/* Lesson 26-1 */
/* File Name = les2601.sas 12/19/02 */
options linesize=72; /* pagesize=20; */
proc format;
value specname
1='Sentosa '
2='Versicolor'
3='Virginica ';
value specchar
1='S'
2='O'
3='V';
run;
data iris;
title "Fisher's Iris Data";
infile 'IRISES.DAT';
do obs=1 to 50;
do species=1 to 3;
input sepallen sepalwid petallen petalwid @@;
format species specname.;
output;
end;
end;
/*
label sepallen='Sepal Length in mm.'
sepalwid='Sepal Width in mm.'
petallen='Petal Length in mm.'
petalwid='Petal Width in mm.';
*/
proc print data=iris(obs=10);
run;
title2 'Correlation of this data';
proc corr data=iris;
var sepallen sepalwid petallen petalwid;
run;
proc sort data=iris;
by species;
run;
proc corr data=iris;
var sepallen sepalwid petallen petalwid;
by species;
run;
title2 'Histogram of petalwid';
proc chart data=iris;
hbar petalwid/subgroup=species midpoints=0.0 to 3.0 by 0.1;
format species specchar.;
run;
title2 'Discriminant Analysis';
proc discrim data=iris;
class species;
var petalwid;
run;
proc discrim data=iris outstat=irisout
method=normal pool=test wcov pcov
distance anova manova listerr crosslisterr;
class species;
var sepallen sepalwid petallen petalwid;
run;
proc print data=irisout;
run;
Fisher's Iris Data 1
13:04 Thursday, December 19, 2002
OBS OBS SPECIES SEPALLEN SEPALWID PETALLEN PETALWID
1 1 Sentosa 5.1 3.5 1.4 0.2
2 1 Versicolor 7.0 3.2 4.7 1.4
3 1 Virginica 6.3 3.3 6.0 2.5
4 2 Sentosa 4.9 3.0 1.4 0.2
5 2 Versicolor 6.4 3.2 4.5 1.5
6 2 Virginica 5.8 2.7 5.1 1.9
7 3 Sentosa 4.7 3.2 1.3 0.2
8 3 Versicolor 6.9 3.1 4.9 1.5
9 3 Virginica 7.1 3.0 5.9 2.1
10 4 Sentosa 4.6 3.1 1.5 0.2
Fisher's Iris Data 2
Correlation of this data
13:04 Thursday, December 19, 2002
Correlation Analysis
4 'VAR' Variables: SEPALLEN SEPALWID PETALLEN PETALWID
Simple Statistics
Variable N Mean Std Dev Sum Minimum Maximum
SEPALLEN 150 5.8433 0.8281 876.5 4.3000 7.9000
SEPALWID 150 3.0573 0.4359 458.6 2.0000 4.4000
PETALLEN 150 3.7580 1.7653 563.7 1.0000 6.9000
PETALWID 150 1.1993 0.7622 179.9 0.1000 2.5000
Pearson Correlation Coefficients / Prob > |R| under Ho: Rho=0 / N = 150
SEPALLEN SEPALWID PETALLEN PETALWID
SEPALLEN 1.00000 -0.11757 0.87175 0.81794
0.0 0.1519 0.0001 0.0001
SEPALWID -0.11757 1.00000 -0.42844 -0.36613
0.1519 0.0 0.0001 0.0001
PETALLEN 0.87175 -0.42844 1.00000 0.96287
0.0001 0.0001 0.0 0.0001
PETALWID 0.81794 -0.36613 0.96287 1.00000
0.0001 0.0001 0.0001 0.0
Fisher's Iris Data 3
Correlation of this data
13:04 Thursday, December 19, 2002
