Examples
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Example 28.1. Fisher s 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 fifty 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, the FASTCLUS procedure is used to find two and, then, three clusters. An output data set is created, and PROC FREQ is invoked to compare the clusters with the species classification. See Output 28.1.1 and Output 28.1.2 for these results. For three clusters, you can use the CANDISC procedure to compute canonical variables for plotting the clusters. See Output 28.1.3 for the results.
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 fastclus data=iris maxc=2 maxiter=10 out=clus; var SepalLength SepalWidth PetalLength PetalWidth; run; proc freq; tables cluster*species; run; proc fastclus data=iris maxc=3 maxiter=10 out=clus; var SepalLength SepalWidth PetalLength PetalWidth; run; proc freq; tables cluster*Species; run; proc candisc anova out=can; class cluster; var SepalLength SepalWidth PetalLength PetalWidth; title2 'Canonical Discriminant Analysis of Iris Clusters'; run; legend1 frame cframe=ligr label=none cborder=black position=center value=(justify=center); axis1 label=(angle=90 rotate=0) minor=none; axis2 minor=none; proc gplot data=Can; plot Can2*Can1=Cluster/frame cframe=ligr legend=legend1 vaxis=axis1 haxis=axis2; title2 'Plot of Canonical Variables Identified by Cluster'; run;
Output 28.1.1: Fisher s Iris Data ”PROC FASTCLUS with MAXC=2 and PROC FREQ
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Fisher (1936) Iris Data The FASTCLUS Procedure Replace=FULL Radius=0 Maxclusters=2 Maxiter=10 Converge=0.02 Initial Seeds Cluster SepalLength SepalWidth PetalLength PetalWidth ------------------------------------------------------------------------------- 1 43.00000000 30.00000000 11.00000000 1.00000000 2 77.00000000 26.00000000 69.00000000 23.00000000 Minimum Distance Between Initial Seeds = 70.85196
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Fisher (1936) Iris Data The FASTCLUS Procedure Replace=FULL Radius=0 Maxclusters=2 Maxiter=10 Converge=0.02 Iteration History Relative Change in Cluster Seeds Iteration Criterion 1 2 ---------------------------------------------- 1 11.0638 0.1904 0.3163 2 5.3780 0.0596 0.0264 3 5.0718 0.0174 0.00766 Convergence criterion is satisfied. Criterion Based on Final Seeds = 5.0417 Cluster Summary Maximum Distance RMS Std from Seed Radius Nearest Distance Between Cluster Frequency Deviation to Observation Exceeded Cluster Cluster Centroids -------------------------------------------------------------------------------------------------- 1 53 3.7050 21.1621 2 39.2879 2 97 5.6779 24.6430 1 39.2879 Statistics for Variables Variable Total STD Within STD R-Square RSQ/(1-RSQ) --------------------------------------------------------------------- SepalLength 8.28066 5.49313 0.562896 1.287784 SepalWidth 4.35866 3.70393 0.282710 0.394137 PetalLength 17.65298 6.80331 0.852470 5.778291 PetalWidth 7.62238 3.57200 0.781868 3.584390 OVER-ALL 10.69224 5.07291 0.776410 3.472463 Pseudo F Statistic = 513.92 Approximate Expected Over-All R-Squared = 0.51539 Cubic Clustering Criterion = 14.806 WARNING: The two values above are invalid for correlated variables.
