The
SAS System 13:03 Friday,
December 1, 2006 233
The
DISCRIM Procedure
Observations 14 DF Total 13
Variables 8
DF Within Classes 11
Classes 3 DF Between Classes 2
Class
Level Information
Variable
Prior
Known Name
Frequency Weight Proportion Probability
E E 5 5.0000 0.357143 0.333333
L L 5
5.0000 0.357143 0.333333
M M 4 4.0000 0.285714 0.333333
The
SAS System 13:03 Friday, December
1, 2006 234
The
DISCRIM Procedure
Within-Class SSCP Matrices
Known = E
Variable FreqA FreqB FreqC ClustA
FreqA 3297.200000 984.200000 2087.400000 -118.688721
FreqB 984.200000 533.200000 1233.400000 -27.377958
FreqC 2087.400000 1233.400000 2934.800000 -48.768093
ClustA -118.688721 -27.377958 -48.768093 5.727342
ClustB 92.028575 32.838097 71.104766 -2.842346
ClustC 97.839659 -76.238331 -211.147553 -5.100506
dbhA 95.738157 17.720863 16.705819 -6.347068
dbhC -3.903879 -2.720325 -3.617752 0.447492
Known = E
Variable ClustB ClustC dbhA dbhC
FreqA 92.028575 97.839659 95.738157 -3.903879
FreqB 32.838097 -76.238331 17.720863 -2.720325
FreqC 71.104766 -211.147553 16.705819 -3.617752
ClustA -2.842346 -5.100506 -6.347068 0.447492
ClustB 2.940590 1.961979 2.102198 -0.138772
ClustC 1.961979 59.474998 5.224842 0.654575
dbhA
2.102198 5.224842 9.180947 -0.947068
dbhC -0.138772 0.654575 -0.947068 0.161735
------------------------------------------------------------------------------------------------------
Known = L
Variable FreqA FreqB FreqC ClustA
FreqA 1576.00000 2004.00000 4806.00000 -48.13521
FreqB 2004.00000 3550.80000 7040.20000 -27.40419
FreqC 4806.00000 7040.20000 28174.80000 -229.55747
ClustA -48.13521 -27.40419 -229.55747 3.69403
ClustB -39.30934 -16.60900 22.35297 1.23448
ClustC -7.27049 8.09570 -53.56219 1.30532
dbhA -176.02143 -100.78225 -676.46634 10.62318
dbhC -1.23434 -1.41882 -37.71696 0.38397
Known = L
Variable ClustB ClustC dbhA dbhC
FreqA -39.30934 -7.27049 -176.02143 -1.23434
FreqB -16.60900 8.09570 -100.78225 -1.41882
FreqC 22.35297 -53.56219 -676.46634 -37.71696
ClustA 1.23448 1.30532 10.62318 0.38397
ClustB 3.20995 0.22919 8.13741 -0.31764
The
SAS System 13:03 Friday,
December 1, 2006 235
The
DISCRIM Procedure
Within-Class SSCP Matrices
Known = L
Variable ClustB ClustC dbhA dbhC
ClustC 0.22919 0.60138 2.38177 0.17919
dbhA 8.13741 2.38177 65.17794 0.11697
dbhC -0.31764 0.17919 0.11697 0.11343
------------------------------------------------------------------------------------------------------
Known = M
Variable FreqA FreqB FreqC ClustA
FreqA 621.00000 -803.50000 -524.00000 41.43094
FreqB -803.50000 6768.75000 7891.50000 -43.63896
FreqC -524.00000 7891.50000 10625.00000 -6.57327
ClustA 41.43094 -43.63896 -6.57327 3.00937
ClustB -41.42910 70.12295 57.35609 -2.71224
ClustC 22.60445 -79.10841 -92.06829 1.27433
dbhA 14.81448 -17.99293 -6.09364 1.06132
dbhC -37.64284 34.41282 40.64867 -2.14940
Known = M
Variable ClustB ClustC dbhA dbhC
FreqA -41.42910 22.60445 14.81448 -37.64284
FreqB 70.12295 -79.10841 -17.99293 34.41282
FreqC 57.35609 -92.06829 -6.09364 40.64867
ClustA -2.71224 1.27433 1.06132 -2.14940
ClustB 2.81383 -1.66664 -0.97777 2.50916
ClustC -1.66664 1.35158 0.48328 -1.49536
dbhA -0.97777 0.48328 0.37572 -0.78057
dbhC 2.50916 -1.49536 -0.78057 2.97413
------------------------------------------------------------------------------------------------------
The
SAS System 13:03 Friday,
December 1, 2006 236
The
DISCRIM Procedure
Pooled
Within-Class SSCP Matrix
Variable FreqA FreqB FreqC ClustA
FreqA 5494.20000 2184.70000 6369.40000 -125.39298
