环境数据分析课件 (30).pdf

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1、Chapter 9 Regression AnalysisSee the future by mathematical model Contents Regression Analysis Linear Regression Goodness of Fit Non-linear Regression Adequancy of the FittingSSeSStSSr yi x x yiyy y yxbyertSSSSSSTotal sum of squares(Total variation)Explained sum of squares(Regression)Unexplained Sum

2、 of Squares(Residual)2222)()()()(xxbxbxbyySSriii222)()()()()(yyyyyyyyyyyyiiiiANOVAAdequancy of the FittingGoodness of Fit-R2A measure of the“goodness of fit”of the estimated equation,R2R2=Explained Variation/Total Variation=Total Variation-unexplained Variation/Total Variation=SST-SSe/SST=SSr/SSTFor

3、 simple linear regression,the R2:teiiiitrSSSSyyyyyyyyyySSSSR1)()()()()(222222102 Rperfectly lousy scorePerfect scoreSST=SSrSSe=0SST=SSeSSr=00Residual y Residual nnyxyxyx,.,2211yyiGoodness of Fit-R211)1(1)1()()1()(1)1/()/(12222knnRnyyknyynSSknSSeRiiTAdjk:the number of independent X;n:the number of sa

4、mplesFor multiple regression,the adjusted R2:even negative22RRAdjGoodness of Fit-R2For example 1,the R2:95.01762.51675.91462506.3926175)91625.5755086(0.796)(95.0938.17602645796.0)()(2222222222iiiiiitryyyxxxbyyxxbSSSSRxy796.0511.6Goodness of Fit-R2545.05.1)697.01(146)0882.00615.01(111)1(1545.06/088.0

5、4/027.01111)1()()1()(12222knnRknnSSSSnyyknyyRtrAdj21082.00.0370.109xxyGoodness of Fit-R2F-testGoodness of Fit-F-test F-test is an important determinant of model fit.0.:210pbbbHThe alternative hypothesis is at least one b differ from 0No statistically significant relationshipStatistically significant

6、 relationship does existpxxxy0.0021Goodness of Fit-F-testF 0.05F0.05(k,n-k-1)F F0.01(k,n-k-1),p F0.01(k,n-k-1),p 0.01,obviously significant linear relationshipModelSSdfMSFP valueRegressionSSrkSSr/k*ResidualSSen-k-1SSe/n-k-1TotalSSTn-1ANOVA table)1(knSSkSSFer)1(knSSkSSFerk:the number of independent X

7、(simple linear regression,k=1);n:the number of samplesF-value calculated as follows:For example 1,the F statistic:761.277141676.440)/-(1760.9381676.44014)()1116(1SSrSStSSrSSSSFerObviously significant linear relationship*862.8)14,1(01.0FGoodness of Fit-F-test For example 2,the F statistic:52.4027.002

8、7.0088.02)127(2erSSSSFMultiple regression not satisfied 6.94(2,4)50.0FGoodness of Fit-F-testT-testGoodness of Fit-T test T-test is an important determinant of model fit-Coefficients.),.2,1(0:),.2,1(0:10pibHpibHiibiiSbt t-value calculated as follows:k:the number of independent X(simple linear refression,k=1);n:the number of samples;Sbi:the standard error of the coefficient bi p valuet(n-k-1)1(kntNo linear relationshipLinear relationship does exist