PCA主成分分析應(yīng)用舉例

PCA主成分分析應(yīng)用舉例例1a=c(177,179,95,96,53,32,-7,-4,-3,179,419,245,131,181,127,-2,1,4,95,245,302,60,109,142,4,4,11,96,131,60,153,102,42,4,3,2,53,181,109,102,137,96,4,5,6,32,127,142,42,96,128,2,2,8,-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)s=matrix(a,ncol=9)S為樣本方差陣求方差陣S的特征值和特征向量c=eigen(s)c樣本前3個(gè)主成分的系數(shù)是：rho=diag(1/(sqrt(diag(s))))%*%s%*%diag(1/(sqrt(diag(s))))rho例2學(xué)生身體各指標(biāo)的主成分分析.隨機(jī)抽取30名某年級(jí)中學(xué)生,測(cè)量其身高(X1)、體重(X2)、胸圍(X3)和坐高(X4)。試對(duì)中學(xué)生身體指標(biāo)數(shù)據(jù)做主成分分析.30名中學(xué)生的四項(xiàng)身體指標(biāo)####用數(shù)據(jù)框形式輸入數(shù)據(jù)student<-data.frame(X1=c(148,139,160,149,159,142,153,150,151,139,140,161,158,140,137,152,149,145,160,156,151,147,157,147,157,151,144,141,139,148),X2=c(41,34,49,36,45,31,43,43,42,31,29,47,49,33,31,35,47,35,47,44,42,38,39,30,48,36,36,30,32,38),X3=c(72,71,77,67,80,66,76,77,77,68,64,78,78,67,66,73,82,70,74,78,73,73,68,65,80,74,68,67,68,70),X4=c(78,76,86,79,86,76,83,79,80,74,74,84,83,77,73,79,79,77,87,85,82,78,80,75,88,80,76,76,73,78))>cor(student)X1X2X3X4X11.00000000.86316210.73211190.9204624X20.86316211.00000000.89650580.8827313X30.73211190.89650581.00000000.7828827X40.92046240.88273130.78288271.0000000>eigen(cor(student))$values[1]3.541098000.313383160.079408950.06610989$vectors[,1][,2][,3][,4][1,]-0.49696610.5432128-0.44962710.5057471[2,]-0.5145705-0.2102455-0.4623300-0.6908436[3,]-0.4809007-0.72462140.17517650.4614884[4,]-0.50692850.36829410.7439083-0.2323433

>####作主成分分析>student.pr<-princomp(student,cor=TRUE)>>####并顯示分析結(jié)果summary(student.pr,loadings=TRUE)Importanceofcomponents:Comp.1Comp.2Comp.3Comp.4Standarddeviation1.88178050.559806360.281795940.25711844ProportionofVariance0.88527450.078345790.019852240.01652747CumulativeProportion0.88527450.963620290.983472531.00000000Loadings:Comp.1Comp.2Comp.3Comp.4X1-0.4970.543-0.4500.506X2-0.515-0.210-0.462-0.691X3-0.481-0.7250.1750.461X4-0.5070.3680.744-0.232PRINCOMP過程由相關(guān)陣出發(fā)進(jìn)行主成分分析.由相關(guān)陣的特征值可以看出，第一主成分的奉獻(xiàn)率已高達(dá)88.53%；且前二個(gè)主成分的累計(jì)奉獻(xiàn)率已達(dá)96.36%.因此只須用兩個(gè)主成分就能很好地概括這組數(shù)據(jù).另由第三和四個(gè)特征值近似為0，可以得出這4個(gè)標(biāo)準(zhǔn)化后的身體指標(biāo)變量(Xi*,i=1,2,3,4)有近似的線性關(guān)系(即所謂共線性),如

0.505747X1*-0.690844X2*+0.461488X3*-0.232343X4*≈c(常數(shù)).由最大的兩個(gè)特征值對(duì)應(yīng)的特征向量可以寫出第一和第二主成分：Z1=-0.4970X1*-0.5146X2*-0.4809X3*-0.5069X4*

Z2=0.5432X1*-0.2102X2*-0.7246X3*+0.3683X4*

第一和第二主成分都是標(biāo)準(zhǔn)化后變Xi*(i=1,2,3,4)的線性組合，且組合系數(shù)就是特征向量的分量.利用特征向量各分量的值可以對(duì)各主成分進(jìn)行解釋.第一大特征值對(duì)應(yīng)的第一個(gè)特征向量的各個(gè)分量值均在0.5附近,它反映學(xué)生身材的魁梧程度.身體高大的學(xué)生,他的4個(gè)部位的尺寸都比較大;而身體矮小的學(xué)生,他的4個(gè)部位的尺寸都比較小.因此我們稱第一主成分為大小因子.第二大特征值對(duì)應(yīng)的特征向量中第一(即身高X1的系數(shù))和第四個(gè)分量(即坐高X4的系數(shù))為正值,而第二(即體重X2的系數(shù))和第三個(gè)分量(即胸圍X3的系數(shù))為負(fù)值,它反映學(xué)生的胖瘦情況,故稱第二主成分為胖瘦因子.###各樣本的主成分的值>predict(student.pr)

