Logistic Regression

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1、LogisticRegressionKwok-LeungTsuiSystemsEngineering&EngineeringManagementCityUniversityofHongKong2/15/20121LinearRegressionModels•Linearregression•Amodelingtechniqueinwhichtheexpectedvalueofaresponsevariableismodeledasalinearcombinationofasetofpredictors.•Simpler

2、egression(onepredictor).YX01•Multipleregression(twoormorepredictors).YXX...X01122nn•Polynomialregression(orderofk)2kYXX...X01121k12/15/20122OtherRegressionModels•Nonlinearregression•Modelsinwhichtheexpectedvalueoftheresponsevariableyi

3、snotexpressedasalinearcombinationbutanonlinearfunctionoftheparameters.•GeneralizedLinearModels:•Modelsinwhichatransformationoftheexpectedvalueoftheresponsevariableisalinearcombinationoftheparameters.•e.g.,Logisticregression,Poissonregression,…•GeneralizedAdditiv

4、eModels:•Modelsinwhichatransformationoftheexpectedvalueoftheresponsevariableisanadditivemodel.2/15/20123OtherRegressionModels1.Classicallinearmodel2.Generalizedlinearmodel3.AdditivemodelSj:smoothnonparametricfunction4.Generalizedadditivemodels2/15/20124Generaliz

5、edAdditiveModels•Additivemodels(Stone,1985).Sj:smoothnonparametricfunction•Thecurseofdimensionalityisavoidedsinceeachoftheindividualadditivetermsisestimatedusingaunivariatesmoother•Estimatesoftheindividualtermsexplainhowthedependentvariablechangeswiththecorrespo

6、ndingindependentvariables.•Generalizedadditivesmodel(HastieandTibshirani,1990).•Themeanofthedependentvariabledependsonanadditivepredictorthroughanonlinearlinkfunction.•Nonparametriclog-linearmodelsforPoissondataijnij,logijlognlogilogj2/15/2

7、0125GeneralizedAdditiveModels•Classicallinkfunctiong•g(E[Y

8、X])=E[Y

9、X];identitylink.•g(E[Y

10、X])=logit[Y

11、X];logitlink.•g(E[Y

12、X])=probit[Y

13、X];probitlink.•g(E[Y

14、X])=log[Y

15、X];log-linearorlog-additivemodelsforPoissoncountdata.•example(Additivelogisticregression)PrY1

16、

17、XlogsX...sX11ppPrY0

18、X2/15/20126GeneralizedAdditiveModelsSource:Snell,E.J.andSimpson,H.R(1991)AHandbookofGenstatAnalysisAfitusingalinearmodelAfitusingagen