Fisher (1936) Iris Data The FASTCLUS Procedure Replace=FULL Radius=0 Maxclusters=2 Maxiter=10 Converge=0.02 Cluster Means Cluster SepalLength SepalWidth PetalLength PetalWidth ------------------------------------------------------------------------------- 1 50.05660377 33.69811321 15.60377358 2.90566038 2 63.01030928 28.86597938 49.58762887 16.95876289 Cluster Standard Deviations Cluster SepalLength SepalWidth PetalLength PetalWidth ------------------------------------------------------------------------------- 1 3.427350930 4.396611045 4.404279486 2.105525249 2 6.336887455 3.267991438 7.800577673 4.155612484
Fisher (1936) Iris Data The FREQ Procedure Table of CLUSTER by Species CLUSTER(Cluster) Species Frequency| Percent | Row Pct | Col Pct |Setosa |Versicol|Virginic| Total | |or |a | ---------+--------+--------+--------+ 1 | 50 | 3 | 0 | 53 | 33.33 | 2.00 | 0.00 | 35.33 | 94.34 | 5.66 | 0.00 | | 100.00 | 6.00 | 0.00 | ---------+--------+--------+--------+ 2 | 0 | 47 | 50 | 97 | 0.00 | 31.33 | 33.33 | 64.67 | 0.00 | 48.45 | 51.55 | | 0.00 | 94.00 | 100.00 | ---------+--------+--------+--------+ Total 50 50 50 150 33.33 33.33 33.33 100.00
Output 28.1.2: Fisher s Iris Data ”PROC FASTCLUS with MAXC=3 and PROC FREQ
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Fisher (1936) Iris Data The FASTCLUS Procedure Replace=FULL Radius=0 Maxclusters=3 Maxiter=10 Converge=0.02 Initial Seeds Cluster SepalLength SepalWidth PetalLength PetalWidth ------------------------------------------------------------------------------- 1 58.00000000 40.00000000 12.00000000 2.00000000 2 77.00000000 38.00000000 67.00000000 22.00000000 3 49.00000000 25.00000000 45.00000000 17.00000000 Minimum Distance Between Initial Seeds = 38.23611
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Fisher (1936) Iris Data The FASTCLUS Procedure Replace=FULL Radius=0 Maxclusters=3 Maxiter=10 Converge=0.02 Iteration History Relative Change in Cluster Seeds Iteration Criterion 1 2 3 ---------------------------------------------------------- 1 6.7591 0.2652 0.3205 0.2985 2 3.7097 0 0.0459 0.0317 3 3.6427 0 0.0182 0.0124 Convergence criterion is satisfied. Criterion Based on Final Seeds = 3.6289 Cluster Summary Maximum Distance RMS Std from Seed Radius Nearest Distance Between Cluster Frequency Deviation to Observation Exceeded Cluster Cluster Centroids -------------------------------------------------------------------------------------------------- 1 50 2.7803 12.4803 3 33.5693 2 38 4.0168 14.9736 3 17.9718 3 62 4.0398 16.9272 2 17.9718 Statistics for Variables Variable Total STD Within STD R-Square RSQ/(1-RSQ) --------------------------------------------------------------------- SepalLength 8.28066 4.39488 0.722096 2.598359 SepalWidth 4.35866 3.24816 0.452102 0.825156 PetalLength 17.65298 4.21431 0.943773 16.784895 PetalWidth 7.62238 2.45244 0.897872 8.791618 OVER-ALL 10.69224 3.66198 0.884275 7.641194 Pseudo F Statistic = 561.63 Approximate Expected Over-All R-Squared = 0.62728 Cubic Clustering Criterion = 25.021 WARNING: The two values above are invalid for correlated variables.
Fisher (1936) Iris Data The FASTCLUS Procedure Replace=FULL Radius=0 Maxclusters=3 Maxiter=10 Converge=0.02 Cluster Means Cluster SepalLength SepalWidth PetalLength PetalWidth ------------------------------------------------------------------------------- 1 50.06000000 34.28000000 14.62000000 2.46000000 2 68.50000000 30.73684211 57.42105263 20.71052632 3 59.01612903 27.48387097 43.93548387 14.33870968 Cluster Standard Deviations Cluster SepalLength SepalWidth PetalLength PetalWidth ------------------------------------------------------------------------------- 1 3.524896872 3.790643691 1.736639965 1.053855894 2 4.941550255 2.900924461 4.885895746 2.798724562 3 4.664100551 2.962840548 5.088949673 2.974997167
