实验八 多元线性回归与逐步回归

1、实验八多元线性回归与逐步回归（2学时）一、实验目的和要求1.掌握逐步回归的思想与方法，掌握Matlab中stepwise命令的使用方法二、实验内容主要语句：逐步回归命令stepwise提供了交互式画面，可自由选择变量，进行统计分析，格式stepwise（X,Y,in,penter,premove）X是自变量数据,Y是因变量数据，分别为矩阵，in是矩阵X列数指标,给出初始模型中包括的子集,缺省时设定为全部自变量不在模型中,penter为变量进入时显著性水平，缺省时=0.05,premove为变量剔除时显著性水平，缺省=0.10.在应用stepwise命令进行运算时，程序不断提醒将某个变量加入（M

2、ovein）回归方程，或提醒将某变量从回归方程中剔除（Moveout）.注意：应用stepwise命令,数据矩阵X第一列不需人工加一个全1向量,程序会自动求出回归方程常数项（intercept）.实验数据与内容选取1989-2003年的全国统计数据，考虑的自变量包括：x1-工业总产值（亿元）；x2-农业总产值（亿元）；x3-建筑业总产值（亿元）；x4-社会商品零售总额（亿元）；x5-全民人口数（万人）；x6-受灾面积；y-国家财政收入（亿元）。数据见表3-20，（1）建立多元回归模型Y=p+px+Px+Px+Px+Px+Px+,求0112233445566回归参数的估计；（2）对上述回归模型和

3、回归系数进行检验（要写出统计量）；（3）用逐步回归求y与6个因素之间的回归关系式.表3-201989-2003年统计数据年份X1X2X3X4X5X6y19896484.004100.60794.008101.40112704.046991.002664.9019906858.004954.30859.408300.10114333.038474.002937.1019918087.105146.401015.109415.60115823.055472.003149.48199210284.505588.001415.0010993.70117171.051333.003483.3719931

4、4143.806605.102284.7012462.10118517.048829.004348.95199419359.609169.203012.6016264.70119850.055043.005218.10199524718.3011884.603819.6020620.00121121.045821.006242.20199629082.6013539.804530.5024774.10122389.046989.007407.99199732412.1013852.504810.6027298.90123626.053429.0086519014241

5、.905231.4029152.50124761.050145.009875.95199935087.2014106.205470.6031134.70125786.049981.0011444.08200039047.3013873.605888.0034152.60126743.054688.0013395.23200142374.6014462.806375.4037595.20127627.052215.0016386.04200245975.2014931.507005.0042027.10128453.047119.0018903.64200353092.9014870.10818

6、1.3045842.00129227.054506.0021715.25解：(1)建立多元回归模型建立多元线性回归模型y=B+Bx+Bx+Bx+Bx+80112233441）程序：data=19896484.004100.60794.008101.40112704.046991.002664.9019906858.004954.30859.408300.10114333.038474.002937.1019918087.105146.401015.109415.60115823.055472.003149.48199210284.505588.001415.0010993.70117171.0

7、51333.003483.37199314143.806605.102284.7012462.10118517.048829.004348.95199419359.609169.203012.6016264.70119850.055043.005218.10199524718.3011884.603819.6020620.00121121.045821.006242.20199629082.6013539.804530.5024774.10122389.046989.007407.99199732412.1013852.504810.6027298.90123626.053429.008651

8、9014241.905231.4029152.50124761.050145.009875.95199935087.2014106.205470.6031134.70125786.049981.0011444.08200039047.3013873.605888.0034152.60126743.054688.0013395.23200142374.6014462.806375.4037595.20127627.052215.0016386.04200245975.2014931.507005.0042027.10128453.047119.0018903.64200

9、353092.9014870.108181.3045842.00129227.054506.0021715.25;n,p=size(data);%读取行数n为样本数,列数p为回归参数个数x=ones(n,1),data(:,2:7);%建立设计矩阵，第一列全是1y=data(:,8);%读取Yb,bint,r,rint,stats=regress(y,x);%建立线性回归模型，输出回归参数b,回归参数b的置信区间，残差r,残差r的置信区间，输出几个统计量stats结果输出：b,bint,r,rint,stats结果：回归参数估计值b=1.0e+03*-6.92260.0001-0.00090.

