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1、SupportVectorMachineForFunctionalDataClassi?cationFabriceRossia,?,NathalieVillab,aProjetAxIS,INRIA-Rocquencourt,DomainedeVoluceau,Rocquencourt,B.P.105,78153LeChesnayCedex,FrancebEquipeGRIMM-Universit′eToulouseLeMirail,5all′eesA.Machado,31058Toulousecedex1-FR
2、ANCE?Correspondingauthor:FabriceRossiProjetAxISINRIARocquencourtDomainedeVoluceau,Rocquencourt,B.P.10578153LECHESNAYCEDEX–FRANCETel:(33)139635445Fax:(33)139635892Emailaddresses:Fabrice.Rossi@inria.fr(FabriceRossi),villa@univ-tlse2.fr(NathalieVilla).Preprints
3、ubmittedtoElsevierScienceJuly1,2005SupportVectorMachineForFunctionalDataClassi?cationAbstractInmanyapplications,inputdataaresampledfunctionstakingtheirvaluesinin?-nitedimensionalspacesratherthanstandardvectors.Thisfacthascomplexcon-sequencesondataanalysisalg
4、orithmsthatmotivatemodi?cationsofthem.Infactmostofthetraditionaldataanalysistoolsforregression,classi?cationandcluster-inghavebeenadaptedtofunctionalinputsunderthegeneralnameofFunctionalDataAnalysis(FDA).Inthispaper,weinvestigatetheuseofSupportVectorMa-chine
5、s(SVMs)forfunctionaldataanalysisandwefocusontheproblemofcurvesdiscrimination.SVMsarelargemarginclassi?ertoolsbasedonimplicitnonlinearmappingsoftheconsidereddataintohighdimensionalspacesthankstokernels.Weshowhowtode?nesimplekernelsthattakeintoaccountthefuncti
6、onalnatureofthedataandleadtoconsistentclassi?cation.Experimentsconductedonrealworlddataemphasizethebene?toftakingintoaccountsomefunctionalaspectsoftheproblems.Keywords:FunctionalDataAnalysis,SupportVectorMachine,Classi?cation,ConsistencyPreprintsubmittedtoEl
7、sevierScienceJuly1,20051INTRODUCTION31IntroductionInmanyrealworldapplications,datashouldbeconsideredasdiscretizedfunc-tionsratherthanasstandardvectors.Intheseapplications,eachobservationcorrespondstoamappingbetweensomeconditions(thatmightbeimplicit)andtheobs
8、ervedresponse.Awellstudiedexampleofthosefunctionaldataisgivenbyspectrometricdata(seesection6.3):eachspectrumisafunctionthatmapsthewavelengthsoftheilluminatinglighttothecorrespondingab-sorbances(
