In this example, four more discriminant analyses of iris data are run with two quantitative variables: petal width and petal length. The following statements produce Output 35.2.1 through Output 35.2.5:
title 'Discriminant Analysis of Fisher (1936) Iris Data'; proc template; define statgraph scatter; begingraph; entrytitle 'Fisher (1936) Iris Data'; layout overlayequated / equatetype=fit; scatterplot x=petallength y=petalwidth / group=species name='iris'; layout gridded / autoalign=(topleft); discretelegend 'iris' / border=false opaque=false; endlayout; endlayout; endgraph; end; run; proc sgrender data=sashelp.iris template=scatter; run;
The scatter plot in Output 35.2.1 shows the joint sample distribution.
Another data set is created for plotting, containing a grid of points suitable for contour plots. The following statements create the data set:
data plotdata; do PetalLength = -2 to 72 by 0.5; do PetalWidth= - 5 to 32 by 0.5; output; end; end; run;
Three macros are defined as follows to make contour plots of density estimates, posterior probabilities, and classification results:
%let close = thresholdmin=0 thresholdmax=0 offsetmin=0 offsetmax=0; %let close = xaxisopts=(&close) yaxisopts=(&close); proc template; define statgraph contour; begingraph; layout overlayequated / equatetype=equate &close; contourplotparm x=petallength y=petalwidth z=z / contourtype=fill nhint=30; scatterplot x=pl y=pw / group=species name='iris' includemissinggroup=false primary=true; layout gridded / autoalign=(topleft); discretelegend 'iris' / border=false opaque=false; endlayout; endlayout; endgraph; end; run; %macro contden; data contour(keep=PetalWidth PetalLength species z pl pw); merge plotd(in=d) sashelp.iris(keep=PetalWidth PetalLength species rename=(PetalWidth=pw PetalLength=pl)); if d then z = max(setosa,versicolor,virginica); run; title3 'Plot of Estimated Densities'; proc sgrender data=contour template=contour; run; %mend; %macro contprob; data posterior(keep=PetalWidth PetalLength species z pl pw into); merge plotp(in=d) sashelp.iris(keep=PetalWidth PetalLength species rename=(PetalWidth=pw PetalLength=pl)); if d then z = max(setosa,versicolor,virginica); into = 1 * (_into_ =: 'Set') + 2 * (_into_ =: 'Ver') + 3 * (_into_ =: 'Vir'); run; title3 'Plot of Posterior Probabilities '; proc sgrender data=posterior template=contour; run; %mend;
%macro contclass; title3 'Plot of Classification Results'; proc sgrender data=posterior(drop=z rename=(into=z)) template=contour; run; %mend;
A normal-theory analysis (METHOD=NORMAL) assuming equal covariance matrices (POOL=YES) illustrates the linearity of the classification boundaries. These statements produce Output 35.2.2:
title2 'Using Normal Density Estimates with Equal Variance'; proc discrim data=sashelp.iris method=normal pool=yes testdata=plotdata testout=plotp testoutd=plotd short noclassify crosslisterr; class Species; var Petal:; run; %contden %contprob %contclass
Output 35.2.2: Normal Density Estimates with Equal Variance
Discriminant Analysis of Fisher (1936) Iris Data |
Using Normal Density Estimates with Equal Variance |
Posterior Probability of Membership in Species | ||||||
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Obs | From Species | Classified into Species |
Setosa | Versicolor | Virginica | |
53 | Versicolor | Virginica | * | 0.0000 | 0.2130 | 0.7870 |
100 | Versicolor | Virginica | * | 0.0000 | 0.3118 | 0.6882 |
103 | Virginica | Versicolor | * | 0.0000 | 0.8453 | 0.1547 |
113 | Virginica | Versicolor | * | 0.0000 | 0.8322 | 0.1678 |
124 | Virginica | Versicolor | * | 0.0000 | 0.8057 | 0.1943 |
136 | Virginica | Versicolor | * | 0.0000 | 0.8903 | 0.1097 |
* Misclassified observation |
Discriminant Analysis of Fisher (1936) Iris Data |
Using Normal Density Estimates with Equal Variance |
Number of Observations and Percent Classified into Species |
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From Species | Setosa | Versicolor | Virginica | Total | ||||||||
Setosa |
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Versicolor |
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Virginica |
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Total |
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Priors |
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Discriminant Analysis of Fisher (1936) Iris Data |
Using Normal Density Estimates with Equal Variance |
Observation Profile for Test Data | |
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Number of Observations Read | 11175 |
Number of Observations Used | 11175 |
A normal-theory analysis assuming unequal covariance matrices (POOL=NO) illustrates quadratic classification boundaries. These statements produce Output 35.2.3:
title2 'Using Normal Density Estimates with Unequal Variance'; proc discrim data=sashelp.iris method=normal pool=no testdata=plotdata testout=plotp testoutd=plotd short noclassify crosslisterr; class Species; var Petal:; run; %contden %contprob %contclass
Output 35.2.3: Normal Density Estimates with Unequal Variance
Discriminant Analysis of Fisher (1936) Iris Data |
Using Normal Density Estimates with Unequal Variance |
Posterior Probability of Membership in Species | ||||||
---|---|---|---|---|---|---|
Obs | From Species | Classified into Species |
