Chapter Contents
Chapter Contents
Previous
Previous
Next
Next
The ACECLUS Procedure

Getting Started

The following example demonstrates how you can use the ACECLUS procedure to obtain approximate estimates of the pooled within-cluster covariance matrix and to compute canonical variables for subsequent analysis. You use PROC ACECLUS to preprocess data before you cluster it using the FASTCLUS or CLUSTER procedure.

Suppose you want to determine whether national figures for birth rates, death rates, and infant death rates can be used to determine certain types or categories of countries. You want to perform a cluster analysis to determine whether the observations can be formed into groups suggested by the data. Previous studies indicate that the clusters computed from this type of data can be elongated and elliptical. Thus, you need to perform a linear transformation on the raw data before the cluster analysis.

The following data* from Rouncefield (1995) are the birth rates, death rates, and infant death rates for 97 countries. The following statements create the SAS data set Poverty:

   data poverty;
      input Birth Death InfantDeath Country $15. @@;
      datalines;
   24.7  5.7  30.8 Albania         12.5 11.9  14.4 Bulgaria            
   13.4 11.7  11.3 Czechoslovakia  12   12.4   7.6 Former_E._Germa     
   11.6 13.4  14.8 Hungary         14.3 10.2    16 Poland              
   13.6 10.7  26.9 Romania           14    9  20.2 Yugoslavia          
   17.7   10    23 USSR            15.2  9.5  13.1 Byelorussia         
   13.4 11.6    13 Ukrainian_SSR   20.7  8.4  25.7 Argentina           
   46.6   18   111 Bolivia         28.6  7.9    63 Brazil              
   23.4  5.8  17.1 Chile           27.4  6.1    40 Columbia            
   32.9  7.4    63 Ecuador         28.3  7.3    56 Guyana              
   34.8  6.6    42 Paraguay        32.9  8.3 109.9 Peru                
     18  9.6  21.9 Uruguay         27.5  4.4  23.3 Venezuela           
     29 23.2    43 Mexico            12 10.6   7.9 Belgium             
   13.2 10.1   5.8 Finland         12.4 11.9   7.5 Denmark             
   13.6  9.4   7.4 France          11.4 11.2   7.4 Germany             
   10.1  9.2    11 Greece          15.1  9.1   7.5 Ireland             
    9.7  9.1   8.8 Italy           13.2  8.6   7.1 Netherlands         
   14.3 10.7   7.8 Norway          11.9  9.5  13.1 Portugal            
   10.7  8.2   8.1 Spain           14.5 11.1   5.6 Sweden              
   12.5  9.5   7.1 Switzerland     13.6 11.5   8.4 U.K.                
   14.9  7.4     8 Austria          9.9  6.7   4.5 Japan               
   14.5  7.3   7.2 Canada          16.7  8.1   9.1 U.S.A.              
   40.4 18.7 181.6 Afghanistan     28.4  3.8    16 Bahrain             
   42.5 11.5 108.1 Iran            42.6  7.8    69 Iraq                
   22.3  6.3   9.7 Israel          38.9  6.4    44 Jordan              
   26.8  2.2  15.6 Kuwait          31.7  8.7    48 Lebanon             
   45.6  7.8    40 Oman            42.1  7.6    71 Saudi_Arabia        
   29.2  8.4    76 Turkey          22.8  3.8    26 United_Arab_Emr     
   42.2 15.5   119 Bangladesh      41.4 16.6   130 Cambodia            
   21.2  6.7    32 China           11.7  4.9   6.1 Hong_Kong           
   30.5 10.2    91 India           28.6  9.4    75 Indonesia           
   23.5 18.1    25 Korea           31.6  5.6    24 Malaysia            
   36.1  8.8    68 Mongolia        39.6 14.8   128 Nepal               
   30.3  8.1 107.7 Pakistan        33.2  7.7    45 Philippines         
   17.8  5.2   7.5 Singapore       21.3  6.2  19.4 Sri_Lanka           
   22.3  7.7    28 Thailand        31.8  9.5    64 Vietnam             
   35.5  8.3    74 Algeria         47.2 20.2   137 Angola              
   48.5 11.6    67 Botswana        46.1 14.6    73 Congo               
   38.8  9.5  49.4 Egypt           48.6 20.7   137 Ethiopia            
   39.4 16.8   103 Gabon           47.4 21.4   143 Gambia              
   44.4 13.1    90 Ghana             47 11.3    72 Kenya               
    44  9.4    82 Libya           48.3   25   130 Malawi               
   35.5  9.8    82 Morocco           45 18.5   141 Mozambique          
     44 12.1   135 Namibia         48.5 15.6   105 Nigeria             
   48.2 23.4   154 Sierra_Leone    50.1 20.2   132 Somalia             
   32.1  9.9    72 South_Africa    44.6 15.8   108 Sudan               
   46.8 12.5   118 Swaziland       31.1  7.3    52 Tunisia             
   52.2 15.6   103 Uganda          50.5   14   106 Tanzania            
   45.6 14.2    83 Zaire           51.1 13.7    80 Zambia              
   41.7 10.3    66 Zimbabwe                                            
   ;

