实验报告-因子分析(多元统计)

PAGEPAGE2实验报告主成分分析（综合性实验）(Principalcomponentanalysis)实验原理：主成分分析利用指标之间的相关性，将多个指标转化为少数几个综合指标，从而达到降维和数据结构简化的目的。这些综合指标反映了原始指标的绝大部分信息，通常表示为原始指标的某种线性组合，且综合指标间不相关。利用矩阵代数的知识可求解主成分。实验题目一：将彩色胶卷在显影液下处理后在不同情形下曝光，然后通过红、绿、蓝三种滤色片并在高、中、低三种密度下进行测量，每个胶卷有高红、高绿、高蓝、中红、…、低蓝等九个指标（分别记为X1－X9九个变量）。试验了108个胶卷，由数据已算得如下协差阵：(S2a1)17717995965332-7-4-3419245131181127-2143026010914244111581024243213796456128228343133393948实验要求：（1）试从协差阵出发进行主成分分析；（2）计算方差累积贡献率；（3）作Scree图，并结合（2）的结果确定主成分的个数；（4）试对结果进行解释。实验题目二：下表中给出了不同国家及地区的男子径赛记录：(t8a6)Country100m(s)200m(s)400m(s)800m(min)1500m(min)5000m(min)10,000m(min)Marathon(mins)Argentina10.3920.8146.841.813.714.0429.36137.72Australia10.3120.0644.841.743.5713.2827.66128.3Austria10.4420.8146.821.793.613.2627.72135.9Belgium10.3420.6845.041.733.613.2227.45129.95Bermuda10.2820.5845.911.83.7514.6830.55146.62Brazil10.2220.4345.211.733.6613.6228.62133.13Burma10.6421.5248.31.83.8514.4530.28139.95Canada10.1720.2245.681.763.6313.5528.09130.15Chile10.3420.846.21.793.7113.6129.3134.03China10.5121.0447.31.813.7313.929.13133.53Colum10.4321.0546.11.823.7413.4927.88131.35CookIslands12.1823.252.942.024.2416.735.38164.7CostaRica10.9421.948.661.873.8414.0328.81136.58Czechoslovakia10.3520.6545.641.763.5813.4228.19134.32Denmark10.5620.5245.891.783.6113.528.11130.78DominicanRepublic10.1420.6546.81.823.8214.9131.45154.12Finland10.4320.6945.491.743.6113.2727.52130.87France10.1120.3845.281.733.5713.3427.97132.3German(D.R.)10.1220.3344.871.733.5613.1727.42129.92German(F.R.)10.1620.3744.51.733.5313.2127.61132.23GreatBrit.&N.Ireland10.1120.2144.931.73.5113.0127.51129.13Greece10.2220.7146.561.783.6414.5928.45134.6Guatemala10.9821.8248.41.893.814.1630.11139.33Hun10.2620.6246.021.773.6213.4928.44132.58India10.621.4245.731.763.7313.7728.81131.98Indonesia10.5921.4947.81.843.9214.7330.79148.83Ireland10.6120.9646.31.793.5613.3227.81132.35Israel10.712147.81.773.7213.6628.93137.55Italy10.0119.7245.261.733.613.2327.52131.08Japan10.3420.8145.861.793.6413.4127.72128.63Kenya10.4620.6644.921.733.5513.127.38129.75Korea10.3420.8946.91.793.7713.9629.23136.25D.P.RKorea10.9121.9447.31.853.7714.1329.67130.87Luxembourg10.3520.7747.41.823.6713.6429.08141.27Malaysia10.420.9246.31.823.814.6431.01154.1Mauritius11.1922.4547.71.883.8315.0631.77152.23Mexico10.4221.346.11.83.6513.4627.95129.2Netherl10.5220.9545.11.743.6213.3627.61129.02NewZealand10.5120.8846.11.743.5413.2127.7128.98Norway10.5521.1646.711.763.6213.3427.69131.48PapuaNewGuinea10.9621.7847.91.94.0114.7231.36148.22Philippines10.7821.6446.241.813.8314.7430.64145.27Poland10.1620.2445.361.763.613.2927.89131.58Portugal10.5321.1746.71.793.6213.1327.38128.65Rumania10.4120.9845.871.763.6413.2527.67132.5Singapore10.3821.2847.41.883.8915.1131.32157.77Sp10.4220.7745.981.763.5513.3127.73131.57Sweden10.2520.6145.631.773.6113.2927.94130.63Switzerland10.3720.4645.781.783.5513.2227.91131.2Taipei10.5921.2946.81.793.7714.0730.07139.27Thailand10.3921.0947.911.833.8415.2332.56149.9Turkey10.7121.4347.61.793.6713.5628.58131.5USA9.9319.7543.861.733.5313.227.43128.22USSR10.072044.61.753.5913.227.53130.55WesternSamoa10.8221.86492.024.2416.2834.71161.83（数据来源：1984年洛杉机奥运会IAAF/AFT径赛与田赛统计手册）实验要求：（1）试求主成分，并对结果进行解释；（2）试用方差累积贡献率和Scree图确定主成分的个数；（3）计算各国第一主成分的得分并排名。