北京工业大学、薛毅、数学模型作业八、作业8、实验八

1、实验八解：假设是一元函数> x<-c(27, 37, 45, + 21, 38, 49, + 22 , 33 , 48 ,+ 24 , 35 , 40 ,+ 25 ,30, 42 ,+ 23 ,31 , 55 ,+ 20 ,37 , 54 ,+ 20, 39 ,57 ,+ 29 , 46, 52 ,+ 24 , 49 , 53 ,+ 25, 40 , 56 ,+ 28 , 42 ,52 ,+ 26 , 43, 50 ,+ 38, 46 , 59 ,+ 32 , 43 , 50 ,+ 33 , 44 ,52 ,+ 31, 46, 58 ,+ 34 , 47 , 57 )> y&

2、lt;-c( 73, 78, 92,+ 66, 87, 80,+ 63 , 76 , 70,+ 75 , 79 ,90,+ 71 ,73, 85,+ 70, 80 , 76,+ 65 , 68 ,71,+ 70 , 75, 99,+ 79 ,89 , 86,+ 72, 101 , 79,+ 68 , 70 ,92,+ 67 , 72, 85,+ 79 ,80 , 71,+ 91, 83 ,90,+ 76 , 75, 91,+ 69 , 71 , 100,+ 66 ,80 ,80,+ 73 ,96, 109)> lm.sol<-lm(y 1+x)> summary(lm.sol

3、)Call:lm(formula = y 1 + x)Residuals: Min 1Q Median 3Q Max -16.4786 -5.7877 -0.0784 5.6117 19.7813 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 56.15693 3.99367 14.061 < 2e-16 *x 0.58003 0.09695 5.983 2.05e-07 *-Signif. codes: 0 * 0.001 * 0.01 * 0.05 . 0.1 1Residual standard

4、error: 8.146 on 52 degrees of freedomMultiple R-squared: 0.4077, Adjusted R-squared: 0.3963 F-statistic: 35.79 on 1 and 52 DF, p-value: 2.05e-07回归方程为Y=56.15693+0.58003X残差分析调用程序influence.measures(lm.sol)op <- par(mfrow=c(2,2), mar=0.4+c(4,4,1,1), oma= c(0,0,2,0)plot(lm.sol, 1:4)par(op)savePlot(&qu

5、ot;diagnoses1", type="eps"求解> influence.measures(lm.sol)Influence measures of lm(formula = y 1 + x) : dfb.1_ dfb.x dffit cov.r cook.d hat inf1 0.027214 -0.02243 0.03032 1.083 4.68e-04 0.0409 2 0.003183 -0.00145 0.00661 1.060 2.22e-05 0.0195 3 -0.029836 0.07939 0.18517 1.005 1.70e-0

6、2 0.0227 4 -0.076295 0.06742 -0.07917 1.111 3.19e-03 0.0674 5 0.061536 -0.02067 0.15157 1.012 1.14e-02 0.0189 6 0.040347 -0.06466 -0.10164 1.060 5.23e-03 0.0311 7 -0.184151 0.16135 -0.19250 1.085 1.87e-02 0.0623 8 0.009884 -0.00685 0.01374 1.066 9.62e-05 0.0246 9 0.105757 -0.18107 -0.30521 0.949 4.4

7、7e-02 0.0286 10 0.136895 -0.11756 0.14584 1.081 1.08e-02 0.0529 11 0.028475 -0.01721 0.04634 1.058 1.09e-03 0.0215 12 0.044116 0.00680 0.18260 0.990 1.64e-02 0.0185 13 0.008942 -0.00759 0.00964 1.093 4.74e-05 0.0486 14 -0.010312 0.00798 -0.01243 1.073 7.88e-05 0.0315 15 0.005591 0.01609 0.07753 1.04

8、8 3.05e-03 0.0194 16 0.014728 -0.01278 0.01553 1.103 1.23e-04 0.0574 17 0.100191 -0.07528 0.12548 1.049 7.95e-03 0.0289 18 0.219347 -0.29053 -0.36164 1.002 6.37e-02 0.0522 19 -0.094697 0.08434 -0.09767 1.116 4.85e-03 0.0728 *20 -0.081263 0.03705 -0.16867 1.003 1.41e-02 0.0195 21 0.279442 -0.37726 -0

