PROC MODECLUS: Cluster Analysis of Samples from Univariate Distributions

Example 59.1 Cluster Analysis of Samples from Univariate Distributions

This example uses pseudo-random samples from a uniform distribution, an exponential distribution, and a bimodal mixture of two normal distributions. Results are presented in Output 59.1.1 through Output 59.1.18 as plots displaying both the true density and the estimated density, as well as cluster membership.

The following statements produce Output 59.1.1 through Output 59.1.4:

title 'Modeclus Example with Univariate Distributions';
title2 'Uniform Distribution';

data uniform;
   drop n;
   true=1;
   do n=1 to 100;
      x=ranuni(123);
      output;
   end;
run;

proc modeclus data=uniform m=1 k=10 20 40 60 out=out short;
   var x;
run;

proc sgplot data=out noautolegend;
   y2axis label='True' values=(0 to 2 by 1.);
   yaxis values=(0 to 3 by 0.5);
   scatter y=density x=x / markerchar=cluster group=cluster;
   pbspline y=true x=x / y2axis nomarkers lineattrs=(thickness= 1);
   by _K_;
run;

proc modeclus data=uniform m=1 r=.05 .10 .20 .30 out=out short;
   var x;
run;

proc sgplot data=out noautolegend;
   y2axis label='True' values=(0 to 2 by 1.);
   yaxis values=(0 to 2 by 0.5);
   scatter y=density x=x / markerchar=cluster group=cluster;
   pbspline y=true x=x / y2axis nomarkers lineattrs=(thickness= 1);
   by _R_;
run;

Output 59.1.1 Cluster Analysis of Sample from a Uniform Distribution

Modeclus Example with Univariate Distributions

Uniform Distribution

The MODECLUS Procedure

Cluster Summary
K	Number of Clusters	Frequency of Unclassified Objects
10	6	0
20	3	0
40	2	0
60	1	0

Output 59.1.2 True Density, Estimated Density, and Cluster Membership by Various _K_ Values

True Density, Estimated Density, and Cluster Membership by Various K Values, continued

Output 59.1.3 Cluster Analysis of Sample from a Uniform Distribution

Modeclus Example with Univariate Distributions

Uniform Distribution

The MODECLUS Procedure

Cluster Summary
R	Number of Clusters	Frequency of Unclassified Objects
0.05	4	0
0.1	2	0
0.2	2	0
0.3	1	0

Output 59.1.4 True Density, Estimated Density, and Cluster Membership by Various _R_ Values

True Density, Estimated Density, and Cluster Membership by Various R Values, continued

The following statements produce Output 59.1.5 through Output 59.1.12:

data expon;
   title2 'Exponential Distribution';
   drop n;
   do n=1 to 100;
      x=ranexp(123);
      true=exp(-x);
      output;
   end;
run;

proc modeclus data=expon m=1 k=10 20 40 out=out short;
   var x;
run;

proc sgplot data=out noautolegend;
   y2axis label='True' values=(0 to 1 by .5);
   yaxis values=(0 to 2 by 0.5);
   scatter y=density x=x / markerchar=cluster group=cluster;
   pbspline y=true x=x / y2axis nomarkers lineattrs=(thickness= 1);
   by _K_;
run;

proc modeclus data=expon m=1 r=.20 .40 .80 out=out short;
   var x;
run;

proc sgplot data=out noautolegend;
   y2axis label='True' values=(0 to 1 by .5);
   yaxis values=(0 to 1 by 0.5);
   scatter y=density x=x / markerchar=cluster group=cluster;
   pbspline y=true x=x / y2axis nomarkers lineattrs=(thickness= 1);
   by _R_;
run;

title3 'Different Density-Estimation and Clustering Windows';

proc modeclus data=expon m=1 r=.20 ck=10 20 40
              out=out short;
   var x;
run;

proc sgplot data=out noautolegend;
   y2axis label='True' values=(0 to 1 by .5);
   yaxis values=(0 to 1 by 0.5);
   scatter y=density x=x / markerchar=cluster group=cluster;
   pbspline y=true x=x / y2axis nomarkers lineattrs=(thickness= 1);
   by _CK_;
run;

title3 'Cascaded Density Estimates Using Arithmetic Means';

proc modeclus data=expon m=1 r=.20 cascade=1 2 4 am out=out short;
   var x;
run;

proc sgplot data=out noautolegend;
   y2axis label='True' values=(0 to 1 by .5);
   yaxis values=(0 to 1 by 0.5);
   scatter y=density x=x / markerchar=cluster group=cluster;
   pbspline y=true x=x / y2axis nomarkers lineattrs=(thickness= 1);
   by _R_ _CASCAD_;
run;

Output 59.1.5 Cluster Analysis of Sample from an Exponential Distribution

Modeclus Example with Univariate Distributions

Exponential Distribution

The MODECLUS Procedure

Cluster Summary
K	Number of Clusters	Frequency of Unclassified Objects
10	5	0
20	3	0
40	1	0

Output 59.1.6 True Density, Estimated Density, and Cluster Membership by Various _K_ Values