FreqB 2184.70000 10852.75000 16165.10000 -98.42111
FreqC 6369.40000 16165.10000 41734.60000 -284.89883
ClustA -125.39298 -98.42111 -284.89883 12.43074
ClustB 11.29013 86.35205 150.81383 -4.32011
ClustC 113.17362 -147.25104 -356.77804 -2.52086
dbhA -65.46879 -101.05432 -665.85417 5.33743
dbhC -42.78106 30.27368 -0.68604 -1.31793
Pooled
Within-Class SSCP Matrix
Variable ClustB ClustC dbhA dbhC
FreqA 11.29013 113.17362 -65.46879 -42.78106
FreqB 86.35205 -147.25104 -101.05432 30.27368
FreqC 150.81383 -356.77804 -665.85417 -0.68604
ClustA -4.32011 -2.52086 5.33743 -1.31793
ClustB 8.96438 0.52453 9.26183 2.05274
ClustC 0.52453 61.42795 8.08989 -0.66159
dbhA 9.26183 8.08989 74.73461 -1.61066
dbhC 2.05274 -0.66159 -1.61066 3.24930
Between-Class
SSCP Matrix
Variable FreqA FreqB FreqC ClustA
FreqA 138.72857 -419.91429 -1755.32857 9.53221
FreqB -419.91429 8101.60714 5049.11429 -200.17627
FreqC -1755.32857 5049.11429 22220.32857 -113.98795
ClustA 9.53221 -200.17627 -113.98795 4.95208
ClustB -4.02457 95.43389 47.70448 -2.36465
ClustC 35.82142 -259.08908 -447.42332 6.24021
dbhA -49.27206 91.27859 625.67474 -1.93426
dbhC 7.84221 -91.15257 -96.62123 2.22974
Between-Class SSCP Matrix
Variable ClustB ClustC dbhA dbhC
FreqA -4.02457 35.82142 -49.27206 7.84221
FreqB 95.43389 -259.08908 91.27859 -91.15257
FreqC 47.70448 -447.42332 625.67474 -96.62123
ClustA -2.36465 6.24021 -1.93426 2.22974
ClustB 1.13144 -2.87548 0.72417 -1.04917
ClustC -2.87548 12.57269 -11.44639 3.51193
dbhA 0.72417 -11.44639 17.99005 -2.21424
dbhC -1.04917 3.51193 -2.21424 1.10868
The
SAS System 13:03 Friday,
December 1, 2006 237
The
DISCRIM Procedure
Total-Sample SSCP Matrix
Variable FreqA FreqB FreqC ClustA
FreqA 5632.92857 1764.78571 4614.07143 -115.86077
FreqB 1764.78571 18954.35714 21214.21429 -298.59738
FreqC 4614.07143 21214.21429 63954.92857 -398.88678
ClustA -115.86077 -298.59738 -398.88678 17.38282
ClustB 7.26557 181.78594 198.51831 -6.68476
ClustC 148.99505 -406.34012 -804.20135 3.71936
dbhA -114.74086 -9.77573 -40.17943 3.40317
dbhC -34.93884 -60.87889 -97.30727 0.91181
Total-Sample SSCP Matrix
Variable ClustB ClustC dbhA dbhC
FreqA 7.26557 148.99505 -114.74086 -34.93884
FreqB 181.78594 -406.34012 -9.77573 -60.87889
FreqC 198.51831 -804.20135 -40.17943 -97.30727
ClustA -6.68476 3.71936 3.40317 0.91181
ClustB 10.09582 -2.35095 9.98601 1.00357
ClustC -2.35095 74.00064 -3.35650 2.85034
dbhA 9.98601 -3.35650 92.72465 -3.82490
dbhC 1.00357 2.85034 -3.82490 4.35797
The
SAS System 13:03 Friday,
December 1, 2006 238
The
DISCRIM Procedure
Within-Class Covariance
Matrices
Known =
E, DF = 4
Variable FreqA FreqB FreqC ClustA
FreqA 824.3000000 246.0500000 521.8500000 -29.6721804
FreqB 246.0500000 133.3000000 308.3500000 -6.8444896
FreqC 521.8500000 308.3500000 733.7000000 -12.1920232
ClustA -29.6721804 -6.8444896 -12.1920232 1.4318356
ClustB 23.0071438 8.2095243 17.7761916 -0.7105865
ClustC 24.4599148 -19.0595827 -52.7868882 -1.2751264
dbhA 23.9345393 4.4302158 4.1764549 -1.5867671
dbhC -0.9759698 -0.6800813 -0.9044381 0.1118730
Known = E, DF = 4
Variable ClustB ClustC dbhA dbhC
FreqA 23.0071438 24.4599148 23.9345393 -0.9759698
FreqB 8.2095243 -19.0595827 4.4302158 -0.6800813
FreqC 17.7761916 -52.7868882 4.1764549 -0.9044381
ClustA -0.7105865 -1.2751264 -1.5867671 0.1118730
ClustB 0.7351475 0.4904948 0.5255495 -0.0346930
ClustC 0.4904948 14.8687496 1.3062104 0.1636438
dbhA 0.5255495 1.3062104 2.2952369 -0.2367671
dbhC -0.0346930 0.1636438 -0.2367671 0.0404338
