> (H, W, C, N)
y = batchnorm(x, moments, params)
# size(y) -> (H, W, C, N)
Keywords
eps=1e-5: The epsilon parameter added to the variance to avoid division by
0.
training: When training is true, the mean and variance of xare used and
moments argument is modified if it is provided. When training is false, mean
and variance stored in the moments argument are used. Default value is true
when at least one of xand params is AutoGrad.Rec,false otherwise.
source
6.4 Knet.bilinear
Knet.bilinear —Method.
Bilinear interpolation filter weights; used for initializing deconvolution layers.
Adapted from https://github.com/shelhamer/fcn.berkeleyvision.org/blob/master/surgery.py#L33
Arguments:
T: Data Type
fw: Width upscale factor
fh: Height upscale factor
IN: Number of input filters
ON: Number of output filters
Example usage:
w = bilinear(Float32,2,2,128,128)
source
6.5 Knet.bnmoments
Knet.bnmoments —Method.
bnmoments(;momentum=0.1, mean=nothing, var=nothing, meaninit=zeros,
varinit=ones) can be used directly load moments from data. meaninit and
varinit are called if mean and var are nothing. Type and size of the mean and
6.6. KNET.BNPARAMS 157
var are determined automatically from the inputs in the batchnorm calls. A
BNMoments object is returned.
BNMoments
A high-level data structure used to store running mean and running variance
of batch normalization with the following fields:
momentum::AbstractFloat: A real number between 0 and 1 to be used as
the scale of last mean and variance. The existing running mean or variance is
multiplied by (1-momentum).
mean: The running mean.
var: The running variance.
meaninit: The function used for initialize the running mean. Should either
be nothing or of the form (eltype, dims...)->data.zeros is a good option.
varinit: The function used for initialize the running variance. Should either
be nothing or (eltype, dims...)->data.ones is a good option.
source
6.6 Knet.bnparams
Knet.bnparams —Method.
bnparams(etype, channels) creates a single 1d array that contains both
scale and bias of batchnorm, where the first half is scale and the second half is
bias.
bnparams(channels) calls bnparams with etype=Float64, following Julia
convention
source
6.7 Knet.conv4
Knet.conv4 —Method.
conv4(w, x; kwargs...)
Execute convolutions or cross-correlations using filters specified with wover
tensor x.
Currently KnetArray{Float32/64,4/5}and Array{Float32/64,4}are sup-
ported as wand x. If whas dimensions (W1,W2,...,I,O) and xhas dimensions
(X1,X2,...,I,N), the result ywill have dimensions (Y1,Y2,...,O,N) where
Yi=1+floor((Xi+2*padding[i]-Wi)/stride[i])
Here Iis the number of input channels, Ois the number of output channels,
Nis the number of instances, and Wi,Xi,Yi are spatial dimensions. padding
and stride are keyword arguments that can be specified as a single number
(in which case they apply to all dimensions), or an array/tuple with entries for
each spatial dimension.
Keywords
158 CHAPTER 6. KNET
•padding=0: the number of extra zeros implicitly concatenated at the start
and at the end of each dimension.
•stride=1: the number of elements to slide to reach the next filtering
window.
•upscale=1: upscale factor for each dimension.
•mode=0: 0 for convolution and 1 for cross-correlation.
•alpha=1: can be used to scale the result.
•handle: handle to a previously created cuDNN context. Defaults to a
Knet allocated handle.
source
6.8 Knet.deconv4
Knet.deconv4 —Method.
y = deconv4(w, x; kwargs...)
Simulate 4-D deconvolution by using transposed convolution operation. Its
forward pass is equivalent to backward pass of a convolution (gradients with
respect to input tensor). Likewise, its backward pass (gradients with respect to
input tensor) is equivalent to forward pass of a convolution. Since it swaps for-
ward and backward passes of convolution operation, padding and stride options
belong to output tensor. See this report for further explanation.
Currently KnetArray{Float32/64,4}and Array{Float32/64,4}are supported
as wand x. If whas dimensions (W1,W2,...,O,I) and xhas dimensions
(X1,X2,...,I,N), the result ywill have dimensions (Y1,Y2,...,O,N) where
Yi = Wi+stride[i](Xi-1)-2 padding[i]
Here I is the number of input channels, O is the number of output channels,
N is the number of instances, and Wi,Xi,Yi are spatial dimensions. padding
and stride are keyword arguments that can be specified as a single number (in
which case they apply to all dimensions), or an array/tuple with entries for each
spatial dimension.
Keywords
•padding=0: the number of extra zeros implicitly concatenated at the start
and at the end of each dimension.
•stride=1: the number of elements to slide to reach the next filtering
window.
•mode=0: 0 for convolution and 1 for cross-correlation.
•alpha=1: can be used to scale the result.
6.9. KNET.DROPOUT 159
•handle: handle to a previously created cuDNN context. Defaults to a
Knet allocated handle.
source
6.9 Knet.dropout
Knet.dropout —Method.
dropout(x, p)
Given an array xand probability 0<=p<=1, just return xif testing, return
an array yin which each element is 0 with probability por x[i]/(1-p) with
probability 1-p if training. Training mode is detected automatically based on
the type of x, which is AutoGrad.Rec during gradient calculation. Use the key-
word argument training::Bool to change the default mode and seed::Number
to set the random number seed for reproducible results. See (Srivastava et al.
2014) for a reference.
source
6.10 Knet.gaussian
Knet.gaussian —Method.
gaussian(a...; mean=0.0, std=0.01)
Return a Gaussian array with a given mean and standard deviation. The a
arguments are passed to randn.
source
6.11 Knet.goldensection
Knet.goldensection —Method.
goldensection(f,n;kwargs) => (fmin,xmin)
Find the minimum of fusing concurrent golden section search in ndimen-
sions. See Knet.goldensection demo() for an example.
fis a function from a Vector{Float64}of length nto a Number. It can
return NaN for out of range inputs. Goldensection will always start with a
zero vector as the initial input to f, and the initial step size will be 1 in each
dimension. The user should define fto scale and shift this input range into a
vector meaningful for their application. For positive inputs like learning rate or
hidden size, you can use a transformation such as x0*exp(x) where xis a value
goldensection passes to fand x0 is your initial guess for this value. This will
effectively start the search at x0, then move with multiplicative steps.
160 CHAPTER 6. KNET
I designed this algorithm combining ideas from Golden Section Search and
Hill Climbing Search. It essentially runs golden section search concurrently in
each dimension, picking the next step based on estimated gain.
Keyword arguments
•dxmin=0.1: smallest step size.
•accel=: acceleration rate. Golden ratio =1.618... is best.
•verbose=false: use true to print individual steps.
•history=[]: cache of [(x,f(x)),...] function evaluations.
source
6.12 Knet.gpu
Knet.gpu —Function.
gpu() returns the id of the active GPU device or -1 if none are active.
gpu(true) resets all GPU devices and activates the one with the most avail-
able memory.
gpu(false) resets and deactivates all GPU devices.
gpu(d::Int) activates the GPU device dif 0<= d <gpuCount(), other-
wise deactivates devices.
gpu(true/false) resets all devices. If there are any allocated KnetArrays
their pointers will be left dangling. Thus gpu(true/false) should only be used
during startup. If you want to suspend GPU use temporarily, use gpu(-1).
gpu(d::Int) does not reset the devices. You can select a previous device
and find allocated memory preserved. However trying to operate on arrays of
an inactive device will result in error.
source
6.13 Knet.hyperband
Knet.hyperband —Function.
hyperband(getconfig, getloss, maxresource=27, reduction=3)
Hyperparameter optimization using the hyperband algorithm from (Lisha et
al. 2016). You can try a simple MNIST example using Knet.hyperband demo().
Arguments
•getconfig() returns random configurations with a user defined type and
distribution.
•getloss(c,n) returns loss for configuration cand number of resources
(e.g. epochs) n.
6.14. KNET.INVX 161
•maxresource is the maximum number of resources any one configuration
should be given.
•reduction is an algorithm parameter (see paper), 3 is a good value.
source
6.14 Knet.invx
Knet.invx —Function.
invx(x) = (1./x)
source
6.15 Knet.knetgc
Knet.knetgc —Function.
knetgc(dev=gpu())
cudaFree all pointers allocated on device dev that were previously allocated
and garbage collected. Normally Knet holds on to all garbage collected pointers
for reuse. Try this if you run out of GPU memory.
source
6.16 Knet.logp
Knet.logp —Method.
logp(x,[dims])
Treat entries in xas as unnormalized log probabilities and return normalized
log probabilities.
dims is an optional argument, if not specified the normalization is over the
whole x, otherwise the normalization is performed over the given dimensions.
In particular, if xis a matrix, dims=1 normalizes columns of xand dims=2
normalizes rows of x.
source
6.17 Knet.logsumexp
Knet.logsumexp —Method.
logsumexp(x,[dims])
162 CHAPTER 6. KNET
Compute log(sum(exp(x),dims)) in a numerically stable manner.
dims is an optional argument, if not specified the summation is over the
whole x, otherwise the summation is performed over the given dimensions. In
particular if xis a matrix, dims=1 sums columns of xand dims=2 sums rows of
x.
source
6.18 Knet.mat
Knet.mat —Method.
mat(x)
Reshape x into a two-dimensional matrix.
This is typically used when turning the output of a 4-D convolution re-
sult into a 2-D input for a fully connected layer. For 1-D inputs returns
reshape(x, (length(x),1)). For inputs with more than two dimensions of
size (X1,X2,...,XD), returns
reshape(x, (X1*X2*...*X[D-1],XD))
source
6.19 Knet.minibatch
Knet.minibatch —Method.
minibatch(x, y, batchsize; shuffle, partial, xtype, ytype)
Return an iterable of minibatches [(xi,yi). . . ] given data tensors x, y and
batchsize. The last dimension of x and y should match and give the number of
instances. Keyword arguments:
•shuffle=false: Shuffle the instances before minibatching.
•partial=false: If true include the last partial minibatch <batchsize.
•xtype=typeof(x): Convert xi in minibatches to this type.
•ytype=typeof(y): Convert yi in minibatches to this type.
source
6.20. KNET.MINIBATCH 163
6.20 Knet.minibatch
Knet.minibatch —Method.
minibatch(x, batchsize; shuffle, partial, xtype, ytype)
Return an iterable of minibatches [x1,x2,. . . ] given data tensor x and batch-
size. The last dimension of x gives the number of instances. Keyword arguments:
•shuffle=false: Shuffle the instances before minibatching.
•partial=false: If true include the last partial minibatch <batchsize.
•xtype=typeof(x): Convert xi in minibatches to this type.
source
6.21 Knet.nll
Knet.nll —Method.
nll(model, data, predict; average=true)
Compute nll(predict(model,x), y) for (x,y) in data and return the
per-instance average (if average=true) or total (if average=false) negative log
likelihood.
source
6.22 Knet.nll
Knet.nll —Method.
nll(scores, answers, d=1; average=true)
Given an unnormalized scores matrix and an Integer array of correct
answers, return the per-instance negative log likelihood. d=1 means instances
are in columns, d=2 means instances are in rows. Use average=false to return
the sum instead of per-instance average.
source
6.23 Knet.optimizers
Knet.optimizers —Method.
optimizers(model, otype; options...)
164 CHAPTER 6. KNET
Given parameters of a model, initialize and return corresponding optimiza-
tion parameters for a given optimization type otype and optimization options
options. This is useful because each numeric array in model needs its own dis-
tinct optimization parameter. optimizers makes the creation of optimization
parameters that parallel model parameters easy when all of them use the same
type and options.
source
6.24 Knet.pool
Knet.pool —Method.
pool(x; kwargs...)
Compute pooling of input values (i.e., the maximum or average of several
adjacent values) to produce an output with smaller height and/or width.
Currently 4 or 5 dimensional KnetArrays with Float32 or Float64 entries
are supported. If xhas dimensions (X1,X2,...,I,N), the result ywill have
dimensions (Y1,Y2,...,I,N) where
Yi=1+floor((Xi+2*padding[i]-window[i])/stride[i])
Here Iis the number of input channels, Nis the number of instances, and
Xi,Yi are spatial dimensions. window,padding and stride are keyword argu-
ments that can be specified as a single number (in which case they apply to all
dimensions), or an array/tuple with entries for each spatial dimension.
Keywords:
•window=2: the pooling window size for each dimension.
•padding=0: the number of extra zeros implicitly concatenated at the start
and at the end of each dimension.
•stride=window: the number of elements to slide to reach the next pooling
window.
•mode=0: 0 for max, 1 for average including padded values, 2 for average
excluding padded values.
•maxpoolingNanOpt=0: Nan numbers are not propagated if 0, they are
propagated if 1.
•alpha=1: can be used to scale the result.
•handle: Handle to a previously created cuDNN context. Defaults to a
Knet allocated handle.
source
6.25. KNET.RELU 165
6.25 Knet.relu
Knet.relu —Function.
relu(x) = max(0,x)
source
6.26 Knet.rnnforw
Knet.rnnforw —Method.
rnnforw(r, w, x[, hx, cx]; batchSizes, hy, cy)
Returns a tuple (y,hyout,cyout,rs) given rnn r, weights w, input xand op-
tionally the initial hidden and cell states hx and cx (cx is only used in LSTMs).
rand wshould come from a previous call to rnninit. Both hx and cx are
optional, they are treated as zero arrays if not provided. The output ycontains
the hidden states of the final layer for each time step, hyout and cyout give the
final hidden and cell states for all layers, rs is a buffer the RNN needs for its
gradient calculation.
The boolean keyword arguments hy and cy control whether hyout and cyout
will be output. By default hy = (hx!=nothing) and cy = (cx!=nothing &&
r.mode==2), i.e. a hidden state will be output if one is provided as input and
for cell state we also require an LSTM. If hy/cy is false,hyout/cyout will be
nothing.batchSizes can be an integer array that specifies non-uniform batch
sizes as explained below. By default batchSizes=nothing and the same batch
size, size(x,2), is used for all time steps.
The input and output dimensions are:
•x: (X,[B,T])
•y: (H/2H,[B,T])
•hx,cx,hyout,cyout: (H,B,L/2L)
•batchSizes:nothing or Vector{Int}(T)
where X is inputSize, H is hiddenSize, B is batchSize, T is seqLength, L is
numLayers. xcan be 1, 2, or 3 dimensional. If batchSizes==nothing, a 1-D x
represents a single instance, a 2-D xrepresents a single minibatch, and a 3-D x
represents a sequence of identically sized minibatches. If batchSizes is an array
of (non-increasing) integers, it gives us the batch size for each time step in the se-
quence, in which case sum(batchSizes) should equal div(length(x),size(x,1)).
yhas the same dimensionality as x, differing only in its first dimension, which
is H if the RNN is unidirectional, 2H if bidirectional. Hidden vectors hx,cx,
hyout,cyout all have size (H,B1,L) for unidirectional RNNs, and (H,B1,2L) for
bidirectional RNNs where B1 is the size of the first minibatch.
source
166 CHAPTER 6. KNET
6.27 Knet.rnninit
Knet.rnninit —Method.
rnninit(inputSize, hiddenSize; opts...)
Return an (r,w) pair where ris a RNN struct and wis a single weight array
that includes all matrices and biases for the RNN. Keyword arguments:
•rnnType=:lstm Type of RNN: One of :relu, :tanh, :lstm, :gru.
•numLayers=1: Number of RNN layers.
•bidirectional=false: Create a bidirectional RNN if true.
•dropout=0.0: Dropout probability. Ignored if numLayers==1.
•skipInput=false: Do not multiply the input with a matrix if true.
•dataType=Float32: Data type to use for weights.
•algo=0: Algorithm to use, see CUDNN docs for details.
•seed=0: Random number seed. Uses time() if 0.
•winit=xavier: Weight initialization method for matrices.
•binit=zeros: Weight initialization method for bias vectors.
•‘usegpu=(gpu()>=0): GPU used by default if one exists.
RNNs compute the output h[t] for a given iteration from the recurrent input
h[t-1] and the previous layer input x[t] given matrices W, R and biases bW, bR
from the following equations:
:relu and :tanh: Single gate RNN with activation function f:
h[t] = f(W * x[t] .+ R * h[t-1] .+ bW .+ bR)
:gru: Gated recurrent unit:
i[t] = sigm(Wi * x[t] .+ Ri * h[t-1] .+ bWi .+ bRi) # input gate
r[t] = sigm(Wr * x[t] .+ Rr * h[t-1] .+ bWr .+ bRr) # reset gate
n[t] = tanh(Wn * x[t] .+ r[t] .* (Rn * h[t-1] .+ bRn) .+ bWn) # new gate
h[t] = (1 - i[t]) .* n[t] .+ i[t] .* h[t-1]
:lstm: Long short term memory unit with no peephole connections:
i[t] = sigm(Wi * x[t] .+ Ri * h[t-1] .+ bWi .+ bRi) # input gate
f[t] = sigm(Wf * x[t] .+ Rf * h[t-1] .+ bWf .+ bRf) # forget gate
o[t] = sigm(Wo * x[t] .+ Ro * h[t-1] .+ bWo .+ bRo) # output gate
n[t] = tanh(Wn * x[t] .+ Rn * h[t-1] .+ bWn .+ bRn) # new gate
c[t] = f[t] .* c[t-1] .+ i[t] .* n[t] # cell output
h[t] = o[t] .* tanh(c[t])
source
6.28. KNET.RNNPARAM 167
6.28 Knet.rnnparam
Knet.rnnparam —Method.
rnnparam{T}(r::RNN, w::KnetArray{T}, layer, id, param)
Return a single weight matrix or bias vector as a slice of w.
Valid layer values:
•For unidirectional RNNs 1:numLayers
•For bidirectional RNNs 1:2*numLayers, forw and back layers alternate.
Valid id values:
•For RELU and TANH RNNs, input = 1, hidden = 2.
•For GRU reset = 1,4; update = 2,5; newmem = 3,6; 1:3 for input, 4:6 for
hidden
•For LSTM inputgate = 1,5; forget = 2,6; newmem = 3,7; output = 4,8;
1:4 for input, 5:8 for hidden
Valid param values:
•Return the weight matrix (transposed!) if param==1.
•Return the bias vector if param==2.
The effect of skipInput: Let I=1 for RELU/TANH, 1:3 for GRU, 1:4 for
LSTM
•For skipInput=false (default), rnnparam(r,w,1,I,1) is a (inputSize,hiddenSize)
matrix.
•For skipInput=true, rnnparam(r,w,1,I,1) is nothing.
•For bidirectional, the same applies to rnnparam(r,w,2,I,1): the first back
layer.
source
6.29 Knet.rnnparams
Knet.rnnparams —Method.
rnnparams(r::RNN, w)
Split w into individual parameters and return them as an array.
The order of params returned (subject to change):
168 CHAPTER 6. KNET
•All weight matrices come before all bias vectors.
•Matrices and biases are sorted lexically based on (layer,id).
•See @doc rnnparam for valid layer and id values.
•Input multiplying matrices are nothing if r.inputMode = 1.
source
6.30 Knet.setseed
Knet.setseed —Method.
setseed(n::Integer)
Run srand(n) on both cpu and gpu.
source
6.31 Knet.sigm
Knet.sigm —Function.
sigm(x) = (1./(1+exp(-x)))
source
6.32 Knet.unpool
Knet.unpool —Method.
Unpooling; reverse of pooling.
x == pool(unpool(x;o...); o...)
source
6.33 Knet.update!
Knet.update! —Function.
update!(weights, gradients, params)
update!(weights, gradients; lr=0.001, gclip=0)
Update the weights using their gradients and the optimization algorithm
parameters specified by params. The 2-arg version defaults to the Sgd algorithm
with learning rate lr and gradient clip gclip.gclip==0 indicates no clipping.
The weights and possibly gradients and params are modified in-place.
6.33. KNET.UPDATE! 169
weights can be an individual numeric array or a collection of arrays repre-
sented by an iterator or dictionary. In the individual case, gradients should
be a similar numeric array of size(weights) and params should be a single
object. In the collection case, each individual weight array should have a cor-
responding params object. This way different weight arrays can have their own
optimization state, different learning rates, or even different optimization algo-
rithms running in parallel. In the iterator case, gradients and params should
be iterators of the same length as weights with corresponding elements. In the
dictionary case, gradients and params should be dictionaries with the same
keys as weights.
Individual optimization parameters can be one of the following types. The
keyword arguments for each type’s constructor and their default values are listed
as well.
•Sgd(;lr=0.001, gclip=0)
•Momentum(;lr=0.001, gclip=0, gamma=0.9)
•Nesterov(;lr=0.001, gclip=0, gamma=0.9)
•Rmsprop(;lr=0.001, gclip=0, rho=0.9, eps=1e-6)
•Adagrad(;lr=0.1, gclip=0, eps=1e-6)
•Adadelta(;lr=0.01, gclip=0, rho=0.9, eps=1e-6)
•Adam(;lr=0.001, gclip=0, beta1=0.9, beta2=0.999, eps=1e-8)
Example:
w = rand(d) # an individual weight array
g = lossgradient(w) # gradient g has the same shape as w
update!(w, g) # update w in-place with Sgd()
update!(w, g; lr=0.1) # update w in-place with Sgd(lr=0.1)
update!(w, g, Sgd(lr=0.1)) # update w in-place with Sgd(lr=0.1)
w = (rand(d1), rand(d2)) # a tuple of weight arrays
g = lossgradient2(w) # g will also be a tuple
p = (Adam(), Sgd()) # p has params for each w[i]
update!(w, g, p) # update each w[i] in-place with g[i],p[i]
w = Any[rand(d1), rand(d2)] # any iterator can be used
g = lossgradient3(w) # g will be similar to w
p = Any[Adam(), Sgd()] # p should be an iterator of same length
update!(w, g, p) # update each w[i] in-place with g[i],p[i]
w = Dict(:a => rand(d1), :b => rand(d2)) # dictionaries can be used
g = lossgradient4(w)
p = Dict(:a => Adam(), :b => Sgd())
update!(w, g, p)
170 CHAPTER 6. KNET
source
6.34 Knet.xavier
Knet.xavier —Method.
xavier(a...)
Xavier initialization. The aarguments are passed to rand. See (Glorot
and Bengio 2010) for a description. Caffe implements this slightly differently.
Lasagne calls it GlorotUniform.
source
Chapter 7
DataFrames
7.1 DataFrames.aggregate
DataFrames.aggregate —Method.
Split-apply-combine that applies a set of functions over columns of an Ab-
stractDataFrame or GroupedDataFrame
[] aggregate(d::AbstractDataFrame, cols, fs) aggregate(gd::GroupedDataFrame,
fs)
Arguments
•d: an AbstractDataFrame
•gd : a GroupedDataFrame
•cols : a column indicator (Symbol, Int, Vector{Symbol}, etc.)
•fs : a function or vector of functions to be applied to vectors within
groups; expects each argument to be a column vector
Each fs should return a value or vector. All returns must be the same
length.
Returns
•::DataFrame
Examples
[] df = DataFrame(a = repeat([1, 2, 3, 4], outer=[2]), b = repeat([2, 1],
outer=[4]), c = randn(8)) aggregate(df, :a, sum) aggregate(df, :a, [sum, mean])
aggregate(groupby(df, :a), [sum, mean]) df —¿ groupby(:a) —¿ [sum, mean]
equivalent
source
171
172 CHAPTER 7. DATAFRAMES
7.2 DataFrames.by
DataFrames.by —Method.
Split-apply-combine in one step; apply fto each grouping in dbased on
columns col
[] by(d::AbstractDataFrame, cols, f::Function) by(f::Function, d::AbstractDataFrame,
cols)
Arguments
•d: an AbstractDataFrame
•cols : a column indicator (Symbol, Int, Vector{Symbol}, etc.)
•f: a function to be applied to groups; expects each argument to be an
AbstractDataFrame
fcan return a value, a vector, or a DataFrame. For a value or vector, these
are merged into a column along with the cols keys. For a DataFrame, cols are
combined along columns with the resulting DataFrame. Returning a DataFrame
is the clearest because it allows column labeling.
A method is defined with fas the first argument, so do-block notation can
be used.
by(d, cols, f) is equivalent to combine(map(f, groupby(d, cols))).
Returns
•::DataFrame
Examples
[] df = DataFrame(a = repeat([1, 2, 3, 4], outer=[2]), b = repeat([2,
1], outer=[4]), c = randn(8)) by(df, :a, d -¿ sum(d[:c])) by(df, :a, d -¿ 2
* d[:c]) by(df, :a, d -¿ DataFrame(csum =sum(d[: c]), cmean =mean(d[:
c])))by(df, :a, d−> DataF rame(c=d[: c], cmean =mean(d[: c])))by(df, [: a, :b])dodDataF rame(m=mean(d[: c]), v =var(d[: c]))end
source
7.3 DataFrames.coefnames
DataFrames.coefnames —Method.
coefnames(mf::ModelFrame)
Returns a vector of coefficient names constructed from the Terms member
and the types of the evaluation columns.
source
7.4. DATAFRAMES.COLWISE 173
7.4 DataFrames.colwise
DataFrames.colwise —Method.
Apply a function to each column in an AbstractDataFrame or Grouped-
DataFrame
[] colwise(f::Function, d) colwise(d)
Arguments
•f: a function or vector of functions
•d: an AbstractDataFrame of GroupedDataFrame
If dis not provided, a curried version of groupby is given.
Returns
•various, depending on the call
Examples
[] df = DataFrame(a = repeat([1, 2, 3, 4], outer=[2]), b = repeat([2, 1],
outer=[4]), c = randn(8)) colwise(sum, df) colwise(sum, groupby(df, :a))
source
7.5 DataFrames.combine
DataFrames.combine —Method.
Combine a GroupApplied object (rudimentary)
[] combine(ga::GroupApplied)
Arguments
•ga : a GroupApplied
Returns
•::DataFrame
Examples
[] df = DataFrame(a = repeat([1, 2, 3, 4], outer=[2]), b = repeat([2, 1],
outer=[4]), c = randn(8)) combine(map(d -¿ mean(d[:c]), gd))
source
7.6 DataFrames.completecases!
DataFrames.completecases! —Method.
Delete rows with NA’s.
[] completecases!(df::AbstractDataFrame)
Arguments
174 CHAPTER 7. DATAFRAMES
•df : the AbstractDataFrame
Result
•::AbstractDataFrame : the updated version
See also completecases.
Examples
[] df = DataFrame(i = 1:10, x = rand(10), y = rand([”a”, ”b”, ”c”], 10))
df[[1,4,5], :x] = NA df[[9,10], :y] = NA completecases!(df)
source
7.7 DataFrames.completecases
DataFrames.completecases —Method.
Indexes of complete cases (rows without NA’s)
[] completecases(df::AbstractDataFrame)
Arguments
•df : the AbstractDataFrame
Result
•::Vector{Bool}: indexes of complete cases
See also completecases!.
Examples
[] df = DataFrame(i = 1:10, x = rand(10), y = rand([”a”, ”b”, ”c”], 10))
df[[1,4,5], :x] = NA df[[9,10], :y] = NA completecases(df)
source
7.8 DataFrames.eltypes
DataFrames.eltypes —Method.
Column elemental types
[] eltypes(df::AbstractDataFrame)
Arguments
•df : the AbstractDataFrame
Result
•::Vector{Type}: the elemental type of each column
Examples
[] df = DataFrame(i = 1:10, x = rand(10), y = rand([”a”, ”b”, ”c”], 10))
eltypes(df)
source
7.9. DATAFRAMES.GROUPBY 175
7.9 DataFrames.groupby
DataFrames.groupby —Method.
A view of an AbstractDataFrame split into row groups
[] groupby(d::AbstractDataFrame, cols) groupby(cols)
Arguments
•d: an AbstractDataFrame
•cols : an
If dis not provided, a curried version of groupby is given.
Returns
•::GroupedDataFrame : a grouped view into d
Details
An iterator over a GroupedDataFrame returns a SubDataFrame view for each
grouping into d. A GroupedDataFrame also supports indexing by groups and
map.
See the following for additional split-apply-combine operations:
•by : split-apply-combine using functions
•aggregate : split-apply-combine; applies functions in the form of a cross
product
•combine : combine (obviously)
•colwise : apply a function to each column in an AbstractDataFrame or
GroupedDataFrame
Piping methods |>are also provided.
See the DataFramesMeta package for more operations on GroupedDataFrames.
Examples
[] df = DataFrame(a = repeat([1, 2, 3, 4], outer=[2]), b = repeat([2, 1],
outer=[4]), c = randn(8)) gd = groupby(df, :a) gd[1] last(gd) vcat([g[:b] for
g in gd]...) for g in gd println(g) end map(d -¿ mean(d[:c]), gd) returns a
GroupApplied object combine(map(d -¿ mean(d[:c]), gd)) df —¿ groupby(:a)
—¿ [sum, length] df —¿ groupby([:a, :b]) —¿ [sum, length]
source
7.10 DataFrames.head
DataFrames.head —Function.
Show the first or last part of an AbstractDataFrame
[] head(df::AbstractDataFrame, r::Int = 6) tail(df::AbstractDataFrame, r::Int
= 6)
Arguments
176 CHAPTER 7. DATAFRAMES
•df : the AbstractDataFrame
•r: the number of rows to show
Result
•::AbstractDataFrame : the first or last part of df
Examples
[] df = DataFrame(i = 1:10, x = rand(10), y = rand([”a”, ”b”, ”c”], 10))
head(df) tail(df)
source
7.11 DataFrames.melt
DataFrames.melt —Method.
Stacks a DataFrame; convert from a wide to long format; see stack.
source
7.12 DataFrames.meltdf
DataFrames.meltdf —Method.
A stacked view of a DataFrame (long format); see stackdf
source
7.13 DataFrames.names!
DataFrames.names! —Method.
Set column names
[] names!(df::AbstractDataFrame, vals)
Arguments
•df : the AbstractDataFrame
•vals : column names, normally a Vector{Symbol}the same length as the
number of columns in df
•allow duplicates : if false (the default), an error will be raised if
duplicate names are found; if true, duplicate names will be suffixed with
i(istarting at 1 for the first duplicate).
Result
•::AbstractDataFrame : the updated result
Examples
[] df = DataFrame(i = 1:10, x = rand(10), y = rand([”a”, ”b”, ”c”], 10))
names!(df, [:a, :b, :c]) names!(df, [:a, :b, :a]) throws ArgumentError names!(df,
[:a, :b, :a], allowduplicates =true)renamessecond :ato :a1
source
7.14. DATAFRAMES.NONUNIQUE 177
7.14 DataFrames.nonunique
DataFrames.nonunique —Method.
Indexes of complete cases (rows without NA’s)
[] nonunique(df::AbstractDataFrame) nonunique(df::AbstractDataFrame, cols)
Arguments
•df : the AbstractDataFrame
•cols : a column indicator (Symbol, Int, Vector{Symbol}, etc.) specifying
the column(s) to compare
Result
•::Vector{Bool}: indicates whether the row is a duplicate of some prior
row
See also unique and unique!.
Examples
[] df = DataFrame(i = 1:10, x = rand(10), y = rand([”a”, ”b”, ”c”], 10)) df
= vcat(df, df) nonunique(df) nonunique(df, 1)
source
7.15 DataFrames.readtable
DataFrames.readtable —Method.
Read data from a tabular-file format (CSV, TSV, . . . )
[] readtable(filename, [keyword options])
Arguments
•filename::AbstractString : the filename to be read
Keyword Arguments
•header::Bool – Use the information from the file’s header line to deter-
mine column names. Defaults to true.
•separator::Char – Assume that fields are split by the separator char-
acter. If not specified, it will be guessed from the filename: .csv defaults
to ,,.tsv defaults to , .wsv defaults to .
•quotemark::Vector{Char}– Assume that fields contained inside of two
quotemark characters are quoted, which disables processing of separators
and linebreaks. Set to Char[] to disable this feature and slightly improve
performance. Defaults to ["].
•decimal::Char – Assume that the decimal place in numbers is written
using the decimal character. Defaults to ..
178 CHAPTER 7. DATAFRAMES
•nastrings::Vector{String}– Translate any of the strings into this vec-
tor into an NA. Defaults to ["", "NA"].
•truestrings::Vector{String}– Translate any of the strings into this
vector into a Boolean true. Defaults to ["T", "t", "TRUE", "true"].
•falsestrings::Vector{String}– Translate any of the strings into this
vector into a Boolean false. Defaults to ["F", "f", "FALSE", "false"].
•makefactors::Bool – Convert string columns into PooledDataVector’s
for use as factors. Defaults to false.
•nrows::Int – Read only nrows from the file. Defaults to -1, which indi-
cates that the entire file should be read.
•names::Vector{Symbol}– Use the values in this array as the names for
all columns instead of or in lieu of the names in the file’s header. Defaults
to [], which indicates that the header should be used if present or that
numeric names should be invented if there is no header.
•eltypes::Vector – Specify the types of all columns. Defaults to [].
•allowcomments::Bool – Ignore all text inside comments. Defaults to
false.
•commentmark::Char – Specify the character that starts comments. De-
faults to #.
•ignorepadding::Bool – Ignore all whitespace on left and right sides of a
field. Defaults to true.
•skipstart::Int – Specify the number of initial rows to skip. Defaults to
0.
•skiprows::Vector{Int}– Specify the indices of lines in the input to
ignore. Defaults to [].
•skipblanks::Bool – Skip any blank lines in input. Defaults to true.
•encoding::Symbol – Specify the file’s encoding as either :utf8 or :latin1.
Defaults to :utf8.
•normalizenames::Bool – Ensure that column names are valid Julia iden-
tifiers. For instance this renames a column named "a b" to "a b" which
can then be accessed with :a b instead of Symbol("a b"). Defaults to
true.
Result
•::DataFrame
7.16. DATAFRAMES.RENAME 179
Examples
[] df = readtable(”data.csv”) df = readtable(”data.tsv”) df = readtable(”data.wsv”)
df = readtable(”data.txt”, separator = ’ ’) df = readtable(”data.txt”, header =
false)
source
7.16 DataFrames.rename
DataFrames.rename —Function.
Rename columns
[] rename!(df::AbstractDataFrame, from::Symbol, to::Symbol) rename!(df::AbstractDataFrame,
d::Associative) rename!(f::Function, df::AbstractDataFrame) rename(df::AbstractDataFrame,
from::Symbol, to::Symbol) rename(f::Function, df::AbstractDataFrame)
Arguments
•df : the AbstractDataFrame
•d: an Associative type that maps the original name to a new name
•f: a function that has the old column name (a symbol) as input and new
column name (a symbol) as output
Result
•::AbstractDataFrame : the updated result
Examples
[] df = DataFrame(i = 1:10, x = rand(10), y = rand([”a”, ”b”, ”c”], 10)) re-
name(x -¿ Symbol(uppercase(string(x))), df) rename(df, Dict(:i=¿:A, :x=¿:X))
rename(df, :y, :Y) rename!(df, Dict(:i=¿:A, :x=¿:X))
source
7.17 DataFrames.rename!
DataFrames.rename! —Function.
Rename columns
[] rename!(df::AbstractDataFrame, from::Symbol, to::Symbol) rename!(df::AbstractDataFrame,
d::Associative) rename!(f::Function, df::AbstractDataFrame) rename(df::AbstractDataFrame,
from::Symbol, to::Symbol) rename(f::Function, df::AbstractDataFrame)
Arguments
•df : the AbstractDataFrame
•d: an Associative type that maps the original name to a new name
•f: a function that has the old column name (a symbol) as input and new
column name (a symbol) as output
180 CHAPTER 7. DATAFRAMES
Result
•::AbstractDataFrame : the updated result
Examples
[] df = DataFrame(i = 1:10, x = rand(10), y = rand([”a”, ”b”, ”c”], 10)) re-
name(x -¿ Symbol(uppercase(string(x))), df) rename(df, Dict(:i=¿:A, :x=¿:X))
rename(df, :y, :Y) rename!(df, Dict(:i=¿:A, :x=¿:X))
source
7.18 DataFrames.stack
DataFrames.stack —Method.