Fisher (1936) Iris Data The FREQ Procedure Table of CLUSTER by Species CLUSTER(Cluster) Species Frequency| Percent | Row Pct | Col Pct |Setosa |Versicol|Virginic| Total | |or |a | ---------+--------+--------+--------+ 1 | 50 | 0 | 0 | 50 | 33.33 | 0.00 | 0.00 | 33.33 | 100.00 | 0.00 | 0.00 | | 100.00 | 0.00 | 0.00 | ---------+--------+--------+--------+ 2 | 0 | 2 | 36 | 38 | 0.00 | 1.33 | 24.00 | 25.33 | 0.00 | 5.26 | 94.74 | | 0.00 | 4.00 | 72.00 | ---------+--------+--------+--------+ 3 | 0 | 48 | 14 | 62 | 0.00 | 32.00 | 9.33 | 41.33 | 0.00 | 77.42 | 22.58 | | 0.00 | 96.00 | 28.00 | ---------+--------+--------+--------+ Total 50 50 50 150 33.33 33.33 33.33 100.00
Output 28.1.3: Fisher s Iris Data ”PROC CANDISC and PROC GPLOT
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Fisher (1936) Iris Data Canonical Discriminant Analysis of Iris Clusters The CANDISC Procedure Observations 150 DF Total 149 Variables 4 DF Within Classes 147 Classes 3 DF Between Classes 2 Class Level Information Variable CLUSTER Name Frequency Weight Proportion 1 _1 50 50.0000 0.333333 2 _2 38 38.0000 0.253333 3 _3 62 62.0000 0.413333
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Fisher (1936) Iris Data Canonical Discriminant Analysis of Iris Clusters The CANDISC Procedure Univariate Test Statistics F Statistics, Num DF=2, Den DF=147 Total Pooled Between Standard Standard Standard R-Square Variable Label Deviation Deviation Deviation R-Square / (1-RSq) F Value Pr > F SepalLength Sepal Length in mm. 8.2807 4.3949 8.5893 0.7221 2.5984 190.98 <.0001 SepalWidth Sepal Width in mm. 4.3587 3.2482 3.5774 0.4521 0.8252 60.65 <.0001 PetalLength Petal Length in mm. 17.6530 4.2143 20.9336 0.9438 16.7849 1233.69 <.0001 PetalWidth Petal Width in mm. 7.6224 2.4524 8.8164 0.8979 8.7916 646.18 <.0001 Average R-Square Unweighted 0.7539604 Weighted by Variance 0.8842753 Multivariate Statistics and F Approximations S=2 M=0.5 N=71 Statistic Value F Value Num DF Den DF Pr > F Wilks' Lambda 0.03222337 164.55 8 288 <.0001 Pillai's Trace 1.25669612 61.29 8 290 <.0001 Hotelling-Lawley Trace 21.06722883 377.66 8 203.4 <.0001 Roy's Greatest Root 20.63266809 747.93 4 145 <.0001 NOTE: F Statistic for Roy's Greatest Root is an upper bound. NOTE: F Statistic for Wilks Lambda is exact.
Fisher (1936) Iris Data Canonical Discriminant Analysis of Iris Clusters The CANDISC Procedure Adjusted Approximate Squared Canonical Canonical Standard Canonical Correlation Correlation Error Correlation 1 0.976613 0.976123 0.003787 0.953774 2 0.550384 0.543354 0.057107 0.302923 Test of H0: The canonical correlations in the Eigenvalues of Inv(E)*H current row and all that follow are zero = CanRsq/(1-CanRsq) Likelihood Approximate Eigenvalue Difference Proportion Cumulative Ratio F Value Num DF Den DF Pr > F 1 20.6327 20.1981 0.9794 0.9794 0.03222337 164.55 8 288 <.0001 2 0.4346 0.0206 1.0000 0.69707749 21.00 3 145 <.0001
Fisher (1936) Iris Data Canonical Discriminant Analysis of Iris Clusters The CANDISC Procedure Total Canonical Structure Variable Label Can1 Can2 SepalLength Sepal Length in mm. 0.831965 0.452137 SepalWidth Sepal Width in mm. 0.515082 0.810630 PetalLength Petal Length in mm. 0.993520 0.087514 PetalWidth Petal Width in mm. 0.966325 0.154745 Between Canonical Structure Variable Label Can1 Can2 SepalLength Sepal Length in mm. 0.956160 0.292846 SepalWidth Sepal Width in mm. 0.748136 0.663545 PetalLength Petal Length in mm. 0.998770 0.049580 PetalWidth Petal Width in mm. 0.995952 0.089883 Pooled Within Canonical Structure Variable Label Can1 Can2 SepalLength Sepal Length in mm. 0.339314 0.716082 SepalWidth Sepal Width in mm. 0.149614 0.914351 PetalLength Petal Length in mm. 0.900839 0.308136 PetalWidth Petal Width in mm. 0.650123 0.404282