------------------------------------------------------------------------------------------------------
Known =
L, DF = 4
Variable FreqA FreqB FreqC ClustA
FreqA 394.000000 501.000000 1201.500000 -12.033802
FreqB 501.000000 887.700000 1760.050000 -6.851046
FreqC 1201.500000 1760.050000 7043.700000 -57.389367
ClustA -12.033802 -6.851046 -57.389367 0.923507
ClustB -9.827336 -4.152250 5.588244 0.308619
ClustC -1.817622 2.023924 -13.390549 0.326330
dbhA -44.005358 -25.195562 -169.116585 2.655795
dbhC -0.308585 -0.354705 -9.429241 0.095994
Known =
L, DF = 4
Variable ClustB ClustC dbhA dbhC
FreqA -9.827336 -1.817622 -44.005358 -0.308585
FreqB -4.152250 2.023924 -25.195562 -0.354705
FreqC 5.588244 -13.390549 -169.116585 -9.429241
ClustA 0.308619 0.326330 2.655795 0.095994
ClustB 0.802488 0.057297 2.034352 -0.079411
The
SAS System 13:03 Friday,
December 1, 2006 239
The
DISCRIM Procedure
Within-Class
Covariance Matrices
Known =
L, DF = 4
Variable ClustB ClustC dbhA dbhC
ClustC 0.057297 0.150344 0.595443 0.044799
dbhA 2.034352 0.595443 16.294486 0.029244
dbhC -0.079411 0.044799 0.029244 0.028357
------------------------------------------------------------------------------------------------------
Known =
M, DF = 3
Variable FreqA FreqB FreqC ClustA
FreqA 207.000000 -267.833333 -174.666667 13.810315
FreqB -267.833333 2256.250000 2630.500000 -14.546321
FreqC -174.666667 2630.500000 3541.666667 -2.191090
ClustA 13.810315 -14.546321 -2.191090 1.003124
ClustB -13.809699 23.374316 19.118696 -0.904081
ClustC 7.534816 -26.369470 -30.689429 0.424777
dbhA 4.938160 -5.997644 -2.031215 0.353774
dbhC -12.547612 11.470941 13.549557 -0.716467
Known = M, DF = 3
Variable ClustB ClustC dbhA dbhC
FreqA -13.809699 7.534816 4.938160 -12.547612
FreqB 23.374316 -26.369470 -5.997644 11.470941
FreqC 19.118696 -30.689429 -2.031215 13.549557
ClustA -0.904081 0.424777 0.353774 -0.716467
ClustB 0.937945 -0.555545 -0.325925 0.836385
ClustC -0.555545 0.450525 0.161093 -0.498454
dbhA -0.325925 0.161093 0.125238 -0.260189
dbhC 0.836385 -0.498454 -0.260189 0.991378
------------------------------------------------------------------------------------------------------
The
SAS System 13:03 Friday,
December 1, 2006 240
The
DISCRIM Procedure
Pooled Within-Class
Covariance Matrix, DF = 11
Variable FreqA FreqB FreqC ClustA
FreqA 499.472727 198.609091 579.036364 -11.399362
FreqB 198.609091 986.613636 1469.554545 -8.947373
FreqC 579.036364 1469.554545 3794.054545 -25.899894
ClustA -11.399362 -8.947373 -25.899894 1.130067
ClustB 1.026376 7.850186 13.710348 -0.392738
ClustC 10.288511 -13.386458 -32.434367 -0.229169
dbhA -5.951708 -9.186756 -60.532197 0.485221
dbhC -3.889187 2.752153 -0.062368 -0.119812
Pooled Within-Class
Covariance Matrix, DF = 11
Variable ClustB ClustC dbhA dbhC
FreqA 1.026376 10.288511 -5.951708 -3.889187
FreqB 7.850186 -13.386458 -9.186756 2.752153
FreqC 13.710348 -32.434367 -60.532197 -0.062368
ClustA -0.392738 -0.229169 0.485221 -0.119812
ClustB 0.814943 0.047685 0.841985 0.186613
ClustC 0.047685 5.584359 0.735445 -0.060145
dbhA 0.841985 0.735445 6.794055 -0.146424
dbhC 0.186613 -0.060145 -0.146424 0.295391
Between-Class
Covariance Matrix, DF = 2
Variable FreqA FreqB FreqC ClustA
FreqA 14.863776 -44.990816 -188.070918 1.021308
FreqB -44.990816 868.029337 540.976531 -21.447458
FreqC -188.070918 540.976531 2380.749490 -12.212995
ClustA 1.021308 -21.447458 -12.212995 0.530580
ClustB -0.431204 10.225060 5.111195 -0.253355