The data set Poverty contains the character variable Country and the numeric variables Birth, Death, and InfantDeath, which represent the birth rate per thousand, death rate per thousand, and infant death rate per thousand. The $15. in the INPUT statement specifies that the variable Country is a character variable with a length of 15. The double trailing at sign (@@) in the INPUT statement specifies that observations are input from each line until all values have been read.

It is often useful when beginning a cluster analysis to look at the data graphically. The following statements use the GPLOT procedure to make a scatter plot of the variables Birth and Death.

   axis1 label=(angle=90 rotate=0) minor=none;
   axis2 minor=none;
   proc gplot data=poverty;
      plot Birth*Death/
      frame cframe=ligr vaxis=axis1 haxis=axis2;
   run;

The plot, displayed in Figure 16.1, indicates the difficulty of dividing the points into clusters. Plots of the other variable pairs (not shown) display similar characteristics. The clusters that comprise these data may be poorly separated and elongated. Data with poorly separated or elongated clusters must be transformed.

aceg1.gif (3745 bytes)

Figure 16.1: Scatter Plot of Original Poverty Data: Birth Rate versus Death Rate

If you know the within-cluster covariances, you can transform the data to make the clusters spherical. However, since you do not know what the clusters are, you cannot calculate exactly the within-cluster covariance matrix. The ACECLUS procedure estimates the within-cluster covariance matrix to transform the data, even when you have no knowledge of cluster membership or the number of clusters.

The following statements perform the ACECLUS procedure transformation using the SAS data set Poverty.

   proc aceclus data=poverty out=ace proportion=.03;
      var Birth Death InfantDeath;
   run;

The OUT= option creates an output data set called Ace to contain the canonical variable scores. The PROPORTION= option specifies that approximately three percent of the pairs are included in the estimation of the within-cluster covariance matrix. The VAR statement specifies that the variables Birth, Death, and InfantDeath are used in computing the canonical variables.

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

Figure 16.2 displays the number of observations, the number of variables, and the settings for the PROPORTION and CONVERGE options. The PROPORTION option is set at 0.03, as specified in the previous statements. The CONVERGE parameter is set at its default value of 0.001.

The ACECLUS Procedure

Approximate Covariance Estimation for Cluster Analysis

Observations 97 Proportion 0.0300
Variables 3 Converge 0.00100

Means and Standard Deviations
Variable Mean Standard
Deviation
Birth 29.2299 13.5467
Death 10.8361 4.6475
InfantDeath 54.9010 45.9926

COV: Total Sample Covariances
  Birth Death InfantDeath
Birth 183.512951 30.610056 534.794969
Death 30.610056 21.599205 139.925900
InfantDeath 534.794969 139.925900 2115.317811

Initial Within-Cluster Covariance Estimate = Full Covariance Matrix

Threshold = 0.292815

Figure 16.2: Means, Standard Deviations, and Covariance Matrix from the ACECLUS Procedure

Figure 16.2 next displays the means, standard deviations, and sample covariance matrix of the analytical variables.

The type of matrix used for the initial within-cluster covariance estimate is displayed in Figure 16.3. In this example, that initial estimate is the full covariance matrix. The threshold value that corresponds to the PROPORTION=0.03 setting is given as 0.292815.

The ACECLUS Procedure

Approximate Covariance Estimation for Cluster Analysis

Initial Within-Cluster Covariance Estimate = Full Covariance Matrix

Iteration History
Iteration RMS
Distance
Distance
Cutoff
Pairs
Within
Cutoff
Convergence
Measure
1 2.449 0.717 385.0 0.552025
2 12.534 3.670 446.0 0.008406
3 12.851 3.763 521.0 0.009655
4 12.882 3.772 591.0 0.011193
5 12.716 3.723 628.0 0.008784
6 12.821 3.754 658.0 0.005553
7 12.774 3.740 680.0 0.003010
8 12.631 3.699 683.0 0.000676

Algorithm converged.

Figure 16.3: Table of Iteration History from the ACECLUS Procedure

Figure 16.3 displays the iteration history. For each iteration, PROC ACECLUS displays the following measures:

Figure 16.4 displays the approximate within-cluster covariance matrix and the table of eigenvalues from the canonical analysis. The first column of the eigenvalues table contains numbers for the eigenvectors. The next column of the table lists the eigenvalues of Inv(ACE)*(COV-ACE).

