試驗八多元線性回歸與逐步回歸

1、實驗八多元線性回歸與逐步回歸（2學時）一、實驗目的和要求1.掌握逐步回歸的思想與方法，掌握Matlab中stepwise命令的使用方法.二、實驗內容1 .主要語句：逐步回歸命令stepwise提供了交互式畫面，可自由選擇變量，進行統(tǒng)計分析，格式stepwise（X,Y,in,penter,premove）X是自變量數(shù)據(jù)，Y是因變量數(shù)據(jù)，分別為矩陣,in是矩陣X列數(shù)指標，給出初始模型中包括的子集，缺省時設定為全部自變量不在模型中，penter為變量進入時顯著性水平，缺省時=0.05,premove為變量剔除時顯著性水平，缺省=0.10.在應用stepwise命令進行運算時，程序不斷提醒將某個變量

2、加入（Movein）回歸方程，或提醒將某變量從回歸方程中剔除（Moveout）.注意：應用stepwise命令，數(shù)據(jù)矩陣X第一列不需人工加一個全1向量，程序會自動求出回歸方程常數(shù)項（intercept）.2 .實驗數(shù)據(jù)與內容選取1989-2003年的全國統(tǒng)計數(shù)據(jù)，考慮的自變量包括：x1-工業(yè)總產(chǎn)值（億元）；x2-農(nóng)業(yè)總產(chǎn)值（億元）；x3-建筑業(yè)總產(chǎn)值（億元）；x4-社會商品零售總額（億元）；x5-全民人口數(shù)（萬人）；x6-受災面積；y-國家財政收入（億元）。數(shù)據(jù)見表3-20,（1）建立多元回歸模型Y=Po+PiXi+P2X2+P3X3+P4X4+P5X5+P6X6+&,求回歸參數(shù)的估計

3、；（2）對上述回歸模型和回歸系數(shù)進行檢驗（要寫出統(tǒng)計量）；（3）用逐步回歸求y與6個因素之間的回歸關系式.表3-201989-2003年統(tǒng)計數(shù)據(jù)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）建立多元回歸模型建立多元線性回歸模型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);%讀取彳T數(shù)n為樣本數(shù)，列數(shù)p為回歸參數(shù)個數(shù)x=ones(n,1),data(:,2:7);%建立設計矩陣，第一列全是1y=data(:,8);%讀取Yb,bint,r,rint,stats=regress(y,x);%建立線性回歸模型，輸出回歸參數(shù)b,回歸參數(shù)b的置信區(qū)間，殘差r,殘差r的置信區(qū)間，輸出幾個統(tǒng)計量stats結果輸出：b,bint,r,rint,stats結果：回歸參數(shù)估計值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回歸參數(shù)置信區(qū)問：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得回歸參數(shù)B的置信區(qū)間如上輸出殘差值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得到殘差向量?=(-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輸出隨機誤差項£=(&述2,，5)T的置信區(qū)問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輸出統(tǒng)計量結果：stats=1.0e+05*0.00000.00620.00001.4152R2=0.99785接近1,相關性強,=SSR/p=62056071.F0056,15-6-1,SSE/(n-p-1)p=PF(p,p-1)F。=3.1585*10-5<0.05，均說明自變量對y線性關系顯著。；?2n14152401

13、212注意：stats轉成長格式數(shù)據(jù)命令和結果：formatlonggstats結果：99785.601208397862056071.9039513.15856319868346e-0514152401212.9742繪制殘差圖命令:ResidualCaseOrderPlot68101214CaseNumber殘差示意圖rcoplot(r,rint)1000800600400sausR2000-200-400-600-800-1000殘差示意圖看出無異常點(2)對上述回歸模型和回歸系數(shù)進行顯著性檢驗%求可決系數(shù)，進彳T相關性檢驗，y是因變量Y數(shù)據(jù)TSS=y'*(eye(n)-1/n*

14、ones(n,n)*y;%總離差平方和H=x*inv(x*x)*x'%帽子矩陣ESS=y'*(eye(n)-H)*y;%計算殘差平方和RSS=y'*(H-1/n*ones(n,n)*y;%計算回歸平方和MSE=RSS/n-p-1;%計算均方殘差R2=RSS/TSS;%計算樣本決定系數(shù)RSS/TSS，相關性檢驗%F僉驗檢驗回歸方程的顯著性F0=(RSS/p)/(ESS/(n-p-1);%計算F0Fa=finv(0.95,p,n-p-1);%F分布時的臨界值%t檢驗檢驗回歸系數(shù)的顯著性S=MSE*inv(x'*x);%計算回歸參數(shù)的協(xié)方差矩陣T0=b./sqrt(d

15、iag(S);%每個回歸參數(shù)的T統(tǒng)計量,b為上述回歸參數(shù)返回結果Ta=tinv(0.975,n-p-1);%t分布的上1-0.05/2分位數(shù)pp=2-2*tcdf(abs(T0),n-p-1);%每個回歸參數(shù)檢驗的T統(tǒng)計量對應的概率pp結果：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,相關性顯著.回歸方程顯著性F檢驗結果：F0=349.065936623445Fa=4.14680416227653表明F0=349.0659>F0.05(P,n-p-1)=4.1680模型顯著回歸參數(shù)檢驗結果：回歸參數(shù)檢驗t統(tǒng)計量值T0=-0.02919355319091510.02818787985

21、54941-0.5449042806666820.002027564977241090.2410645568925540.0411337557415811-0.0973456131842114T統(tǒng)計量上0.05分位數(shù)Ta=2.44691185114497T統(tǒng)計量檢驗P值Pp=0.97770.97840.60550.99840.81750.96850.9256檢驗P值P0k,表明每個變量對Y回歸均不顯著，需要進行模型改進。(3)逐步回歸逐步回歸方法1：初始模型選擇全部自變量xx=data(:,2:7);%輸入原始自變量數(shù)據(jù)，為x1-x7,第一列不需要1y=data(:,8);%輸入因變量Y數(shù)據(jù)s

22、tepwise(xx,y,1,2,3,4,5,6,0.05,0.1)%初始模型當前集為全部自變量集第一步：先將所有變量放入模型中，第一步結果將x3移除;x5移除;E1寫屈中。聃叼府爾on-n3S44RliEiMm-SwpwiieiUDiWftWitprficiertssrthE>wBa打TL白訕ifej-137?4ifl觀T耳-I2WHrad;-1«277叩BE-3M有。.鰭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輸出結果：HStefmiM!口egfEMionsw百m工me時一，unpj叩陽»&hiQxiowtiwihEnrrEm.Cm樽LrELn"”l沒有移除和引入的，最優(yōu)自變量集為X2,X4最優(yōu)回歸方程為？=519.678-0.812016x20.72372x4H0為真F(1,12)檢驗假設：Ho：-2=N=0選取統(tǒng)計量F=一邳華一SSE/（15-2-1）R2=0.996923，修正的復相關系數(shù)平方（擬合優(yōu)度判別公式）R；=0.99641均接近1,自變量與因變量y線性關系顯著，F(xiàn)統(tǒng)計量觀測值Fo=194