Stacks a DataFrame; convert from a wide to long format
[] stack(df::AbstractDataFrame, measurevars, idvars)stack(df :: AbstractDataF rame, measurevars)stack(df :: AbstractDataF rame)melt(df :: AbstractDataF rame, idvars, measurevars)melt(df :: AbstractDataF rame, idvars)
Arguments
•df : the AbstractDataFrame to be stacked
•measure vars : the columns to be stacked (the measurement variables),
a normal column indexing type, like a Symbol, Vector{Symbol}, Int, etc.;
for melt, defaults to all variables that are not id vars
•id vars : the identifier columns that are repeated during stacking, a
normal column indexing type; for stack defaults to all variables that are
not measure vars
If neither measure vars or id vars are given, measure vars defaults to all
floating point columns.
Result
•::DataFrame : the long-format dataframe with column :value holding the
values of the stacked columns (measure vars), with column :variable
a Vector of Symbols with the measure vars name, and with columns for
each of the id vars.
See also stackdf and meltdf for stacking methods that return a view into
the original DataFrame. See unstack for converting from long to wide format.
Examples
[] d1 = DataFrame(a = repeat([1:3;], inner = [4]), b = repeat([1:4;], inner
= [3]), c = randn(12), d = randn(12), e = map(string, ’a’:’l’))
d1s = stack(d1, [:c, :d]) d1s2 = stack(d1, [:c, :d], [:a]) d1m = melt(d1, [:a,
:b, :e])
source
7.19. DATAFRAMES.STACKDF 181
7.19 DataFrames.stackdf
DataFrames.stackdf —Method.
A stacked view of a DataFrame (long format)
Like stack and melt, but a view is returned rather than data copies.
[] stackdf(df::AbstractDataFrame, measurevars, idvars)stackdf (df :: AbstractDataF rame, measurevars)meltdf (df :: AbstractDataF rame, idvars, measurevars)meltdf(df :: AbstractDataF rame, idvars)
Arguments
•df : the wide AbstractDataFrame
•measure vars : the columns to be stacked (the measurement variables),
a normal column indexing type, like a Symbol, Vector{Symbol}, Int, etc.;
for melt, defaults to all variables that are not id vars
•id vars : the identifier columns that are repeated during stacking, a
normal column indexing type; for stack defaults to all variables that are
not measure vars
Result
•::DataFrame : the long-format dataframe with column :value holding the
values of the stacked columns (measure vars), with column :variable
a Vector of Symbols with the measure vars name, and with columns for
each of the id vars.
The result is a view because the columns are special AbstractVectors that
return indexed views into the original DataFrame.
Examples
[] d1 = DataFrame(a = repeat([1:3;], inner = [4]), b = repeat([1:4;], inner
= [3]), c = randn(12), d = randn(12), e = map(string, ’a’:’l’))
d1s = stackdf(d1, [:c, :d]) d1s2 = stackdf(d1, [:c, :d], [:a]) d1m = meltdf(d1,
[:a, :b, :e])
source
7.20 DataFrames.tail
DataFrames.tail —Function.
Show the first or last part of an AbstractDataFrame
[] head(df::AbstractDataFrame, r::Int = 6) tail(df::AbstractDataFrame, r::Int
= 6)
Arguments
•df : the AbstractDataFrame
•r: the number of rows to show
Result
182 CHAPTER 7. DATAFRAMES
•::AbstractDataFrame : the first or last part of df
Examples
[] df = DataFrame(i = 1:10, x = rand(10), y = rand([”a”, ”b”, ”c”], 10))
head(df) tail(df)
source
7.21 DataFrames.unique!
DataFrames.unique! —Function.
Delete duplicate rows
[] unique(df::AbstractDataFrame) unique(df::AbstractDataFrame, cols) unique!(df::AbstractDataFrame)
unique!(df::AbstractDataFrame, cols)
Arguments
•df : the AbstractDataFrame
•cols : column indicator (Symbol, Int, Vector{Symbol}, etc.)
specifying the column(s) to compare.
Result
•::AbstractDataFrame : the updated version of df with unique rows.
When cols is specified, the return DataFrame contains complete rows, re-
taining in each case the first instance for which df[cols] is unique.
See also nonunique.
Examples
[] df = DataFrame(i = 1:10, x = rand(10), y = rand([”a”, ”b”, ”c”], 10)) df
= vcat(df, df) unique(df) doesn’t modify df unique(df, 1) unique!(df) modifies
df
source
7.22 DataFrames.unstack
DataFrames.unstack —Method.
Unstacks a DataFrame; convert from a long to wide format
[] unstack(df::AbstractDataFrame, rowkey, colkey, value) unstack(df::AbstractDataFrame,
colkey, value) unstack(df::AbstractDataFrame)
Arguments
•df : the AbstractDataFrame to be unstacked
•rowkey : the column with a unique key for each row, if not given, find a
key by grouping on anything not a colkey or value
7.23. DATAFRAMES.WRITETABLE 183
•colkey : the column holding the column names in wide format, defaults
to :variable
•value : the value column, defaults to :value
Result
•::DataFrame : the wide-format dataframe
Examples
[] wide = DataFrame(id = 1:12, a = repeat([1:3;], inner = [4]), b = re-
peat([1:4;], inner = [3]), c = randn(12), d = randn(12))
long = stack(wide) wide0 = unstack(long) wide1 = unstack(long, :variable,
:value) wide2 = unstack(long, :id, :variable, :value)
Note that there are some differences between the widened results above.
source
7.23 DataFrames.writetable
DataFrames.writetable —Method.
Write data to a tabular-file format (CSV, TSV, . . . )
[] writetable(filename, df, [keyword options])
Arguments
•filename::AbstractString : the filename to be created
•df::AbstractDataFrame : the AbstractDataFrame to be written
Keyword Arguments
•separator::Char – The separator character that you would like to use.
Defaults to the output of getseparator(filename), which uses commas
for files that end in .csv, tabs for files that end in .tsv and a single space
for files that end in .wsv.
•quotemark::Char – The character used to delimit string fields. Defaults
to ".
•header::Bool – Should the file contain a header that specifies the column
names from df. Defaults to true.
•nastring::AbstractString – What to write in place of missing data.
Defaults to "NA".
Result
•::DataFrame
Examples
[] df = DataFrame(A = 1:10) writetable(”output.csv”, df) writetable(”output.dat”,
df, separator = ’,’, header = false) writetable(”output.dat”, df, quotemark = ”,
separator = ’,’) writetable(”output.dat”, df, header = false)
source
184 CHAPTER 7. DATAFRAMES
7.24 StatsBase.describe
StatsBase.describe —Method.
Summarize the columns of an AbstractDataFrame
[] describe(df::AbstractDataFrame) describe(io, df::AbstractDataFrame)
Arguments
•df : the AbstractDataFrame
•io : optional output descriptor
Result
•nothing
Details
If the column’s base type derives from Number, compute the minimum, first
quantile, median, mean, third quantile, and maximum. NA’s are filtered and
reported separately.
For boolean columns, report trues, falses, and NAs.
For other types, show column characteristics and number of NAs.
Examples
[] df = DataFrame(i = 1:10, x = rand(10), y = rand([”a”, ”b”, ”c”], 10))
describe(df)
source
7.25 StatsBase.model response
StatsBase.model response —Method.
StatsBase.model_response(mf::ModelFrame)
Extract the response column, if present. DataVector or PooledDataVector
columns are converted to Arrays
source
Chapter 8
DataStructures
8.1 Base.pop!
Base.pop! —Method.
pop!(sc, k)
Deletes the item with key kin SortedDict or SortedSet sc and returns the
value that was associated with kin the case of SortedDict or kitself in the case
of SortedSet. A KeyError results if kis not in sc. Time: O(clog n)
source
8.2 Base.pop!
Base.pop! —Method.
pop!(sc, k)
Deletes the item with key kin SortedDict or SortedSet sc and returns the
value that was associated with kin the case of SortedDict or kitself in the case
of SortedSet. A KeyError results if kis not in sc. Time: O(clog n)
source
8.3 Base.pop!
Base.pop! —Method.
pop!(ss)
Deletes the item with first key in SortedSet ss and returns the key. A
BoundsError results if ss is empty. Time: O(clog n)
source
185
186 CHAPTER 8. DATASTRUCTURES
8.4 Base.push!
Base.push! —Method.
push!(sc, k)
Argument sc is a SortedSet and kis a key. This inserts the key into the
container. If the key is already present, this overwrites the old value. (This is
not necessarily a no-op; see below for remarks about the customizing the sort
order.) The return value is sc. Time: O(clog n)
source
8.5 Base.push!
Base.push! —Method.
push!(sc, k=>v)
Argument sc is a SortedDict or SortedMultiDict and k=>vis a key-value
pair. This inserts the key-value pair into the container. If the key is already
present, this overwrites the old value. The return value is sc. Time: O(clog n)
source
8.6 Base.push!
Base.push! —Method.
push!(sc, k=>v)
Argument sc is a SortedDict or SortedMultiDict and k=>vis a key-value
pair. This inserts the key-value pair into the container. If the key is already
present, this overwrites the old value. The return value is sc. Time: O(clog n)
source
8.7 DataStructures.back
DataStructures.back —Method.
back(q::Deque)
Returns the last element of the deque q.
source
8.8. DATASTRUCTURES.COMPARE 187
8.8 DataStructures.compare
DataStructures.compare —Method.
compare(m::SAContainer, s::IntSemiToken, t::IntSemiToken)
Determines the relative positions of the data items indexed by (m,s) and
(m,t) in the sorted order. The return value is -1 if (m,s) precedes (m,t),0if
they are equal, and 1if (m,s) succeeds (m,t).sand tare semitokens for the
same container m.
source
8.9 DataStructures.counter
DataStructures.counter —Method.
counter(seq)
Returns an Accumulator object containing the elements from seq.
source
8.10 DataStructures.dec!
DataStructures.dec! —Method.
dec!(ct, x, [v=1])
Decrements the count for xby v(defaulting to one)
source
8.11 DataStructures.deque
DataStructures.deque —Method.
deque(T)
Create a deque of type T.
source
8.12 DataStructures.dequeue!
DataStructures.dequeue! —Method.
dequeue!(pq)
Remove and return the lowest priority key from a priority queue.
188 CHAPTER 8. DATASTRUCTURES
julia> a = PriorityQueue(["a","b","c"],[2,3,1],Base.Order.Forward)
PriorityQueue{String,Int64,Base.Order.ForwardOrdering} with 3 entries:
"c" => 1
"b" => 3
"a" => 2
julia> dequeue!(a)
"c"
julia> a
PriorityQueue{String,Int64,Base.Order.ForwardOrdering} with 2 entries:
"b" => 3
"a" => 2
source
8.13 DataStructures.dequeue!
DataStructures.dequeue! —Method.
dequeue!(s::Queue)
Removes an element from the front of the queue sand returns it.
source
8.14 DataStructures.dequeue pair!
DataStructures.dequeue pair! —Method.
dequeue_pair!(pq)
Remove and return a the lowest priority key and value from a priority queue
as a pair.
julia> a = PriorityQueue(["a","b","c"],[2,3,1],Base.Order.Forward)
PriorityQueue{String,Int64,Base.Order.ForwardOrdering} with 3 entries:
"c" => 1
"b" => 3
"a" => 2
julia> dequeue_pair!(a)
"c" => 1
julia> a
PriorityQueue{String,Int64,Base.Order.ForwardOrdering} with 2 entries:
"b" => 3
"a" => 2
source
8.15. DATASTRUCTURES.ENQUEUE! 189
8.15 DataStructures.enqueue!
DataStructures.enqueue! —Method.
enqueue!(pq, k, v)
Insert the a key kinto a priority queue pq with priority v.
source
8.16 DataStructures.enqueue!
DataStructures.enqueue! —Method.
enqueue!(s::Queue, x)
Inserts the value xto the end of the queue s.
source
8.17 DataStructures.enqueue!
DataStructures.enqueue! —Method.
enqueue!(pq, k=>v)
Insert the a key kinto a priority queue pq with priority v.
julia> a = PriorityQueue(PriorityQueue("a"=>1, "b"=>2, "c"=>3))
PriorityQueue{String,Int64,Base.Order.ForwardOrdering} with 3 entries:
"c" => 3
"b" => 2
"a" => 1
julia> enqueue!(a, "d"=>4)
PriorityQueue{String,Int64,Base.Order.ForwardOrdering} with 4 entries:
"c" => 3
"b" => 2
"a" => 1
"d" => 4
source
8.18 DataStructures.find root
DataStructures.find root —Method.
find_root{T}(s::DisjointSets{T}, x::T)
Finds the root element of the subset in swhich has the element xas a
member.
source
190 CHAPTER 8. DATASTRUCTURES
8.19 DataStructures.front
DataStructures.front —Method.
front(q::Deque)
Returns the first element of the deque q.
source
8.20 DataStructures.heapify
DataStructures.heapify —Function.
heapify(v, ord::Ordering=Forward)
Returns a new vector in binary heap order, optionally using the given order-
ing.
julia> a = [1,3,4,5,2];
julia> heapify(a)
5-element Array{Int64,1}:
1
2
4
5
3
julia> heapify(a, Base.Order.Reverse)
5-element Array{Int64,1}:
5
3
4
1
2
source
8.21 DataStructures.heapify!
DataStructures.heapify! —Function.
heapify!(v, ord::Ordering=Forward)
In-place heapify.
source
8.22. DATASTRUCTURES.HEAPPOP! 191
8.22 DataStructures.heappop!
DataStructures.heappop! —Function.
heappop!(v, [ord])
Given a binary heap-ordered array, remove and return the lowest ordered
element. For efficiency, this function does not check that the array is indeed
heap-ordered.
source
8.23 DataStructures.heappush!
DataStructures.heappush! —Function.
heappush!(v, x, [ord])
Given a binary heap-ordered array, push a new element x, preserving the
heap property. For efficiency, this function does not check that the array is
indeed heap-ordered.
source
8.24 DataStructures.in same set
DataStructures.in same set —Method.
in_same_set(s::IntDisjointSets, x::Integer, y::Integer)
Returns true if xand ybelong to the same subset in sand false otherwise.
source
8.25 DataStructures.inc!
DataStructures.inc! —Method.
inc!(ct, x, [v=1])
Increments the count for xby v(defaulting to one)
source
192 CHAPTER 8. DATASTRUCTURES
8.26 DataStructures.isheap
DataStructures.isheap —Function.
isheap(v, ord::Ordering=Forward)
Return true if an array is heap-ordered according to the given order.
julia> a = [1,2,3]
3-element Array{Int64,1}:
1
2
3
julia> isheap(a,Base.Order.Forward)
true
julia> isheap(a,Base.Order.Reverse)
false
source
8.27 DataStructures.nlargest
DataStructures.nlargest —Method.
Returns the nlargest elements of arr.
Equivalent to sort(arr, lt = >)[1:min(n, end)]
source
8.28 DataStructures.nsmallest
DataStructures.nsmallest —Method.
Returns the nsmallest elements of arr.
Equivalent to sort(arr, lt = <)[1:min(n, end)]
source
8.29 DataStructures.orderobject
DataStructures.orderobject —Method.
orderobject(sc)
Returns the order object used to construct the container. Time: O(1)
source
8.30. DATASTRUCTURES.ORDEROBJECT 193
8.30 DataStructures.orderobject
DataStructures.orderobject —Method.
orderobject(sc)
Returns the order object used to construct the container. Time: O(1)
source
8.31 DataStructures.orderobject
DataStructures.orderobject —Method.
orderobject(sc)
Returns the order object used to construct the container. Time: O(1)
source
8.32 DataStructures.ordtype
DataStructures.ordtype —Method.
ordtype(sc)
Returns the order type for SortedDict, SortedMultiDict and SortedSet. This
function may also be applied to the type itself. Time: O(1)
source
8.33 DataStructures.ordtype
DataStructures.ordtype —Method.
ordtype(sc)
Returns the order type for SortedDict, SortedMultiDict and SortedSet. This
function may also be applied to the type itself. Time: O(1)
source
8.34 DataStructures.ordtype
DataStructures.ordtype —Method.
ordtype(sc)
Returns the order type for SortedDict, SortedMultiDict and SortedSet. This
function may also be applied to the type itself. Time: O(1)
source
194 CHAPTER 8. DATASTRUCTURES
8.35 DataStructures.packcopy
DataStructures.packcopy —Method.
packcopy(sc)
This returns a copy of sc in which the data is packed. When deletions take
place, the previously allocated memory is not returned. This function can be
used to reclaim memory after many deletions. Time: O(cn log n)
source
8.36 DataStructures.packcopy
DataStructures.packcopy —Method.
packcopy(sc)
This returns a copy of sc in which the data is packed. When deletions take
place, the previously allocated memory is not returned. This function can be
used to reclaim memory after many deletions. Time: O(cn log n)
source
8.37 DataStructures.packcopy
DataStructures.packcopy —Method.
packcopy(sc)
This returns a copy of sc in which the data is packed. When deletions take
place, the previously allocated memory is not returned. This function can be
used to reclaim memory after many deletions. Time: O(cn log n)
source
8.38 DataStructures.packdeepcopy
DataStructures.packdeepcopy —Method.
packdeepcopy(sc)
This returns a packed copy of sc in which the keys and values are deep-
copied. This function can be used to reclaim memory after many deletions.
Time: O(cn log n)
source
8.39. DATASTRUCTURES.PACKDEEPCOPY 195
8.39 DataStructures.packdeepcopy
DataStructures.packdeepcopy —Method.
packdeepcopy(sc)
This returns a packed copy of sc in which the keys and values are deep-
copied. This function can be used to reclaim memory after many deletions.
Time: O(cn log n)
source
8.40 DataStructures.packdeepcopy
DataStructures.packdeepcopy —Method.
packdeepcopy(sc)
This returns a packed copy of sc in which the keys and values are deep-
copied. This function can be used to reclaim memory after many deletions.
Time: O(cn log n)
source
8.41 DataStructures.peek
DataStructures.peek —Method.
peek(pq)
Return the lowest priority key from a priority queue without removing that
key from the queue.
source
8.42 DataStructures.reset!
DataStructures.reset! —Method.
reset!(ct::Accumulator, x)
Resets the count of xto zero. Returns its former count.
source
8.43 DataStructures.top
DataStructures.top —Method.
top(h::BinaryHeap)
Returns the element at the top of the heap h.
source
196 CHAPTER 8. DATASTRUCTURES
8.44 DataStructures.top with handle
DataStructures.top with handle —Method.
top_with_handle(h::MutableBinaryHeap)
Returns the element at the top of the heap hand its handle.
source
8.45 DataStructures.update!
DataStructures.update! —Method.
update!{T}(h::MutableBinaryHeap{T}, i::Int, v::T)
Replace the element at index iin heap hwith v. This is equivalent to
h[i]=v.
source
Chapter 9
JDBC
9.1 Base.close
Base.close —Method.
Closes the JDBCConnection conn. Throws a JDBCError if connection is
null.
Returns nothing.
source
9.2 Base.close
Base.close —Method.
Close the JDBCCursor csr. Throws a JDBCError if cursor is not initialized.
Returns nothing.
source
9.3 Base.connect
Base.connect —Method.
Open a JDBC Connection to the specified host. The username and password
can be optionally passed as a Dictionary props of the form Dict("user" =>
"username", "passwd" =>"password"). The JDBC connector location can
be optionally passed as connectorpath, if it is not added to the java class path.
Returns a JDBCConnection instance.
source
9.4 Base.isopen
Base.isopen —Method.
Returns a boolean indicating whether connection conn is open.
197
198 CHAPTER 9. JDBC
source
9.5 DBAPI.DBAPIBase.connection
DBAPI.DBAPIBase.connection —Method.
Return the corresponding connection for a given cursor.
source
9.6 DBAPI.DBAPIBase.cursor
DBAPI.DBAPIBase.cursor —Method.
Create a new database cursor.
Returns a JDBCCursor instance.
source
9.7 DBAPI.DBAPIBase.execute!
DBAPI.DBAPIBase.execute! —Method.
Run a query on a database.
The results of the query are not returned by this function but are accessible
through the cursor.
parameters can be any iterable of positional parameters, or of some T<:Associative
for keyword/named parameters.
Throws a JDBCError if query caused an error, cursor is not initialized or
connection is null.
Returns nothing.
source
9.8 DBAPI.DBAPIBase.rows
DBAPI.DBAPIBase.rows —Method.
Create a row iterator.
This method returns an instance of an iterator type which returns one row
on each iteration. Each row returnes a Tuple{. . . }.
Throws a JDBCError if execute! was not called on the cursor or connection
is null.
Returns a JDBCRowIterator instance.
source
9.9 JDBC.commit
JDBC.commit —Method.
9.10. JDBC.COMMIT 199
Commit any pending transaction to the database. Throws a JDBCError if
connection is null.
Returns nothing.
source
9.10 JDBC.commit
JDBC.commit —Method.
commit(connection::JConnection)
Commits the transaction
Args
•connection: The connection object
Returns
None
source
9.11 JDBC.createStatement
JDBC.createStatement —Method.
createStatement(connection::JConnection)
Initializes a Statement
Args
•connection: The connection object
Returns
The JStatement object
source
9.12 JDBC.execute
JDBC.execute —Method.
execute(stmt::JStatement, query::AbstractString)
Executes the auery based on JStatement or any of its sub-types
Args
•stmt: The JStatement object or any of its sub-types
•query: The query to be executed
Returns
A boolean indicating whether the execution was successful or not
source
200 CHAPTER 9. JDBC
9.13 JDBC.execute
JDBC.execute —Method.
execute(stmt::@compat(Union{JPreparedStatement, JCallableStatement}))
Executes the auery based on the Prepared Statement or Callable Statement
Args
•stmt: The Prepared Statement or the Callable Statement object
Returns
A boolean indicating whether the execution was successful or not
source
9.14 JDBC.executeQuery
JDBC.executeQuery —Method.
executeQuery(stmt::JStatement, query::AbstractString)
Executes the auery and returns the results as a JResultSet object.
Args
•stmt: The Statement object
•query: The query to be executed
Returns
The result set as a JResultSet object
source
9.15 JDBC.executeQuery
JDBC.executeQuery —Method.
executeQuery(stmt::@compat(Union{JPreparedStatement, JCallableStatement}))
Executes the auery based on a JPreparedStatement object or a JCallableState-
ment object
Args
•stmt: The JPreparedStatement object or JCallableStatement object
Returns
The result set as a JResultSet object
source
9.16. JDBC.EXECUTEUPDATE 201
9.16 JDBC.executeUpdate
JDBC.executeUpdate —Method.
executeUpdate(stmt::JStatement, query::AbstractString)
Executes the update auery and returns the status of the execution of the
query
Args
•stmt: The Statement object
•query: The query to be executed
Returns
An integer representing the status of the execution
source
9.17 JDBC.executeUpdate
JDBC.executeUpdate —Method.
executeUpdate(stmt::@compat(Union{JPreparedStatement, JCallableStatement}))
Executes the update auery based on a JPreparedStatement object or a
JCallableStatement object
Args
•stmt: The JPreparedStatement object or JCallableStatement object
Returns
An integer indicating the status of the execution of the query
source
9.18 JDBC.getColumnCount
JDBC.getColumnCount —Method.
getColumnCount(rsmd::JResultSetMetaData)
Returns the number of columns based on the JResultSetMetaData object
Args
•rsmd: The JResultSetMetaData object
Returns
The number of columns.
source
202 CHAPTER 9. JDBC
9.19 JDBC.getColumnName
JDBC.getColumnName —Method.
getColumnName(rsmd::JResultSetMetaData, col::Integer)
Returns the column’s name based on the JResultSetMetaData object and
the column number
Args
•rsmd: The JResultSetMetaData object
•col: The column number
Returns
The column name
source
9.20 JDBC.getColumnType
JDBC.getColumnType —Method.
getColumnType(rsmd::JResultSetMetaData, col::Integer)
Returns the column’s data type based on the JResultSetMetaData object
and the column number
Args
•rsmd: The JResultSetMetaData object
•col: The column number
Returns
The column type as an integer
source
9.21 JDBC.getDate
JDBC.getDate —Method.
getDate(rs::@compat(Union{JResultSet, JCallableStatement}), fld::AbstractString)
Returns the Date object based on the result set or a callable statement. The
value is extracted based on the column name.
Args
•stmt: The JResultSet or JCallableStatement object
•fld: The column name
Returns
The Date object.
source
9.22. JDBC.GETDATE 203
9.22 JDBC.getDate
JDBC.getDate —Method.
getDate(rs::@compat(Union{JResultSet, JCallableStatement}), fld::Integer)
Returns the Date object based on the result set or a callable statement. The
value is extracted based on the column number.
Args
•stmt: The JResultSet or JCallableStatement object
•fld: The column number
Returns
The Date object.
source
9.23 JDBC.getMetaData
JDBC.getMetaData —Method.
getMetaData(rs::JResultSet)
Returns information about the types and properties of the columns in the
ResultSet object
Args
•stmt: The JResultSet object
Returns
The JResultSetMetaData object.
source
9.24 JDBC.getResultSet
JDBC.getResultSet —Method.
getResultSet(stmt::JStatement)
Returns the result set based on the previous execution of the query based
on a JStatement
Args
•stmt: The JStatement object
Returns
The JResultSet object.
source
204 CHAPTER 9. JDBC
9.25 JDBC.getTableMetaData
JDBC.getTableMetaData —Method.
Get the metadata (column name and type) for each column of the table in
the result set rs.
Returns an array of (column name, column type) tuples.
source
9.26 JDBC.getTime
JDBC.getTime —Method.
getTime(rs::@compat(Union{JResultSet, JCallableStatement}), fld::AbstractString)
Returns the Time object based on the result set or a callable statement. The
value is extracted based on the column name.
Args
•stmt: The JResultSet or JCallableStatement object
•fld: The column name
Returns
The Time object.
source
9.27 JDBC.getTime
JDBC.getTime —Method.
getTime(rs::@compat(Union{JResultSet, JCallableStatement}), fld::Integer)
Returns the Time object based on the result set or a callable statement. The
value is extracted based on the column number.
Args
•stmt: The JResultSet or JCallableStatement object
•fld: The column number
Returns
The Time object.
source
9.28. JDBC.GETTIMESTAMP 205
9.28 JDBC.getTimestamp
JDBC.getTimestamp —Method.
getTimestamp(rs::@compat(Union{JResultSet, JCallableStatement}), fld::AbstractString)
Returns the Timestamp object based on the result set or a callable state-
ment. The value is extracted based on the column name.
Args
•stmt: The JResultSet or JCallableStatement object
•fld: The column name
Returns
The Timestamp object.
source
9.29 JDBC.getTimestamp
JDBC.getTimestamp —Method.
getTimestamp(rs::@compat(Union{JResultSet, JCallableStatement}), fld::Integer)
Returns the Timestamp object based on the result set or a callable state-
ment. The value is extracted based on the column number.
Args
•stmt: The JResultSet or JCallableStatement object
•fld: The column number
Returns
The Timestamp object.
source
9.30 JDBC.prepareCall
JDBC.prepareCall —Method.
prepareCall(connection::JConnection, query::AbstractString)
Prepares the Callable Statement for the given query
Args
•connection: The connection object
•query: The query string
Returns
The JCallableStatement object
source
206 CHAPTER 9. JDBC
9.31 JDBC.prepareStatement
JDBC.prepareStatement —Method.
prepareStatement(connection::JConnection, query::AbstractString)
Prepares the Statement for the given query
Args
•connection: The connection object
•query: The query string
Returns
The JPreparedStatement object
source
9.32 JDBC.rollback
JDBC.rollback —Method.
Roll back to the start of any pending transaction. Throws a JDBCError if
connection is null.
Returns nothing.
source
9.33 JDBC.rollback
JDBC.rollback —Method.
rollback(connection::JConnection)
Rolls back the transactions.
Args
•connection: The connection object
Returns
None
source
9.34 JDBC.setAutoCommit
JDBC.setAutoCommit —Method.
setAutoCommit(connection::JConnection, x::Bool)
9.35. JDBC.SETFETCHSIZE 207
Set the Auto Commit flag to either true or false. If set to false, commit has
to be called explicitly
Args
•connection: The connection object
Returns
None
source
9.35 JDBC.setFetchSize
JDBC.setFetchSize —Method.
setFetchSize(stmt::@compat(Union{JStatement, JPreparedStatement, JCallableStatement }), x::Integer)
Sets the fetch size in a JStatement or a JPreparedStatement object or a
JCallableStatement object. The number of records that are returned in subse-
quent query executions are determined by what is set here.
Args
•stmt: The JPreparedStatement object or JCallableStatement object
•x: The number of records to be returned
Returns
None
source
Chapter 10
NNlib
10.1 NNlib.elu
NNlib.elu —Function.
elu(x, = 1) =
x > 0 ? x : * (exp(x) - 1)
Exponential Linear Unit activation function. See Fast and Accurate Deep
Network Learning by Exponential Linear Units. You can also specify the coef-
ficient explicitly, e.g. elu(x, 1).
3
-1
-3 0 3
source
208
10.2. NNLIB.LEAKYRELU 209
10.2 NNlib.leakyrelu
NNlib.leakyrelu —Function.
leakyrelu(x) = max(0.01x, x)
Leaky Rectified Linear Unit activation function. You can also specify the
coefficient explicitly, e.g. leakyrelu(x, 0.01).
3
-1
-3 0 3
source
10.3 NNlib.logsoftmax
NNlib.logsoftmax —Function.
logsoftmax(xs) = log.(exp.(xs) ./ sum(exp.(xs)))
logsoftmax computes the log of softmax(xs) and it is more numerically stable
than softmax function in computing the cross entropy loss.
source
10.4 NNlib.log
NNlib.log —Method.
log(x)
Return log((x)) which is computed in a numerically stable way.
210 CHAPTER 10. NNLIB
julia> log(0.)
-0.6931471805599453
julia> log.([-100, -10, 100.])
3-element Array{Float64,1}:
-100.0
-10.0
-0.0
source
10.5 NNlib.relu
NNlib.relu —Method.
relu(x) = max(0, x)
Rectified Linear Unit activation function.
3
0
-3 0 3
source
10.6 NNlib.selu
NNlib.selu —Method.
selu(x) = * (x 0 ? x : * (exp(x) - 1))
1.0507
1.6733
Scaled exponential linear units. See Self-Normalizing Neural Networks.
4 -2 -3 0 3
source
10.7. NNLIB.SOFTMAX 211
10.7 NNlib.softmax
NNlib.softmax —Function.
softmax(xs) = exp.(xs) ./ sum(exp.(xs))
Softmax takes log-probabilities (any real vector) and returns a probability
distribution that sums to 1.
If given a matrix it will treat it as a batch of vectors, with each column
independent.
julia> softmax([1,2,3.])
3-element Array{Float64,1}:
0.0900306
0.244728
0.665241
source
10.8 NNlib.softplus
NNlib.softplus —Method.
softplus(x) = log(exp(x) + 1)
See Deep Sparse Rectifier Neural Networks.
4
0
-3 0 3
source
212 CHAPTER 10. NNLIB
10.9 NNlib.softsign
NNlib.softsign —Method.
softsign(x) = x / (1 + |x|)
See Quadratic Polynomials Learn Better Image Features.
1
-1
-3 0 3
source
10.10 NNlib.swish
NNlib.swish —Method.
swish(x) = x * (x)
Self-gated actvation function. See Swish: a Self-Gated Activation Function.
3
10.11. NNLIB. 213
-1
-3 0 3
source
10.11 NNlib.
NNlib. —Method.
(x) = 1 / (1 + exp(-x))
Classic sigmoid activation function.
1
0
-3 0 3
source
Chapter 11
ImageCore
11.1 ImageCore.assert timedim last
ImageCore.assert timedim last —Method.
assert_timedim_last(img)
Throw an error if the image has a time dimension that is not the last di-
mension.
source
11.2 ImageCore.channelview
ImageCore.channelview —Method.
channelview(A)
returns a view of A, splitting out (if necessary) the color channels of Ainto a
new first dimension. This is almost identical to ChannelView(A), except that if
Ais a ColorView, it will simply return the parent of A, or will use reinterpret
when appropriate. Consequently, the output may not be a ChannelView array.
Of relevance for types like RGB and BGR, the channels of the returned array
will be in constructor-argument order, not memory order (see reinterpret if
you want to use memory order).
source
11.3 ImageCore.clamp01
ImageCore.clamp01 —Method.
clamp01(x) -> y
214
11.4. IMAGECORE.CLAMP01NAN 215
Produce a value ythat lies between 0 and 1, and equal to xwhen xis already
in this range. Equivalent to clamp(x, 0, 1) for numeric values. For colors,
this function is applied to each color channel separately.
See also: clamp01nan.
source
11.4 ImageCore.clamp01nan
ImageCore.clamp01nan —Method.
clamp01nan(x) -> y
Similar to clamp01, except that any NaN values are changed to 0.
See also: clamp01.
source
11.5 ImageCore.colorsigned
ImageCore.colorsigned —Method.
colorsigned()
colorsigned(colorneg, colorpos) -> f
colorsigned(colorneg, colorcenter, colorpos) -> f
Define a function that maps negative values (in the range [-1,0]) to the linear
colormap between colorneg and colorcenter, and positive values (in the range
[0,1]) to the linear colormap between colorcenter and colorpos.
The default colors are:
•colorcenter: white
•colorneg: green1
•colorpos: magenta
See also: scalesigned.
source
11.6 ImageCore.colorview
ImageCore.colorview —Method.
colorview(C, gray1, gray2, ...) -> imgC
216 CHAPTER 11. IMAGECORE
Combine numeric/grayscale images gray1,gray2, etc., into the separate
color channels of an array imgC with element type C<:Colorant.
As a convenience, the constant zeroarray fills in an array of matched size
with all zeros.
Example
[] imgC = colorview(RGB, r, zeroarray, b)
creates an image with rin the red chanel, bin the blue channel, and nothing
in the green channel.
See also: StackedView.
source
11.7 ImageCore.colorview
ImageCore.colorview —Method.
colorview(C, A)
returns a view of the numeric array A, interpreting successive elements of Aas
if they were channels of Colorant C. This is almost identical to ColorView{C}(A),
except that if Ais a ChannelView, it will simply return the parent of A, or
use reinterpret when appropriate. Consequently, the output may not be a
ColorView array.
Of relevance for types like RGB and BGR, the elements of Aare interpreted
in constructor-argument order, not memory order (see reinterpret if you want
to use memory order).
Example
[] A = rand(3, 10, 10) img = colorview(RGB, A)
source
11.8 ImageCore.coords spatial
ImageCore.coords spatial —Method.
coords spatial(img)
Return a tuple listing the spatial dimensions of img.
Note that a better strategy may be to use ImagesAxes and take slices along
the time axis.
source
11.9 ImageCore.float32
ImageCore.float32 —Function.
float32.(img)
11.10. IMAGECORE.FLOAT64 217
converts the raw storage type of img to Float32, without changing the color
space.
source
11.10 ImageCore.float64
ImageCore.float64 —Function.
float64.(img)
converts the raw storage type of img to Float64, without changing the color
space.
source
11.11 ImageCore.indices spatial
ImageCore.indices spatial —Method.
indices_spatial(img)
Return a tuple with the indices of the spatial dimensions of the image. De-
faults to the same as indices, but using ImagesAxes you can mark some axes
as being non-spatial.
source
11.12 ImageCore.n0f16
ImageCore.n0f16 —Function.
n0f16.(img)
converts the raw storage type of img to N0f16, without changing the color
space.
source
11.13 ImageCore.n0f8
ImageCore.n0f8 —Function.
n0f8.(img)
converts the raw storage type of img to N0f8, without changing the color
space.
source
218 CHAPTER 11. IMAGECORE
11.14 ImageCore.n2f14
ImageCore.n2f14 —Function.
n2f14.(img)
converts the raw storage type of img to N2f14, without changing the color
space.
source
11.15 ImageCore.n4f12
ImageCore.n4f12 —Function.
n4f12.(img)
converts the raw storage type of img to N4f12, without changing the color
space.
source
11.16 ImageCore.n6f10
ImageCore.n6f10 —Function.
n6f10.(img)
converts the raw storage type of img to N6f10, without changing the color
space.
source
11.17 ImageCore.nimages
ImageCore.nimages —Method.
nimages(img)
Return the number of time-points in the image array. Defaults to
1. Use ImagesAxes if you want to use an explicit time dimension.
source
11.18. IMAGECORE.NORMEDVIEW 219
11.18 ImageCore.normedview
ImageCore.normedview —Method.
normedview([T], img::AbstractArray{Unsigned})
returns a “view” of img where the values are interpreted in terms of Normed
number types. For example, if img is an Array{UInt8}, the view will act like an
Array{N0f8}. Supply Tif the element type of img is UInt16, to specify whether
you want a N6f10,N4f12,N2f14, or N0f16 result.
source
11.19 ImageCore.permuteddimsview
ImageCore.permuteddimsview —Method.
permuteddimsview(A, perm)
returns a “view” of Awith its dimensions permuted as specified by perm.
