應(yīng)用回歸分析課后題答案詳細(xì)版

課后題題目要改的地方：1.P265 10.9改成：對(duì)例10.7，根據(jù)輸出結(jié)果10.8，手工算出以下兩個(gè)樣品的預(yù)測(cè)概率：樣品號(hào)modeincome樣品號(hào)modeincome1charter7083.3322boat3750.0002.P26610.1110.910.8數(shù)據(jù)中兩個(gè)樣品的預(yù)測(cè)概率。2個(gè)樣品的取值如下：樣品號(hào)InflTypeCont1LowTowerLow2LowAtriumLow3.P266 10.12刪掉答案：第二章2.2

=

nnxiyii1 x21i1i12.7提示：n ny??ye?x?xi i i i 1i 1i1 i1n n?ex?xe02.9提示：vari

vari?i

1 iii1

1 ii10 1i ivarivar?i2covi,?i0 1i ii1ivaryvar?i1i

x2covy,y

x

x2

121

i

2 x 1x 1

i

x2n Lxx n 1 x

x2 i

2 n Lxx 2.14R代碼及部分輸出：>x=c(1:5)>x=c(1:5)>y=c(10,10,20,20,40)>#散點(diǎn)圖>plot(x,y,main="銷售收入與廣告費(fèi)用關(guān)系圖",xlab="廣告費(fèi)用",ylab="銷售收入")fit=lm(y~x) #summary(fit) #pCall:lm(formula=y~x)Residuals:1 2 3 4 54.000e+00-3.000e+00 5.004e-16-7.000e+00 6.000e+00Coefficients:EstimateErrortvaluePr(>|t|)(Intercept) -1.000 6.351 -0.157 x 7.000 1.915 3.656 0.0354*Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:6.0553degreesoffreedomMultipleR-squared: 0.8167, AdjustedR-squared: 0.7556F-statistic:13.36on1and3p-value:0.03535confint(fit) #2.5% 97.5%(Intercept)-21.211248519.21125x 0.906079313.09392anova(fit) #Response:yDfSumSqMeanSqFvalue Pr(>F)x 1 490 490.00 13.3640.03535*Residuals 3 36.67Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1>="two.side") #correlationdata: xandyt=3.6556,df=3,p-value=0.03535alternativehypothesis:truecorrelationisnotequalto095percentconfidenceinterval:>#prediction>#predictionyconfidencey的期望值的預(yù)測(cè)>new=data.frame(x=4.2)>pred_y=predict(fit,new)>pred_plim=predict(fit,new,interval="prediction")>pred_clim=predict(fit,new,interval="confidence")#殘差圖0.10572160.9936915sampleestimates:cor0.9036961>plot(x,fit$residuals,main="殘差圖")由輸出可知，估計(jì)方程為?17x，回歸標(biāo)準(zhǔn)誤差?

6.06，?置信水平10為95%的置信區(qū)間為置信水平為95%的置信區(qū)間為1010.91,13.09，決定系數(shù)為r2=0.82，調(diào)整后r2=0.760.05的顯著性水平下顯著不為0xy在0.05的顯著性水平下有高度顯著的線性依賴關(guān)系，14.20y028.40y0置信水平為95%的區(qū)間估計(jì)為6.06,50.74，E0置信水平為95%的區(qū)間估計(jì)為.0。本例樣本量n所以區(qū)間估計(jì)的誤差很大。2.15

5很小，R代碼與2.14類似。由輸出可知，估計(jì)方程為?