Output 59.1.7 Cluster Analysis of Sample from an Exponential Distribution

Modeclus Example with Univariate Distributions

Exponential Distribution

The MODECLUS Procedure

Cluster Summary
R	Number of Clusters	Frequency of Unclassified Objects
0.2	8	0
0.4	6	0
0.8	1	0

Output 59.1.8 True Density, Estimated Density, and Cluster Membership by Various _R_ Values

Output 59.1.9 Cluster Analysis of Sample from an Exponential Distribution

Modeclus Example with Univariate Distributions

Exponential Distribution

Different Density-Estimation and Clustering Windows

The MODECLUS Procedure

Cluster Summary
R	CK	Number of Clusters	Frequency of Unclassified Objects
0.2	10	3	0
0.2	20	2	0
0.2	40	1	0

Output 59.1.10 True Density, Estimated Density, and Cluster Membership by _R_=0.2 with Various _CK_ Values

True Density, Estimated Density, and Cluster Membership by R=0.2 with Various CK Values, continued

Output 59.1.11 Cluster Analysis of Sample from an Exponential Distribution

Modeclus Example with Univariate Distributions

Exponential Distribution

Cascaded Density Estimates Using Arithmetic Means

The MODECLUS Procedure

Cluster Summary
R	Cascade	Number of Clusters	Frequency of Unclassified Objects
0.2	1	8	0
0.2	2	8	0
0.2	4	7	0

Output 59.1.12 True Density, Estimated Density, and Cluster Membership by _R_=0.2 with Various _CASCAD_ Values

True Density, Estimated Density, and Cluster Membership by R=0.2 with Various CASCAD Values, continued

The following statements produce Output 59.1.13 through Output 59.1.18:

title2 'Normal Mixture Distribution';

data normix;
   drop n sigma;
   sigma=.125;
   do n=1 to 100;
      x=rannor(456)*sigma+mod(n,2)/2;
      true=exp(-.5*(x/sigma)**2)+exp(-.5*((x-.5)/sigma)**2);
      true=.5*true/(sigma*sqrt(2*3.1415926536));
      output;
   end;
run;

proc modeclus data=normix m=1 k=10 20 40 60 out=out short;
   var x;
run;

proc sgplot data=out noautolegend;
   y2axis label='True' values=(0 to 1.6 by .1);
   yaxis values=(0 to 3 by 0.5);
   scatter y=density x=x / markerchar=cluster group=cluster;
   pbspline y=true x=x / y2axis nomarkers lineattrs=(thickness= 1);
   by _K_;
run;

proc modeclus data=normix m=1 r=.05 .10 .20 .30 out=out short;
   var x;
run;

proc sgplot data=out noautolegend;
   y2axis label='True' values=(0 to 1.6 by .1);
   yaxis values=(0 to 3 by 0.5);
   scatter y=density x=x / markerchar=cluster group=cluster;
   pbspline y=true x=x / y2axis nomarkers lineattrs=(thickness= 1);
   by _R_;
run;

title3 'Cascaded Density Estimates Using Arithmetic Means';

proc modeclus data=normix m=1 r=.05 cascade=1 2 4 am out=out short;
   var x;
run;

proc sgplot data=out noautolegend;
   y2axis label='True' values=(0 to 1.6 by .1);
   yaxis values=(0 to 2 by 0.5);
   scatter y=density x=x / markerchar=cluster group=cluster;
   pbspline y=true x=x / y2axis nomarkers lineattrs=(thickness= 1);
   by _R_ _CASCAD_;
run;

Output 59.1.13 Cluster Analysis of Sample from a Bimodal Mixture of Two Normal Distributions

Modeclus Example with Univariate Distributions

Normal Mixture Distribution

The MODECLUS Procedure

Cluster Summary
K	Number of Clusters	Frequency of Unclassified Objects
10	7	0
20	2	0
40	2	0
60	1	0

Output 59.1.14 True Density, Estimated Density, and Cluster Membership by Various _K_ Values

Output 59.1.15 Cluster Analysis of Sample from a Bimodal Mixture of Two Normal Distributions

Modeclus Example with Univariate Distributions

Normal Mixture Distribution

The MODECLUS Procedure

Cluster Summary
R	Number of Clusters	Frequency of Unclassified Objects
0.05	5	0
0.1	2	0
0.2	2	0
0.3	1	0

Output 59.1.16 True Density, Estimated Density, and Cluster Membership by Various _R_= Values

Output 59.1.17 Cluster Analysis of Sample from a Bimodal Mixture of Two Normal Distributions

Modeclus Example with Univariate Distributions

Normal Mixture Distribution

Cascaded Density Estimates Using Arithmetic Means

The MODECLUS Procedure

Cluster Summary
R	Cascade	Number of Clusters	Frequency of Unclassified Objects
0.05	1	5	0
0.05	2	4	0
0.05	4	4	0

Output 59.1.18 True Density, Estimated Density, and Cluster Membership by _R_=0.05 with Various _CASCAD_ Values

True Density, Estimated Density, and Cluster Membership by R=0.05 with Various CASCAD Values, continued