--------------------------- SPECIES=Sentosa ----------------------------
Correlation Analysis
4 'VAR' Variables: SEPALLEN SEPALWID PETALLEN PETALWID
Simple Statistics
Variable N Mean Std Dev Sum Minimum Maximum
SEPALLEN 50 5.0060 0.3525 250.3 4.3000 5.8000
SEPALWID 50 3.4280 0.3791 171.4 2.3000 4.4000
PETALLEN 50 1.4620 0.1737 73.1000 1.0000 1.9000
PETALWID 50 0.2460 0.1054 12.3000 0.1000 0.6000
Pearson Correlation Coefficients / Prob > |R| under Ho: Rho=0 / N = 50
SEPALLEN SEPALWID PETALLEN PETALWID
SEPALLEN 1.00000 0.74255 0.26718 0.27810
0.0 0.0001 0.0607 0.0505
SEPALWID 0.74255 1.00000 0.17770 0.23275
0.0001 0.0 0.2170 0.1038
PETALLEN 0.26718 0.17770 1.00000 0.33163
0.0607 0.2170 0.0 0.0186
PETALWID 0.27810 0.23275 0.33163 1.00000
0.0505 0.1038 0.0186 0.0
Fisher's Iris Data 4
Correlation of this data
13:04 Thursday, December 19, 2002
-------------------------- SPECIES=Versicolor --------------------------
Correlation Analysis
4 'VAR' Variables: SEPALLEN SEPALWID PETALLEN PETALWID
Simple Statistics
Variable N Mean Std Dev Sum Minimum Maximum
SEPALLEN 50 5.9360 0.5162 296.8 4.9000 7.0000
SEPALWID 50 2.7700 0.3138 138.5 2.0000 3.4000
PETALLEN 50 4.2600 0.4699 213.0 3.0000 5.1000
PETALWID 50 1.3260 0.1978 66.3000 1.0000 1.8000
Pearson Correlation Coefficients / Prob > |R| under Ho: Rho=0 / N = 50
SEPALLEN SEPALWID PETALLEN PETALWID
SEPALLEN 1.00000 0.52591 0.75405 0.54646
0.0 0.0001 0.0001 0.0001
SEPALWID 0.52591 1.00000 0.56052 0.66400
0.0001 0.0 0.0001 0.0001
PETALLEN 0.75405 0.56052 1.00000 0.78667
0.0001 0.0001 0.0 0.0001
PETALWID 0.54646 0.66400 0.78667 1.00000
0.0001 0.0001 0.0001 0.0
Fisher's Iris Data 5
Correlation of this data
13:04 Thursday, December 19, 2002
-------------------------- SPECIES=Virginica ---------------------------
Correlation Analysis
4 'VAR' Variables: SEPALLEN SEPALWID PETALLEN PETALWID
Simple Statistics
Variable N Mean Std Dev Sum Minimum Maximum
SEPALLEN 50 6.5880 0.6359 329.4 4.9000 7.9000
SEPALWID 50 2.9740 0.3225 148.7 2.2000 3.8000
PETALLEN 50 5.5520 0.5519 277.6 4.5000 6.9000
PETALWID 50 2.0260 0.2747 101.3 1.4000 2.5000
Pearson Correlation Coefficients / Prob > |R| under Ho: Rho=0 / N = 50
SEPALLEN SEPALWID PETALLEN PETALWID
SEPALLEN 1.00000 0.45723 0.86422 0.28111
0.0 0.0008 0.0001 0.0480
SEPALWID 0.45723 1.00000 0.40104 0.53773
0.0008 0.0 0.0039 0.0001
PETALLEN 0.86422 0.40104 1.00000 0.32211
0.0001 0.0039 0.0 0.0225
PETALWID 0.28111 0.53773 0.32211 1.00000
0.0480 0.0001 0.0225 0.0
Fisher's Iris Data 6
Histogram of petalwid
13:04 Thursday, December 19, 2002
PETALWID Cum. Cum.