0.120.0036x，回歸標(biāo)準(zhǔn)10誤差?0.48，?置信水平為95%的置信區(qū)間為.4，?置信水平為10195%的置信區(qū)間為6，決定系數(shù)為r2=0，調(diào)整后r2=9，1y1000.00y03.70置信水平為95%的區(qū)間估計(jì)為9，0置信水平為95%的近似區(qū)間估計(jì)為6，E0置信水平為95%的區(qū)間估計(jì)為2。2.16R代碼：fit=lm(y~x) #fit=lm(y~x) #最小二乘估計(jì)hist(fit$residuals)qqnorm(fit$residuals,main="殘差qq圖")qqline(fit$residuals)（1）散點(diǎn)圖（略（2）?2x（3）第三章3.11R代碼及部分輸出：>x1=c(70,75,65,74,72,68,78,66,70,65)>x2=c(35,40,40,42,38,45,42,36,44,42)>x3=c(1.0,2.4,2.0,3.0,1.2,1.5,4.0,2.0,3.2,3.0)>y=c(160,260,210,265,240,220,275,160,275,250)>data1=data.frame(x1,x2,x3,y)>cor(data1) #fit1=lm(y~x1+x2+x3) #summary(fit1) #pCall:lm(formula=y~x1+x2+x3)Residuals:Min 1Q Median 3Q Max-25.198-17.035 2.627 11.677 33.225Coefficients:EstimateErrortvaluePr(>|t|)(Intercept)-348.280 176.459 -1.974 0.0959.x13.7541.9331.9420.1002x27.1012.8802.4650.0488*x312.44710.5691.1780.2835Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:23.446degreesoffreedomMultipleR-squared: 0.8055, AdjustedR-squared: 0.7083F-statistic:8.283on3and6p-value:0.01487fit2=lm(y~x1+x2) summary(fit2) #pCall:lm(formula=y~x1+x2)Residuals:Min 1Q Median 3Q Max-42.012-10.656 4.358 11.984 28.927Coefficients:EstimateStd.ErrortvaluePr(>|t|)(Intercept)-459.624 153.058 -3.003 0.01986*x14.6761.8162.575 0.03676*x28.9712.4683.634 0.00835**Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:24.087degreesoffreedomMultipleR-squared: 0.7605, AdjustedR-squared: 0.6921F-statistic:11.12on2and7p-value:0.006718confint(fit1) #2.5% 97.5%(Intercept)-780.060328783.499990x1-0.9766149 x20.052917914.148507x3-13.414748838.309689>data2=scale(data1)/sqrt(length(y)-1)>data2=data.frame(data2)>names(data2)=c("x1","x2","x3","y")fit3=lm(y~x1+x2+x3,data=data2) #summary(fit3) #pCall:lm(formula=y~x1+x2+x3,data=data2)Residuals:Min 1Q Median 3Q Max-0.19353-0.13083 0.02017 0.08969 0.25518Coefficients:ErrortvaluePr(>|t|)(Intercept)-1.928e-16 5.693e-02 0.000 x13.848e-011.982e-011.9420.1002x25.355e-012.172e-012.4650.0488*x32.771e-012.353e-011.1780.2835Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:0.186degreesfreedomMultipleR-squared: 0.8055, AdjustedR-squared: 0.7083F-statistic:8.283on3and6p-value:0.01487>newdata=data.frame(75,42,3.1)>names(newdata)=c("x1","x2","x3")>names(newdata)=c("x1","x2","x3")>pred_y=predict(fit2,newdata)>pred_plim=predict(fit2,newdata,interval="prediction")>pred_clim=predict(fit2,newdata,interval="confidence")>pred_y+2*24.08>pred_y-2*24.08RP0.28x017542時(shí)?0267.83，0置信水平為95%的區(qū)間估計(jì)是,.2，0置信水平95%的近似區(qū)間估計(jì)是1.9，本例樣本量n10E0置信水平為95%的區(qū)間估計(jì)為.9。3.12Call:lm(formula=y~x1+x2)Residuals:MinCall:lm(formula=y~x1+x2)Residuals:Min1Q Median3QMax-6589.0-2504.9 123.3 2110.110961.5Coefficients:EstimateStd.ErrortvaluePr(>|t|)(Intercept)-4490.3824 3797.2636 -1.1830.251Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:4055on20degreesoffreedomMultipleR-squared: 0.9993,AdjustedR-squared: 0.9993F-statistic:1.468e+04on2and20p-value:<2.2e-16x10.79830.6475 1.233 0.232x22.00140.1254 15.963 7.6e-13***x1系數(shù)0.80明顯不合理。第四章4.9R代碼及部分輸出：>>cor.test(x,abs(fit1$residuals),alternative="two.side",method="spearman") #相關(guān)系數(shù)顯著性檢驗(yàn)#殘差圖fit1=lm(y~x) #最小二乘估計(jì)>plot(x,fit1$residuals,main="殘差圖")Spearman'srankcorrelationrhodata: xandabs(fit1$residuals)S=16928,p-value=0.02091alternativehypothesis:truerhoisnotequalto0sampleestimates:rho0.3175294>parm=c(-2.0,-1.5,-1.0,-0.5,0,0.5,1.0,1.5,2.0)>loglike=-999999>for(iinparm){+ fit2=lm(y~x,weights=x^(-i))+ loglike1=logLik(fit2)+ if(loglike1>loglike){+ + loglike=loglike1}+}>m[1]1.5>fit2=lm(y~x,weights=x^(-m))>summary(fit2)Call:lm(formula=y~x,weights=x^(-m))WeightedResiduals:Min 1Q Median 3Q Max-0.017626-0.005384-0.002158 0.006423 0.015820Coefficients:ErrortvaluePr(>|t|)(Intercept)-0.6834635 0.2976891 -2.296 0.0258*x 0.0035571 0.0003582 9.9301.64e-13***Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:0.00791451degreesfreedomMultipleR-squared: 0.6591, AdjustedR-squared: 0.6524F-statistic: 98.6on1and51p-value:1.638e-13>plot(x,sqrt(x^(-m))*fit2$residuals,main="殘差圖") #殘差圖>cor.test(x,abs(sqrt(x^(-m))*fit2$residuals),alternative="two.side",method="spearman") #相關(guān)系數(shù)顯著性檢驗(yàn)Spearman'srankcorrelationrhodata: xandabs(sqrt(x^(-m))*fit2$residuals)S=26678,p-value=0.5898alternativehypothesis:truerhoisnotequalto0sampleestimates:rho-0.07555233>newy=sqrt(y)fit3=lm(newy~x) #>summary(fit3)Call:lm(formula=newy~x)Residuals:Min 1Q Median 3Q Max-1.39185-0.30576-0.03875 0.25378 0.81027Coefficients:ErrortvaluePr(>|t|)(Intercept)5.822e-01 1.299e-01 4.4814.22e-05x 9.529e-04 9.824e-05 9.6993.61e-13***Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:0.46451degreesoffreedomMultipleR-squared: 0.6485, AdjustedR-squared: 0.6416F-statistic:94.08on1and51p-value:3.614e-13>cor.test(x,abs(fit3$residuals),alternative="two.side",method="spearman")Spearman'srankcorrelationrhodata: xandabs(fit3$residuals)S=29120,p-value=0.2121alternativehypothesis:truerhoisnotequalto0sampleestimates:rho-0.1740042>fity=matrix(as.numeric(fit3$fitted.values),nrow=length(y))>newresi=y-fity^2>newresi=y-fity^2>newrsq=1-sum(newresi*newresi)/((length(y)-1)*(sd(y))^2)普通最小二乘?

