Multivariate Latent Variable

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1、TowardsSmoothParticleFiltersforLikelihoodEstimationwithMultivariateLatentVariablesbyAnthonyLeeB.Sc.,TheUniversityofBritishColumbia,2006ATHESISSUBMITTEDINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFMASTEROFSCIENCEinTheFacultyofGraduateStudies(ComputerScience)THEUNIVERSITYOFBRITISHCOLU

2、MBIA(Vancouver)August,2008cAnthonyLee2008AbstractInparametrizedcontinuousstate-spacemodels,onecanobtainestimatesofthelikelihoodofthedataforfixedparametersviatheSequentialMonteCarlomethodology.Unfortunately,evenifthelikelihoodiscontinuousintheparameters,theestimatesproducedbypracticalpartic

3、lefiltersarenot,evenwhencommonrandomnumbersareusedforeachfilter.Thisisbecausethesameresamplingstepwhichdrasticallyreducesthevarianceoftheestimatesalsointroducesdiscontinuitiesintheparticlesthatareselectedacrossfilterswhentheparameterschange.Whenthestatevariablesareunivariate,themethodologyof

4、[23]givesanestimatorofthelog-likelihoodthatiscontinuousintheparameters.Wepresentanon-trivialgeneralizationofthismethodusingtree-basedo(N2)(andaslowasO(NlogN))resamplingschemesthatinducesignifi-cantcorrelationamongsttheselectedparticlesacrossfilters.Inturn,thisreducesthevarianceofthedifferenc

5、ebetweenthelikelihoodevaluatedfordifferentvaluesoftheparametersandtheresultingestimatorisconsiderablysmootherthannaivelyrunningthefilterswithcommonrandomnumbers.Importantly,inpracticeourmethodsrequireonlyachangetotheresampleoperationintheSMCframeworkwithouttheadditionofanyextraparametersand

6、canthereforebeusedforanyapplica-tioninwhichparticlefiltersarealreadyused.Inaddition,exceptingtheoptionaluseofinterpolationintheschemes,therearenoregularityconditionsfortheirusealthoughcertainconditionsmakethemmoreadvantageous.Inthisthesis,wefirstintroducetherelevantaspectsoftheSMCmethodolog

7、ytothetaskoflikelihoodestimationincontinuousstate-spacemodelsandpresentanoverviewofworkrelatedtothetaskofsmoothlikelihoodestimation.Followingthis,weintroducetheoreticallycorrectresamplingschemesthatcannotbeimplementedandthepracticaltree-basedresamplingschemesthatwer