ClustC 3.838010 -27.759544 -47.938213 0.668594
dbhA -5.279150 9.779849 67.036579 -0.207242
dbhC 0.840237 -9.766347 -10.352274 0.238901
Between-Class
Covariance Matrix, DF = 2
Variable ClustB ClustC dbhA dbhC
FreqA -0.431204 3.838010 -5.279150 0.840237
FreqB 10.225060 -27.759544 9.779849 -9.766347
FreqC 5.111195 -47.938213 67.036579 -10.352274
ClustA -0.253355 0.668594 -0.207242 0.238901
ClustB 0.121226 -0.308087 0.077590 -0.112411
ClustC -0.308087 1.347073 -1.226399 0.376278
dbhA 0.077590 -1.226399 1.927505 -0.237239
dbhC -0.112411 0.376278 -0.237239 0.118787
The
SAS System 13:03 Friday,
December 1, 2006 241
The DISCRIM Procedure
Total-Sample
Covariance Matrix, DF = 13
Variable FreqA FreqB FreqC ClustA
FreqA 433.302198 135.752747 354.928571 -8.912367
FreqB 135.752747 1458.027473 1631.862637 -22.969029
FreqC 354.928571 1631.862637 4919.609890 -30.683599
ClustA -8.912367 -22.969029 -30.683599 1.337140
ClustB 0.558890 13.983534 15.270639 -0.514212
ClustC 11.461157 -31.256933 -61.861643 0.286105
dbhA -8.826220 -0.751979 -3.090725 0.261782
dbhC -2.687603 -4.682992 -7.485175 0.070139
Total-Sample Covariance Matrix, DF = 13
Variable ClustB ClustC dbhA dbhC
FreqA 0.558890 11.461157 -8.826220 -2.687603
FreqB 13.983534 -31.256933 -0.751979 -4.682992
FreqC 15.270639 -61.861643 -3.090725 -7.485175
ClustA -0.514212 0.286105 0.261782 0.070139
ClustB 0.776601 -0.180843 0.768154 0.077198
ClustC -0.180843 5.692357 -0.258192 0.219257
dbhA 0.768154 -0.258192 7.132666 -0.294223
dbhC 0.077198 0.219257 -0.294223 0.335229
The
SAS System 13:03 Friday,
December 1, 2006 242
The
DISCRIM Procedure
Within-Class Correlation
Coefficients / Pr > |r|
Known = E
Variable
FreqA FreqB FreqC
ClustA ClustB ClustC dbhA dbhC
FreqA
1.00000 0.74228 0.67103
-0.86369 0.93462 0.22094
0.55026 -0.16905
0.1508 0.2150
0.0592 0.0199 0.7210
0.3366 0.7858
FreqB
0.74228 1.00000 0.98598
-0.49543 0.82931 -0.42812
0.25328 -0.29294
0.1508 0.0020 0.3961
0.0825 0.4720 0.6810
0.6324
FreqC
0.67103 0.98598 1.00000
-0.37616 0.76541 -0.50539
0.10177 -0.16605
0.2150 0.0020 0.5326 0.1315
0.3851 0.8706 0.7896
ClustA
-0.86369 -0.49543 -0.37616
1.00000 -0.69260 -0.27636
-0.87529 0.46495
0.0592 0.3961
0.5326 0.1949 0.6527
0.0519 0.4301
ClustB
0.93462 0.82931 0.76541
-0.69260 1.00000 0.14836
0.40459 -0.20123
0.0199 0.0825
0.1315 0.1949 0.8118 0.4993
0.7455
ClustC
0.22094 -0.42812 -0.50539
-0.27636 0.14836 1.00000
0.22360 0.21105
0.7210 0.4720
0.3851 0.6527 0.8118 0.7177 0.7333
dbhA
0.55026 0.25328 0.10177
-0.87529 0.40459 0.22360
1.00000 -0.77720
0.3366 0.6810 0.8706
0.0519 0.4993 0.7177 0.1219
dbhC
-0.16905 -0.29294 -0.16605
0.46495 -0.20123 0.21105
-0.77720 1.00000
0.7858 0.6324
0.7896 0.4301 0.7455
0.7333 0.1219
------------------------------------------------------------------------------------------------------
The
SAS System 13:03 Friday,
December 1, 2006 243
The
DISCRIM Procedure
Within-Class
Correlation Coefficients / Pr > |r|
Known = L
Variable
FreqA FreqB FreqC
ClustA ClustB ClustC dbhA dbhC
FreqA
1.00000 0.84714 0.72123
-0.63086 -0.55267 -0.23616
-0.54921 -0.09232
0.0701 0.1691
0.2538 0.3340 0.7021
0.3377 0.8826
FreqB
0.84714 1.00000 0.70387
-0.23928 -0.15557 0.17519
-0.20949 -0.07070
0.0701 0.1846 0.6983
0.8027 0.7781 0.7352
0.9101
FreqC
0.72123 0.70387 1.00000
-0.71156 0.07433 -0.41149
-0.49919 -0.66718
0.1691 0.1846
0.1777 0.9054
0.4913 0.3919 0.2186
ClustA
-0.63086 -0.23928 -0.71156
1.00000 0.35849 0.87578
0.68463 0.59318
0.2538 0.6983
0.1777 0.5535 0.0516
0.2022 0.2917
ClustB
