第五章習(xí)題
1.習(xí)題5.1
解：假定兩總體服從正態(tài)分布，且協(xié)方差矩陣，誤判損失相同又先驗(yàn)概率按比例分配，通過(guò)SAS計(jì)算得到先驗(yàn)概率如表：
Class
Level
Information
group
Variable
Name
Frequency
Weight
Proportion
Prior
Probability
G1
G1
6.0000
0.428571
0.428571
G2
G2
8.0000
0.571429
0.571429
即：
又計(jì)算可得：
有計(jì)算的總體協(xié)防差距矩陣S為：
Pooled
Within-Class
Covariance
Matrix,DF
=
Variable
x1
x2
x1
1.081944444
-0.310902778
x2
-0.310902778
0.174756944
并且：
計(jì)算廣義平方距離函數(shù)：
并計(jì)算后驗(yàn)概率：
回代判別結(jié)果如下:
Posterior
Probability
of
Membership
in
group
Obs
From
group
Classified
into
group
G1
G2
G1
G1
0.9387
0.0613
G1
G1
0.9303
0.0697
G1
G1
0.9999
0.0001
G1
G2
*
0.4207
0.5793
G1
G1
0.9893
0.0107
G1
G1
1.0000
0.0000
G2
G2
0.0007
0.9993
G2
G2
0.0026
0.9974
G2
G2
0.0008
0.9992
G2
G2
0.0586
0.9414
G2
G2
0.0350
0.9650
G2
G2
0.0006
0.9994
G2
G2
0.0038
0.9962
G2
G2
0.0012
0.9988
由此可見誤判的回代估計(jì)：
若按照交叉確認(rèn)法，定義廣義平方距離如下：
逐個(gè)剔除,交叉判別，后驗(yàn)概率按下式計(jì)算：
通過(guò)SAS計(jì)算得到表所示結(jié)果。發(fā)現(xiàn)同樣也是屬于G1的4號(hào)被誤判為G2，因此誤判率的交叉確認(rèn)估計(jì)為
Posterior
Probability
of
Membership
in
group
Obs
From
group
Classified
into
group
G1
G2
G1
G1
0.9060
0.0940
G1
G1
0.7641
0.2359
G1
G1
1.0000
0.0000
G1
G2
*
0.1950
0.8050
G1
G1
0.9743
0.0257
G1
G1
1.0000
0.0000
G2
G2
0.0012
0.9988
G2
G2
0.0051
0.9949
G2
G2
0.0014
0.9986
G2
G2
0.0713
0.9287
G2
G2
0.0422
0.9578
G2
G2
0.0009
0.9991
G2
G2
0.0059
0.9941
G2
G2
0.0022
0.9978
其中=12.1138，又因?yàn)椋裕詈罂傻煤篁?yàn)概率p為：0.048709
習(xí)題5.3
解：（1）在并且先驗(yàn)概率相同的的假設(shè)前提下，建立矩離判別的線性判別函數(shù)。利用SAS的proc
discrim過(guò)程首先計(jì)算得到總體的協(xié)方差矩陣，如表：
Pooled
Within-Class
Covariance
Matrix,DF
=
Variable
x1
x2
x3
x4
x5
x6
x7
x8
x1
2.25705591
-0.91513311
0.34259974
-0.6084399
-0.9576508
-0.8929719
-0.0539445
-0.2192724
x2
-0.9151331
25.2318255
-0.3390873
-2.5515272
-5.0966371
0.78571637
-0.0835586
4.37529806
x3
0.34259974
-0.33908734
3.30063123
1.42276017
1.78692343
0.40208409
-0.0676655
-0.0732213
x4
-0.6084399
-2.55152726
1.42276017
6.07845863
5.78100857
2.32039331
-0.3205116
0.48605897
x5
-0.9576508
-5.09663714
1.78692343
5.78100857
8.15854743
3.44983429
-0.1096651
0.08904743
x6
-0.8929719
0.78571637
0.40208409
2.32039331
3.44983429
4.16657066
-0.2236278
0.87862549
x7
-0.0539445
-0.08355869
-0.0676655
-0.3205116
-0.1096651
-0.2236278
0.26009291
-0.0767347
x8
-0.2192724
4.37529806
-0.0732213
0.48605897
0.08904743
0.87862549
-0.0767347
2.51054423
各個(gè)總體的馬氏平方距離見表：
Generalized
Squared
Distance
to
group
From
group
G1
G2
G1
0
24.61468
G2
24.61468
0
線性判別函數(shù)為：
得到訓(xùn)練樣本回判法判別結(jié)果如表：
Error
Count
Estimates
for
group
G1
G2
Total
Rate
0.0000
0.0000
0.0000
Priors
0.5000
0.5000
訓(xùn)練樣本的交叉確認(rèn)判別結(jié)果：
Posterior
Probability
of
Membership
in
group
Obs
From
group
Classified
into
group
G1
G2
G1
G2
*
0.4501
0.5499
G1
G2
*
0.0920
0.9080
Error
Count
Estimates
for
group
G1
G2
Total
Rate
0.1000
0.0000
0.0500
Priors
0.5000
0.5000
（2）假設(shè)兩總體服從正態(tài)分布，先驗(yàn)概率按比例分配且誤判損失相同，在兩總體協(xié)方差矩陣相同，即的條件下進(jìn)行Bayes判別分析，通過(guò)SAS
discrim過(guò)程得到結(jié)果：
Error
Count
Estimates
for
group
G1
G2
Total
Rate
0.0000
0.0000
0.0000
Priors
0.7407
0.2593
交叉確認(rèn)判別結(jié)果：
Posterior
Probability
of
Membership
in
group
Obs
From
group
Classified
