Regression Trees and Adaptive Splines for a Continuous Response.pdf

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1、10RegressionTreesandAdaptiveSplinesforaContinuousResponseThethemeofthischapteristomodeltherelationshipbetweenacontinuousresponsevariableYandasetofppredictors,x1,...,xp,basedonobser-vations{x,...,x,Y}N.Weassumethattheunderlyingdatastructurei1ipi1canbedescribedbyY=f(x1,...,

2、xp)+ε,(10.1)wherefisanunknownsmoothfunctionandεisthemeasurementerrorwithmeanzerobutunknowndistribution.Inordinarylinearregression,fisassumedtobeoftheformf(x)=p1xiβi.Then,theestimationofthefunctionfbecomesaproblemofes-timatingparametersβ.Thankstoitssimplicity,linearregres

3、sionisamongthemostfrequentlyusedstatisticaltechniques.Inapplications,however,theunderlyingdatastructurecannotalwaysbesummarizedbyasimplemodel,andhencetherestrictiveassumptionsbehindthesimplicitymayresultinpoorfits.Accordingly,alternativestolinearregressionareofcon-siderabl

4、einterestforunderstandingtherelationshipbetweenthecovariatesandtheresponse.Nonparametricproceduresusingsplinesofferonesolu-tiontothisproblem.Theyarebasedontheideathatasmoothfunctioncanbewellapproximatedbypiecewisepolynomials(see,e.g.,DeBoor1978).Here,wefocusontwoclassesofm

5、odelsthatarebuiltupontherecursivepartitioningtechnique.Oneisregressiontrees,andtheotherismultivariateadaptiveregressionsplines(MARS,Friedman1991).Regressiontreesfitaconstanttotheresponsewithineveryterminalnode,whereasadaptivesplinesusepiecewiselinearfunctionsasthebasisfunc

6、tions.ThekeydifferencebetweenMARSandCARTliesinthefactH.ZhangandB.H.Singer,RecursivePartitioningandApplications,133SpringerSeriesinStatistics,DOI10.1007/978-1-4419-6824-1_10,©SpringerScience+BusinessMedia,LLC201013410.RegressionTreesandAdaptiveSplinesforaContinuousResponsey

7、xFIGURE10.1.One-dimensionalMARS(thethinnerpiecewiseline)andCART(thestepfunction)models.ThedottedcurveistheunderlyingsmoothfunctionthattheregressionfunctioniscontinuousinMARSwithrespecttoacon-tinuouscovariate,butnotinCART.Therefore,MARSmodelsaremoreappealingwhencontinuity

8、isaconcern.Insomeapplicationssuchasimagecompression(e.g.,GershoandGray1992,andPoggiandOlshen1995