Midpoint Freq Freq Percent Percent
|
0.0 | 0 0 0.00 0.00
0.1 |SSSSS 5 5 3.33 3.33
0.2 |SSSSSSSSSSSSSSSSSSSSSSSSSSSSS 29 34 19.33 22.67
0.3 |SSSSSSS 7 41 4.67 27.33
0.4 |SSSSSSS 7 48 4.67 32.00
0.5 |S 1 49 0.67 32.67
0.6 |S 1 50 0.67 33.33
0.7 | 0 50 0.00 33.33
0.8 | 0 50 0.00 33.33
0.9 | 0 50 0.00 33.33
1.0 |OOOOOOO 7 57 4.67 38.00
1.1 |OOO 3 60 2.00 40.00
1.2 |OOOOO 5 65 3.33 43.33
1.3 |OOOOOOOOOOOOO 13 78 8.67 52.00
1.4 |OOOOOOOV 8 86 5.33 57.33
1.5 |OOOOOOOOOOVV 12 98 8.00 65.33
1.6 |OOOV 4 102 2.67 68.00
1.7 |OV 2 104 1.33 69.33
1.8 |OVVVVVVVVVVV 12 116 8.00 77.33
1.9 |VVVVV 5 121 3.33 80.67
2.0 |VVVVVV 6 127 4.00 84.67
2.1 |VVVVVV 6 133 4.00 88.67
2.2 |VVV 3 136 2.00 90.67
2.3 |VVVVVVVV 8 144 5.33 96.00
2.4 |VVV 3 147 2.00 98.00
2.5 |VVV 3 150 2.00 100.00
2.6 | 0 150 0.00 100.00
2.7 | 0 150 0.00 100.00
2.8 | 0 150 0.00 100.00
2.9 | 0 150 0.00 100.00
3.0 | 0 150 0.00 100.00
|
-----+----+----+----+----+----
5 10 15 20 25
Frequency
記号 SPECIES 記号 SPECIES 記号 SPECIES
S S O O V V
Fisher's Iris Data 7
Discriminant Analysis
13:04 Thursday, December 19, 2002
Discriminant Analysis
150 Observations 149 DF Total
1 Variables 147 DF Within Classes
3 Classes 2 DF Between Classes
Class Level Information
Prior
SPECIES Frequency Weight Proportion Probability
Sentosa 50 50.0000 0.333333 0.333333
Versicolor 50 50.0000 0.333333 0.333333
Virginica 50 50.0000 0.333333 0.333333
Fisher's Iris Data 8
Discriminant Analysis
13:04 Thursday, December 19, 2002
Discriminant Analysis Pooled Covariance Matrix Information
Covariance Natural Log of the Determinant
Matrix Rank of the Covariance Matrix
1 -3.1729079
Fisher's Iris Data 9
Discriminant Analysis
13:04 Thursday, December 19, 2002
Discriminant Analysis
Pairwise Generalized Squared Distances Between Groups
2 _ _ -1 _ _
D (i|j) = (X - X )' COV (X - X )
i j i j
Generalized Squared Distance to SPECIES
From
SPECIES Sentosa Versicolor Virginica
Sentosa 0 27.84992 75.65130
Versicolor 27.84992 0 11.69964
Virginica 75.65130 11.69964 0
Fisher's Iris Data 10
Discriminant Analysis
13:04 Thursday, December 19, 2002
Discriminant Analysis Linear Discriminant Function
_ -1 _ -1 _
Constant = -.5 X' COV X Coefficient Vector = COV X
j j j
SPECIES
Sentosa Versicolor Virginica
CONSTANT -0.72246 -20.99102 -49.00330
PETALWID 5.87370 31.66066 48.37443
Fisher's Iris Data 11
Discriminant Analysis
13:04 Thursday, December 19, 2002
Discriminant Analysis
Classification Summary for Calibration Data: WORK.IRIS
Resubstitution Summary using Linear Discriminant Function
Generalized Squared Distance Function:
2 _ -1 _
D (X) = (X-X )' COV (X-X )
j j j
Posterior Probability of Membership in each SPECIES:
2 2
Pr(j|X) = exp(-.5 D (X)) / SUM exp(-.5 D (X))
j k k
Number of Observations and Percent Classified into SPECIES:
From SPECIES Sentosa Versicolor Virginica Total
Sentosa 50 0 0 50
100.00 0.00 0.00 100.00
Versicolor 0 48 2 50
0.00 96.00 4.00 100.00
Virginica 0 4 46 50
0.00 8.00 92.00 100.00
Total 50 52 48 150
Percent 33.33 34.67 32.00 100.00
Priors 0.3333 0.3333 0.3333
Error Count Estimates for SPECIES:
Sentosa Versicolor Virginica Total
Rate 0.0000 0.0400 0.0800 0.0400
Priors 0.3333 0.3333 0.3333
Fisher's Iris Data 12
Discriminant Analysis