-0.55267 -0.15557 0.07433
0.35849 1.00000 0.16496
0.56258 -0.52641
0.3340 0.8027
0.9054 0.5535 0.7909 0.3235
0.3621
ClustC
-0.23616 0.17519 -0.41149
0.87578 0.16496 1.00000
0.38043 0.68610
0.7021 0.7781
0.4913 0.0516 0.7909 0.5276 0.2009
dbhA
-0.54921 -0.20949 -0.49919
0.68463 0.56258 0.38043
1.00000 0.04302
0.3377 0.7352
0.3919 0.2022 0.3235
0.5276 0.9452
dbhC
-0.09232 -0.07070 -0.66718
0.59318 -0.52641 0.68610
0.04302 1.00000
0.8826 0.9101
0.2186 0.2917 0.3621
0.2009 0.9452
------------------------------------------------------------------------------------------------------
The
SAS System 13:03 Friday,
December 1, 2006 244
The DISCRIM Procedure
Within-Class
Correlation Coefficients / Pr > |r|
Known = M
Variable
FreqA FreqB
FreqC ClustA ClustB
ClustC dbhA dbhC
FreqA
1.00000 -0.39191 -0.20400
0.95839 -0.99108 0.78024
0.96986 -0.87590
0.6081 0.7960
0.0416 0.0089 0.2198
0.0301 0.1241
FreqB
-0.39191 1.00000 0.93055
-0.30576 0.50811 -0.82708
-0.35679 0.24254
0.6081 0.0694 0.6942
0.4919 0.1729 0.6432
0.7575
FreqC
-0.20400 0.93055 1.00000
-0.03676 0.33172 -0.76829
-0.09645 0.22867
0.7960 0.0694 0.9632 0.6683
0.2317 0.9036 0.7713
ClustA
0.95839 -0.30576 -0.03676
1.00000 -0.93205 0.63186
0.99811 -0.71845
0.0416 0.6942
0.9632 0.0679 0.3681
0.0019 0.2815
ClustB
-0.99108 0.50811 0.33172
-0.93205 1.00000 -0.85462
-0.95096 0.86736
0.0089
0.4919 0.6683 0.0679 0.1454 0.0490
0.1326
ClustC
0.78024 -0.82708 -0.76829
0.63186 -0.85462 1.00000
0.67819 -0.74584
0.2198 0.1729
0.2317 0.3681 0.1454 0.3218 0.2542
dbhA
0.96986 -0.35679
-0.09645 0.99811 -0.95096
0.67819 1.00000 -0.73841
0.0301 0.6432
0.9036 0.0019 0.0490
0.3218 0.2616
dbhC
-0.87590 0.24254 0.22867
-0.71845 0.86736 -0.74584
-0.73841 1.00000
0.1241 0.7575
0.7713 0.2815 0.1326
0.2542 0.2616
------------------------------------------------------------------------------------------------------
The
SAS System 13:03 Friday,
December 1, 2006 245
The
DISCRIM Procedure
Pooled Within-Class
Correlation Coefficients / Pr > |r|
Variable
FreqA FreqB FreqC
ClustA ClustB ClustC dbhA dbhC
FreqA
1.00000 0.28292 0.42063
-0.47981 0.05087 0.19481
-0.10217 -0.32019
0.3729 0.1733
0.1144 0.8752 0.5440
0.7520 0.3103
FreqB
0.28292 1.00000 0.75956
-0.26796 0.27685 -0.18035
-0.11221 0.16121
0.3729 0.0042 0.3998
0.3837 0.5749 0.7284
0.6167
FreqC
0.42063 0.75956 1.00000
-0.39554 0.24657 -0.22283
-0.37702 -0.00186
0.1733 0.0042 0.2031 0.4398
0.4864 0.2270 0.9954
ClustA
-0.47981 -0.26796 -0.39554
1.00000 -0.40925 -0.09123
0.17511 -0.20737
0.1144 0.3998
0.2031 0.1865 0.7780
0.5862 0.5178
ClustB
0.05087 0.27685 0.24657
-0.40925 1.00000 0.02235
0.35783 0.38035
0.8752 0.3837
0.4398 0.1865 0.9450 0.2534
0.2226
ClustC
0.19481 -0.18035 -0.22283
-0.09123 0.02235 1.00000
0.11940 -0.04683
0.5440 0.5749
0.4864 0.7780 0.9450 0.7117 0.8851
dbhA
-0.10217 -0.11221 -0.37702
0.17511 0.35783 0.11940
1.00000 -0.10336
0.7520 0.7284
0.2270 0.5862 0.2534
0.7117 0.7492
dbhC
-0.32019 0.16121 -0.00186
-0.20737 0.38035 -0.04683
-0.10336 1.00000
0.3103 0.6167
0.9954 0.5178 0.2226
0.8851 0.7492
The
SAS System 13:03 Friday,
December 1, 2006 246
The
DISCRIM Procedure
Between-Class
Correlation Coefficients / Pr > |r|
Variable
FreqA FreqB FreqC
ClustA ClustB ClustC dbhA dbhC
FreqA
1.00000 -0.39609 -0.99977
0.36368 -0.32123 0.85772
-0.98628 0.63234
0.7407 0.0136
0.7630 0.7918 0.3438
0.1056 0.5642
FreqB
-0.39609 1.00000 0.37632
-0.99939 0.99678 -0.81180
0.23909 -0.96179
0.7407 0.7544 0.0223
0.0511 0.3970 0.8463
0.1765
FreqC
-0.99977 0.37632 1.00000
