The ACECLUS Procedure |
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 data, and the clustering is performed by PROC FASTCLUS. Compare this with the example in Chapter 34, The FASTCLUS Procedure. The results from the FREQ procedure display fewer misclassifications when PROC ACECLUS is used. The following statements produce Output 22.1.1 through Output 22.1.5.
proc format; value specname 1='Setosa ' 2='Versicolor' 3='Virginica '; run; data iris; title 'Fisher (1936) Iris Data'; input SepalLength SepalWidth PetalLength PetalWidth Species @@; format Species specname.; label SepalLength='Sepal Length in mm.' SepalWidth ='Sepal Width in mm.' PetalLength='Petal Length in mm.' PetalWidth ='Petal Width in mm.'; symbol = put(species, specname10.); datalines; 50 33 14 02 1 64 28 56 22 3 65 28 46 15 2 67 31 56 24 3 63 28 51 15 3 46 34 14 03 1 69 31 51 23 3 62 22 45 15 2 59 32 48 18 2 46 36 10 02 1 61 30 46 14 2 60 27 51 16 2 65 30 52 20 3 56 25 39 11 2 65 30 55 18 3 58 27 51 19 3 68 32 59 23 3 51 33 17 05 1 57 28 45 13 2 62 34 54 23 3 77 38 67 22 3 63 33 47 16 2 67 33 57 25 3 76 30 66 21 3 49 25 45 17 3 55 35 13 02 1 67 30 52 23 3 70 32 47 14 2 64 32 45 15 2 61 28 40 13 2 48 31 16 02 1 59 30 51 18 3 55 24 38 11 2 63 25 50 19 3 64 32 53 23 3 52 34 14 02 1 49 36 14 01 1 54 30 45 15 2 79 38 64 20 3 44 32 13 02 1 67 33 57 21 3 50 35 16 06 1 58 26 40 12 2 44 30 13 02 1 77 28 67 20 3 63 27 49 18 3 47 32 16 02 1 55 26 44 12 2 50 23 33 10 2 72 32 60 18 3 48 30 14 03 1 51 38 16 02 1 61 30 49 18 3 48 34 19 02 1 50 30 16 02 1 50 32 12 02 1 61 26 56 14 3 64 28 56 21 3 43 30 11 01 1 58 40 12 02 1 51 38 19 04 1 67 31 44 14 2 62 28 48 18 3 49 30 14 02 1 51 35 14 02 1 56 30 45 15 2 58 27 41 10 2 50 34 16 04 1 46 32 14 02 1 60 29 45 15 2 57 26 35 10 2 57 44 15 04 1 50 36 14 02 1 77 30 61 23 3 63 34 56 24 3 58 27 51 19 3 57 29 42 13 2 72 30 58 16 3 54 34 15 04 1 52 41 15 01 1 71 30 59 21 3 64 31 55 18 3 60 30 48 18 3 63 29 56 18 3 49 24 33 10 2 56 27 42 13 2 57 30 42 12 2 55 42 14 02 1 49 31 15 02 1 77 26 69 23 3 60 22 50 15 3 54 39 17 04 1 66 29 46 13 2 52 27 39 14 2 60 34 45 16 2 50 34 15 02 1 44 29 14 02 1 50 20 35 10 2 55 24 37 10 2 58 27 39 12 2 47 32 13 02 1 46 31 15 02 1 69 32 57 23 3 62 29 43 13 2 74 28 61 19 3 59 30 42 15 2 51 34 15 02 1 50 35 13 03 1 56 28 49 20 3 60 22 40 10 2 73 29 63 18 3 67 25 58 18 3 49 31 15 01 1 67 31 47 15 2 63 23 44 13 2 54 37 15 02 1 56 30 41 13 2 63 25 49 15 2 61 28 47 12 2 64 29 43 13 2 51 25 30 11 2 57 28 41 13 2 65 30 58 22 3 69 31 54 21 3 54 39 13 04 1 51 35 14 03 1 72 36 61 25 3 65 32 51 20 3 61 29 47 14 2 56 29 36 13 2 69 31 49 15 2 64 27 53 19 3 68 30 55 21 3 55 25 40 13 2 48 34 16 02 1 48 30 14 01 1 45 23 13 03 1 57 25 50 20 3 57 38 17 03 1 51 38 15 03 1 55 23 40 13 2 66 30 44 14 2 68 28 48 14 2 54 34 17 02 1 51 37 15 04 1 52 35 15 02 1 58 28 51 24 3 67 30 50 17 2 63 33 60 25 3 53 37 15 02 1 ;
proc aceclus data=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"; format Species specname. ; run;
proc fastclus data=ace maxc=3 maxiter=10 conv=0 out=clus; var can:; run;
proc freq; tables cluster*Species; run;
Observations | 150 | Proportion | 0.0200 |
---|---|---|---|
Variables | 4 | Converge | 0.00100 |
Means and Standard Deviations | |||
---|---|---|---|
Variable | Mean | Standard Deviation |
Label |
SepalLength | 58.4333 | 8.2807 | Sepal Length in mm. |
SepalWidth | 30.5733 | 4.3587 | Sepal Width in mm. |
PetalLength | 37.5800 | 17.6530 | Petal Length in mm. |
PetalWidth | 11.9933 | 7.6224 | Petal Width in 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 |
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 |
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 in mm. | -.012009 | -.098074 | -.059852 | 0.402352 |
SepalWidth | Sepal Width in mm. | -.211068 | -.000072 | 0.402391 | -.225993 |
PetalLength | Petal Length in mm. | 0.324705 | -.328583 | 0.110383 | -.321069 |
PetalWidth | Petal Width in mm. | 0.266239 | 0.870434 | -.085215 | 0.320286 |
Standardized Canonical Coefficients | |||||
---|---|---|---|---|---|
Can1 | Can2 | Can3 | Can4 | ||
SepalLength | Sepal Length in mm. | -0.09944 | -0.81211 | -0.49562 | 3.33174 |
SepalWidth | Sepal Width in mm. | -0.91998 | -0.00031 | 1.75389 | -0.98503 |
PetalLength | Petal Length in mm. | 5.73200 | -5.80047 | 1.94859 | -5.66782 |
PetalWidth | Petal Width in mm. | 2.02937 | 6.63478 | -0.64954 | 2.44134 |
Cluster Summary | ||||||
---|---|---|---|---|---|---|
Cluster | Frequency | RMS Std Deviation | Maximum Distance from Seed to Observation |
Radius Exceeded |
Nearest Cluster | Distance Between Cluster Centroids |
1 | 50 | 1.1016 | 5.2768 | 3 | 13.2845 | |
2 | 50 | 1.8880 | 6.8298 | 3 | 5.8580 | |
3 | 50 | 1.4138 | 5.3152 | 2 | 5.8580 |
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 |
Cluster Means | ||||
---|---|---|---|---|
Cluster | Can1 | Can2 | Can3 | Can4 |
1 | -10.67516964 | 0.06706906 | 0.27068819 | 0.11164209 |
2 | 8.12988211 | 0.52566663 | 0.51836499 | 0.14915404 |
3 | 2.54528754 | -0.59273569 | -0.78905317 | -0.26079612 |
Cluster Standard Deviations | ||||
---|---|---|---|---|
Cluster | Can1 | Can2 | Can3 | Can4 |
1 | 0.953761025 | 0.931943571 | 1.398456061 | 1.058217627 |
2 | 1.799159552 | 2.743869556 | 1.270344142 | 1.370523175 |
3 | 1.572366584 | 1.393565864 | 1.303411851 | 1.372050319 |
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