Fisher (1936) Iris Data Canonical Discriminant Analysis of Iris Clusters The CANDISC Procedure Total-Sample Standardized Canonical Coefficients Variable Label Can1 Can2 SepalLength Sepal Length in mm. 0.047747341 1.021487262 SepalWidth Sepal Width in mm. 0.577569244 0.864455153 PetalLength Petal Length in mm. 3.341309573 1.283043758 PetalWidth Petal Width in mm. 0.996451144 0.900476563 Pooled Within-Class Standardized Canonical Coefficients Variable Label Can1 Can2 SepalLength Sepal Length in mm. 0.0253414487 0.5421446856 SepalWidth Sepal Width in mm. .4304161258 0.6442092294 PetalLength Petal Length in mm. 0.7976741592 .3063023132 PetalWidth Petal Width in mm. 0.3205998034 0.2897207865 Raw Canonical Coefficients Variable Label Can1 Can2 SepalLength Sepal Length in mm. 0.0057661265 0.1233581748 SepalWidth Sepal Width in mm. .1325106494 0.1983303556 PetalLength Petal Length in mm. 0.1892773419 .0726814163 PetalWidth Petal Width in mm. 0.1307270927 0.1181359305 Class Means on Canonical Variables CLUSTER Can1 Can2 1 6.131527227 0.244761516 2 4.931414018 0.861972277 3 1.922300462 0.725693908
Example 28.2. Outliers
The second example involves data artificially generated to contain two clusters and several severe outliers. A preliminary analysis specifies twenty clusters and outputs an OUTSEED= data set to be used for a diagnostic plot. The exact number of initial clusters is not important; similar results could be obtained with ten or fifty initial clusters. Examination of the plot suggests that clusters with more than five (again, the exact number is not important) observations may yield good seeds for the main analysis. A DATA step deletes clusters with five or fewer observations, and the remaining cluster means provide seeds for the next PROC FASTCLUS analysis.
Two clusters are requested ; the LEAST= option specifies the mean absolute deviation criterion (LEAST=1) . Values of the LEAST= option less than 2 reduce the effect of outliers on cluster centers.
The next analysis also requests two clusters; the STRICT= option is specified to prevent outliers from distorting the results. The STRICT= value is chosen to be close to the _GAP_ and _RADIUS_ values of the larger clusters in the diagnostic plot; the exact value is not critical.
A final PROC FASTCLUS run assigns the outliers to clusters. The results are displayed in Output 28.2.1 through Output 28.2.4.
/* Create artificial data set with two clusters */ /* and some outliers. */ data x; title 'Using PROC FASTCLUS to Analyze Data with Outliers'; drop n; do n=1 to 100; x=rannor(12345)+2; y=rannor(12345); output; end; do n=1 to 100; x=rannor(12345)-2; y=rannor(12345); output; end; do n=1 to 10; x=10*rannor(12345); y=10*rannor(12345); output; end; run; /* Run PROC FASTCLUS with many clusters and OUTSEED= output */ /* data set for diagnostic plot. */ title2 'Preliminary PROC FASTCLUS Analysis with 20 Clusters'; proc fastclus data=x outseed=mean1 maxc=20 maxiter=0 summary; var x y; run; legend1 frame cframe=ligr label=none cborder=black position=center value=(justify=center); axis1 label=(angle=90 rotate=0) minor=none order=(0 to 10 by 2); axis2 minor=none ; proc gplot data=mean1; plot _gap_*_freq_ _radius_*_freq_ /overlay frame cframe=ligr vaxis=axis1 haxis=axis2 legend=legend1; run;
Output 28.2.1: Preliminary Analysis of Data with Outliers ”PROC FASTCLUS and PROC GPLOT
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Using PROC FASTCLUS to Analyze Data with Outliers Preliminary PROC FASTCLUS Analysis with 20 Clusters The FASTCLUS Procedure Replace=FULL Radius=0 Maxclusters=20 Maxiter=0 Criterion Based on Final Seeds = 0.6873 Cluster Summary Maximum Distance RMS Std from Seed Radius Nearest Distance Between Cluster Frequency Deviation to Observation Exceeded Cluster Cluster Centroids -------------------------------------------------------------------------------------------------- 1 8 0.4753 1.1924 19 1.7205 2 1 . 0 6 6.2847 3 44 0.6252 1.6774 5 1.4386 4 1 . 0 20 5.2130 5 38 0.5603 1.4528 3 1.4386 6 2 0.0542 0.1085 2 6.2847 7 1 . 0 14 2.5094 8 2 0.6480 1.2961 1 1.8450 9 1 . 0 7 9.4534 10 1 . 0 18 4.2514 11 1 . 0 16 4.7582 12 20 0.5911 1.6291 16 1.5601 13 5 0.6682 1.4244 3 1.9553 14 1 . 0 7 2.5094 15 5 0.4074 1.2678 3 1.7609 16 22 0.4168 1.5139 19 1.4936 17 8 0.4031 1.4794 5 1.5564 18 1 . 0 10 4.2514 19 45 0.6475 1.6285 16 1.4936 20 3 0.5719 1.3642 15 1.8999 Pseudo F Statistic = 207.58 Approximate Expected Over-All R-Squared = 0.96103 Cubic Clustering Criterion = -2.503 WARNING: The two values above are invalid for correlated variables.
