GCV广义交叉验证

InnehllGeneralsWhatRegularizationParameterisOptimal?GeneralizedCross-ValidationDiscussionGeneralizedCrossValidationMrtenMarcus26november2009MrtenMarcusGeneralizedCrossValidationInnehllGeneralsWhatRegularizationParameterisOptimal?GeneralizedCross-ValidationDiscussionPlan1Generals2WhatRegularizationParameterisOptimal?ExamplesDeningtheOptimal3GeneralizedCross-ValidationCrossValidationGeneralizedCrossValidation(GCV)ConvergenceResult4DiscussionMrtenMarcusGeneralizedCrossValidationInnehllGeneralsWhatRegularizationParameterisOptimal?GeneralizedCross-ValidationDiscussionPlan1Generals2WhatRegularizationParameterisOptimal?ExamplesDeningtheOptimal3GeneralizedCross-ValidationCrossValidationGeneralizedCrossValidation(GCV)ConvergenceResult4DiscussionMrtenMarcusGeneralizedCrossValidationInnehllGeneralsWhatRegularizationParameterisOptimal?GeneralizedCross-ValidationDiscussionReferencesChapter4ofSplinemodelsforObservationalData(1990)-GraceWahbaOptimalEstimationofContourPropertiesbyCross-ValidatedRegularization(1989)-BehzadShahraray,DavidAndersonSmoothingNoisyDatawithSplineFunction(1979)-PeterCraven,GraceWahbaMrtenMarcusGeneralizedCrossValidationInnehllGeneralsWhatRegularizationParameterisOptimal?GeneralizedCross-ValidationDiscussionConditionalExpectationsFromprobabilitytheory,wehaveforX2L2f;F;PgE[X]=argmin2RE(X)2(1)TheleastsquaresestimatethereforegivesdiscretizedestimateofandanaturalnormtochoosewhensearchingfordatadisturbedbywhitenoiseMrtenMarcusGeneralizedCrossValidationInnehllGeneralsWhatRegularizationParameterisOptimal?GeneralizedCross-ValidationDiscussionIll-posednessConsiderthemodelyi=g(ti)+i;i=1;2;:::;n;ti2[0;1]whereg2W(m)2=ffjf0;f00;:::;f(m1)gabs:cont:;f(m)2L2[0;1]andfiWN(0;2)gwhereisunknown.Theleastsquaresestimategivesmin^gi2W(m)2nXi=1(yi^gi)2=08fyigni=1(2)Theminimumdoesnotdependondataandisclearlyill-posed.MrtenMarcusGeneralizedCrossValidationInnehllGeneralsWhatRegularizationParameterisOptimal?GeneralizedCross-ValidationDiscussionRegularizationWemaychoosetoregularize,orsmooth,thedataas^gn;=argmin^gi2W(m)2nXi=1(yi^g(ti))2+Z10(^g(m)(t))2dt0(3)whichhasauniquesolution.MrtenMarcusGeneralizedCrossValidationInnehllGeneralsWhatRegularizationParameterisOptimal?GeneralizedCross-ValidationDiscussionPlan1Generals2WhatRegularizationParameterisOptimal?ExamplesDeningtheOptimal3GeneralizedCross-ValidationCrossValidationGeneralizedCrossValidation(GCV)ConvergenceResult4DiscussionMrtenMarcusGeneralizedCrossValidationInnehllGeneralsWhatRegularizationParameterisOptimal?GeneralizedCross-ValidationDiscussionPropertiesof^gn;ThereasonforthetermsmoothingsplineisthefollowingFor=0,^gn;canbeseenasaninterpolatingsplineFor=1,^gn;isasinglepolynomialofdegreem1(optimalintheleastsquaressense)For01,itcanbeshownthat^gn;iscomposedbypolynomialsofdegreeatmost2m1ontheintevals[ti;ti+1];i2f1;2;:::;n1gsuchthatthefunctionanditsderivativesuptoandincludingthe2m2derivativearecontinuousattheknots,andf(k)(t1)=f(k)(tn)=0fork=fm;m+1;:::;2m2g,i.e.naturalconditionsintheendpoints.Fromwhichwecanseethatthereisastrongdependencebetweenthequalityoftheresultandagoodchoiceof.MrtenMarcusGeneralizedCrossValidationInnehllGeneralsWhatRegularizationParameterisOptimal?GeneralizedCross-ValidationDiscussionExamplesPlan1Generals2WhatRegularizationParameterisOptimal?ExamplesDeningtheOptimal3GeneralizedCross-ValidationCrossValidationGeneralizedCrossValidation(GCV)ConvergenceResult4DiscussionMrtenMarcusGeneralizedCrossValidationInnehllGeneralsWhatRegularizationParameterisOptimal?GeneralizedCross-ValidationDiscussionExamplesExampleExampleofimpactofdierenton^gn;and^g0n;foryi=g(ti)+iwhereg(t)=sin(t180);t2[0;360]MrtenMarcusGeneralizedCrossValidationInnehllGeneralsWhatRegularizationParameterisOptimal?GeneralizedCross-ValidationDiscussionDeningtheOptimalPlan1Generals2WhatRegularizationParameterisOptimal?ExamplesDeningtheOptimal3GeneralizedCross-ValidationCrossValidationGeneralizedCrossValidation(GCV)ConvergenceResult4DiscussionMrtenMarcusGeneralizedCrossValidationInnehllGeneralsWhatRegularizationParameterisOptimal?GeneralizedCross-ValidationDiscussionDeningtheOptimalTrueMeanSquareErrorR()andOptimalUsingthenotationabovewedenethetruemeansquareerror,R(),asR():=1nnXi=1(^gn;(ti)gti)2(4)Theoptimalisthendenedas=argmin2R+R()(5)MrtenMarcusGeneralizedCrossValidationInnehllGeneralsWhatRegularizationParameterisOptimal?GeneralizedCross-ValidationDiscussionPlan1Generals2WhatRegularizationParameterisOptimal?ExamplesDeningtheOptimal3GeneralizedCross-ValidationCrossValidationGeneralizedCrossValidation(GCV)ConvergenceResult4DiscussionMrtenMarcusGeneralizedCrossValidationInnehllGeneralsWhatRegularizationParameterisOptimal?GeneralizedCross-ValidationDiscussionCrossValidationPlan1Generals2WhatRegularizationParameterisOptimal?ExamplesDeningtheOptimal3GeneralizedCross-ValidationCrossValidationGeneralizedCrossValidation(GCV)ConvergenceResult4DiscussionMrtenMarcusGeneralizedCrossValidationInnehl