-0.34363 0.30086 -0.84650
0.98959 -0.61560
0.0136 0.7544 0.7767 0.8055
0.3574 0.0919 0.5778
ClustA
0.36368 -0.99939 -0.34363
1.00000 -0.99898 0.79085
-0.20493 0.95161
0.7630 0.0223
0.7767 0.0288 0.4193
0.8686 0.1989
ClustB
-0.32123 0.99678 0.30086
-0.99898 1.00000 -0.76240
0.16051 -0.93676
0.7918 0.0511
0.8055 0.0288 0.4480 0.8974
0.2276
ClustC
0.85772 -0.81180 -0.84650
0.79085 -0.76240 1.00000
-0.76109 0.94065
0.3438 0.3970
0.3574 0.4193 0.4480 0.4493 0.2204
dbhA
-0.98628 0.23909 0.98959
-0.20493 0.16051 -0.76109
1.00000 -0.49580
0.1056 0.8463
0.0919 0.8686 0.8974
0.4493 0.6698
dbhC
0.63234 -0.96179 -0.61560
0.95161 -0.93676 0.94065
-0.49580 1.00000
0.5642 0.1765
0.5778 0.1989 0.2276
0.2204 0.6698
The
SAS System 13:03 Friday,
December 1, 2006 247
The
DISCRIM Procedure
Total-Sample
Correlation Coefficients / Pr > |r|
Variable
FreqA FreqB FreqC
ClustA ClustB ClustC dbhA dbhC
FreqA
1.00000 0.17079 0.24310
-0.37026 0.03047 0.23077
-0.15876 -0.22300
0.5594 0.4023
0.1925 0.9177 0.4273
0.5877 0.4435
FreqB
0.17079 1.00000 0.60931
-0.52020 0.41556 -0.34310
-0.00737 -0.21182
0.5594 0.0207 0.0565
0.1395 0.2298 0.9800
0.4672
FreqC
0.24310 0.60931 1.00000
-0.37831 0.24705 -0.36967
-0.01650 -0.18432
0.4023 0.0207 0.1823 0.3945
0.1933 0.9554 0.5282
ClustA
-0.37026 -0.52020 -0.37831
1.00000 -0.50461 0.10370
0.08477 0.10476
0.1925
0.0565 0.1823 0.0657 0.7242
0.7733 0.7215
ClustB
0.03047 0.41556 0.24705
-0.50461 1.00000 -0.08601
0.32638 0.15130
0.9177 0.1395
0.3945 0.0657 0.7700 0.2548
0.6056
ClustC
0.23077 -0.34310 -0.36967
0.10370 -0.08601 1.00000
-0.04052 0.15872
0.4273 0.2298
0.1933 0.7242 0.7700 0.8906 0.5878
dbhA
-0.15876 -0.00737 -0.01650
0.08477 0.32638 -0.04052
1.00000 -0.19027
0.5877 0.9800
0.9554 0.7733 0.2548
0.8906 0.5147
dbhC
-0.22300 -0.21182 -0.18432
0.10476 0.15130 0.15872
-0.19027 1.00000
0.4435 0.4672
0.5282 0.7215 0.6056
0.5878 0.5147
The
SAS System 13:03 Friday,
December 1, 2006 248
The
DISCRIM Procedure
Simple Statistics
Total-Sample
Standard
Variable N Sum Mean Variance Deviation
FreqA 14
447.00000 31.92857 433.30220 20.8159
FreqB 14
487.00000 34.78571 1458 38.1841
FreqC 14 1009 72.07143 4920 70.1399
ClustA 14
30.61405 2.18672 1.33714 1.1563
ClustB 14
12.42322 0.88737 0.77660 0.8812
ClustC 14
39.01811 2.78701 5.69236 2.3859
dbhA 14 4.12101 0.29436 7.13267 2.6707
dbhC 14 -3.34209 -0.23872 0.33523 0.5790
------------------------------------------------------------------------------------------------------
Known = E
Standard
Variable N Sum Mean Variance Deviation
FreqA 5
177.00000 35.40000 824.30000 28.7106
FreqB 5 37.00000 7.40000 133.30000 11.5456
FreqC 5
144.00000 28.80000 733.70000 27.0869
ClustA 5
14.24289 2.84858 1.43184 1.1966
ClustB 5 2.90476 0.58095 0.73515 0.8574
ClustC 5
20.27827 4.05565 14.86875 3.8560
dbhA 5 -3.97805 -0.79561 2.29524 1.5150
dbhC 5
0.62048 0.12410 0.04043 0.2011
------------------------------------------------------------------------------------------------------
Known = L
Standard
Variable N Sum Mean Variance Deviation
FreqA 5
140.00000 28.00000 394.00000 19.8494
FreqB 5
179.00000 35.80000 887.70000 29.7943
FreqC 5
611.00000 122.20000 7044 83.9268
ClustA 5
10.94798 2.18960 0.92351 0.9610
ClustB 5 4.34386 0.86877 0.80249 0.8958
ClustC 5
10.06258 2.01252 0.15034 0.3877
dbhA 5 8.90902 1.78180 16.29449 4.0366
dbhC 5 -1.76724 -0.35345 0.02836 0.1684
------------------------------------------------------------------------------------------------------
