第五章数据分析（梅长林）习题

第五章习题 1.习题5.1 解：假定两总体服从正态分布，且协方差矩阵，误判损失相同又先验概率按比例分配，通过SAS计算得到先验概率如表：
Class Level Information group Variable Name Frequency Weight Proportion Prior Probability G1 G1 6 6.0000 0.428571 0.428571 G2 G2 8 8.0000 0.571429 0.571429 即：
又计算可得：
有计算的总体协防差距矩阵S为：
Pooled Within-Class Covariance Matrix, DF = 12 Variable x1 x2 x1 1.081944444 -0.310902778 x2 -0.310902778 0.174756944 并且：
计算广义平方距离函数：
并计算后验概率：
回代判别结果如下: Posterior Probability of Membership in group Obs From group Classified into group G1 G2 1 G1 G1 0.9387 0.0613 2 G1 G1 0.9303 0.0697 3 G1 G1 0.9999 0.0001 4 G1 G2 * 0.4207 0.5793 5 G1 G1 0.9893 0.0107 6 G1 G1 1.0000 0.0000 7 G2 G2 0.0007 0.9993 8 G2 G2 0.0026 0.9974 9 G2 G2 0.0008 0.9992 10 G2 G2 0.0586 0.9414 11 G2 G2 0.0350 0.9650 12 G2 G2 0.0006 0.9994 13 G2 G2 0.0038 0.9962 14 G2 G2 0.0012 0.9988 由此可见误判的回代估计：
若按照交叉确认法，定义广义平方距离如下：
逐个剔除, 交叉判别，后验概率按下式计算：
通过SAS计算得到表所示结果。发现同样也是属于G1的4号被误判为G2，因此误判率的交叉确认估计为 Posterior Probability of Membership in group Obs From group Classified into group G1 G2 1 G1 G1 0.9060 0.0940 2 G1 G1 0.7641 0.2359 3 G1 G1 1.0000 0.0000 4 G1 G2 * 0.1950 0.8050 5 G1 G1 0.9743 0.0257 6 G1 G1 1.0000 0.0000 7 G2 G2 0.0012 0.9988 8 G2 G2 0.0051 0.9949 9 G2 G2 0.0014 0.9986 10 G2 G2 0.0713 0.9287 11 G2 G2 0.0422 0.9578 12 G2 G2 0.0009 0.9991 13 G2 G2 0.0059 0.9941 14 G2 G2 0.0022 0.9978 其中=12.1138， ，又因为，所以， 最后可得后验概率p为：0.048709 习题5.3 解：（1）在并且先验概率相同的的假设前提下，建立矩离判别的线性判别函数。利用SAS的proc discrim过程首先计算得到总体的协方差矩阵，如表：
Pooled Within-Class Covariance Matrix, DF = 25 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 各个总体的马氏平方距离见表：
Generalized Squared Distance to group From group G1 G2 G1 0 24.61468 G2 24.61468 0 线性判别函数为：
得到训练样本回判法判别结果如表：
Error Count Estimates for group G1 G2 Total Rate 0.0000 0.0000 0.0000 Priors 0.5000 0.5000 训练样本的交叉确认判别结果：
Posterior Probability of Membership in group Obs From group Classified into group G1 G2 17 G1 G2 * 0.4501 0.5499 19 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）假设两总体服从正态分布，先验概率按比例分配且误判损失相同，在两总体协方差矩阵相同，即的条件下进行Bayes判别分析，通过SAS discrim过程得到结果：
Error Count Estimates for group G1 G2 Total Rate 0.0000 0.0000 0.0000 Priors 0.7407 0.2593 交叉确认判别结果：
Posterior Probability of Membership in group Obs From group Classified into group G1 G2 19 G1 G2 * 0.2246 0.7754 25 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，并且先验概率按比例分配的假设前提下利用SAS的proc discrim过程进行Bays判别分析，这时以个总体的训练样本单独估计各总体的协方差矩阵，可到的训练样本的回判和交叉确认结果：
回判结果：
Error Count Estimates for group G1 G2 Total Rate 0.0000 0.0000 0.0000 Priors 0.7407 0.2593 交叉确认判别结果：
Posterior Probability of Membership in group Obs From group Classified into group G1 G2 21 G2 G1 * 1.0000 0.0000 22 G2 G1 * 1.0000 0.0000 23 G2 G1 * 1.0000 0.0000 24 G2 G1 * 1.0000 0.0000 25 G2 G1 * 1.0000 0.0000 26 G2 G1 * 1.0000 0.0000 27 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）在不同的假设前提，采用不同判别方法得到待判样本的判别结果：
1.距离判别分析得到西藏、上海、广东的判别结果：
Posterior Probability of Membership in group Obs Classified into group G1 G2 1 G2 0.0000 1.0000 2 G2 0.0000 1.0000 3 G2 0.0000 1.0000 2.在协方差矩阵相同的前提下，Bayes对西藏、上海、广东的判别结果：
Posterior Probability of Membership in group Obs Classified into group G1 G2 1 G2 0.0000 1.0000 2 G2 0.0000 1.0000 3 G2 0.0000 1.0000 3在协方差不同矩阵相同的前提下，Bayes对西藏、上海、广东的判别结果：
Posterior Probability of Membership in group Obs Classified into group G1 G2 1 G1 1.0000 0.0000 2 G1 1.0000 0.0000 3 G1 1.0000 0.0000 3.习题5.4 解：（1）假设两总体服从正态分布且在两总体协方差矩阵相同，即，先验概率按相同的条件下进行Bayes判别分析，通过SAS discrim过程得到结果：
首先得到线性判别函数：
回代误判结果：
Posterior Probability of Membership in group Obs From group Classified into group G1 G2 9 G1 G2 * 0.3401 0.6599 29 G2 G1 * 0.8571 0.1429 由计算结果发现，第9号样本被误判到G2,29号样本被误判到G1.误判率为6.34% Error Count Estimates for group G1 G2 Total Rate 0.0833 0.0435 0.0634 Priors 0.5000 0.5000 交叉确认判别结果：由计算发现总共有四个样本被判错，分别是9、28、29、35号样品。累计误判率为10.69% Posterior Probability of Membership in group Obs From group Classified into group G1 G2 9 G1 G2 * 0.0973 0.9027 28 G2 G1 * 0.6130 0.3870 29 G2 G1 * 0.9643 0.0357 35 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）假设两总体服从正态分布且在两总体协方差矩阵相同，即，先验概率按比例分配且误判损失相同的条件下进行Bayes判别分析，通过SAS discrim过程得到结果：
首先得到线性判别函数：
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 回代误判结果 Posterior Probability of Membership in group Obs From group Classified into group G1 G2 9 G1 G2 * 0.2119 0.7881 29 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 交叉确认误判结果：
Posterior Probability of Membership in group Obs From group Classified into group G1 G2 5 G1 G2 * 0.3436 0.6564 9 G1 G2 * 0.0532 0.9468 11 G1 G2 * 0.4052 0.5948 12 G1 G2 * 0.3519 0.6481 29 G2 G1 * 0.9338 0.0662 35 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