主成分分析和因子分析习题及答案

..主成分分析和因子分析教材习题答案11.1下表是2007年30家能源类上市公司的有关经营数据。其中：X1=主营业务利润；X2=净资产收益率；X3=每股收益；X4=总资产周转率；X5=资产负债率；X6=流动比率；X7=主营业务收入增长率；X8=资本积累率。进行主成分分析并确定主成分的数量。股票简称X1X2X3X4X5X6X7X8海油工程19.75127.0101.1320.92250.4691.23725.49510.620中海油服33.73312.9900.4980.51025.3983.37846.990-1.576中国石化13.07918.2600.6341.83554.5840.67455.04343.677中国石油33.44119.9000.7350.92328.0681.04342.68245.593广聚能源6.79015.6500.4411.18813.2573.60238.44617.262鲁润股份5.3150.5000.0111.87952.5931.222207.37333.721海越股份3.35715.4800.5380.62648.8300.80733.43854.972国际实业29.33210.3400.2990.66253.1401.21816.5797.622靖远煤电29.96116.0400.2550.66236.5960.70020.902-3.682美锦能源23.34218.5800.4970.92360.9630.9921.27112.128神火股份26.04242.5001.6400.99069.7760.51050.13852.066金牛能源35.02215.7300.7250.94439.2670.9539.002-3.877煤气化25.80914.9800.6770.92845.7680.949-3.85124.881西山煤电39.50617.8200.8680.70345.4501.5259.162-85.430露天煤业29.89522.4500.7090.80040.9771.3213.3104.369郑州煤电18.16012.7400.2991.37452.9621.240-100.00085.688兰花科创41.40220.0701.4140.61752.9161.0606.78914.259..黑化股份8.7831.4300.0330.75348.0610.545-11.6596.856兖州煤业45.59213.7300.5480.68822.3502.15821.19921.953国阳新能16.06114.9201.0301.62348.3860.97315.34220.860盘江股份11.0036.6600.2601.18730.2011.68241.65775.804上海能源24.87617.9500.7090.96848.6740.51020.54814.526山西焦化12.8254.4500.3310.84948.4761.41743.67629.419恒源煤电32.22817.8201.0700.44972.0790.5159.872149.837开滦股份24.42320.6701.1020.84554.1981.10273.28526.542大同煤业44.00512.9900.5970.66747.5541.84330.62115.668中国神华48.18015.4000.9940.40837.6872.09727.81346.229潞安环能28.56721.7101.5341.02354.2611.59048.31529.610中煤能源41.21416.6800.4410.66940.9322.05829.90311.350国投新集30.0159.6800.2220.35064.4710.63024.27836.437详细答案：SPSS输出的各主成分分析结果如下表：主成分的方差贡献率和累计方差贡献率TotalVarianceExplainedComponentInitialEigenvaluesExtractionSumsofSquaredLoadingsTotal%ofVarianceCumulative%Total%ofVarianceCumulative%12.34629.32129.3212.34629.32129.32122.05925.73255.0532.05925.73255.05331.24915.61470.6671.24915.61470.6674.84310.53981.2065.7549.42190.6286.3374.21594.843..7.2493.11497.9578.1632.043100.000ExtractionMethod:PrincipalComponentAnalysis.主成分的因子载荷矩阵ComponentMatrix(a)Component123X1.490-.698-.113X2.804-.066.442X3.824-.049.464X4-.363.603.498X5.573.643-.219X6-.434-.672.332X7-.329.248.610X8.147.524-.219ExtractionMethod:PrincipalComponentAnalysis.a3componentsextracted.主成分方差贡献率表中前3个主成分的累计方差贡献率为70.667%.虽然没有达到80%以上.但第四个主成分的特征根小于1。因此.按着主成分的选择要求.选择3个主成分比较合适。从因子载荷矩阵看.第一主成分主要解释了X2（净资产收益率）和X3（每股收益）两个变量；第二个主成分主要解释了X1（主营业务利润）、X4（总资产周转率）、X5（资产负债率）、X6（流动比率）和X8（资本积累率）这5个变量；而第三个主成分只解释了X7（主营业务收入增长率）一个变量。