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/* Remove low frequency clusters. */ data seed; set mean1; if _freq_>5; run; /* Run PROC FASTCLUS again, selecting seeds from the */ /* high frequency clusters in the previous analysis */ /* using LEAST=1 Clustering Criterion */ title2 'PROC FASTCLUS Analysis Using LEAST= Clustering Criterion'; title3 'Values < 2 Reduce Effect of Outliers on Cluster Centers'; proc fastclus data=x seed=seed maxc=2 least=1 out=out; var x y; run; legend1 frame cframe=ligr label=none cborder=black position=center value=(justify=center); axis1 label=(angle=90 rotate=0) minor=none; axis2 minor=none; proc gplot data=out; plot y*x=cluster/frame cframe=ligr legend=legend1 vaxis=axis1 haxis=axis2; run;
Output 28.2.2: Analysis of Data with Outliers using the LEAST= Option
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Using PROC FASTCLUS to Analyze Data with Outliers PROC FASTCLUS Analysis Using LEAST= Clustering Criterion Values < 2 Reduce Effect of Outliers on Cluster Centers The FASTCLUS Procedure Replace=FULL Radius=0 Maxclusters=2 Maxiter=20 Converge=0.0001 Least=1 Initial Seeds Cluster x y ------------------------------------------- 1 2.794174248 0.065970836 2 2.027300384 2.051208579 Minimum Distance Between Initial Seeds = 6.806712 Preliminary L(1) Scale Estimate = 2.796579
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Using PROC FASTCLUS to Analyze Data with Outliers PROC FASTCLUS Analysis Using LEAST= Clustering Criterion Values < 2 Reduce Effect of Outliers on Cluster Centers The FASTCLUS Procedure Replace=FULL Radius=0 Maxclusters=2 Maxiter=20 Converge=0.0001 Least=1 Number of Bins = 100 Iteration History Relative Change Maximum in Cluster Seeds Iteration Criterion Bin Size 1 2 ---------------------------------------------------------- 1 1.3983 0.2263 0.4091 0.6696 2 1.0776 0.0226 0.00511 0.0452 3 1.0771 0.00226 0.00229 0.00234 4 1.0771 0.000396 0.000253 0.000144 5 1.0771 0.000396 0 0 Convergence criterion is satisfied.
Using PROC FASTCLUS to Analyze Data with Outliers PROC FASTCLUS Analysis Using LEAST= Clustering Criterion Values < 2 Reduce Effect of Outliers on Cluster Centers The FASTCLUS Procedure Replace=FULL Radius=0 Maxclusters=2 Maxiter=20 Converge=0.0001 Least=1 Criterion Based on Final Seeds = 1.0771 Cluster Summary Mean Maximum Distance Absolute from Seed Radius Nearest Distance Between Cluster Frequency Deviation to Observation Exceeded Cluster Cluster Medians ------------------------------------------------------------------------------------------------- 1 102 1.1278 24.1622 2 4.2585 2 108 1.0494 14.8292 1 4.2585 Cluster Medians Cluster x y ------------------------------------------- 1 1.923023887 0.222482918 2 1.826721743 0.286253041 Mean Absolute Deviations from Final Seeds Cluster x y ------------------------------------------- 1 1.113465261 1.142120480 2 0.890331835 1.208370913
/* Run PROC FASTCLUS again, selecting seeds from the */ /* high frequency clusters in the previous analysis */ /* STRICT= prevents outliers from distorting the results. */ title2 'PROC FASTCLUS Analysis Using STRICT= to Omit Outliers'; proc fastclus data=x seed=seed maxc=2 strict=3.0 out=out outseed=mean2; var x y; run; proc gplot data=out; plot y*x=cluster/frame cframe=ligr legend=legend1 vaxis=axis1 haxis=axis2; run;
Output 28.2.3: Cluster Analysis with Outliers Omitted ”PROC FASTCLUS and PROC GPLOT