0.830.0037x,R2

0.71，殘差圖略。0.32P0.021，存在異方差。m

m1.5

，得：?w

0.680.0036x計(jì)算出加權(quán)變換殘差w iw繪制加權(quán)變換殘差（略，w與i的等級(jí)相關(guān)系數(shù)為-0.076，P值為0.59，說明異方差已經(jīng)消除。但是加權(quán)最小二乘的R2

R2

0.705，說明加權(quán)最小二乘的效果并不好。y對(duì)因變量做變換y 得回歸方程?y

0.580.00095x，保存預(yù)測(cè)值?y-0.17，i iPR2優(yōu)于普通最小二乘的效果。4.13R代碼及部分輸出：

/SSTR2

0.710，#DW#DW檢驗(yàn)p-value:<2.2e-16#殘差圖F-statistic:1.165e+04on1and18DF,>plot(x,fit1$residuals,main="殘差圖")>library(lmtest)>dwtest(fit1,alternative="two.side")AdjustedR-squared: 0.9984MultipleR-squared: 0.9985,Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:0.09744on18degreesoffreedom0.176163 0.001632 107.93 <2e-16***x-0.151659-0.068633-0.003432 0.046715 0.184384Coefficients:ErrortvaluePr(>|t|)(Intercept)-1.434832 0.241956 -5.93 1.3e-05Max3QMedian1QMinfit1=lm(y~x) #最小二乘估計(jì)>summary(fit1)Call:lm(formula=y~x)Residuals:Durbin-Watsontestdata: fit1DW=0.66325,p-value=0.0001257alternativehypothesis:trueautocorrelationisnot0>rhohat=1-0.66325/2>newy=y[2:20]-rhohat*y[1:19]>newx=x[2:20]-rhohat*x[1:19]>fit2=lm(newy~newx)dwtest(fit2,alternative="two.side") #DW檢驗(yàn)Durbin-Watsontestdata: fit2DW=1.3597,p-value=0.0862alternativehypothesis:trueautocorrelationisnot0>summary(fit2)Call:lm(formula=newy~newx)Residuals:Min 1Q Median 3Q Max-0.107113-0.051913-0.000163 0.036810 0.130277Coefficients:ErrortvaluePr(>|t|)(Intercept)-0.300278 0.177646 -1.69 newx 0.172686 0.003475 49.69 <2e-16***Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:0.0729617offreedomMultipleR-squared: 0.9932, AdjustedR-squared: 0.9928F-statistic: 2469on1and17p-value:<2.2e-16>deltax=diff(x)>deltay=diff(y)>fit3=lm(deltay~deltax-1)>summary(fit3)Call:lm(formula=deltay~deltax-1)Residuals:alternativehypothesis:trueautocorrelationisnot0alternativehypothesis:trueautocorrelationisnot0#DW檢驗(yàn)>dwtest(fit3,alternative="two.side")Durbin-Watsontestdata: fit3DW=1.4617,p-value=0.2716F-statistic:927.9on1and18p-value:<2.2e-16AdjustedR-squared: 0.9799MultipleR-squared: 0.981,-0.10701-0.05247 0.01545 0.06961 0.15038Coefficients:EstimateStd.ErrortvaluePr(>|t|)deltax0.168833 0.005543 30.46 <2e-16***Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:0.0757618offreedomMax3Q1Q MedianMin普通最小二乘?

-1.430.18x,R2

0.999。DW=0.66<dL關(guān)。

1.120，P值為0.00013，存在正的序列相迭代法?

10.66325/2

0.668375，對(duì)自變量與因變量進(jìn)行變換后建立回歸模型?07DW1.36，P值為0.090.05的顯著性水平下可以認(rèn)為已不存在序列相關(guān)。還原為原始方程：?t

07t1+7t

0.67xt1差分法對(duì)自變量與因變量進(jìn)行變換后建立回歸模型?7x，此時(shí)DWP0.270.05的顯著性水平下可以認(rèn)為已不存在序列相關(guān)。還原為原始方程：?t

t1+7t

xt1在都消除了自相關(guān)的前提下，迭代法的擬合優(yōu)度更大，故迭代法較優(yōu)。4.14R4.13=0.75dL的序列相關(guān)。各種自回歸方法主要結(jié)果見下表：