13:04 Thursday, December 19, 2002
Discriminant Analysis
150 Observations 149 DF Total
4 Variables 147 DF Within Classes
3 Classes 2 DF Between Classes
Class Level Information
Prior
SPECIES Frequency Weight Proportion Probability
Sentosa 50 50.0000 0.333333 0.333333
Versicolor 50 50.0000 0.333333 0.333333
Virginica 50 50.0000 0.333333 0.333333
Fisher's Iris Data 13
Discriminant Analysis
13:04 Thursday, December 19, 2002
Discriminant Analysis Within-Class Covariance Matrices
SPECIES = Sentosa DF = 49
Variable SEPALLEN SEPALWID PETALLEN PETALWID
SEPALLEN 0.1242489796 0.0992163265 0.0163551020 0.0103306122
SEPALWID 0.0992163265 0.1436897959 0.0116979592 0.0092979592
PETALLEN 0.0163551020 0.0116979592 0.0301591837 0.0060693878
PETALWID 0.0103306122 0.0092979592 0.0060693878 0.0111061224
------------------------------------------------------------------------
SPECIES = Versicolor DF = 49
Variable SEPALLEN SEPALWID PETALLEN PETALWID
SEPALLEN 0.2664326531 0.0851836735 0.1828979592 0.0557795918
SEPALWID 0.0851836735 0.0984693878 0.0826530612 0.0412040816
PETALLEN 0.1828979592 0.0826530612 0.2208163265 0.0731020408
PETALWID 0.0557795918 0.0412040816 0.0731020408 0.0391061224
------------------------------------------------------------------------
SPECIES = Virginica DF = 49
Variable SEPALLEN SEPALWID PETALLEN PETALWID
SEPALLEN 0.4043428571 0.0937632653 0.3032897959 0.0490938776
SEPALWID 0.0937632653 0.1040040816 0.0713795918 0.0476285714
PETALLEN 0.3032897959 0.0713795918 0.3045877551 0.0488244898
PETALWID 0.0490938776 0.0476285714 0.0488244898 0.0754326531
Fisher's Iris Data 14
Discriminant Analysis
13:04 Thursday, December 19, 2002
Discriminant Analysis
Pooled Within-Class Covariance Matrix DF = 147
Variable SEPALLEN SEPALWID PETALLEN PETALWID
SEPALLEN 0.2650081633 0.0927210884 0.1675142857 0.0384013605
SEPALWID 0.0927210884 0.1153877551 0.0552435374 0.0327102041
PETALLEN 0.1675142857 0.0552435374 0.1851877551 0.0426653061
PETALWID 0.0384013605 0.0327102041 0.0426653061 0.0418816327
Fisher's Iris Data 15
Discriminant Analysis
13:04 Thursday, December 19, 2002
Discriminant Analysis Within Covariance Matrix Information
Covariance Natural Log of the Determinant
SPECIES Matrix Rank of the Covariance Matrix
Sentosa 4 -13.06736
Versicolor 4 -10.87433
Virginica 4 -8.92706
Pooled 4 -9.95854
Fisher's Iris Data 16
Discriminant Analysis
13:04 Thursday, December 19, 2002
Discriminant Analysis
Test of Homogeneity of Within Covariance Matrices
Notation: K = Number of Groups
P = Number of Variables
N = Total Number of Observations - Number of Groups
N(i) = Number of Observations in the i'th Group - 1
__ N(i)/2
|| |Within SS Matrix(i)|
V = -----------------------------------
N/2
|Pooled SS Matrix|
_ _ 2
| 1 1 | 2P + 3P - 1
RHO = 1.0 - | SUM ----- - --- | -------------
|_ N(i) N _| 6(P+1)(K-1)
DF = .5(K-1)P(P+1)
_ _
| PN/2 |
| N V |
Under null hypothesis: -2 RHO ln | ------------------ |
| __ PN(i)/2 |
|_ || N(i) _|
is distributed approximately as chi-square(DF)
Test Chi-Square Value = 140.943050
with 20 DF Prob > Chi-Sq = 0.0001
Since the chi-square value is significant at the 0.1 level,
the within covariance matrices will be used in
the discriminant function.
