# The ACECLUS Procedure

### Example 24.1 Transformation and Cluster Analysis of Fisher Iris Data

The iris data published by Fisher (1936) have been widely used for examples in discriminant analysis 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 analyses of the iris data.

In this example PROC ACECLUS is used to transform the iris data, which is available from the `Sashelp` library, and the clustering is performed by PROC FASTCLUS. Compare this with the example in Chapter 38: The FASTCLUS Procedure. The results from the FREQ procedure display fewer misclassifications when PROC ACECLUS is used.

The following statements produce Output 24.1.1 through Output 24.1.5:

```title 'Fisher (1936) Iris Data';

proc aceclus data=sashelp.iris out=ace p=.02 outstat=score;
var SepalLength SepalWidth PetalLength PetalWidth ;
run;
```
```proc sgplot data=ace;
scatter y=can2 x=can1 / group=Species;
keylegend / title="Species";
run;
```
```proc fastclus data=ace maxc=3 maxiter=10 conv=0 out=clus;
var can:;
run;
```
```proc freq;
tables cluster*Species;
run;
```

Output 24.1.1: Using PROC ACECLUS to Transform Fisher’s Iris Data

 Fisher (1936) Iris Data

The ACECLUS Procedure

Approximate Covariance Estimation for Cluster Analysis

 Observations Proportion 150 0.02 4 0.001

Means and Standard Deviations
Variable Mean Standard
Deviation
Label
SepalLength 58.4333 8.2807 Sepal Length (mm)
SepalWidth 30.5733 4.3587 Sepal Width (mm)
PetalLength 37.5800 17.6530 Petal Length (mm)
PetalWidth 11.9933 7.6224 Petal Width (mm)

COV: Total Sample Covariances
SepalLength SepalWidth PetalLength PetalWidth
SepalLength 68.5693512 -4.2434004 127.4315436 51.6270694
SepalWidth -4.2434004 18.9979418 -32.9656376 -12.1639374
PetalLength 127.4315436 -32.9656376 311.6277852 129.5609396
PetalWidth 51.6270694 -12.1639374 129.5609396 58.1006264

 Initial Within-Cluster Covariance Estimate = Full Covariance Matrix

 Threshold = 0.334211

Iteration History
Iteration RMS
Distance
Distance
Cutoff
Pairs
Within
Cutoff
Convergence
Measure
1 2.828 0.945 408.0 0.465775
2 11.905 3.979 559.0 0.013487
3 13.152 4.396 940.0 0.029499
4 13.439 4.491 1506.0 0.046846
5 13.271 4.435 2036.0 0.046859
6 12.591 4.208 2285.0 0.025027
7 12.199 4.077 2366.0 0.009559
8 12.121 4.051 2402.0 0.003895
9 12.064 4.032 2417.0 0.002051
10 12.047 4.026 2429.0 0.000971

 Algorithm converged.

Output 24.1.2: Eigenvalues, Raw Canonical Coefficients, and Standardized Canonical Coefficients

ACE: Approximate Covariance Estimate Within Clusters
SepalLength SepalWidth PetalLength PetalWidth
SepalLength 11.73342939 5.47550432 4.95389049 2.02902429
SepalWidth 5.47550432 6.91992590 2.42177851 1.74125154
PetalLength 4.95389049 2.42177851 6.53746398 2.35302594
PetalWidth 2.02902429 1.74125154 2.35302594 2.05166735

Eigenvalues of Inv(ACE)*(COV-ACE)
Eigenvalue Difference Proportion Cumulative
1 63.7716 61.1593 0.9367 0.9367
2 2.6123 1.5561 0.0384 0.9751
3 1.0562 0.4167 0.0155 0.9906
4 0.6395   0.00939 1.0000

Eigenvectors (Raw Canonical Coefficients)
Can1 Can2 Can3 Can4
SepalLength Sepal Length (mm) -.012009 -.098074 -.059852 0.402352
SepalWidth Sepal Width (mm) -.211068 -.000072 0.402391 -.225993
PetalLength Petal Length (mm) 0.324705 -.328583 0.110383 -.321069
PetalWidth Petal Width (mm) 0.266239 0.870434 -.085215 0.320286

Standardized Canonical Coefficients
Can1 Can2 Can3 Can4
SepalLength Sepal Length (mm) -0.09944 -0.81211 -0.49562 3.33174
SepalWidth Sepal Width (mm) -0.91998 -0.00031 1.75389 -0.98503
PetalLength Petal Length (mm) 5.73200 -5.80047 1.94859 -5.66782
PetalWidth Petal Width (mm) 2.02937 6.63478 -0.64954 2.44134

Output 24.1.3: Plot of Transformed Iris Data: PROC SGPLOT

Output 24.1.4: Clustering of Transformed Iris Data: Partial Output from PROC FASTCLUS

 Fisher (1936) Iris Data

The FASTCLUS Procedure

Cluster Summary
Cluster Frequency RMS Std Deviation Maximum Distance
from Seed
to Observation
Exceeded
Nearest Cluster Distance Between
Cluster Centroids
1 50 1.4138 5.3152   2 5.8580
2 50 1.8880 6.8298   1 5.8580
3 50 1.1016 5.2768   1 13.2845

Statistics for Variables
Variable Total STD Within STD R-Square RSQ/(1-RSQ)
Can1 8.04808 1.48537 0.966394 28.756658
Can2 1.90061 1.85646 0.058725 0.062389
Can3 1.43395 1.32518 0.157417 0.186826
Can4 1.28044 1.27550 0.021025 0.021477
OVER-ALL 4.24499 1.50298 0.876324 7.085666

 Pseudo F Statistic = 520.8

 Approximate Expected Over-All R-Squared = 0.80391

 Cubic Clustering Criterion = 5.179

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

Cluster Means
Cluster Can1 Can2 Can3 Can4
1 2.54528754 -0.59273569 -0.78905317 -0.26079612
2 8.12988211 0.52566663 0.51836499 0.14915404
3 -10.67516964 0.06706906 0.27068819 0.11164209

Cluster Standard Deviations
Cluster Can1 Can2 Can3 Can4
1 1.572366584 1.393565864 1.303411851 1.372050319
2 1.799159552 2.743869556 1.270344142 1.370523175
3 0.953761025 0.931943571 1.398456061 1.058217627

Output 24.1.5: Crosstabulation of Cluster by Species for Fisher’s Iris Data: PROC FREQ

 Fisher (1936) Iris Data

The FREQ Procedure

Frequency
Percent
Row Pct
Col Pct
Table of CLUSTER by Species
CLUSTER(Cluster) Species(Iris Species)
Setosa Versicolor Virginica Total
1
 0 0 0 0
 48 32 96 96
 2 1.33 4 4
 50 33.33
2
 0 0 0 0
 2 1.33 4 4
 48 32 96 96
 50 33.33
3
 50 33.33 100 100
 0 0 0 0
 0 0 0 0
 50 33.33
Total
 50 33.33
 50 33.33
 50 33.33
 150 100