SAS/STAT 9.2 User's Guide: The FASTCLUS Procedure (Book Excerpt) SAS Users Guide

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SAS/STAT®9.2 User’s Guide

The FASTCLUS Procedure

(Book Excerpt)

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Chapter 34

The FASTCLUS Procedure

Contents

Overview: FASTCLUS Procedure .......................... 1621

Background .................................. 1622

Getting Started: FASTCLUS Procedure ....................... 1624

Syntax: FASTCLUS Procedure ............................ 1632

PROC FASTCLUS Statement ......................... 1632

BY Statement ................................. 1640

FREQ Statement ................................ 1640

ID Statement .................................. 1641

VAR Statement ................................. 1641

WEIGHT Statement .............................. 1641

Details: FASTCLUS Procedure ........................... 1642

Updates in the FASTCLUS Procedure ..................... 1642

Missing Values ................................. 1642

Output Data Sets ................................ 1643

Computational Resources ........................... 1647

Using PROC FASTCLUS ........................... 1647

Displayed Output ................................ 1649

ODS Table Names ............................... 1652

Examples: FASTCLUS Procedure .......................... 1653

Example 34.1: Fisher’s Iris Data ....................... 1653

Example 34.2: Outliers ............................ 1662

References ...................................... 1673

Overview: FASTCLUS Procedure

The FASTCLUS procedure performs a disjoint cluster analysis on the basis of distances computed

from one or more quantitative variables. The observations are divided into clusters such that every

observation belongs to one and only one cluster; the clusters do not form a tree structure as they do

in the CLUSTER procedure. If you want separate analysis for different numbers of clusters, you

can run PROC FASTCLUS once for each analysis. Alternatively, to do hierarchical clustering on

a large data set, use PROC FASTCLUS to ﬁnd initial clusters, and then use those initial clusters as

input to PROC CLUSTER.

1622 FChapter 34: The FASTCLUS Procedure

By default, the FASTCLUS procedure uses Euclidean distances, so the cluster centers are based

on least squares estimation. This kind of clustering method is often called a k-means model, since

the cluster centers are the means of the observations assigned to each cluster when the algorithm is

run to complete convergence. Each iteration reduces the least squares criterion until convergence is

achieved.

Often there is no need to run the FASTCLUS procedure to convergence. PROC FASTCLUS is

designed to ﬁnd good clusters (but not necessarily the best possible clusters) with only two or three

passes through the data set. The initialization method of PROC FASTCLUS guarantees that, if

there exist clusters such that all distances between observations in the same cluster are less than all

distances between observations in different clusters, and if you tell PROC FASTCLUS the correct

number of clusters to ﬁnd, it can always ﬁnd such a clustering without iterating. Even with clusters

that are not as well separated, PROC FASTCLUS usually ﬁnds initial seeds that are sufﬁciently

good that few iterations are required. Hence, by default, PROC FASTCLUS performs only one

iteration.

The initialization method used by the FASTCLUS procedure makes it sensitive to outliers. PROC

FASTCLUS can be an effective procedure for detecting outliers because outliers often appear as

clusters with only one member.

The FASTCLUS procedure can use an Lp(least pth powers) clustering criterion (Spath 1985,

pp. 62–63) instead of the least squares (L2) criterion used in k-means clustering methods. The

LEAST=poption speciﬁes the power pto be used. Using the LEAST= option increases execution

time since more iterations are usually required, and the default iteration limit is increased when

you specify LEAST=p. Values of pless than 2 reduce the effect of outliers on the cluster centers

compared with least squares methods; values of pgreater than 2 increase the effect of outliers.

The FASTCLUS procedure is intended for use with large data sets, with 100 or more observations.

With small data sets, the results can be highly sensitive to the order of the observations in the data

set.

PROC FASTCLUS uses algorithms that place a larger inﬂuence on variables with larger variance,

so it might be necessary to standardize the variables before performing the cluster analysis. See the

“Using PROC FASTCLUS” section for standardization details.

PROC FASTCLUS produces brief summaries of the clusters it ﬁnds. For more extensive examina-

tion of the clusters, you can request an output data set containing a cluster membership variable.

Background

The FASTCLUS procedure combines an effective method for ﬁnding initial clusters with a standard

iterative algorithm for minimizing the sum of squared distances from the cluster means. The result

is an efﬁcient procedure for disjoint clustering of large data sets. PROC FASTCLUS was directly

inspired by Hartigan’s (1975) leader algorithm and MacQueen’s (1967) k-means algorithm. PROC

FASTCLUS uses a method that Anderberg (1973) calls nearest centroid sorting. A set of points

called cluster seeds is selected as a ﬁrst guess of the means of the clusters. Each observation is

assigned to the nearest seed to form temporary clusters. The seeds are then replaced by the means

Background F1623

of the temporary clusters, and the process is repeated until no further changes occur in the clusters.

Similar techniques are described in most references on clustering (Anderberg 1973; Hartigan 1975;

Everitt 1980; Spath 1980).

The FASTCLUS procedure differs from other nearest centroid sorting methods in the way the initial

cluster seeds are selected. The importance of initial seed selection is demonstrated by Milligan

(1980).

The clustering is done on the basis of Euclidean distances computed from one or more numeric

variables. If there are missing values, PROC FASTCLUS computes an adjusted distance by using

the nonmissing values. Observations that are very close to each other are usually assigned to the

same cluster, while observations that are far apart are in different clusters.

The FASTCLUS procedure operates in four steps:

1. Observations called cluster seeds are selected.

2. If you specify the DRIFT option, temporary clusters are formed by assigning each observation

to the cluster with the nearest seed. Each time an observation is assigned, the cluster seed is

updated as the current mean of the cluster. This method is sometimes called incremental,

on-line, or adaptive training.

3. If the maximum number of iterations is greater than zero, clusters are formed by assigning

each observation to the nearest seed. After all observations are assigned, the cluster seeds are

replaced by either the cluster means or other location estimates (cluster centers) appropriate

to the LEAST=poption. This step can be repeated until the changes in the cluster seeds

become small or zero (MAXITER=n1).

4. Final clusters are formed by assigning each observation to the nearest seed.

If PROC FASTCLUS runs to complete convergence, the ﬁnal cluster seeds will equal the cluster

means or cluster centers. If PROC FASTCLUS terminates before complete convergence, which

often happens with the default settings, the ﬁnal cluster seeds might not equal the cluster means or

cluster centers. If you want complete convergence, specify CONVERGE=0 and a large value for

the MAXITER= option.

The initial cluster seeds must be observations with no missing values. You can specify the maximum

number of seeds (and, hence, clusters) by using the MAXCLUSTERS= option. You can also specify

a minimum distance by which the seeds must be separated by using the RADIUS= option.

PROC FASTCLUS always selects the ﬁrst complete (no missing values) observation as the ﬁrst

seed. The next complete observation that is separated from the ﬁrst seed by at least the distance

speciﬁed in the RADIUS= option becomes the second seed. Later observations are selected as new

seeds if they are separated from all previous seeds by at least the radius, as long as the maximum

number of seeds is not exceeded.

If an observation is complete but fails to qualify as a new seed, PROC FASTCLUS considers using

it to replace one of the old seeds. Two tests are made to see if the observation can qualify as a new

seed.

First, an old seed is replaced if the distance between the observation and the closest seed is greater

than the minimum distance between seeds. The seed that is replaced is selected from the two

1624 FChapter 34: The FASTCLUS Procedure

seeds that are closest to each other. The seed that is replaced is the one of these two with the

shortest distance to the closest of the remaining seeds when the other seed is replaced by the current

observation.

If the observation fails the ﬁrst test for seed replacement, a second test is made. The observation

replaces the nearest seed if the smallest distance from the observation to all seeds other than the

nearest one is greater than the shortest distance from the nearest seed to all other seeds. If the

observation fails this test, PROC FASTCLUS goes on to the next observation.

You can specify the REPLACE= option to limit seed replacement. You can omit the second test

for seed replacement (REPLACE=PART), causing PROC FASTCLUS to run faster, but the seeds

selected might not be as widely separated as those obtained by the default method. You can also

suppress seed replacement entirely by specifying REPLACE=NONE. In this case, PROC FAST-

CLUS runs much faster, but you must choose a good value for the RADIUS= option in order to

get good clusters. This method is similar to Hartigan’s (1975, pp. 74–78) leader algorithm and the

simple cluster seeking algorithm described by Tou and Gonzalez (1974, pp. 90–92).

Getting Started: FASTCLUS Procedure

The following example demonstrates how to use the FASTCLUS procedure to compute disjoint

clusters of observations in a SAS data set.

The data in this example are measurements taken on 159 freshwater ﬁsh caught from the same lake

(Laengelmavesi) near Tampere in Finland. This data set is available from the Data Archive of the

Journal of Statistics Education. The complete data set is displayed in Chapter 82, “The STEPDISC

Procedure.”

The species (bream, parkki, pike, perch, roach, smelt, and whiteﬁsh), weight, three different length

measurements (measured from the nose of the ﬁsh to the beginning of its tail, the notch of its tail,

and the end of its tail), height, and width of each ﬁsh are tallied. The height and width are recorded

as percentages of the third length variable.

Suppose that you want to group empirically the ﬁsh measurements into clusters and that you want to

associate the clusters with the species. You can use the FASTCLUS procedure to perform a cluster

analysis.

The following DATA step creates the SAS data set Fish:

proc format;

value specfmt

1=’Bream’

2=’Roach’

3=’Whitefish’

4=’Parkki’

5=’Perch’

6=’Pike’

7=’Smelt’;

run;

Getting Started: FASTCLUS Procedure F1625

data fish (drop=HtPct WidthPct);

title ’Fish Measurement Data’;

input Species Weight Length1 Length2 Length3 HtPct WidthPct @@;

*** transform variables;

if Weight <= 0 or Weight =. then delete;

Weight3=Weight**(1/3);

Height=HtPct*Length3/(Weight3*100);

Width=WidthPct*Length3/(Weight3*100);

Length1=Length1/Weight3;

Length2=Length2/Weight3;

Length3=Length3/Weight3;

logLengthRatio=log(Length3/Length1);

format Species specfmt.;

symbol = put(Species, specfmt2.);

datalines;

1 242.0 23.2 25.4 30.0 38.4 13.4 1 290.0 24.0 26.3 31.2 40.0 13.8

1 340.0 23.9 26.5 31.1 39.8 15.1 1 363.0 26.3 29.0 33.5 38.0 13.3

1 430.0 26.5 29.0 34.0 36.6 15.1 1 450.0 26.8 29.7 34.7 39.2 14.2

1 500.0 26.8 29.7 34.5 41.1 15.3 1 390.0 27.6 30.0 35.0 36.2 13.4

1 450.0 27.6 30.0 35.1 39.9 13.8 1 500.0 28.5 30.7 36.2 39.3 13.7

1 475.0 28.4 31.0 36.2 39.4 14.1 1 500.0 28.7 31.0 36.2 39.7 13.3

1 500.0 29.1 31.5 36.4 37.8 12.0 1 . 29.5 32.0 37.3 37.3 13.6

1 600.0 29.4 32.0 37.2 40.2 13.9 1 600.0 29.4 32.0 37.2 41.5 15.0

... more lines ...

7 9.8 11.4 12.0 13.2 16.7 8.7 7 12.2 11.5 12.2 13.4 15.6 10.4

7 13.4 11.7 12.4 13.5 18.0 9.4 7 12.2 12.1 13.0 13.8 16.5 9.1

7 19.7 13.2 14.3 15.2 18.9 13.6 7 19.9 13.8 15.0 16.2 18.1 11.6

;

The double trailing at sign (@@) in the INPUT statement speciﬁes that observations are input from

each line until all values are read. The variables are rescaled in order to adjust for dimensionality.

Because the new variables Weight3–logLengthRatio depend on the variable Weight, observations with

missing values for Weight are not added to the data set. Consequently, there are 157 observations in

the SAS data set Fish.

In the Fish data set, the variables are not measured in the same units and cannot be assumed to have

equal variance. Therefore, it is necessary to standardize the variables before performing the cluster

analysis.

The following statements standardize the variables and perform a cluster analysis on the standard-

ized data:

proc standard data=Fish out=Stand mean=0 std=1;

var Length1 logLengthRatio Height Width Weight3;

proc fastclus data=Stand out=Clust

maxclusters=7 maxiter=100 ;

var Length1 logLengthRatio Height Width Weight3;

run;

1626 FChapter 34: The FASTCLUS Procedure

The STANDARD procedure is ﬁrst used to standardize all the analytical variables to a mean of 0 and

standard deviation of 1. The procedure creates the output data set Stand to contain the transformed

variables.