This is like permutedims, except that it produces a view rather than a copy of
A; consequently, any manipulations you make to the output will be mirrored in
A. Compared to the copy, the view is much faster to create, but generally slower
to use.
source
11.20 ImageCore.pixelspacing
ImageCore.pixelspacing —Method.
pixelspacing(img) -> (sx, sy, ...)
Return a tuple representing the separation between adjacent pixels along
each axis of the image. Defaults to (1,1,. . . ). Use ImagesAxes for images with
anisotropic spacing or to encode the spacing using physical units.
source
11.21 ImageCore.rawview
ImageCore.rawview —Method.
rawview(img::AbstractArray{FixedPoint})
returns a “view” of img where the values are interpreted in terms of their
raw underlying storage. For example, if img is an Array{N0f8}, the view will
act like an Array{UInt8}.
source
220 CHAPTER 11. IMAGECORE
11.22 ImageCore.scaleminmax
ImageCore.scaleminmax —Method.
scaleminmax(min, max) -> f
scaleminmax(T, min, max) -> f
Return a function fwhich maps values less than or equal to min to 0, values
greater than or equal to max to 1, and uses a linear scale in between. min and
max should be real values.
Optionally specify the return type T. If Tis a colorant (e.g., RGB), then
scaling is applied to each color channel.
Examples
Example 1
[] julia¿ f = scaleminmax(-10, 10) (::9) (generic function with 1 method)
julia¿ f(10) 1.0
julia¿ f(-10) 0.0
julia¿ f(5) 0.75
Example 2
[] julia¿ c = RGB(255.0,128.0,0.0) RGB{Float64}(255.0,128.0,0.0)
julia¿ f = scaleminmax(RGB, 0, 255) (::13) (generic function with 1 method)
julia¿ f(c) RGB{Float64}(1.0,0.5019607843137255,0.0)
See also: takemap.
source
11.23 ImageCore.scalesigned
ImageCore.scalesigned —Method.
scalesigned(maxabs) -> f
Return a function fwhich scales values in the range [-maxabs, maxabs]
(clamping values that lie outside this range) to the range [-1, 1].
See also: colorsigned.
source
11.24 ImageCore.scalesigned
ImageCore.scalesigned —Method.
scalesigned(min, center, max) -> f
Return a function fwhich scales values in the range [min, center] to
[-1,0] and [center,max] to [0,1]. Values smaller than min/max get clamped
to min/max, respectively.
See also: colorsigned.
source
11.25. IMAGECORE.SDIMS 221
11.25 ImageCore.sdims
ImageCore.sdims —Method.
sdims(img)
Return the number of spatial dimensions in the image. Defaults to the same
as ndims, but with ImagesAxes you can specify that some axes correspond to
other quantities (e.g., time) and thus not included by sdims.
source
11.26 ImageCore.size spatial
ImageCore.size spatial —Method.
size_spatial(img)
Return a tuple listing the sizes of the spatial dimensions of the image. De-
faults to the same as size, but using ImagesAxes you can mark some axes as
being non-spatial.
source
11.27 ImageCore.spacedirections
ImageCore.spacedirections —Method.
spacedirections(img) -> (axis1, axis2, ...)
Return a tuple-of-tuples, each axis[i] representing the displacement vector
between adjacent pixels along spatial axis iof the image array, relative to some
external coordinate system (“physical coordinates”).
By default this is computed from pixelspacing, but you can set this man-
ually using ImagesMeta.
source
11.28 ImageCore.takemap
ImageCore.takemap —Function.
takemap(f, A) -> fnew
takemap(f, T, A) -> fnew
Given a value-mapping function fand an array A, return a “concrete” map-
ping function fnew. When applied to elements of A,fnew should return valid
values for storage or display, for example in the range from 0 to 1 (for grayscale)
222 CHAPTER 11. IMAGECORE
or valid colorants. fnew may be adapted to the actual values present in A, and
may not produce valid values for any inputs not in A.
Optionally one can specify the output type Tthat fnew should produce.
Example:
[] julia¿ A = [0, 1, 1000];
julia¿ f = takemap(scaleminmax, A) (::7) (generic function with 1 method)
julia¿ f.(A) 3-element Array{Float64,1}: 0.0 0.001 1.0
source
Chapter 12
Reactive
12.1 Base.filter
Base.filter —Method.
filter(f, default, signal)
remove updates from the signal where freturns false. The filter will hold
the value default until f(value(signal)) returns true, when it will be updated to
value(signal).
source
12.2 Base.map
Base.map —Method.
map(f, s::Signal...) -> signal
Transform signal sby applying fto each element. For multiple signal argu-
ments, apply felementwise.
source
12.3 Base.merge
Base.merge —Method.
merge(inputs...)
Merge many signals into one. Returns a signal which updates when any of
the inputs update. If many signals update at the same time, the value of the
youngest (most recently created) input signal is taken.
source
223
224 CHAPTER 12. REACTIVE
12.4 Base.push!
Base.push! —Function.
push!(signal, value, onerror=Reactive.print error)
Queue an update to a signal. The update will be propagated when all
currently queued updates are done processing.
The third (optional) argument, onerror, is a callback triggered when the
update ends in an error. The callback receives 4 arguments, onerror(sig,
val, node, capex), where sig and val are the Signal and value that push!
was called with, respectively, node is the Signal whose action triggered the error,
and capex is a CapturedException with the fields ex which is the original
exception object, and processed bt which is the backtrace of the exception.
The default error callback will print the error and backtrace to STDERR.
source
12.5 Reactive.async map
Reactive.async map —Method.
tasks, results = async_map(f, init, input...;typ=typeof(init), onerror=Reactive.print_error)
Spawn a new task to run a function when input signal updates. Returns
a signal of tasks and a results signal which updates asynchronously with the
results. init will be used as the default value of results.onerror is the
callback to be called when an error occurs, by default it is set to a callback
which prints the error to STDERR. It’s the same as the onerror argument to
push! but is run in the spawned task.
source
12.6 Reactive.bind!
Reactive.bind! —Function.
‘bind!(dest, src, twoway=true; initial=true)‘
for every update to src also update dest with the same value and, if twoway
is true, vice-versa. If initial is false, dest will only be updated to src’s value
when src next updates, otherwise (if initial is true) both dest and src will
take src’s value immediately.
source
12.7 Reactive.bound dests
Reactive.bound dests —Method.
12.8. REACTIVE.BOUND SRCS 225
bound dests(src::Signal) returns a vector of all signals that will update
when src updates, that were bound using bind!(dest, src)
source
12.8 Reactive.bound srcs
Reactive.bound srcs —Method.
bound srcs(dest::Signal) returns a vector of all signals that will cause an
update to dest when they update, that were bound using bind!(dest, src)
source
12.9 Reactive.debounce
Reactive.debounce —Method.
debounce(dt, input, f=(acc,x)->x, init=value(input), reinit=x->x;
typ=typeof(init), name=auto_name!(string("debounce ",dt,"s"), input))
Creates a signal that will delay updating until dt seconds have passed since
the last time input has updated. By default, the debounce signal holds the last
update of the input signal since the debounce signal last updated.
This behavior can be changed by the f,init and reinit arguments. The
init and ffunctions are similar to init and fin foldp.reinit is called after
the debounce sends an update, to reinitialize the initial value for accumulation,
it gets one argument, the previous accumulated value.
For example y = debounce(0.2, x, push!, Int[], ->Int[]) will accu-
mulate a vector of updates to the integer signal xand push it after xis inactive
(doesn’t update) for 0.2 seconds.
source
12.10 Reactive.delay
Reactive.delay —Method.
delay(input, default=value(input))
Schedule an update to happen after the current update propagates through-
out the signal graph.
Returns the delayed signal.
source
226 CHAPTER 12. REACTIVE
12.11 Reactive.droprepeats
Reactive.droprepeats —Method.
droprepeats(input)
Drop updates to input whenever the new value is the same as the previous
value of the signal.
source
12.12 Reactive.every
Reactive.every —Method.
every(dt)
A signal that updates every dt seconds to the current timestamp. Consider
using fpswhen or fps if you want specify the timing signal by frequency, rather
than delay.
source
12.13 Reactive.filterwhen
Reactive.filterwhen —Method.
filterwhen(switch::Signal{Bool}, default, input)
Keep updates to input only when switch is true.
If switch is false initially, the specified default value is used.
source
12.14 Reactive.flatten
Reactive.flatten —Method.
flatten(input::Signal{Signal}; typ=Any)
Flatten a signal of signals into a signal which holds the value of the current
signal. The typ keyword argument specifies the type of the flattened signal. It
is Any by default.
source
12.15. REACTIVE.FOLDP 227
12.15 Reactive.foldp
Reactive.foldp —Method.
foldp(f, init, inputs...)
Fold over past values.
Accumulate a value as the input signals change. init is the initial value of
the accumulator. fshould take 1 + length(inputs) arguments: the first is
the current accumulated value and the rest are the current input signal values.
fwill be called when one or more of the inputs updates. It should return the
next accumulated value.
source
12.16 Reactive.fps
Reactive.fps —Method.
fps(rate)
Same as fpswhen(Input(true), rate)
source
12.17 Reactive.fpswhen
Reactive.fpswhen —Method.
fpswhen(switch, rate)
returns a signal which when switch signal is true, updates rate times every
second. If rate is not possible to attain because of slowness in computing
dependent signal values, the signal will self adjust to provide the best possible
rate.
source
12.18 Reactive.preserve
Reactive.preserve —Method.
preserve(signal::Signal)
prevents signal from being garbage collected as long as any of its parents
are around. Useful for when you want to do some side effects in a signal. e.g.
preserve(map(println, x)) - this will continue to print updates to x, until x
goes out of scope. foreach is a shorthand for map with preserve.
source
228 CHAPTER 12. REACTIVE
12.19 Reactive.previous
Reactive.previous —Method.
previous(input, default=value(input))
Create a signal which holds the previous value of input. You can optionally
specify a different initial value.
source
12.20 Reactive.remote map
Reactive.remote map —Method.
remoterefs, results = remote_map(procid, f, init, input...;typ=typeof(init), onerror=Reactive.print_error)
Spawn a new task on process procid to run a function when input signal
updates. Returns a signal of remote refs and a results signal which updates
asynchronously with the results. init will be used as the default value of
results.onerror is the callback to be called when an error occurs, by default
it is set to a callback which prints the error to STDERR. It’s the same as the
onerror argument to push! but is run in the spawned task.
source
12.21 Reactive.rename!
Reactive.rename! —Method.
rename!(s::Signal, name::String)
Change a Signal’s name
source
12.22 Reactive.sampleon
Reactive.sampleon —Method.
sampleon(a, b)
Sample the value of bwhenever aupdates.
source
12.23 Reactive.throttle
Reactive.throttle —Method.
throttle(dt, input, f=(acc,x)->x, init=value(input), reinit=x->x;
typ=typeof(init), name=auto_name!(string("throttle ",dt,"s"), input), leading=false)
12.24. REACTIVE.UNBIND! 229
Throttle a signal to update at most once every dt seconds. By default, the
throttled signal holds the last update of the input signal during each dt second
time window.
This behavior can be changed by the f,init and reinit arguments. The
init and ffunctions are similar to init and fin foldp.reinit is called when a
new throttle time window opens to reinitialize the initial value for accumulation,
it gets one argument, the previous accumulated value.
For example y = throttle(0.2, x, push!, Int[], ->Int[]) will cre-
ate vectors of updates to the integer signal xwhich occur within 0.2 second time
windows.
If leading is true, the first update from input will be sent immediately by
the throttle signal. If it is false, the first update will happen dt seconds after
input’s first update
New in v0.4.1: throttle’s behaviour from previous versions is now available
with the debounce signal type.
source
12.24 Reactive.unbind!
Reactive.unbind! —Function.
‘unbind!(dest, src, twoway=true)‘
remove a link set up using bind!
source
12.25 Reactive.unpreserve
Reactive.unpreserve —Method.
unpreserve(signal::Signal)
allow signal to be garbage collected. See also preserve.
source
Chapter 13
JuliaDB
13.1 Dagger.compute
Dagger.compute —Method.
compute(t::DNDSparse; allowoverlap, closed)
Computes any delayed-evaluations in the DNDSparse. The computed data
is left on the worker processes. Subsequent operations on the results will reuse
the chunks.
If allowoverlap is false then the computed data is re-sorted if required to
have no chunks with overlapping index ranges if necessary.
If closed is true then the computed data is re-sorted if required to have no
chunks with overlapping OR continuous boundaries.
See also collect.
!!! warning compute(t) requires at least as much memory as the size of
the result of the computing t. You usually don’t need to do this for the whole
dataset. If the result is expected to be big, try compute(save(t, "output dir"))
instead. See save for more.
source
13.2 Dagger.distribute
Dagger.distribute —Function.
distribute(itable::NDSparse, nchunks::Int=nworkers())
Distributes an NDSparse object into a DNDSparse of nchunks chunks of
approximately equal size.
Returns a DNDSparse.
source
230
13.3. DAGGER.DISTRIBUTE 231
13.3 Dagger.distribute
Dagger.distribute —Method.
distribute(t::Table, chunks)
Distribute a table in chunks pieces. Equivalent to table(t, chunks=chunks).
source
13.4 Dagger.distribute
Dagger.distribute —Method.
distribute(itable::NDSparse, rowgroups::AbstractArray)
Distributes an NDSparse object into a DNDSparse by splitting it up into
chunks of rowgroups elements. rowgroups is a vector specifying the number of
rows in the chunks.
Returns a DNDSparse.
source
13.5 Dagger.load
Dagger.load —Method.
load(dir::AbstractString; tomemory)
Load a saved DNDSparse from dir directory. Data can be saved using the
save function.
source
13.6 Dagger.save
Dagger.save —Method.
save(t::Union{DNDSparse, DNDSparse}, outputdir::AbstractString)
Saves a distributed dataset to disk. Saved data can be loaded with load.
source
13.7 IndexedTables.convertdim
IndexedTables.convertdim —Method.
convertdim(x::DNDSparse, d::DimName, xlate; agg::Function, name)
232 CHAPTER 13. JULIADB
Apply function or dictionary xlate to each index in the specified dimension.
If the mapping is many-to-one, agg is used to aggregate the results. name op-
tionally specifies a name for the new dimension. xlate must be a monotonically
increasing function.
See also reducedim and aggregate
source
13.8 IndexedTables.leftjoin
IndexedTables.leftjoin —Method.
leftjoin(left::DNDSparse, right::DNDSparse, [op::Function])
Keeps only rows with indices in left. If rows of the same index are present
in right, then they are combined using op.op by default picks the value from
right.
source
13.9 IndexedTables.naturaljoin
IndexedTables.naturaljoin —Method.
naturaljoin(op, left::DNDSparse, right::DNDSparse, ascolumns=false)
Returns a new DNDSparse containing only rows where the indices are present
both in left AND right tables. The data columns are concatenated. The data
of the matching rows from left and right are combined using op. If op returns
a tuple or NamedTuple, and ascolumns is set to true, the output table will
contain the tuple elements as separate data columns instead as a single column
of resultant tuples.
source
13.10 IndexedTables.naturaljoin
IndexedTables.naturaljoin —Method.
naturaljoin(left::DNDSparse, right::DNDSparse, [op])
Returns a new DNDSparse containing only rows where the indices are present
both in left AND right tables. The data columns are concatenated.
source
13.11. INDEXEDTABLES.REDUCEDIM VEC 233
13.11 IndexedTables.reducedim vec
IndexedTables.reducedim vec —Method.
reducedim_vec(f::Function, t::DNDSparse, dims)
Like reducedim, except uses a function mapping a vector of values to a scalar
instead of a 2-argument scalar function.
See also reducedim and aggregate vec.
source
13.12 JuliaDB.loadndsparse
JuliaDB.loadndsparse —Method.
loadndsparse(files::Union{AbstractVector,String};<options>)
Load an NDSparse from CSV files.
files is either a vector of file paths, or a directory name.
Options:
•indexcols::Vector – columns to use as indexed columns. (by default a
1:n implicit index is used.)
•datacols::Vector – non-indexed columns. (defaults to all columns but
indexed columns). Specify this to only load a subset of columns. In place
of the name of a column, you can specify a tuple of names – this will treat
any column with one of those names as the same column, but use the first
name in the tuple. This is useful when the same column changes name
between CSV files. (e.g. vendor id and VendorId)
All other options are identical to those in loadtable
source
13.13 JuliaDB.loadtable
JuliaDB.loadtable —Method.
loadtable(files::Union{AbstractVector,String};<options>)
Load a table from CSV files.
files is either a vector of file paths, or a directory name.
Options:
•output::AbstractString – directory name to write the table to. By
default data is loaded directly to memory. Specifying this option will
allow you to load data larger than the available memory.
•indexcols::Vector – columns to use as primary key columns. (defaults
to [])
234 CHAPTER 13. JULIADB
•datacols::Vector – non-indexed columns. (defaults to all columns but
indexed columns). Specify this to only load a subset of columns. In place
of the name of a column, you can specify a tuple of names – this will treat
any column with one of those names as the same column, but use the first
name in the tuple. This is useful when the same column changes name
between CSV files. (e.g. vendor id and VendorId)
•distributed::Bool – should the output dataset be loaded as a dis-
tributed table? If true, this will use all available worker processes to
load the data. (defaults to true if workers are available, false if not)
•chunks::Int – number of chunks to create when loading distributed. (de-
faults to number of workers)
•delim::Char – the delimiter character. (defaults to ,). Use spacedelim=true
to split by spaces.
•spacedelim::Bool: parse space-delimited files. delim has no effect if
true.
•quotechar::Char – quote character. (defaults to ")
•escapechar::Char – escape character. (defaults to ")
•filenamecol::Union{Symbol, Pair}– create a column containing the
file names from where each row came from. This argument gives a name
to the column. By default, basename(name) of the name is kept, and
“.csv” suffix will be stripped. To provide a custom function to apply
on the names, use a name =>Function pair. By default, no file name
column will be created.
•header exists::Bool – does header exist in the files? (defaults to true)
•colnames::Vector{String}– specify column names for the files, use this
with (header exists=false, otherwise first row is discarded). By default
column names are assumed to be present in the file.
•samecols – a vector of tuples of strings where each tuple contains alterna-
tive names for the same column. For example, if some files have the name
“vendor id” and others have the name “VendorID”, pass samecols=[("VendorID",
"vendor id")].
•colparsers – either a vector or dictionary of data types or an AbstractToken
object from TextParse package. By default, these are inferred automati-
cally. See type detect rows option below.
•type detect rows: number of rows to use to infer the initial colparsers
defaults to 20.
•nastrings::Vector{String}– strings that are to be considered NA. (de-
faults to TextParse.NA STRINGS)
13.14. JULIADB.PARTITIONPLOT 235
•skiplines begin::Char – skip some lines in the beginning of each file.
(doesn’t skip by default)
•usecache::Bool: (vestigial)
source
13.14 JuliaDB.partitionplot
JuliaDB.partitionplot —Function.
partitionplot(table, y; stat=Extrema(), nparts=100, by=nothing, dropmissing=false)
partitionplot(table, x, y; stat=Extrema(), nparts=100, by=nothing, dropmissing=false)
Plot a summary of variable yagainst x(1:length(y) if not specified). Using
nparts approximately-equal sections along the x-axis, the data in yover each
section is summarized by stat.
source
13.15 JuliaDB.rechunk
JuliaDB.rechunk —Function.
rechunk(t::Union{DNDSparse, DNDSparse}[, by[, select]]; <options>)
Reindex and sort a distributed dataset by keys selected by by.
Optionally select specifies which non-indexed fields are kept. By default
this is all fields not mentioned in by for Table and the value columns for
NDSparse.
Options:
•chunks – how to distribute the data. This can be:
1. An integer – number of chunks to create
2. An vector of kintegers – number of elements in each of the kchunks.
sum(k) must be same as length(t)
3. The distribution of another array. i.e. vec.subdomains where vec
is a distributed array.
•merge::Function – a function which merges two sub-table or sub-ndsparse
into one NDSparse. They may have overlaps in their indices.
•splitters::AbstractVector – specify keys to split by. To create n
chunks you would need to pass n-1 splitters and also the chunks=n option.
•chunks sorted::Bool – are the chunks sorted locally? If true, this skips
sorting or re-indexing them.
236 CHAPTER 13. JULIADB
•affinities::Vector{<:Integer}– which processes (Int pid) should each
output chunk be created on. If unspecified all workers are used.
•closed::Bool – if true, the same key will not be present in multiple
chunks (although sorted). true by default.
•nsamples::Integer – number of keys to randomly sample from each
chunk to estimate splitters in the sorting process. (See samplesort). De-
faults to 2000.
•batchsize::Integer – how many chunks at a time from the input should
be loaded into memory at any given time. This will essentially sort in
batches of batchsize chunks.
source
13.16 JuliaDB.tracktime
JuliaDB.tracktime —Method.
tracktime(f)
Track the time spent on different processes in different categories in running
f.
source
Chapter 14
Combinatorics
14.1 Base.factorial
Base.factorial —Method.
computes n!/k!
source
14.2 Combinatorics.bellnum
Combinatorics.bellnum —Method.
Returns the n-th Bell number
source
14.3 Combinatorics.catalannum
Combinatorics.catalannum —Method.
Returns the n-th Catalan number
source
14.4 Combinatorics.character
Combinatorics.character —Method.
Computes character () of the partition in the th irrep of the symmetric
group Sn
Implements the Murnaghan-Nakayama algorithm as described in: Dan Bern-
stein, “The computational complexity of rules for the character table of Sn”,
Journal of Symbolic Computation, vol. 37 iss. 6 (2004), pp 727-748. doi:10.1016/j.jsc.2003.11.001
source
237
238 CHAPTER 14. COMBINATORICS
14.5 Combinatorics.combinations
Combinatorics.combinations —Method.
Generate all combinations of nelements from an indexable object. Because
the number of combinations can be very large, this function returns an iterator
object. Use collect(combinations(array,n)) to get an array of all combina-
tions.
source
14.6 Combinatorics.combinations
Combinatorics.combinations —Method.
generate combinations of all orders, chaining of order iterators is eager, but
sequence at each order is lazy
source
14.7 Combinatorics.derangement
Combinatorics.derangement —Method.
The number of permutations of n with no fixed points (subfactorial)
source
14.8 Combinatorics.integer partitions
Combinatorics.integer partitions —Method.
Lists the partitions of the number n, the order is consistent with GAP
source
14.9 Combinatorics.isrimhook
Combinatorics.isrimhook —Method.
Checks if skew diagram is a rim hook
source
14.10 Combinatorics.isrimhook
Combinatorics.isrimhook —Method.
Takes two elements of a partition sequence, with a to the left of b
source
14.11. COMBINATORICS.LASSALLENUM 239
14.11 Combinatorics.lassallenum
Combinatorics.lassallenum —Method.
Computes Lassalle’s sequence OEIS entry A180874
source
14.12 Combinatorics.leglength
Combinatorics.leglength —Method.
Strictly speaking, defined for rim hook only, but here we define it for all
skew diagrams
source
14.13 Combinatorics.levicivita
Combinatorics.levicivita —Method.
Levi-Civita symbol of a permutation.
Returns 1 is the permutation is even, -1 if it is odd and 0 otherwise.
The parity is computed by using the fact that a permutation is odd if and
only if the number of even-length cycles is odd.
source
14.14 Combinatorics.multiexponents
Combinatorics.multiexponents —Method.
multiexponents(m, n)
Returns the exponents in the multinomial expansion (x + x + . . . + x).
For example, the expansion (x + x + x) = x + xx + xx + . . . has the
exponents:
julia> collect(multiexponents(3, 2))
6-element Array{Any,1}:
[2, 0, 0]
[1, 1, 0]
[1, 0, 1]
[0, 2, 0]
[0, 1, 1]
[0, 0, 2]
source
240 CHAPTER 14. COMBINATORICS
14.15 Combinatorics.multinomial
Combinatorics.multinomial —Method.
Multinomial coefficient where n = sum(k)
source
14.16 Combinatorics.multiset combinations
Combinatorics.multiset combinations —Method.
generate all combinations of size t from an array a with possibly duplicated
elements.
source
14.17 Combinatorics.multiset permutations
Combinatorics.multiset permutations —Method.
generate all permutations of size t from an array a with possibly duplicated
elements.
source
14.18 Combinatorics.nthperm!
Combinatorics.nthperm! —Method.
In-place version of nthperm.
source
14.19 Combinatorics.nthperm
Combinatorics.nthperm —Method.
Compute the kth lexicographic permutation of the vector a.
source
14.20 Combinatorics.nthperm
Combinatorics.nthperm —Method.
Return the kthat generated permutation p. Note that nthperm(nthperm([1:n],
k)) == k for 1<= k <= factorial(n).
source
14.21. COMBINATORICS.PARITY 241
14.21 Combinatorics.parity
Combinatorics.parity —Method.
Computes the parity of a permutation using the levicivita function, so you
can ask iseven(parity(p)). If p is not a permutation throws an error.
source
14.22 Combinatorics.partitions
Combinatorics.partitions —Method.
Generate all set partitions of the elements of an array into exactly m subsets,
represented as arrays of arrays. Because the number of partitions can be very
large, this function returns an iterator object. Use collect(partitions(array,m))
to get an array of all partitions. The number of partitions into m subsets is equal
to the Stirling number of the second kind and can be efficiently computed using
length(partitions(array,m)).
source
14.23 Combinatorics.partitions
Combinatorics.partitions —Method.
Generate all set partitions of the elements of an array, represented as arrays
of arrays. Because the number of partitions can be very large, this function
returns an iterator object. Use collect(partitions(array)) to get an array of
all partitions. The number of partitions to generate can be efficiently computed
using length(partitions(array)).
source
14.24 Combinatorics.partitions
Combinatorics.partitions —Method.
Generate all arrays of mintegers that sum to n. Because the number of
partitions can be very large, this function returns an iterator object. Use
collect(partitions(n,m)) to get an array of all partitions. The number of
partitions to generate can be efficiently computed using length(partitions(n,m)).
source
14.25 Combinatorics.partitions
Combinatorics.partitions —Method.
Generate all integer arrays that sum to n. Because the number of partitions
can be very large, this function returns an iterator object. Use collect(partitions(n))
to get an array of all partitions. The number of partitions to generate can be
efficiently computed using length(partitions(n)).
242 CHAPTER 14. COMBINATORICS
source
14.26 Combinatorics.partitionsequence
Combinatorics.partitionsequence —Method.
Computes essential part of the partition sequence of lambda
source
14.27 Combinatorics.permutations
Combinatorics.permutations —Method.
Generate all size t permutations of an indexable object.
source
14.28 Combinatorics.permutations
Combinatorics.permutations —Method.
Generate all permutations of an indexable object. Because the number of
permutations can be very large, this function returns an iterator object. Use
collect(permutations(array)) to get an array of all permutations.
source
14.29 Combinatorics.prevprod
Combinatorics.prevprod —Method.
Previous integer not greater than nthat can be written as Qkpi
ifor integers
p1,p2, etc.
For a list of integers i1, i2, i3, find the largest i1ˆn1 * i2ˆn2 * i3ˆn3 <= x
for integer n1, n2, n3
source
14.30 Combinatorics.with replacement combinations
Combinatorics.with replacement combinations —Method.
generate all combinations with replacement of size t from an array a.
source
Chapter 15
HypothesisTests
15.1 HypothesisTests.ChisqTest
HypothesisTests.ChisqTest —Method.
ChisqTest(x[, y][, theta0 = ones(length(x))/length(x)])
Perform a PowerDivergenceTest with = 1, i.e. in the form of Pearson’s
chi-squared statistic.
If yis not given and xis a matrix with one row or column, or xis a vector,
then a goodness-of-fit test is performed (xis treated as a one-dimensional con-
tingency table). In this case, the hypothesis tested is whether the population
probabilities equal those in theta0, or are all equal if theta0 is not given.
If xis a matrix with at least two rows and columns, it is taken as a two-
dimensional contingency table. Otherwise, xand ymust be vectors of the same
length. The contingency table is calculated using counts function from the
StatsBase package. Then the power divergence test is conducted under the
null hypothesis that the joint distribution of the cell counts in a 2-dimensional
contingency table is the product of the row and column marginals.
Note that the entries of x(and yif provided) must be non-negative integers.
Implements: pvalue,confint
source
15.2 HypothesisTests.MannWhitneyUTest
HypothesisTests.MannWhitneyUTest —Method.
MannWhitneyUTest(x::AbstractVector{<:Real}, y::AbstractVector{<:Real})
Perform a Mann-Whitney U test of the null hypothesis that the probability
that an observation drawn from the same population as xis greater than an ob-
servation drawn from the same population as yis equal to the probability that
243
244 CHAPTER 15. HYPOTHESISTESTS
an observation drawn from the same population as yis greater than an obser-
vation drawn from the same population as xagainst the alternative hypothesis
that these probabilities are not equal.
The Mann-Whitney U test is sometimes known as the Wilcoxon rank-sum
test.
When there are no tied ranks and 50 samples, or tied ranks and 10 samples,
MannWhitneyUTest performs an exact Mann-Whitney U test. In all other cases,
MannWhitneyUTest performs an approximate Mann-Whitney U test. Behavior
may be further controlled by using ExactMannWhitneyUTest or ApproximateMannWhitneyUTest
directly.
Implements: pvalue
source
15.3 HypothesisTests.MultinomialLRT
HypothesisTests.MultinomialLRT —Method.
MultinomialLRT(x[, y][, theta0 = ones(length(x))/length(x)])
Perform a PowerDivergenceTest with = 0, i.e. in the form of the likelihood
ratio test statistic.
If yis not given and xis a matrix with one row or column, or xis a vector,
then a goodness-of-fit test is performed (xis treated as a one-dimensional con-
tingency table). In this case, the hypothesis tested is whether the population
probabilities equal those in theta0, or are all equal if theta0 is not given.
If xis a matrix with at least two rows and columns, it is taken as a two-
dimensional contingency table. Otherwise, xand ymust be vectors of the same
length. The contingency table is calculated using counts function from the
StatsBase package. Then the power divergence test is conducted under the
null hypothesis that the joint distribution of the cell counts in a 2-dimensional
contingency table is the product of the row and column marginals.
Note that the entries of x(and yif provided) must be non-negative integers.
Implements: pvalue,confint
source
15.4 HypothesisTests.SignedRankTest
HypothesisTests.SignedRankTest —Method.
SignedRankTest(x::AbstractVector{<:Real})
SignedRankTest(x::AbstractVector{<:Real}, y::AbstractVector{T<:Real})
Perform a Wilcoxon signed rank test of the null hypothesis that the distri-
bution of x(or the difference x-yif yis provided) has zero median against
the alternative hypothesis that the median is non-zero.
15.5. HYPOTHESISTESTS.PVALUE 245
When there are no tied ranks and 50 samples, or tied ranks and 15 sam-
ples, SignedRankTest performs an exact signed rank test. In all other cases,
SignedRankTest performs an approximate signed rank test. Behavior may be
further controlled by using ExactSignedRankTest or ApproximateSignedRankTest
directly.
Implements: pvalue,confint
source
15.5 HypothesisTests.pvalue
HypothesisTests.pvalue —Function.
pvalue(test::HypothesisTest; tail = :both)
Compute the p-value for a given significance test.
If tail is :both (default), then the p-value for the two-sided test is returned.
If tail is :left or :right, then a one-sided test is performed.
source
15.6 HypothesisTests.pvalue
HypothesisTests.pvalue —Method.
pvalue(x::FisherExactTest; tail = :both, method = :central)
Compute the p-value for a given Fisher exact test.
The one-sided p-values are based on Fisher’s non-central hypergeometric
distribution f(i) with odds ratio :
p(left) =X
ia
f(i)
p(right) =X
ia
f(i)
For tail = :both, possible values for method are:
•:central (default): Central interval, i.e. the p-value is two times the
minimum of the one-sided p-values.
•:minlike: Minimum likelihood interval, i.e. the p-value is computed by
summing all tables with the same marginals that are equally or less prob-
able:
p=X
f(i)f(a)
f(i)
References
246 CHAPTER 15. HYPOTHESISTESTS
•Gibbons, J.D., Pratt, J.W., P-values: Interpretation and Methodology,
American Statistican, 29(1):20-25, 1975.
•Fay, M.P., Supplementary material to “Confidence intervals that match
Fisher’s exact or Blaker’s exact tests”. Biostatistics, Volume 11, Issue 2,
1 April 2010, Pages 373–374, link
source
15.7 HypothesisTests.testname
HypothesisTests.testname —Method.
testname(::HypothesisTest)
Returns the string value, e.g. “Binomial test” or “Sign Test”.
source
15.8 StatsBase.confint
StatsBase.confint —Function.
confint(test::HypothesisTest, alpha = 0.05; tail = :both)
Compute a confidence interval C with coverage 1-alpha.
If tail is :both (default), then a two-sided confidence interval is returned.
If tail is :left or :right, then a one-sided confidence interval is returned.
!!! note Most of the implemented confidence intervals are strongly consistent,
that is, the confidence interval with coverage 1-alpha does not contain the test
statistic under h0if and only if the corresponding test rejects the null hypothesis
h0: = 0:
$$
C (x, 1 ) = \{ : p_ (x) > \},
$$
where $p_$ is the [‘pvalue‘](HypothesisTests.md#HypothesisTests.pvalue) of the corresponding test.
source
15.9 StatsBase.confint
StatsBase.confint —Function.
confint(test::BinomialTest, alpha = 0.05; tail = :both, method = :clopper_pearson)
15.10. STATSBASE.CONFINT 247
Compute a confidence interval with coverage 1-alpha for a binomial propor-
tion using one of the following methods. Possible values for method are:
•:clopper pearson (default): Clopper-Pearson interval is based on the bi-
nomial distribution. The empirical coverage is never less than the nominal
coverage of 1-alpha; it is usually too conservative.
•:wald: Wald (or normal approximation) interval relies on the standard
approximation of the actual binomial distribution by a normal distribu-
tion. Coverage can be erratically poor for success probabilities close to
zero or one.
•:wilson: Wilson score interval relies on a normal approximation. In con-
trast to :wald, the standard deviation is not approximated by an empirical
estimate, resulting in good empirical coverages even for small numbers of
draws and extreme success probabilities.
•:jeffrey: Jeffreys interval is a Bayesian credible interval obtained by
using a non-informative Jeffreys prior. The interval is very similar to the
Wilson interval.
•:agresti coull: Agresti-Coull interval is a simplified version of the Wil-
son interval; both are centered around the same value. The Agresti Coull
interval has higher or equal coverage.
•:arcsine: Confidence interval computed using the arcsine transformation
to make var(p) independent of the probability p.
References
•Brown, L.D., Cai, T.T., and DasGupta, A. Interval estimation for a bino-
mial proportion. Statistical Science, 16(2):101–117, 2001.
External links
•Binomial confidence interval on Wikipedia
source
15.10 StatsBase.confint
StatsBase.confint —Function.
confint(x::FisherExactTest, alpha::Float64=0.05; tail=:both, method=:central)
Compute a confidence interval with coverage 1 - alpha. One-sided inter-
vals are based on Fisher’s non-central hypergeometric distribution. For tail =
:both, the only method implemented yet is the central interval (:central).
