（约翰逊版）实用多元统计分析 第八章课后答案.docx

第八章作业8.10 解：首先对数据进行标准化处理，消除不同的度量带来的差异标准化的数据如下表：JP MorganCitibankWells FargoRoyal Dutch ShellExxon Moball-0.71441-0.75227-0.49227-1.77750.1236660.1268320.573819-0.651510.3824790.569489-0.90557-0.795520.205325-0.07289-0.488730.637755-0.517110.595515-1.16145-0.53240.217914-0.12734-0.858211.0385052.0514320.213614-0.98797-0.765930.4491090.003450.2300331.1852370.101451.0899050.0163881.690092-0.606850.6382250.1681190.256281-1.55331-1.07447-0.74853-1.21133-0.74049-0.387220.806277-0.64466-0.84924-0.840761.2538331.6139331.0089062.1426690.757090.725319-0.16031-0.686840.393521.3608650.7988112.5113281.821979-1.27834-1.17391-2.787420.138096-0.016131.6237121.4913240.6447910.9127751.2156020.2484561.563561-0.07332-1.1561-1.58006-0.13648-1.08658-0.07371-0.17167-0.48489-0.11934-0.680650.4598890.1856690.635589-0.68894-0.565280.201105-0.472242.0186560.021531-1.04453-0.70503-1.10528-1.3122-0.26251-1.04022表1对处理的数据做主成分分析样本相关系数矩阵即为相应的样本协方差矩阵S即相应的协方差矩阵为：协方差矩阵JP MorganCitibankWells FargoRoyal Dutch ShellExxon MoballJP Morgan1.000.334.392.063.036Citibank.3341.000.545.232.169Wells Fargo.392 .5451.000.022-.044Royal Dutch Shell.063.232.0221.000.621Exxon Moball.036.169-.044.6211.000表2 从表3可以得出，五个主因子解释的总体方差比重总方差解释因子初始特征根被提取的载荷平方和特征值解释的方差比重累计方差比重特征值解释的方差比重累计方差比重11.97539.50239.5021.97539.50239.50221.54430.87970.3821.54430.87970.3823.69313.85684.238.69313.85684.2384.4138.25592.493.4138.25592.4935.3757.507100.000.3757.507100.000表3五个主因子间的协方差矩阵如下表4：主成分的协方差矩阵因子1234511.000.000.000.000.0002.0001.000.000.000.0003.000.0001.000.000.0004.000.000.0001.000.0005.000.000.000.0001.000表4从表4可以看出，这两个因子之间的相关程度比较低计算相应的因子载荷表因子载荷矩阵因子12345JP Morgan.302-.2451.016-.163-.044Citibank.403-.140-.517-1.058.056Wells Fargo.342-.339-.3651.096.100Royal Dutch Shell.277.460.005.216-1.157Exxon Moball.242.492.102.1691.144表5从表5可以得出五个主成分的表达式：F1=0.302X1+0.403X2+0.342X3+0.277X4+0.242X5F2=(-0.245)X1+(-0.14)X2+(-0.339)X3+0.46X4+0.492X5F3=1.016X1+(-0.517)X2+(-0.365)X3+0.005X4+0.102X5F4=(-0.163)X1+(-1.058)X2+1.096X3+0.216X4+0.169X5F5=(-0.044)X1+0.056X2+0.1X3+(-1.157)X4+1.144X5(b)五个特征值分别为：1，2，3，4，5，从表三可以得出：第一主成分的总方差贡献为：11+2+3+4+5=39.502%第二主成分的总方差贡献为：21+2+3+4+5=30.879%第三主成分的总方差贡献为：31+2+3+4+5=13.856%（c）第一主成分的特征值1对应的庞弗罗尼联合置信区间为【0.00106，0.00195】第二主成分的特征值2对应的庞弗罗尼联合置信区间为【0.00054，0.001】第三主成分的特征值3对应的庞弗罗尼联合置信区间为【0.00019，0.00036】（d）从（a）（c）的结果，前三个主成分的方差贡献超过80%，我们可以得出，综合股票回报率数据能在小于五维的空间中得到解释。8.13 （a）变量的相关系数矩阵如下表：相关系数X1X2X3X4X5X6X11.000.843.322.412.766.348X2.8431.000.451.610.843-.116X3.322.4511.000.466.641.005X4.412.610.4661.000.811.067X5.766.843.641.8111.000.102X6.348-.116.005.067.1021.000(b)有相应的相关系数表可以求出相应的特征值及特征向量相应的总方差解释表如下表因子初始特征根被提取的载荷平方和特征值解释的方差累积解释的方差特征值解释的方差累积的解释方差13.53158.84658.8463.53158.84658.84621.13518.92577.7711.13518.92577.7713.74612.43390.204.74612.43390.2044.5188.64198.845.5188.64198.8455.0611.01099.855.0611.01099.8556.009.145100.000.009.145100.000表1从表1可以得出相应的特征值特征值为：1=3.531，2=1.135，3=0.746，4=0.518，5=0.061，5=0.009因子载荷矩阵因子123456x1.827.361-.382.189-.016-.061x2.903-.151-.372.071.128.049x3.659-.230.576.423.042-.010x4.790-.128.246-.541.065-.028x5.977-.037.044-.068-.191.032x6.134.955.259-.033.038.026表2从上表可以得出相应的特征向量e1=(0.872,0.903,0.659,0.79,0.977,0.134)e2=(0.361,-0.151,-0.23,-0.128,-0.037,0.955)e3=(-0.382, -0.372,0.576, 0.246,0.044, 0.259)e4=(0.189,0.071,0.423,-0.541,-0.068,-0.033)e5=(-0.016,0.128,0.042,0.065,-0.191,0.038)e6=(-0.061,0.049,-0.01,-0.028,0.032,0.026)第一主成分的总方差贡献为：11+2+3+4+5+6=58.846%第二主成分的总方差贡献为：21+2+3+4+5+6=18.925%第三主成分的总方差贡献为：31+2+3+4+5+6=12.433%第四主成分的总方差贡献为：41+2+3+4+5+6=8.641%第五主成分的总方差贡献为：51+2+3+4+5+6=1.010%第六主成分的总方差贡献为：61+2+3+4+5+6=0.145%（c）从（b）的结果可以看出，第一个主成解释了总方差的58.846%，低于80%，所以用一个指标来反映综合放射法数据是不合理的（d）从（b的结果可以得出，提取前三个主成分比较合适，前三个主成分的的累积方差贡献超过80%，前三个主成分与x1,x2,x3,x4.x5及x6的相关系数表如下：相关系数F1F2F3X10.753-0.0280.299X20.727-0.3030.440X30.6070.394-0.554X40.783-0.022-0.110X50.882-0.076-0.179X60.0950.9070.379表3第九章作业9.20（a）空气污染变量X1,X2,X5,X6的样本协方差矩阵如表1：协方差X1X2X5X6X12.5-2.77-0.59-2.23X2-2.77300.526.7830.78X5-0.596.7811.363