1.50，存在正0 0 1 1 2 2 u自回歸方法 ? ?0 0 1 1 2 2 u

DW 第五章5.9

迭代法 0.63 — -178.84 1.44 1.72 257.90差分法 — — 0 210.12 1.40 2.04 281.00部分R代碼及輸出：>fit1=lm(y~x1+x2+x3+x4+x5+x6,data=data)>fit2=step(fit1,direction="backward")>summary(fit2)Call:lm(formula=y~x1+x2+x5,data=data)Residuals:Min 1Q Median 3Q Max-372.26-102.79 -7.77 157.98 313.69Coefficients:ErrortvaluePr(>|t|)(Intercept)874.60021 106.86563 8.1842.67e-07***x1-0.611190.12382-4.9360.000125***x2-0.353050.08840-3.9940.000940***x50.636710.089147.1431.65e-06***Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:183.117degreesoffreedomMultipleR-squared: 0.9958, AdjustedR-squared: 0.9951F-statistic: 1356on3and17p-value:<2.2e-16>fit3=step(fit1,direction="both")>summary(fit3)Call:lm(formula=y~x1+x2+x5,data=data)Residuals:Min 1Q Median 3Q Max-372.26-102.79 -7.77 157.98 313.69Coefficients:ErrortvaluePr(>|t|)(Intercept)874.60021 106.86563 8.1842.67e-07***x1-0.611190.12382-4.9360.000125***x2-0.353050.08840-3.9940.000940***x50.636710.089147.1431.65e-06***Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:183.117degreesoffreedomF-statistic: 1356on3andF-statistic: 1356on3and17p-value:<2.2e-16AdjustedR-squared: 0.9951MultipleR-squared: 0.9958,x4,x3,x6x1,x2,x5Rstep函數(shù)進(jìn)行逐步回歸，其結(jié)果與后退法一致。兩個(gè)方法的最終模型是：?

874.60-

0.35x2

0.64x5但是回歸系數(shù)的解釋不合理。5.10R代碼與5.9類似，不再贅述。略。x2,x3,x4,x6作為最終模型。Rstep函數(shù)進(jìn)行逐步回歸，其結(jié)果與后退法一致。RstepAICAIC信息SPSSFSPSSR軟件逐步回歸的第六章6.6R代碼及部分輸出：x11.00000000.99434690.99184900.90893530.99691160.5062195x11.00000000.99434690.99184900.90893530.99691160.5062195x20.99434691.00000000.99880770.87146540.99890100.5018967x30.99184900.99880771.00000000.86667410.99725510.4943296x40.90893530.87146540.86667411.00000000.89040460.5714781x50.99691160.99890100.99725510.89040461.00000000.5113394x60.50621950.50189670.49432960.57147810.51133941.0000000>kappa(cor(data[2:7]),exact=TRUE)x6x5x4x3x2x1x61.74265127.1773371860.726476319.4844772636.564359>cor(data[2:7])x5x4x3479.287849x2x1>library(car)>fit1=lm(y~x1+x2+x3+x4+x5+x6,data=data)>vif(fit1)[1]21642.62>fit2=lm(y~x1+x3+x4+x5+x6,data=data)>vif(fit2)x1 x3 x4 x6276.968819306.617361 11.605489632.895698 1.645146>fit3=lm(y~x1+x3+x4+x6,data=data)>vif(fit3)x1 x3 x4 160.5125801.539699>fit4=lm(y~x3+x4+x6,data=data)>vif(fit4)x3 x4 x64.0180874.5087061.484981>summary(fit4)Call:lm(formula=y~x3+x4+x6,data=data)Residuals:Min 1Q Median 3Q Max-628.53-109.44 -0.65 165.51 913.41Coefficients:ErrortvaluePr(>|t|)(Intercept)-2.296e+03 1.870e+03 -1.228 0.236x31.359e+009.681e-0214.0368.84e-11***x43.144e-021.906e-021.6490.117x63.701e-031.446e-020.2560.801Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:369.817degreesoffreedomMultipleR-squared: 0.983, AdjustedR-squared: 0.98F-statistic:328.2on3and17p-value:3.055e-15>fit5=lm(y~x3+x4,data=data)>vif(fit5)x3 x44.0180654.018065>summary(fit5)F-statistic:519.2on2andF-statistic:519.2on2and18p-value:<2.2e-16AdjustedR-squared: 0.9811MultipleR-squared: 0.983,Residualstandarderror:360.1on18degreesoffreedom0‘***’0.0010.01‘*’0.05‘.’0.1‘’10.0755.1.8863.304e-02 1.752e-021.359e+00 9.426e-02 14.415 ***x3x4Signif.codes:-643.15-105.66 -4.27 168.59 908.95Coefficients:ErrortvaluePr(>|t|)(Intercept)-2.307e+03 1.820e+03 -1.267 Max3Q1Q MedianMinCall:lm(formula=y~x3+x4,data=data)Residuals:方差擴(kuò)大因子=2636.5621642.62，x2，重新做回歸；剔除變量與常規(guī)的后退法及逐步回歸法剔除變量的結(jié)果會(huì)有較大的差別。重新做x3,x4兩x4P0.076x4只有較弱的顯著性。第七章7.5R代碼及部分輸出：data=read.table("question5_9.csv",head=TRUE,sep=",")data=data[-22,]data=read.table("question5_9.csv",head=TRUE,sep=",")data=data[-22,]data1=data[,c(2,3,6,8)]library(MASS)datas=data.frame(scale(data1)) #對(duì)樣本數(shù)據(jù)進(jìn)行標(biāo)準(zhǔn)化處理并轉(zhuǎn)換為數(shù)據(jù)框的格式存儲(chǔ)ridge1=lm.ridge(y~.-1,data=datas,lambda=seq(0,3,0.1)) #做嶺回歸對(duì)于標(biāo)準(zhǔn)化后的數(shù)據(jù)模型不包含截距項(xiàng)，其中l(wèi)ambda為嶺參數(shù)k的所有取值beta=coef(ridge1) #將所有不同嶺參數(shù)所對(duì)應(yīng)的回歸系數(shù)的結(jié)果賦給betabeta #beta###畫嶺跡線legend(2.5,3,inset=0.5,legend=c("x1","x2","x5"),cex=1,pch=char,lty=linetype)添加圖例#創(chuàng)建沒有plot(k,k,type="n",xlab="k",ylab="嶺回歸系數(shù)",ylim=c(-2,4))任何點(diǎn)和線的圖形區(qū)域linetype<-c(1:3)char<-c(18:20)for(iin1:3)lines(k,beta[,i],type="o",lty=linetype[i],pch=char[i],cex=0.75)#k#繪制嶺跡圖k=ridge1$lambda用標(biāo)準(zhǔn)化嶺回歸系數(shù)繪制嶺跡圖，可以看到當(dāng)嶺參數(shù)取k量的嶺估計(jì)已經(jīng)基本平穩(wěn)。則此時(shí)一般的嶺回歸方程為：