!!! note Since the p-value is not necessarily unimodal, the corresponding
confidence region might not be an interval.
References
248 CHAPTER 15. HYPOTHESISTESTS
•Gibbons, J.D, Pratt, J.W. P-values: Interpretation and Methodology,
American Statistican, 29(1):20-25, 1975.
•Fay, M.P., Supplementary material to “Confidence intervals that match
Fisher’s exact or Blaker’s exact tests”. Biostatistics, Volume 11, Issue 2,
1 April 2010, Pages 373–374, link
source
15.11 StatsBase.confint
StatsBase.confint —Function.
confint(test::PowerDivergenceTest, alpha = 0.05; tail = :both, method = :sison_glaz)
Compute a confidence interval with coverage 1-alpha for multinomial pro-
portions using one of the following methods. Possible values for method are:
•:sison glaz (default): Sison-Glaz intervals
•:bootstrap: Bootstrap intervals
•:quesenberry hurst: Quesenberry-Hurst intervals
•:gold: Gold intervals (asymptotic simultaneous intervals)
References
•Agresti, Alan. Categorical Data Analysis, 3rd Edition. Wiley, 2013.
•Sison, C.P and Glaz, J. Simultaneous confidence intervals and sample
size determination for multinomial proportions. Journal of the American
Statistical Association, 90:366-369, 1995.
•Quesensberry, C.P. and Hurst, D.C. Large Sample Simultaneous Confi-
dence Intervals for Multinational Proportions. Technometrics, 6:191-195,
1964.
•Gold, R. Z. Tests Auxiliary to 2Tests in a Markov Chain. Annals of
Mathematical Statistics, 30:56-74, 1963.
source
Chapter 16
DataArrays
16.1 DataArrays.PooledDataVecs
DataArrays.PooledDataVecs —Method.
PooledDataVecs(v1, v2) -> (pda1, pda2)
Return a tuple of PooledDataArrays created from the data in v1 and v2,
respectively, but sharing a common value pool.
source
16.2 DataArrays.compact
DataArrays.compact —Method.
compact(d::PooledDataArray)
Return a PooledDataArray with the smallest possible reference type for the
data in d.
!!! note If the reference type is already the smallest possible for the data,
the input array is returned, i.e. the function aliases the input.
Examples
julia> p = @pdata(repeat(["A", "B"], outer=4))
8-element DataArrays.PooledDataArray{String,UInt32,1}:
"A"
"B"
"A"
"B"
"A"
"B"
"A"
249
250 CHAPTER 16. DATAARRAYS
"B"
julia> compact(p) # second type parameter compacts to UInt8 (only need 2 unique values)
8-element DataArrays.PooledDataArray{String,UInt8,1}:
"A"
"B"
"A"
"B"
"A"
"B"
"A"
"B"
source
16.3 DataArrays.cut
DataArrays.cut —Method.
cut(x::AbstractVector, breaks::Vector) -> PooledDataArray
cut(x::AbstractVector, ngroups::Integer) -> PooledDataArray
Divide the range of xinto intervals based on the cut points specified in
breaks, or into ngroups intervals of approximately equal length.
Examples
julia> cut([1, 2, 3, 4], [1, 3])
4-element DataArrays.PooledDataArray{String,UInt32,1}:
"[1,3]"
"[1,3]"
"[1,3]"
"(3,4]"
source
16.4 DataArrays.data
DataArrays.data —Method.
data(a::AbstractArray) -> DataArray
Convert ato a DataArray.
Examples
16.5. DATAARRAYS.DROPNA 251
julia> data([1, 2, 3])
3-element DataArrays.DataArray{Int64,1}:
1
2
3
julia> data(@data [1, 2, NA])
3-element DataArrays.DataArray{Int64,1}:
1
2
NA
source
16.5 DataArrays.dropna
DataArrays.dropna —Method.
dropna(v::AbstractVector) -> AbstractVector
Return a copy of vwith all NA elements removed.
Examples
julia> dropna(@data [NA, 1, NA, 2])
2-element Array{Int64,1}:
1
2
julia> dropna([4, 5, 6])
3-element Array{Int64,1}:
4
5
6
source
16.6 DataArrays.getpoolidx
DataArrays.getpoolidx —Method.
getpoolidx(pda::PooledDataArray, val)
Return the index of val in the value pool for pda. If val is not already in
the value pool, pda is modified to include it in the pool.
source
252 CHAPTER 16. DATAARRAYS
16.7 DataArrays.gl
DataArrays.gl —Method.
gl(n::Integer, k::Integer, l::Integer = n*k) -> PooledDataArray
Generate a PooledDataArray with nlevels and kreplications, optionally
specifying an output length l. If specified, lmust be a multiple of n*k.
Examples
julia> gl(2, 1)
2-element DataArrays.PooledDataArray{Int64,UInt8,1}:
1
2
julia> gl(2, 1, 4)
4-element DataArrays.PooledDataArray{Int64,UInt8,1}:
1
2
1
2
source
16.8 DataArrays.isna
DataArrays.isna —Method.
isna(x) -> Bool
Determine whether xis missing, i.e. NA.
Examples
julia> isna(1)
false
julia> isna(NA)
true
source
16.9 DataArrays.isna
DataArrays.isna —Method.
isna(a::AbstractArray, i) -> Bool
16.10. DATAARRAYS.LEVELS 253
Determine whether the element of aat index iis missing, i.e. NA.
Examples
julia> X = @data [1, 2, NA];
julia> isna(X, 2)
false
julia> isna(X, 3)
true
source
16.10 DataArrays.levels
DataArrays.levels —Method.
levels(da::DataArray) -> DataVector
Return a vector of the unique values in da, excluding any NAs.
levels(a::AbstractArray) -> Vector
Equivalent to unique(a).
Examples
julia> levels(@data [1, 2, NA])
2-element DataArrays.DataArray{Int64,1}:
1
2
source
16.11 DataArrays.padna
DataArrays.padna —Method.
padna(dv::AbstractDataVector, front::Integer, back::Integer) -> DataVector
Pad dv with NA values. front is an integer number of NAs to add at the
beginning of the array and back is the number of NAs to add at the end.
Examples
julia> padna(@data([1, 2, 3]), 1, 2)
6-element DataArrays.DataArray{Int64,1}:
NA
1
254 CHAPTER 16. DATAARRAYS
2
3
NA
NA
source
16.12 DataArrays.reorder
DataArrays.reorder —Method.
reorder(x::PooledDataArray) -> PooledDataArray
Return a PooledDataArray containing the same data as xbut with the value
pool sorted.
source
16.13 DataArrays.replace!
DataArrays.replace! —Method.
replace!(x::PooledDataArray, from, to)
Replace all occurrences of from in xwith to, modifying xin place.
source
16.14 DataArrays.setlevels!
DataArrays.setlevels! —Method.
setlevels!(x::PooledDataArray, newpool::Union{AbstractVector, Dict})
Set the value pool for the PooledDataArray xto newpool, modifying xin
place. The values can be replaced using a mapping specified in a Dict or with
an array, since the order of the levels is used to identify values. The pool can
be enlarged to contain values not present in the data, but it cannot be reduced
to exclude present values.
Examples
julia> p = @pdata repeat(["A", "B"], inner=3)
6-element DataArrays.PooledDataArray{String,UInt32,1}:
"A"
"A"
"A"
"B"
"B"
16.15. DATAARRAYS.SETLEVELS 255
"B"
julia> setlevels!(p, Dict("A"=>"C"));
julia> p # has been modified
6-element DataArrays.PooledDataArray{String,UInt32,1}:
"C"
"C"
"C"
"B"
"B"
"B"
source
16.15 DataArrays.setlevels
DataArrays.setlevels —Method.
setlevels(x::PooledDataArray, newpool::Union{AbstractVector, Dict})
Create a new PooledDataArray based on xbut with the new value pool
specified by newpool. The values can be replaced using a mapping specified in
aDict or with an array, since the order of the levels is used to identify values.
The pool can be enlarged to contain values not present in the data, but it cannot
be reduced to exclude present values.
Examples
julia> p = @pdata repeat(["A", "B"], inner=3)
6-element DataArrays.PooledDataArray{String,UInt32,1}:
"A"
"A"
"A"
"B"
"B"
"B"
julia> p2 = setlevels(p, ["C", "D"]) # could also be Dict("A"=>"C", "B"=>"D")
6-element DataArrays.PooledDataArray{String,UInt32,1}:
"C"
"C"
"C"
"D"
"D"
"D"
256 CHAPTER 16. DATAARRAYS
julia> p3 = setlevels(p2, ["C", "D", "E"])
6-element DataArrays.PooledDataArray{String,UInt32,1}:
"C"
"C"
"C"
"D"
"D"
"D"
julia> p3.pool # the pool can contain values not in the array
3-element Array{String,1}:
"C"
"D"
"E"
source
Chapter 17
GLM
17.1 GLM.canonicallink
GLM.canonicallink —Function.
canonicallink(D::Distribution)
Return the canonical link for distribution D, which must be in the exponential
family.
Examples
julia> canonicallink(Bernoulli())
GLM.LogitLink()
source
17.2 GLM.devresid
GLM.devresid —Method.
devresid(D, y, )
Return the squared deviance residual of from yfor distribution D
The deviance of a GLM can be evaluated as the sum of the squared de-
viance residuals. This is the principal use for these values. The actual deviance
residual, say for plotting, is the signed square root of this value
[] sign(y - ) * sqrt(devresid(D, y, ))
Examples
julia> showcompact(devresid(Normal(), 0, 0.25)) # abs2(y - )
0.0625
julia> showcompact(devresid(Bernoulli(), 1, 0.75)) # -2log() when y == 1
0.575364
257
258 CHAPTER 17. GLM
julia> showcompact(devresid(Bernoulli(), 0, 0.25)) # -2log1p(-) = -2log(1-) when y == 0
0.575364
source
17.3 GLM.ftest
GLM.ftest —Method.
ftest(mod::LinearModel...; atol=0::Real)
For each sequential pair of linear predictors in mod, perform an F-test to
determine if the first one fits significantly better than the next.
A table is returned containing residual degrees of freedom (DOF), degrees of
freedom, difference in DOF from the preceding model, sum of squared residuals
(SSR), difference in SSR from the preceding model, R, difference in R from the
preceding model, and F-statistic and p-value for the comparison between the
two models.
!!! note This function can be used to perform an ANOVA by testing the
relative fit of two models to the data
Optional keyword argument atol controls the numerical tolerance when test-
ing whether the models are nested.
Examples
Suppose we want to compare the effects of two or more treatments on some
result. Because this is an ANOVA, our null hypothesis is that Result ~ 1 fits
the data as well as Result ~ 1 + Treatment.
julia> dat = DataFrame(Treatment=[1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2.],
Result=[1.1, 1.2, 1, 2.2, 1.9, 2, .9, 1, 1, 2.2, 2, 2],
Other=[1, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 1]);
julia> mod = lm(@formula(Result ~ 1 + Treatment), dat);
julia> nullmod = lm(@formula(Result ~ 1), dat);
julia> bigmod = lm(@formula(Result ~ 1 + Treatment + Other), dat);
julia> ft = ftest(mod.model, nullmod.model)
Res. DOF DOF DOF SSR SSR R R F* p(>F)
Model 1 10 3 0.1283 0.9603
Model 2 11 2 -1 3.2292 -3.1008 0.0000 0.9603 241.6234 <1e-7
julia> ftest(bigmod.model, mod.model, nullmod.model)
Res. DOF DOF DOF SSR SSR R R F* p(>F)
Model 1 9 4 0.1038 0.9678
17.4. GLM.GLMVAR 259
Model 2 10 3 -1 0.1283 -0.0245 0.9603 0.0076 2.1236 0.1790
Model 3 11 2 -1 3.2292 -3.1008 0.0000 0.9603 241.6234 <1e-7
source
17.4 GLM.glmvar
GLM.glmvar —Function.
glmvar(D::Distribution, )
Return the value of the variance function for Dat
The variance of Dat is the product of the dispersion parameter, , which
does not depend on and the value of glmvar. In other words glmvar returns
the factor of the variance that depends on .
Examples
julia> = inv(6):inv(3):1; showcompact(collect())
[0.166667, 0.5, 0.833333]
julia> showcompact(glmvar.(Normal(), )) # constant for Normal()
[1.0, 1.0, 1.0]
julia> showcompact(glmvar.(Bernoulli(), )) # * (1 - ) for Bernoulli()
[0.138889, 0.25, 0.138889]
julia> showcompact(glmvar.(Poisson(), )) # for Poisson()
[0.166667, 0.5, 0.833333]
source
17.5 GLM.inverselink
GLM.inverselink —Function.
inverselink(L::Link, )
Return a 3-tuple of the inverse link, the derivative of the inverse link, and
when appropriate, the variance function *(1 - ).
The variance function is returned as NaN unless the range of is (0, 1)
Examples
julia> showcompact(inverselink(LogitLink(), 0.0))
(0.5, 0.25, 0.25)
julia> showcompact(inverselink(CloglogLink(), 0.0))
(0.632121, 0.367879, 0.232544)
julia> showcompact(inverselink(LogLink(), 2.0))
(7.38906, 7.38906, NaN)
source
260 CHAPTER 17. GLM
17.6 GLM.linkfun
GLM.linkfun —Function.
linkfun(L::Link, )
Return , the value of the linear predictor for link Lat mean .
Examples
julia> = inv(10):inv(5):1
0.1:0.2:0.9
julia> show(linkfun.(LogitLink(), ))
[-2.19722, -0.847298, 0.0, 0.847298, 2.19722]
source
17.7 GLM.linkinv
GLM.linkinv —Function.
linkinv(L::Link, )
Return , the mean value, for link Lat linear predictor value .
Examples
julia> = inv(10):inv(5):1; showcompact(collect())
[0.1, 0.3, 0.5, 0.7, 0.9]
julia> = logit.(); showcompact()
[-2.19722, -0.847298, 0.0, 0.847298, 2.19722]
julia> showcompact(linkinv.(LogitLink(), ))
[0.1, 0.3, 0.5, 0.7, 0.9]
source
17.8 GLM.mueta
GLM.mueta —Function.
mueta(L::Link, )
Return the derivative of linkinv,d/d, for link Lat linear predictor value .
Examples
julia> showcompact(mueta(LogitLink(), 0.0))
0.25
julia> showcompact(mueta(CloglogLink(), 0.0))
0.367879
julia> showcompact(mueta(LogLink(), 2.0))
7.38906
source
17.9. GLM.MUSTART 261
17.9 GLM.mustart
GLM.mustart —Function.
mustart(D::Distribution, y, wt)
Return a starting value for .
For some distributions it is appropriate to set = y to initialize the IRLS
algorithm but for others, notably the Bernoulli, the values of yare not allowed
as values of and must be modified.
Examples
julia> showcompact(mustart(Bernoulli(), 0.0, 1))
0.25
julia> showcompact(mustart(Bernoulli(), 1.0, 1))
0.75
julia> showcompact(mustart(Binomial(), 0.0, 10))
0.0454545
julia> showcompact(mustart(Normal(), 0.0, 1))
0.0
source
17.10 GLM.update!
GLM.update! —Method.
update!{T<:FPVector}(r::GlmResp{T}, linPr::T)
Update the mean, working weights and working residuals, in rgiven a value
of the linear predictor, linPr.
source
17.11 GLM.wrkresp
GLM.wrkresp —Method.
wrkresp(r::GlmResp)
The working response, r.eta + r.wrkresid - r.offset.
source
17.12 StatsBase.deviance
StatsBase.deviance —Method.
deviance(obj::LinearModel)
For linear models, the deviance is equal to the residual sum of squares (RSS).
source
262 CHAPTER 17. GLM
17.13 StatsBase.nobs
StatsBase.nobs —Method.
nobs(obj::LinearModel)
nobs(obj::GLM)
For linear and generalized linear models, returns the number of rows, or,
when prior weights are specified, the sum of weights.
source
17.14 StatsBase.nulldeviance
StatsBase.nulldeviance —Method.
nulldeviance(obj::LinearModel)
For linear models, the deviance of the null model is equal to the total sum
of squares (TSS).
source
17.15 StatsBase.predict
StatsBase.predict —Function.
predict(mm::LinearModel, newx::AbstractMatrix, interval_type::Symbol, level::Real = 0.95)
Specifying interval type will return a 3-column matrix with the prediction
and the lower and upper confidence bounds for a given level (0.95 equates
alpha = 0.05). Valid values of interval type are :confint delimiting the
uncertainty of the predicted relationship, and :predint delimiting estimated
bounds for new data points.
source
17.16 StatsBase.predict
StatsBase.predict —Method.
predict(mm::AbstractGLM, newX::AbstractMatrix; offset::FPVector=Vector{eltype(newX)}(0))
Form the predicted response of model mm from covariate values newX and,
optionally, an offset.
source
Chapter 18
Documenter
18.1 Documenter.deploydocs
Documenter.deploydocs —Method.
deploydocs(
root = "<current-directory>",
target = "site",
repo = "<required>",
branch = "gh-pages",
latest = "master",
osname = "linux",
julia = "nightly",
deps = <Function>,
make = <Function>,
)
Converts markdown files generated by makedocs to HTML and pushes them
to repo. This function should be called from within a package’s docs/make.jl
file after the call to makedocs, like so
[] using Documenter, PACKAGENAMEmakedocs(options...)deploydocs(repo =”github.com/...”)
Keywords
root has the same purpose as the root keyword for makedocs.
target is the directory, relative to root, where generated HTML content
should be written to. This directory must be added to the repository’s .gitignore
file. The default value is "site".
repo is the remote repository where generated HTML content should be
pushed to. Do not specify any protocol - “https://” or “git@” should not be
present. This keyword must be set and will throw an error when left undefined.
For example this package uses the following repo value:
[] repo = ”github.com/JuliaDocs/Documenter.jl.git”
263
264 CHAPTER 18. DOCUMENTER
branch is the branch where the generated documentation is pushed. If the
branch does not exist, a new orphaned branch is created automatically. It
defaults to "gh-pages".
latest is the branch that “tracks” the latest generated documentation. By
default this value is set to "master".
osname is the operating system which will be used to deploy generated docu-
mentation. This defaults to "linux". This value must be one of those specified
in the os: section of the .travis.yml configuration file.
julia is the version of Julia that will be used to deploy generated documen-
tation. This defaults to "nightly". This value must be one of those specified
in the julia: section of the .travis.yml configuration file.
deps is the function used to install any dependencies needed to build the
documentation. By default this function installs pygments and mkdocs using
the Deps.pip function:
[] deps = Deps.pip(”pygments”, ”mkdocs”)
make is the function used to convert the markdown files to HTML. By default
this just runs mkdocs build which populates the target directory.
See Also
The Hosting Documentation section of the manual provides a step-by-step
guide to using the deploydocs function to automatically generate docs and
push then to GitHub.
source
18.2 Documenter.hide
Documenter.hide —Method.
[] hide(root, children)
Allows a subsection of pages to be hidden from the navigation menu. root
will be linked to in the navigation menu, with the title determined as usual.
children should be a list of pages (note that it can not be hierarchical).
Usage
[] makedocs( ..., pages = [ ..., hide(”Hidden section” =¿ ”hiddenindex.md”,[”hidden1.md”,”Hidden2”=>”hidden2.md”]),hide(”hiddenindex.md”,[...])])
source
18.3 Documenter.hide
Documenter.hide —Method.
[] hide(page)
Allows a page to be hidden in the navigation menu. It will only show up if
it happens to be the current page. The hidden page will still be present in the
linear page list that can be accessed via the previous and next page links. The
title of the hidden page can be overriden using the =>operator as usual.
Usage
18.4. DOCUMENTER.MAKEDOCS 265
[] makedocs( ..., pages = [ ..., hide(”page1.md”), hide(”Title” =¿
”page2.md”) ] )
source
18.4 Documenter.makedocs
Documenter.makedocs —Method.
makedocs(
root = "<current-directory>",
source = "src",
build = "build",
clean = true,
doctest = true,
modules = Module[],
repo = "",
)
Combines markdown files and inline docstrings into an interlinked document.
In most cases makedocs should be run from a make.jl file:
[] using Documenter makedocs( keywords... )
which is then run from the command line with:
[] juliamake.jl
The folder structure that makedocs expects looks like:
docs/
build/
src/
make.jl
Keywords
root is the directory from which makedocs should run. When run from a
make.jl file this keyword does not need to be set. It is, for the most part,
needed when repeatedly running makedocs from the Julia REPL like so:
julia> makedocs(root = Pkg.dir("MyPackage", "docs"))
source is the directory, relative to root, where the markdown source files
are read from. By convention this folder is called src. Note that any non-
markdown files stored in source are copied over to the build directory when
makedocs is run.
build is the directory, relative to root, into which generated files and folders
are written when makedocs is run. The name of the build directory is, by
convention, called build, though, like with source, users are free to change
this to anything else to better suit their project needs.
266 CHAPTER 18. DOCUMENTER
clean tells makedocs whether to remove all the content from the build
folder prior to generating new content from source. By default this is set to
true.
doctest instructs makedocs on whether to try to test Julia code blocks
that are encountered in the generated document. By default this keyword is
set to true. Doctesting should only ever be disabled when initially setting
up a newly developed package where the developer is just trying to get their
package and documentation structure correct. After that, it’s encouraged to
always make sure that documentation examples are runnable and produce the
expected results. See the Doctests manual section for details about running
doctests.
modules specifies a vector of modules that should be documented in source.
If any inline docstrings from those modules are seen to be missing from the
generated content then a warning will be printed during execution of makedocs.
By default no modules are passed to modules and so no warnings will appear.
This setting can be used as an indicator of the “coverage” of the generated
documentation. For example Documenter’s make.jl file contains:
[] makedocs( modules = [Documenter], ... )
and so any docstring from the module Documenter that is not spliced into
the generated documentation in build will raise a warning.
repo specifies a template for the “link to source” feature. If you are using
GitHub, this is automatically generated from the remote. If you are using a
different host, you can use this option to tell Documenter how URLs should be
generated. The following placeholders will be replaced with the respective value
of the generated link:
• {commit}Git branch or tag name, or commit hash
• {path}Path to the file in the repository
• {line}Line (or range of lines) in the source file
For example if you are using GitLab.com, you could use
[] makedocs(repo = ”https://gitlab.com/user/project/blob/{commit}{path}L{line}”)
Experimental keywords
In addition to standard arguments there is a set of non-finalized experimental
keyword arguments. The behaviour of these may change or they may be removed
without deprecation when a minor version changes (i.e. except in patch releases).
checkdocs instructs makedocs to check whether all names within the mod-
ules defined in the modules keyword that have a docstring attached have the
docstring also listed in the manual (e.g. there’s a @docs blocks with that doc-
string). Possible values are :all (check all names) and :exports (check only
exported names). The default value is :none, in which case no checks are per-
formed. If strict is also enabled then the build will fail if any missing docstrings
are encountered.
18.4. DOCUMENTER.MAKEDOCS 267
linkcheck – if set to true makedocs uses curl to check the status codes of
external-pointing links, to make sure that they are up-to-date. The links and
their status codes are printed to the standard output. If strict is also enabled
then the build will fail if there are any broken (400+ status code) links. Default:
false.
linkcheck ignore allows certain URLs to be ignored in linkcheck. The
values should be a list of strings (which get matched exactly) or Regex objects.
By default nothing is ignored.
strict –makedocs fails the build right before rendering if it encountered
any errors with the document in the previous build phases.
Non-MkDocs builds
Documenter also has (experimental) support for native HTML and LaTeX
builds. These can be enabled using the format keyword and they generally re-
quire additional keywords be defined, depending on the format. These keywords
are also currently considered experimental.
format allows the output format to be specified. Possible values are :html,
:latex and :markdown (default).
Other keywords related to non-MkDocs builds (assets,sitename,analytics,
authors,pages,version) should be documented at the respective *Writer
modules (Writers.HTMLWriter,Writers.LaTeXWriter).
See Also
A guide detailing how to document a package using Documenter’s makedocs
is provided in the Usage section of the manual.
source
Chapter 19
ColorTypes
19.1 ColorTypes.alpha
ColorTypes.alpha —Method.
alpha(p) extracts the alpha component of a color. For a color without an
alpha channel, it will always return 1.
source
19.2 ColorTypes.alphacolor
ColorTypes.alphacolor —Function.
alphacolor(RGB) returns ARGB, i.e., the corresponding transparent color
type with storage order (alpha, color).
source
19.3 ColorTypes.base color type
ColorTypes.base color type —Method.
base color type is similar to color type, except it “strips off” the element
type. For example,
color_type(RGB{N0f8}) == RGB{N0f8}
base_color_type(RGB{N0f8}) == RGB
This can be very handy if you want to switch element types. For example:
c64 = base_color_type(c){Float64}(color(c))
converts cinto a Float64 representation (potentially discarding any alpha-
channel information).
source
268
19.4. COLORTYPES.BASE COLORANT TYPE 269
19.4 ColorTypes.base colorant type
ColorTypes.base colorant type —Method.
base colorant type is similar to base color type, but it preserves the
“alpha” portion of the type.
For example,
base_color_type(ARGB{N0f8}) == RGB
base_colorant_type(ARGB{N0f8}) == ARGB
If you just want to switch element types, this is the safest default and the
easiest to use:
c64 = base_colorant_type(c){Float64}(c)
source
19.5 ColorTypes.blue
ColorTypes.blue —Method.
blue(c) returns the blue component of an AbstractRGB opaque or trans-
parent color.
source
19.6 ColorTypes.ccolor
ColorTypes.ccolor —Method.
ccolor (“concrete color”) helps write flexible methods. The idea is that
users may write convert(HSV, c) or even convert(Array{HSV}, A) without
specifying the element type explicitly (e.g., convert(Array{HSV{Float32}},
A)). ccolor implements the logic “choose the user’s eltype if specified, otherwise
retain the eltype of the source object.” However, when the source object has
FixedPoint element type, and the destination only supports AbstractFloat, we
choose Float32.
Usage:
ccolor(desttype, srctype) -> concrete desttype
Example:
convert{C<:Colorant}(::Type{C}, p::Colorant) = cnvt(ccolor(C,typeof(p)), p)
where cnvt is the function that performs explicit conversion.
source
270 CHAPTER 19. COLORTYPES
19.7 ColorTypes.color
ColorTypes.color —Method.
color(c) extracts the opaque color component from a Colorant (e.g., omits
the alpha channel, if present).
source
19.8 ColorTypes.color type
ColorTypes.color type —Method.
color type(c) or color type(C) (cbeing a color instance and Cbeing the
type) returns the type of the Color object (without alpha channel). This, and
related functions like base color type,base colorant type, and ccolor are
useful for manipulating types for writing generic code.
For example,
color_type(RGB) == RGB
color_type(RGB{Float32}) == RGB{Float32}
color_type(ARGB{N0f8}) == RGB{N0f8}
source
19.9 ColorTypes.coloralpha
ColorTypes.coloralpha —Function.
coloralpha(RGB) returns RGBA, i.e., the corresponding transparent color
type with storage order (color, alpha).
source
19.10 ColorTypes.comp1
ColorTypes.comp1 —Method.
comp1(c) extracts the first component you’d pass to the constructor of the
corresponding object. For most color types without an alpha channel, this is
just the first field, but for types like BGR that reverse the internal storage order
this provides the value that you’d use to reconstruct the color.
Specifically, for any Color{T,3},
c == typeof(c)(comp1(c), comp2(c), comp3(c))
returns true.
source
19.11. COLORTYPES.COMP2 271
19.11 ColorTypes.comp2
ColorTypes.comp2 —Method.
comp2(c) extracts the second constructor argument (see comp1).
source
19.12 ColorTypes.comp3
ColorTypes.comp3 —Method.
comp3(c) extracts the third constructor argument (see comp1).
source
19.13 ColorTypes.gray
ColorTypes.gray —Method.
gray(c) returns the gray component of a grayscale opaque or transparent
color.
source
19.14 ColorTypes.green
ColorTypes.green —Method.
green(c) returns the green component of an AbstractRGB opaque or trans-
parent color.
source
19.15 ColorTypes.mapc
ColorTypes.mapc —Method.
mapc(f, rgb) -> rgbf
mapc(f, rgb1, rgb2) -> rgbf
mapc applies the function fto each color channel of the input color(s), re-
turning an output color in the same colorspace.
Examples:
julia> mapc(x->clamp(x,0,1), RGB(-0.2,0.3,1.2))
RGB{Float64}(0.0,0.3,1.0)
julia> mapc(max, RGB(0.1,0.8,0.3), RGB(0.5,0.5,0.5))
RGB{Float64}(0.5,0.8,0.5)
julia> mapc(+, RGB(0.1,0.8,0.3), RGB(0.5,0.5,0.5))
RGB{Float64}(0.6,1.3,0.8)
272 CHAPTER 19. COLORTYPES
source
19.16 ColorTypes.mapreducec
ColorTypes.mapreducec —Method.
mapreducec(f, op, v0, c)
Reduce across color channels of cwith the binary operator op, first applying
fto each channel. v0 is the neutral element used to initiate the reduction. For
grayscale,
mapreducec(f, op, v0, c::Gray) = op(v0, f(comp1(c)))
whereas for RGB
mapreducec(f, op, v0, c::RGB) = op(f(comp3(c)), op(f(comp2(c)), op(v0, f(comp1(c)))))
If chas an alpha channel, it is always the last one to be folded into the
reduction.
source
19.17 ColorTypes.red
ColorTypes.red —Method.
red(c) returns the red component of an AbstractRGB opaque or transparent
color.
source
19.18 ColorTypes.reducec
ColorTypes.reducec —Method.
reducec(op, v0, c)
Reduce across color channels of cwith the binary operator op.v0 is the
neutral element used to initiate the reduction. For grayscale,
reducec(op, v0, c::Gray) = op(v0, comp1(c))
whereas for RGB
reducec(op, v0, c::RGB) = op(comp3(c), op(comp2(c), op(v0, comp1(c))))
If chas an alpha channel, it is always the last one to be folded into the
reduction.
source
Chapter 20
Primes
20.1 Primes.factor
Primes.factor —Method.
factor(ContainerType, n::Integer) -> ContainerType
Return the factorization of nstored in a ContainerType, which must be a
subtype of Associative or AbstractArray, a Set, or an IntSet.
[] julia¿ factor(DataStructures.SortedDict, 100) DataStructures.SortedDict{Int64,Int64,Base.Order.ForwardOrdering}
with 2 entries: 2 =¿ 2 5 =¿ 2
When ContainerType <: AbstractArray, this returns the list of all prime
factors of nwith multiplicities, in sorted order.
[] julia¿ factor(Vector, 100) 4-element Array{Int64,1}: 2 2 5 5
julia¿ prod(factor(Vector, 100)) == 100 true
When ContainerType == Set, this returns the distinct prime factors as a
set.
[] julia¿ factor(Set, 100) Set([2,5])
source
20.2 Primes.factor
Primes.factor —Method.
factor(n::Integer) -> Primes.Factorization
Compute the prime factorization of an integer n. The returned object, of
type Factorization, is an associative container whose keys correspond to the
factors, in sorted order. The value associated with each key indicates the mul-
tiplicity (i.e. the number of times the factor appears in the factorization).
[] julia¿ factor(100) 2252
273
274 CHAPTER 20. PRIMES
For convenience, a negative number nis factored as -1*(-n) (i.e. -1 is
considered to be a factor), and 0is factored as 0^1:
[] julia¿ factor(-9) -1 32
julia¿ factor(0) 0
julia¿ collect(factor(0)) 1-element Array{Pair{Int64,Int64},1}: 0=¿1
source
20.3 Primes.ismersenneprime
Primes.ismersenneprime —Method.
ismersenneprime(M::Integer; [check::Bool = true]) -> Bool
Lucas-Lehmer deterministic test for Mersenne primes. Mmust be a Mersenne
number, i.e. of the form M=2^p-1, where pis a prime number. Use the key-
word argument check to enable/disable checking whether Mis a valid Mersenne
number; to be used with caution. Return true if the given Mersenne number
is prime, and false otherwise.
julia> ismersenneprime(2^11 - 1)
false
julia> ismersenneprime(2^13 - 1)
true
source
20.4 Primes.isprime
Primes.isprime —Function.
isprime(x::BigInt, [reps = 25]) -> Bool
Probabilistic primality test. Returns true if xis prime with high probability
(pseudoprime); and false if xis composite (not prime). The false positive rate
is about 0.25^reps.reps = 25 is considered safe for cryptographic applications
(Knuth, Seminumerical Algorithms).
[] julia¿ isprime(big(3)) true
source
20.5 Primes.isprime
Primes.isprime —Method.
isprime(n::Integer) -> Bool
Returns true if nis prime, and false otherwise.
[] julia¿ isprime(3) true
source
20.6. PRIMES.ISRIESELPRIME 275
20.6 Primes.isrieselprime
Primes.isrieselprime —Method.
isrieselprime(k::Integer, Q::Integer) -> Bool
Lucas-Lehmer-Riesel deterministic test for N of the form N=k*Q, with
0<k<Q,Q=2^n-1and n>0, also known as Riesel primes. Returns
true if Ris prime, and false otherwise or if the combination of k and n is not
supported.
julia> isrieselprime(1, 2^11 - 1) # == ismersenneprime(2^11 - 1)
false
julia> isrieselprime(3, 2^607 - 1)
true
source
20.7 Primes.nextprime
Primes.nextprime —Function.
nextprime(n::Integer, i::Integer=1)
The i-th smallest prime not less than n(in particular, nextprime(p) == p
if pis prime). If i<0, this is equivalent to prevprime(n, -i). Note that for
n::BigInt, the returned number is only a pseudo-prime (the function isprime
is used internally). See also prevprime.
julia> nextprime(4)
5
julia> nextprime(5)
5
julia> nextprime(4, 2)
7
julia> nextprime(5, 2)
7
source
276 CHAPTER 20. PRIMES
20.8 Primes.prevprime
Primes.prevprime —Function.
prevprime(n::Integer, i::Integer=1)
The i-th largest prime not greater than n(in particular prevprime(p) == p
if pis prime). If i<0, this is equivalent to nextprime(n, -i). Note that for
n::BigInt, the returned number is only a pseudo-prime (the function isprime
is used internally). See also nextprime.
julia> prevprime(4)
3
julia> prevprime(5)
5
julia> prevprime(5, 2)
3
source
20.9 Primes.prime
Primes.prime —Method.
prime{T}(::Type{T}=Int, i::Integer)
The i-th prime number.
julia> prime(1)
2
julia> prime(3)
5
source
20.10 Primes.primes
Primes.primes —Method.
primes([lo,] hi)
Returns a collection of the prime numbers (from lo, if specified) up to hi.
source
20.11. PRIMES.PRIMESMASK 277
20.11 Primes.primesmask
Primes.primesmask —Method.
primesmask([lo,] hi)
Returns a prime sieve, as a BitArray, of the positive integers (from lo,
if specified) up to hi. Useful when working with either primes or composite
numbers.
source
20.12 Primes.prodfactors
Primes.prodfactors —Function.
prodfactors(factors)
Compute n(or the radical of nwhen factors is of type Set or IntSet)
where factors is interpreted as the result of factor(typeof(factors), n).
Note that if factors is of type AbstractArray or Primes.Factorization, then
prodfactors is equivalent to Base.prod.
julia> prodfactors(factor(100))
100
source
20.13 Primes.radical
Primes.radical —Method.
radical(n::Integer)
Compute the radical of n, i.e. the largest square-free divisor of n. This is
equal to the product of the distinct prime numbers dividing n.
julia> radical(2*2*3)
6
source
278 CHAPTER 20. PRIMES
20.14 Primes.totient
Primes.totient —Method.
totient(n::Integer) -> Integer
Compute the Euler totient function (n), which counts the number of positive
integers less than or equal to nthat are relatively prime to n(that is, the number
of positive integers m n with gcd(m, n) == 1). The totient function of nwhen
nis negative is defined to be totient(abs(n)).
source
20.15 Primes.totient
Primes.totient —Method.
totient(f::Factorization{T}) -> T
Compute the Euler totient function of the number whose prime factorization
is given by f. This method may be preferable to totient(::Integer) when
the factorization can be reused for other purposes.
source
Chapter 21
Roots
21.1 Roots.D
Roots.D —Function.