9、.48138 0.919 1.08e-01 0.0480 22 0.076977 -0.06856 0.07939 1.118 3.21e-03 0.0728 *23 -0.020794 0.00320 -0.06391 1.050 2.07e-03 0.0186 24 -0.207717 0.26679 0.31908 1.043 5.04e-02 0.0615 25 0.120973 -0.09595 0.14131 1.053 1.01e-02 0.0344 26 -0.027571 0.05906 0.12054 1.042 7.32e-03 0.0244 27 0.004210 -0

10、.00597 -0.00811 1.083 3.35e-05 0.0404 28 0.053294 -0.04577 0.05678 1.095 1.64e-03 0.0529 29 -0.150454 0.24110 0.37902 0.907 6.73e-02 0.0311 30 0.117387 -0.16206 -0.21286 1.047 2.27e-02 0.0441 31 -0.069500 0.05898 -0.07496 1.088 2.86e-03 0.0486 32 -0.038643 -0.00595 -0.15994 1.005 1.27e-02 0.0185 33

11、-0.065121 0.08485 0.10338 1.094 5.43e-03 0.0567 34 -0.116498 0.09434 -0.13261 1.061 8.89e-03 0.0375 35 -0.010710 -0.03082 -0.14851 1.015 1.10e-02 0.0194 36 0.017430 -0.02471 -0.03358 1.082 5.75e-04 0.0404 37 0.192393 -0.16107 0.21059 1.049 2.22e-02 0.0446 38 0.000191 -0.00556 -0.01936 1.060 1.91e-04

12、 0.0202 39 0.149684 -0.22807 -0.33848 0.951 5.49e-02 0.0339 40 0.090694 -0.03047 0.22339 0.959 2.42e-02 0.0189 41 -0.000719 0.00154 0.00314 1.066 5.04e-06 0.0244 42 0.009147 -0.01147 -0.01331 1.120 9.03e-05 0.0720 *43 0.019922 -0.01444 0.02615 1.067 3.48e-04 0.0266 44 0.001067 -0.03103 -0.10809 1.03

13、8 5.89e-03 0.0202 45 -0.060180 0.09169 0.13608 1.054 9.34e-03 0.0339 46 -0.089190 0.06179 -0.12396 1.041 7.74e-03 0.0246 47 0.017322 -0.07109 -0.19692 0.992 1.91e-02 0.0213 48 -0.186149 0.26392 0.35865 0.964 6.19e-02 0.0404 49 -0.139760 0.10502 -0.17504 1.029 1.53e-02 0.0289 50 0.012644 -0.02708 -0.

14、05528 1.060 1.55e-03 0.0244 51 0.224064 -0.28417 -0.33444 1.048 5.53e-02 0.0666 52 -0.036395 0.02379 -0.05428 1.059 1.50e-03 0.0229 53 -0.075865 0.14224 0.26115 0.969 3.31e-02 0.0263 54 -0.441352 0.56686 0.67798 0.856 2.06e-01 0.0615 *> op <- par(mfrow=c(2,2), mar=0.4+c(4,4,1,1), oma= c(0,0,2,

15、0)> plot(lm.sol, 1:4)> par(op)> savePlot("diagnoses1", type="eps")由结果可以看出残差分布不满足齐性要求，但满足正态性要求。BOX-COX变换运行结果从结果中可以看出变换后满足方差齐次性要求。解：（1）R程序计算，结果如下多元线性回归方程为Y=3.2022+0.9626X1-2.6317X2-0.1305X3（2）从（1）的结果可以看出X2、X3是不显著的（3）调用influence.measures()并做回归诊断图Influence measures of lm(for

16、mula = Y X1 + X2 + X3, data = CO) : dfb.1_ dfb.X1 dfb.X2 dfb.X3 dffit cov.r cook.d hat1 -0.01011 -0.229978 0.227795 -0.026603 -0.265129 1.369 1.81e-02 0.17042 -0.18606 0.111515 -0.116101 0.200001 0.285155 1.298 2.09e-02 0.14703 1.35169 0.385890 -1.006688 -0.708960 -3.607345 0.348 2.09e+00 0.50754 0.

