實(shí)驗(yàn)報(bào)告-因子分析(多元統(tǒng)計(jì))

PAGEPAGE2實(shí)驗(yàn)報(bào)告主成分分析（綜合性實(shí)驗(yàn)）(Principalcomponentanalysis)實(shí)驗(yàn)原理：主成分分析利用指標(biāo)之間的相關(guān)性，將多個(gè)指標(biāo)轉(zhuǎn)化為少數(shù)幾個(gè)綜合指標(biāo)，從而達(dá)到降維和數(shù)據(jù)結(jié)構(gòu)簡(jiǎn)化的目的。這些綜合指標(biāo)反映了原始指標(biāo)的絕大部分信息，通常表示為原始指標(biāo)的某種線性組合，且綜合指標(biāo)間不相關(guān)。利用矩陣代數(shù)的知識(shí)可求解主成分。實(shí)驗(yàn)題目一：將彩色膠卷在顯影液下處理后在不同情形下曝光，然后通過紅、綠、藍(lán)三種濾色片并在高、中、低三種密度下進(jìn)行測(cè)量，每個(gè)膠卷有高紅、高綠、高藍(lán)、中紅、…、低藍(lán)等九個(gè)指標(biāo)（分別記為X1－X9九個(gè)變量）。試驗(yàn)了108個(gè)膠卷，由數(shù)據(jù)已算得如下協(xié)差陣：(S2a1)17717995965332-7-4-3419245131181127-2143026010914244111581024243213796456128228343133393948實(shí)驗(yàn)要求：（1）試從協(xié)差陣出發(fā)進(jìn)行主成分分析；（2）計(jì)算方差累積貢獻(xiàn)率；（3）作Scree圖，并結(jié)合（2）的結(jié)果確定主成分的個(gè)數(shù)；（4）試對(duì)結(jié)果進(jìn)行解釋。實(shí)驗(yàn)題目二：下表中給出了不同國(guó)家及地區(qū)的男子徑賽記錄：(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（數(shù)據(jù)來源：1984年洛杉機(jī)奧運(yùn)會(huì)IAAF/AFT徑賽與田賽統(tǒng)計(jì)手冊(cè)）實(shí)驗(yàn)要求：（1）試求主成分，并對(duì)結(jié)果進(jìn)行解釋；（2）試用方差累積貢獻(xiàn)率和Scree圖確定主成分的個(gè)數(shù)；（3）計(jì)算各國(guó)第一主成分的得分并排名。實(shí)驗(yàn)題目一分析報(bào)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)#相關(guān)系數(shù)矩陣>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：根據(jù)相關(guān)系數(shù)矩陣可以看出:相關(guān)性較大,可以做主成分分析.（2）>sj.cov#協(xié)方差矩陣[,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#載荷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：由結(jié)果可得前三個(gè)樣本主成分提取的信息達(dá)83.44%（3）>screeplot(princomp.ap,type="lines")#Scree圖答：由Scree圖可以看出：坡度從第三個(gè)主成分開始逐漸變緩；由（2）的結(jié)果可得：從第三個(gè)主成分后開始累計(jì)貢獻(xiàn)率為83.44%>80%，結(jié)合兩者可以得出主成分的個(gè)數(shù)為3個(gè)。答：第一主成分表達(dá)式：y1=0.305x1+0.653x2+0.483x3+0.261x4+0.324x5+0.271x6第二主成分表達(dá)式：y2=0.486x1+0.151x2-0.588x3+0.491x4-0.495x5-0.373x6第三主成分表達(dá)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紅綠藍(lán)濾色片在膠卷曝光中應(yīng)用較少；第二主成分為對(duì)比成分，反映了中高密度的紅色濾色片對(duì)彩色膠卷曝光影響較大，運(yùn)用在風(fēng)景照或建筑攝影時(shí)，可以制作出“風(fēng)暴似”的云彩，月光及樹木，營(yíng)造出超寫實(shí)的特效，這種濾色片是絕對(duì)必要的，而中高密度的藍(lán)色濾色片強(qiáng)調(diào)山谷中的山嵐及穿透水面與煙霧的光線等方式，來達(dá)到凸顯天空層次的效果對(duì)彩色膠卷；第三主成分為綜合成分，反映了同樣是紅綠藍(lán)三種顏色濾色片，只有在高密度下才對(duì)膠卷曝光起到?jīng)Q定作用，密度越高，說明對(duì)紅綠藍(lán)三種顏色吸收越多，從而影響到照片的色彩。

實(shí)驗(yàn)題目二分析報(bào)告：所有的表格經(jīng)過Excel表格處理及保留小數(shù)點(diǎn)三位。（1）>AFT=read.csv("D:AFT.csv",head=T)#導(dǎo)入數(shù)據(jù)>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)#算出數(shù)據(jù)的方差>P.ap=cor(AFT1)#算出數(shù)據(jù)的協(xié)方差>P.ap>princomp.ap=princomp(AFT1,cor=T)#用相關(guān)系數(shù)做主成分分析>summary(princomp.ap)#匯總統(tǒng)計(jì)>princomp.ap$loadings#主成分分析Comp.1為綜合能力Comp.2為極限能力（即瞬間爆發(fā)力和耐力）

（2）>summary(princomp.ap)#匯總統(tǒng)計(jì)最后一列為方差累計(jì)貢獻(xiàn)率，可以取Comp.1和Comp.2為主成分。>screeplot(princomp.ap,type="lines")#畫出了scree圖根據(jù)圖可以得出，取Comp.1和Comp.2為主成分

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