Take derivative of order kof a function.
Arguments:
•f::Function: a mathematical function from R to R.
•k::Int=1: A non-negative integer specifying the order of the derivative.
Values larger than 8 can be slow to compute.
Wrapper around derivative function in ForwardDiff
source
21.2 Roots.find zero
Roots.find zero —Method.
Find a zero of a univariate function using one of several different methods.
Positional arugments:
•fa function, callable object, or tuple of same. A tuple is used to pass in
derivatives, as desired. Most methods are derivative free. Some (Newton,
Halley) may have derivative(s) computed using the ForwardDiff pacakge.
•x0 an initial starting value. Typically a scalar, but may be a two-element
tuple or array for bisection methods. The value float.(x0) is passed on.
•method one of several methods, see below.
Keyword arguments:
•xabstol=zero(): declare convergence if |x n - x {n-1}| <= max(xabstol,
max(1, |x n|) * xreltol)
279
280 CHAPTER 21. ROOTS
•xreltol=eps():
•abstol=zero(): declare convergence if |f(x n)|<= max(abstol, max(1,
|x n|) * reltol)
•reltol:
•bracket: Optional. A bracketing interval for the sought after root. If
given, a hybrid algorithm may be used where bisection is utilized for steps
that would go out of bounds. (Using a FalsePosition method instead
would be suggested.)
•maxevals::Int=40: stop trying after maxevals steps
•maxfnevals::Int=typemax(Int): stop trying after maxfnevals function
evaluations
•verbose::Bool=false: If true show information about algorithm and a
trace.
Returns:
Returns xn if the algorithm converges. If the algorithm stops, returns xn if
|f(xn)|ˆ(2/3), where = reltol, otherwise a ConvergenceFailed error is thrown.
Exported methods:
Bisection();Order0() (heuristic, slow more robust); Order1() (also Secant());
Order2() (also Steffensen()); Order5() (KSS); Order8() (Thukral); Order16()
(Thukral); FalsePosition(i) (false position, i in 1..12);
Not exported:
Secant(), use Order1() Steffensen() use Order2() Newton() (use newton()
function) Halley() (use halley() function)
The order 0 method is more robust to the initial starting point, but can
utilize many more function calls. The higher order methods may be of use
when greater precision is desired.‘
Examples:
f(x) = x^5 - x - 1
find_zero(f, 1.0, Order5())
find_zero(f, 1.0, Steffensen()) # also Order2()
find_zero(f, (1.0, 2.0), FalsePosition())
source
21.3 Roots.fzero
Roots.fzero —Method.
Find zero of a function within a bracket
Uses a modified bisection method for non big arguments
Arguments:
21.4. ROOTS.FZERO 281
•fA scalar function or callable object
•aleft endpont of interval
•bright endpont of interval
•xtol optional additional tolerance on sub-bracket size.
For a bracket to be valid, it must be that f(a)*f(b) <0.
For Float64 values, the answer is such that f(prevfloat(x)) * f(nextfloat(x))
<0unless a non-zero value of xtol is specified in which case, it stops when the
sub-bracketing produces an bracket with length less than max(xtol, abs(x1)*xtolrel).
For Big values, which defaults to the algorithm of Alefeld, Potra, and Shi,
a default tolerance is used for the sub-bracket length that can be enlarged with
xtol.
If a==-Inf it is replaced with nextfloat(a); if b==Inf it is replaced with
prevfloat(b).
Example:
‘fzero(sin, 3, 4)‘ # find pi
‘fzero(sin, [big(3), 4]) find pi with more digits
source
21.4 Roots.fzero
Roots.fzero —Method.
Find zero of a function using an iterative algorithm
•f: a scalar function or callable object
•x0: an initial guess, finite valued.
Keyword arguments:
•ftol: tolerance for a guess abs(f(x)) <ftol
•xtol: stop if abs(xold - xnew) <= xtol + max(1, |xnew|)*xtolrel
•xtolrel: see xtol
•maxeval: maximum number of steps
•verbose: Boolean. Set true to trace algorithm
•order: Can specify order of algorithm. 0 is most robust, also 1, 2, 5, 8,
16.
•kwargs... passed on to different algorithms. There are maxfneval when
order is 1,2,5,8, or 16 and beta for orders 2,5,8,16,
This is a polyalgorithm redirecting different algorithms based on the value
of order.
source
282 CHAPTER 21. ROOTS
21.5 Roots.fzero
Roots.fzero —Method.
Find zero using Newton’s method.
source
21.6 Roots.fzero
Roots.fzero —Method.
Find a zero with bracket specified via [a,b], as fzero(sin, [3,4]).
source
21.7 Roots.fzero
Roots.fzero —Method.
Find a zero within a bracket with an initial guess to possibly speed things
along.
source
21.8 Roots.fzeros
Roots.fzeros —Method.
fzeros(f, a, b)
Attempt to find all zeros of fwithin an interval [a,b].
Simple algorithm that splits [a,b] into subintervals and checks each for a
root. For bracketing subintervals, bisection is used. Otherwise, a derivative-
free method is used. If there are a large number of zeros found relative to the
number of subintervals, the number of subintervals is increased and the process
is re-run.
There are possible issues with close-by zeros and zeros which do not cross the
origin (non-simple zeros). Answers should be confirmed graphically, if possible.
source
21.9 Roots.halley
Roots.halley —Method.
Implementation of Halley’s method. xn1 = xn - 2f(xn)*f(xn) / (2*f(xn)^2
- f(xn) * f(xn))
Arguments:
•f::Function – function to find zero of
•fp::Function=D(f) – derivative of f. Defaults to automatic derivative
•fpp:Function=D(f,2) – second derivative of f.
21.10. ROOTS.NEWTON 283
•x0::Real – initial guess
Keyword arguments:
•ftol. Stop iterating when |f(xn)|<= max(1, |xn|) * ftol.
•xtol. Stop iterating when |xn+1 - xn|<= xtol + max(1, |xn|) * xtolrel
•xtolrel. Stop iterating when |xn+1 - xn|<= xtol + max(1, |xn|) * xtolrel
•maxeval. Stop iterating if more than this many steps, throw error.
•verbose::Bool=false Set to true to see trace.
source
21.10 Roots.newton
Roots.newton —Method.
Implementation of Newton’s method: x n1 = x n - f(x n)/ f(x n)
Arguments:
•f::Function – function to find zero of
•fp::Function=D(f) – derivative of f. Defaults to automatic derivative
•x0::Number – initial guess. For Newton’s method this may be complex.
Keyword arguments:
•ftol. Stop iterating when |f(xn)|<= max(1, |xn|) * ftol.
•xtol. Stop iterating when |xn+1 - xn|<= xtol + max(1, |xn|) * xtolrel
•xtolrel. Stop iterating when |xn+1 - xn|<= xtol + max(1, |xn|) * xtolrel
•maxeval. Stop iterating if more than this many steps, throw error.
•maxfneval. Stop iterating if more than this many function calls, throw
error.
•verbose::Bool=false Set to true to see trace.
source
21.11 Roots.secant method
Roots.secant method —Method.
secant_method(f, x0, x1; [kwargs...])
Solve for zero of f(x) = 0 using the secant method.
Not exported. Use find zero with Order1().
source
Chapter 22
ImageTransformations
22.1 ImageTransformations.imresize
ImageTransformations.imresize —Method.
imresize(img, sz) -> imgr
imresize(img, inds) -> imgr
Change img to be of size sz (or to have indices inds). This interpolates the
values at sub-pixel locations. If you are shrinking the image, you risk aliasing
unless you low-pass filter img first. For example:
= map((o,n)->0.75*o/n, size(img), sz)
kern = KernelFactors.gaussian() # from ImageFiltering
imgr = imresize(imfilter(img, kern, NA()), sz)
See also restrict.
source
22.2 ImageTransformations.invwarpedview
ImageTransformations.invwarpedview —Method.
invwarpedview(img, tinv, [indices], [degree = Linear()], [fill = NaN]) -> wv
Create a view of img that lazily transforms any given index Ipassed to
wv[I] to correspond to img[inv(tinv)(I)]. While technically this approach
is known as backward mode warping, note that InvWarpedView is created by
supplying the forward transformation. The given transformation tinv must
accept a SVector as input and support inv(tinv). A useful package to create
a wide variety of such transformations is CoordinateTransformations.jl.
When invoking wv[I], values for img must be reconstructed at arbitrary lo-
cations inv(tinv)(I).InvWarpedView serves as a wrapper around WarpedView
284
22.3. IMAGETRANSFORMATIONS.RESTRICT 285
which takes care of interpolation and extrapolation. The parameters degree and
fill can be used to specify the b-spline degree and the extrapolation scheme
respectively.
The optional parameter indices can be used to specify the domain of the
resulting wv. By default the indices are computed in such a way that wv contains
all the original pixels in img.
source
22.3 ImageTransformations.restrict
ImageTransformations.restrict —Method.
restrict(img[, region]) -> imgr
Reduce the size of img by two-fold along the dimensions listed in region, or
all spatial coordinates if region is not specified. It anti-aliases the image as it
goes, so is better than a naive summation over 2x2 blocks.
See also imresize.
source
22.4 ImageTransformations.warp
ImageTransformations.warp —Method.
warp(img, tform, [indices], [degree = Linear()], [fill = NaN]) -> imgw
Transform the coordinates of img, returning a new imgw satisfying imgw[I]
= img[tform(I)]. This approach is known as backward mode warping. The
transformation tform must accept a SVector as input. A useful package to
create a wide variety of such transformations is CoordinateTransformations.jl.
Reconstruction scheme
During warping, values for img must be reconstructed at arbitrary locations
tform(I) which do not lie on to the lattice of pixels. How this reconstruction
is done depends on the type of img and the optional parameter degree.
When img is a plain array, then on-grid b-spline interpolation will be used.
It is possible to configure what degree of b-spline to use with the parameter
degree. For example one can use degree = Linear() for linear interpola-
tion, degree = Constant() for nearest neighbor interpolation, or degree =
Quadratic(Flat()) for quadratic interpolation.
In the case tform(I) maps to indices outside the original img, those locations
are set to a value fill (which defaults to NaN if the element type supports it,
and 0otherwise). The parameter fill also accepts extrapolation schemes, such
as Flat(),Periodic() or Reflect().
For more control over the reconstruction scheme — and how beyond-the-edge
points are handled — pass img as an AbstractInterpolation or AbstractExtrapolation
from Interpolations.jl.
286 CHAPTER 22. IMAGETRANSFORMATIONS
The meaning of the coordinates
The output array imgw has indices that would result from applying inv(tform)
to the indices of img. This can be very handy for keeping track of how pixels
in imgw line up with pixels in img.
If you just want a plain array, you can “strip” the custom indices with
parent(imgw).
Examples: a 2d rotation (see JuliaImages documentation for pic-
tures)
julia> using Images, CoordinateTransformations, TestImages, OffsetArrays
julia> img = testimage("lighthouse");
julia> indices(img)
(Base.OneTo(512),Base.OneTo(768))
# Rotate around the center of ‘img‘
julia> tfm = recenter(RotMatrix(-pi/4), center(img))
AffineMap([0.707107 0.707107; -0.707107 0.707107], [-196.755,293.99])
julia> imgw = warp(img, tfm);
julia> indices(imgw)
(-196:709,-68:837)
# Alternatively, specify the origin in the image itself
julia> img0 = OffsetArray(img, -30:481, -384:383); # origin near top of image
julia> rot = LinearMap(RotMatrix(-pi/4))
LinearMap([0.707107 -0.707107; 0.707107 0.707107])
julia> imgw = warp(img0, rot);
julia> indices(imgw)
(-293:612,-293:611)
julia> imgr = parent(imgw);
julia> indices(imgr)
(Base.OneTo(906),Base.OneTo(905))
source
22.5 ImageTransformations.warpedview
ImageTransformations.warpedview —Method.
22.5. IMAGETRANSFORMATIONS.WARPEDVIEW 287
warpedview(img, tform, [indices], [degree = Linear()], [fill = NaN]) -> wv
Create a view of img that lazily transforms any given index Ipassed to
wv[I] to correspond to img[tform(I)]. This approach is known as backward
mode warping. The given transformation tform must accept a SVector as
input. A useful package to create a wide variety of such transformations is
CoordinateTransformations.jl.
When invoking wv[I], values for img must be reconstructed at arbitrary
locations tform(I) which do not lie on to the lattice of pixels. How this re-
construction is done depends on the type of img and the optional parameter
degree. When img is a plain array, then on-grid b-spline interpolation will
be used, where the pixel of img will serve as the coeficients. It is possible to
configure what degree of b-spline to use with the parameter degree. The two
possible values are degree = Linear() for linear interpolation, or degree =
Constant() for nearest neighbor interpolation.
In the case tform(I) maps to indices outside the domain of img, those
locations are set to a value fill (which defaults to NaN if the element type
supports it, and 0otherwise). Additionally, the parameter fill also accepts
extrapolation schemes, such as Flat(),Periodic() or Reflect().
The optional parameter indices can be used to specify the domain of the
resulting WarpedView. By default the indices are computed in such a way that
the resulting WarpedView contains all the original pixels in img. To do this
inv(tform) has to be computed. If the given transformation tform does not
support inv, then the parameter indices has to be specified manually.
warpedview is essentially a non-coping, lazy version of warp. As such,
the two functions share the same interface, with one important difference.
warpedview will insist that the resulting WarpedView will be a view of img
(i.e. parent(warpedview(img, ...)) === img). Consequently, warpedview
restricts the parameter degree to be either Linear() or Constant().
source
Chapter 23
PyCall
23.1 PyCall.PyNULL
PyCall.PyNULL —Method.
PyNULL()
Return a PyObject that has a NULL underlying pointer, i.e. it doesn’t actually
refer to any Python object.
This is useful for initializing PyObject global variables and array elements
before an actual Python object is available. For example, you might do const
myglobal = PyNULL() and later on (e.g. in a module init function), reas-
sign myglobal to point to an actual object with copy!(myglobal, someobject).
This procedure will properly handle Python’s reference counting (so that the
Python object will not be freed until you are done with myglobal).
source
23.2 PyCall.PyReverseDims
PyCall.PyReverseDims —Method.
PyReverseDims(array)
Passes a Julia array to Python as a NumPy row-major array (rather than
Julia’s native column-major order) with the dimensions reversed (e.g. a 234 Julia
array is passed as a 432 NumPy row-major array). This is useful for Python
libraries that expect row-major data.
source
23.3 PyCall.PyTextIO
PyCall.PyTextIO —Method.
288
23.4. PYCALL.PYBUILTIN 289
PyTextIO(io::IO)
PyObject(io::IO)
Julia IO streams are converted into Python objects implementing the Raw-
IOBase interface, so they can be used for binary I/O in Python
source
23.4 PyCall.pybuiltin
PyCall.pybuiltin —Method.
pybuiltin(s::AbstractString)
Look up a string or symbol samong the global Python builtins. If sis
a string it returns a PyObject, while if sis a symbol it returns the builtin
converted to PyAny.
source
23.5 PyCall.pybytes
PyCall.pybytes —Method.
pybytes(b::Union{String,Vector{UInt8}})
Convert bto a Python bytes object. This differs from the default PyObject(b)
conversion of String to a Python string (which may fail if bdoes not con-
tain valid Unicode), or from the default conversion of a Vector{UInt8}to a
bytearray object (which is mutable, unlike bytes).
source
23.6 PyCall.pycall
PyCall.pycall —Method.
pycall(o::Union{PyObject,PyPtr}, returntype::TypeTuple, args...; kwargs...)
Call the given Python function (typically looked up from a module) with
the given args. . . (of standard Julia types which are converted automatically
to the corresponding Python types if possible), converting the return value to
returntype (use a returntype of PyObject to return the unconverted Python
object reference, or of PyAny to request an automated conversion)
source
290 CHAPTER 23. PYCALL
23.7 PyCall.pyeval
PyCall.pyeval —Function.
pyeval(s::AbstractString, returntype::TypeTuple=PyAny, locals=PyDict{AbstractString, PyObject}(),
input_type=Py_eval_input; kwargs...)
This evaluates sas a Python string and returns the result converted to rtype
(which defaults to PyAny). The remaining arguments are keywords that define
local variables to be used in the expression.
For example, pyeval("x + y", x=1, y=2) returns 3.
source
23.8 PyCall.pygui
PyCall.pygui —Method.
pygui()
Return the current GUI toolkit as a symbol.
source
23.9 PyCall.pygui start
PyCall.pygui start —Function.
pygui_start(gui::Symbol = pygui())
Start the event loop of a certain toolkit.
The argument gui defaults to the current default GUI, but it could be :wx,
:gtk,:gtk3,:tk, or :qt.
source
23.10 PyCall.pygui stop
PyCall.pygui stop —Function.
pygui_stop(gui::Symbol = pygui())
Stop any running event loop for gui. The gui argument defaults to current
default GUI.
source
23.11. PYCALL.PYIMPORT 291
23.11 PyCall.pyimport
PyCall.pyimport —Method.
pyimport(s::AbstractString)
Import the Python module s(a string or symbol) and return a pointer to it
(a PyObject). Functions or other symbols in the module may then be looked
up by s[name] where name is a string (for the raw PyObject) or symbol (for
automatic type-conversion). Unlike the @pyimport macro, this does not define
a Julia module and members cannot be accessed with s.name
source
23.12 PyCall.pyimport conda
PyCall.pyimport conda —Function.
pyimport_conda(modulename, condapkg, [channel])
Returns the result of pyimport(modulename) if possible. If the module is not
found, and PyCall is configured to use the Conda Python distro (via the Julia
Conda package), then automatically install condapkg via Conda.add(condapkg)
and then re-try the pyimport. Other Anaconda-based Python installations are
also supported as long as their conda program is functioning.
If PyCall is not using Conda and the pyimport fails, throws an exception
with an error message telling the user how to configure PyCall to use Conda for
automated installation of the module.
The third argument, channel is an optional Anaconda “channel” to use for
installing the package; this is useful for packages that are not included in the
default Anaconda package listing.
source
23.13 PyCall.pytype mapping
PyCall.pytype mapping —Method.
pytype_mapping(pytype, jltype)
Given a Python type object pytype, tell PyCall to convert it to jltype in
PyAny(object) conversions.
source
292 CHAPTER 23. PYCALL
23.14 PyCall.pytype query
PyCall.pytype query —Function.
pytype_query(o::PyObject, default=PyObject)
Given a Python object o, return the corresponding native Julia type (de-
faulting to default) that we convert oto in PyAny(o) conversions.
source
23.15 PyCall.pywrap
PyCall.pywrap —Function.
pywrap(o::PyObject)
This returns a wrapper wthat is an anonymous module which provides (read)
access to converted versions of o’s members as w.member.
For example, @pyimport module as name is equivalent to const name =
pywrap(pyimport("module"))
If the Python module contains identifiers that are reserved words in Julia
(e.g. function), they cannot be accessed as w.member; one must instead use
w.pymember(:member) (for the PyAny conversion) or w.pymember(“member”)
(for the raw PyObject).
source
Chapter 24
Gadfly
24.1 Compose.draw
Compose.draw —Method.
draw(backend::Compose.Backend, p::Plot)
A convenience version of Compose.draw without having to call render
Args
•backend: The Compose.Backend object
•p: The Plot object
source
24.2 Compose.hstack
Compose.hstack —Method.
hstack(ps::Union{Plot,Context}...)
hstack(ps::Vector)
Arrange plots into a horizontal row. Use context() as a placeholder for
an empty panel. Heterogeneous vectors must be typed. See also vstack,
gridstack,subplot grid.
Examples
“‘ p1 = plot(x=[1,2], y=[3,4], Geom.line); p2 = Compose.context(); hstack(p1,
p2) hstack(Union{Plot,Compose.Context}[p1, p2])
source
293
294 CHAPTER 24. GADFLY
24.3 Compose.vstack
Compose.vstack —Method.
vstack(ps::Union{Plot,Context}...)
vstack(ps::Vector)
Arrange plots into a vertical column. Use context() as a placeholder for
an empty panel. Heterogeneous vectors must be typed. See also hstack,
gridstack,subplot grid.
Examples
“‘ p1 = plot(x=[1,2], y=[3,4], Geom.line); p2 = Compose.context(); vs-
tack(p1, p2) vstack(Union{Plot,Compose.Context}[p1, p2])
source
24.4 Gadfly.layer
Gadfly.layer —Method.
layer(data_source::@compat(Union{AbstractDataFrame, (@compat Void)}),
elements::ElementOrFunction...; mapping...)
Creates layers based on elements
Args
•data source: The data source as a dataframe
•elements: The elements
•mapping: mapping
Returns
An array of layers
source
24.5 Gadfly.plot
Gadfly.plot —Method.
function plot(data_source::@compat(Union{(@compat Void), AbstractMatrix, AbstractDataFrame}),
mapping::Dict, elements::ElementOrFunctionOrLayers...)
The old fashioned (pre named arguments) version of plot.
This version takes an explicit mapping dictionary, mapping aesthetics sym-
bols to expressions or columns in the data frame.
Args:
•data source: Data to be bound to aesthetics.
24.6. GADFLY.PLOT 295
•mapping: Dictionary of aesthetics symbols (e.g. :x, :y, :color) to names of
columns in the data frame or other expressions.
•elements: Geometries, statistics, etc.
Returns:
A Plot object.
source
24.6 Gadfly.plot
Gadfly.plot —Method.
function plot(data_source::@compat(Union{AbstractMatrix, AbstractDataFrame}),
elements::ElementOrFunctionOrLayers...; mapping...)
Create a new plot.
Grammar of graphics style plotting consists of specifying a dataset, one
or more plot elements (scales, coordinates, geometries, etc), and binding of
aesthetics to columns or expressions of the dataset.
For example, a simple scatter plot would look something like:
plot(my data, Geom.point, x=“time”, y=“price”)
Where “time” and “price” are the names of columns in my data.
Args:
•data source: Data to be bound to aesthetics.
•elements: Geometries, statistics, etc.
•mapping: Aesthetics symbols (e.g. :x, :y, :color) mapped to names of
columns in the data frame or other expressions.
source
24.7 Gadfly.render
Gadfly.render —Method.
render(plot::Plot)
Render a plot based on the Plot object
Args
•plot: Plot to be rendered.
Returns
A Compose context containing the rendered plot.
source
296 CHAPTER 24. GADFLY
24.8 Gadfly.set default plot format
Gadfly.set default plot format —Method.
set_default_plot_format(fmt::Symbol)
Sets the default plot format
source
24.9 Gadfly.set default plot size
Gadfly.set default plot size —Method.
set_default_plot_size(width::Compose.MeasureOrNumber, height::Compose.MeasureOrNumber)
Sets preferred canvas size when rendering a plot without an explicit call to
draw
source
24.10 Gadfly.spy
Gadfly.spy —Method.
spy(M::AbstractMatrix, elements::ElementOrFunction...; mapping...)
Simple heatmap plots of matrices.
It is a wrapper around the plot() function using the rectbin geometry.
It also applies a sane set of defaults to make sure that the plots look nice by
default. Specifically
•the aspect ratio of the coordinate system is fixed Coord.cartesian(fixed=true),
so that the rectangles become squares
•the axes run from 0.5 to N+0.5, because the first row/column is drawn to
(0.5, 1.5) and the last one to (N-0.5, N+0.5).
•the y-direction is flipped, so that the [1,1] of a matrix is in the top
left corner, as is customary
•NaNs are not drawn. spy leaves “holes” instead into the heatmap.
Args:
•M: A matrix.
24.11. GADFLY.STYLE 297
Returns:
A plot object.
Known bugs:
•If the matrix is only NaNs, then it throws an ArgumentError, because an
empty collection gets passed to the plot function / rectbin geometry.
source
24.11 Gadfly.style
Gadfly.style —Method.
Set some attributes in the current Theme. See Theme for available field.
source
24.12 Gadfly.title
Gadfly.title —Method.
title(ctx::Context, str::String, props::Property...) -> Context
Add a title string to a group of plots, typically created with vstack,hstack,
or gridstack.
Examples
p1 = plot(x=[1,2], y=[3,4], Geom.line);
p2 = plot(x=[1,2], y=[4,3], Geom.line);
title(hstack(p1,p2), "my latest data", Compose.fontsize(18pt), fill(colorant"red"))
source
Chapter 25
IterTools
25.1 IterTools.chain
IterTools.chain —Method.
chain(xs...)
Iterate through any number of iterators in sequence.
julia> for i in chain(1:3, [’a’, ’b’, ’c’])
@show i
end
i=1
i=2
i=3
i = ’a’
i = ’b’
i = ’c’
source
25.2 IterTools.distinct
IterTools.distinct —Method.
distinct(xs)
Iterate through values skipping over those already encountered.
julia> for i in distinct([1,1,2,1,2,4,1,2,3,4])
@show i
end
i=1
298
25.3. ITERTOOLS.GROUPBY 299
i=2
i=4
i=3
source
25.3 IterTools.groupby
IterTools.groupby —Method.
groupby(f, xs)
Group consecutive values that share the same result of applying f.
julia> for i in groupby(x -> x[1], ["face", "foo", "bar", "book", "baz", "zzz"])
@show i
end
i = String["face","foo"]
i = String["bar","book","baz"]
i = String["zzz"]
source
25.4 IterTools.imap
IterTools.imap —Method.
imap(f, xs1, [xs2, ...])
Iterate over values of a function applied to successive values from one or
more iterators.
julia> for i in imap(+, [1,2,3], [4,5,6])
@show i
end
i=5
i=7
i=9
source
25.5 IterTools.iterate
IterTools.iterate —Method.
iterate(f, x)
300 CHAPTER 25. ITERTOOLS
Iterate over successive applications of f, as in x,f(x),f(f(x)),f(f(f(x))),
. . .
Use Base.take() to obtain the required number of elements.
julia> for i in take(iterate(x -> 2x, 1), 5)
@show i
end
i=1
i=2
i=4
i=8
i = 16
julia> for i in take(iterate(sqrt, 100), 6)
@show i
end
i = 100
i = 10.0
i = 3.1622776601683795
i = 1.7782794100389228
i = 1.333521432163324
i = 1.1547819846894583
source
25.6 IterTools.ncycle
IterTools.ncycle —Method.
ncycle(xs, n)
Cycle through iter n times.
julia> for i in ncycle(1:3, 2)
@show i
end
i=1
i=2
i=3
i=1
i=2
i=3
source
25.7. ITERTOOLS.NTH 301
25.7 IterTools.nth
IterTools.nth —Method.
nth(xs, n)
Return the nth element of xs. This is mostly useful for non-indexable col-
lections.
julia> mersenne = Set([3, 7, 31, 127])
Set([7,31,3,127])
julia> nth(mersenne, 3)
3
source
25.8 IterTools.partition
IterTools.partition —Method.
partition(xs, n, [step])
Group values into n-tuples.
julia> for i in partition(1:9, 3)
@show i
end
i = (1,2,3)
i = (4,5,6)
i = (7,8,9)
If the step parameter is set, each tuple is separated by step values.
julia> for i in partition(1:9, 3, 2)
@show i
end
i = (1,2,3)
i = (3,4,5)
i = (5,6,7)
i = (7,8,9)
julia> for i in partition(1:9, 3, 3)
@show i
end
i = (1,2,3)
i = (4,5,6)
302 CHAPTER 25. ITERTOOLS
i = (7,8,9)
julia> for i in partition(1:9, 2, 3)
@show i
end
i = (1,2)
i = (4,5)
i = (7,8)
source
25.9 IterTools.peekiter
IterTools.peekiter —Method.
peekiter(xs)
Lets you peek at the head element of an iterator without updating the state.
julia> it = peekiter(["face", "foo", "bar", "book", "baz", "zzz"])
IterTools.PeekIter{Array{String,1}}(String["face","foo","bar","book","baz","zzz"])
julia> s = start(it)
(2,Nullable{String}("face"))
julia> @show peek(it, s)
peek(it,s) = Nullable{String}("face")
Nullable{String}("face")
julia> @show peek(it, s)
peek(it,s) = Nullable{String}("face")
Nullable{String}("face")
julia> x, s = next(it, s)
("face",(3,Nullable{String}("foo"),false))
julia> @show x
x = "face"
"face"
julia> @show peek(it, s)
peek(it,s) = Nullable{String}("foo")
Nullable{String}("foo")
source
25.10. ITERTOOLS.PRODUCT 303
25.10 IterTools.product
IterTools.product —Method.
product(xs...)
Iterate over all combinations in the Cartesian product of the inputs.
julia> for p in product(1:3,4:5)
@show p
end
p = (1,4)
p = (2,4)
p = (3,4)
p = (1,5)
p = (2,5)
p = (3,5)
source
25.11 IterTools.repeatedly
IterTools.repeatedly —Method.
repeatedly(f, n)
Call function f n times, or infinitely if nis omitted.
[] julia¿ t() = (sleep(0.1); Dates.millisecond(now())) t (generic function with
1 method)
julia¿ collect(repeatedly(t, 5)) 5-element Array{Any,1}: 993 97 200 303 408
source
25.12 IterTools.subsets
IterTools.subsets —Method.
subsets(xs)
subsets(xs, k)
Iterate over every subset of the collection xs. You can restrict the subsets
to a specific size k.
julia> for i in subsets([1, 2, 3])
@show i
end
i = Int64[]
i = [1]
304 CHAPTER 25. ITERTOOLS
i = [2]
i = [1,2]
i = [3]
i = [1,3]
i = [2,3]
i = [1,2,3]
julia> for i in subsets(1:4, 2)
@show i
end
i = [1,2]
i = [1,3]
i = [1,4]
i = [2,3]
i = [2,4]
i = [3,4]
source
25.13 IterTools.takenth
IterTools.takenth —Method.
takenth(xs, n)
Iterate through every nth element of xs.
julia> collect(takenth(5:15,3))
3-element Array{Int64,1}:
7
10
13
source
25.14 IterTools.takestrict
IterTools.takestrict —Method.
takestrict(xs, n::Int)
Like take(), an iterator that generates at most the first nelements of xs,
but throws an exception if fewer than nitems are encountered in xs.
25.14. ITERTOOLS.TAKESTRICT 305
julia> a = :1:2:11
1:2:11
julia> collect(takestrict(a, 3))
3-element Array{Int64,1}:
1
3
5
source
Chapter 26
Iterators
26.1 Iterators.chain
Iterators.chain —Method.
chain(xs...)
Iterate through any number of iterators in sequence.
julia> for i in chain(1:3, [’a’, ’b’, ’c’])
@show i
end
i=1
i=2
i=3
i = ’a’
i = ’b’
i = ’c’
source
26.2 Iterators.distinct
Iterators.distinct —Method.
distinct(xs)
Iterate through values skipping over those already encountered.
julia> for i in distinct([1,1,2,1,2,4,1,2,3,4])
@show i
end
i=1
306
26.3. ITERATORS.GROUPBY 307
i=2
i=4
i=3
source
26.3 Iterators.groupby
Iterators.groupby —Method.
groupby(f, xs)
Group consecutive values that share the same result of applying f.
julia> for i in groupby(x -> x[1], ["face", "foo", "bar", "book", "baz", "zzz"])
@show i
end
i = String["face","foo"]
i = String["bar","book","baz"]
i = String["zzz"]
source
26.4 Iterators.imap
Iterators.imap —Method.
imap(f, xs1, [xs2, ...])
Iterate over values of a function applied to successive values from one or
more iterators.
julia> for i in imap(+, [1,2,3], [4,5,6])
@show i
end
i=5
i=7
i=9
source
26.5 Iterators.iterate
Iterators.iterate —Method.
iterate(f, x)
308 CHAPTER 26. ITERATORS
Iterate over successive applications of f, as in f(x),f(f(x)),f(f(f(x))),
. . . .
Use Base.take() to obtain the required number of elements.
julia> for i in take(iterate(x -> 2x, 1), 5)
@show i
end
i=1
i=2
i=4
i=8
i = 16
julia> for i in take(iterate(sqrt, 100), 6)
@show i
end
i = 100
i = 10.0
i = 3.1622776601683795
i = 1.7782794100389228
i = 1.333521432163324
i = 1.1547819846894583
source
26.6 Iterators.ncycle
Iterators.ncycle —Method.
ncycle(xs, n)
Cycle through iter n times.
julia> for i in ncycle(1:3, 2)
@show i
end
i=1
i=2
i=3
i=1
i=2
i=3
source
26.7. ITERATORS.NTH 309
26.7 Iterators.nth
Iterators.nth —Method.
nth(xs, n)
Return the nth element of xs. This is mostly useful for non-indexable col-
lections.
julia> mersenne = Set([3, 7, 31, 127])
Set([7,31,3,127])
julia> nth(mersenne, 3)
3
source
26.8 Iterators.partition
Iterators.partition —Method.
partition(xs, n, [step])
Group values into n-tuples.
julia> for i in partition(1:9, 3)
@show i
end
i = (1,2,3)
i = (4,5,6)
i = (7,8,9)
If the step parameter is set, each tuple is separated by step values.
julia> for i in partition(1:9, 3, 2)
@show i
end
i = (1,2,3)
i = (3,4,5)
i = (5,6,7)
i = (7,8,9)
julia> for i in partition(1:9, 3, 3)
@show i
end
i = (1,2,3)
i = (4,5,6)
310 CHAPTER 26. ITERATORS
i = (7,8,9)
julia> for i in partition(1:9, 2, 3)
@show i
end
i = (1,2)
i = (4,5)
i = (7,8)
source
26.9 Iterators.peekiter
Iterators.peekiter —Method.
peekiter(xs)
Lets you peek at the head element of an iterator without updating the state.
julia> it = peekiter(["face", "foo", "bar", "book", "baz", "zzz"])
Iterators.PeekIter{Array{String,1}}(String["face","foo","bar","book","baz","zzz"])
julia> s = start(it)
(2,Nullable{String}("face"))
julia> @show peek(it, s)
peek(it,s) = Nullable{String}("face")
Nullable{String}("face")
julia> @show peek(it, s)
peek(it,s) = Nullable{String}("face")
Nullable{String}("face")
julia> x, s = next(it, s)
("face",(3,Nullable{String}("foo"),false))
julia> @show x
x = "face"
"face"
julia> @show peek(it, s)
peek(it,s) = Nullable{String}("foo")
Nullable{String}("foo")
source
26.10. ITERATORS.PRODUCT 311
26.10 Iterators.product
Iterators.product —Method.
product(xs...)
Iterate over all combinations in the Cartesian product of the inputs.
julia> for p in product(1:3,4:5)
@show p
end
p = (1,4)
p = (2,4)
p = (3,4)
p = (1,5)
p = (2,5)
p = (3,5)
source
26.11 Iterators.repeatedly
Iterators.repeatedly —Method.
repeatedly(f, n)
Call function f n times, or infinitely if nis omitted.
[] julia¿ t() = (sleep(0.1); Dates.millisecond(now())) t (generic function with
1 method)
julia¿ collect(repeatedly(t, 5)) 5-element Array{Any,1}: 993 97 200 303 408
source
26.12 Iterators.subsets
Iterators.subsets —Method.
subsets(xs)
subsets(xs, k)
Iterate over every subset of the collection xs. You can restrict the subsets
to a specific size k.
julia> for i in subsets([1, 2, 3])
@show i
end
i = Int64[]
i = [1]
312 CHAPTER 26. ITERATORS
i = [2]
i = [1,2]
i = [3]
i = [1,3]
i = [2,3]
i = [1,2,3]
julia> for i in subsets(1:4, 2)
@show i
end
i = [1,2]
i = [1,3]
i = [1,4]
i = [2,3]
i = [2,4]
i = [3,4]
source
26.13 Iterators.takenth
Iterators.takenth —Method.
takenth(xs, n)
Iterate through every nth element of xs.
julia> collect(takenth(5:15,3))
3-element Array{Int64,1}:
7
10
13
source
26.14 Iterators.takestrict
Iterators.takestrict —Method.
takestrict(xs, n::Int)
Like take(), an iterator that generates at most the first nelements of xs,
but throws an exception if fewer than nitems are encountered in xs.