The FASTCLUS procedure then uses the data set Stand as input and creates the data set Clust.

This output data set contains the original variables and two new variables, Cluster and Distance.

The variable Cluster contains the cluster number to which each observation has been assigned. The

variable Distance gives the distance from the observation to its cluster seed.

It is usually desirable to try several values of the MAXCLUSTERS= option. A reasonable beginning

for this example is to use MAXCLUSTERS=7, since there are seven species of ﬁsh represented in

the data set Fish.

The VAR statement speciﬁes the variables used in the cluster analysis.

The results from this analysis are displayed in the following ﬁgures.

Figure 34.1 Initial Seeds Used in the FASTCLUS Procedure

Fish Measurement Data

The FASTCLUS Procedure

Replace=FULL Radius=0 Maxclusters=7 Maxiter=100 Converge=0.02

Initial Seeds

logLength

Cluster Length1 Ratio Height Width Weight3

-----------------------------------------------------------------------------

1 1.388338414 -0.979577858 -1.594561848 -2.254050655 2.103447062

2 -1.117178039 -0.877218192 -0.336166276 2.528114070 1.170706464

3 2.393997461 -0.662642015 -0.930738701 -2.073879107 -1.839325419

4 -0.495085516 -0.964041012 -0.265106856 -0.028245072 1.536846394

5 -0.728772773 0.540096664 1.130501398 -1.207930053 -1.107018207

6 -0.506924177 0.748211648 1.762482687 0.211507596 1.368987826

7 1.573996573 -0.796593995 -0.824217424 1.561715851 -1.607942726

Criterion Based on Final Seeds = 0.3979

Figure 34.1 displays the table of initial seeds used for each variable and cluster. The ﬁrst line in the

ﬁgure displays the option settings for REPLACE, RADIUS, MAXCLUSTERS, and MAXITER.

These options, with the exception of MAXCLUSTERS and MAXITER, are set at their respective

default values (REPLACE=FULL, RADIUS=0). Both the MAXCLUSTERS= and MAXITER=

options are set in the PROC FASTCLUS statement.

Next, PROC FASTCLUS produces a table of summary statistics for the clusters. Figure 34.2 dis-

plays the number of observations in the cluster (frequency) and the root mean squared standard

deviation. The next two columns display the largest Euclidean distance from the cluster seed to any

observation within the cluster and the number of the nearest cluster.

The last column of the table displays the distance between the centroid of the nearest cluster and

the centroid of the current cluster. A centroid is the point having coordinates that are the means of

all the observations in the cluster.

Getting Started: FASTCLUS Procedure F1627

Figure 34.2 Cluster Summary Table from the FASTCLUS Procedure

Cluster Summary

Maximum Distance

RMS Std from Seed Radius Nearest

Cluster Frequency Deviation to Observation Exceeded Cluster

-----------------------------------------------------------------------------

1 17 0.5064 1.7781 4

2 19 0.3696 1.5007 4

3 13 0.3803 1.7135 1

4 13 0.4161 1.3976 7

5 11 0.2466 0.6966 6

6 34 0.3563 1.5443 5

7 50 0.4447 2.3915 4

Cluster Summary

Distance Between

Cluster Cluster Centroids

-----------------------------

1 2.5106

2 1.5510

3 2.6704

4 1.4266

5 1.7301

6 1.7301

7 1.4266

Figure 34.3 displays the table of statistics for the variables. The table lists for each variable the

total standard deviation, the pooled within-cluster standard deviation and the R-square value for

predicting the variable from the cluster. The ratio of between-cluster variance to within-cluster

variance (R2to 1R2) appears in the last column.

Figure 34.3 Statistics for Variables Used in the FASTCLUS Procedure

Statistics for Variables

Variable Total STD Within STD R-Square RSQ/(1-RSQ)

------------------------------------------------------------------------

Length1 1.00000 0.31428 0.905030 9.529606

logLengthRatio 1.00000 0.39276 0.851676 5.741989

Height 1.00000 0.20917 0.957929 22.769295

Width 1.00000 0.55558 0.703200 2.369270

Weight3 1.00000 0.47251 0.785323 3.658162

OVER-ALL 1.00000 0.40712 0.840631 5.274764

Pseudo F Statistic = 131.87

Approximate Expected Over-All R-Squared = 0.57420

1628 FChapter 34: The FASTCLUS Procedure

The pseudo Fstatistic, approximate expected overall R square, and cubic clustering criterion (CCC)

are listed at the bottom of the ﬁgure. You can compare values of these statistics by running PROC

FASTCLUS with different values for the MAXCLUSTERS= option. The R square and CCC values

are not valid for correlated variables.

Values of the cubic clustering criterion greater than 2 or 3 indicate good clusters. Values between

0 and 2 indicate potential clusters, but they should be taken with caution; large negative values can

indicate outliers.

PROC FASTCLUS next produces the within-cluster means and standard deviations of the variables,

displayed in Figure 34.4.

Figure 34.4 Cluster Means and Standard Deviations from the FASTCLUS Procedure

Cluster Means

logLength

Cluster Length1 Ratio Height Width Weight3

-----------------------------------------------------------------------------

1 1.747808245 -0.868605685 -1.327226832 -1.128760946 0.806373599

2 -0.405231510 -0.979113021 -0.281064162 1.463094486 1.060450065

3 2.006796315 -0.652725165 -1.053213440 -1.224020795 -1.826752838

4 -0.136820952 -1.039312574 -0.446429482 0.162596336 0.278560318

5 -0.850130601 0.550190242 1.245156076 -0.836585750 -0.567022647

6 -0.843912827 1.522291347 1.511408739 -0.380323563 0.763114370

7 -0.165570970 -0.048881276 -0.353723615 0.546442064 -0.668780782

Cluster Standard Deviations

logLength

Cluster Length1 Ratio Height Width Weight3

-----------------------------------------------------------------------------

1 0.3418476428 0.3544065543 0.1666302451 0.6172880027 0.7944227150

2 0.3129902863 0.3592350778 0.1369052680 0.5467406493 0.3720119097

3 0.2962504486 0.1740941675 0.1736086707 0.7528475622 0.0905232968

4 0.3254364840 0.2836681149 0.1884592934 0.4543390702 0.6612055341

5 0.1781837609 0.0745984121 0.2056932592 0.2784540794 0.3832002850

6 0.2273744242 0.3385584051 0.2046010964 0.5143496067 0.4025849044

7 0.3734733622 0.5275768119 0.2551130680 0.5721303628 0.4223181710

It is useful to study further the clusters calculated by the FASTCLUS procedure. One method is

to look at a frequency tabulation of the clusters with other classiﬁcation variables. The following

statements invoke the FREQ procedure to crosstabulate the empirical clusters with the variable

Species:

proc freq data=Clust;

tables Species*Cluster;

run;

Getting Started: FASTCLUS Procedure F1629

Figure 34.5 displays the marked division between clusters.

Figure 34.5 Frequency Table of Cluster versus Species

Fish Measurement Data

The FREQ Procedure

Table of Species by CLUSTER

Species CLUSTER(Cluster)

Frequency |

Percent |

Row Pct |

Col Pct | 1| 2| 3| 4| Total

----------+--------+--------+--------+--------+

Bream | 0 | 0 | 0 | 0 | 34

| 0.00 | 0.00 | 0.00 | 0.00 | 21.66

| 0.00 | 0.00 | 0.00 | 0.00 |

----------+--------+--------+--------+--------+

Roach | 0 | 0 | 0 | 0 | 19

| 0.00 | 0.00 | 0.00 | 0.00 | 12.10

| 0.00 | 0.00 | 0.00 | 0.00 |

----------+--------+--------+--------+--------+

Whitefish | 0 | 2 | 0 | 1 | 6

| 0.00 | 1.27 | 0.00 | 0.64 | 3.82

| 0.00 | 33.33 | 0.00 | 16.67 |

| 0.00 | 10.53 | 0.00 | 7.69 |

----------+--------+--------+--------+--------+

Parkki | 0 | 0 | 0 | 0 | 11

| 0.00 | 0.00 | 0.00 | 0.00 | 7.01

| 0.00 | 0.00 | 0.00 | 0.00 |

----------+--------+--------+--------+--------+

Perch | 0 | 17 | 0 | 12 | 56

| 0.00 | 10.83 | 0.00 | 7.64 | 35.67

| 0.00 | 30.36 | 0.00 | 21.43 |

| 0.00 | 89.47 | 0.00 | 92.31 |

----------+--------+--------+--------+--------+

Pike | 17 | 0 | 0 | 0 | 17

| 10.83 | 0.00 | 0.00 | 0.00 | 10.83

| 100.00 | 0.00 | 0.00 | 0.00 |

----------+--------+--------+--------+--------+

Smelt | 0 | 0 | 13 | 0 | 14

| 0.00 | 0.00 | 8.28 | 0.00 | 8.92

| 0.00 | 0.00 | 92.86 | 0.00 |

| 0.00 | 0.00 | 100.00 | 0.00 |

----------+--------+--------+--------+--------+

Total 17 19 13 13 157

10.83 12.10 8.28 8.28 100.00

(Continued)

1630 FChapter 34: The FASTCLUS Procedure

Figure 34.5 continued

Fish Measurement Data

The FREQ Procedure

Table of Species by CLUSTER

Species CLUSTER(Cluster)

Frequency |

Percent |

Row Pct |

Col Pct | 5| 6| 7| Total

----------+--------+--------+--------+

Bream | 0 | 34 | 0 | 34

| 0.00 | 21.66 | 0.00 | 21.66

| 0.00 | 100.00 | 0.00 |

----------+--------+--------+--------+

Roach | 0 | 0 | 19 | 19

| 0.00 | 0.00 | 12.10 | 12.10

| 0.00 | 0.00 | 100.00 |

| 0.00 | 0.00 | 38.00 |

----------+--------+--------+--------+

Whitefish | 0 | 0 | 3 | 6

| 0.00 | 0.00 | 1.91 | 3.82

| 0.00 | 0.00 | 50.00 |

| 0.00 | 0.00 | 6.00 |

----------+--------+--------+--------+

Parkki | 11 | 0 | 0 | 11

| 7.01 | 0.00 | 0.00 | 7.01

| 100.00 | 0.00 | 0.00 |

----------+--------+--------+--------+

Perch | 0 | 0 | 27 | 56

| 0.00 | 0.00 | 17.20 | 35.67

| 0.00 | 0.00 | 48.21 |

| 0.00 | 0.00 | 54.00 |

----------+--------+--------+--------+

Pike | 0 | 0 | 0 | 17

| 0.00 | 0.00 | 0.00 | 10.83

| 0.00 | 0.00 | 0.00 |

----------+--------+--------+--------+

Smelt | 0 | 0 | 1 | 14

| 0.00 | 0.00 | 0.64 | 8.92

| 0.00 | 0.00 | 7.14 |

| 0.00 | 0.00 | 2.00 |

----------+--------+--------+--------+

Total 11 34 50 157

7.01 21.66 31.85 100.00

Getting Started: FASTCLUS Procedure F1631

For cases in which you have three or more clusters, you can use the CANDISC and SGPLOT proce-

dures to obtain a graphical check on the distribution of the clusters. In the following statements, the

CANDISC and SGPLOT procedures are used to compute canonical variables and plot the clusters:

proc candisc data=Clust out=Can noprint;

class Cluster;

var Length1 logLengthRatio Height Width Weight3;

proc sgplot data=Can;

scatter y=Can2 x=Can1 / group=Cluster ;

run;

First, the CANDISC procedure is invoked to perform a canonical discriminant analysis by using the

data set Clust and creating the output SAS data set Can. The NOPRINT option suppresses display

of the output. The CLASS statement speciﬁes the variable Cluster to deﬁne groups for the analysis.

The VAR statement speciﬁes the variables used in the analysis.

Next, the SGPLOT procedure plots the two canonical variables from PROC CANDISC, Can1 and

Can2. The PLOT statement speciﬁes the variable Cluster as the identiﬁcation variable. The resulting

plot (Figure 34.6) illustrates the spatial separation of the clusters calculated in the FASTCLUS

procedure.

