Presentation - Introduction to Kernel Smoothing

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1、IntroductiontoKernelSmoothing0.0060.004Density0.0020.0007008009001000110012001300WilcoxonscoreM.P.Wand&M.C.JonesKernelSmoothingMonographsonStatisticsandAppliedProbabilityChapman&Hall,1995.StefanieScheid-IntroductiontoKernelSmoothing-January5,20041IntroductionHistogramofsomep−values121086D

2、ensity4200.00.20.40.60.81.0p−valuesStefanieScheid-IntroductiontoKernelSmoothing-January5,20042Introduction{Estimationoffunctionssuchasregressionfunctionsorprobabilitydensityfunctions.{Kernel-basedmethodsaremostpopularnon-parametricestimators.{Canuncoverstructuralfeaturesinthedatawhichapar

3、ametricapproachmightnotreveal.StefanieScheid-IntroductiontoKernelSmoothing-January5,20043UnivariatekerneldensityestimatorGivenarandomsampleX1;:::;Xnwithacontinuous,univariatedensityf.Thekerneldensityestimatorisnf^(x;h)=1XxXiKnhhi=1withkernelKandbandwidthh.Undermildconditions(hmustdecre

4、asewithincreasingn)thekernelestimateconvergesinprobabilitytothetruedensity.StefanieScheid-IntroductiontoKernelSmoothing-January5,20044ThekernelK{Canbeaproperpdf.Usuallychosentobeunimodalandsymmetricaboutzero.)Centerofkernelisplacedrightovereachdatapoint.)In uenceofeachdatapointisspreadabo

5、utitsneighborhood.)Contributionfromeachpointissummedtooverallestimate.StefanieScheid-IntroductiontoKernelSmoothing-January5,20045Gaussiankerneldensityestimate1.51.00.5●●●●●●●●●●0.0051015StefanieScheid-IntroductiontoKernelSmoothing-January5,20046Thebandwidthh{Scalingfactor.{Controlshowwide

6、theprobabilitymassisspreadaroundapoint.{Controlsthesmoothnessorroughnessofadensityestimate.)Bandwidthselectionbearsdangerofunder-oroversmoothing.StefanieScheid-IntroductiontoKernelSmoothing-January5,20047Fromover-toundersmoothingKDEwithb=0.1121086Density4200.00.20.40.60.81.0p−valuesStefan

7、ieScheid-IntroductiontoKernelSmoothing-January5,20048Fromover-toundersmoothingKDEwithb=0.05121086Density4200.00.20.40.60.81.0p−valuesStefanieScheid-IntroductiontoKernelSmoothing-January5,20049Fromover-toundersmoothingKDEwithb=0.02121086Density4200.00.20.40.60.81.0p−