0.20時(shí)，三個(gè)自變?

752.540.051x1+0.081x2

0.10x5各系數(shù)的估計(jì)合理。7.6普通最小二乘?

5377.001.22x0.98x，其中回歸系數(shù)?=0.98明顯不合理。當(dāng)嶺參數(shù)取k

2 3 3時(shí)，兩個(gè)自變量的嶺估計(jì)已經(jīng)基本平穩(wěn)，且各系數(shù)的估計(jì)合理，此時(shí)嶺回歸方程為：7.7

?

6981.701.091x2

1.093x3采用后退法與逐步回歸法，得回歸方程?

-0.970.04x1

0.15x2-0.029x4，其中x4的系數(shù)是負(fù)數(shù)不合理，說明仍然存在共線性。當(dāng)嶺參數(shù)取k

20時(shí)，三個(gè)自變量的嶺估計(jì)已經(jīng)基本平穩(wěn)，且各系數(shù)的估計(jì)合理，此時(shí)嶺回歸方程為：?

0.0740.015x1

0.15x2

0.0066x4用y對(duì)x1,x2,x3做嶺回歸，當(dāng)嶺參數(shù)取k15時(shí)，三個(gè)自變量的嶺估計(jì)已經(jīng)基本平穩(wěn)，且各系數(shù)的估計(jì)合理，此時(shí)嶺回歸方程為：?

-0.540.015x1

0.15x2

0.072x3回歸系數(shù)都能有合理解釋。第八章8.3R代碼及部分輸出如下：>pr1=princomp(~x1+x2+x3+x4,data=data,cor=T)>pr1=princomp(~x1+x2+x3+x4,data=data,cor=T)>summary(pr1,loadings=TRUE)Importanceofcomponents:Comp.2 Comp.3 Standarddeviation 1.4952271.25541470.431979340.0402957285ProportionofVariance0.5589260.39401650.046651540.0004059364CumulativeProportion 0.5589260.95294250.999594061.0000000000Loadings:Comp.2Comp.3Comp.4x1 0.476 0.509 0.676 0.241x2 0.564-0.414-0.3140.642x3-0.394-0.605 0.6380.268x4-0.548 0.451-0.1950.677>score=pr1$scores[,1:4]>scoreComp.1Comp.2 Comp.3 Comp.41-1.52714951.9807424-0.55164170 0.0401032542-2.22304100.2480864-0.30203549-0.03105098731.17600650.1913854-0.01115003-0.0975269224-0.68684101.6411586 0.18652107-0.03446860150.37341400.5032822-0.77034369 0.01997058061.00611040.1768834 0.08920187-0.01266390070.9687086-2.2219875-0.18004962 0.0086341048-2.3232829-0.7199137 0.47849150 0.0235288039-0.3658690-1.4907279-0.03285325-0.046824608101.7304296 1.9027433 0.88594988 0.020646667-1.7071534-1.3479961 0.514357240.03267074112 1.7617078-0.4082655-0.020618970.03870358913 1.8169600-0.4553910-0.285828800.038277280>fit1=lm(y~score[,1]+score[,2])>fit1Call:lm(formula=y~score[,1]+score[,2])Coefficients:(Intercept) score[,1] score[,2]95.4231 9.4954 -0.1201>fit2=lm(score[,1]~x1+x2+x3+x4)>fit2Call:lm(formula=score[,1]~x1+x2+x3+x4)Coefficients:(Intercept) x1 x2 x4-0.66874 0.08422 0.03772 -0.06404 -0.03407>fit3=lm(score[,2]~x1+x2+x3+x4)>fit3Call:lm(formula=score[,2]~x1+x2+x3+x4)Coefficients:(Intercept) x1 x2 x40.97648 0.09006 -0.02769 -0.09831 0.02806>fit4=lm(y~x1+x2+x3+x4)>fit5=step(fit4,direction="both")>summary(fit5)Call:lm(formula=y~x1+x2+x4)Residuals:Min 1Q Median 3Q Max-3.0919-1.8016 0.2562 1.2818 3.8982Coefficients:EstimateStd.ErrortvaluePr(>|t|)(Intercept)71.648314.14245.0660.000675***x11.45190.117012.4105.78e-07***x20.41610.18562.2420.051687.x4-0.23650.1733-1.3650.205395Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:2.3099degreesoffreedomMultipleR-squared: 0.9823, AdjustedR-squared: 0.9764F-statistic:166.8on3and9p-value:3.323e-08y對(duì)前兩個(gè)主成分做最小二乘回歸，得主成分回歸的方程：?21-2分別以兩個(gè)主成分Factor1和Factor2做因變量，以四個(gè)原始變量為自變量做線性回歸，所得的回歸系數(shù)就是所需要的線性組合的系數(shù)。Factor1Factor2