Figure 34.6 Plot of Canonical Variables and Cluster Value

1632 FChapter 34: The FASTCLUS Procedure

Syntax: FASTCLUS Procedure

The following statements are available in the FASTCLUS procedure:

PROC FASTCLUS <DATA=SAS-data-set >

<MAXCLUSTERS=n>

<RADIUS=t>;

VAR variables ;

ID variables ;

FREQ variable ;

WEIGHT variable ;

BY variables ;

Usually you need only the VAR statement in addition to the PROC FASTCLUS statement. The

BY, FREQ, ID, VAR, and WEIGHT statements are described in alphabetical order after the PROC

FASTCLUS statement.

PROC FASTCLUS Statement

PROC FASTCLUS MAXCLUSTERS= n| RADIUS=t<options >;

You must specify the MAXCLUSTERS= option or RADIUS= option or both in the PROC FAST-

CLUS statement.

MAXCLUSTERS=n

MAXC=n

speciﬁes the maximum number of clusters permitted. If you omit the MAXCLUSTERS=

option, a value of 100 is assumed.

RADIUS=t

R=t

establishes the minimum distance criterion for selecting new seeds. No observation is con-

sidered as a new seed unless its minimum distance to previous seeds exceeds the value given

by the RADIUS= option. The default value is 0. If you specify the REPLACE=RANDOM

option, the RADIUS= option is ignored.

PROC FASTCLUS Statement F1633

You can specify the following options in the PROC FASTCLUS statement. Table 34.1 summarizes

the options.

Table 34.1 PROC FASTCLUS Statement Options

Option Description

Specify input and output data sets

DATA= speciﬁes input data set

INSTAT= speciﬁes input SAS data set previously created by

the OUTSTAT= option

SEED= speciﬁes input SAS data set for selecting initial

cluster seeds

VARDEF= speciﬁes divisor for variances

Output Data Processing

CLUSTER= speciﬁes name for cluster membership variable in

OUTSEED= and OUT= data sets

CLUSTERLABEL= speciﬁes label for cluster membership variable in

OUTSEED= and OUT= data sets

OUT= speciﬁes output SAS data set containing original

data and cluster assignments

OUTITER speciﬁes writing to OUTSEED= data set on every

iteration

OUTSEED= or MEAN= speciﬁes output SAS data set containing cluster

centers

OUTSTAT= speciﬁes output SAS data set containing statistics

Initial Clusters

DRIFT permits cluster to seeds to drift during initializa-

tion

MAXCLUSTERS= speciﬁes maximum number of clusters

RADIUS= speciﬁes minimum distance for selecting new

seeds

RANDOM= speciﬁes seed to initializes pseudo-random num-

ber generator

REPLACE= speciﬁes seed replacement method

Clustering Methods

CONVERGE= speciﬁes convergence criterion

DELETE= deletes cluster seeds with few observations

LEAST= optimizes an Lpcriterion, where 1p 1

MAXITER= speciﬁes maximum number of iterations

STRICT prevents an observation from being assigned to a

cluster if its distance to the nearest cluster seed is

large

1634 FChapter 34: The FASTCLUS Procedure

Table 34.1 continued

Option Description

Arcane Algorithmic Options

BINS= speciﬁes number of bins used for computing me-

dians for LEAST=1

HC= speciﬁes criterion for updating the homotopy pa-

rameter

HP= speciﬁes initial value of the homotopy parameter

IRLS uses an iteratively reweighted least squares

method instead of the modiﬁed Ekblom-Newton

method for 1<p<2

Missing Values

IMPUTE imputes missing values after ﬁnal cluster assign-

ment

NOMISS excludes observations with missing values

Control Displayed Output

DISTANCE displays distances between cluster centers

LIST displays cluster assignments for all observations

NOPRINT suppresses displayed output

SHORT suppresses display of large matrices

SUMMARY suppresses display of all results except for the

cluster summary

The following list provides details on these options. The list is in alphabetical order.

BINS=n

speciﬁes the number of bins used in the bin-sort algorithm for computing medians for

LEAST=1. By default, PROC FASTCLUS uses from 10 to 100 bins, depending on the

amount of memory available. Larger values use more memory and make each iteration some-

what slower, but they can reduce the number of iterations. Smaller values have the opposite

effect. The minimum value of nis 5.

CLUSTER=name

speciﬁes a name for the variable in the OUTSEED= and OUT= data sets that indicates cluster

membership. The default name for this variable is CLUSTER.

CLUSTERLABEL=name

speciﬁes a label for the variable CLUSTER in the OUTSEED= and OUT= data sets. By

default this variable has no label.

CONVERGE=c

CONV=c

speciﬁes the convergence criterion. Any nonnegative value is permitted. The default value is

0.0001 for all values of pif LEAST=pis explicitly speciﬁed; otherwise, the default value is

0.02. Iterations stop when the maximum relative change in the cluster seeds is less than or

PROC FASTCLUS Statement F1635

equal to the convergence criterion and additional conditions on the homotopy parameter, if

any, are satisﬁed (see the HP= option). The relative change in a cluster seed is the distance

between the old seed and the new seed divided by a scaling factor. If you do not specify the

LEAST= option, the scaling factor is the minimum distance between the initial seeds. If you

specify the LEAST= option, the scaling factor is an L1scale estimate and is recomputed on

each iteration. Specify the CONVERGE= option only if you specify a MAXITER= value

greater than 1.

DATA=SAS-data-set

speciﬁes the input data set containing observations to be clustered. If you omit the DATA=

option, the most recently created SAS data set is used. The data must be coordinates, not

distances, similarities, or correlations.

DELETE=n

deletes cluster seeds to which nor fewer observations are assigned. Deletion occurs after

processing for the DRIFT option is completed and after each iteration speciﬁed by the MAX-

ITER= option. Cluster seeds are not deleted after the ﬁnal assignment of observations to

clusters, so in rare cases a ﬁnal cluster might not have more than nmembers. The DELETE=

option is ineffective if you specify MAXITER=0 and do not specify the DRIFT option. By

default, no cluster seeds are deleted.

DISTANCE | DIST

computes distances between the cluster means.

DRIFT

executes the second of the four steps described in the section “Background” on page 1622.

After initial seed selection, each observation is assigned to the cluster with the nearest seed.

After an observation is processed, the seed of the cluster to which it is assigned is recalculated

as the mean of the observations currently assigned to the cluster. Thus, the cluster seeds drift

about rather than remaining ﬁxed for the duration of the pass.

HC=c

HP=p1<p2>

pertains to the homotopy parameter for LEAST=p, where 1 < p < 2. You should specify

these options only if you encounter convergence problems when you use the default values.

For 1<p<2, PROC FASTCLUS tries to optimize a perturbed variant of the Lpclustering

criterion (Gonin and Money 1989, pp. 5–6). When the homotopy parameter is 0, the opti-

mization criterion is equivalent to the clustering criterion. For a large homotopy parameter,

the optimization criterion approaches the least squares criterion and is therefore easy to opti-

mize. Beginning with a large homotopy parameter, PROC FASTCLUS gradually decreases

it by a factor in the range [0.01,0.5] over the course of the iterations. When both the homo-

topy parameter and the convergence measure are sufﬁciently small, the optimization process

is declared to have converged.

If the initial homotopy parameter is too large or if it is decreased too slowly, the optimization

can require many iterations. If the initial homotopy parameter is too small or if it is decreased

too quickly, convergence to a local optimum is likely. The following list gives details on

setting the homotopy parameter.

1636 FChapter 34: The FASTCLUS Procedure

HC=cspeciﬁes the criterion for updating the homotopy parameter. The homotopy

parameter is updated when the maximum relative change in the cluster seeds

is less than or equal to c. The default is the minimum of 0.01 and 100 times

the value of the CONVERGE= option.

HP=p1speciﬁes p1as the initial value of the homotopy parameter. The default is 0.05

if the modiﬁed Ekblom-Newton method is used; otherwise, it is 0.25.

HP=p1p2also speciﬁes p2as the minimum value for the homotopy parameter, which

must be reached for convergence. The default is the minimum of p1and 0.01

times the value of the CONVERGE= option.

IMPUTE

requests imputation of missing values after the ﬁnal assignment of observations to clusters. If

an observation that is assigned (or would have been assigned) to a cluster has a missing value

for variables used in the cluster analysis, the missing value is replaced by the corresponding

value in the cluster seed to which the observation is assigned (or would have been assigned).

If the observation cannot be assigned to a cluster, missing value replacement depends on

whether or not the NOMISS option is speciﬁed. If NOMISS is not speciﬁed, missing values

are replaced by the mean of all observations in the DATA= data set having a value for that

variable. If NOMISS is speciﬁed, missing values are replace by the mean of only observations

used in the analysis. (A weighted mean is used if a variable is speciﬁed in the WEIGHT

statement.) For information about cluster assignment see the section “OUT= Data Set” on

page 1643. If you specify the IMPUTE option, the imputed values are not used in computing

cluster statistics.

If you also request an OUT= data set, it contains the imputed values.

INSTAT=SAS-data-set

reads a SAS data set previously created with the FASTCLUS procedure by using the OUT-

STAT= option. If you specify the INSTAT= option, no clustering iterations are performed and

no output is displayed. Only cluster assignment and imputation are performed as an OUT=

data set is created.

IRLS

causes PROC FASTCLUS to use an iteratively reweighted least squares method instead of

the modiﬁed Ekblom-Newton method. If you specify the IRLS option, you must also specify

LEAST=p, where 1<p<2. Use the IRLS option only if you encounter convergence

problems with the default method.

LEAST=p| MAX

L=p| MAX

causes PROC FASTCLUS to optimize an Lpcriterion, where 1p 1 (Spath 1985,

pp. 62–63). Inﬁnity is indicated by LEAST=MAX. The value of this clustering criterion is

displayed in the iteration history.

If you do not specify the LEAST= option, PROC FASTCLUS uses the least squares (L2)

criterion. However, the default number of iterations is only 1 if you omit the LEAST= option,

so the optimization of the criterion is generally not completed. If you specify the LEAST=

option, the maximum number of iterations is increased to permit the optimization process a

chance to converge. See the MAXITER= option for details.

PROC FASTCLUS Statement F1637

Specifying the LEAST= option also changes the default convergence criterion from 0.02 to

0.0001. See the CONVERGE= option for details.

When LEAST=2, PROC FASTCLUS tries to minimize the root mean squared difference

between the data and the corresponding cluster means.

When LEAST=1, PROC FASTCLUS tries to minimize the mean absolute difference between

the data and the corresponding cluster medians.

When LEAST=MAX, PROC FASTCLUS tries to minimize the maximum absolute difference

between the data and the corresponding cluster midranges.

For general values of p, PROC FASTCLUS tries to minimize the pth root of the mean of the

pth powers of the absolute differences between the data and the corresponding cluster seeds.

The divisor in the clustering criterion is either the number of nonmissing data used in the

analysis or, if there is a WEIGHT statement, the sum of the weights corresponding to all

the nonmissing data used in the analysis (that is, an observation with nnonmissing data

contributes ntimes the observation weight to the divisor). The divisor is not adjusted for

degrees of freedom.

The method for updating cluster seeds during iteration depends on the LEAST= option, as

follows (Gonin and Money 1989).

LEAST=pAlgorithm for Computing Cluster Seeds

pD1bin sort for median

1<p<2 modiﬁed Merle-Spath if you specify IRLS;

otherwise modiﬁed Ekblom-Newton

pD2arithmetic mean

2 < p < 1Newton

pD 1 midrange

During the ﬁnal pass, a modiﬁed Merle-Spath step is taken to compute the cluster centers for

1p < 2 or 2 < p < 1.

If you specify the LEAST=poption with a value other than 2, PROC FASTCLUS computes

pooled scale estimates analogous to the root mean squared standard deviation but based on

pth power deviations instead of squared deviations.

LEAST=pScale Estimate

pD1mean absolute deviation

1 < p < 1root mean pth-power absolute deviation

pD 1 maximum absolute deviation

The divisors for computing the mean absolute deviation or the root mean pth-power absolute

deviation are adjusted for degrees of freedom just like the divisors for computing standard

deviations. This adjustment can be suppressed by the VARDEF= option.

LIST

lists all observations, giving the value of the ID variable (if any), the number of the cluster

to which the observation is assigned, and the distance between the observation and the ﬁnal

cluster seed.

