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A Comparative Study of Adaptive MCMC Based Particle Filtering Methods

Permanent Link: http://ufdc.ufl.edu/UFE0044219/00001

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Title: A Comparative Study of Adaptive MCMC Based Particle Filtering Methods
Physical Description: 1 online resource (49 p.)
Language: english
Creator: Yoon, Jae Myung
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: filter -- filtering -- mcmc -- particle -- tracking
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this thesis, we present a comparative study of conventional particle filtering (PF) algorithms for tracking applications. Through the review from the generic PF to more recent Markov chain Monte Carlo (MCMC) based PFs, we will revisit the sample impoverishment problem. For all PF methods using resampling process, maintaining appropriate sample diversity is a big problem. Although Gilks et al. proposed an MCMC based PF to avoid this problem, their method sometimes fails due to small process noise. Therefore, we propose an improved MCMC move PF method which employs an adaptive MCMC move. This adaptive MCMC process elastically manages the MCMC proposal density function to circumvent the sample impoverishment problem efficiently and gives better sample diversity for posterior approximation.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Jae Myung Yoon.
Thesis: Thesis (M.S.)--University of Florida, 2012.
Local: Adviser: Kumar, Mrinal.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044219:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044219/00001

Material Information

Title: A Comparative Study of Adaptive MCMC Based Particle Filtering Methods
Physical Description: 1 online resource (49 p.)
Language: english
Creator: Yoon, Jae Myung
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: filter -- filtering -- mcmc -- particle -- tracking
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this thesis, we present a comparative study of conventional particle filtering (PF) algorithms for tracking applications. Through the review from the generic PF to more recent Markov chain Monte Carlo (MCMC) based PFs, we will revisit the sample impoverishment problem. For all PF methods using resampling process, maintaining appropriate sample diversity is a big problem. Although Gilks et al. proposed an MCMC based PF to avoid this problem, their method sometimes fails due to small process noise. Therefore, we propose an improved MCMC move PF method which employs an adaptive MCMC move. This adaptive MCMC process elastically manages the MCMC proposal density function to circumvent the sample impoverishment problem efficiently and gives better sample diversity for posterior approximation.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Jae Myung Yoon.
Thesis: Thesis (M.S.)--University of Florida, 2012.
Local: Adviser: Kumar, Mrinal.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044219:00001


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ACOMPARATIVESTUDYOFADAPTIVEMCMCBASEDPARTICLEFILTERINGMETHODSByJAEMYUNGYOONATHESISPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFMASTEROFSCIENCEUNIVERSITYOFFLORIDA2012

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c2012JaeMyungYoon 2

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Tomywife,JungeunPark,myson,AndrewJ.Yoon,myparents,HyungsubYoonandHeejaYang,myin-laws,WonrakParkandYoungheeBae,andtherestofmyfamilymembersfortheirunwaveringsupportandconstantencouragement 3

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ACKNOWLEDGMENTS Iwouldliketosincerelythankmyadvisor,MrinalKumar,whoseinnitemotivationandinspirationcarriedmetothesuccessfulcompletionofmyMasterofSciencedegree.Hisconstantguidanceandthepatienceoverthepastyearshelpedmematureacademically,professionally,andpersonally.IwouldalsoliketoexpressmysenseofgratitudetoallmycommitteemembersNormanFitz-Coy,andPrabirBarooahforthetimeandhelptheyprovided.Iwouldliketothankmywifeforherloveandpatience.Also,Iwouldliketoextendimmeasurablegratitudetomyfamilyforsupportingandbelievinginme. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................... 4 LISTOFTABLES ....................................... 7 LISTOFFIGURES ....................................... 8 ABSTRACT ........................................... 9 CHAPTER 1INTRODUCTION .................................... 10 1.1Motivation ...................................... 10 1.2Background ..................................... 11 2NONLINEARBAYESIANTRACKING ......................... 16 2.1ProblemFormulationforStateEstimation ..................... 16 2.1.1PredictionStep ............................... 17 2.1.2UpdateStep ................................. 17 2.2ResearchIssues ................................... 17 2.2.1Nonlinearity ................................ 17 2.2.2Dimensionality ............................... 18 2.2.3Real-time .................................. 18 3PARTICLEFILTERINGMETHODS .......................... 19 3.1ImportanceSamplingMonteCarloIntegration ................... 19 3.2SequentialImportanceSampling(SIS) ....................... 21 3.2.1SequentialImportanceSampling ...................... 21 3.2.2ProblemofSIS:SampleDegeneracy .................... 24 3.3GenericParticleFilter(PF) ............................. 26 3.3.1Resampling ................................. 26 3.3.2SequentialImportanceResampling(SIR) ................. 27 3.3.3AdaptiveSIR ................................ 28 3.4MarkovChainMonteCarlo(MCMC)Move .................... 30 3.4.1DerivationofMetropolis-Hastings(MH)AlgorithmBasedMCMCMove 31 3.4.2MCMCMoveParticleFilter ........................ 32 3.5AdaptiveMCMCMove ............................... 33 4RESULTS ......................................... 39 4.1SyntheticExperiment ................................ 39 4.1.1Model .................................... 39 4.1.2ComparativeResults ............................ 39 4.2BearingOnlyTracking ............................... 41 5