10、00000.00060.0001-0.0000得0(-6.9226,0.0001,-0.0009,0.0000,0.0006,0.0001,-0.0000)t回归参数置信区间：bint=1.0e+04*-4.16302.7785-0.00010.0001-0.0001-0.0001-0.00030.00030.00000.0001-0.00000.0000-0.00000.0000得回归参数0的置信区间如上输出残差值r=-228.1801132.3052382.5207-382.0969-164.3261413.2697235.0416-64.6531-215.4275-83.5491-101

11、.2389-476.3158462.6145结果：99785.601208397862056071.90395192.4558-2.4199得到残差向量二(-228.1801,132.3052382.5207,-382.0969,-164.3261,413.2697,235.0416,-64.6531，-215.4275，-83.5491，-101.2389，-476.3158，462.6145，92.4558，-2.4199T输出随机误差项=(S,8,8)T的置信区间12nrint=1.0e+03*-0.70470.2484-0.38350.6481-0.17040.9354-1.06510

12、.3009-0.71590.3872-0.25251.0791-0.48820.9583-0.80890.6795-0.79990.3690-0.81920.6521-0.85530.6528-1.15400.2014-0.20461.1299-0.55190.7368-0.40230.3974输出统计量结果：stats=1.0e+05*0.00000.00620.00001.4152R2=0.99785接近1,相关性强，F=SSR/p=62056071F(6,1561),0SSE/(np1)0.05p=PF(p,p1)F=3.1585*10-5F（p,n-p-1）二4.1680模型显著00.

13、05回归参数检验结果：回归参数检验t统计量值T0=-0.02919355319091510.0281878798554941-0.5449042806666820.002027564977241090.2410645568925540.0411337557415811-0.0973456131842114T统计量上0.05分位数Ta=2.44691185114497T统计量检验P值Pp=0.97770.97840.60550.99840.81750.96850.9256检验P值p，表明每个变量对Y回归均不显著，需要进行模型改进。0k（3）逐步回归逐步回归方法1：初始模型选择全部自变量xx二da

14、ta（：,2：7）；%输入原始自变量数据，为xl-x7,第一列不需要1y=data（：,8）；%输入因变量Y数据stepwise(xx,y,1,2,3,4,5,6,0.05,0.1)%初始模型当前集为全部自变量集输出结果：先将所有变量放入模型中，第一步结果将x3移除;TH一步：第二步：x5移除;第三步：x6移除；没有移除和引入的，最优自变量集为x,x24最优回归方程为y=519.678-0.812016x+0.72372x24检验假设：H:p=p=0024选取统计量F=SSR/2SSE/(15-2-1)h0为真F(1,12)R2=0.996923，修正的复相关系数平方(拟合优度判别公式)R2=

15、0.99641均接a近1,自变量与因变量y线性关系显著,F统计量观测值F=1943.91F(p,n-p-1)=F(1,12)，检验p值00.050.05p=PF(p,n-p-1)F=8.48825-10-16ltI)2P(t(15-6-1)11I)0kH0k0kH00kp=0.00000.05,p=0.00000.05,0204拒绝H,H，认为Y与(x,x)线性回归显著.020424注意：C为(XX)-1对角元,pn(BQ2c),s(p)=5厂kkkkkkkkk逐步回归方法2：初始自变量集选定为空集程序及结果如下：stepwise(xx,y,0.05,0.1)%初始模型当前集为全部自变量集结果

16、：nX1X2X3X4X6X6StepwiseRegression文件(F)隅(E)TMCDStepwise黨面(D)窗口(W)轄助(H)CoefficientswithErrorBarsCoeff.t-statp-vaI0.38622712.14660.00001.178555.16690.00022.4448511.15430.00000.47089915.84120.00001.067948.82190.00000.3971531.11900.2834Nextstep:MoveX4inNextStepIntercept=9054.89R-square=0FNaNRMSE-6141.65Ad

17、jR-sq=0p=NaN123ModelHistory6143116142-*6141-6140111第一步X4进入当前集StepwiseRegression1416141514141413:-0.661951-2.95900.0119I-0.812016-13.41930.0000-2.87202-2.66350.02070.47089915.84120.0000-0.705312-2.48270.0288.*-0.010829-0.11990.9065CoefficientswithErrorBarsIntercept=-2188.13R-square=0.950747F=250.943RMSE=1414.47AdjR-sq=0.946958p=7.02947e/0第二步X2进入当前集QStepwiseRegressionAdjR-sq=0.99641p=8.48825e-16文件(F)爾(E)TM(T)Stepwise桌面(D)窗口(W)耕助(H)CoefficientswithErrorBarsCoeff.t-statp-va1X1*0.07497950.70420.4960Nextstep:Movenoterms2-0.812016-13.41930.00