17、05858 -0.146496 0.124351 -0.037965 0.255995 1.152 1.66e-02 0.08355 0.01317 0.051085 -0.017883 -0.042885 -0.166624 1.361 7.22e-03 0.13726 0.04372 0.006790 0.002269 -0.045017 0.058121 1.384 8.86e-04 0.12587 0.07955 0.116438 -0.096514 -0.083547 -0.169070 1.392 7.44e-03 0.15368 -0.02218 0.025875 -0.0295

18、93 0.024407 -0.048721 1.331 6.22e-04 0.09089 0.02304 0.013969 -0.006324 -0.024218 0.044365 1.343 5.16e-04 0.098410 0.13980 0.039331 -0.011771 -0.140748 0.182836 1.333 8.68e-03 0.128311 -0.02071 0.043869 -0.052965 0.025914 -0.106701 1.247 2.96e-03 0.059712 -0.12842 -0.317944 0.295547 0.087973 -0.3617

19、76 1.451 3.36e-02 0.231513 -0.02611 0.041803 -0.074973 0.069632 0.199859 1.221 1.03e-02 0.085514 0.65773 -0.186247 0.293378 -0.749715 -0.850437 1.086 1.72e-01 0.253415 -0.06777 0.043641 -0.046433 0.066325 -0.095610 1.420 2.39e-03 0.152616 -0.82651 -0.180981 0.316932 0.578157 -1.252465 0.861 3.46e-01

20、 0.293317 0.44679 -0.435454 0.596855 -0.576211 0.975134 0.757 2.10e-01 0.199918 -0.00860 -0.021747 0.021597 0.006152 0.025316 1.639 1.68e-04 0.258519 0.16940 0.581731 -0.536291 -0.102562 0.667438 1.053 1.07e-01 0.192520 -0.07659 0.108255 -0.043715 0.007981 -0.382218 1.117 3.64e-02 0.118821 -0.05395

21、0.036015 -0.023968 0.060163 0.249927 1.015 1.55e-02 0.049922 -0.00025 -0.000407 0.000244 0.000376 0.000975 1.341 2.50e-07 0.093423 -0.02481 -0.092551 0.066026 0.047544 0.205327 1.167 1.08e-02 0.069124 -0.09465 -0.104258 0.075679 0.087361 -0.225293 1.156 1.29e-02 0.073225 -0.86523 0.250664 -0.380502

22、0.978918 1.120363 0.721 2.71e-01 0.2261 inf1 2 3 *4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 *19 20 21 22 23 24 25 分析回归诊断结果， 3号点和18号点是强影响点。第1张残差图，可以认为残差的方差满足齐性。第2张正态QQ图，除3号点基本都在一条直线上，除2号点外残差满足正态性。第3张标准的平方根与预测值的散点图，3号点的值大于1.5，说明3号点可能是异常值点。第4张图Cook距离值，3号点Cook距离最大，说明3号点是强影响点。剔除3号点后的回归诊断结果：可以看到所有结果均有所改善（4）逐

23、步回归分析> lm.sol<-lm(Y X1+X2+X3, data=CO)> summary(lm.sol)Call:lm(formula = Y X1 + X2 + X3, data = CO)Residuals: Min 1Q Median 3Q Max -2.89261 -0.78269 0.00428 0.92891 2.45082 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 3.2022 3.4618 0.925 0.365464 X1 0.9626 0.2422 3.974

24、0.000692 *X2 -2.6317 3.9006 -0.675 0.507234 X3 -0.1305 3.8853 -0.034 0.973527 -Signif. codes: 0 * 0.001 * 0.01 * 0.05 . 0.1 1Residual standard error: 1.446 on 21 degrees of freedomMultiple R-squared: 0.9186, Adjusted R-squared: 0.907 F-statistic: 78.98 on 3 and 21 DF, p-value: 1.329e-11> > lm.step<-step(lm.sol)Start: AIC=22.07Y X1 + X2 + X3 Df Sum of Sq RSS AIC- X3 1 0.002 43.895 20