多元统计正交因子分析实验报告

WordWord文档正交因子分析［役计性卖验丿(Orthogonalfactoranalysis)实菠点理：因子分析是主成分分析的推/•和发展，其目的是用*数儿个不可观测的隐变量，即因子，来解粹原始变量之间的和关关糸，它&是厲于多元分析中处理吟维的一种统计方冻。因子分析的基本思•想是通过变量间的协方差矩阵(戎相关糸数矩阵丿部结构的研•死，寻找能技制所有变量的少救几个因子去描述.多个变量之间的和关关糸。因子分析中最常用的数学楼更是正交因子栈型，其特点是棋型中的因子相互之间正交。下表中给岀了二战以来典运会运动员十顼运动成绩的相关糸数矩阵：(E9a6)100卷1.00■•跳遗0.591.00••铅球0.350.421.00•跳壽0.340.510.381.00400耒0.630.490.190.29110卷跨栏0.400.520.360.46铁饼0.280.310.730.27擇竿跳壽0.200.360.240.39标枪0.110.210.440.171500*-0.070.09-0.080」8

1.00 TOC\o"1-5"\h\z0.341.00 . ^ ^ .0」70.321.00 . ^ .0.230.330.241.00 . .0.130.180340.241.00 .0.390.00-0.020.17-0.001.00卖絵要求:(1J试由相关糸数矩阵作因子分析；covmat(2)试根据因子戟椅，并结合題目背景知识，对公共因子进行命名。卖檢题目二：下在中给出了不同国家及地区的女子径躱记录：(t1a7)100m200m400m800m1500m3000mMarathonCountry(s)(min)(min)(min)(min)argentin11.6122.9454.52」54.439.79178.52australi11.222.3551.081.989.08152.37austria11.4323.0950.621.994.229.34159.37belgium11.4123.045224」48.88157.85bermuda11.4623.055332」64.589.81169.98brazil11.3123.1752.82」4.499.77168.75burma12.1424.47552」84.459.51191.02Canada1122.2550.0624.068.81149.45chile1224.5254.92.054.239.37171.38china11.9524.4154.972.084.339.31168.48Columbia11.62453.262」14.359.46165.42cookis12.927」60.42.34.8411」233.22costa11.9624.658.252.214.6810.43171.8czech11.0921.9747.991.894」48.92158.85denmark11.4223.5253.62.034」88.71151.75domrep11.7924.0556.052.244.749.89203.88finland223950.142.034」8.92154.23france11」522.5951.7324」48.98155.27gdr10.8121.7148.161.933.968.75157.68他11.0122.3949.751.954.038.59148.53gbni1122.1350.461.984.038.62149.72greece11.7924.0854.932.074.359.87182.2guatemal11.8424.5456.092.284.8610.54215.08hungary11.4523.0651.52.014」48.98156.37india11.9524.2853.62」4.329.98188.03indonesi11.8524.2455.342.224.6110.02201.28Ireland11.4323.5153.242.054.118.89149.38israel11.4523.5754.92」4.259.37160.48italy11.292352.011.963.988.63151.82japan11.732453.732.094.359.2150.5kenya11.7323.8852.724」59.2181.05korea11.9624.4955.72」54.429.62164.65dprkorea12.2525.7851.21.974.259.35179.17luxembou12.0324.9656.12.074389.64174.68malaysia12.2324.2155.092」94.6910.46182.17mauritiu11.7625.0858.12.274.7910.9261.13mexico11.8923.6253.762.044.259.59158.53

netherla11.2522.8152.381.994.069.01152.48nz11.5523.1351.62.024.188.76145.48norway11.5823.3153.122.034.018.53145.48png12.2525.0756.962.244.8410.69233philippi11.7623.5454.62」94.610.16200.37poland11.1322.2149.291.953.998.97160.82Portugal11.8124.2254.32.094」68.84151.2rumania11.4423.4651.21.923.968.53165.45singapor12.32555.082」24.529.94182.77spain11.823.9853.592.054」49.02162.6Sweden11.1622.8251.792.024」28.84154.48switzerl11.4523.3153.112.024.078.77153.42taipei11.2222.6252.52」4.389.63177.87thailand11.7524.4655.82.24.7210.28168.45turkey11.9824.4456.452」54.37938201.08usa10.7921.8350.621.963.958.5142.72ussr11.0622.1949.191.893.878.45151.22wsamoa12.7425.8558.732.335.8113.04306（数据来源：1984年洛杉机典运会IAAF/AFT徳球与田球统计手册丿ussr11.0622.1949.191.893.878.45151.22rumania11.4423.4651.21.923.968.53165.45卖絵要求:

(1丿很据以上数据对女子後赛顼目作因子分析；对亦共因孑进行解释；计算各个国家的笫一因子得分并进行排名。要求列出排名前10的国家或地区，并给岀中国的名次。卖蠢题目一分析掖告：R«4：record<-read.table(Hdata4.txf\head=F) #导入救据record<-record[,-1] #刪除笫一列record<-as.matrix(record) #将原救据矩阵化option$(digit$=2) #保Q两住小教pca.datal<-princomp(covmat=record)#以和关糸数矩阵作为基础，建立主成分分析summary(pca.datal) #输出主成分分析报表factl.$t<-factanal(covmat=record,factor$=5,rotation=”none”)#作因子分析,不淡转factl.stfactl.rofactl.ro<-factan8l(covm8t=recorcLfdctors=5、rotdtion=\8rimax”)#作因子分析,淡携factl.stfactl.ro#输出不炎转的结果#输出淡转的结黑#计算共同度apply((factl.ro$loadings)八2J』um)#计算共同度fact2.ro<-factan8l(covm8t=recorcLfdctors=4、rot8tion=”vdrimdx”)#作因孑分析,淡转fact2.ro#输出淡转的结果#计算共同度apply((fact2.ro$loadings)^2J,sum)#计算共同度输岀姑系及分析：(1丿试由柑关糸数矩阵作因子分析；record<-read.table(ndata4.txt\head=F)#导入数据record<-record[,-1]#刪徐第一列record<-as.matrix(record)#将療教据矩阵化option$(digit$=2)#保倒两住小数pca.datal<-princomp(covmat=record)#以和关糸数矩阵作为基础，建立主成分分析summary(pca.datal) #输出主成分分析圾表为了确主因子分析中因子的数目,我们先对柑关糸数矩阵做主成分分析叔1主成分分析掖在Comp.1Comp.2Comp.3Comp.4Comp.5Comp.6Comp.7Comp.8Comp.9Comp」0Standarddeviation1.951.231.060.9560.8490.7710.7260.6190.4850.456ProportionofVariance0.380.150.110.0910.0720.0590.0530.0380.0240.021CumulativeProportion0.380.530.640.7330.8050.8650.9170.9560.9791.000

由方差累计贾伙率得刊，柱笫五主成分，累积贾故率达到了80%以上，并趨于稔定。我们确定因子分析中因子数目为5.factl.st<-factanal(covmat=record,factor$=5,rotation=MnoneM)#作因子分析,不淡转factl.stfactl.rofactl.ro<-factanal(covmat=record,factor$=5,rotation=HvarimaxM)#factl.stfactl.ro#输出不炎转的结果#输出淡转的结果apply((factl.ro$loadings)^2J,sum) #计算共同度做因子分析，得到未淡转的因子栽持以及淡转的因子栽希表2未炎转的因孑载荷FactorFactorlFactor2Factor3Factor4Factor5100耒0.2080.7910.301-0.167跳迄0.3780.5950.2460.242鉛球0.6440.761跳需0.41503440.1570.471-0.139400*0.4460.688-0.113-0.2030.116110耒跨栏0.2650.4350.2610.343铁饼0.5030.534撐竿跳需0.3070.2400.4020.214标枪0.3130.引40.3781500*0.707-0.704