Reference: Morrison, D.F. (1976) Multivariate Statistical Methods p252.
Fisher's Iris Data 17
Discriminant Analysis
13:04 Thursday, December 19, 2002
Discriminant Analysis Pairwise Squared Distances Between Groups
2 _ _ -1 _ _
D (i|j) = (X - X )' COV (X - X )
i j j i j
Squared Distance to SPECIES
From
SPECIES Sentosa Versicolor Virginica
Sentosa 0 103.19382 168.76759
Versicolor 323.06203 0 13.83875
Virginica 706.08494 17.86670 0
Fisher's Iris Data 18
Discriminant Analysis
13:04 Thursday, December 19, 2002
Discriminant Analysis
Pairwise Generalized Squared Distances Between Groups
2 _ _ -1 _ _
D (i|j) = (X - X )' COV (X - X ) + ln |COV |
i j j i j j
Generalized Squared Distance to SPECIES
From
SPECIES Sentosa Versicolor Virginica
Sentosa -13.06736 92.31949 159.84053
Versicolor 309.99467 -10.87433 4.91170
Virginica 693.01757 6.99238 -8.92706
Fisher's Iris Data 19
Discriminant Analysis
13:04 Thursday, December 19, 2002
Discriminant Analysis
Univariate Test Statistics
F Statistics, Num DF= 2 Den DF= 147
Total Pooled Between RSQ/
Variable STD STD STD R-Squared (1-RSQ)
SEPALLEN 0.8281 0.5148 0.7951 0.618706 1.6226
SEPALWID 0.4359 0.3397 0.3368 0.400783 0.6688
PETALLEN 1.7653 0.4303 2.0907 0.941372 16.0566
PETALWID 0.7622 0.2047 0.8967 0.928883 13.0613
Univariate Test Statistics
Variable F Pr > F
SEPALLEN 119.2645 0.0001
SEPALWID 49.1600 0.0001
PETALLEN 1180.1612 0.0001
PETALWID 960.0071 0.0001
Average R-Squared: Unweighted = 0.7224358
Weighted by Variance = 0.8689444
Multivariate Statistics and F Approximations
S=2 M=0.5 N=71
Statistic Value F Num DF Den DF Pr > F
Wilks' Lambda 0.023438631 199.145 8 288 0.0001
Pillai's Trace 1.191898825 53.4665 8 290 0.0001
Hotelling-Lawley Trace 32.47732024 580.532 8 286 0.0001
Roy's Greatest Root 32.1919292 1166.96 4 145 0.0001
NOTE: ROY'S GREATEST ROOT のＦ統計量は上側限界です．
NOTE: WILKS のラムダのＦ統計量は正確です．
Fisher's Iris Data 20
Discriminant Analysis
13:04 Thursday, December 19, 2002
Discriminant Analysis
Classification Results for Calibration Data: WORK.IRIS
Resubstitution Results using Quadratic Discriminant Function
Generalized Squared Distance Function:
2 _ -1 _
D (X) = (X-X )' COV (X-X ) + ln |COV |
j j j j j
Posterior Probability of Membership in each SPECIES:
2 2
Pr(j|X) = exp(-.5 D (X)) / SUM exp(-.5 D (X))
j k k
Posterior Probability of Membership in SPECIES:
Obs From Classified
SPECIES into SPECIES Sentosa Versicolor Virginica
71 Versicolor Virginica * 0.0000 0.3359 0.6641
84 Versicolor Virginica * 0.0000 0.1543 0.8457
134 Virginica Versicolor * 0.0000 0.6050 0.3950