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4.2.1Model .................................... 42 4.2.2ComparativeResults ............................ 43 5CONCLUSION ...................................... 46 5.1ClosingStatement .................................. 46 5.2FutureWork ..................................... 46 REFERENCES ......................................... 47 BIOGRAPHICALSKETCH .................................. 49 6

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LISTOFTABLES Table page 4-1Averageofperformancesfromeachlterkindsover50runs .............. 40 4-2Averageofperformancesfromeachlterkindsover100runs .............. 44 7

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LISTOFFIGURES Figure page 3-1Graphicalrepresentationofthesequentialimportancesampling(SIS)particlelter(PF)(1cycle) ....................................... 25 3-2SISPFalgorithm ..................................... 25 3-3Graphicalrepresentationofthesampledegeneracyproblem ............... 26 3-4Graphicalrepresentationofthemultinomialresamplingprocess ............ 27 3-5Systematicresamplingalgorithm ............................. 28 3-6Graphicalrepresentationofthesequentialimportanceresampling(SIR)PF(1cycle) 29 3-7SIRPFalgorithm ..................................... 29 3-8AdaptiveSIRparticlelteralgorithm .......................... 30 3-9GraphicalrepresentationoftheMarkovchainMonteCarlo(MCMC)movePF(1cycle) 33 3-10MCMCmovealgorithm .................................. 34 3-11MCMCmovePFalgorithm ................................ 35 3-12GraphicalrepresentationoftheadaptiveMCMCmovePF(1cycle) ........... 37 3-13AdaptiveMCMCmovealgorithm ............................ 38 4-1Trajectoriesofthesyntheticexperimentmodel. ..................... 41 4-2Samplediversitycomparisonofthebearingonlytrackingmodel. ............ 44 4-3Trajectoriesofthebearingonlytrackingmodelwithvariousltermethods. ...... 45 8

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AbstractofThesisPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofMasterofScienceACOMPARATIVESTUDYOFADAPTIVEMCMCBASEDPARTICLEFILTERINGMETHODSByJaeMyungYoonMay2012Chair:MrinalKumarMajor:MechanicalEngineering Inthisthesis,wepresentacomparativestudyofconventionalparticleltering(PF)algorithmsfortrackingapplications.ThroughthereviewfromthegenericPFtomorerecentMarkovChainMonteCarlo(MCMC)basedPFs,wewillrevisitthesampleimpoverishmentproblem.ForallPFmethodsusingresamplingprocess,maintainingappropriatesamplediversityisabigproblem.AlthoughGilksetal.proposedaMCMCbasedPFtoavoidthisproblem,theirmethodsometimesfailsduetosmallprocessnoise.Therefore,weproposeanimprovedMCMCmovePFmethodwhichemploysanadaptiveMCMCmove.ThisadaptiveMCMCprocesselasticallymanagestheMCMCproposaldensityfunctiontocircumventthesampleimpoverishmentproblemefcientlyandgivesbettersamplediversityforposteriorapproximation. 9