1638 FChapter 34: The FASTCLUS Procedure

MAXITER=n

speciﬁes the maximum number of iterations for recomputing cluster seeds. When the value of

the MAXITER= option is greater than zero, PROC FASTCLUS executes the third of the four

steps described in the section “Background” on page 1622. In each iteration, each observation

is assigned to the nearest seed, and the seeds are recomputed as the means of the clusters.

The default value of the MAXITER= option depends on the LEAST=poption.

LEAST=pMAXITER=

not speciﬁed 1

pD120

1 < p < 1:5 50

1:5 p < 2 20

pD210

2<p 1 20

MEAN=SAS-data-set

creates an output data set to contain the cluster means and other statistics for each cluster. If

you want to create a permanent SAS data set, you must specify a two-level name. See “SAS

Data Files” in SAS Language Reference: Concepts for more information about permanent

data sets.

NOMISS

excludes observations with missing values from the analysis. However, if you also specify

the IMPUTE option, observations with missing values are included in the ﬁnal cluster assign-

ments.

NOPRINT

suppresses the display of all output. Note that this option temporarily disables the Output

Delivery System (ODS). For more information, see Chapter 20, “Using the Output Delivery

System.”

OUT=SAS-data-set

creates an output data set to contain all the original data, plus the new variables CLUSTER and

DISTANCE. See “SAS Data Files” in SAS Language Reference: Concepts for more informa-

tion about permanent data sets.

OUTITER

outputs information from the iteration history to the OUTSEED= data set, including the clus-

ter seeds at each iteration.

OUTSEED=SAS-data-set

OUTS=SAS-data-set

is another name for the MEAN= data set, provided because the data set can contain location

estimates other than means. The MEAN= option is still accepted.

OUTSTAT=SAS-data-set

creates an output data set to contain various statistics, especially those not included in the

OUTSEED= data set. Unlike the OUTSEED= data set, the OUTSTAT= data set is not suitable

for use as a SEED= data set in a subsequent PROC FASTCLUS step.

PROC FASTCLUS Statement F1639

RANDOM=n

speciﬁes a positive integer as a starting value for the pseudo-random number generator for

use with REPLACE=RANDOM. If you do not specify the RANDOM= option, the time of

day is used to initialize the pseudo-random number sequence.

REPLACE=FULL | PART | NONE | RANDOM

speciﬁes how seed replacement is performed, as follows:

FULL requests default seed replacement as described in the section

“Background” on page 1622.

PART requests seed replacement only when the distance between the observation

and the closest seed is greater than the minimum distance between seeds.

NONE suppresses seed replacement.

RANDOM selects a simple pseudo-random sample of complete observations as initial

cluster seeds.

SEED=SAS-data-set

speciﬁes an input data set from which initial cluster seeds are to be selected. If you do not

specify the SEED= option, initial seeds are selected from the DATA= data set. The SEED=

data set must contain the same variables that are used in the data analysis.

SHORT

suppresses the display of the initial cluster seeds, cluster means, and standard deviations.

STRICT

STRICT=s

prevents an observation from being assigned to a cluster if its distance to the nearest cluster

seed exceeds the value of the STRICT= option. If you specify the STRICT option without a

numeric value, you must also specify the RADIUS= option, and its value is used instead. In

the OUT= data set, observations that are not assigned due to the STRICT= option are given

a negative cluster number, the absolute value of which indicates the cluster with the nearest

seed.

SUMMARY

suppresses the display of the initial cluster seeds, statistics for variables, cluster means, and

standard deviations.

VARDEF=DF | N | WDF | WEIGHT | WGT

speciﬁes the divisor to be used in the calculation of variances and covariances. The default

value is VARDEF=DF. The possible values of the VARDEF= option and associated divisors

are as follows.

Value Description Divisor

DF error degrees of freedom nc

N number of observations n

WDF sum of weights DF .Piwi/c

WEIGHT | WGT sum of weights Piwi

In the preceding deﬁnitions, crepresents the number of clusters.

1640 FChapter 34: The FASTCLUS Procedure

BY Statement

BY variables ;

You can specify a BY statement with PROC FASTCLUS to obtain separate analysis on observations

in groups deﬁned by the BY variables. When a BY statement appears, the procedure expects the

input data set to be sorted in order of the BY variables.

If your input data set is not sorted in ascending order, use one of the following alternatives:

Sort the data by using the SORT procedure with a similar BY statement.

Specify the BY statement option NOTSORTED or DESCENDING in the BY statement for

the FASTCLUS procedure. The NOTSORTED option does not mean that the data are un-

sorted but rather that the data are arranged in groups (according to values of the BY variables)

and that these groups are not necessarily in alphabetical or increasing numeric order.

Create an index on the BY variables by using the DATASETS procedure.

If you specify the SEED= option and the SEED= data set does not contain any of the BY variables,

then the entire SEED= data set is used to obtain initial cluster seeds for each BY group in the

DATA= data set.

If the SEED= data set contains some but not all of the BY variables, or if some BY variables do

not have the same type or length in the SEED= data set as in the DATA= data set, then PROC

FASTCLUS displays an error message and stops.

If all the BY variables appear in the SEED= data set with the same type and length as in the DATA=

data set, then each BY group in the SEED= data set is used to obtain initial cluster seeds for the

corresponding BY group in the DATA= data set. All BY groups in the DATA= data set must also

appear in the SEED= data set. The BY groups in the SEED= data set must be in the same order as

in the DATA= data set. If you specify the NOTSORTED option in the BY statement, both data sets

must contain exactly the same BY groups in the same order. If you do not specify NOTSORTED,

some BY groups can appear in the SEED= data set but not in the DATA= data set; such BY groups

are not used in the analysis.

For more information about the BY statement, see SAS Language Reference: Concepts. For more

information about the DATASETS procedure, see the Base SAS Procedures Guide.

FREQ Statement

FREQ variable ;

If a variable in the data set represents the frequency of occurrence for the other values in the obser-

vation, include the variable’s name in a FREQ statement. The procedure then treats the data set as

if each observation appears ntimes, where nis the value of the FREQ variable for the observation.

ID Statement F1641

If the value of the FREQ variable is missing or less than or equal to zero, the observation is not

used in the analysis. The exact values of the FREQ variable are used in computations: frequency

values are not truncated to integers. The total number of observations is considered to be equal to

the sum of the FREQ variable when the procedure determines degrees of freedom for signiﬁcance

probabilities.

The WEIGHT and FREQ statements have a similar effect, except in determining the number of

observations for signiﬁcance tests.

ID Statement

ID variable ;

The ID variable, which can be character or numeric, identiﬁes observations on the output when you

specify the LIST option.

VAR Statement

VAR variables ;

The VAR statement lists the numeric variables to be used in the cluster analysis. If you omit the

VAR statement, all numeric variables not listed in other statements are used.

WEIGHT Statement

WEIGHT variable ;

The values of the WEIGHT variable are used to compute weighted cluster means. The WEIGHT

and FREQ statements have a similar effect, except the WEIGHT statement does not alter the degrees

of freedom or the number of observations. The WEIGHT variable can take nonintegral values. An

observation is used in the analysis only if the value of the WEIGHT variable is greater than zero.

1642 FChapter 34: The FASTCLUS Procedure

Details: FASTCLUS Procedure

Updates in the FASTCLUS Procedure

Some FASTCLUS procedure options and statements have changed from previous versions. The

differences are as follows:

Values of the FREQ variable are no longer truncated to integers. Noninteger variables speci-

ﬁed in the FREQ statement produce results different from those in previous releases.

The IMPUTE option produces different cluster standard deviations and related statistics.

When you specify the IMPUTE option, imputed values are no longer used in computing clus-

ter statistics. This change causes the cluster standard deviations and other statistics computed

from the standard deviations to be different from those in previous releases.

The INSTAT= option reads a SAS data set previously created with the FASTCLUS proce-

dure by using the OUTSTAT= option. If you specify the INSTAT= option, no clustering

iterations are performed and no output is produced. Only cluster assignment and imputation

are performed as an OUT= data set is created.

The OUTSTAT= data set contains additional information used for imputation.

_TYPE_=SEED corresponds to values that are cluster seeds. Observations previously desig-

nated _TYPE_=’SCALE’ are now _TYPE_=’DISPERSION’.

Missing Values

Observations with all missing values are excluded from the analysis. If you specify the NOMISS

option, observations with any missing values are excluded. Observations with missing values cannot

be cluster seeds.

The distance between an observation with missing values and a cluster seed is obtained by comput-

ing the squared distance based on the nonmissing values, multiplying by the ratio of the number of

variables, n, to the number of variables having nonmissing values, m, and taking the square root:

rn

mX.xisi/2

where

nDnumber of variables

mDnumber of variables with nonmissing values

xiDvalue of the ith variable for the observation

siDvalue of the ith variable for the seed

Output Data Sets F1643

If you specify the LEAST=poption with a power pother than 2(the default), the distance is

computed using

n

mX.xisi/p1

The summation is taken over variables with nonmissing values.

The IMPUTE option ﬁlls in missing values in the OUT= output data set.

Output Data Sets

OUT= Data Set

The OUT= data set contains the following:

the original variables

a new variable indicating the cluster assignment status of each observation. The value will

be less than the permitted number of clusters (see the MAXCLUSTERS= option) if the pro-

cedure detects fewer clusters than the maximum. A positive value indicates the cluster to

which the observation was assigned. A negative value indicates that the observation was not

assigned to a cluster (see the STRICT option), and the absolute value indicates the cluster

to which the observation would have been assigned. If the value is missing, the observation

cannot be assigned to any cluster. You can specify the variable name with the CLUSTER=

option. The default name is CLUSTER.

a new variable, DISTANCE, giving the distance from the observation to its cluster seed

If you specify the IMPUTE option, the OUT= data set also contains a new variable, _IMPUTE_,

giving the number of imputed values in each observation.

OUTSEED= Data Set

The OUTSEED= data set contains one observation for each cluster. The variables are as follows:

the BY variables, if any

a new variable giving the cluster number. You can specify the variable name with the CLUS-

TER= option. The default name is CLUSTER.

either the FREQ variable or a new variable called _FREQ_ giving the number of observations

in the cluster

the WEIGHT variable, if any

1644 FChapter 34: The FASTCLUS Procedure

a new variable, _RMSSTD_, giving the root mean squared standard deviation for the cluster.

See Chapter 29, “The CLUSTER Procedure,” for details.

a new variable, _RADIUS_, giving the maximum distance between any observation in the

cluster and the cluster seed

a new variable, _GAP_, containing the distance between the current cluster mean and the

nearest other cluster mean. The value is the centroid distance given in the output.

a new variable, _NEAR_, specifying the cluster number of the nearest cluster

the VAR variables giving the cluster means

If you specify the LEAST=poption with a value other than 2, the _RMSSTD_ variable is replaced by

the _SCALE_ variable, which contains the pooled scale estimate analogous to the root mean squared

standard deviation but based on pth-power deviations instead of squared deviations:

LEAST=1 mean absolute deviation

LEAST=proot mean p-th-power absolute deviation

LEAST=MAX maximum absolute deviation

If you specify the OUTITER option, there is one set of observations in the OUTSEED= data set for

each pass through the data set (that is, one set for initial seeds, one for each iteration, and one for

the ﬁnal clusters). Also, several additional variables appear:

_ITER_ is the iteration number. For the initial seeds, the value is 0. For the ﬁnal clus-

ter means or centers, the _ITER_ variable is one greater than the last iteration

reported in the iteration history.

_CRIT_ is the clustering criterion as described under the LEAST= option.

_CHANGE_ is the maximum over clusters of the relative change in the cluster seed from the

previous iteration. The relative change in a cluster seed is the distance between

the old seed and the new seed divided by a scaling factor. If you do not specify

the LEAST= option, the scaling factor is the minimum distance between the

initial seeds. If you specify the LEAST= option, the scaling factor is an L1scale

estimate and is recomputed on each iteration.

_HOMPAR_ is the value of the homotopy parameter. This variable appears only for

LEAST=pwith 1<p<2.

_BINSIZ_ is the maximum bin size used for estimating medians. This variable appears only

for LEAST=1.

If you specify the OUTITER option, the variables _SCALE_ or _RMSSTD_,_RADIUS_,_NEAR_, and

_GAP_ have missing values except for the last pass.

You can use the OUTSEED= data set as a SEED= input data set for a subsequent analysis.