26.14. ITERATORS.TAKESTRICT 313
julia> a = :1:2:11
1:2:11
julia> collect(takestrict(a, 3))
3-element Array{Int64,1}:
1
3
5
source
Chapter 27
Polynomials
27.1 Polynomials.coeffs
Polynomials.coeffs —Method.
coeffs(p::Poly)
Return the coefficient vector [a 0, a 1, ..., a n] of a polynomial p.
source
27.2 Polynomials.degree
Polynomials.degree —Method.
degree(p::Poly)
Return the degree of the polynomial p, i.e. the highest exponent in the
polynomial that has a nonzero coefficient.
source
27.3 Polynomials.poly
Polynomials.poly —Method.
poly(r)
Construct a polynomial from its roots. Compare this to the Poly type
constructor, which constructs a polynomial from its coefficients.
If ris a vector, the constructed polynomial is (x−r1)(x−r2)···(x−rn).
If ris a matrix, the constructed polynomial is (x−e1)···(x−en), where eiis
the ith eigenvalue of r.
Examples
314
27.4. POLYNOMIALS.POLYDER 315
[] julia¿ poly([1, 2, 3]) The polynomial (x - 1)(x - 2)(x - 3) Poly(-6 + 11x -
6x2+x3)
julia¿ poly([1 2; 3 4]) The polynomial (x - 5.37228)(x + 0.37228) Poly(-
1.9999999999999998 - 5.0x + 1.0x2)
source
27.4 Polynomials.polyder
Polynomials.polyder —Method.
polyder(p::Poly, k=1)
Compute the kth derivative of the polynomial p.
Examples
[] julia¿ polyder(Poly([1, 3, -1])) Poly(3 - 2x)
julia¿ polyder(Poly([1, 3, -1]), 2) Poly(-2)
source
27.5 Polynomials.polyfit
Polynomials.polyfit —Function.
polyfit(x, y, n=length(x)-1, sym=:x)
Fit a polynomial of degree nthrough the points specified by xand y, where
n<= length(x) - 1, using least squares fit. When n=length(x)-1 (the de-
fault), the interpolating polynomial is returned. The optional fourth argument
can be used to specify the symbol for the returned polynomial.
Examples
[] julia¿ xs = linspace(0, pi, 5);
julia¿ ys = map(sin, xs);
julia¿ polyfit(xs, ys, 2) Poly(-0.004902082150108854 + 1.242031920509868x
- 0.39535103925413095x2)
source
27.6 Polynomials.polyint
Polynomials.polyint —Method.
polyint(p::Poly, a::Number, b::Number)
Compute the definite integral of the polynomial pover the interval [a,b].
Examples
[] julia¿ polyint(Poly([1, 0, -1]), 0, 1) 0.6666666666666667
source
316 CHAPTER 27. POLYNOMIALS
27.7 Polynomials.polyint
Polynomials.polyint —Method.
polyint(p::Poly, k::Number=0)
Integrate the polynomial pterm by term, optionally adding a constant term
k. The order of the resulting polynomial is one higher than the order of p.
Examples
[] julia¿ polyint(Poly([1, 0, -1])) Poly(1.0x - 0.3333333333333333x3)
julia¿ polyint(Poly([1, 0, -1]), 2) Poly(2.0 + 1.0x - 0.3333333333333333x3)
source
27.8 Polynomials.polyval
Polynomials.polyval —Method.
polyval(p::Poly, x::Number)
Evaluate the polynomial pat xusing Horner’s method. Poly objects are
callable, using this function.
Examples
[] julia¿ p = Poly([1, 0, -1]) Poly(1 - x2)
julia¿ polyval(p, 1) 0
julia¿ p(1) 0
source
27.9 Polynomials.printpoly
Polynomials.printpoly —Method.
printpoly(io::IO, p::Poly, mimetype = MIME"text/plain"(); descending_powers=false)
Print a human-readable representation of the polynomial pto io. The MIME
types “text/plain” (default), “text/latex”, and “text/html” are supported. By
default, the terms are in order of ascending powers, matching the order in
coeffs(p); specifying descending powers=true reverses the order.
Examples
julia> printpoly(STDOUT, Poly([1,2,3], :y))
1 + 2*y + 3*y^2
julia> printpoly(STDOUT, Poly([1,2,3], :y), descending_powers=true)
3*y^2 + 2*y + 1
source
27.10. POLYNOMIALS.ROOTS 317
27.10 Polynomials.roots
Polynomials.roots —Method.
roots(p::Poly)
Return the roots (zeros) of p, with multiplicity. The number of roots re-
turned is equal to the order of p. The returned roots may be real or complex.
Examples
[] julia¿ roots(Poly([1, 0, -1])) 2-element Array{Float64,1}: -1.0 1.0
julia¿ roots(Poly([1, 0, 1])) 2-element Array{Complex{Float64},1}: 0.0+1.0im
0.0-1.0im
julia¿ roots(Poly([0, 0, 1])) 2-element Array{Float64,1}: 0.0 0.0
julia¿ roots(poly([1,2,3,4])) 4-element Array{Float64,1}: 4.0 3.0 2.0 1.0
source
27.11 Polynomials.variable
Polynomials.variable —Method.
variable(p::Poly)
variable([T::Type,] var)
variable()
Return the indeterminate of a polynomial, i.e. its variable, as a Poly object.
When passed no arguments, this is equivalent to variable(Float64, :x).
Examples
[] julia¿ variable(Poly([1, 2], :x)) Poly(x)
julia¿ variable(:y) Poly(1.0y)
julia¿ variable() Poly(1.0x)
julia¿ variable(Float32, :x) Poly(1.0f0x)
source
Chapter 28
Colors
28.1 Base.hex
Base.hex —Method.
hex(c)
Print a color as a RGB hex triple, or a transparent paint as an ARGB hex
quadruplet.
source
28.2 Colors.MSC
Colors.MSC —Method.
MSC(h)
MSC(h, l)
Calculates the most saturated color for any given hue hby finding the cor-
responding corner in LCHuv space. Optionally, the lightness lmay also be
specified.
source
28.3 Colors.colordiff
Colors.colordiff —Method.
colordiff(a, b)
colordiff(a, b, metric)
318
28.4. COLORS.COLORMAP 319
Compute an approximate measure of the perceptual difference between col-
ors aand b. Optionally, a metric may be supplied, chosen among DE 2000
(the default), DE 94,DE JPC79,DE CMC,DE BFD,DE AB,DE DIN99,DE DIN99d,
DE DIN99o.
source
28.4 Colors.colormap
Colors.colormap —Function.
colormap(cname, [N; mid, logscale, kvs...])
Returns a predefined sequential or diverging colormap computed using the
algorithm by Wijffelaars, M., et al. (2008). Sequential colormaps cname choices
are Blues,Greens,Grays,Oranges,Purples, and Reds. Diverging colormap
choices are RdBu. Optionally, you can specify the number of colors N(default
100). Keyword arguments include the position of the middle point mid (default
0.5) and the possibility to switch to log scaling with logscale (default false).
source
28.5 Colors.colormatch
Colors.colormatch —Method.
colormatch(wavelength)
colormatch(matchingfunction, wavelength)
Evaluate the CIE standard observer color match function.
Args:
•matchingfunction (optional): a type used to specify the matching func-
tion. Choices include CIE1931 CMF (the default, the CIE 1931 2 match-
ing function), CIE1964 CMF (the CIE 1964 10 color matching function),
CIE1931J CMF (Judd adjustment to CIE1931 CMF), CIE1931JV CMF (Judd-
Vos adjustment to CIE1931 CMF).
•wavelen: Wavelength of stimulus in nanometers.
Returns: XYZ value of perceived color.
source
28.6 Colors.deuteranopic
Colors.deuteranopic —Method.
320 CHAPTER 28. COLORS
deuteranopic(c)
deuteranopic(c, p)
Convert a color to simulate deuteranopic color deficiency (lack of the middle-
wavelength photopigment). See the description of protanopic for detail about
the arguments.
source
28.7 Colors.distinguishable colors
Colors.distinguishable colors —Method.
distinguishable_colors(n, [seed]; [transform, lchoices, cchoices, hchoices])
Generate n maximally distinguishable colors.
This uses a greedy brute-force approach to choose n colors that are maxi-
mally distinguishable. Given seed color(s), and a set of possible hue, chroma,
and lightness values (in LCHab space), it repeatedly chooses the next color as
the one that maximizes the minimum pairwise distance to any of the colors
already in the palette.
Args:
•n: Number of colors to generate.
•seed: Initial color(s) included in the palette. Default is Vector{RGB{N0f8}}(0).
Keyword arguments:
•transform: Transform applied to colors before measuring distance. De-
fault is the identity; other choices include deuteranopic to simulate color-
blindness.
•lchoices: Possible lightness values (default linspace(0,100,15))
•cchoices: Possible chroma values (default linspace(0,100,15))
•hchoices: Possible hue values (default linspace(0,340,20))
Returns: A Vector of colors of length n, of the type specified in seed.
source
28.8 Colors.protanopic
Colors.protanopic —Method.
protanopic(c)
protanopic(c, p)
Convert a color to simulate protanopic color deficiency (lack of the long-
wavelength photopigment). cis the input color; the optional argument pis the
fraction of photopigment loss, in the range 0 (no loss) to 1 (complete loss).
source
28.9. COLORS.TRITANOPIC 321
28.9 Colors.tritanopic
Colors.tritanopic —Method.
tritanopic(c)
tritanopic(c, p)
Convert a color to simulate tritanopic color deficiency (lack of the short-
wavelength photogiment). See protanopic for more detail about the arguments.
source
28.10 Colors.weighted color mean
Colors.weighted color mean —Method.
weighted_color_mean(w1, c1, c2)
Returns the color w1*c1 + (1-w1)*c2 that is the weighted mean of c1 and
c2, where c1 has a weight 0 w1 1.
source
28.11 Colors.whitebalance
Colors.whitebalance —Method.
whitebalance(c, src_white, ref_white)
Whitebalance a color.
Input a source (adopted) and destination (reference) white. E.g., if you have
a photo taken under florencent lighting that you then want to appear correct
under regular sunlight, you might do something like whitebalance(c, WP F2,
WP D65).
Args:
•c: An observed color.
•src white: Adopted or source white corresponding to c
•ref white: Reference or destination white.
Returns: A whitebalanced color.
source
Chapter 29
FileIO
29.1 Base.info
Base.info —Method.
info(fmt) returns the magic bytes/extension information for DataFormat
fmt.
source
29.2 FileIO.add format
FileIO.add format —Method.
add format(fmt, magic, extention) registers a new DataFormat. For ex-
ample:
add_format(format"PNG", (UInt8[0x4d,0x4d,0x00,0x2b], UInt8[0x49,0x49,0x2a,0x00]), [".tiff", ".tif"])
add_format(format"PNG", [0x89,0x50,0x4e,0x47,0x0d,0x0a,0x1a,0x0a], ".png")
add_format(format"NRRD", "NRRD", [".nrrd",".nhdr"])
Note that extensions, magic numbers, and format-identifiers are case-sensitive.
source
29.3 FileIO.add loader
FileIO.add loader —Function.
add loader(fmt, :Package) triggers using Package before loading format
fmt
source
322
29.4. FILEIO.ADD SAVER 323
29.4 FileIO.add saver
FileIO.add saver —Function.
add saver(fmt, :Package) triggers using Package before saving format
fmt
source
29.5 FileIO.del format
FileIO.del format —Method.
del format(fmt::DataFormat) deletes fmt from the format registry.
source
29.6 FileIO.file extension
FileIO.file extension —Method.
file extension(file) returns the file extension associated with File file.
source
29.7 FileIO.file extension
FileIO.file extension —Method.
file extension(file) returns a nullable-string for the file extension asso-
ciated with Stream stream.
source
29.8 FileIO.filename
FileIO.filename —Method.
filename(file) returns the filename associated with File file.
source
29.9 FileIO.filename
FileIO.filename —Method.
filename(stream) returns a nullable-string of the filename associated with
Stream stream.
source
324 CHAPTER 29. FILEIO
29.10 FileIO.load
FileIO.load —Method.
•load(filename) loads the contents of a formatted file, trying to infer
the format from filename and/or magic bytes in the file.
•load(strm) loads from an IOStream or similar object. In this case,
the magic bytes are essential.
•load(File(format"PNG",filename)) specifies the format directly, and
bypasses inference.
•load(f; options...) passes keyword arguments on to the loader.
source
29.11 FileIO.magic
FileIO.magic —Method.
magic(fmt) returns the magic bytes of format fmt
source
29.12 FileIO.query
FileIO.query —Method.
query(filename) returns a File object with information about the format
inferred from the file’s extension and/or magic bytes.
source
29.13 FileIO.query
FileIO.query —Method.
query(io, [filename]) returns a Stream object with information about
the format inferred from the magic bytes.
source
29.14 FileIO.save
FileIO.save —Method.
•save(filename, data...) saves the contents of a formatted file,
trying to infer the format from filename.
29.15. FILEIO.SKIPMAGIC 325
•save(Stream(format"PNG",io), data...) specifies the format directly,
and bypasses inference.
•save(f, data...; options...) passes keyword arguments on to the
saver.
source
29.15 FileIO.skipmagic
FileIO.skipmagic —Method.
skipmagic(s) sets the position of Stream s to be just after the magic bytes.
For a plain IO object, you can use skipmagic(io, fmt).
source
29.16 FileIO.stream
FileIO.stream —Method.
stream(s) returns the stream associated with Stream s
source
29.17 FileIO.unknown
FileIO.unknown —Method.
unknown(f) returns true if the format of fis unknown.
source
Chapter 30
Interact
30.1 Interact.button
Interact.button —Method.
button(label; value=nothing, signal)
Create a push button. Optionally specify the label, the value emitted
when then button is clicked, and/or the (Reactive.jl) signal coupled to this
button.
source
30.2 Interact.checkbox
Interact.checkbox —Method.
checkbox(value=false; label="", signal)
Provide a checkbox with the specified starting (boolean) value. Optional
provide a label for this widget and/or the (Reactive.jl) signal coupled to this
widget.
source
30.3 Interact.dropdown
Interact.dropdown —Method.
dropdown(choices; label="", value, typ, icons, tooltips, signal)
Create a “dropdown” widget. choices can be a vector of options. Option-
ally specify the starting value (defaults to the first choice), the typ of elements
326
30.4. INTERACT.RADIOBUTTONS 327
in choices, supply custom icons, provide tooltips, and/or specify the (Re-
active.jl) signal coupled to this widget.
Examples
a = dropdown(["one", "two", "three"])
To link a callback to the dropdown, use
f = dropdown(["turn red"=>colorize_red, "turn green"=>colorize_green])
map(g->g(image), signal(f))
source
30.4 Interact.radiobuttons
Interact.radiobuttons —Method.
radiobuttons: see the help for dropdown
source
30.5 Interact.selection
Interact.selection —Method.
selection: see the help for dropdown
source
30.6 Interact.selection slider
Interact.selection slider —Method.
selection slider: see the help for dropdown If the slider has numeric (<:Real)
values, and its signal is updated, it will update to the nearest value from the
range/choices provided. To disable this behaviour, so that the widget state will
only update if an exact match for signal value is found in the range/choice, use
syncnearest=false.
source
30.7 Interact.set!
Interact.set! —Method.
set!(w::Widget, fld::Symbol, val)
Set the value of a widget property and update all displayed instances of the
widget. If val is a Signal, then updates to that signal will be reflected in
widget instances/views.
If fld is :value,val is also push!ed to signal(w)
source
328 CHAPTER 30. INTERACT
30.8 Interact.slider
Interact.slider —Method.
slider(range; value, signal, label="", readout=true, continuous_update=true)
Create a slider widget with the specified range. Optionally specify the
starting value (defaults to the median of range), provide the (Reactive.jl)
signal coupled to this slider, and/or specify a string label for the widget.
source
30.9 Interact.textarea
Interact.textarea —Method.
textarea(value=""; label="", signal)
Creates an extended text-entry area. Optionally provide a label and/or the
(Reactive.jl) signal associated with this widget. The signal updates when you
type.
source
30.10 Interact.textbox
Interact.textbox —Method.
textbox(value=""; label="", typ=typeof(value), range=nothing, signal)
Create a box for entering text. value is the starting value; if you don’t want
to provide an initial value, you can constrain the type with typ. Optionally pro-
vide a label, specify the allowed range (e.g., -10.0:10.0) for numeric entries,
and/or provide the (Reactive.jl) signal coupled to this text box.
source
30.11 Interact.togglebutton
Interact.togglebutton —Method.
togglebutton(label=""; value=false, signal)
Create a toggle button. Optionally specify the label, the initial state
(value=false is off, value=true is on), and/or provide the (Reactive.jl) signal
coupled to this button.
source
30.12. INTERACT.TOGGLEBUTTONS 329
30.12 Interact.togglebuttons
Interact.togglebuttons —Method.
togglebuttons: see the help for dropdown
source
30.13 Interact.vselection slider
Interact.vselection slider —Method.
vselection slider(args...; kwargs...)
Shorthand for selection slider(args...; orientation="vertical", kwargs...)
source
30.14 Interact.vslider
Interact.vslider —Method.
vslider(args...; kwargs...)
Shorthand for slider(args...; orientation="vertical", kwargs...)
source
Chapter 31
FFTW
31.1 Base.DFT.FFTW.dct
Base.DFT.FFTW.dct —Function.
dct(A [, dims])
Performs a multidimensional type-II discrete cosine transform (DCT) of the
array A, using the unitary normalization of the DCT. The optional dims argu-
ment specifies an iterable subset of dimensions (e.g. an integer, range, tuple, or
array) to transform along. Most efficient if the size of Aalong the transformed
dimensions is a product of small primes; see nextprod. See also plan dct for
even greater efficiency.
source
31.2 Base.DFT.FFTW.dct!
Base.DFT.FFTW.dct! —Function.
dct!(A [, dims])
Same as dct!, except that it operates in-place on A, which must be an array
of real or complex floating-point values.
source
31.3 Base.DFT.FFTW.idct
Base.DFT.FFTW.idct —Function.
idct(A [, dims])
330
31.4. BASE.DFT.FFTW.IDCT! 331
Computes the multidimensional inverse discrete cosine transform (DCT) of
the array A(technically, a type-III DCT with the unitary normalization). The
optional dims argument specifies an iterable subset of dimensions (e.g. an inte-
ger, range, tuple, or array) to transform along. Most efficient if the size of A
along the transformed dimensions is a product of small primes; see nextprod.
See also plan idct for even greater efficiency.
source
31.4 Base.DFT.FFTW.idct!
Base.DFT.FFTW.idct! —Function.
idct!(A [, dims])
Same as idct!, but operates in-place on A.
source
31.5 Base.DFT.FFTW.plan dct
Base.DFT.FFTW.plan dct —Function.
plan_dct(A [, dims [, flags [, timelimit]]])
Pre-plan an optimized discrete cosine transform (DCT), similar to plan fft
except producing a function that computes dct. The first two arguments have
the same meaning as for dct.
source
31.6 Base.DFT.FFTW.plan dct!
Base.DFT.FFTW.plan dct! —Function.
plan_dct!(A [, dims [, flags [, timelimit]]])
Same as plan dct, but operates in-place on A.
source
31.7 Base.DFT.FFTW.plan idct
Base.DFT.FFTW.plan idct —Function.
plan_idct(A [, dims [, flags [, timelimit]]])
Pre-plan an optimized inverse discrete cosine transform (DCT), similar to
plan fft except producing a function that computes idct. The first two argu-
ments have the same meaning as for idct.
source
332 CHAPTER 31. FFTW
31.8 Base.DFT.FFTW.plan idct!
Base.DFT.FFTW.plan idct! —Function.
plan_idct!(A [, dims [, flags [, timelimit]]])
Same as plan idct, but operates in-place on A.
source
31.9 Base.DFT.FFTW.plan r2r
Base.DFT.FFTW.plan r2r —Function.
plan_r2r(A, kind [, dims [, flags [, timelimit]]])
Pre-plan an optimized r2r transform, similar to plan fft except that the
transforms (and the first three arguments) correspond to r2r and r2r!, respec-
tively.
source
31.10 Base.DFT.FFTW.plan r2r!
Base.DFT.FFTW.plan r2r! —Function.
plan_r2r!(A, kind [, dims [, flags [, timelimit]]])
Similar to plan fft, but corresponds to r2r!.
source
31.11 Base.DFT.FFTW.r2r
Base.DFT.FFTW.r2r —Function.
r2r(A, kind [, dims])
Performs a multidimensional real-input/real-output (r2r) transform of type
kind of the array A, as defined in the FFTW manual. kind specifies either
a discrete cosine transform of various types (FFTW.REDFT00,FFTW.REDFT01,
FFTW.REDFT10, or FFTW.REDFT11), a discrete sine transform of various types
(FFTW.RODFT00,FFTW.RODFT01,FFTW.RODFT10, or FFTW.RODFT11), a real-input
DFT with halfcomplex-format output (FFTW.R2HC and its inverse FFTW.HC2R),
or a discrete Hartley transform (FFTW.DHT). The kind argument may be an array
or tuple in order to specify different transform types along the different dimen-
sions of A;kind[end] is used for any unspecified dimensions. See the FFTW
manual for precise definitions of these transform types, at http://www.fftw.org/doc.
31.12. BASE.DFT.FFTW.R2R! 333
The optional dims argument specifies an iterable subset of dimensions (e.g. an
integer, range, tuple, or array) to transform along. kind[i] is then the trans-
form type for dims[i], with kind[end] being used for i>length(kind).
See also plan r2r to pre-plan optimized r2r transforms.
source
31.12 Base.DFT.FFTW.r2r!
Base.DFT.FFTW.r2r! —Function.
r2r!(A, kind [, dims])
Same as r2r, but operates in-place on A, which must be an array of real or
complex floating-point numbers.
source
Chapter 32
AxisArrays
32.1 AxisArrays.axes
AxisArrays.axes —Method.
axes(A::AxisArray) -> (Axis...)
axes(A::AxisArray, ax::Axis) -> Axis
axes(A::AxisArray, dim::Int) -> Axis
Returns the tuple of axis vectors for an AxisArray. If an specific Axis is
specified, then only that axis vector is returned. Note that when extracting a
single axis vector, axes(A, Axis{1})) is type-stable and will perform better
than axes(A)[1].
For an AbstractArray without Axis information, axes returns the default
axes, i.e., those that would be produced by AxisArray(A).
source
32.2 AxisArrays.axisdim
AxisArrays.axisdim —Method.
axisdim(::AxisArray, ::Axis) -> Int
axisdim(::AxisArray, ::Type{Axis}) -> Int
Given an AxisArray and an Axis, return the integer dimension of the Axis
within the array.
source
32.3 AxisArrays.axisnames
AxisArrays.axisnames —Method.
334
32.4. AXISARRAYS.AXISVALUES 335
axisnames(A::AxisArray) -> (Symbol...)
axisnames(::Type{AxisArray{...}}) -> (Symbol...)
axisnames(ax::Axis...) -> (Symbol...)
axisnames(::Type{Axis{...}}...) -> (Symbol...)
Returns the axis names of an AxisArray or list of Axises as a tuple of Sym-
bols.
source
32.4 AxisArrays.axisvalues
AxisArrays.axisvalues —Method.
axisvalues(A::AxisArray) -> (AbstractVector...)
axisvalues(ax::Axis...) -> (AbstractVector...)
Returns the axis values of an AxisArray or list of Axises as a tuple of vectors.
source
32.5 AxisArrays.collapse
AxisArrays.collapse —Method.
collapse(::Type{Val{N}}, As::AxisArray...) -> AxisArray
collapse(::Type{Val{N}}, labels::Tuple, As::AxisArray...) -> AxisArray
collapse(::Type{Val{N}}, ::Type{NewArrayType}, As::AxisArray...) -> AxisArray
collapse(::Type{Val{N}}, ::Type{NewArrayType}, labels::Tuple, As::AxisArray...) -> AxisArray
Collapses AxisArrays with Nequal leading axes into a single AxisArray. All
additional axes in any of the arrays are collapsed into a single additional axis
of type Axis{:collapsed, CategoricalVector{Tuple}}.
Arguments
•::Type{Val{N}}: the greatest common dimension to share between all
input arrays. The remaining axes are collapsed. All Naxes must be
common to each input array, at the same dimension. Values from 0up to
the minimum number of dimensions across all input arrays are allowed.
•labels::Tuple: (optional) an index for each array in As used as the
leading element in the index tuples in the :collapsed axis. Defaults to
1:length(As).
•::Type{NewArrayType<:AbstractArray{, N+1}}: (optional) the desired
underlying array type for the returned AxisArray.
•As::AxisArray...:AxisArrays to be collapsed together.
336 CHAPTER 32. AXISARRAYS
Examples
julia> price_data = AxisArray(rand(10), Axis{:time}(Date(2016,01,01):Day(1):Date(2016,01,10)))
1-dimensional AxisArray{Float64,1,...} with axes:
:time, 2016-01-01:1 day:2016-01-10
And data, a 10-element Array{Float64,1}:
0.885014
0.418562
0.609344
0.72221
0.43656
0.840304
0.455337
0.65954
0.393801
0.260207
julia> size_data = AxisArray(rand(10,2), Axis{:time}(Date(2016,01,01):Day(1):Date(2016,01,10)), Axis{:measure}([:area, :volume]))
2-dimensional AxisArray{Float64,2,...} with axes:
:time, 2016-01-01:1 day:2016-01-10
:measure, Symbol[:area, :volume]
And data, a 102 Array{Float64,2}:
0.159434 0.456992
0.344521 0.374623
0.522077 0.313256
0.994697 0.320953
0.95104 0.900526
0.921854 0.729311
0.000922581 0.148822
0.449128 0.761714
0.650277 0.135061
0.688773 0.513845
julia> collapsed = collapse(Val{1}, (:price, :size), price_data, size_data)
2-dimensional AxisArray{Float64,2,...} with axes:
:time, 2016-01-01:1 day:2016-01-10
:collapsed, Tuple{Symbol,Vararg{Symbol,N} where N}[(:price,), (:size, :area), (:size, :volume)]
And data, a 103 Array{Float64,2}:
0.885014 0.159434 0.456992
0.418562 0.344521 0.374623
0.609344 0.522077 0.313256
0.72221 0.994697 0.320953
0.43656 0.95104 0.900526
0.840304 0.921854 0.729311
0.455337 0.000922581 0.148822
0.65954 0.449128 0.761714
32.5. AXISARRAYS.COLLAPSE 337
0.393801 0.650277 0.135061
0.260207 0.688773 0.513845
julia> collapsed[Axis{:collapsed}(:size)] == size_data
true
source
Chapter 33
QuadGK
33.1 Base.quadgk
Base.quadgk —Method.
quadgk(f, a,b,c...; reltol=sqrt(eps), abstol=0, maxevals=10^7, order=7, norm=vecnorm)
Numerically integrate the function f(x) from ato b, and optionally over
additional intervals bto cand so on. Keyword options include a relative error
tolerance reltol (defaults to sqrt(eps) in the precision of the endpoints), an
absolute error tolerance abstol (defaults to 0), a maximum number of function
evaluations maxevals (defaults to 10^7), and the order of the integration rule
(defaults to 7).
Returns a pair (I,E) of the estimated integral Iand an estimated up-
per bound on the absolute error E. If maxevals is not exceeded then E<=
max(abstol, reltol*norm(I)) will hold. (Note that it is useful to specify a
positive abstol in cases where norm(I) may be zero.)
The endpoints aet cetera can also be complex (in which case the integral is
performed over straight-line segments in the complex plane). If the endpoints
are BigFloat, then the integration will be performed in BigFloat precision as
well.
!!! note It is advisable to increase the integration order in rough proportion
to the precision, for smooth integrands.
More generally, the precision is set by the precision of the integration end-
points (promoted to floating-point types).
The integrand f(x) can return any numeric scalar, vector, or matrix type,
or in fact any type supporting +,-, multiplication by real values, and a norm
(i.e., any normed vector space). Alternatively, a different norm can be specified
by passing a norm-like function as the norm keyword argument (which defaults
to vecnorm).
!!! note Only one-dimensional integrals are provided by this function. For
multi-dimensional integration (cubature), there are many different algorithms
338
33.2. QUADGK.GAUSS 339
(often much better than simple nested 1d integrals) and the optimal choice
tends to be very problem-dependent. See the Julia external-package listing for
available algorithms for multidimensional integration or other specialized tasks
(such as integrals of highly oscillatory or singular functions).
The algorithm is an adaptive Gauss-Kronrod integration technique: the in-
tegral in each interval is estimated using a Kronrod rule (2*order+1 points) and
the error is estimated using an embedded Gauss rule (order points). The inter-
val with the largest error is then subdivided into two intervals and the process
is repeated until the desired error tolerance is achieved.
These quadrature rules work best for smooth functions within each interval,
so if your function has a known discontinuity or other singularity, it is best to
subdivide your interval to put the singularity at an endpoint. For example, if f
has a discontinuity at x=0.7 and you want to integrate from 0 to 1, you should
use quadgk(f, 0,0.7,1) to subdivide the interval at the point of discontinuity.
The integrand is never evaluated exactly at the endpoints of the intervals, so it
is possible to integrate functions that diverge at the endpoints as long as the
singularity is integrable (for example, a log(x) or 1/sqrt(x) singularity).
For real-valued endpoints, the starting and/or ending points may be infinite.
(A coordinate transformation is performed internally to map the infinite interval
to a finite one.)
source
33.2 QuadGK.gauss
QuadGK.gauss —Method.
gauss([T,] N)
Return a pair (x, w) of Nquadrature points x[i] and weights w[i] to
integrate functions on the interval (-1, 1), i.e. sum(w .* f.(x)) approximates
the integral. Uses the method described in Trefethen & Bau, Numerical Linear
Algebra, to find the N-point Gaussian quadrature in O(N) operations.
Tis an optional parameter specifying the floating-point type, defaulting to
Float64. Arbitrary precision (BigFloat) is also supported.
source
33.3 QuadGK.kronrod
QuadGK.kronrod —Method.
kronrod([T,] n)
Compute 2n+1 Kronrod points xand weights wbased on the description
in Laurie (1997), appendix A, simplified for a=0, for integrating on [-1,1].
Since the rule is symmetric, this only returns the n+1 points with x<= 0. The
340 CHAPTER 33. QUADGK
function Also computes the embedded n-point Gauss quadrature weights gw
(again for x<= 0), corresponding to the points x[2:2:end]. Returns (x,w,wg)
in O(n) operations.
Tis an optional parameter specifying the floating-point type, defaulting to
Float64. Arbitrary precision (BigFloat) is also supported.
Given these points and weights, the estimated integral Iand error Ecan be
computed for an integrand f(x) as follows:
x, w, wg = kronrod(n)
fx = f(x[end]) # f(0)
x = x[1:end-1] # the x < 0 Kronrod points
fx = f.(x) .+ f.((-).(x)) # f(x < 0) + f(x > 0)
I = sum(fx .* w[1:end-1]) + fx * w[end]
if isodd(n)
E = abs(sum(fx[2:2:end] .* wg[1:end-1]) + fx*wg[end] - I)
else
E = abs(sum(fx[2:2:end] .* wg[1:end])- I)
end
source
Chapter 34
BusinessDays
34.1 BusinessDays.advancebdays
BusinessDays.advancebdays —Method.
advancebdays(calendar, dt, bdays_count)
Increments given date dt by bdays count. Decrements it if bdays count is
negative. bdays count can be a Int,Vector{Int}or a UnitRange.
Computation starts by next Business Day if dt is not a Business Day.
source
34.2 BusinessDays.bdayscount
BusinessDays.bdayscount —Method.
bdays(calendar, dt0, dt1)
Counts the number of Business Days between dt0 and dt1. Returns in-
stances of Dates.Day.
Computation is always based on next Business Day if given dates are not
Business Days.
source
34.3 BusinessDays.firstbdayofmonth
BusinessDays.firstbdayofmonth —Method.
firstbdayofmonth(calendar, dt)
firstbdayofmonth(calendar, yy, mm)
Returns the first business day of month.
source
341
342 CHAPTER 34. BUSINESSDAYS
34.4 BusinessDays.isbday
BusinessDays.isbday —Method.
isbday(calendar, dt)
Returns false for weekends or holidays. Returns true otherwise.
source
34.5 BusinessDays.isholiday
BusinessDays.isholiday —Method.
isholiday(calendar, dt)
Checks if dt is a holiday based on a given calendar of holidays.
calendar can be an instance of HolidayCalendar, a Symbol or an AbstractString.
Returns boolean values.
source
34.6 BusinessDays.isweekday
BusinessDays.isweekday —Method.
isweekday(dt)
Returns true for Monday to Friday. Returns false otherwise.
source
34.7 BusinessDays.isweekend
BusinessDays.isweekend —Method.
isweekend(dt)
Returns true for Saturdays or Sundays. Returns false otherwise.
source
34.8 BusinessDays.lastbdayofmonth
BusinessDays.lastbdayofmonth —Method.
lastbdayofmonth(calendar, dt)
lastbdayofmonth(calendar, yy, mm)
Returns the last business day of month.
source
34.9. BUSINESSDAYS.LISTBDAYS 343
34.9 BusinessDays.listbdays
BusinessDays.listbdays —Method.
listbdays(calendar, dt0::Date, dt1::Date) Vector{Date}
Returns the list of business days between dt0 and dt1.
source
34.10 BusinessDays.listholidays
BusinessDays.listholidays —Method.
listholidays(calendar, dt0::Date, dt1::Date) Vector{Date}
Returns the list of holidays between dt0 and dt1.
source
34.11 BusinessDays.tobday
BusinessDays.tobday —Method.
tobday(calendar, dt; [forward=true])
Adjusts dt to next Business Day if it’s not a Business Day. If isbday(dt),
returns dt.
source
Chapter 35
NLSolversBase
35.1 Base.LinAlg.gradient
Base.LinAlg.gradient —Method.
Evaluates the gradient value at x
This does not update obj.DF.
source
35.2 Base.LinAlg.gradient
Base.LinAlg.gradient —Method.
Get the ith element of the most recently evaluated gradient of obj.
source
35.3 Base.LinAlg.gradient
Base.LinAlg.gradient —Method.
Get the most recently evaluated gradient of obj.
source
35.4 NLSolversBase.complex to real
NLSolversBase.complex to real —Method.
Convert a complex array of size dims to a real array of size 2 x dims
source
35.5 NLSolversBase.gradient!
NLSolversBase.gradient! —Method.
344
35.6. NLSOLVERSBASE.HESSIAN 345
Evaluates the gradient value at x.
Stores the value in obj.DF.
source
35.6 NLSolversBase.hessian
NLSolversBase.hessian —Method.
Get the most recently evaluated Hessian of obj
source
35.7 NLSolversBase.jacobian
NLSolversBase.jacobian —Method.
Get the most recently evaluated Jacobian of obj.
source
35.8 NLSolversBase.real to complex
NLSolversBase.real to complex —Method.
Convert a real array of size 2 x dims to a complex array of size dims
source
35.9 NLSolversBase.value!!
NLSolversBase.value!! —Method.
Force (re-)evaluation of the objective value at x.
Returns f(x) and stores the value in obj.F
source
35.10 NLSolversBase.value!
NLSolversBase.value! —Method.
Evaluates the objective value at x.
Returns f(x) and stores the value in obj.F
source
35.11 NLSolversBase.value
NLSolversBase.value —Method.
Evaluates the objective value at x.
Returns f(x), but does not store the value in obj.F
source
346 CHAPTER 35. NLSOLVERSBASE
35.12 NLSolversBase.value
NLSolversBase.value —Method.
