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 The MODECLUS Procedure

## Example 42.3: Cluster Analysis with Significance Tests

This example uses artificial data containing two clusters. One cluster is from a circular bivariate normal distribution. The other is a ring-shaped cluster that completely surrounds the first cluster. Without significance tests, the ring is divided into several sample clusters for any degree of smoothing that yields reasonable density estimates. The JOIN= option puts the ring back together. Output 42.3.1 displays a short summary generated from the first PROC MODECLUS statement. Output 42.3.2 contains a series of tables produced from the second PROC MODECLUS statement. The lack of p-value in the JOIN= option makes joining continue until only one cluster remains (see the description of the JOIN= option). The cluster memberships are then plotted as displayed in Output 42.3.3.

```   title  'Modeclus Analysis with the JOIN= option';
title2 'A Normal Cluster Surrounded by a Ring Cluster';
options ls=120 ps=38;

data circle; keep x y;
c=1;
do n=1 to 30;
x=rannor(5);
y=rannor(5);
output;
end;

c=2;
do n=1 to 300;
x=rannor(5);
y=rannor(5);
z=rannor(5)+8;
l=z/sqrt(x**2+y**2);
x=x*l;
y=y*l;
output;
end;

axis1 label=(angle=90 rotate=0) minor=none
order=(-10 to 10 by 5);
axis2 minor=none order=(-15 to 15 by 5);

proc modeclus data=circle m=1 r=1 to 3.5 by .25 join=20 short;
proc modeclus data=circle m=1 r=2.5 join out=out;

proc gplot data=out;
plot y*x=cluster/frame cframe=ligr
vzero nolegend
vaxis=axis1 haxis=axis2 ;
by _NJOIN_;
run;
```

Output 42.3.1: Significance Tests with the JOIN=20 and SHORT Options

 Modeclus Analysis with the JOIN= option A Normal Cluster Surrounded by a Ring Cluster

 The MODECLUS Procedure

 Cluster Summary R Number ofClustersJoined MaximumP-value Number ofClusters Frequency ofUnclassifiedObjects 1 36 0.9339 1 301 1.25 20 0.7131 1 301 1.5 10 0.3296 1 300 1.75 5 0.1990 2 0 2 5 0.0683 2 0 2.25 3 0.0504 2 0 2.5 4 0.0301 2 0 2.75 3 0.0585 2 0 3 5 0.0003 1 0 3.25 4 0.1923 2 0 3.5 4 0.0000 1 0

Output 42.3.2: Significance Tests with the JOIN Option

 Modeclus Analysis with the JOIN= option A Normal Cluster Surrounded by a Ring Cluster

 The MODECLUS Procedure R=2.5 METHOD=1

 Cluster Statistics -Saddle Test: Version 92.7- Cluster Frequency MaximumEstimatedDensity BoundaryFrequency EstimatedSaddleDensity ModeCount SaddleCount OverlapCount Z ApproxP-value 1 103 0.00617328 22 0.00308664 39 19 0 2.495 0.5055 2 71 0.00571029 20 0.0043213 36 27 9 1.193 0.999 3 53 0.00509296 18 0.00401263 32 25 10 0.986 0.9999 4 45 0.00478429 19 0.00354964 30 22 14 1.429 0.9924 5 30 0.00462996 0 . 29 0 . 3.611 0.0301 6 28 0.00370397 17 0.00354964 23 22 9 0.000 1

 Modeclus Analysis with the JOIN= option A Normal Cluster Surrounded by a Ring Cluster

 The MODECLUS Procedure R=2.5 METHOD=1

 Cluster Statistics -Saddle Test: Version 92.7- Cluster Frequency MaximumEstimatedDensity BoundaryFrequency EstimatedSaddleDensity ModeCount SaddleCount OverlapCount Z ApproxP-value 1 103 0.00617328 22 0.00308664 39 19 0 2.495 0.5055 2 71 0.00571029 20 0.0043213 36 27 9 1.193 0.999 3 53 0.00509296 18 0.00401263 32 25 10 0.986 0.9999 4 73 0.00478429 13 0.00293231 30 18 0 1.588 0.9778 5 30 0.00462996 0 . 29 0 . 3.611 0.0301

 Modeclus Analysis with the JOIN= option A Normal Cluster Surrounded by a Ring Cluster

 The MODECLUS Procedure R=2.5 METHOD=1

 Cluster Statistics -Saddle Test: Version 92.7- Cluster Frequency MaximumEstimatedDensity BoundaryFrequency EstimatedSaddleDensity ModeCount SaddleCount OverlapCount Z ApproxP-value 1 156 0.00617328 17 0.00246931 39 15 0 3.130 0.1318 2 71 0.00571029 20 0.0043213 36 27 9 1.193 0.999 3 73 0.00478429 13 0.00293231 30 18 0 1.588 0.9778 4 30 0.00462996 0 . 29 0 . 3.611 0.0301

 Modeclus Analysis with the JOIN= option A Normal Cluster Surrounded by a Ring Cluster

 The MODECLUS Procedure R=2.5 METHOD=1

 Cluster Statistics -Saddle Test: Version 92.7- Cluster Frequency MaximumEstimatedDensity BoundaryFrequency EstimatedSaddleDensity ModeCount SaddleCount OverlapCount Z ApproxP-value 1 156 0.00617328 17 0.00246931 39 15 0 3.130 0.1318 2 144 0.00571029 14 0.00293231 36 18 0 2.313 0.6447 3 30 0.00462996 0 . 29 0 . 3.611 0.0301

 Modeclus Analysis with the JOIN= option A Normal Cluster Surrounded by a Ring Cluster

 The MODECLUS Procedure R=2.5 METHOD=1

 Cluster Statistics -Saddle Test: Version 92.7- Cluster Frequency MaximumEstimatedDensity BoundaryFrequency EstimatedSaddleDensity ModeCount SaddleCount OverlapCount Z ApproxP-value 1 300 0.00617328 0 . 39 0 . 4.246 0.0026 2 30 0.00462996 0 . 29 0 . 3.611 0.0301

 Modeclus Analysis with the JOIN= option A Normal Cluster Surrounded by a Ring Cluster

 The MODECLUS Procedure R=2.5 METHOD=1

 Cluster Statistics -Saddle Test: Version 92.7- Cluster Frequency MaximumEstimatedDensity BoundaryFrequency EstimatedSaddleDensity ModeCount SaddleCount OverlapCount Z ApproxP-value 1 300 0.00617328 0 . 39 0 . 4.246 0.0026

 Modeclus Analysis with the JOIN= option A Normal Cluster Surrounded by a Ring Cluster

 The MODECLUS Procedure

 Cluster Summary R Number ofClustersJoined MaximumP-value Number ofClusters Frequency ofUnclassifiedObjects 2.5 0 1.0000 6 0 2.5 1 0.9999 5 0 2.5 2 0.9990 4 0 2.5 3 0.6447 3 0 2.5 4 0.0301 2 0 2.5 5 0.0026 1 30

Output 42.3.3: Scatter Plots of Cluster Memberships by _NJOIN_

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