The SAS System 13:03 Friday, December 1, 2006 249
The
DISCRIM Procedure
Simple Statistics
Known = M
Standard
Variable N Sum Mean Variance Deviation
FreqA 4
130.00000 32.50000 207.00000 14.3875
FreqB 4
271.00000 67.75000 2256 47.5000
FreqC 4
254.00000 63.50000 3542
59.5119
ClustA 4 5.42318 1.35579 1.00312 1.0016
ClustB 4 5.17460 1.29365 0.93794 0.9685
ClustC 4 8.67726 2.16931 0.45053 0.6712
dbhA 4 -0.80995 -0.20249 0.12524 0.3539
dbhC 4 -2.19534 -0.54883
0.99138 0.9957
------------------------------------------------------------------------------------------------------
The SAS
System 13:03 Friday, December 1,
2006 250
The
DISCRIM Procedure
Total-Sample
Standardized Class Means
Variable E L M
FreqA 0.1667680258 -.1887292473 0.0274515269
FreqB -.7172015497 0.0265630204 0.8632981617
FreqC -.6169300153 0.7146937682 -.1222046910
ClustA 0.5723716230 0.0024893528 -.7185762197
ClustB -.3477111035 -.0211077452 0.4610235609
ClustC 0.5317341387 -.3246163873 -.2588971892
dbhA -.4081201106 0.5569485556 -.1860355562
dbhC 0.6266393930 -.1981504140 -.5356112238
Pooled
Within-Class Standardized Class Means
Variable E L M
FreqA 0.155328928 -0.175783766 0.025568548
FreqB -0.871867554 0.032291391 1.049470204
FreqC -0.702505046 0.813829715 -0.139155836
ClustA 0.622607485 0.002707838 -0.781644154
ClustB -0.339432896 -0.020605218 0.450047642
ClustC 0.536851193 -0.327740279 -0.261388643
dbhA -0.418166656 0.570658757 -0.190615126
dbhC 0.667559315 -0.211089753 -0.570586952
Pooled Covariance
Matrix Information
Natural Log of the
Covariance Determinant of
the
Matrix
Rank Covariance Matrix
8 20.58002
The
SAS System 13:03 Friday, December
1, 2006 251
The
DISCRIM Procedure
Pairwise Squared
Distances Between Groups
2 _
_ -1 _
_
D (i|j) = (X
- X )' COV (X - X )
i
j i j
Squared
Distance to Known
From
Known E L M
E 0 11.17767 19.97527
L 11.17767 0 23.82850
M 19.97527 23.82850 0
F Statistics, NDF=8,
DDF=4 for Squared Distance to Known
From
Known E L M
E 0 1.27019 2.01770
L 1.27019 0 2.40692
M 2.01770 2.40692 0
Prob > Mahalanobis
Distance for Squared Distance to Known
From
Known E L M
E 1.0000 0.4354 0.2598
L 0.4354 1.0000 0.2066
M 0.2598 0.2066 1.0000
Pairwise Generalized
Squared Distances Between Groups
2 _
_ -1 _
_
D (i|j) = (X - X )'
COV (X - X )
i j i
j
Generalized
Squared Distance to Known
From
Known E L
M
E 0 11.17767 19.97527
L 11.17767 0 23.82850
M 19.97527 23.82850 0
The
SAS System 13:03 Friday,
December 1, 2006 252
The
DISCRIM Procedure
Univariate
Test Statistics
F
Statistics, Num DF=2, Den DF=11
Total Pooled Between
Standard Standard Standard R-Square
Variable
Deviation Deviation Deviation
R-Square / (1-RSq) F Value
Pr > F
FreqA 20.8159 22.3489 3.8554 0.0246 0.0253 0.14
0.8718
FreqB 38.1841 31.4104 29.4623 0.4274 0.7465 4.11
0.0466
FreqC 70.1399 61.5959 48.7929 0.3474 0.5324 2.93
0.0956
ClustA 1.1563 1.0630 0.7284 0.2849 0.3984 2.19
0.1582
ClustB 0.8812 0.9027 0.3482 0.1121 0.1262 0.69
0.5201
ClustC 2.3859 2.3631 1.1606 0.1699 0.2047 1.13
0.3591
dbhA 2.6707 2.6065 1.3883 0.1940 0.2407 1.32
0.3053
dbhC 0.5790 0.5435 0.3447 0.2544 0.3412 1.88
0.1990
Average R-Square
Unweighted
0.2268454
Weighted by
Variance 0.3436799
Multivariate
Statistics and F Approximations
S=2
M=2.5 N=1
Statistic Value F Value
Num DF Den DF Pr > F
Wilks' Lambda 0.04786501 1.79 16
8 0.2051
Pillai's Trace 1.54512169 2.12 16 10
0.1147
Hotelling-Lawley Trace 7.50335734 1.88 16
4 0.2866
Roy's Greatest Root 5.05027910 3.16 8
5 0.1105
NOTE: F Statistic for Roy's
Greatest Root is an upper bound.