Get the most recently evaluated objective value of obj.
source
Chapter 36
Dagger
36.1 Dagger.alignfirst
Dagger.alignfirst —Method.
alignfirst(a)
Make a subdomain a standalone domain. For example,
alignfirst(ArrayDomain(11:25, 21:100))
# => ArrayDomain((1:15), (1:80))
source
36.2 Dagger.cached stage
Dagger.cached stage —Method.
A memoized version of stage. It is important that the tasks generated for
the same DArray have the same identity, for example:
A = rand(Blocks(100,100), Float64, 1000, 1000)
compute(A+A’)
must not result in computation of A twice.
source
36.3 Dagger.compute
Dagger.compute —Method.
A DArray object may contain a thunk in it, in which case we first turn it
into a Thunk object and then compute it.
source
347
348 CHAPTER 36. DAGGER
36.4 Dagger.compute
Dagger.compute —Method.
Compute a Thunk - creates the DAG, assigns ranks to nodes for tie breaking
and runs the scheduler.
source
36.5 Dagger.domain
Dagger.domain —Function.
domain(x::T)
Returns metadata about x. This metadata will be in the domain field of a
Chunk object when an object of type Tis created as the result of evaluating a
Thunk.
source
36.6 Dagger.domain
Dagger.domain —Method.
The domain of an array is a ArrayDomain
source
36.7 Dagger.domain
Dagger.domain —Method.
If no domain method is defined on an object, then we use the UnitDomain
on it. A UnitDomain is indivisible.
source
36.8 Dagger.load
Dagger.load —Method.
load(ctx, file_path)
Load an Union{Chunk, Thunk}from a file.
source
36.9. DAGGER.LOAD 349
36.9 Dagger.load
Dagger.load —Method.
load(ctx, ::Type{Chunk}, fpath, io)
Load a Chunk object from a file, the file path is required for creating a
FileReader object
source
36.10 Dagger.save
Dagger.save —Method.
save(io::IO, val)
Save a value into the IO buffer. In the case of arrays and sparse matrices,
this will save it in a memory-mappable way.
load(io::IO, t::Type, domain) will load the object given its domain
source
36.11 Dagger.save
Dagger.save —Method.
save(ctx, chunk::Union{Chunk, Thunk}, file_path::AbsractString)
Save a chunk to a file at file path.
source
36.12 Dagger.save
Dagger.save —Method.
special case distmem writing - write to disk on the process with the chunk.
source
Chapter 37
ShowItLikeYouBuildIt
37.1 ShowItLikeYouBuildIt.showarg
ShowItLikeYouBuildIt.showarg —Method.
showarg(stream::IO, x)
Show xas if it were an argument to a function. This function is used in the
printing of “type summaries” in terms of sequences of function calls on objects.
The fallback definition is to print xas ::$(typeof(x)), representing argu-
ment xin terms of its type. However, you can specialize this function for specific
types to customize printing.
Example
A SubArray created as view(a, :, 3, 2:5), where ais a 3-dimensional
Float64 array, has type
SubArray{Float64,2,Array{Float64,3},Tuple{Colon,Int64,UnitRange{Int64}},false}
and this type will be printed in the summary. To change the printing of this
object to
view(::Array{Float64,3}, Colon(), 3, 2:5)
you could define
function ShowItLikeYouBuildIt.showarg(io::IO, v::SubArray)
print(io, "view(")
showarg(io, parent(v))
print(io, ", ", join(v.indexes, ", "))
print(io, ’)’)
end
350
37.2. SHOWITLIKEYOUBUILDIT.SUMMARY BUILD 351
Note that we’re calling showarg recursively for the parent array type. Print-
ing the parent as ::Array{Float64,3}is the fallback behavior, assuming no
specialized method for Array has been defined. More generally, this would
display as
view(<a>, Colon(), 3, 2:5)
where <a>is the output of showarg for a.
This printing might be activated any time vis a field in some other container,
or if you specialize Base.summary for SubArray to call summary build.
See also: summary build.
source
37.2 ShowItLikeYouBuildIt.summary build
ShowItLikeYouBuildIt.summary build —Function.
summary_build(A::AbstractArray, [cthresh])
Return a string representing Ain terms of the sequence of function calls that
might be used to create A, along with information about A’s size or indices and
element type. This function should never be called directly, but instead used to
specialize Base.summary for specific AbstractArray subtypes. For example, if
you want to change the summary of SubArray, you might define
Base.summary(v::SubArray) = summary_build(v)
This function goes hand-in-hand with showarg. If you have defined a showarg
method for SubArray as in the documentation for showarg, then the summary
of a SubArray might look like this:
34 view(::Array{Float64,3}, :, 3, 2:5) with element type Float64
instead of this:
34 SubArray{Float64,2,Array{Float64,3},Tuple{Colon,Int64,UnitRange{Int64}},false}
The optional argument cthresh is a “complexity threshold”; objects with
type descriptions that are less complex than the specified threshold will be
printed using the traditional type-based summary. The default value is n+1,
where nis the number of parameters in typeof(A). The complexity is calculated
with type complexity. You can choose a cthresh of 0 if you want to ensure
that your showarg version is always used.
See also: showarg, type complexity.
source
352 CHAPTER 37. SHOWITLIKEYOUBUILDIT
37.3 ShowItLikeYouBuildIt.type complexity
ShowItLikeYouBuildIt.type complexity —Method.
type_complexity(T::Type) -> c
Return an integer crepresenting a measure of the “complexity” of type T.
For unnested types, c = n+1 where nis the number of parameters in type T.
However, type complexity calls itself recursively on the parameters, so if the
parameters have their own parameters then the complexity of the type will be
(potentially much) higher than this.
Examples
# Array{Float64,2}
julia> a = rand(3,5);
julia> type_complexity(typeof(a))
3
julia> length(typeof(a).parameters)+1
3
# Create an object that has type:
# SubArray{Int64,2,Base.ReshapedArray{Int64,2,UnitRange{Int64},Tuple{}},Tuple{StepRange{Int64,Int64},Colon},false}
julia> r = reshape(1:9, 3, 3);
julia> v = view(r, 1:2:3, :);
julia> type_complexity(typeof(v))
15
julia> length(typeof(v).parameters)+1
6
The second example indicates that the total complexity of v’s type is con-
siderably higher than the complexity of just its “outer” SubArray type.
source
Chapter 38
CoordinateTransformations
38.1 CoordinateTransformations.cameramap
CoordinateTransformations.cameramap —Method.
cameramap()
cameramap(scale)
cameramap(scale, offset)
Create a transformation that takes points in real space (e.g. 3D) and projects
them through a perspective transformation onto the focal plane of an ideal
(pinhole) camera with the given properties.
The scale sets the scale of the screen. For a standard digital camera, this
would be scale = focal length / pixel size. Non-square pixels are sup-
ported by providing a pair of scales in a tuple, scale = (scale x, scale y).
Positive scales represent a camera looking in the +z axis with a virtual screen
in front of the camera (the x,y coordinates are not inverted compared to 3D
space). Note that points behind the camera (with negative z component) will
be projected (and inverted) onto the image coordinates and it is up to the user
to cull such points as necessary.
The offset = (offset x, offset y) is used to define the origin in the
imaging plane. For instance, you may wish to have the point (0,0) represent the
top-left corner of your imaging sensor. This measurement is in the units after
applying scale (e.g. pixels).
(see also PerspectiveMap)
source
38.2 CoordinateTransformations.compose
CoordinateTransformations.compose —Method.
compose(trans1, trans2)
353
354 CHAPTER 38. COORDINATETRANSFORMATIONS
trans1 trans2
Take two transformations and create a new transformation that is equivalent
to successively applying trans2 to the coordinate, and then trans1. By default
will create a ComposedTransformation, however this method can be overloaded
for efficiency (e.g. two affine transformations naturally compose to a single affine
transformation).
source
38.3 CoordinateTransformations.recenter
CoordinateTransformations.recenter —Method.
recenter(trans::Union{AbstractMatrix,Transformation}, origin::AbstractVector) -> ctrans
Return a new transformation ctrans such that point origin serves as the
origin-of-coordinates for trans. Translation by origin occurs both before and
after applying trans, so that if trans is linear we have
ctrans(origin) == origin
As a consequence, recenter only makes sense if the output space of trans
is isomorphic with the input space.
For example, if trans is a rotation matrix, then ctrans rotates space around
origin.
source
38.4 CoordinateTransformations.transform deriv
CoordinateTransformations.transform deriv —Method.
transform_deriv(trans::Transformation, x)
A matrix describing how differentials on the parameters of xflow through
to the output of transformation trans.
source
38.5 CoordinateTransformations.transform deriv params
CoordinateTransformations.transform deriv params —Method.
transform_deriv_params(trans::AbstractTransformation, x)
A matrix describing how differentials on the parameters of trans flow through
to the output of transformation trans given input x.
source
Chapter 39
Graphics
39.1 Graphics.aspect ratio
Graphics.aspect ratio —Method.
aspect_ratio(bb::BoundingBox) -> r
Compute the ratio rof the height and width of bb.
source
39.2 Graphics.deform
Graphics.deform —Method.
deform(bb::BoundingBox, l, r, t, b) -> bbnew
Add l(left), r(right), t(top), and b(bottom) to the edges of a Bounding-
Box.
source
39.3 Graphics.inner canvas
Graphics.inner canvas —Method.
inner_canvas(c::GraphicsContext, device::BoundingBox, user::BoundingBox)
inner_canvas(c::GraphicsContext, x, y, w, h, l, r, t, b)
Create a rectangular drawing area inside device (represented in device-
coordinates), giving it user-coordinates user. Any drawing that occurs outside
this box is clipped.
x,y,w, and hare an alternative parametrization of device, and l,r,t,b
parametrize user.
See also: set coordinates.
source
355
356 CHAPTER 39. GRAPHICS
39.4 Graphics.isinside
Graphics.isinside —Method.
isinside(bb::BoundingBox, p::Point) -> tf::Bool
isinside(bb::BoundingBox, x, y) -> tf::Bool
Determine whether the point lies within bb.
source
39.5 Graphics.rotate
Graphics.rotate —Method.
rotate(bb::BoundingBox, angle, o) -> bbnew
Rotate bb around oby angle, returning the BoundingBox that encloses the
vertices of the rotated box.
source
39.6 Graphics.rotate
Graphics.rotate —Method.
rotate(p::Vec2, angle::Real, o::Vec2) -> pnew
Rotate paround oby angle.
source
39.7 Graphics.set coordinates
Graphics.set coordinates —Method.
set_coordinates(c::GraphicsContext, device::BoundingBox, user::BoundingBox)
set_coordinates(c::GraphicsContext, user::BoundingBox)
Set the device->user coordinate transformation of cso that device, ex-
pressed in “device coordinates” (pixels), is equivalent to user as expressed in
“user coordinates”. If device is omitted, it defaults to the full span of c,
BoundingBox(0, width(c), 0, height(c)).
See also get matrix,set matrix.
source
39.8. GRAPHICS.SHIFT 357
39.8 Graphics.shift
Graphics.shift —Method.
shift(bb::BoundingBox, x, y) -> bbnew
Shift center by (x,y), keeping width & height fixed.
source
Chapter 40
MacroTools
40.1 MacroTools.inexpr
MacroTools.inexpr —Method.
inexpr(expr, x)
Simple expression match; will return true if the expression xcan be found
inside expr.
inexpr(:(2+2), 2) == true
source
40.2 MacroTools.isdef
MacroTools.isdef —Method.
Test for function definition expressions.
source
40.3 MacroTools.isexpr
MacroTools.isexpr —Method.
isexpr(x, ts...)
Convenient way to test the type of a Julia expression. Expression heads and
types are supported, so for example you can call
isexpr(expr, String, :string)
to pick up on all string-like expressions.
source
358
40.4. MACROTOOLS.NAMIFY 359
40.4 MacroTools.namify
MacroTools.namify —Method.
An easy way to get pull the (function/type) name out of expressions like
foo{T}or Bar{T}<: Vector{T}.
source
40.5 MacroTools.prettify
MacroTools.prettify —Method.
prettify(ex)
Makes generated code generaly nicer to look at.
source
40.6 MacroTools.rmlines
MacroTools.rmlines —Method.
rmlines(x)
Remove the line nodes from a block or array of expressions.
Compare quote end vs rmlines(quote end)
source
40.7 MacroTools.splitarg
MacroTools.splitarg —Method.
splitarg(arg)
Match function arguments (whether from a definition or a function call) such
as x::Int=2 and return (arg name, arg type, is splat, default).arg name
and default are nothing when they are absent. For example:
[] ¿ map(splitarg, (:(f(a=2, x::Int=nothing, y, args...))).args[2:end]) 4-element
Array{Tuple{Symbol,Symbol,Bool,Any},1}: (:a, :Any, false, 2) (:x, :Int, false,
:nothing) (:y, :Any, false, nothing) (:args, :Any, true, nothing)
source
360 CHAPTER 40. MACROTOOLS
40.8 MacroTools.splitdef
MacroTools.splitdef —Method.
splitdef(fdef)
Match any function definition
[] function name{params}(args; kwargs)::rtype where {whereparams}body
end
and return Dict(:name=>..., :args=>..., etc.). The definition can be
rebuilt by calling MacroTools.combinedef(dict), or explicitly with
rtype = get(dict, :rtype, :Any)
all_params = [get(dict, :params, [])..., get(dict, :whereparams, [])...]
:(function $(dict[:name]){$(all_params...)}($(dict[:args]...);
$(dict[:kwargs]...))::$rtype
$(dict[:body])
end)
source
40.9 MacroTools.unblock
MacroTools.unblock —Method.
unblock(expr)
Remove outer begin blocks from an expression, if the block is redundant
(i.e. contains only a single expression).
source
Chapter 41
ImageMorphology
41.1 ImageMorphology.bothat
ImageMorphology.bothat —Function.
imgbh = bothat(img, [region]) performs bottom hat of an image, which
is defined as its morphological closing minus the original image. region allows
you to control the dimensions over which this operation is performed.
source
41.2 ImageMorphology.closing
ImageMorphology.closing —Function.
imgc = closing(img, [region]) performs the closing morphology oper-
ation, equivalent to erode(dilate(img)).region allows you to control the
dimensions over which this operation is performed.
source
41.3 ImageMorphology.dilate
ImageMorphology.dilate —Function.
imgd = dilate(img, [region])
perform a max-filter over nearest-neighbors. The default is 8-connectivity
in 2d, 27-connectivity in 3d, etc. You can specify the list of dimensions that
you want to include in the connectivity, e.g., region = [1,2] would exclude
the third dimension from filtering.
source
361
362 CHAPTER 41. IMAGEMORPHOLOGY
41.4 ImageMorphology.erode
ImageMorphology.erode —Function.
imge = erode(img, [region])
perform a min-filter over nearest-neighbors. The default is 8-connectivity in
2d, 27-connectivity in 3d, etc. You can specify the list of dimensions that you
want to include in the connectivity, e.g., region = [1,2] would exclude the
third dimension from filtering.
source
41.5 ImageMorphology.morphogradient
ImageMorphology.morphogradient —Function.
imgmg = morphogradient(img, [region]) returns morphological gradient
of the image, which is the difference between the dilation and the erosion of
a given image. region allows you to control the dimensions over which this
operation is performed.
source
41.6 ImageMorphology.morpholaplace
ImageMorphology.morpholaplace —Function.
imgml = morpholaplace(img, [region]) performs Morphological Laplacian
of an image, which is defined as the arithmetic difference between the internal
and the external gradient. region allows you to control the dimensions over
which this operation is performed.
source
41.7 ImageMorphology.opening
ImageMorphology.opening —Function.
imgo = opening(img, [region]) performs the opening morphology oper-
ation, equivalent to dilate(erode(img)).region allows you to control the
dimensions over which this operation is performed.
source
41.8 ImageMorphology.tophat
ImageMorphology.tophat —Function.
imgth = tophat(img, [region]) performs top hat of an image, which is
defined as the image minus its morphological opening. region allows you to
control the dimensions over which this operation is performed.
source
Chapter 42
Contour
42.1 Contour.contour
Contour.contour —Method.
contour(x, y, z, level::Number) Trace a single contour level, indicated
by the argument level.
You’ll usually call lines on the output of contour, and then iterate over
the result.
source
42.2 Contour.contours
Contour.contours —Method.
contours(x,y,z,levels) Trace the contour levels indicated by the levels
argument.
source
42.3 Contour.contours
Contour.contours —Method.
contours returns a set of isolines.
You’ll usually call levels on the output of contours.
source
42.4 Contour.contours
Contour.contours —Method.
contours(x,y,z,Nlevels::Integer) Trace Nlevels contour levels at heights
chosen by the library (using the contourlevels function).
source
363
364 CHAPTER 42. CONTOUR
42.5 Contour.contours
Contour.contours —Method.
contours(x,y,z) Trace 10 automatically chosen contour levels.
source
42.6 Contour.coordinates
Contour.coordinates —Method.
coordinates(c) Returns the coordinates of the vertices of the contour line
as a tuple of lists.
source
42.7 Contour.level
Contour.level —Method.
level(c) Indicates the z-value at which the contour level cwas traced.
source
42.8 Contour.levels
Contour.levels —Method.
Turns the output of contours into an iterable with each of the traced contour
levels. Each of the objects support level and coordinates.
source
42.9 Contour.lines
Contour.lines —Method.
lines(c) Extracts an iterable collection of isolines from a contour level. Use
coordinates to get the coordinates of a line.
source
Chapter 43
Compose
43.1 Compose.circle
Compose.circle —Function.
circle(xs, ys, rs)
Arguments can be passed in arrays in order to perform multiple drawing
operations.
source
43.2 Compose.circle
Compose.circle —Function.
circle(x, y, r)
Define a circle with its center at (x,y) and a radius of r.
source
43.3 Compose.circle
Compose.circle —Method.
circle()
Define a circle in the center of the current context with a diameter equal to
the width of the context.
source
365
366 CHAPTER 43. COMPOSE
43.4 Compose.polygon
Compose.polygon —Method.
polygon(points)
Define a polygon. points is an array of (x,y) tuples that specify the corners
of the polygon.
source
43.5 Compose.rectangle
Compose.rectangle —Function.
rectangle(x0s, y0s, widths, heights)
Arguments can be passed in arrays in order to perform multiple drawing
operations at once.
source
43.6 Compose.rectangle
Compose.rectangle —Function.
rectangle(x0, y0, width, height)
Define a rectangle of size widthxheight with its top left corner at the point
(x,y).
source
43.7 Compose.rectangle
Compose.rectangle —Method.
rectangle()
Define a rectangle that fills the current context completely.
source
43.8 Compose.text
Compose.text —Function.
text(x, y, value [,halign::HAlignment [,valign::VAlignment [,rot::Rotation]]])
43.9. COMPOSE.TEXT 367
Draw the text value at the position (x,y) relative to the current context.
The default alignment of the text is hleft vbottom. The vertical and hor-
izontal alignment is specified by passing hleft,hcenter or hright and vtop,
vcenter or vbottom as values for halgin and valgin respectively.
source
43.9 Compose.text
Compose.text —Function.
text(xs, ys, values [,haligns::HAlignment [,valigns::VAlignment [,rots::Rotation]]])
Arguments can be passed in arrays in order to perform multiple drawing
operations at once.
source
Chapter 44
CoupledFields
44.1 CoupledFields.CVfn
CoupledFields.CVfn —Method.
CVfn{T<:Matrix{Float64}}(parm::T, X::T, Y::T, modelfn::Function, kerneltype::DataType; verbose::Bool=true, dcv::Int64=2)
Cross-validation function
source
44.2 CoupledFields.Rsq adj
CoupledFields.Rsq adj —Method.
Rsq_adj{T<:Array{Float64}}(Tx::T, Ty::T, df::Int):
Cross-validation metric
source
44.3 CoupledFields.bf
CoupledFields.bf —Method.
bf(x::Vector{Float64}, df::Int):
Compute a piecewise linear basis matrix for the vector x.
source
368
44.4. COUPLEDFIELDS.CCA 369
44.4 CoupledFields.cca
CoupledFields.cca —Method.
cca{T<:Matrix{Float64}}(v::Array{Float64}, X::T,Y::T):
Regularized Canonical Correlation Analysis using SVD.
source
44.5 CoupledFields.gKCCA
CoupledFields.gKCCA —Method.
gKCCA(par::Array{Float64}, X::Matrix{Float64}, Y::Matrix{Float64}, kpars::KernelParameters):
Compute the projection matrices and components for gKCCA.
source
44.6 CoupledFields.gradvecfield
CoupledFields.gradvecfield —Method.
gradvecfield{N<:Float64, T<:Matrix{Float64}}(par::Array{N}, X::T, Y::T, kpars::KernelParameters ):
Compute the gradient vector or gradient matrix at each instance of the X
and Yfields, by making use of a kernel feature space.
source
44.7 CoupledFields.whiten
CoupledFields.whiten —Method.
whiten(x::Matrix{Float64}, d::Float64; lat=nothing): Whiten matrix
d(0-1) Percentage variance of components to retain.
lat Latitudinal area-weighting.
source
Chapter 45
AxisAlgorithms
45.1 AxisAlgorithms.A ldiv B md!
AxisAlgorithms.A ldiv B md! —Method.
A ldiv B md!(dest, F, src, dim) solves a tridiagonal system along di-
mension dim of src, storing the result in dest. Currently, Fmust be an LU-
factorized tridiagonal matrix. If desired, you may safely use the same array for
both src and dest, so that this becomes an in-place algorithm.
source
45.2 AxisAlgorithms.A ldiv B md
AxisAlgorithms.A ldiv B md —Method.
A ldiv B md(F, src, dim) solves Ffor slices bof src along dimension dim,
storing the result along the same dimension of the output. Currently, Fmust
be an LU-factorized tridiagonal matrix or a Woodbury matrix.
source
45.3 AxisAlgorithms.A mul B md!
AxisAlgorithms.A mul B md! —Method.
A mul B md!(dest, M, src, dim) computes M*x for slices xof src along
dimension dim, storing the result in dest.Mmust be an AbstractMatrix. This
uses an in-place naive algorithm.
source
45.4 AxisAlgorithms.A mul B md
AxisAlgorithms.A mul B md —Method.
370
45.5. AXISALGORITHMS.A MUL B PERM! 371
A mul B md(M, src, dim) computes M*x for slices xof src along dimension
dim, storing the resulting vector along the same dimension of the output. M
must be an AbstractMatrix. This uses an in-place naive algorithm.
source
45.5 AxisAlgorithms.A mul B perm!
AxisAlgorithms.A mul B perm! —Method.
A mul B perm!(dest, M, src, dim) computes M*x for slices xof src along
dimension dim, storing the result in dest.Mmust be an AbstractMatrix. This
uses permutedims to make dimension dim into the first dimension, performs a
standard matrix multiplication, and restores the original dimension ordering.
In many cases, this algorithm exhibits the best cache behavior.
source
45.6 AxisAlgorithms.A mul B perm
AxisAlgorithms.A mul B perm —Method.
A mul B perm(M, src, dim) computes M*x for slices xof src along dimen-
sion dim, storing the resulting vector along the same dimension of the output.
Mmust be an AbstractMatrix. This uses permutedims to make dimension dim
into the first dimension, performs a standard matrix multiplication, and restores
the original dimension ordering. In many cases, this algorithm exhibits the best
cache behavior.
source
Chapter 46
Libz
46.1 Libz.ZlibDeflateInputStream
Libz.ZlibDeflateInputStream —Method.
ZlibDeflateInputStream(input[; <keyword arguments>])
Construct a zlib deflate input stream to compress gzip/zlib data.
Arguments
•input: a byte vector, IO object, or BufferedInputStream containing data
to compress.
•bufsize::Integer=8192: input and output buffer size.
•gzip::Bool=true: if true, data is gzip compressed; if false, zlib com-
pressed.
•level::Integer=6: compression level in 1-9.
•mem level::Integer=8: memory to use for compression in 1-9.
•strategy=Z DEFAULT STRATEGY: compression strategy; see zlib documen-
tation.
source
46.2 Libz.ZlibDeflateOutputStream
Libz.ZlibDeflateOutputStream —Method.
ZlibDeflateOutputStream(output[; <keyword arguments>])
Construct a zlib deflate output stream to compress gzip/zlib data.
Arguments
372
46.3. LIBZ.ZLIBINFLATEINPUTSTREAM 373
•output: a byte vector, IO object, or BufferedInputStream to which com-
pressed data should be written.
•bufsize::Integer=8192: input and output buffer size.
•gzip::Bool=true: if true, data is gzip compressed; if false, zlib com-
pressed.
•level::Integer=6: compression level in 1-9.
•mem level::Integer=8: memory to use for compression in 1-9.
•strategy=Z DEFAULT STRATEGY: compression strategy; see zlib documen-
tation.
source
46.3 Libz.ZlibInflateInputStream
Libz.ZlibInflateInputStream —Method.
ZlibInflateInputStream(input[; <keyword arguments>])
Construct a zlib inflate input stream to decompress gzip/zlib data.
Arguments
•input: a byte vector, IO object, or BufferedInputStream containing com-
pressed data to inflate.
•bufsize::Integer=8192: input and output buffer size.
•gzip::Bool=true: if true, data is gzip compressed; if false, zlib com-
pressed.
•reset on end::Bool=true: on stream end, try to find the start of another
stream.
source
46.4 Libz.ZlibInflateOutputStream
Libz.ZlibInflateOutputStream —Method.
ZlibInflateOutputStream(output[; <keyword arguments>])
Construct a zlib inflate output stream to decompress gzip/zlib data.
Arguments
•output: a byte vector, IO object, or BufferedInputStream to which de-
compressed data should be written.
374 CHAPTER 46. LIBZ
•bufsize::Integer=8192: input and output buffer size.
•gzip::Bool=true: if true, data is gzip compressed; if false, zlib com-
pressed.
source
46.5 Libz.adler32
Libz.adler32 —Function.
adler32(data)
Compute the Adler-32 checksum over the data input. data can be BufferedInputStream
or Vector{UInt8}.
source
46.6 Libz.crc32
Libz.crc32 —Function.
crc32(data)
Compute the CRC-32 checksum over the data input. data can be BufferedInputStream
or Vector{UInt8}.
source
Chapter 47
NullableArrays
47.1 NullableArrays.dropnull!
NullableArrays.dropnull! —Method.
dropnull!(X::NullableVector)
Remove null entries of Xin-place and return a Vector view of the unwrapped
Nullable entries.
source
47.2 NullableArrays.dropnull!
NullableArrays.dropnull! —Method.
dropnull!(X::AbstractVector)
Remove null entries of Xin-place and return a Vector view of the unwrapped
Nullable entries. If no nulls are present, this is a no-op and Xis returned.
source
47.3 NullableArrays.dropnull
NullableArrays.dropnull —Method.
dropnull(X::AbstractVector)
Return a vector containing only the non-null entries of X, unwrapping Nullable
entries. A copy is always returned, even when Xdoes not contain any null values.
source
375
376 CHAPTER 47. NULLABLEARRAYS
47.4 NullableArrays.nullify!
NullableArrays.nullify! —Method.
nullify!(X::NullableArray, I...)
This is a convenience method to set the entry of Xat index Ito be null
source
47.5 NullableArrays.padnull!
NullableArrays.padnull! —Method.
padnull!(X::NullableVector, front::Integer, back::Integer)
Insert front null entries at the beginning of Xand add back null entries at
the end of X. Returns X.
source
47.6 NullableArrays.padnull
NullableArrays.padnull —Method.
padnull(X::NullableVector, front::Integer, back::Integer)
return a copy of Xwith front null entries inserted at the beginning of the
copy and back null entries inserted at the end.
source
Chapter 48
ImageMetadata
48.1 ImageAxes.data
ImageAxes.data —Method.
data(img::ImageMeta) -> array
Extract the data from img, omitting the properties dictionary. array shares
storage with img, so changes to one affect the other.
See also: properties.
source
48.2 ImageMetadata.copyproperties
ImageMetadata.copyproperties —Method.
copyproperties(img::ImageMeta, data) -> imgnew
Create a new “image,” copying the properties dictionary of img but using the
data of the AbstractArray data. Note that changing the properties of imgnew
does not affect the properties of img.
See also: shareproperties.
source
48.3 ImageMetadata.properties
ImageMetadata.properties —Method.
properties(imgmeta) -> props
Extract the properties dictionary props for imgmeta.props shares storage
with img, so changes to one affect the other.
See also: data.
source
377
378 CHAPTER 48. IMAGEMETADATA
48.4 ImageMetadata.shareproperties
ImageMetadata.shareproperties —Method.
shareproperties(img::ImageMeta, data) -> imgnew
Create a new “image,” reusing the properties dictionary of img but using the
data of the AbstractArray data. The two images have synchronized properties;
modifying one also affects the other.
See also: copyproperties.
source
48.5 ImageMetadata.spatialproperties
ImageMetadata.spatialproperties —Method.
spatialproperties(img)
Return a vector of strings, containing the names of properties that have been
declared “spatial” and hence should be permuted when calling permutedims.
Declare such properties like this:
img["spatialproperties"] = ["spacedirections"]
source
Chapter 49
LegacyStrings
49.1 LegacyStrings.utf16
LegacyStrings.utf16 —Function.
utf16(::Union{Ptr{UInt16},Ptr{Int16}} [, length])
Create a string from the address of a NUL-terminated UTF-16 string. A
copy is made; the pointer can be safely freed. If length is specified, the string
does not have to be NUL-terminated.
source
49.2 LegacyStrings.utf16
LegacyStrings.utf16 —Method.
utf16(s)
Create a UTF-16 string from a byte array, array of UInt16, or any other
string type. (Data must be valid UTF-16. Conversions of byte arrays check for
a byte-order marker in the first two bytes, and do not include it in the resulting
string.)
Note that the resulting UTF16String data is terminated by the NUL code-
point (16-bit zero), which is not treated as a character in the string (so that
it is mostly invisible in Julia); this allows the string to be passed directly to
external functions requiring NUL-terminated data. This NUL is appended au-
tomatically by the utf16(s) conversion function. If you have a UInt16 array A
that is already NUL-terminated valid UTF-16 data, then you can instead use
UTF16String(A) to construct the string without making a copy of the data and
treating the NUL as a terminator rather than as part of the string.
source
379
380 CHAPTER 49. LEGACYSTRINGS
49.3 LegacyStrings.utf32
LegacyStrings.utf32 —Function.
utf32(::Union{Ptr{Char},Ptr{UInt32},Ptr{Int32}} [, length])
Create a string from the address of a NUL-terminated UTF-32 string. A
copy is made; the pointer can be safely freed. If length is specified, the string
does not have to be NUL-terminated.
source
49.4 LegacyStrings.utf32
LegacyStrings.utf32 —Method.
utf32(s)
Create a UTF-32 string from a byte array, array of Char or UInt32, or any
other string type. (Conversions of byte arrays check for a byte-order marker in
the first four bytes, and do not include it in the resulting string.)
Note that the resulting UTF32String data is terminated by the NUL code-
point (32-bit zero), which is not treated as a character in the string (so that
it is mostly invisible in Julia); this allows the string to be passed directly to
external functions requiring NUL-terminated data. This NUL is appended au-
tomatically by the utf32(s) conversion function. If you have a Char or UInt32
array Athat is already NUL-terminated UTF-32 data, then you can instead use
UTF32String(A) to construct the string without making a copy of the data and
treating the NUL as a terminator rather than as part of the string.
source
Chapter 50
BufferedStreams
50.1 BufferedStreams.anchor!
BufferedStreams.anchor! —Method.
Set the buffer’s anchor to its current position.
source
50.2 BufferedStreams.fillbuffer!
BufferedStreams.fillbuffer! —Method.
Refill the buffer, optionally moving and retaining part of the data.
source
50.3 BufferedStreams.isanchored
BufferedStreams.isanchored —Method.
Return true if the stream is anchored.
source
50.4 BufferedStreams.peek
BufferedStreams.peek —Method.
Return the next byte from the input stream without advancing the position.
source
50.5 BufferedStreams.peekbytes!
BufferedStreams.peekbytes! —Function.
Fills buffer with bytes from stream’s buffer without advancing the position.
381
382 CHAPTER 50. BUFFEREDSTREAMS
Unless the buffer is empty, we do not re-fill it. Therefore the number of bytes
read is limited to the minimum of nb and the remaining bytes in the buffer.
source
50.6 BufferedStreams.takeanchored!
BufferedStreams.takeanchored! —Method.
Copy and return a byte array from the anchor up to, but not including the
current position, also removing the anchor.
source
50.7 BufferedStreams.upanchor!
BufferedStreams.upanchor! —Method.
Remove and return a buffer’s anchor.
source
Chapter 51
MbedTLS
51.1 MbedTLS.decrypt
MbedTLS.decrypt —Function.
decrypt(cipher, key, msg, [iv]) ->Vector{UInt8}
Decrypt a message using the given cipher. The cipher can be specified as
•a generic cipher (like CIPHER AES)
•a specific cipher (like CIPHER AES 256 CBC)
•a Cipher object
key is the symmetric key used for cryptography, given as either a String or a
Vector{UInt8}. It must be the right length for the chosen cipher; for example,
CIPHER AES 256 CBC requires a 32-byte (256-bit) key.
msg is the message to be encoded. It should either be convertible to a String
or be a Vector{UInt8}.
iv is the initialization vector, whose size must match the block size of the
cipher (eg, 16 bytes for AES) and correspond to the iv used by the encryptor.
By default, it will be set to all zeros.
source
51.2 MbedTLS.digest
MbedTLS.digest —Function.
digest(kind::MDKind, msg::Vector{UInt8}, [key::Vector{UInt8}]) ->
Vector{UInt8}
Perform a digest of the given type on the given message (a byte array),
return a byte array with the digest.
If an optional key is given, perform an HMAC digest.
source
383
384 CHAPTER 51. MBEDTLS
51.3 MbedTLS.digest!
MbedTLS.digest! —Function.
digest!(kind::MDKind, msg::Vector{UInt8}, [key::Vector{UInt8}, ],
buffer::Vector{UInt8})
In-place version of digest that stores the digest to buffer.
It is the user’s responsibility to ensure that buffer is long enough to contain
the digest. get size(kind::MDKind) returns the appropriate size.
source
51.4 MbedTLS.encrypt
MbedTLS.encrypt —Function.
encrypt(cipher, key, msg, [iv]) ->Vector{UInt8}
Encrypt a message using the given cipher. The cipher can be specified as
•a generic cipher (like CIPHER AES)
•a specific cipher (like CIPHER AES 256 CBC)
•a Cipher object
key is the symmetric key used for cryptography, given as either a String or a
Vector{UInt8}. It must be the right length for the chosen cipher; for example,
CIPHER AES 256 CBC requires a 32-byte (256-bit) key.
msg is the message to be encoded. It should either be convertible to a String
or be a Vector{UInt8}.
iv is the initialization vector, whose size must match the block size of the
cipher (eg, 16 bytes for AES). By default, it will be set to all zeros, which is
not secure. For security reasons, it should be set to a different value for each
encryption operation.
source
Chapter 52
DataValues
52.1 DataValues.dropna!
DataValues.dropna! —Method.
dropna!(X::DataValueVector)
Remove missing entries of Xin-place and return a Vector view of the un-
wrapped DataValue entries.
source
52.2 DataValues.dropna!
DataValues.dropna! —Method.
dropna!(X::AbstractVector)
Remove missing entries of Xin-place and return a Vector view of the un-
wrapped DataValue entries. If no missing values are present, this is a no-op
and Xis returned.
source
52.3 DataValues.dropna
DataValues.dropna —Method.
dropna(X::AbstractVector)
Return a vector containing only the non-missing entries of X, unwrapping
DataValue entries. A copy is always returned, even when Xdoes not contain
any missing values.
source
385
386 CHAPTER 52. DATAVALUES
52.4 DataValues.padna!
DataValues.padna! —Method.
padna!(X::DataValueVector, front::Integer, back::Integer)
Insert front null entries at the beginning of Xand add back null entries at
the end of X. Returns X.
source
52.5 DataValues.padna
DataValues.padna —Method.
padna(X::DataValueVector, front::Integer, back::Integer)
return a copy of Xwith front null entries inserted at the beginning of the
copy and back null entries inserted at the end.
source
Chapter 53
OnlineStats
53.1 OnlineStats.mapblocks
OnlineStats.mapblocks —Function.
mapblocks(f::Function, b::Int, data, dim::ObsDimension = Rows())
Map data in batches of size bto the function f. If data includes an Ab-
stractMatrix, the batches will be based on rows or columns, depending on dim.