Output Data Sets F1645

OUTSTAT= Data Set

The variables in the OUTSTAT= data set are as follows:

BY variables, if any

a new character variable, _TYPE_, specifying the type of statistic given by other variables (see

Table 34.2 and Table 34.3)

a new numeric variable giving the cluster number. You can specify the variable name with

the CLUSTER= option. The default name is CLUSTER.

a new numeric variable, OVER_ALL, containing statistics that apply over all of the VAR vari-

ables

the VAR variables giving statistics for particular variables

The values of _TYPE_ for all LEAST= options are given in Table 34.2.

Table 34.2 _TYPE_

_TYPE_ Contents of VAR Variables Contents of OVER_ALL

INITIAL Initial seeds Missing

CRITERION Missing Optimization criterion (see the

LEAST= option); this value is

displayed just before the “Cluster

Summary” table.

CENTER Cluster centers (see the LEAST= op-

tion)

Missing

SEED Cluster seeds: additional information

used for imputation

DISPERSION Dispersion estimates for each cluster

(see the LEAST= option); these val-

ues are displayed in a separate row

with title depending on the LEAST=

option

Dispersion estimates pooled over

variables (see the LEAST= op-

tion); these values are displayed

in the “Cluster Summary” ta-

ble with label depending on the

LEAST= option.

FREQ Frequency of each cluster omitting

observations with missing values for

the VAR variable; these values are not

displayed

Frequency of each cluster based

on all observations with any non-

missing value; these values are

displayed in the “Cluster Sum-

mary” table.

1646 FChapter 34: The FASTCLUS Procedure

Table 34.2 continued

_TYPE_ Contents of VAR variables Contents of OVER_ALL

WEIGHT Sum of weights for each cluster omit-

ting observations with missing values

for the VAR variable; these values are

not displayed

Sum of weights for each clus-

ter based on all observations with

any nonmissing value; these val-

ues are displayed in the “Cluster

Summary” table.

Observations with _TYPE_=’WEIGHT’ are included only if you specify the WEIGHT statement.

The _TYPE_ values included only for least squares clustering are given Table 34.3. Least squares

clustering is obtained by omitting the LEAST= option or by specifying LEAST=2.

Table 34.3 _TYPE_

_TYPE_ Contents of VAR Variables Contents of OVER_ALL

MEAN Mean for the total sample; this is not

displayed

Missing

STD Standard deviation for the total sam-

ple; labeled “Total STD” in the output

Standard deviation pooled over

all the VAR variables; labeled

“Total STD” in the output

WITHIN_STD Pooled within-cluster standard

deviation

Within cluster standard deviation

pooled over clusters and all the

VAR variables

RSQ R square for predicting the vari-

able from the clusters; labeled “R-

Squared” in the output

R square pooled over all the VAR

variables; labeled “R-Squared”

in the output

RSQ_RATIO R2

1R2; labeled “RSQ/(1-RSQ)” in the

output

1R2; labeled “RSQ/(1-RSQ)”

in the output

PSEUDO_F Missing Pseudo Fstatistic

ESRQ Missing Approximate expected value of

R square under the null hypoth-

esis of a single uniform cluster

CCC Missing Cubic clustering criterion

Computational Resources F1647

Computational Resources

Let

nDnumber of observations

vDnumber of variables

cDnumber of clusters

pDnumber of passes over the data set

Memory

The memory required is approximately 4.19v C12cv C10c C2max.c C1; v// bytes.

If you request the DISTANCE option, an additional 4c.c C1/ bytes of space is needed.

Time

The overall time required by PROC FASTCLUS is roughly proportional to nvcp if cis small with

respect to n.

Initial seed selection requires one pass over the data set. If the observations are in random order, the

time required is roughly proportional to

nvc Cvc2

unless you specify REPLACE=NONE. In that case, a complete pass might not be necessary, and

the time is roughly proportional to mvc, where cmn.

The DRIFT option, each iteration, and the ﬁnal assignment of cluster seeds each require one pass,

with time for each pass roughly proportional to nvc.

For greatest efﬁciency, you should list the variables in the VAR statement in order of decreasing

variance.

Using PROC FASTCLUS

Before using PROC FASTCLUS, decide whether your variables should be standardized in some

way, since variables with large variances tend to have more effect on the resulting clusters than

those with small variances. If all variables are measured in the same units, standardization might

not be necessary. Otherwise, some form of standardization is strongly recommended. The STAN-

DARD procedure can standardize all variables to mean zero and variance one. The FACTOR or

1648 FChapter 34: The FASTCLUS Procedure

PRINCOMP procedure can compute standardized principal component scores. The ACECLUS

procedure can transform the variables according to an estimated within-cluster covariance matrix.

Nonlinear transformations of the variables can change the number of population clusters and should

therefore be approached with caution. For most applications, the variables should be transformed

so that equal differences are of equal practical importance. An interval scale of measurement is

required. Ordinal or ranked data are generally not appropriate.

PROC FASTCLUS produces relatively little output. In most cases you should create an output data

set and use another procedure such as PRINT, PLOT, CHART, MEANS, DISCRIM, or CANDISC

to study the clusters. It is usually desirable to try several values of the MAXCLUSTERS= option.

Macros are useful for running PROC FASTCLUS repeatedly with other procedures.

A simple application of PROC FASTCLUS with two variables to examine the 2- and 3-cluster

solutions can proceed as follows:

proc standard mean=0 std=1 out=stan;

var v1 v2;

run;

proc fastclus data=stan out=clust maxclusters=2;

var v1 v2;

run;

proc plot;

plot v2*v1=cluster;

run;

proc fastclus data=stan out=clust maxclusters=3;

var v1 v2;

run;

proc plot;

plot v2*v1=cluster;

run;

If you have more than two variables, you can use the CANDISC procedure to compute canonical

variables for plotting the clusters. For example:

proc standard mean=0 std=1 out=stan;

var v1-v10;

run;

proc fastclus data=stan out=clust maxclusters=3;

var v1-v10;

run;

proc candisc out=can;

var v1-v10;

class cluster;

run;

Displayed Output F1649

proc plot;

plot can2*can1=cluster;

run;

If the data set is not too large, it might also be helpful to use the following to list the clusters:

proc sort;

by cluster distance;

run;

proc print;

by cluster;

run;

By examining the values of DISTANCE, you can determine if any observations are unusually far

from their cluster seeds.

It is often advisable, especially if the data set is large or contains outliers, to make a preliminary

PROC FASTCLUS run with a large number of clusters, perhaps 20 to 100. Use MAXITER=0 and

OUTSEED=SAS-data-set. You can save time on subsequent runs if you select cluster seeds from

this output data set by using the SEED= option.

You should check the preliminary clusters for outliers, which often appear as clusters with only one

member. Use a DATA step to delete outliers from the data set created by the OUTSEED= option

before using it as a SEED= data set in later runs. If there are severe outliers, you should specify the

STRICT option in the subsequent PROC FASTCLUS runs to prevent the outliers from distorting

the clusters.

You can use the OUTSEED= data set with the PLOT procedure to plot _GAP_ by _FREQ_. An

overlay of _RADIUS_ by _FREQ_ provides a baseline against which to compare the values of _GAP_.

Outliers appear in the upper-left area of the plot, with large _GAP_ values and small _FREQ_ values.

Good clusters appear in the upper-right area, with large values of both _GAP_ and _FREQ_. Good

potential cluster seeds appear in the lower right, as well as in the upper-right, since large _FREQ_

values indicate high-density regions. Small _FREQ_ values in the left part of the plot indicate poor

cluster seeds because the points are in low-density regions. It often helps to remove all clusters with

small frequencies even though the clusters might not be remote enough to be considered outliers.

Removing points in low-density regions improves cluster separation and provides visually sharper

cluster outlines in scatter plots.

Displayed Output

Unless the SHORT or SUMMARY option is speciﬁed, PROC FASTCLUS displays the following:

Initial Seeds, cluster seeds selected after one pass through the data

Change in Cluster Seeds for each iteration, if you specify MAXITER=n>1

1650 FChapter 34: The FASTCLUS Procedure

If you specify the LEAST=poption, with .1 < p < 2/, and you omit the IRLS option, an additional

column is displayed in the Iteration History table. This column contains a character to identify the

method used in each iteration. PROC FASTCLUS chooses the most efﬁcient method to cluster

the data at each iterative step, given the condition of the data. Thus, the method chosen is data

dependent. The possible values are described as follows:

Value Method

N Newton’s Method

I or L iteratively weighted least squares (IRLS)

1 IRLS step, halved once

2 IRLS step, halved twice

3 IRLS step, halved three times

PROC FASTCLUS displays a Cluster Summary, giving the following for each cluster:

Cluster number

Frequency, the number of observations in the cluster

Weight, the sum of the weights of the observations in the cluster, if you specify the WEIGHT

statement

RMS Std Deviation, the root mean squared across variables of the cluster standard deviations,

which is equal to the root mean square distance between observations in the cluster

Maximum Distance from Seed to Observation, the maximum distance from the cluster seed

to any observation in the cluster

Nearest Cluster, the number of the cluster with mean closest to the mean of the current cluster

Centroid Distance, the distance between the centroids (means) of the current cluster and the

nearest other cluster

A table of statistics for each variable is displayed unless you specify the SUMMARY option. The

table contains the following:

Total STD, the total standard deviation

Within STD, the pooled within-cluster standard deviation

R-Squared, the R square for predicting the variable from the cluster

RSQ/(1 - RSQ), the ratio of between-cluster variance to within-cluster variance .R2=.1R2//

OVER-ALL, all of the previous quantities pooled across variables

PROC FASTCLUS also displays the following:

Displayed Output F1651

Pseudo FStatistic,

c1

1R2

nc

where R square is the observed overall R square, cis the number of clusters, and nis the

number of observations. The pseudo Fstatistic was suggested by Calinski and Harabasz

(1974). See Milligan and Cooper (1985) and Cooper and Milligan (1988) regarding the use of

the pseudo Fstatistic in estimating the number of clusters. See Example 29.2 in Chapter 29,

“The CLUSTER Procedure,” for a comparison of pseudo Fstatistics.

Observed Overall R-Squared, if you specify the SUMMARY option

Approximate Expected Overall R-Squared, the approximate expected value of the overall R

square under the uniform null hypothesis assuming that the variables are uncorrelated. The

value is missing if the number of clusters is greater than one-ﬁfth the number of observations.

Cubic Clustering Criterion, computed under the assumption that the variables are uncorre-

lated. The value is missing if the number of clusters is greater than one-ﬁfth the number of

observations.

If you are interested in the approximate expected R square or the cubic clustering criterion but

your variables are correlated, you should cluster principal component scores from the PRIN-

COMP procedure. Both of these statistics are described by Sarle (1983). The performance

of the cubic clustering criterion in estimating the number of clusters is examined by Milligan

and Cooper (1985) and Cooper and Milligan (1988).

Distances Between Cluster Means, if you specify the DISTANCE option

Unless you specify the SHORT or SUMMARY option, PROC FASTCLUS displays the following:

Cluster Means for each variable

Cluster Standard Deviations for each variable

1652 FChapter 34: The FASTCLUS Procedure

ODS Table Names

PROC FASTCLUS assigns a name to each table it creates. You can use these names to reference

the table when using the Output Delivery System (ODS) to select tables and create output data sets.

These names are listed in Table 34.4. For more information on ODS, see Chapter 20, “Using the

Output Delivery System.”

Table 34.4 ODS Tables Produced by PROC FASTCLUS

ODS Table Name Description Statement Option

ApproxExpOverAllRSq Approximate expected overall R-

squared, single number

PROC default

CCC CCC, Cubic Clustering Crite-

rion, single number

PROC default

ClusterList Cluster listing, obs, id, and dis-

tances

PROC LIST

ClusterSum Cluster summary, cluster num-

ber, distances

PROC PRINTALL

ClusterCenters Cluster centers PROC default

ClusterDispersion Cluster dispersion PROC default

ConvergenceStatus Convergence status PROC PRINTALL

Criterion Criterion based on ﬁnal seeds,

single number

PROC default

DistBetweenClust Distance between clusters PROC default

InitialSeeds Initial seeds PROC default

IterHistory Iteration history, various statis-

tics for each iteration

PROC PRINTALL

MinDist Minimum distance between ini-

tial seeds, single number

PROC PRINTALL

NumberOfBins Number of bins PROC default

ObsOverAllRSquare Observed overall R-squared, sin-

gle number

PROC SUMMARY

PrelScaleEst Preliminary L(1) scale estimate,

single number

PROC PRINTALL

PseudoFStat Pseudo Fstatistic, single num-

ber

PROC default

SimpleStatistics Simple statistics for input vari-

ables

PROC default

VariableStat Statistics for variables within

clusters

PROC default

Examples: FASTCLUS Procedure F1653

Examples: FASTCLUS Procedure

Example 34.1: Fisher’s Iris Data

The iris data published by Fisher (1936) have been widely used for examples in discriminant anal-

ysis and cluster analysis. The sepal length, sepal width, petal length, and petal width are measured

in millimeters on 50 iris specimens from each of three species, Iris setosa, I. versicolor, and I.

virginica. Mezzich and Solomon (1980) discuss a variety of cluster analysis of the iris data.

