试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1 .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

2、加入（Movein）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）.2 .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=Po+PiXi+P2X2+P3X3+P4X4+P5X5+P6X6+&,求回归参数的估计

3、；（2）对上述回归模型和回归系数进行检验（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X1X2X3X4X5X6y19896484.004100.60794.008101.40112704.046991.002664.9019906858.004954.30859.408300.10114333.038474.002937.1019918087.105146.401015.109415.60115823.055472.003149.48199210284.505588.001415.0010993.70117171.051333.00

4、3483.37199314143.806605.102284.7012462.10118517.048829.004348.95199419359.609169.203012.6016264.70119850.055043.005218.10199524718.3011884.603819.6020620.00121121.045821.006242.20199629082.6013539.804530.5024774.10122389.046989.007407.99199732412.1013852.504810.6027298.90123626.053429.008651.1419983

5、3387.9014241.905231.4029152.50124761.050145.009875.95199935087.2014106.205470.6031134.70125786.049981.0011444.08200039047.3013873.605888.0034152.60126743.054688.0013395.23200142374.6014462.806375.4037595.20127627.052215.0016386.04200245975.2014931.507005.0042027.10128453.047119.0018903.64200353092.9

6、014870.108181.3045842.00129227.054506.0021715.25解：（1）建立多元回归模型建立多元线性回归模型Y=：0.：lXi2X2-3X3.：4乂.；1)程序：data=19896484.004100.60794.008101.40112704.046991.002664.9019906858.004954.30859.408300.10114333.038474.002937.1019918087.105146.401015.109415.60115823.055472.003149.48199210284.505588.001415.0010993.70

7、117171.051333.003483.37199314143.806605.102284.7012462.10118517.048829.004348.95199419359.609169.203012.6016264.70119850.055043.005218.10199524718.3011884.603819.6020620.00121121.045821.006242.20199629082.6013539.804530.5024774.10122389.046989.007407.99199732412.1013852.504810.6027298.90123626.05342

8、9.0086519014241.905231.4029152.50124761.050145.009875.95199935087.2014106.205470.6031134.70125786.049981.0011444.08200039047.3013873.605888.0034152.60126743.054688.0013395.23200142374.6014462.806375.4037595.20127627.052215.0016386.04200245975.2014931.507005.0042027.10128453.047119.00189

9、03.64200353092.9014870.108181.3045842.00129227.054506.0021715.25;n,p=size(data);%读取彳T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

10、-0.00090.00000.00060.0001-0.0000得？=(-6.9226,0.0001,-0.0009,0.0000,0.0006,0.0001,-0.0000)T回归参数置信区问：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B的置信区间如上输出残差值r=-228.1801132.3052382.5207-382.0969-164.3261413.2697235.0416-64.6531-215.4275-8

11、3.5491-101.2389-476.3158462.614592.4558-2.4199得到残差向量?=(-228.1801,132.3052,382.5207,-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£=(&述2,，5)T的置信区问rint=1.0e+03*-0.70470.2484-0.38350.6481-0.17040.9354-1.06510.3009-0.71590.

12、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,=SSR/p=62056071.F0056,15-6-1,SSE/(n-p-1)p=PF(p,p-1)F。=3.1585*10-5<0.05，均说明自变量对y线性关系显著。；?2n14152401

13、212注意：stats转成长格式数据命令和结果：formatlonggstats结果：99785.601208397862056071.9039513.15856319868346e-0514152401212.9742绘制残差图命令:ResidualCaseOrderPlot68101214CaseNumber残差示意图rcoplot(r,rint)1000800600400sausR2000-200-400-600-800-1000残差示意图看出无异常点(2)对上述回归模型和回归系数进行显著性检验%求可决系数，进彳T相关性检验，y是因变量Y数据TSS=y'*(eye(n)-1/n*

14、ones(n,n)*y;%总离差平方和H=x*inv(x*x)*x'%帽子矩阵ESS=y'*(eye(n)-H)*y;%计算残差平方和RSS=y'*(H-1/n*ones(n,n)*y;%计算回归平方和MSE=RSS/n-p-1;%计算均方残差R2=RSS/TSS;%计算样本决定系数RSS/TSS，相关性检验%F佥验检验回归方程的显著性F0=(RSS/p)/(ESS/(n-p-1);%计算F0Fa=finv(0.95,p,n-p-1);%F分布时的临界值%t检验检验回归系数的显著性S=MSE*inv(x'*x);%计算回归参数的协方差矩阵T0=b./sqrt(d