Comp.1Comp.2Comp.3Comp.4[1,]0.06990950-0.23813701-0.35509248-0.266120139[2,]1.59526340-0.718473990.32813232-0.118056646[3,]-2.847931510.38956679-0.09731731-0.279482487[4,]0.759969880.80604335-0.04945722-0.162949298[5,]-2.739667770.017180870.360126150.358653044[6,]2.105831680.322843930.18600422-0.036456084[7,]-1.42105591-0.060531650.21093321-0.044223092[8,]-0.82583977-0.78102576-0.275577980.057288572[9,]-0.93464402-0.58469242-0.088141360.181037746[10,]2.36463820-0.365321990.088404760.045520127[11,]2.837419160.348758410.03310423-0.031146930[12,]-2.608512240.21278728-0.333980370.210157574[13,]-2.44253342-0.16769496-0.46918095-0.162987830[14,]1.866306690.050213840.37720280-0.358821916[15,]2.81347421-0.31790107-0.03291329-0.222035112[16,]0.063929830.207184480.043343400.703533624[17,]-1.55561022-1.70439674-0.331264060.007551879[18,]1.07392251-0.067634180.022836480.048606680[19,]-2.521742120.972743010.12164633-0.390667991[20,]-2.140723770.022178810.374109720.129548960[21,]-0.796244220.163078870.12781270-0.294140762[22,]0.28708321-0.35744666-0.039621160.080991989[23,]-0.251510751.25555188-0.556173250.109068939[24,]2.057060320.78894494-0.265521090.388088643[25,]-3.08596855-0.057753180.62110421-0.218939612[26,]-0.163675550.043179320.244818500.560248997[27,]1.372650530.02220972-0.23378320-0.257399715[28,]2.160977780.137332330.355897390.093123683[29,]2.40434827-0.48613137-0.16154441-0.007914021[30,]0.502874680.14734317-0.20590831-0.122078819------------------------------聚類分析biplot(student.pr)##畫出第一主成分和第二主成分的樣本散點(diǎn)圖可以看出那些學(xué)生屬于高大魁魁梧，如25；哪些同學(xué)屬于身材瘦小的，如11和15。細(xì)高的同學(xué)，如23;矮胖的同學(xué)，如17.####畫碎石圖screeplot(student.pr)

####用數(shù)據(jù)框的形式輸入數(shù)據(jù)conomy<-data.frame(x1=c(149.3,161.2,171.5,175.5,180.8,190.7,202.1,212.4,226.1,231.9,239.0),x2=c(4.2,4.1,3.1,3.1,1.1,2.2,2.1,5.6,5.0,5.1,0.7),x3=c(108.1,114.8,123.2,126.9,132.1,137.7,146.0,154.1,162.3,164.3,167.6),y=c(15.9,16.4,19.0,19.1,18.8,20.4,22.7,26.5,28.1,27.6,26.3))

>conomyx1x2x3y1149.34.2108.115.92161.24.1114.816.43171.53.1123.219.04175.53.1126.919.15180.81.1132.118.86190.72.2137.720.47202.12.1146.022.78212.45.6154.126.59226.15.0162.328.110231.95.1164.327.611239.00.7167.626.3>####作線性回歸>lm.sol<-lm(y~x1+x2+x3,data=conomy)>summary(lm.sol)Call:lm(formula=y~x1+x2+x3,data=conomy)Residuals:Min1QMedian3QMax-0.52367-0.389530.054240.226440.78313Coefficients:EstimateStd.ErrortvaluePr(>|t|)(Intercept)-10.127991.21216-8.3556.9e-05***x1-0.051400.07028-0.7310.488344x20.586950.094626.2030.000444***x30.286850.102212.8070.026277*---Signif.codes:0‘***’0.001‘**’0.01‘*’0.05‘.’0.1‘’1Residualstandarderror:0.4889on7degreesoffreedomMultipleR-squared:0.9919,AdjustedR-squared:0.9884F-statistic:285.6on3and7DF,p-value:1.112e-07得到回歸方程Y=-10.12799-0.05140X1+0.58695X2+0.28685X3Y是進(jìn)口量，X1是國內(nèi)總產(chǎn)值，而對(duì)應(yīng)系數(shù)的符號(hào)卻是負(fù)的，也就是說，國內(nèi)生產(chǎn)總值越高，其進(jìn)口量越少，這與實(shí)際情況是不相符的。其原因是三個(gè)變量存在共線性關(guān)系。####作主成分分析conomy.pr<-princomp(~x1+x2+x3,data=conomy,cor=T)summary(conomy.pr,loadings=TRUE)Importanceofcomponents:Comp.1Comp.2Comp.3Standarddeviation1.4139150.99907670.0518737839ProportionofVariance0.6663850.33271810.0008969632CumulativeProportion0.6663850.99910301.0000000000Loadings:Comp.1Comp.2Comp.3x1-0.7060.707x2-0.999x3-0.707-0.707前兩個(gè)主成分已經(jīng)到達(dá)99%的累計(jì)奉獻(xiàn)率。第一主成分椒關(guān)于國內(nèi)總產(chǎn)值和總消費(fèi)，因此稱為產(chǎn)銷因子，第二主成分只與存儲(chǔ)量有關(guān)，稱為存儲(chǔ)因子。

pre<-predict(conomy.pr)conomy$z1<-pre[,1]conomy$z2<-