In this example, the FASTCLUS procedure is used to ﬁnd two and then three clusters. In the

following code, an output data set is created, and PROC FREQ is invoked to compare the clusters

with the species classiﬁcation. See Output 34.1.1 and Output 34.1.2 for these results.

For three clusters, you can use the CANDISC procedure to compute canonical variables for plotting

the clusters. See Output 34.1.3 and Output 34.1.4 for the results.

proc format;

value specname

1=’Setosa ’

2=’Versicolor’

3=’Virginica ’;

run;

data iris;

title ’Fisher (1936) Iris Data’;

input SepalLength SepalWidth PetalLength PetalWidth Species @@;

format Species specname.;

label SepalLength=’Sepal Length in mm.’

SepalWidth =’Sepal Width in mm.’

PetalLength=’Petal Length in mm.’

PetalWidth =’Petal Width in mm.’;

symbol = put(species, specname10.);

datalines;

503314021642856223652846152673156243

632851153463414031693151233622245152

593248182463610021613046142602751162

653052203562539112653055183582751193

683259233513317051572845132623454233

... more lines ...

552340132663044142682848142543417021

513715041523515021582851243673050172

63 33 60 25 3 53 37 15 02 1

;

proc fastclus data=iris maxc=2 maxiter=10 out=clus;

var SepalLength SepalWidth PetalLength PetalWidth;

run;

1654 FChapter 34: The FASTCLUS Procedure

proc freq;

tables cluster*species;

run;

proc fastclus data=iris maxc=3 maxiter=10 out=clus;

var SepalLength SepalWidth PetalLength PetalWidth;

run;

proc freq;

tables cluster*Species;

run;

proc candisc anova out=can;

class cluster;

var SepalLength SepalWidth PetalLength PetalWidth;

title2 ’Canonical Discriminant Analysis of Iris Clusters’;

run;

proc sgplot data=Can;

scatter y=Can2 x=Can1 /group=Cluster ;

title2 ’Plot of Canonical Variables Identified by Cluster’;

run;

Output 34.1.1 Fisher’s Iris Data: PROC FASTCLUS with MAXC=2 andPROC FREQ

Fisher (1936) Iris Data

The FASTCLUS Procedure

Replace=FULL Radius=0 Maxclusters=2 Maxiter=10 Converge=0.02

Initial Seeds

Cluster SepalLength SepalWidth PetalLength PetalWidth

-------------------------------------------------------------------------------

1 43.00000000 30.00000000 11.00000000 1.00000000

2 77.00000000 26.00000000 69.00000000 23.00000000

Minimum Distance Between Initial Seeds = 70.85196

Iteration History

Relative Change

in Cluster Seeds

Iteration Criterion 1 2

----------------------------------------------

1 11.0638 0.1904 0.3163

2 5.3780 0.0596 0.0264

3 5.0718 0.0174 0.00766

Convergence criterion is satisfied.

Criterion Based on Final Seeds = 5.0417

Example 34.1: Fisher’s Iris Data F1655

Output 34.1.1 continued

Cluster Summary

Maximum Distance

RMS Std from Seed Radius Nearest

Cluster Frequency Deviation to Observation Exceeded Cluster

-----------------------------------------------------------------------------

1 53 3.7050 21.1621 2

2 97 5.6779 24.6430 1

Cluster Summary

Distance Between

Cluster Cluster Centroids

-----------------------------

1 39.2879

2 39.2879

Statistics for Variables

Variable Total STD Within STD R-Square RSQ/(1-RSQ)

---------------------------------------------------------------------

SepalLength 8.28066 5.49313 0.562896 1.287784

SepalWidth 4.35866 3.70393 0.282710 0.394137

PetalLength 17.65298 6.80331 0.852470 5.778291

PetalWidth 7.62238 3.57200 0.781868 3.584390

OVER-ALL 10.69224 5.07291 0.776410 3.472463

Pseudo F Statistic = 513.92

Approximate Expected Over-All R-Squared = 0.51539

Cubic Clustering Criterion = 14.806

WARNING: The two values above are invalid for correlated variables.

Cluster Means

Cluster SepalLength SepalWidth PetalLength PetalWidth

-------------------------------------------------------------------------------

1 50.05660377 33.69811321 15.60377358 2.90566038

2 63.01030928 28.86597938 49.58762887 16.95876289

Cluster Standard Deviations

Cluster SepalLength SepalWidth PetalLength PetalWidth

-------------------------------------------------------------------------------

1 3.427350930 4.396611045 4.404279486 2.105525249

2 6.336887455 3.267991438 7.800577673 4.155612484

1656 FChapter 34: The FASTCLUS Procedure

Output 34.1.1 continued

Fisher (1936) Iris Data

The FREQ Procedure

Table of CLUSTER by Species

CLUSTER(Cluster) Species

Frequency|

Percent |

Row Pct |

Col Pct |Setosa |Versicol|Virginic| Total

| |or |a |

---------+--------+--------+--------+

1 | 50 | 3 | 0 | 53

| 33.33 | 2.00 | 0.00 | 35.33

| 94.34 | 5.66 | 0.00 |

| 100.00 | 6.00 | 0.00 |

---------+--------+--------+--------+

2 | 0 | 47 | 50 | 97

| 0.00 | 31.33 | 33.33 | 64.67

| 0.00 | 48.45 | 51.55 |

| 0.00 | 94.00 | 100.00 |

---------+--------+--------+--------+

Total 50 50 50 150

33.33 33.33 33.33 100.00

Output 34.1.2 Fisher’s Iris Data: PROC FASTCLUS with MAXC=3 and PROC FREQ

Fisher (1936) Iris Data

The FASTCLUS Procedure

Replace=FULL Radius=0 Maxclusters=3 Maxiter=10 Converge=0.02

Initial Seeds

Cluster SepalLength SepalWidth PetalLength PetalWidth

-------------------------------------------------------------------------------

1 58.00000000 40.00000000 12.00000000 2.00000000

2 77.00000000 38.00000000 67.00000000 22.00000000

3 49.00000000 25.00000000 45.00000000 17.00000000

Minimum Distance Between Initial Seeds = 38.23611

Iteration History

Relative Change in Cluster Seeds

Iteration Criterion 1 2 3

----------------------------------------------------------

1 6.7591 0.2652 0.3205 0.2985

2 3.7097 0 0.0459 0.0317

3 3.6427 0 0.0182 0.0124

Example 34.1: Fisher’s Iris Data F1657

Output 34.1.2 continued

Convergence criterion is satisfied.

Criterion Based on Final Seeds = 3.6289

Cluster Summary

Maximum Distance

RMS Std from Seed Radius Nearest

Cluster Frequency Deviation to Observation Exceeded Cluster

-----------------------------------------------------------------------------

1 50 2.7803 12.4803 3

2 38 4.0168 14.9736 3

3 62 4.0398 16.9272 2

Cluster Summary

Distance Between

Cluster Cluster Centroids

-----------------------------

1 33.5693

2 17.9718

3 17.9718

Statistics for Variables

Variable Total STD Within STD R-Square RSQ/(1-RSQ)

---------------------------------------------------------------------

SepalLength 8.28066 4.39488 0.722096 2.598359

SepalWidth 4.35866 3.24816 0.452102 0.825156

PetalLength 17.65298 4.21431 0.943773 16.784895

PetalWidth 7.62238 2.45244 0.897872 8.791618

OVER-ALL 10.69224 3.66198 0.884275 7.641194

Pseudo F Statistic = 561.63

Approximate Expected Over-All R-Squared = 0.62728

Cubic Clustering Criterion = 25.021

WARNING: The two values above are invalid for correlated variables.

Cluster Means

Cluster SepalLength SepalWidth PetalLength PetalWidth

-------------------------------------------------------------------------------

1 50.06000000 34.28000000 14.62000000 2.46000000

2 68.50000000 30.73684211 57.42105263 20.71052632

3 59.01612903 27.48387097 43.93548387 14.33870968

1658 FChapter 34: The FASTCLUS Procedure

Output 34.1.2 continued

Cluster Standard Deviations

Cluster SepalLength SepalWidth PetalLength PetalWidth

-------------------------------------------------------------------------------

1 3.524896872 3.790643691 1.736639965 1.053855894

2 4.941550255 2.900924461 4.885895746 2.798724562

3 4.664100551 2.962840548 5.088949673 2.974997167

Fisher (1936) Iris Data

The FREQ Procedure

Table of CLUSTER by Species

CLUSTER(Cluster) Species

Frequency|

Percent |

Row Pct |

Col Pct |Setosa |Versicol|Virginic| Total

| |or |a |

---------+--------+--------+--------+

1 | 50 | 0 | 0 | 50

| 33.33 | 0.00 | 0.00 | 33.33

| 100.00 | 0.00 | 0.00 |

---------+--------+--------+--------+

2 | 0 | 2 | 36 | 38

| 0.00 | 1.33 | 24.00 | 25.33

| 0.00 | 5.26 | 94.74 |

| 0.00 | 4.00 | 72.00 |

---------+--------+--------+--------+

3 | 0 | 48 | 14 | 62

| 0.00 | 32.00 | 9.33 | 41.33

| 0.00 | 77.42 | 22.58 |

| 0.00 | 96.00 | 28.00 |

---------+--------+--------+--------+

Total 50 50 50 150

33.33 33.33 33.33 100.00

Output 34.1.3 Fisher’s Iris Data using PROC CANDISC

Fisher (1936) Iris Data

Canonical Discriminant Analysis of Iris Clusters

The CANDISC Procedure

Total Sample Size 150 DF Total 149

Variables 4 DF Within Classes 147

Classes 3 DF Between Classes 2

Number of Observations Read 150

Number of Observations Used 150

Example 34.1: Fisher’s Iris Data F1659

Output 34.1.3 continued

Class Level Information

Variable

CLUSTER Name Frequency Weight Proportion

1 _1 50 50.0000 0.333333

2 _2 38 38.0000 0.253333

3 _3 62 62.0000 0.413333

Fisher (1936) Iris Data

Canonical Discriminant Analysis of Iris Clusters

The CANDISC Procedure

Univariate Test Statistics

F Statistics, Num DF=2, Den DF=147

Total Pooled Between

Standard Standard Standard R-Square

Variable Label Deviation Deviation Deviation R-Square / (1-RSq) F Value Pr > F

Sepal Sepal 8.2807 4.3949 8.5893 0.7221 2.5984 190.98 <.0001

Length Length

in mm.

Sepal Sepal 4.3587 3.2482 3.5774 0.4521 0.8252 60.65 <.0001

Width Width

in mm.

Petal Petal 17.6530 4.2143 20.9336 0.9438 16.7849 1233.69 <.0001

Length Length

in mm.

Petal Petal 7.6224 2.4524 8.8164 0.8979 8.7916 646.18 <.0001

Width Width

in mm.

Average R-Square

Unweighted 0.7539604

Weighted by Variance 0.8842753

Multivariate Statistics and F Approximations

S=2 M=0.5 N=71

Statistic Value F Value Num DF Den DF Pr > F

Wilks’ Lambda 0.03222337 164.55 8 288 <.0001

Pillai’s Trace 1.25669612 61.29 8 290 <.0001

Hotelling-Lawley Trace 21.06722883 377.66 8 203.4 <.0001

Roy’s Greatest Root 20.63266809 747.93 4 145 <.0001

NOTE: F Statistic for Roy’s Greatest Root is an upper bound.

NOTE: F Statistic for Wilks’ Lambda is exact.