-0.670.084x10.980.090x1

0.037x2-0.028x2

-0.064x3-0.098x3

-0.034x40.028x4還原后的主成分回歸方程為?=88.960.791逐步回歸法得到的回歸方程為

0.36x2

-0.60x3

-0.33x4?=71.651451

0.42x2

-0.24x48.4>datanew=scale(data)>datanewx1 x2>datanew=scale(data)>datanewx1 x2x3x4y[1,]-0.07846099-1.42368840-0.9007209 1.7923096-1.12492615[2,]-1.09845379-1.23089726 0.5044037 1.3143603-1.40411237[3,] 0.60153422 0.50422298-0.5884710-0.5974365 0.59007490[4,] 0.60153422-1.10236984-0.5884710 1.0156421-0.52002268[8,]-1.09845379-1.10236984 1.5972783 -1.52376360[9,]-0.92845499 0.37569555 0.9727785-0.4779492-0.15442168[10,] 2.30152223-0.07415044-1.2129708-0.2389746 [11,]-1.09845379-0.52399643 1.7534033 0.2389746-0.77261973[12,] 0.60153422 1.14686010-0.4323460-1.0753857 1.18833108[13,] 0.43153542 1.27538753-0.5884710-1.0753857 0.92908673attr(,"scaled:center")[5,]-0.078460990.24716813-0.9007209 0.03170246[6,] 0.601534220.43995926-0.4323460-0.47794920.91579215[7,]-0.758456191.46817866 0.8166536-1.43384770.48371824x1 x2 x3 x4 y7.46153848.15384611.76923130.00000095.423077attr(,"scaled:scale")x1 x2 x3 x4 y5.88239415.560881 6.40512616.73818015.043723>datanew=data.frame(datanew)>library(pls)>fit1=lm(y~.,data=datanew)>summary(fit1)Call:lm(formula=y~.,data=datanew)Residuals:Min 1Q Median 3Q Max-0.211050.01667 0.09162 0.26094Coefficients:EstimateStd.ErrortvaluePr(>|t|)(Intercept)2.641e-16 4.510e-020.0001.0000x16.065e-01 2.0830.0708.x25.277e-01 0.7050.5009x34.339e-02 0.1350.8959x4-1.603e-01 -0.2030.8441Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:0.16268degreesoffreedomMultipleR-squared: 0.9824, AdjustedR-squared: 0.9736F-statistic:4and8p-value:4.756e-07>pls1=plsr(y~.,data=datanew,validation="LOO",jackknife=TRUE,method="widekernelpls")summary(pls1,what="all") #RMSEPData: X134Ydimension:131Fitmethod:widekernelplsNumberofcomponentsconsidered:4VALIDATION:RMSEPCross-validatedusing13leave-one-outsegments.(Intercept) 1comps 2comps 3comps 4compsCV1.0410.26440.2239 0.17510.1937adjCV1.0410.25610.2059 0.17320.1910TRAINING:%varianceexplained1comps2comps3comps4compsX55.8962.1299.96100.00y96.7898.1698.2198.24>pls2=plsr(y~.,data=datanew,ncomp=3,validation="LOO",jackknife=TRUE)>coef(pls2)#>pls2=plsr(y~.,data=datanew,ncomp=3,validation="LOO",jackknife=TRUE)>coef(pls2)#得到方程的回歸系數(shù),,3compsyx1 0.51398740x2 0.28126396x3-0.05967267x4-0.42014716程為?=62.411551

0.51x2+0.10x3

-0.14x4從系數(shù)上看可以發(fā)現(xiàn)明顯不合理的地方，x3y從使用了所有主成分進(jìn)行回歸所得到的結(jié)果可以看出，主成分個(gè)數(shù)為3個(gè)RMSEP3均高于98%，因此將回歸的主成分個(gè)數(shù)定為m準(zhǔn)化后的數(shù)據(jù)的回歸方程為：

3。由以上結(jié)果可知，對(duì)于標(biāo)y*=0.51x*0.28x*-0.060x*-0.42x*還原為原始變量為：

1 2 3 4?