Setosa | Versicolor | Virginica | |
53 | Versicolor | Virginica | * | 0.0000 | 0.0903 | 0.9097 |
100 | Versicolor | Virginica | * | 0.0000 | 0.4675 | 0.5325 |
103 | Virginica | Versicolor | * | 0.0000 | 0.7288 | 0.2712 |
113 | Virginica | Versicolor | * | 0.0000 | 0.5196 | 0.4804 |
136 | Virginica | Versicolor | * | 0.0000 | 0.8335 | 0.1665 |
* Misclassified observation |
Discriminant Analysis of Fisher (1936) Iris Data |
Using Normal Density Estimates with Unequal Variance |
Number of Observations and Percent Classified into Species |
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From Species | Setosa | Versicolor | Virginica | Total | ||||||||
Setosa |
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Versicolor |
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Virginica |
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Total |
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Priors |
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Discriminant Analysis of Fisher (1936) Iris Data |
Using Normal Density Estimates with Unequal Variance |
Observation Profile for Test Data | |
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Number of Observations Read | 11175 |
Number of Observations Used | 11175 |
A nonparametric analysis (METHOD=NPAR) follows, using normal kernels (KERNEL=NORMAL) and equal bandwidths (POOL=YES) in each class. The value of the radius parameter r that, assuming normality, minimizes an approximate mean integrated square error is 0.50 (see the section Nonparametric Methods). These statements produce Output 35.2.4:
title2 'Using Kernel Density Estimates with Equal Bandwidth'; proc discrim data=sashelp.iris method=npar kernel=normal r=.5 pool=yes testoutd=plotd testdata=plotdata testout=plotp short noclassify crosslisterr; class Species; var Petal:; run; %contden %contprob %contclass
Output 35.2.4: Kernel Density Estimates with Equal Bandwidth
Discriminant Analysis of Fisher (1936) Iris Data |
Using Kernel Density Estimates with Equal Bandwidth |
Posterior Probability of Membership in Species | ||||||
---|---|---|---|---|---|---|
Obs | From Species | Classified into Species |
Setosa | Versicolor | Virginica | |
53 | Versicolor | Virginica | * | 0.0000 | 0.0800 | 0.9200 |
100 | Versicolor | Virginica | * | 0.0000 | 0.4123 | 0.5877 |
103 | Virginica | Versicolor | * | 0.0000 | 0.7474 | 0.2526 |
113 | Virginica | Versicolor | * | 0.0000 | 0.5863 | 0.4137 |
136 | Virginica | Versicolor | * | 0.0000 | 0.8358 | 0.1642 |
* Misclassified observation |
Discriminant Analysis of Fisher (1936) Iris Data |
Using Kernel Density Estimates with Equal Bandwidth |
Number of Observations and Percent Classified into Species |
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From Species | Setosa | Versicolor | Virginica | Total | ||||||||
Setosa |
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Versicolor |
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Virginica |
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Total |
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Priors |
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Discriminant Analysis of Fisher (1936) Iris Data |
Using Kernel Density Estimates with Equal Bandwidth |
Observation Profile for Test Data | |
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Number of Observations Read | 11175 |
Number of Observations Used | 11175 |
Another nonparametric analysis is run with unequal bandwidths (POOL=NO). These statements produce Output 35.2.5:
title2 'Using Kernel Density Estimates with Unequal Bandwidth'; proc discrim data=sashelp.iris method=npar kernel=normal r=.5 pool=no testoutd=plotd testdata=plotdata testout=plotp short noclassify crosslisterr; class Species; var Petal:; run; %contden %contprob %contclass
Output 35.2.5: Kernel Density Estimates with Unequal Bandwidth
Discriminant Analysis of Fisher (1936) Iris Data |
Using Kernel Density Estimates with Unequal Bandwidth |
Posterior Probability of Membership in Species | ||||||
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Obs | From Species | Classified into Species |
Setosa | Versicolor | Virginica | |
53 | Versicolor | Virginica | * | 0.0000 | 0.0516 | 0.9484 |
100 | Versicolor | Virginica | * | 0.0000 | 0.3773 | 0.6227 |
103 | Virginica | Versicolor | * | 0.0000 | 0.7826 | 0.2174 |
136 | Virginica | Versicolor | * | 0.0000 | 0.8802 | 0.1198 |
* Misclassified observation |
Discriminant Analysis of Fisher (1936) Iris Data |
Using Kernel Density Estimates with Unequal Bandwidth |
Number of Observations and Percent Classified into Species |
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From Species | Setosa | Versicolor | Virginica | Total | ||||||||
Setosa |
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Versicolor |
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Virginica |
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Total |
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Priors |
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Discriminant Analysis of Fisher (1936) Iris Data |
Using Kernel Density Estimates with Unequal Bandwidth |
Observation Profile for Test Data | |
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Number of Observations Read | 11175 |
Number of Observations Used | 11175 |