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CHAPTER1INTRODUCTION 1.1Motivation Recently,variousestimationstrategiesfornonlinearsystemshavebeenutilizedinvariouseldsofscience,nanceandengineering.Theapplicationdomainofestimationtechniqueishighlyvariedandsometimesalsocloselyconnectedwithourdailylives,forexample: 1. Fromsecuritydevicessuchassurveillancecamerastothearticialintelligencefocus(AIfocus)featuresinthelatesthandhelddigitalcameras,targettrackingisheavilyused.TheAIfocusfeatureautomaticallyadjuststhefocallengthtomaintainfocusonapossiblymovingtarget. 2. Visionsystemsandautonomousnavigationsystemsfoundingroundvehicles,aircraft,spacecraftandrobotsusevarioustargettrackingalgorithmsandarecurrentlywidelyresearched. 3. Trackingalgorithmsalsondwideuseinthestockmarket,futurestradingmarket,aswellasforecastingofvariouseconomicindicators. 4. Tracking/lteringalsondswidespreaduseinweatherforecasting;aswellaspredictionoflifeexpectancyofmanufacturingtools/products. Despiteitswidespreadapplicabilityasenumeratedabove,itisextremelydifcultinrealitytoobtainaccurateestimatesofthestateofnonlinearsystems.Wedeneoptimalstateestimation(alsoknownasltering)astheproblemofndingthebestestimateofthestatesofanevolvingdynamicsystem.Twomainsourcesprovidestateinformation: 1. Physicallawsbelievedtogoverndynamicalbehaviorofthesystemand; 2. Sensormeasurementsthatprovidedirectinformationaboutsystembehavior. Itisimportanttomentionthatboththeabovesourcesofinformationareimperfectandmaybecorruptedbyvaryingdegreesofnoise(knownasprocessnoiseandmeasurementnoiserespectively).Basedontherequiredlevelofaccuracy,anappropriatetrackingalgorithmcanbeconstructedthatperformsafusionofinformationobtainedfromtheabovetwosources.Intheabovestructureofstatetracking,numerouspoorlyknownparametersappearaswell.However,wewillnotbeconcernedwiththeproblemofestimatingsuchparametersinthisthesis. 10

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Physicallawsgoverningsystembehavioraretypicallymodeledasstochasticdifferentialequations(SDEs).Theseequationscompriseofadeterministicpartandarandompart(typicallycalledprocessnoise).Dependingonthelevelofaccuracyrequiredorcomputingcapabilityavailable,thedeterministicpartcanvaryincomplexity.Ifsimpliedmodelsareusedforthedeterministicportion,thentheprocessnoisecanbeappropriatelyadjusted(increased)toaccountforlossofinformation.Inadditiontophysicallaws,trackingalgorithmsutilizesensorinformation.Themeasurementmodelscanalsovaryincomplexityandisinvariablycorruptedbysensornoise. Basedonthedynamicalandmeasurementmodelsused,awidevarietyoftrackingalgorithmscanbeutilized.Itmusthoweverbenotedthateachalgorithmisbasedoncertainassumptionsaboutthenatureofunderlyingmodels.Ifanyoftheassumptionsbreakdown,thetrackingalgorithmcannotguaranteethedesiredaccuracy.Forexample,theKalmanlteristheoptimaltrackingalgorithmonlyifinadditiontootherassumptions,boththedynamicalmodelandmeasurementmodelarelinear.DespiteofthefactthattheKalmanlteristhemostwidelyusedfamilyoftrackingalgorithms,itsunderlyingassumptionscanmakeitvulnerable.Inthisthesis,adifferentfamilyoftrackingalgorithmscalledparticleltersareexplored.Theideaistoincreaseaccuracywhileminimizingassumptionsonthedynamicalandmeasurementmodels. 1.2Background Eversinceitsinception,Kalmanlter(KF)[ 16 ]hasbeenthemostwidelyusedestimationstrategyduetoitselegantrecursiveform,computationalefciencyandeaseofimplementation.Therefore,eventhoughKFonlytracksthersttwomomentsofthestate(meanandcovariance),itisextremelypowerful.Furthermore,KFisoptimalifthefollowingassumptionshold: 1. Thedynamicsystemmodelisalinearfunctionofthestateandprocessnoise. 2. Themeasurementmodelisalsolinearfunctionofthestateandmeasurementnoise. 3. Theprocessnoiseandthemeasurementnoisearemutuallyindependentandzero-meanWienerprocesswithknowncovariance. 4. TheposteriordensityisGaussian. 11