85.501.31x1

0.27x2-0.14x3-0.38x4x1,x2yx3,x4y第九章9.2R代碼及部分輸出如下：>x=c(1000,2000,3000,3500,4000,4500,5000)>x=c(1000,2000,3000,3500,4000,4500,5000)(Intercept)5.843e+00 1.324e+004.4140.0116*x-8.651e-04 9.710e-04-0.8910.4233I(x^2)4.468e-07 1.589e-072.8120.0482*>y=c(5.2,6.5,6.8,8.1,10.2,10.3,13.0)>plot(x,y,main="廢品率與生產(chǎn)率關(guān)系圖",xlab="生產(chǎn)率",ylab="廢品率>y=c(5.2,6.5,6.8,8.1,10.2,10.3,13.0)>plot(x,y,main="廢品率與生產(chǎn)率關(guān)系圖",xlab="生產(chǎn)率",ylab="廢品率")>fit=lm(y~x+I(x^2))>summary(fit)Call:lm(formula=y~x+I(x^2))Residuals:1234567-0.2246 0.6000-0.4690-0.1886 0.6683-0.6981 0.3120Coefficients:EstimateStd.ErrortvaluePr(>|t|)Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:0.65144degreesoffreedomMultipleR-squared: 0.9617, AdjustedR-squared: 0.9425F-statistic:50.16on2and4p-value:0.001479.3R代碼及部分輸出：

5.840.0087x4.47107x2，也可以使用指數(shù)曲線。<2e-16***<2e-16***69.12<2e-16***0.03714-103.830.087956.07991xnew1-0.104434-0.025434 0.000215 0.035054 0.058886Coefficients:EstimateStd.ErrortvaluePr(>|t|)(Intercept)-3.85598Max3QMedian1QMin>ynew1=log(data[,2])>xnew1=1/data[,1]>fit1=lm(ynew1~xnew1)>summary(fit1)Call:lm(formula=ynew1~xnew1)Residuals:F-statistic: 4778on1andF-statistic: 4778on1and13p-value:<2.2e-16>fit2=nls(y~a*exp(b/x),data=data,start=list(a=1,b=1))>summary(fit2)Formula:y~a*exp(b/x)Parameters:EstimateStd.ErrortvaluePr(>|t|)a0.0213415 0.0006207 34.383.77e-14***b6.0605474 0.0444621 136.31 <2e-16***Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:0.00963113degreesfreedomNumberofiterationstoconvergence:8Achievedconvergencetolerance:3.153e-08AdjustedR-squared: 0.9971MultipleR-squared: 0.9973,Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:0.04783on13degreesoffreedom乘性誤差項(xiàng)：加性誤差項(xiàng)：9.4R代碼及部分輸出：

6.08。6.06。Signif.Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:0.1682on17degreesoffreedom0.080327 -23.042.93e-14-0.264374 0.007045 -37.53 <2e-16***t-0.33166-0.05046-0.00611 0.10499 0.34985Coefficients:EstimateStd.ErrortvaluePr(>|t|)(Intercept)-1.850663Max3Q1Q MedianMin>t=data[,2]>newy1=log(1/data[,3]-0.01)>fit1=lm(newy1~t)>summary(fit1)Call:lm(formula=newy1~t)Residuals:u91.062002.0353144.741 <2e-16***b00.211250.027957.5581.15e-06***b10.726750.0122059.561 <2e-16***MultipleR-squared: 0.9881,AdjustedR-squared: 0.9874F-statistic: 1408on1andMultipleR-squared: 0.9881,AdjustedR-squared: 0.9874F-statistic: 1408on1and17p-value:<2.2e-16>fit2=nls(y~1/(1/u+b0*(b1^t)),data=data,start=list(u=100,b0=1,b1=0.5))>summary(fit2)Formula:y~1/(1/u+b0*(b1^t))Parameters:EstimateStd.ErrortvaluePr(>|t|)Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:2.3116degreesoffreedomNumberofiterationstoconvergence:8Achievedconvergencetolerance:1.759e-06（2）u9.5