into
group
G1
G2
G1
G2
*
0.2246
0.7754
G2
G1
*
0.5282
0.4718
Error
Count
Estimates
for
group
G1
G2
Total
Rate
0.0500
0.1429
0.0741
Priors
0.7407
0.2593
在，并且先驗(yàn)概率按比例分配的假設(shè)前提下利用SAS的proc
discrim過(guò)程進(jìn)行Bays判別分析，這時(shí)以個(gè)總體的訓(xùn)練樣本單獨(dú)估計(jì)各總體的協(xié)方差矩陣，可到的訓(xùn)練樣本的回判和交叉確認(rèn)結(jié)果：
回判結(jié)果：
Error
Count
Estimates
for
group
G1
G2
Total
Rate
0.0000
0.0000
0.0000
Priors
0.7407
0.2593
交叉確認(rèn)判別結(jié)果：
Posterior
Probability
of
Membership
in
group
Obs
From
group
Classified
into
group
G1
G2
G2
G1
*
1.0000
0.0000
G2
G1
*
1.0000
0.0000
G2
G1
*
1.0000
0.0000
G2
G1
*
1.0000
0.0000
G2
G1
*
1.0000
0.0000
G2
G1
*
1.0000
0.0000
G2
G1
*
1.0000
0.0000
Error
Count
Estimates
for
group
G1
G2
Total
Rate
0.0000
1.0000
0.2593
Priors
0.7407
0.2593
（3）在不同的假設(shè)前提，采用不同判別方法得到待判樣本的判別結(jié)果：
1.距離判別分析得到西藏、上海、廣東的判別結(jié)果：
Posterior
Probability
of
Membership
in
group
Obs
Classified
into
group
G1
G2
G2
0.0000
1.0000
G2
0.0000
1.0000
G2
0.0000
1.0000
2.在協(xié)方差矩陣相同的前提下，Bayes對(duì)西藏、上海、廣東的判別結(jié)果：
Posterior
Probability
of
Membership
in
group
Obs
Classified
into
group
G1
G2
G2
0.0000
1.0000
G2
0.0000
1.0000
G2
0.0000
1.0000
3在協(xié)方差不同矩陣相同的前提下，Bayes對(duì)西藏、上海、廣東的判別結(jié)果：
Posterior
Probability
of
Membership
in
group
Obs
Classified
into
group
G1
G2
G1
1.0000
0.0000
G1
1.0000
0.0000
G1
1.0000
0.0000
3.習(xí)題5.4
解：（1）假設(shè)兩總體服從正態(tài)分布且在兩總體協(xié)方差矩陣相同，即，先驗(yàn)概率按相同的條件下進(jìn)行Bayes判別分析，通過(guò)SAS
discrim過(guò)程得到結(jié)果：
首先得到線性判別函數(shù)：
回代誤判結(jié)果：
Posterior
Probability
of
Membership
in
group
Obs
From
group
Classified
into
group
G1
G2
G1
G2
*
0.3401
0.6599
G2
G1
*
0.8571
0.1429
由計(jì)算結(jié)果發(fā)現(xiàn)，第9號(hào)樣本被誤判到G2,29號(hào)樣本被誤判到G1.誤判率為6.34%
Error
Count
Estimates
for
group
G1
G2
Total
Rate
0.0833
0.0435
0.0634
Priors
0.5000
0.5000
交叉確認(rèn)判別結(jié)果：由計(jì)算發(fā)現(xiàn)總共有四個(gè)樣本被判錯(cuò)，分別是9、28、29、35號(hào)樣品。累計(jì)誤判率為10.69%
Posterior
Probability
of
Membership
in
group
Obs
From
group
Classified
into
group
G1
G2
G1
G2
*
0.0973
0.9027
G2
G1
*
0.6130
0.3870
G2
G1
*
0.9643
0.0357
G2
G1
*
0.8470
0.1530
Error
Count
Estimates
for
group
G1
G2
Total
Rate
0.0833
0.1304
0.1069
Priors
0.5000
0.5000
（1）假設(shè)兩總體服從正態(tài)分布且在兩總體協(xié)方差矩陣相同，即，先驗(yàn)概率按比例分配且誤判損失相同的條件下進(jìn)行Bayes判別分析，通過(guò)SAS
discrim過(guò)程得到結(jié)果：
首先得到線性判別函數(shù)：
Linear
Discriminant
Function
for
group
Variable
G1
G2
Constant
-99.91796
-95.41991
x1
30.35060
29.87680
x2
-0.15214
-0.15210
x3
-0.78868
-0.22662
x4
1.95176
1.39528
x5
0.58964
0.06490
x6
-108.10195
-85.33735
x7
-0.31156
-0.25957
回代誤判結(jié)果
Posterior
Probability
of
Membership
in
group
Obs
From
group
Classified
into
group
G1
G2
G1
G2
*
0.2119
0.7881
G2
G1
*
0.7579
0.2421
Error
Count
Estimates
for
group
G1
G2
Total
Rate
0.0833
0.0435
0.0571
Priors
0.3429
0.6571
交叉確認(rèn)誤判結(jié)果：
Posterior
Probability
of
Membership
in
group
Obs
From
group
Classified
into
group
G1
G2
G1
G2
*
0.3436
0.6564
G1
G2
*
0.0532
0.9468
G1
G2
*
0.4052
0.5948
G1
G2
*
0.3519
0.6481
G2
G1
*
0.9338
0.0662
G2
G1
*
0.7428
0.2572
Error
Count
Estimates
for
group
G1
G2
Total
Rate
0.3333
0.0870
0.1714
Priors
0.3429
0.6571
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