15、iag(S);%每个回归参数的T统计量,b为上述回归参数返回结果Ta=tinv(0.975,n-p-1);%t分布的上1-0.05/2分位数pp=2-2*tcdf(abs(T0),n-p-1);%每个回归参数检验的T统计量对应的概率pp结果：TSS=5.2808e+08H=Columns1through60.69000.18140.25720.0049-0.12320.02950.18140.6753-0.08380.14150.1402-0.16300.2572-0.08380.51530.2866-0.04370.14920.00490.14150.28660.32870.19570.09

16、46-0.12320.1402-0.04370.19570.62250.26070.0295-0.16300.14920.09460.26070.3338-0.02670.2168-0.1760-0.05130.13700.09370.13070.0307-0.0756-0.1369-0.04020.0776-0.00360.00810.0637-0.0101-0.20790.09260.0338-0.13530.0506-0.04090.00550.0927-0.0757-0.07670.05210.04420.06690.0422-0.1563-0.03440.09510.1268-0.0

17、0190.0568-0.13310.06940.04310.1084-0.0573-0.05620.01270.1461-0.02020.0376-0.0762-0.21050.1783-0.1163-0.1138Columns7through12-0.13000.12170.1064-0.02670.1307-0.00360.0338-0.0757-0.15630.21680.03070.0081-0.1353-0.0767-0.0344-0.1760-0.07560.06370.05060.05210.0951-0.0513-0.1369-0.0101-0.04090.04420.1268

18、0.1370-0.0402-0.20790.00550.0669-0.00190.09370.07760.09260.09270.04220.05680.34300.23330.22340.05980.01060.02590.23330.35240.13910.26550.1600-0.05700.22340.13910.5619-0.0361-0.10060.23280.05980.2655-0.03610.36460.3135-0.02370.01060.1600-0.10060.31350.32980.03180.0259-0.05700.2328-0.02370.03180.26570

19、.0122-0.06150.1857-0.02990.04870.2537-0.08540.0132-0.14400.14250.23350.0884-0.0165-0.0315-0.0051Columns13through15-0.13310.01270.17830.06940.1461-0.11630.0431-0.0202-0.11380.10840.0376-0.1300-0.0573-0.07620.1217-0.0562-0.21050.10640.0122-0.0854-0.0165-0.06150.0132-0.03150.1857-0.1440-0.0051-0.02990.

20、1425-0.06270.04870.2335-0.08040.25370.08840.0962-0.0627-0.08040.09620.29330.21810.10520.21810.50980.13430.10520.13430.8141R2=0.997856015345558接近1,相关性显著.回归方程显著性F检验结果：F0=349.065936623445Fa=4.14680416227653表明F0=349.0659>F0.05(P,n-p-1)=4.1680模型显著回归参数检验结果：回归参数检验t统计量值T0=-0.02919355319091510.02818787985

21、54941-0.5449042806666820.002027564977241090.2410645568925540.0411337557415811-0.0973456131842114T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P0k,表明每个变量对Y回归均不显著，需要进行模型改进。(3)逐步回归逐步回归方法1：初始模型选择全部自变量xx=data(:,2:7);%输入原始自变量数据，为x1-x7,第一列不需要1y=data(:,8);%输入因变量Y数据s

22、tepwise(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第一步：先将所有变量放入模型中，第一步结果将x3移除;x5移除;E1写屈中。聃叼府尔on-n3S44RliEiMm-SwpwiieiUDiWftWitprficiertssrthE>wBa打TL白讪ifej-137?4ifl观T耳-I2WHrad;-1«277叩BE-3M有。.鳍HW1英由1Q|!»11IM】I皆3st20M1¥Cm"trUtp-nl矍77?WJ4W«Qlf打IQUMACa4N6b-tall口XE&ilE&呻第

23、三步：x6移除;“dHHislory物|J1第3«-i+J4jIJl输出结果：HStefmiM!口egfEMionsw百m工me时一，unpj叩阳»&hiQxiowtiwihEnrrEm.Cm樽LrELn"”l没有移除和引入的，最优自变量集为X2,X4最优回归方程为？=519.678-0.812016x20.72372x4H0为真F(1,12)检验假设：Ho：-2=N=0选取统计量F=一邳华一SSE/（15-2-1）R2=0.996923，修正的复相关系数平方（拟合优度判别公式）R；=0.99641均接近1,自变量与因变量y线性关系显著，F统计量观测值Fo=194