0.16,b1

0.77

0.73R代碼及部分輸出：0.05571 14.3781.15e-120.05571 14.3781.15e-12***0.80094newk1.43706 -1.244 0.2265(Intercept)-1.78798-0.126800-0.022682 0.004224 0.036238 0.077387Coefficients:EstimateStd.ErrortvaluePr(>|t|)Max3QMedian1QMin>data=read.table("question9_5.csv",head=TRUE,sep=",")>data=data[-26,]>newgdp=log(data[,4])>newk=log(data[,5])>newl=log(data[,6])>fit1=lm(newgdp~newk+newl)>summary(fit1)Call:lm(formula=newgdp~newk+newl)Residuals:newl 0.40197 0.17044 2.358 0.0277*Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:0.0483422offreedomMultipleR-squared: 0.9939, AdjustedR-squared: 0.9934F-statistic: 1807on2and22p-value:<2.2e-16>fit2=nls(GDP~a*(K^b)*(L^c),data=data,start=list(a=1,b=0.5,c=0.5))>summary(fit2)Formula:GDP~a*(K^b)*(L^c)Parameters:EstimateStd.ErrortvaluePr(>|t|)a 0.40742 0.88525 0.46 0.650b 0.86834 0.06625 13.117.16e-12***c 0.26984 0.24299 0.279Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:627on22degreesfreedomNumberofiterationstoconvergence:12Achievedconvergencetolerance:5.681e-07>library(lmtest)dwtest(fit1,alternative="two.side") #DW檢驗(yàn)Durbin-Watsontestdata: fit1DW=0.71527,p-value=3.09e-05alternativehypothesis:trueautocorrelationisnot0>rhohat=1-0.71527/2>newgdp1=newgdp[2:25]-rhohat*newgdp[1:24]>newk1=newk[2:25]-rhohat*newk[1:24]>newl1=newl[2:25]-rhohat*newl[1:24]>fit3=lm(newgdp1~newk1+newl1)>summary(fit3)Call:lm(formula=newgdp1~newk1+newl1)Residuals:Min 1Q Median 3Q Max-0.110842-0.010241 0.004829 0.018324 0.055785>>plot(k,k,type="n",xlab="嶺數(shù)k",ylab="嶺回歸系",ylim=c(0.4,1.15)) 創(chuàng)建沒任何點(diǎn)和線的圖形區(qū)域>linetype=c(1:2)>char=c(18:19)>for(iin1:2)lines(k,beta1[,i+1],type="o",lty=linetype[i],pch=char[i],cex=0.75) #畫嶺跡線legend(6.5,0.8,inset=0.5,legend=c("ln(K)","ln(L)"),cex=1,pch=char,lty=linetype) #添加圖例#k>k=ridge1$lambda13.0247213.02472>kappa(cor(data.frame(newgdp,newk,newl)),exact=TRUE)[1]870.646>library(MASS)>ridge1=lm.ridge(newgdp~newk+newl,lambda=seq(0,8,0.2))>beta1=coef(ridge1)newlnewkF-statistic:393.2on2and21p-value:<2.2e-16>vif(fit1)AdjustedR-squared: 0.9715MultipleR-squared: 0.974,Residualstandarderror:0.03491on21degreesoffreedom0‘***’0.0010.01‘*’0.05‘.’0.1‘’10.0144*2.6690.197140.05849 12.4263.81e-11***0.726820.52613newk1newl1Signif.codes:0.63538 -1.423 0.1693(Intercept)-0.90439EstimateStd.ErrortvaluePr(>|t|)Coefficients:（1）A

0.40。（2）A

0.27。（3）DW

0.72A

0.53。（4）兩個(gè)自變量的方差擴(kuò)大因子皆大于13，且條件數(shù)大于870，存在嚴(yán)重多重共線性。取嶺回歸參數(shù)=5A9.6R代碼及部分輸出：

1.13。>library(car)>library(car)>data=read.table("question9_5.csv",head=TRUE,sep=",")>data=data[-26,]>t=data[,2]>newgdp=log(data[,4])>newk=log(data[,5])>newl=log(data[,6])>fit1=lm(newgdp~newk+newl+t)>summary(fit1)Call:lm(formula=newgdp~newk+newl+t)Residuals:Min 1Q Median 3Q Max-0.036638-0.010917 0.001029 0.016708 0.044192Coefficients:ErrortvaluePr(>|t|)(Intercept) 5.154834 1.014673 5.0804.96e-05***newk 0.460063 0.045620 10.0851.67e-09***newl -0.027374 0.091963 -0.298 0.769t 0.041839 0.004622 9.0521.08e-08***Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:0.0223521offreedomMultipleR-squared: 0.9988, AdjustedR-squared: 0.9986F-statistic: 5664on3and21p-value:<2.2e-16>fit2=nls(GDP~a*(K^b)*(L^c)*exp(d*t),data=data,start=list(a=1,b=0.5,c=0.5,d=0.1),control=list(maxiter=150))>summary(fit2)Formula:GDP~a*(K^b)*(L^c)*exp(d*t)Parameters:EstimateStd.ErrortvaluePr(>|t|)a2.024e+02 1.820e+02 1.112 0.279b3.951e-01 4.401e-02 8.9771.24e-08***c2.452e-03 8.593e-02 0.029 0.978d4.577e-02 3.695e-03 12.388***Signif.codes: 0‘***’0.0010.01‘*’0.05‘.’0.1‘’1Residualstandarderror:218.621degreesoffreedomNumberofiterationstoconvergence:125Achievedconvergencetolerance:5.418e-06>library(lmtest)dwtest(fit1,alternative="two.side") #DW檢驗(yàn)Durbin-Watsontestdata: fit1DW=1.2847,p-value=0.01241alternativehypothesis:trueautocorrelationisnot0>rhohat=1-1.2847/2>newgdp1=newgdp[2:25]-rhohat*newgdp[1:24]>newt1=data[2:25,2]-rhohat*data[1:24,2]>newk1=newk[2:25]-rhohat*newk[1:24]>newl1