1660 FChapter 34: The FASTCLUS Procedure

Output 34.1.3 continued

Fisher (1936) Iris Data

Canonical Discriminant Analysis of Iris Clusters

The CANDISC Procedure

Adjusted Approximate Squared

Canonical Canonical Standard Canonical

Correlation Correlation Error Correlation

1 0.976613 0.976123 0.003787 0.953774

2 0.550384 0.543354 0.057107 0.302923

Eigenvalues of Inv(E)*H

= CanRsq/(1-CanRsq)

Eigenvalue Difference Proportion Cumulative

1 20.6327 20.1981 0.9794 0.9794

2 0.4346 0.0206 1.0000

Test of H0: The canonical correlations in the

current row and all that follow are zero

Likelihood Approximate

Ratio F Value Num DF Den DF Pr > F

1 0.03222337 164.55 8 288 <.0001

2 0.69707749 21.00 3 145 <.0001

Fisher (1936) Iris Data

Canonical Discriminant Analysis of Iris Clusters

The CANDISC Procedure

Total Canonical Structure

Variable Label Can1 Can2

SepalLength Sepal Length in mm. 0.831965 0.452137

SepalWidth Sepal Width in mm. -0.515082 0.810630

PetalLength Petal Length in mm. 0.993520 0.087514

PetalWidth Petal Width in mm. 0.966325 0.154745

Between Canonical Structure

Variable Label Can1 Can2

SepalLength Sepal Length in mm. 0.956160 0.292846

SepalWidth Sepal Width in mm. -0.748136 0.663545

PetalLength Petal Length in mm. 0.998770 0.049580

PetalWidth Petal Width in mm. 0.995952 0.089883

Example 34.1: Fisher’s Iris Data F1661

Output 34.1.3 continued

Pooled Within Canonical Structure

Variable Label Can1 Can2

SepalLength Sepal Length in mm. 0.339314 0.716082

SepalWidth Sepal Width in mm. -0.149614 0.914351

PetalLength Petal Length in mm. 0.900839 0.308136

PetalWidth Petal Width in mm. 0.650123 0.404282

Fisher (1936) Iris Data

Canonical Discriminant Analysis of Iris Clusters

The CANDISC Procedure

Total-Sample Standardized Canonical Coefficients

Variable Label Can1 Can2

SepalLength Sepal Length in mm. 0.047747341 1.021487262

SepalWidth Sepal Width in mm. -0.577569244 0.864455153

PetalLength Petal Length in mm. 3.341309573 -1.283043758

PetalWidth Petal Width in mm. 0.996451144 0.900476563

Pooled Within-Class Standardized Canonical Coefficients

Variable Label Can1 Can2

SepalLength Sepal Length in mm. 0.0253414487 0.5421446856

SepalWidth Sepal Width in mm. -.4304161258 0.6442092294

PetalLength Petal Length in mm. 0.7976741592 -.3063023132

PetalWidth Petal Width in mm. 0.3205998034 0.2897207865

Raw Canonical Coefficients

Variable Label Can1 Can2

SepalLength Sepal Length in mm. 0.0057661265 0.1233581748

SepalWidth Sepal Width in mm. -.1325106494 0.1983303556

PetalLength Petal Length in mm. 0.1892773419 -.0726814163

PetalWidth Petal Width in mm. 0.1307270927 0.1181359305

Class Means on Canonical Variables

CLUSTER Can1 Can2

1 -6.131527227 0.244761516

2 4.931414018 0.861972277

3 1.922300462 -0.725693908

1662 FChapter 34: The FASTCLUS Procedure

Output 34.1.4 Plot of Fisher’s Iris Data using PROC CANDISC

Example 34.2: Outliers

This example involves data artiﬁcially generated to contain two clusters and several severe outliers.

A preliminary analysis speciﬁes 20 clusters and outputs an OUTSEED= data set to be used for

a diagnostic plot. The exact number of initial clusters is not important; similar results could be

obtained with 10 or 50 initial clusters. Examination of the plot suggests that clusters with more

than ﬁve (again, the exact number is not important) observations can yield good seeds for the main

analysis. A DATA step deletes clusters with ﬁve or fewer observations, and the remaining cluster

means provide seeds for the next PROC FASTCLUS analysis.

Two clusters are requested; the LEAST= option speciﬁes the mean absolute deviation criterion

(LEAST=1). Values of the LEAST= option less than 2 reduce the effect of outliers on cluster

centers.

The next analysis also requests two clusters; the STRICT= option is speciﬁed to prevent outliers

from distorting the results. The STRICT= value is chosen to be close to the _GAP_ and _RADIUS_

values of the larger clusters in the diagnostic plot; the exact value is not critical.

A ﬁnal PROC FASTCLUS run assigns the outliers to clusters.

Example 34.2: Outliers F1663

The following SAS statements implement these steps, and the results are displayed in Output 34.2.3

through Output 34.2.8. First, an artiﬁcial data set is created with two clusters and some outliers.

Then PROC FASTCLUS is run with many clusters to produce an OUTSEED= data set. A diagnostic

plot using the variables _GAP_ and _RADIUS_ is then produced using the SGSCATTER procedure.

The results from these steps are shown in Output 34.2.1 and Output 34.2.2.

data x;

title ’Using PROC FASTCLUS to Analyze Data with Outliers’;

drop n;

do n=1 to 100;

x=rannor(12345)+2;

y=rannor(12345);

output;

end;

do n=1 to 100;

x=rannor(12345)-2;

y=rannor(12345);

output;

end;

do n=1 to 10;

x=10*rannor(12345);

y=10*rannor(12345);

output;

end;

run;

title2 ’Preliminary PROC FASTCLUS Analysis with 20 Clusters’;

proc fastclus data=x outseed=mean1 maxc=20 maxiter=0 summary;

var x y;

run;

proc sgscatter data=mean1 ;

compare y=(_gap_ _radius_) x=_freq_ ;

run;

Output 34.2.1 Preliminary Analysis of Data with Outliers Using PROC FASTCLUS

Using PROC FASTCLUS to Analyze Data with Outliers

Preliminary PROC FASTCLUS Analysis with 20 Clusters

The FASTCLUS Procedure

Replace=FULL Radius=0 Maxclusters=20 Maxiter=0

Criterion Based on Final Seeds = 0.6873

1664 FChapter 34: The FASTCLUS Procedure

Output 34.2.1 continued

Cluster Summary

Maximum Distance

RMS Std from Seed Radius Nearest

Cluster Frequency Deviation to Observation Exceeded Cluster

-----------------------------------------------------------------------------

1 8 0.4753 1.1924 19

2 1 . 0 6

3 44 0.6252 1.6774 5

4 1 . 0 20

5 38 0.5603 1.4528 3

6 2 0.0542 0.1085 2

7 1 . 0 14

8 2 0.6480 1.2961 1

9 1 . 0 7

10 1 . 0 18

11 1 . 0 16

12 20 0.5911 1.6291 16

13 5 0.6682 1.4244 3

14 1 . 0 7

15 5 0.4074 1.2678 3

16 22 0.4168 1.5139 19

17 8 0.4031 1.4794 5

18 1 . 0 10

19 45 0.6475 1.6285 16

20 3 0.5719 1.3642 15

Cluster Summary

Distance Between

Cluster Cluster Centroids

-----------------------------

1 1.7205

2 6.2847

3 1.4386

4 5.2130

5 1.4386

6 6.2847

7 2.5094

8 1.8450

9 9.4534

10 4.2514

11 4.7582

12 1.5601

13 1.9553

14 2.5094

15 1.7609

16 1.4936

17 1.5564

18 4.2514

19 1.4936

20 1.8999

Pseudo F Statistic = 207.58

Example 34.2: Outliers F1665

Output 34.2.1 continued

Observed Over-All R-Squared = 0.95404

Approximate Expected Over-All R-Squared = 0.96103

Cubic Clustering Criterion = -2.503

WARNING: The two values above are invalid for correlated variables.

Output 34.2.2 Preliminary Analysis of Data with Outliers: Plot Using and PROC SGSCATGTER

1666 FChapter 34: The FASTCLUS Procedure

In the following SAS statements, a DATA step is used to remove low frequency clusters, then

the FASTCLUS procedure is run again, selecting seeds from the high frequency clusters in the

previous analysis using LEAST=1 clustering criterion. The results are shown in Output 34.2.3 and

Output 34.2.4.

data seed;

set mean1;

if _freq_>5;

run;

title2 ’PROC FASTCLUS Analysis Using LEAST= Clustering Criterion’;

title3 ’Values < 2 Reduce Effect of Outliers on Cluster Centers’;

proc fastclus data=x seed=seed maxc=2 least=1 out=out;

var x y;

run;

proc sgplot data=out;

scatter y=y x=x /group=cluster;

run;

Output 34.2.3 Analysis of Data with Outliers Using the LEAST= Option

Using PROC FASTCLUS to Analyze Data with Outliers

PROC FASTCLUS Analysis Using LEAST= Clustering Criterion

Values < 2 Reduce Effect of Outliers on Cluster Centers

The FASTCLUS Procedure

Replace=FULL Radius=0 Maxclusters=2 Maxiter=20 Converge=0.0001 Least=1

Initial Seeds

Cluster x y

-------------------------------------------

1 2.794174248 -0.065970836

2 -2.027300384 -2.051208579

Minimum Distance Between Initial Seeds = 6.806712

Preliminary L(1) Scale Estimate = 2.796579

Number of Bins = 100

Iteration History

Relative Change

Maximum in Cluster Seeds

Iteration Criterion Bin Size 1 2

----------------------------------------------------------

1 1.3983 0.2263 0.4091 0.6696

2 1.0776 0.0226 0.00511 0.0452

3 1.0771 0.00226 0.00229 0.00234

4 1.0771 0.000396 0.000253 0.000144

5 1.0771 0.000396 0 0

Example 34.2: Outliers F1667

Output 34.2.3 continued

Convergence criterion is satisfied.

Criterion Based on Final Seeds = 1.0771

Cluster Summary

Mean Maximum Distance

Absolute from Seed Radius Nearest

Cluster Frequency Deviation to Observation Exceeded Cluster

-----------------------------------------------------------------------------

1 102 1.1278 24.1622 2

2 108 1.0494 14.8292 1

Cluster Summary

Distance Between

Cluster Cluster Medians

----------------------------

1 4.2585

2 4.2585

Cluster Medians

Cluster x y

-------------------------------------------

1 1.923023887 0.222482918

2 -1.826721743 -0.286253041

Mean Absolute Deviations from Final Seeds

Cluster x y

-------------------------------------------

1 1.113465261 1.142120480

2 0.890331835 1.208370913

1668 FChapter 34: The FASTCLUS Procedure

Output 34.2.4 Analysis Plot of Data with Outliers

The FASTCLUS procedure is run again, selecting seeds from high frequency clusters in the pre-

vious analysis. STRICT= prevents outliers from distorting the results. The results are shown in

Output 34.2.5 and Output 34.2.6.

title2 ’PROC FASTCLUS Analysis Using STRICT= to Omit Outliers’;

proc fastclus data=x seed=seed

maxc=2 strict=3.0 out=out outseed=mean2;

var x y;

run;

proc sgplot data=out;

scatter y=y x=x /group=cluster ;

run;

Example 34.2: Outliers F1669

Output 34.2.5 Cluster Analysis with Outliers Omitted: PROC FASTCLUS SGPLOT

Using PROC FASTCLUS to Analyze Data with Outliers

PROC FASTCLUS Analysis Using STRICT= to Omit Outliers

The FASTCLUS Procedure

Replace=FULL Radius=0 Strict=3 Maxclusters=2 Maxiter=1

Initial Seeds

Cluster x y

-------------------------------------------

1 2.794174248 -0.065970836

2 -2.027300384 -2.051208579

Criterion Based on Final Seeds = 0.9515

Cluster Summary

Maximum Distance

RMS Std from Seed Radius Nearest

Cluster Frequency Deviation to Observation Exceeded Cluster

-----------------------------------------------------------------------------

1 99 0.9501 2.9589 2

2 99 0.9290 2.8011 1

Cluster Summary

Distance Between

Cluster Cluster Centroids

-----------------------------

1 3.7666

2 3.7666

12 Observation(s) were not assigned to a cluster

because the minimum distance to a cluster seed exceeded the STRICT= value.

Statistics for Variables

Variable Total STD Within STD R-Square RSQ/(1-RSQ)

------------------------------------------------------------------

x 2.06854 0.87098 0.823609 4.669219

y 1.02113 1.00352 0.039093 0.040683

OVER-ALL 1.63119 0.93959 0.669891 2.029303

Pseudo F Statistic = 397.74

Approximate Expected Over-All R-Squared = 0.60615

Cubic Clustering Criterion = 3.197

1670 FChapter 34: The FASTCLUS Procedure

Output 34.2.5 continued

WARNING: The two values above are invalid for correlated variables.