NOTE: F Statistic
for Wilks' Lambda is exact.
The
SAS System 13:03 Friday,
December 1, 2006 253
The
DISCRIM Procedure
Linear Discriminant Function
_ -1 _ -1 _
Constant = -.5 X' COV X
Coefficient Vector = COV X
j j j
Linear
Discriminant Function for Known
Variable E L M
Constant -17.45676 -12.56128 -12.80423
FreqA 0.23043 0.11294 0.10949
FreqB -0.03371 -0.07345 0.14592
FreqC 0.01324 0.08519 -0.06972
ClustA 7.14465 5.24883 4.11349
ClustB 3.87164 1.71830 5.73590
ClustC 0.70912 0.64270 0.40157
dbhA -0.82441 0.41465 -1.50089
dbhC 3.95869 2.37264 -4.40804
The SAS System 13:03 Friday, December 1, 2006 254
The
DISCRIM Procedure
Classification Summary
for Calibration Data: WORK.SPECIES
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 Known
2 2
Pr(j|X) = exp(-.5 D (X)) /
SUM exp(-.5 D (X))
j k k
Number of Observations
and Percent Classified into Known
From
Known E L M Total
6 4 11 21
28.57 19.05 52.38 100.00
E 5 0
0 5
100.00 0.00 0.00 100.00
L 1 4 0 5
20.00 80.00 0.00 100.00
M 1 0 3 4
25.00 0.00 75.00 100.00
Total 13 8 14 35
37.14 22.86
40.00 100.00
Priors 0.33333 0.33333 0.33333
Error Count
Estimates for Known
E L M Total
Rate 0.0000 0.2000 0.2500
0.1500
Priors 0.3333 0.3333 0.3333
The
SAS System 13:03 Friday,
December 1, 2006 255
The
DISCRIM Procedure
Classification Results
for Calibration Data: WORK.SPECIES
Resubstitution Results
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 Known
2 2
Pr(j|X) = exp(-.5 D
(X)) / SUM exp(-.5 D (X))
j k k
Number of Observations and Average
Posterior Probabilities Classified into Known
From
Known E L M
6 4 11
0.8308 0.8026 0.8871
E 5 0 0
0.9596 . .
L 1 4 0
0.6722 0.9960 .
M 1 0 3
0.7464 . 1.0000
Total 13 8 14
0.8617 0.8993 0.9113
Priors 0.33333 0.33333 0.33333
Posterior Probability
Error Rate Estimates for Known
Estimate E L M Total
Stratified -0.2807 0.2032 0.2500 0.0575
Unstratified 0.0399 0.3833
-0.0936 0.1099
Priors 0.3333 0.3333 0.3333
The SAS
System 13:03 Friday, December 1,
2006 256
The
DISCRIM Procedure
Classification
Summary for Test Data: WORK.SPECIES
Classification 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 Known
2 2
Pr(j|X) = exp(-.5 D
(X)) / SUM exp(-.5 D (X))
j k k
Number of Observations
and Percent Classified into Known
From
Known E L M Total
6 4 11 21
28.57 19.05 52.38 100.00
E 5 0 0 5
100.00 0.00 0.00 100.00
L 1 4 0 5
20.00 80.00 0.00 100.00
M 1 0 3 4
25.00 0.00 75.00 100.00
Total 13 8 14 35
37.14 22.86 40.00 100.00
Priors 0.33333 0.33333 0.33333
Error Count
Estimates for Known
E L M Total
Rate 0.0000 0.2000 0.2500
0.1500
Priors 0.3333 0.3333 0.3333
The
SAS System 13:03 Friday,
December 1, 2006 257
The
DISCRIM Procedure
Classification
Results for Test Data: WORK.SPECIES
Classification Results
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 Known
2 2
Pr(j|X) = exp(-.5 D
(X)) / SUM exp(-.5 D (X))
j k k
Number of Observations and Average
Posterior Probabilities Classified into Known
From
Known E L M
6 4 11
0.8308 0.8026 0.8871
E 5 0 0
0.9596 . .
L 1 4 0
0.6722 0.9960 .
M 1 0 3
0.7464 . 1.0000
Total 13 8
14
0.8617 0.8993 0.9113
Priors 0.33333 0.33333 0.33333
Posterior Probability
Error Rate Estimates for Known
Estimate E L M Total
Stratified -0.2807 0.2032 0.2500 0.0575
Unstratified 0.0399 0.3833
-0.0936 0.1099
Priors 0.3333 0.3333 0.3333