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 . @@; 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=white legend=legend1 vaxis=axis1 haxis=axis2; run; quit;
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.
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 Standard Variable Mean 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
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.
Initial Within-Cluster Covariance Estimate = Full Covariance Matrix Threshold = 0.292815 Iteration History Pairs RMS Distance Within Convergence Iteration Distance Cutoff Cutoff 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 displays the iteration history. For each iteration, PROC ACECLUS displays the following measures:
root mean square distance between all pairs of observations
distance cutoff for including pairs of observations in the estimate of within-cluster covariances (equal to RMS*Threshold)
number of pairs within the cutoff
convergence measure
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).
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
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 .
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
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=white 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=white 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.
Figure 16.6 and Figure 16.7 display the separation of the clusters when three clusters are calculated.
^{ [*] } These data have been compiled from the United Nations Demographic Yearbook 1990 (United Nations publications , Sales No. E/F.91.XII.1, copyright 1991, United Nations, New York) and are reproduced with the permission of the United Nations.