Most usage is through Julia’s do block syntax.
Examples
s = Series(Mean())
mapblocks(10, randn(100)) do yi
fit!(s, yi)
info("nobs: $(nobs(s))")
end
x=[1234;
1234;
1234;
1234]
mapblocks(println, 2, x)
mapblocks(println, 2, x, Cols())
source
53.2 OnlineStats.series
OnlineStats.series —Method.
387
388 CHAPTER 53. ONLINESTATS
series(o::OnlineStat...; kw...)
series(wt::Weight, o::OnlineStat...; kw...)
series(data, o::OnlineStat...; kw...)
series(data, wt::Weight, o::OnlineStat...; kw...)
Create a Series or AugmentedSeries based on whether keyword arguments
filter and transform are present.
Example
series(-rand(100), Mean(), Variance(); filter = isfinite, transform = abs)
source
53.3 StatsBase.confint
StatsBase.confint —Function.
confint(b::Bootstrap, coverageprob = .95)
Return a confidence interval for a Bootstrap b.
source
53.4 StatsBase.fit!
StatsBase.fit! —Method.
fit!(s::Series, data)
Update a Series with more data.
Examples
# Univariate Series
s = Series(Mean())
fit!(s, randn(100))
# Multivariate Series
x = randn(100, 3)
s = Series(CovMatrix(3))
fit!(s, x) # Same as fit!(s, x, Rows())
fit!(s, x’, Cols())
# Model Series
x, y = randn(100, 10), randn(100)
s = Series(LinReg(10))
fit!(s, (x, y))
source
Chapter 54
NearestNeighbors
54.1 NearestNeighbors.injectdata
NearestNeighbors.injectdata —Method.
injectdata(datafreetree, data) -> tree
Returns the KDTree/BallTree wrapped by datafreetree, set up to use
data for the points data.
source
54.2 NearestNeighbors.inrange
NearestNeighbors.inrange —Method.
inrange(tree::NNTree, points, radius [, sortres=false]) -> indices
Find all the points in the tree which is closer than radius to points. If
sortres = true the resulting indices are sorted.
source
54.3 NearestNeighbors.knn
NearestNeighbors.knn —Method.
knn(tree::NNTree, points, k [, sortres=false]) -> indices, distances
Performs a lookup of the knearest neigbours to the points from the data
in the tree. If sortres = true the result is sorted such that the results are in
the order of increasing distance to the point. skip is an optional predicate to
determine if a point that would be returned should be skipped.
source
389
Chapter 55
IJulia
55.1 IJulia.installkernel
IJulia.installkernel —Method.
installkernel(name, options...; specname=replace(lowercase(name), " ", "-")
Install a new Julia kernel, where the given options are passed to the julia
executable, and the user-visible kernel name is given by name followed by the
Julia version.
Internally, the Jupyter name for the kernel (for the jupyter kernelspec
command is given by the optional keyword specname (which defaults to name,
converted to lowercase with spaces replaced by hyphens), followed by the Julia
version number.
Both the kernelspec command (a Cmd object) and the new kernel name are
returned by installkernel. For example:
kernelspec, kernelname = installkernel("Julia O3", "-O3")
creates a new Julia kernel in which julia is launched with the -O3 optimiza-
tion flag. The returned kernelspec command will be something like jupyter
kernelspec (perhaps with different path), and kernelname will be something
like julia-O3-0.6 (in Julia 0.6). You could uninstall the kernel by running e.g.
run(‘$kernelspec remove -f $kernelname‘)
source
55.2 IJulia.notebook
IJulia.notebook —Method.
notebook(; dir=homedir(), detached=false)
390
55.2. IJULIA.NOTEBOOK 391
The notebook() function launches the Jupyter notebook, and is equivalent
to running jupyter notebook at the operating-system command-line. The ad-
vantage of launching the notebook from Julia is that, depending on how Jupyter
was installed, the user may not know where to find the jupyter executable.
By default, the notebook server is launched in the user’s home directory, but
this location can be changed by passing the desired path in the dir keyword
argument. e.g. notebook(dir=pwd()) to use the current directory.
By default, notebook() does not return; you must hit ctrl-c or quit Ju-
lia to interrupt it, which halts Jupyter. So, you must leave the Julia termi-
nal open for as long as you want to run Jupyter. Alternatively, if you run
notebook(detached=true), the jupyter notebook will launch in the back-
ground, and will continue running even after you quit Julia. (The only way to
stop Jupyter will then be to kill it in your operating system’s process manager.)
source
Chapter 56
WebSockets
56.1 Base.close
Base.close —Method.
close(ws::WebSocket)
Send a close message.
source
56.2 Base.read
Base.read —Method.
read(ws::WebSocket)
Read one non-control message from a WebSocket. Any control messages that
are read will be handled by the handle control frame function. This function
will not return until a full non-control message has been read. If the other side
doesn’t ever complete its message, this function will never return. Only the data
(contents/body/payload) of the message will be returned from this function.
source
56.3 Base.write
Base.write —Method.
Write binary data; will be sent as one frame.
source
392
56.4. BASE.WRITE 393
56.4 Base.write
Base.write —Method.
Write text data; will be sent as one frame.
source
56.5 WebSockets.send ping
WebSockets.send ping —Method.
Send a ping message, optionally with data.
source
56.6 WebSockets.send pong
WebSockets.send pong —Method.
Send a pong message, optionally with data.
source
Chapter 57
AutoGrad
57.1 AutoGrad.getval
AutoGrad.getval —Method.
getval(x)
Unbox xif it is a boxed value (Rec), otherwise return x.
source
57.2 AutoGrad.grad
AutoGrad.grad —Function.
grad(fun, argnum=1)
Take a function fun(X...)->Yand return another function gfun(X...)->dXi
which computes its gradient with respect to positional argument number argnum.
The function fun should be scalar-valued. The returned function gfun takes the
same arguments as fun, but returns the gradient instead. The gradient has the
same type and size as the target argument which can be a Number, Array,
Tuple, or Dict.
source
57.3 AutoGrad.gradcheck
AutoGrad.gradcheck —Method.
gradcheck(f, w, x...; kwargs...)
394
57.4. AUTOGRAD.GRADLOSS 395
Numerically check the gradient of f(w,x...;o...) with respect to its first
argument wand return a boolean result.
The argument wcan be a Number, Array, Tuple or Dict which in turn can
contain other Arrays etc. Only the largest 10 entries in each numerical gradient
array are checked by default. If the output of f is not a number, gradcheck
constructs and checks a scalar function by taking its dot product with a random
vector.
Keywords
•gcheck=10: number of largest entries from each numeric array in gradient
dw=(grad(f))(w,x...;o...) compared to their numerical estimates.
•verbose=false: print detailed messages if true.
•kwargs=[]: keyword arguments to be passed to f.
•delta=atol=rtol=cbrt(eps(w)): tolerance parameters. See isapprox
for their meaning.
source
57.4 AutoGrad.gradloss
AutoGrad.gradloss —Function.
gradloss(fun, argnum=1)
Another version of grad where the generated function returns a (gradi-
ent,value) pair.
source
Chapter 58
ComputationalResources
58.1 ComputationalResources.addresource
ComputationalResources.addresource —Method.
addresource(T)
Add Tto the list of available resources. For example, addresource(OpenCLLibs)
would indicate that you have a GPU and the OpenCL libraries installed.
source
58.2 ComputationalResources.haveresource
ComputationalResources.haveresource —Method.
haveresource(T)
Returns true if Tis an available resource. For example, haveresource(OpenCLLibs)
tests whether the OpenCLLibs have been added as an available resource. This
function is typically used inside a module’s init function.
Example:
[] The initfunctionf orM yP ackage:f unctioninit()...otherinitializationcode,possiblysettingtheLOADPAT Hif haver esource(OpenCLLibs)@evalusingM yP ackageCLaseparatemodulecontainingOpenCLimplementationsendP utadditionalresourcecheckshere...otherinitializationcode,possiblycleaninguptheLOADPAT Hend
source
58.3 ComputationalResources.rmresource
ComputationalResources.rmresource —Method.
rmresource(T)
396
58.3. COMPUTATIONALRESOURCES.RMRESOURCE 397
Remove Tfrom the list of available resources. For example, rmresource(OpenCLLibs)
would indicate that any future package loads should avoid loading their special-
izations for OpenCL.
source
Chapter 59
Clustering
59.1 Clustering.dbscan
Clustering.dbscan —Method.
dbscan(points, radius ; leafsize = 20, min_neighbors = 1, min_cluster_size = 1) -> clusters
Cluster points using the DBSCAN (density-based spatial clustering of ap-
plications with noise) algorithm.
Arguments
•points: matrix of points
•radius::Real: query radius
Keyword Arguments
•leafsize::Int: number of points binned in each leaf node in the KDTree
•min neighbors::Int: minimum number of neighbors to be a core point
•min cluster size::Int: minimum number of points to be a valid cluster
Output
•Vector{DbscanCluster}: an array of clusters with the id, size core indices
and boundary indices
Example:
[] points = randn(3, 10000) clusters = dbscan(points, 0.05, minneighbors =
3, minclustersize =20)clusterswithlessthan20pointswillbediscarded
source
398
59.2. CLUSTERING.MCL 399
59.2 Clustering.mcl
Clustering.mcl —Method.
mcl(adj::Matrix; [keyword arguments])::MCLResult
Identify clusters in the weighted graph using Markov Clustering Algorithm
(MCL).
Arguments
•adj::Matrix{Float64}: adjacency matrix that defines the weighted graph
to cluster
•add loops::Bool: whether edges of weight 1.0 from the node to itself
should be appended to the graph (enabled by default)
•expansion::Number: MCL expansion constant (2)
•inflation::Number: MCL inflation constant (2.0)
•save final matrix::Bool: save final equilibrium state in the result, oth-
erwise leave it empty; disabled by default, could be useful if MCL doesn’t
converge
•max iter::Integer: max number of MCL iterations
•tol::Number: MCL adjacency matrix convergence threshold
•prune tol::Number: pruning threshold
•display::Symbol::none for no output or :verbose for diagnostic mes-
sages
See original MCL implementation.
Ref: Stijn van Dongen, “Graph clustering by flow simulation”, 2001
source
Chapter 60
JuliaWebAPI
60.1 JuliaWebAPI.apicall
JuliaWebAPI.apicall —Method.
Calls a remote api cmd with args... and data.... The response is format-
ted as specified by the formatter specified in conn.
source
60.2 JuliaWebAPI.fnresponse
JuliaWebAPI.fnresponse —Method.
extract and return the response data as a direct function call would have
returned but throw error if the call was not successful.
source
60.3 JuliaWebAPI.httpresponse
JuliaWebAPI.httpresponse —Method.
construct an HTTP Response object from the API response
source
60.4 JuliaWebAPI.process
JuliaWebAPI.process —Method.
start processing as a server
source
400
60.5. JULIAWEBAPI.REGISTER 401
60.5 JuliaWebAPI.register
JuliaWebAPI.register —Method.
Register a function as API call. TODO: validate method belongs to module?
source
Chapter 61
DecisionTree
61.1 DecisionTree.apply adaboost stumps proba
DecisionTree.apply adaboost stumps proba —Method.
apply_adaboost_stumps_proba(stumps::Ensemble, coeffs, features, labels::Vector)
computes P(L=label|X) for each row in features. It returns a N row x
n labels matrix of probabilities, each row summing up to 1.
col labels is a vector containing the distinct labels (eg. [“versicolor”, “vir-
ginica”, “setosa”]). It specifies the column ordering of the output matrix.
source
61.2 DecisionTree.apply forest proba
DecisionTree.apply forest proba —Method.
apply_forest_proba(forest::Ensemble, features, col_labels::Vector)
computes P(L=label|X) for each row in features. It returns a N row x
n labels matrix of probabilities, each row summing up to 1.
col labels is a vector containing the distinct labels (eg. [“versicolor”, “vir-
ginica”, “setosa”]). It specifies the column ordering of the output matrix.
source
61.3 DecisionTree.apply tree proba
DecisionTree.apply tree proba —Method.
apply_tree_proba(::Node, features, col_labels::Vector)
402
61.3. DECISIONTREE.APPLY TREE PROBA 403
computes P(L=label|X) for each row in features. It returns a N row x
n labels matrix of probabilities, each row summing up to 1.
col labels is a vector containing the distinct labels (eg. [“versicolor”, “vir-
ginica”, “setosa”]). It specifies the column ordering of the output matrix.
source
Chapter 62
Blosc
62.1 Blosc.compress
Blosc.compress —Function.
compress(data; level=5, shuffle=true, itemsize)
Return a Vector{UInt8}of the Blosc-compressed data, where data is an
array or a string.
The level keyword indicates the compression level (between 0=no compres-
sion and 9=max), shuffle indicates whether to use Blosc’s shuffling precondi-
tioner, and the shuffling preconditioner is optimized for arrays of binary items
of size (in bytes) itemsize (defaults to sizeof(eltype(data)) for arrays and
the size of the code units for strings).
source
62.2 Blosc.compress!
Blosc.compress! —Function.
compress!(dest::Vector{UInt8}, src; kws...)
Like compress(src; kws...), but writes to a pre-allocated array dest of
bytes. The return value is the size in bytes of the data written to dest, or 0if
the buffer was too small.
source
62.3 Blosc.decompress!
Blosc.decompress! —Method.
decompress!(dest::Vector{T}, src::Vector{UInt8})
404
62.4. BLOSC.DECOMPRESS 405
Like decompress, but uses a pre-allocated destination buffer dest, which is
resized as needed to store the decompressed data from src.
source
62.4 Blosc.decompress
Blosc.decompress —Method.
decompress(T::Type, src::Vector{UInt8})
Return the compressed buffer src as an array of element type T.
source
Chapter 63
Missings
63.1 Missings.allowmissing
Missings.allowmissing —Method.
allowmissing(x::AbstractArray)
Return an array equal to xallowing for missing values, i.e. with an element
type equal to Union{eltype(x), Missing}.
When possible, the result will share memory with x(as with convert).
See also: disallowmissing
source
63.2 Missings.disallowmissing
Missings.disallowmissing —Method.
disallowmissing(x::AbstractArray)
Return an array equal to xnot allowing for missing values, i.e. with an
element type equal to Missings.T(eltype(x)).
When possible, the result will share memory with x(as with convert). If x
contains missing values, a MethodError is thrown.
See also: allowmissing
source
63.3 Missings.levels
Missings.levels —Method.
levels(x)
406
63.3. MISSINGS.LEVELS 407
Return a vector of unique values which occur or could occur in collection x,
omitting missing even if present. Values are returned in the preferred order for
the collection, with the result of sort as a default.
Contrary to unique, this function may return values which do not actually
occur in the data, and does not preserve their order of appearance in x.
source
Chapter 64
Parameters
64.1 Parameters.reconstruct
Parameters.reconstruct —Method.
Make a new instance of a type with the same values as the input type except
for the fields given in the associative second argument or as keywords.
[] struct A; a; b end a = A(3,4) b = reconstruct(a, [(:b, 99)]) ==A(3,99)
source
64.2 Parameters.type2dict
Parameters.type2dict —Method.
Transforms a type-instance into a dictionary.
julia> type T
a
b
end
julia> type2dict(T(4,5))
Dict{Symbol,Any} with 2 entries:
:a => 4
:b => 5
source
64.3 Parameters.with kw
Parameters.with kw —Function.
This function is called by the @with kw macro and does the syntax transfor-
mation from:
408
64.3. PARAMETERS.WITH KW 409
[] @withkwstructM M {R}r:: R=1000.a :: Rend
into
[] struct MM{R}r::R a::R MM{R}(r,a) where {R}= new(r,a) MM{R}(;r=1000.,
a=error(”no default for a”)) where {R}= MM{R}(r,a) inner kw, type-paras
are required when calling end MM(r::R,a::R) where {R}= MM{R}(r,a) de-
fault outer positional constructor MM(;r=1000,a=error(”no default for a”)) =
MM(r,a) outer kw, so no type-paras are needed when calling MM(m::MM;
kws...) = reconstruct(mm,kws) MM(m::MM, di::Union{Associative, Tuple{Symbol,Any}})
= reconstruct(mm, di) macro unpackMM(varname)esc(quoter =varname.ra =varname.aend)endmacropackMM(varname)esc(quotevarname =P arameters.reconstruct(varname, r =r, a =a)end)end
source
Chapter 65
HDF5
65.1 HDF5.h5open
HDF5.h5open —Function.
h5open(filename::AbstractString, mode::AbstractString="r"; swmr=false)
Open or create an HDF5 file where mode is one of:
•“r” read only
•“r+” read and write
•“w” read and write, create a new file (destroys any existing contents)
Pass swmr=true to enable (Single Writer Multiple Reader) SWMR write
access for “w” and “r+”, or SWMR read access for “r”.
source
65.2 HDF5.h5open
HDF5.h5open —Method.
function h5open(f::Function, args...; swmr=false)
Apply the function f to the result of h5open(args...;kwargs...) and close
the resulting HDF5File upon completion. For example with a do block:
h5open("foo.h5","w") do h5
h5["foo"]=[1,2,3]
end
source
410
65.3. HDF5.ISHDF5 411
65.3 HDF5.ishdf5
HDF5.ishdf5 —Method.
ishdf5(name::AbstractString)
Returns true if name is a path to a valid hdf5 file, false otherwise.
source
65.4 HDF5.set dims!
HDF5.set dims! —Method.
set_dims!(dset::HDF5Dataset, new_dims::Dims)
Change the current dimensions of a dataset to new dims, limited by max dims
= get dims(dset)[2]. Reduction is possible and leads to loss of truncated
data.
source
Chapter 66
HttpServer
66.1 Base.run
Base.run —Method.
run starts server
Functionality:
•Accepts incoming connections and instantiates each Client.
•Manages the client.id pool.
•Spawns a new Task for each connection.
•Blocks forever.
Method accepts following keyword arguments:
•host - binding address
•port - binding port
•ssl - SSL configuration. Use this argument to enable HTTPS support.
Can be either an MbedTLS.SSLConfig object that already has associated
certificates, or a tuple of an MbedTLS.CRT (certificate) and MbedTLS.PKContext
(private key)). In the latter case, a configuration with reasonable defaults will
be used.
•socket - named pipe/domain socket path. Use this argument to enable
Unix socket support.
It’s available only on Unix. Network options are ignored.
Compatibility:
•for backward compatibility use run(server::Server, port::Integer)
412
66.2. BASE.WRITE 413
Example:
server = Server() do req, res
"Hello world"
end
# start server on localhost
run(server, host=IPv4(127,0,0,1), port=8000)
# or
run(server, 8000)
source
66.2 Base.write
Base.write —Method.
Converts a Response to an HTTP response string
source
66.3 HttpServer.setcookie!
HttpServer.setcookie! —Function.
Sets a cookie with the given name, value, and attributes on the given re-
sponse object.
source
Chapter 67
MappedArrays
67.1 MappedArrays.mappedarray
MappedArrays.mappedarray —Method.
mappedarray(f, A)
creates a view of the array Athat applies fto every element of A. The view
is read-only (you can get values but not set them).
source
67.2 MappedArrays.mappedarray
MappedArrays.mappedarray —Method.
mappedarray((f, finv), A)
creates a view of the array Athat applies fto every element of A. The inverse
function, finv, allows one to also set values of the view and, correspondingly,
the values in A.
source
67.3 MappedArrays.of eltype
MappedArrays.of eltype —Method.
of_eltype(T, A)
of_eltype(val::T, A)
creates a view of Athat lazily-converts the element type to T.
source
414
Chapter 68
TextParse
68.1 TextParse.csvread
TextParse.csvread —Function.
csvread(file::Union{String,IO}, delim=’,’; <arguments>...)
Read CSV from file. Returns a tuple of 2 elements:
1. A tuple of columns each either a Vector,DataValueArray or PooledArray
2. column names if header exists=true, empty array otherwise
Arguments:
•file: either an IO object or file name string
•delim: the delimiter character
•spacedelim: (Bool) parse space-delimited files. delim has no effect if
true.
•quotechar: character used to quote strings, defaults to "
•escapechar: character used to escape quotechar in strings. (could be the
same as quotechar)
•pooledstrings: whether to try and create PooledArray of strings
•nrows: number of rows in the file. Defaults to 0in which case we try to
estimate this.
•skiplines begin: skips specified number of lines at the beginning of the
file
•header exists: boolean specifying whether CSV file contains a header
415
416 CHAPTER 68. TEXTPARSE
•nastrings: strings that are to be considered NA. Defaults to TextParse.NA STRINGS
•colnames: manually specified column names. Could be a vector or a
dictionary from Int index (the column) to String column name.
•colparsers: Parsers to use for specified columns. This can be a vector
or a dictionary from column name / column index (Int) to a “parser”.
The simplest parser is a type such as Int, Float64. It can also be a
dateformat"...", see CustomParser if you want to plug in custom pars-
ing behavior
•type detect rows: number of rows to use to infer the initial colparsers
defaults to 20.
source
Chapter 69
LossFunctions
69.1 LearnBase.scaled
LearnBase.scaled —Method.
scaled(loss::SupervisedLoss, K)
Returns a version of loss that is uniformly scaled by K. This function dis-
patches on the type of loss in order to choose the appropriate type of scaled
loss that will be used as the decorator. For example, if typeof(loss) <:
DistanceLoss then the given loss will be boxed into a ScaledDistanceLoss.
Note: If typeof(K) <: Number, then this method will poison the type-
inference of the calling scope. This is because Kwill be promoted to a type
parameter. For a typestable version use the following signature: scaled(loss,
Val{K})
source
69.2 LossFunctions.weightedloss
LossFunctions.weightedloss —Method.
weightedloss(loss, weight)
Returns a weighted version of loss for which the value of the positive class
is changed to be weight times its original, and the negative class 1 - weight
times its original respectively.
Note: If typeof(weight) <: Number, then this method will poison the
type-inference of the calling scope. This is because weight will be promoted
to a type parameter. For a typestable version use the following signature:
weightedloss(loss, Val{weight})
source
417
Chapter 70
LearnBase
70.1 LearnBase.grad
LearnBase.grad —Function.
Return the gradient of the learnable parameters w.r.t. some objective
source
70.2 LearnBase.grad!
LearnBase.grad! —Function.
Do a backward pass, updating the gradients of learnable parameters and/or
inputs
source
70.3 LearnBase.inputdomain
LearnBase.inputdomain —Function.
Returns an AbstractSet representing valid input values
source
70.4 LearnBase.targetdomain
LearnBase.targetdomain —Function.
Returns an AbstractSet representing valid output/target values
source
70.5 LearnBase.transform!
LearnBase.transform! —Function.
418
70.5. LEARNBASE.TRANSFORM! 419
Do a forward pass, and return the output
source
Chapter 71
Juno
71.1 Juno.clearconsole
Juno.clearconsole —Method.
clearconsole()
Clear the console if Juno is used; does nothing otherwise.
source
71.2 Juno.input
Juno.input —Function.
input(prompt = "") -> "..."
Prompt the user to input some text, and return it. Optionally display a
prompt.
source
71.3 Juno.selector
Juno.selector —Method.
selector([xs...]) -> x
Allow the user to select one of the xs.
xs should be an iterator of strings. Currently there is no fallback in other
environments.
source
420
71.4. JUNO.STRUCTURE 421
71.4 Juno.structure
Juno.structure —Method.
structure(x)
Display x’s underlying representation, rather than using its normal display
method.
For example, structure(:(2x+1)) displays the Expr object with its head
and args fields instead of printing the expression.
source
Chapter 72
HttpParser
72.1 HttpParser.http method str
HttpParser.http method str —Method.
Returns a string version of the HTTP method.
source
72.2 HttpParser.http parser execute
HttpParser.http parser execute —Method.
Run a request through a parser with specific callbacks on the settings in-
stance.
source
72.3 HttpParser.http parser init
HttpParser.http parser init —Function.
Intializes the Parser object with the correct memory.
source
72.4 HttpParser.parse url
HttpParser.parse url —Method.
Parse a URL
source
422
Chapter 73
ImageAxes
73.1 ImageAxes.istimeaxis
ImageAxes.istimeaxis —Method.
istimeaxis(ax)
Test whether the axis ax corresponds to time.
source
73.2 ImageAxes.timeaxis
ImageAxes.timeaxis —Method.
timeaxis(A)
Return the time axis, if present, of the array A, and nothing otherwise.
source
73.3 ImageAxes.timedim
ImageAxes.timedim —Method.
timedim(img) -> d::Int
Return the dimension of the array used for encoding time, or 0 if not using
an axis for this purpose.
Note: if you want to recover information about the time axis, it is generally
better to use timeaxis.
source
423
Chapter 74
Flux
74.1 Flux.GRU
Flux.GRU —Method.
GRU(in::Integer, out::Integer, = tanh)
Gated Recurrent Unit layer. Behaves like an RNN but generally exhibits a
longer memory span over sequences.
See this article for a good overview of the internals.
source
74.2 Flux.LSTM
Flux.LSTM —Method.
LSTM(in::Integer, out::Integer, = tanh)
Long Short Term Memory recurrent layer. Behaves like an RNN but gener-
ally exhibits a longer memory span over sequences.
See this article for a good overview of the internals.
source
74.3 Flux.RNN
Flux.RNN —Method.
RNN(in::Integer, out::Integer, = tanh)
The most basic recurrent layer; essentially acts as a Dense layer, but with
the output fed back into the input each time step.
source
424
Chapter 75
IntervalSets
75.1 IntervalSets.:..
IntervalSets.:.. —Method.
iv = l..r
Construct a ClosedInterval iv spanning the region from lto r.
source
75.2 IntervalSets.:
IntervalSets.: —Method.
iv = centerhalfwidth
Construct a ClosedInterval iv spanning the region from center - halfwidth
to center + halfwidth.
source
75.3 IntervalSets.width
IntervalSets.width —Method.
w = width(iv)
Calculate the width (max-min) of interval iv. Note that for integers land
r,width(l..r) = length(l:r) - 1.
source
425
Chapter 76
Media
76.1 Media.getdisplay
Media.getdisplay —Method.
getdisplay(T)
Find out what output device Twill display on.
source
76.2 Media.media
Media.media —Method.
media(T) gives the media type of the type T. The default is Textual.
media(Gadfly.Plot) == Media.Plot
source
76.3 Media.setdisplay
Media.setdisplay —Method.
setdisplay([input], T, output)
Display Tobjects using output when produced by input.
Tis an object type or media type, e.g. Gadfly.Plot or Media.Graphical.
display(Editor(), Image, Console())
source
426
Chapter 77
Rotations
77.1 Rotations.isrotation
Rotations.isrotation —Method.
isrotation(r)
isrotation(r, tol)
Check whether ris a 33 rotation matrix, where r*ris within tol of the
identity matrix (using the Frobenius norm). (tol defaults to 1000 * eps(eltype(r)).)
source
77.2 Rotations.rotation between
Rotations.rotation between —Method.
rotation_between(from, to)
Compute the quaternion that rotates vector from so that it aligns with vector
to, along the geodesic (shortest path).
source
427
Chapter 78
Mustache
78.1 Mustache.render
Mustache.render —Method.
Render a set of tokens with a view, using optional io object to print or store.
Arguments
•io::IO: Optional IO object.
•tokens: Either Mustache tokens, or a string to parse into tokens
•view: A view provides a context to look up unresolved symbols demar-
cated by mustache braces. A view may be specified by a dictionary, a
module, a composite type, a vector, or keyword arguments.
source
78.2 Mustache.render from file
Mustache.render from file —Method.
Renders a template from filepath and view. If it has seen the file before
then it finds the compiled MustacheTokens in TEMPLATES rather than calling
parse a second time.
source
428
Chapter 79
TiledIteration
79.1 TiledIteration.padded tilesize
TiledIteration.padded tilesize —Method.
padded_tilesize(T::Type, kernelsize::Dims, [ncache=2]) -> tilesize::Dims
Calculate a suitable tile size to approximately maximize the amount of pro-
ductive work, given a stencil of size kernelsize. The element type of the array
is T. Optionally specify ncache, the number of such arrays that you’d like to
have fit simultaneously in L1 cache.
This favors making the first dimension larger, since the first dimension cor-
responds to individual cache lines.
Examples
julia>padded tilesize(UInt8, (3,3)) (768,18)
julia>padded tilesize(UInt8, (3,3), 4) (512,12)
julia>padded tilesize(Float64, (3,3)) (96,18)
julia>padded tilesize(Float32, (3,3,3)) (64,6,6)
source
429
Chapter 80
Distances
80.1 Distances.renyi divergence
Distances.renyi divergence —Function.
RenyiDivergence(::Real)
renyi_divergence(P, Q, ::Real)
Create a Rnyi premetric of order .
Rnyi defined a spectrum of divergence measures generalising the Kullback–
Leibler divergence (see KLDivergence). The divergence is not a semimetric
as it is not symmetric. It is parameterised by a parameter , and is equal to
Kullback–Leibler divergence at = 1:
At = 0, R0(P|Q) = −log(sumi:pi>0(qi))
At = 1, R1(P|Q) = sumi:pi>0(pilog(pi/qi))
At = , R(P|Q) = log(supi:pi>0(pi/qi))
Otherwise R(P|Q) = log(sumi:pi>0((pi)/(q(
i−1)))/(−1)
Example:
julia> x = reshape([0.1, 0.3, 0.4, 0.2], 2, 2);
julia> pairwise(RenyiDivergence(0), x, x)
22 Array{Float64,2}:
0.0 0.0
0.0 0.0
julia> pairwise(Euclidean(2), x, x)
22 Array{Float64,2}:
0.0 0.577315
0.655407 0.0
source
430
Chapter 81
HttpCommon
81.1 HttpCommon.escapeHTML
HttpCommon.escapeHTML —Method.
escapeHTML(i::String)
Returns a string with special HTML characters escaped: &, <,>, “, ’
source
81.2 HttpCommon.parsequerystring
HttpCommon.parsequerystring —Method.
parsequerystring(query::String)
Convert a valid querystring to a Dict:
q = "foo=bar&baz=%3Ca%20href%3D%27http%3A%2F%2Fwww.hackershool.com%27%3Ehello%20world%21%3C%2Fa%3E"
parsequerystring(q)
# Dict{String,String} with 2 entries:
# "baz" => "<a href=’http://www.hackershool.com’>hello world!</a>"
# "foo" => "bar"
source
431
Chapter 82
StaticArrays
82.1 StaticArrays.similar type
StaticArrays.similar type —Function.
similar_type(static_array)
similar_type(static_array, T)
similar_type(array, ::Size)
similar_type(array, T, ::Size)
Returns a constructor for a statically-sized array similar to the input array
(or type) static array/array, optionally with different element type Tor size
Size. If the input array is not a StaticArray then the Size is mandatory.
This differs from similar() in that the resulting array type may not be
mutable (or define setindex!()), and therefore the returned type may need to
be constructed with its data.
Note that the (optional) size must be specified as a static Size object (so
the compiler can infer the result statically).
New types should define the signature similar type{A<:MyType,T,S}(::Type{A},::Type{T},::Size{S})
if they wish to overload the default behavior.
source
432
Chapter 83
SweepOperator
83.1 SweepOperator.sweep!
SweepOperator.sweep! —Method.
Symmetric sweep operator
Symmetric sweep operator of the matrix Aon element k.Ais overwritten.
inv = true will perform the inverse sweep. Only the upper triangle is read and
swept.
sweep!(A, k, inv = false)
Providing a Range, rather than an Integer, sweeps on each element in the
range.
sweep!(A, first:last, inv = false)
Example:
[] x = randn(100, 10) xtx = x’x sweep!(xtx, 1) sweep!(xtx, 1, true)
source
433
Chapter 84
PaddedViews
84.1 PaddedViews.paddedviews
PaddedViews.paddedviews —Method.
Aspad = paddedviews(fillvalue, A1, A2, ....)
Pad the arrays A1,A2, . . . , to a common size or set of axes, chosen as the
span of axes enclosing all of the input arrays.
Example:
[] julia¿ a1 = reshape([1,2], 2, 1) 21 Array{Int64,2}: 1 2
julia¿ a2 = [1.0,2.0]’ 12 Array{Float64,2}: 1.0 2.0
julia¿ a1p, a2p = paddedviews(0, a1, a2);
julia¿ a1p 22 PaddedViews.PaddedView{Int64,2,Tuple{Base.OneTo{Int64},Base.OneTo{Int64}},Array{Int64,2}}:
1 0 2 0
julia¿ a2p 22 PaddedViews.PaddedView{Float64,2,Tuple{Base.OneTo{Int64},Base.OneTo{Int64}},Array{Float64,2}}:
1.0 2.0 0.0 0.0
source
434
Chapter 85
SpecialFunctions
85.1 SpecialFunctions.cosint
SpecialFunctions.cosint —Function.
cosint(x)
Compute the cosine integral function of x, defined by Ci(x) := γ+ log x+
Rx
0
cos t−1
tdt for real x>0, where γis the Euler-Mascheroni constant.
source
85.2 SpecialFunctions.sinint
SpecialFunctions.sinint —Function.
sinint(x)
Compute the sine integral function of x, defined by Si(x) := Rx
0
sin t
tdt for
real x.
source
435
Chapter 86
NamedTuples
86.1 NamedTuples.delete
NamedTuples.delete —Method.
Create a new NamedTuple with the specified element removed.
source
86.2 NamedTuples.setindex
NamedTuples.setindex —Method.
Create a new NamedTuple with the new value set on it, either overwriting
the old value or appending a new value. This copies the underlying data.
source
436
Chapter 87
Loess
87.1 Loess.loess
Loess.loess —Method.
loess(xs, ys, normalize=true, span=0.75, degreee=2)
Fit a loess model.
Args: xs: A nby mmatrix with nobservations from mindependent predic-
tors ys: A length nresponse vector. normalize: Normalize the scale of each
predicitor. (default true when m>1)span: The degree of smoothing, typi-
cally in [0,1]. Smaller values result in smaller local context in fitting. degree:
Polynomial degree.
Returns: A fit LoessModel.
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Chapter 88
Nulls
88.1 Nulls.levels
Nulls.levels —Method.
levels(x)
Return a vector of unique values which occur or could occur in collection x,
omitting null even if present. Values are returned in the preferred order for
the collection, with the result of sort as a default.
Contrary to unique, this function may return values which do not actually
occur in the data, and does not preserve their order of appearance in x.
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438
Chapter 89
WoodburyMatrices
89.1 WoodburyMatrices.liftFactor
WoodburyMatrices.liftFactor —Method.
liftFactor(A)
More stable version of inv(A). Returns a function which computs the inverse
on evaluation, i.e. liftFactor(A)(x) is the same as inv(A)*x.
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439
Chapter 90
Requests
90.1 Requests.save
Requests.save —Function.
save(r::Response, path=".")
Saves the data in the response in the directory path. If the path is a direc-
tory, then the filename is automatically chosen based on the response headers.
Returns the full pathname of the saved file.
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440
Chapter 91
MemPool
91.1 MemPool.savetodisk
MemPool.savetodisk —Method.
Allow users to specifically save something to disk. This does not free the
data from memory, nor does it affect size accounting.
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441
Chapter 92
SimpleTraits
92.1 SimpleTraits.istrait
SimpleTraits.istrait —Method.
This function checks whether a trait is fulfilled by a specific set of types.
istrait(Tr1{Int,Float64}) => return true or false
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442
Chapter 93
BinDeps
93.1 BinDeps.glibc version
BinDeps.glibc version —Method.
glibc_version()
For Linux-based systems, return the version of glibc in use. For non-glibc
Linux and other platforms, returns nothing.
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