实验题目一分析报告：（1）>sj.cov<-matrix(c(177,179,95,96,53,32,-7,-4,-3,179,419,245,131,181,127,-2,1,4,95,245,302,60,109,142,4,4,11,96,131,60,158,102,42,4,3,2,53,181,109,102,137,96,4,5,6,32,127,142,42,96,128,2,2,8,-7,-2,4,4,4,2,34,31,33,-4,1,4,3,5,2,31,39,39,-3,4,11,2,6,8,33,39,48),9,9)>sj.cor<-cor(sj.cov)#相关系数矩阵>round(sj.cor,2)[,1][,2][,3][,4][,5][,6][,7][,8][,9][1,]1.000.840.620.780.660.44-0.84-0.80-0.80[2,]0.841.000.860.720.910.78-0.79-0.77-0.73[3,]0.620.861.000.430.780.91-0.68-0.68-0.59[4,]0.780.720.431.000.790.42-0.80-0.80-0.85[5,]0.660.910.780.791.000.85-0.79-0.79-0.77[6,]0.440.780.910.420.851.00-0.70-0.71-0.62[7,]-0.84-0.79-0.68-0.80-0.79-0.701.000.980.96[8,]-0.80-0.77-0.68-0.80-0.79-0.710.981.000.98[9,]-0.80-0.73-0.59-0.85-0.77-0.620.960.981.00答：根据相关系数矩阵可以看出:相关性较大,可以做主成分分析.（2）>sj.cov#协方差矩阵[,1][,2][,3][,4][,5][,6][,7][,8][,9][1,]17717995965332-7-4-3[2,]179419245131181127-214[3,]95245302601091424411[4,]961316015810242432[5,]5318110910213796456[6,]321271424296128228[7,]-7-24442343133[8,]-414352313939[9,]-3411268333948>pca.sj<-princomp(covmat=sj.cov)#主成分分析>pca.sj$loadings#载荷Loadings:Comp.1Comp.2Comp.3Comp.4Comp.5Comp.6Comp.7Comp.8Comp.9[1,]0.3050.4860.4120.2870.3580.4640.263[2,]0.6530.1510.183-0.648-0.105-0.270[3,]0.483-0.5880.2360.1150.425-0.3900.126[4,]0.2610.491-0.457-0.1080.461-0.350-0.346-0.121[5,]0.324-0.495-0.280-0.1590.1460.6740.259[6,]0.271-0.373-0.269-0.2340.1540.696-0.130-0.331-0.158[7,]-0.2560.438-0.8180.252[8,]-0.2660.5050.1470.257-0.761[9,]-0.2820.5580.1030.516-0.2960.483>summary(pca.sj)Importanceofcomponents:Comp.1Comp.2Comp.3Comp.4Standarddeviation29.639825014.003424211.3420944110.17006418ProportionOfVariance0.60923660.13598880.089211580.07172691CumulativeProportion0.60923660.74522550.834437050.90616396Comp.5Comp.6Comp.7Comp.8Standarddeviation9.014492656.152140602.6412843322.388822159ProportionOfVariance0.056353040.026247460.0048379910.003957331CumulativeProportion0.962516990.988764450.9936024430.997559774Comp.9Standarddeviation1.875847965ProportionOfVariance0.002440226CumulativeProportion1.000000000答：由结果可得前三个样本主成分提取的信息达83.44%（3）>screeplot(princomp.ap,type="lines")#Scree图答：由Scree图可以看出：坡度从第三个主成分开始逐渐变缓；由（2）的结果可得：从第三个主成分后开始累计贡献率为83.44%>80%，结合两者可以得出主成分的个数为3个。答：第一主成分表达式：y1=0.305x1+0.653x2+0.483x3+0.261x4+0.324x5+0.271x6第二主成分表达式：y2=0.486x1+0.151x2-0.588x3+0.491x4-0.495x5-0.373x6第三主成分表达式：y3=0.412x1+0.183x2+0.236x3-0.457x4-0.280x5-0.269x6-0.256x7-0.266x8-0.282x9第一主成分反映了低密度的红绿蓝滤色片在胶卷曝光中应用较少；第二主成分为对比成分，反映了中高密度的红色滤色片对彩色胶卷曝光影响较大，运用在风景照或建筑摄影时，可以制作出“风暴似”的云彩，月光及树木，营造出超写实的特效，这种滤色片是绝对必要的，而中高密度的蓝色滤色片强调山谷中的山岚及穿透水面与烟雾的光线等方式，来达到凸显天空层次的效果对彩色胶卷；第三主成分为综合成分，反映了同样是红绿蓝三种颜色滤色片，只有在高密度下才对胶卷曝光起到决定作用，密度越高，说明对红绿蓝三种颜色吸收越多，从而影响到照片的色彩。

实验题目二分析报告：所有的表格经过Excel表格处理及保留小数点三位。（1）>AFT=read.csv("D:AFT.csv",head=T)#导入数据>AFT=AFT[-1,]#去掉第一行缺失值>AFT1=AFT[,-1]#去掉第一列名字>Sigma.ap=var(AFT1)#算出数据的方差>P.ap=cor(AFT1)#算出数据的协方差>P.ap>princomp.ap=princomp(AFT1,cor=T)#用相关系数做主成分分析>summary(princomp.ap)#汇总统计>princomp.ap$loadings#主成分分析Comp.1为综合能力Comp.2为极限能力（即瞬间爆发力和耐力）

（2）>summary(princomp.ap)#汇总统计最后一列为方差累计贡献率，可以取Comp.1和Comp.2为主成分。>screeplot(princomp.ap,type="lines")#画出了scree图根据图可以得出，取Comp.1和Comp.2为主成分

（3）>y=princomp.ap$scores[,1]#计算主成分得分>y在Excel中整理并排序：CountryPointCookIslands10