* Misclassified observation
Fisher's Iris Data 21
Discriminant Analysis
13:04 Thursday, December 19, 2002
Discriminant Analysis
Classification Summary for Calibration Data: WORK.IRIS
Resubstitution Summary using Quadratic Discriminant Function
Generalized Squared Distance Function:
2 _ -1 _
D (X) = (X-X )' COV (X-X ) + ln |COV |
j j j j j
Posterior Probability of Membership in each SPECIES:
2 2
Pr(j|X) = exp(-.5 D (X)) / SUM exp(-.5 D (X))
j k k
Number of Observations and Percent Classified into SPECIES:
From SPECIES Sentosa Versicolor Virginica Total
Sentosa 50 0 0 50
100.00 0.00 0.00 100.00
Versicolor 0 48 2 50
0.00 96.00 4.00 100.00
Virginica 0 1 49 50
0.00 2.00 98.00 100.00
Total 50 49 51 150
Percent 33.33 32.67 34.00 100.00
Priors 0.3333 0.3333 0.3333
Error Count Estimates for SPECIES:
Sentosa Versicolor Virginica Total
Rate 0.0000 0.0400 0.0200 0.0200
Priors 0.3333 0.3333 0.3333
Fisher's Iris Data 22
Discriminant Analysis
13:04 Thursday, December 19, 2002
Discriminant Analysis
Classification Results for Calibration Data: WORK.IRIS
Cross-validation Results using Quadratic Discriminant Function
Generalized Squared Distance Function:
2 _ -1 _
D (X) = (X-X )' COV (X-X ) + ln |COV |
j (X)j (X)j (X)j (X)j
Posterior Probability of Membership in each SPECIES:
2 2
Pr(j|X) = exp(-.5 D (X)) / SUM exp(-.5 D (X))
j k k
Posterior Probability of Membership in SPECIES:
Obs From Classified
SPECIES into SPECIES Sentosa Versicolor Virginica
69 Versicolor Virginica * 0.0000 0.3134 0.6866
71 Versicolor Virginica * 0.0000 0.1616 0.8384
84 Versicolor Virginica * 0.0000 0.0713 0.9287
134 Virginica Versicolor * 0.0000 0.6632 0.3368
* Misclassified observation
Fisher's Iris Data 23
Discriminant Analysis
13:04 Thursday, December 19, 2002
Discriminant Analysis
Classification Summary for Calibration Data: WORK.IRIS
Cross-validation Summary using Quadratic Discriminant Function
Generalized Squared Distance Function:
2 _ -1 _
D (X) = (X-X )' COV (X-X ) + ln |COV |
j (X)j (X)j (X)j (X)j
Posterior Probability of Membership in each SPECIES:
2 2
Pr(j|X) = exp(-.5 D (X)) / SUM exp(-.5 D (X))
j k k
Number of Observations and Percent Classified into SPECIES:
From SPECIES Sentosa Versicolor Virginica Total
Sentosa 50 0 0 50
100.00 0.00 0.00 100.00
Versicolor 0 47 3 50
0.00 94.00 6.00 100.00
Virginica 0 1 49 50
0.00 2.00 98.00 100.00
Total 50 48 52 150
Percent 33.33 32.00 34.67 100.00
Priors 0.3333 0.3333 0.3333
Error Count Estimates for SPECIES:
Sentosa Versicolor Virginica Total
Rate 0.0000 0.0600 0.0200 0.0267
Priors 0.3333 0.3333 0.3333