11.2根据11.1题的数据：（1）检验该数据是否适合进行因子分析？（2）进行因子分析.并对30家上市公司的因子综合得分进行排序。详细答案：..SPSS输出的因子分析结果如下表：（1）KMO检验和Bartlett球度检验表如下：KMOandBartlett'sTestKaiser-Meyer-OlkinMeasureofSamplingAdequacy..554Bartlett'sTestofSphericityApprox.Chi-Square75.082df28Sig..000从检验表中可见.Bartlett球度检验统计量为75.082。检验的值接近0。表明8个变量之间有较强的相关关系。而KMO统计量为0.554.小于0.7。进行因子分析的效果不一定很好。（2）旋转后的因子载荷矩阵如下：RotatedComponentMatrix(a)Component123X1.404-.313-.693X2.912.094-.075X3.940.106-.058X4-.066.126.850X5.264.848-.023X6-.082-.862.025X7.065-.192.707X8-.066.575.090ExtractionMethod:PrincipalComponentAnalysis.RotationMethod:VarimaxwithKaiserNormalization...aRotationconvergedin5iterations.因子1与X2（净资产收益率）和X3（每股收益）的载荷系数较大.这两个变量主要与上市公司盈利能力有关.因此可命名为“盈利能力”。因子2与X5（资产负债率）、X6（流动比率）、X8（资本积累率）这3个变量的载荷系数较大.这三个变量主要涉及企业的偿债能力.因此可命名为“偿债能力因子”。因子3与X1（主营业务利润）、X4（总资产周转率）、X7（主营业务收入增长率）这三个变量的载荷系数较大.这三个变量分别涉及了盈利能力、资产管理水平、企业成长能力等.因此.这个因子的命名比较困难。各公所的因子综合得分和排名如下：11.3对下表中的50名学生成绩进行主成分分析.可以选择几个综合变量来代表这些学生的六门课程成绩？学生代码数学物理化学语文历史英语171649452615227896818089763695667759480477908068666058467756070636626783718577..7746575729073891749762716697287727983761082708368778511637060918582127479955974591366617762736414908298477160157790856873761691828454626017788410051606018907878597266198010083537370205851677991852172898877808322645550686865237789807375702472687783927925726761929288267372708886792777816285908728616581989495297995838989793081907973858031857775527359..3268857084898633859195637666349185100706576357474846180693688100854971663763826689788038878410074817639819884576569406479647276744160516078747642758476657673435975818277734464595671796745646149100999546564861858280476245677876824886789287877749667279818766506166489810096详细答案：SPSS输出的主成分分析结果如下表：主成分的方差贡献率和累计方差贡献率TotalVarianceExplainedInitialEigenvaluesExtractionSumsofSquaredLoadings..ComponentTotal%ofVarianceCumulative%Total%ofVarianceCumulative%13.72962.14662.1463.72962.14662.14621.20620.09682.2421.20620.09682.2423.4036.72488.9664.3255.41494.3805.2043.39597.7756.1342.225100.000ExtractionMethod:PrincipalComponentAnalysis.主成分载荷矩阵ComponentMatrix(a)Component12数学-.778.430物理-.580.682化学-.784.318语文.893.312历史.826.406英语.833.438ExtractionMethod:PrincipalComponentAnalysis.a2componentsextracted.头两个主成分能够解释总方差的82.242%.所以可以选择这两个主成分来代表原来的六门课程成绩。由主成分载荷矩阵来看.第一个主成分既充分解释了数学、物理、化学三门课程成绩.也充分解释了语文、历史、英语三门课程成绩.但前三门课程的主成分载荷为均为负值.后三门课程的主成分载荷恰好相反.均为正值.这可能是由于文理科课程的性质不同而导致的。第二主成分则与六门课程成绩均表现出一定的正相关关系。..11.4如果事先确定选择两个因子来代表习题11.3中50名学生的六门课程成绩.试对该数据进行因子分析.得到的两个因子有没有合理的直观意义？详细答案：SPSS输出的因子分析结果如下表：旋转后的因子载荷矩阵RotatedComponentMatrix(a)Component12数学-.341.821