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Using PROC FASTCLUS to Analyze Data with Outliers PROC FASTCLUS Analysis Using STRICT= to Omit Outliers The FASTCLUS Procedure Replace=FULL Radius=0 Strict=3 Maxclusters=2 Maxiter=1 Initial Seeds Cluster x y ------------------------------------------- 1 2.794174248 0.065970836 2 2.027300384 2.051208579 Criterion Based on Final Seeds = 0.9515 Cluster Summary Maximum Distance RMS Std from Seed Radius Nearest Distance Between Cluster Frequency Deviation to Observation Exceeded Cluster Cluster Centroids -------------------------------------------------------------------------------------------------- 1 99 0.9501 2.9589 2 3.7666 2 99 0.9290 2.8011 1 3.7666 12 Observation(s) were not assigned to a cluster because the minimum distance to a cluster seed exceeded the STRICT= value. Statistics for Variables Variable Total STD Within STD R-Square RSQ/(1-RSQ) ------------------------------------------------------------------ x 2.06854 0.87098 0.823609 4.669219 y 1.02113 1.00352 0.039093 0.040683 OVER-ALL 1.63119 0.93959 0.669891 2.029303 Pseudo F Statistic = 397.74 Approximate Expected Over-All R-Squared = 0.60615 Cubic Clustering Criterion = 3.197 WARNING: The two values above are invalid for correlated variables.
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Using PROC FASTCLUS to Analyze Data with Outliers PROC FASTCLUS Analysis Using STRICT= to Omit Outliers The FASTCLUS Procedure Replace=FULL Radius=0 Strict=3 Maxclusters=2 Maxiter=1 Cluster Means Cluster x y ------------------------------------------- 1 1.825111432 0.141211701 2 1.919910712 0.261558725 Cluster Standard Deviations Cluster x y ------------------------------------------- 1 0.889549271 1.006965219 2 0.852000588 1.000062579
/* Run PROC FASTCLUS one more time with zero iterations */ /* to assign outliers and tails to clusters. */ title2 'Final PROC FASTCLUS Analysis Assigning Outliers to ' 'Clusters'; proc fastclus data=x seed=mean2 maxc=2 maxiter=0 out=out; var x y; run; proc gplot data=out; plot y*x=cluster/frame cframe=ligr legend=legend1 vaxis=axis1 haxis=axis2; run;
Output 28.2.4: Final Analysis with Outliers Assigned to Clusters ”PROC FASTCLUS and PROC GPLOT
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Using PROC FASTCLUS to Analyze Data with Outliers Final PROC FASTCLUS Analysis Assigning Outliers to Clusters The FASTCLUS Procedure Replace=FULL Radius=0 Maxclusters=2 Maxiter=0 Initial Seeds Cluster x y ------------------------------------------- 1 1.825111432 0.141211701 2 1.919910712 0.261558725 Criterion Based on Final Seeds = 2.0594 Cluster Summary Maximum Distance RMS Std from Seed Radius Nearest Distance Between Cluster Frequency Deviation to Observation Exceeded Cluster Cluster Centroids -------------------------------------------------------------------------------------------------- 1 103 2.2569 17.9426 2 4.3753 2 107 1.8371 11.7362 1 4.3753 Statistics for Variables Variable Total STD Within STD R-Square RSQ/(1-RSQ) ------------------------------------------------------------------ x 2.92721 1.95529 0.555950 1.252000 y 2.15248 2.14754 0.009347 0.009435 OVER-ALL 2.56922 2.05367 0.364119 0.572621 Pseudo F Statistic = 119.11 Approximate Expected Over-All R-Squared = 0.49090 Cubic Clustering Criterion = 5.338 WARNING: The two values above are invalid for correlated variables.
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Using PROC FASTCLUS to Analyze Data with Outliers Final PROC FASTCLUS Analysis Assigning Outliers to Clusters The FASTCLUS Procedure Replace=FULL Radius=0 Maxclusters=2 Maxiter=0 Cluster Means Cluster x y ------------------------------------------- 1 2.280017469 0.263940765 2 2.075547895 0.151348765 Cluster Standard Deviations Cluster x y ------------------------------------------- 1 2.412264861 2.089922815 2 1.379355878 2.201567557
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