The ACECLUS Procedure

Approximate Covariance Estimation for Cluster Analysis

Initial Within-Cluster Covariance Estimate = Full Covariance Matrix

ACE: Approximate Covariance Estimate Within Clusters
  Birth Death InfantDeath
Birth 5.94644949 -0.63235725 6.28151537
Death -0.63235725 2.33464129 1.59005857
InfantDeath 6.28151537 1.59005857 35.10327233

Eigenvalues of Inv(ACE)*(COV-ACE)
  Eigenvalue Difference Proportion Cumulative
1 63.5500 54.7313 0.8277 0.8277
2 8.8187 4.4038 0.1149 0.9425
3 4.4149   0.0575 1.0000

Figure 16.4: Approximate Within -Cluster Covariance Estimates

The next three columns of the eigenvalue table (Figure 16.4) display measures of the relative size and importance of the eigenvalues. The first column lists the difference between each eigenvalue and its successor. The last two columns display the individual and cumulative proportions that each eigenvalue contributes to the total sum of eigenvalues.

The raw and standardized canonical coefficients are displayed in Figure 16.5. The coefficients are standardized by multiplying the raw coefficients with the standard deviation of the associated variable. The ACECLUS procedure uses these standardized canonical coefficients to create the transformed canonical variables, which are the linear transformations of the original input variables, Birth, Death, and InfantDeath.

The ACECLUS Procedure

Approximate Covariance Estimation for Cluster Analysis

Initial Within-Cluster Covariance Estimate = Full Covariance Matrix

Eigenvectors (Raw Canonical Coefficients)
  Can1 Can2 Can3
Birth 0.125610 0.457037 0.003875
Death 0.108402 0.163792 0.663538
InfantDeath 0.134704 -.133620 -.046266

Standardized Canonical Coefficients
  Can1 Can2 Can3
Birth 1.70160 6.19134 0.05249
Death 0.50380 0.76122 3.08379
InfantDeath 6.19540 -6.14553 -2.12790

Figure 16.5: Raw and Standardized Canonical Coefficients from the ACECLUS Procedure

The following statements invoke the CLUSTER procedure, using the SAS data set Ace created in the previous ACECLUS procedure.

   proc cluster data=ace outtree=tree noprint method=ward;
      var can1 can2 can3 ;
      copy Birth--Country;
   run;

The OUTTREE= option creates the output SAS data set Tree that is used in subsequent statements to draw a tree diagram. The NOPRINT option suppresses the display of the output. The METHOD= option specifies Ward's minimum-variance clustering method.

The VAR statement specifies that the canonical variables computed in the ACECLUS procedure are used in the cluster analysis. The COPY statement specifies that all the variables from the SAS data set Poverty (Birth - Country) are added to the output data set Tree.

The following statements use the TREE procedure to create an output SAS data set called New. The NCLUSTERS= option specifies the number of clusters desired in the SAS data set New. The NOPRINT option suppresses the display of the output.

   proc tree data=tree out=new nclusters=3 noprint;
      copy Birth Death InfantDeath can1 can2 ;
      id Country;
   run;

The COPY statement copies the canonical variables CAN1 and CAN2 (computed in the preceding ACECLUS procedure) and the original analytical variables Birth, Death, and InfantDeath into the output SAS data set New.

The following statements invoke the GPLOT procedure, using the SAS data set created by PROC TREE:

   legend1 frame cframe=ligr cborder=black 
           position=center value=(justify=center);
   axis1 label=(angle=90 rotate=0) minor=none;
   axis2 minor=none;
   proc gplot data=new;
      plot Birth*Death=cluster/
           frame cframe=ligr legend=legend1 vaxis=axis1 haxis=axis2;
      plot can2*can1=cluster/
           frame cframe=ligr legend=legend1 vaxis=axis1 haxis=axis2;
   run;

The first plot statement requests a scatter plot of the two variables Birth and Death, using the variable CLUSTER as the identification variable.

The second PLOT statement requests a plot of the two canonical variables, using the value of the variable CLUSTER as the identification variable.

aceg6.gif (4302 bytes)

Figure 16.6: Scatter Plot of Poverty Data, Identified by Cluster

Figure 16.6 and Figure 16.7 display the separation of the clusters when three clusters are calculated.

aceg7.gif (4336 bytes)

Figure 16.7: Scatter Plot of Canonical Variables

Chapter Contents
Chapter Contents
Previous
Previous
Next
Next
Top
Top

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.