Cluster Means

Cluster x y

-------------------------------------------

1 1.825111432 0.141211701

2 -1.919910712 -0.261558725

Cluster Standard Deviations

Cluster x y

-------------------------------------------

1 0.889549271 1.006965219

2 0.852000588 1.000062579

Output 34.2.6 Cluster Analysis with Outliers Omitted: Plot Using PROC SGPLOT

Finally, the FASTCLUS procedure is run one more time with zero iterations to assign outliers and

tails to clusters. The results are show in Output 34.2.7 and Output 34.2.8.

Example 34.2: Outliers F1671

title2 ’Final PROC FASTCLUS Analysis Assigning Outliers to ’

’Clusters’;

proc fastclus data=x seed=mean2 maxc=2 maxiter=0 out=out;

var x y;

run;

proc sgplot data=out;

scatter y=y x=x /group=cluster ;

run;

Output 34.2.7 Cluster Analysis with Outliers Omitted: PROC FASTCLUS

Using PROC FASTCLUS to Analyze Data with Outliers

Final PROC FASTCLUS Analysis Assigning Outliers to Clusters

The FASTCLUS Procedure

Replace=FULL Radius=0 Maxclusters=2 Maxiter=0

Initial Seeds

Cluster x y

-------------------------------------------

1 1.825111432 0.141211701

2 -1.919910712 -0.261558725

Criterion Based on Final Seeds = 2.0594

Cluster Summary

Maximum Distance

RMS Std from Seed Radius Nearest

Cluster Frequency Deviation to Observation Exceeded Cluster

-----------------------------------------------------------------------------

1 103 2.2569 17.9426 2

2 107 1.8371 11.7362 1

Cluster Summary

Distance Between

Cluster Cluster Centroids

-----------------------------

1 4.3753

2 4.3753

Statistics for Variables

Variable Total STD Within STD R-Square RSQ/(1-RSQ)

------------------------------------------------------------------

x 2.92721 1.95529 0.555950 1.252000

y 2.15248 2.14754 0.009347 0.009435

OVER-ALL 2.56922 2.05367 0.364119 0.572621

Pseudo F Statistic = 119.11

1672 FChapter 34: The FASTCLUS Procedure

Output 34.2.7 continued

Approximate Expected Over-All R-Squared = 0.49090

Cubic Clustering Criterion = -5.338

WARNING: The two values above are invalid for correlated variables.

Cluster Means

Cluster x y

-------------------------------------------

1 2.280017469 0.263940765

2 -2.075547895 -0.151348765

Cluster Standard Deviations

Cluster x y

-------------------------------------------

1 2.412264861 2.089922815

2 1.379355878 2.201567557

Output 34.2.8 Cluster Analysis with Outliers Omitted: Plot Using PROC SGPLOT

References F1673

References

Anderberg, M. R. (1973), Cluster Analysis for Applications, New York: Academic Press.

Bock, H. H. (1985), “On Some Signiﬁcance Tests in Cluster Analysis,” Journal of Classiﬁcation, 2,

77–108.

Calinski, T. and Harabasz, J. (1974), “A Dendrite Method for Cluster Analysis,” Communications

in Statistics, 3, 1–27.

Cooper, M. C. and Milligan, G. W. (1988), “The Effect of Error on Determining the Number of

Clusters,” Proceedings of the International Workshop on Data Analysis, Decision Support, and

Expert Knowledge Representation in Marketing and Related Areas of Research, 319–328.

Everitt, B. S. (1980), Cluster Analysis, Second Edition, London: Heineman Educational Books.

Fisher, R. A. (1936), “The Use of Multiple Measurements in Taxonomic Problems,” Annals of

Eugenics, 7, 179–188.

Gonin, R. and Money, A. H. (1989), Nonlinear Lp-Norm Estimation, New York: Marcel Dekker.

Hartigan, J. A. (1975), Clustering Algorithms, New York: John Wiley & Sons.

Hartigan, J. A. (1985), “Statistical Theory in Clustering,” Journal of Classiﬁcation, 2, 63–76.

Journal of Statistics Education, “Fish Catch Data Set,” http://www.amstat.org/

publications/jse/jse_data_archive.html.

MacQueen, J. B. (1967), “Some Methods for Classiﬁcation and Analysis of Multivariate Observa-

tions,” Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, 1,

281–297.

McLachlan, G. J. and Basford, K. E. (1988), Mixture Models, New York: Marcel Dekker.

Mezzich, J. E. and Solomon, H. (1980), Taxonomy and Behavioral Science, New York: Academic

Press.

Milligan, G. W. (1980), “An Examination of the Effect of Six Types of Error Perturbation on Fifteen

Clustering Algorithms,” Psychometrika, 45, 325–342.

Milligan, G. W. and Cooper, M. C. (1985), “An Examination of Procedures for Determining the

Number of Clusters in a Data Set,” Psychometrika, 50, 159–179.

Pollard, D. (1981), “Strong Consistency of k-Means Clustering,” Annals of Statistics, 9, 135–140.

Sarle, W. S. (1983), “The Cubic Clustering Criterion,” SAS Technical Report A-108, Cary, NC:

SAS Institute Inc.

Spath, H. (1980), Cluster Analysis Algorithms, Chichester, Eng.: Ellis Horwood.

1674 FChapter 34: The FASTCLUS Procedure

Spath, H. (1985), Cluster Dissection and Analysis, Chichester, Eng.: Ellis Horwood.

Titterington, D. M., Smith, A. F. M., and Makov, U. E. (1985), Statistical Analysis of Finite Mixture

Distributions, New York: John Wiley & Sons.

Tou, J. T. and Gonzalez, R. C. (1974), Pattern Recognition Principles, Reading, MA: Addison-

Wesley.

Subject Index

analyzing data in groups

FASTCLUS procedure, 1640

bin-sort algorithm, 1634

cluster

centers, 1623,1637

deletion, 1635

ﬁnal, 1623

initial, 1622,1623

mean, 1637

median, 1634,1637

midrange, 1637

minimum distance separating, 1623

seeds, 1622

cluster analysis

disjoint, 1621

large data sets, 1621

robust, 1622,1637

clustering criterion

FASTCLUS procedure, 1622,1636,1637

clustering methods

FASTCLUS procedure, 1622,1624

computational problems

convergence (FASTCLUS), 1635

computational resources

FASTCLUS procedure, 1647

disjoint clustering, 1621,1622,1624

distance

between clusters (FASTCLUS), 1642

data (FASTCLUS), 1622

Euclidean (FASTCLUS), 1623

Ekblom-Newton algorithm

FASTCLUS procedure, 1637

Euclidean distances, 1623

FASTCLUS procedure

algorithm for updating cluster seeds, 1637

bin-sort algorithm, 1634

cluster deletion, 1635

clustering criterion, 1622,1636,1637

clustering methods, 1622,1624

compared to other procedures, 1647

computational problems, convergence, 1635

computational resources, 1647

controlling iterations, 1638

convergence criterion, 1634

distance, 1622,1623,1642

DRIFT option, 1623

Ekblom-Newton algorithm, 1637

homotopy parameter, 1635

imputation of missing values, 1636

incompatibilities, 1642

iteratively reweighted least squares, 1636

Lpclustering, 1622,1636

MEAN= data sets, 1638

memory requirements, 1647

Merle-Spath algorithm, 1637

missing values, 1623,1636,1638,1642

Newton algorithm, 1637

OUT= data sets, 1643

outliers, 1622

output data sets, 1638,1643

output table names, 1652

OUTSTAT= data set, 1638,1645

random number generator, 1639

scale estimates, 1635,1637,1642,1644

seed replacement, 1623,1639

weighted cluster means, 1639

homotopy parameter

FASTCLUS procedure, 1635

imputation of missing values

FASTCLUS procedure, 1636

initial seeds

FASTCLUS procedure, 1622,1623,1639

k-means clustering, 1622

leader algorithm, 1622

Lpclustering

FASTCLUS procedure, 1622

Lpclustering

FASTCLUS procedure, 1636

MEAN= data sets

FASTCLUS procedure, 1638

median

cluster, 1634,1637

memory requirements

FASTCLUS procedure, 1647

Merle-Spath algorithm

FASTCLUS procedure, 1637

missing values

FASTCLUS procedure, 1623,1636,1638,

1642

nearest centroid sorting, 1622,1623

Newton algorithm

FASTCLUS procedure, 1637

OUT= data sets

FASTCLUS procedure, 1643

outliers

FASTCLUS procedure, 1622

output data sets

FASTCLUS procedure, 1638,1643

output table names

FASTCLUS procedure, 1652

robust

cluster analysis, 1622,1637

scale estimates

FASTCLUS procedure, 1635,1637,1642,

1644

simple cluster-seeking algorithm, 1624

Syntax Index

BINS= option

PROC FASTCLUS statement, 1634

CLUSTER= option

PROC FASTCLUS statement, 1634

CLUSTERLABEL= option

PROC FASTCLUS statement, 1634

CONVERGE= option

PROC FASTCLUS statement, 1634

DATA= option

PROC FASTCLUS statement, 1635

DELETE= option

PROC FASTCLUS statement, 1635

DISTANCE option

PROC FASTCLUS statement, 1635

DRIFT option

PROC FASTCLUS statement, 1635

FASTCLUS procedure

MAXCLUSTERS= option, 1623

RADIUS= option, 1623

syntax, 1632

FASTCLUS procedure, BY statement, 1640

FASTCLUS procedure, FREQ statement, 1640

FASTCLUS procedure, ID statement, 1641

FASTCLUS procedure, PROC FASTCLUS

statement, 1632

BINS= option, 1634

CLUSTER= option, 1634

CLUSTERLABEL= option, 1634

CONVERGE= option, 1634

DATA= option, 1635

DELETE= option, 1635

DISTANCE option, 1635

DRIFT option, 1635

HC= option, 1635

HP= option, 1635

IMPUTE option, 1636

INSTAT= option, 1636

IRLS option, 1636

L= option, 1636

LEAST= option, 1636

LIST option, 1637

MAXCLUSTERS= option, 1632

MAXITER= option, 1638

MEAN= option, 1638

NOMISS option, 1638

NOPRINT option, 1638

OUT= option, 1638

OUTITER option, 1638

OUTS= option, 1638

OUTSEED= option, 1638

OUTSTAT= option, 1638

RADIUS= option, 1632

RANDOM= option, 1639

REPLACE= option, 1639

SEED= option, 1639

SHORT option, 1639

STRICT= option, 1639

SUMMARY option, 1639

VARDEF= option, 1639

FASTCLUS procedure, VAR statement, 1641

FASTCLUS procedure, WEIGHT statement,

1641

HC= option

PROC FASTCLUS statement, 1635

HP= option

PROC FASTCLUS statement, 1635

IMPUTE option

PROC FASTCLUS statement, 1636

INSTAT= option

PROC FASTCLUS statement, 1636

IRLS option

PROC FASTCLUS statement, 1636

L= option

PROC FASTCLUS statement, 1636

LEAST= option

PROC FASTCLUS statement, 1636

LIST option

PROC FASTCLUS statement, 1637

MAXCLUSTERS= option

PROC FASTCLUS statement, 1632

MAXITER= option

PROC FASTCLUS statement, 1638

MEAN= option

PROC FASTCLUS statement, 1638

NOMISS option

PROC FASTCLUS statement, 1638

NOPRINT option

PROC FASTCLUS statement, 1638

OUT= option

PROC FASTCLUS statement, 1638

OUTITER option

PROC FASTCLUS statement, 1638

OUTS= option

PROC FASTCLUS statement, 1638

OUTSEED= option

PROC FASTCLUS statement, 1638

OUTSTAT= option

PROC FASTCLUS statement, 1638

PROC FASTCLUS statement, see FASTCLUS

procedure

RADIUS= option

PROC FASTCLUS statement, 1632

RANDOM= option

PROC FASTCLUS statement, 1639

REPLACE= option

PROC FASTCLUS statement, 1639

SEED= option

PROC FASTCLUS statement, 1639

SHORT option

PROC FASTCLUS statement, 1639

STRICT= option

PROC FASTCLUS statement, 1639

SUMMARY option

PROC FASTCLUS statement, 1639

VARDEF= option

PROC FASTCLUS statement, 1639

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