累积贡伙率0.20.380.550.6160.640表3炎转的因孑我椅FactorFactorlFactor2Factor3Factor4Factor5Communalities100*0.1710.8150.276-0.1410.79跳遗0.2230.4800.5800.62铅球0.9550.1390.2411.00跳需0.2110.1520.6870.1170.56400*0.7600.19303260.1260.74110耒跨栏0.1870.2780.5650.45铁饼0.6930.1250.1940.1110.55撐竿跳壽0.1120.5010.1190.2820.36标枪0.4080.1400.4010.351500卷0.9891.00累积贡故率0」70.340.500.610.640观疼表格中枚标注为绿色的两个因子我満(标枪项目一行丿,在Factor!中的因子栽'荷为0.408,A.Factor5中的因子我持％0.401,比较两个因子我椅，0.408>0.401,因此我们最终选取0.408。这样一来，我们做因子分肘，只谢要4个因子即可。因此.我们下面冉做4个因子的簸桔因子分析。fact2.rov・factanal(covmat=record,factor$=4,rotation=HvarimaxM)#作因孑分析,淡转

fact2.ro#输出淡转的结果fact2.roapply((fact2.ro$loadings)八2J,sum) #计算共同度在4<44的因孑我椅FactorFactorlFactor2Factor3Factor4Communalities100耒0.1670.8570.246-0.1380.84跳远0.2390.4760.5810.62铅球0.9630.1530.2011.00跳需0.2420.1720.6320.1130.50400耒0.7100.23603310.67110卷跨栏0.2050.2610.5880.46铁饼0.6990.1330.1790.54撐华跳壽0.1380.5120.1170.30标枪0.4180.1750.211500耒0.1130.9881.00累积贾秋率0」80340.500.61(2)试根据因子戟椅，并结今題目背景知帜，对公共因子进行令名由淡转后的栽持可发现，第一因子中，鉛球、铁饼和标枪的栽持较大，可令名为仪掷因子；笫二因子中，100卷和400耒的我椅较大，可命名为短跑因子；笫三因子中，跳迄、跳需、110卷跨栏、擇竿跳壽较丸，可令名为弹跳因子；第四因子中，1500耒的我荷较丸，可令名为长跑因子。

卖絵题目二分析掖告:R程序:bv-read.csvfdata42.csvv)#导入数据b1<-b[,-l]#刪除笫一列pc.b1<-princomp(b1,cor=T)#做主成分分析summary(pc.bl)#主成分分析结果fact.b1<-factanal(bl,factor=2,method=umleM,rotation=HnoneH)#未淡转的因子分析fact.b1$loadings#输出不炎转的结果fact.b2<-factanal(bljactor=2,method=vmleM,rotation=Mvarimax,\scores=Hregression11)#淡转的因子分析fact.b2$loadings#输出炎转的结果apply((fact.b2$loadings)2J,sum)#计算共同度shapiro.test(fact.b2$scores)#检验正态性fact.b3<-factanal(bLfactor=2,method=,,mle,\rotation=Mvarimax,\scores=MBartlett1')b[order(fact.b3$scores[J],decreasing=F)J] #排名输岀结果及分析：(1J根据以上数据对女子径瘵项目作因子分析；bv-read.csvfdata42.c$vu) #4■入数据bl<-b[,-1]pc.b1<・princomp(b1,cor=T)summary(pc.bl)裹4主成分分析结果CompCompCompCompCompCompComp

.1.2.3.4.5.6.7Standard0.2320」970.1492.410.8080.5480.354deviation068Proportionof0.0070.0050.0030.830.0930.0430.018Variance762Cumulative0.9910.9961.0000.830.9230.9660.984Proportion280根据主成分分析的结果可以看出，在笫2个特征根处，累计贾伙率就已经达到了92.3%。因此，我们选用2个因孑进行因孑分析。fact.b1<-factanal(b1,factor=2,method=nmleM,rotation=”none”)fact.bl$loadingsfact.b2<-factanal(blJactor=2,method=nmleH,rotation=MvarimaxK,scores=nregression")fact.b2$loadingsapply((fact.b2$loadings)2J,sum)表5未炎转的因孑我椅Factor!Factor2X100.m..s.0.95-0.13X200.m..$.0.97-0.22X400.m..$.0.900

X800.m..min.0.830.38X1500.m..min.0.840.53X3000.m..min.0.840.49Marathon..min・0.800.40表6<44的因孑我椅Factor!Factor2CommunalitiesX100.m..