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Unfortunately,mostreallifesituationsdonotupholdalloftheaboveassumptions.Asaresult,anonlinearversionofKFwasrequired.ThelocallinearizedversionofKF,knownastheExtendedKalmanFilter(EKF)[ 13 ]wasproposed.IntheEKF,thenonlinearsystemisreplacedwitharstorderperturbationmodelobtainedviaTaylorSeriesexpansionaboutareferencetrajectory.WhileEKFextendstheapplicabilityoftheKalmanltertononlineardynamicsystemsandmeasurementmodels,itperformspoorlywhenthehigherordertermsstarttodominate(e.g.whenperturbationgrowsinmagnitudeovertimeduetononlineareffects).Toovercomethisweakness,higher-orderEKF[ 22 ]algorithmsusingthesecondorevenhigherordertermsintheTaylorseriesexpansionwereintroduced.However,thefundamentalproblemremainsthesame,i.e.divergenceduetononlineareffectscanonlybedelayedbutnotavoided.Overall,theclassofEKFalgorithmshavethefollowinggeneralshortcomings. 1. Linearizationorhighordersystemapproximationsarepronetodivergencedependentondegreeofsystemnonlinearity. 2. LinearizedtransformationsometimescannotbeobtainedthroughJacobianmatrix,e.g.ifthesystemcontainsadiscontinuity,theJacobiandoesnotexist. 3. ComputingtheJacobianmatrixcanbetedious,ofteninvolvingexcessiveandcomplexalgebraicmanipulations.Also,thecomputationprocessispronetohumanerror. Inthehigher-orderEKFalgorithms,theseproblemscanbecomemoreserious. Recently,JulierandUhlmannproposedanotherlteringstrategybasedonthesocalledunscentedtransformation(UT)samplingandtheKalmanlterframework.ThislterisknownastheunscentedKalmanlter(UKF)[ 15 ].AsopposedtoEKF,whichperformslocallinearization,theUKFisbasedontheprincipleofstatisticallinearization.Byvirtueofstatisticallinearization,thersttwomomentsofthesystemderivedfromthesamplepointsmatchexactlywiththersttwomomentsoftheactualsystem.Toperformtheunscentedtransformation,aminimalsetofsamplepoints(calledsigmapoints)aredeterministicallyselectedaroundthecurrentlyknownmeanofthestate.Thesigmapointsarethenpropagatedthroughthenonlinearsystemdynamics(i.e.withoutanalyticlocallinearization).Asaconsequence,theUKFalgorithmisapplicableto 12

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systemswithdiscontinuities.However,ifnonlineareffectsarestrong,theUKFstrategymaynotbeadequatetodescribestateuncertainty. AnotherpopularlteringstrategyisthesequentialMonteCarlo(SMC)method(alsoknownasparticlelter(PF))[ 1 4 7 26 ].ThebasicideabehindtheSMCistocharacterizestateuncertaintyintermsofanitenumberofparticles.AsopposedtotheUKF,theSMCmethodscancapturehigherorderstatisticalinformationofthesystemstate(e.g.themean,variance,skewness,kurtosisetc.),byanalysisoftheparticlestatistics. EventhoughtherstversionsoftheSMCmethodscanbefoundinpapersdatingbacktothe1950s[ 10 ],theycouldnotbecomepopularimmediatelyduetothefollowingdrawbacks: 1. TheSMCmethodsgenerallyrequireshighcomputingpower,whichhaveonlyrecentlybecamereadilyavailable. 2. TheearlySMCmethodswerebasedonthesequentialimportancesampling(SIS)technique,whichsuffersfromaseriousproblemcalledsampledegeneracy(a.k.a.particledepletion). In1993,Gordon,SalmondandSmithproposedthesequentialimportanceresampling(SIR)[ 9 ]algorithmtoovercomesampledegeneracy.SinceintroductionoftheSIRlter,researchintheSMCmethodshasgrownvigorouslyandresultedsignicanttheoreticalprogress.Inaddition,thankstotherecentcomputerrevolution,theSMCmethodsbecameincreasinglyamenabletotheonlinedemandsofthelteringproblem.AftertheSIRlterwasintroducedmanyotherresamplingideaswereproposedandimplementedtoimprovetheSIRlter.Forexample,multinomialresampling[ 9 ],residualresampling[ 20 ],strariedresampling[ 17 ],andsystematicresampling[ 17 ]algorithmshavebeenproposedbymanyauthors.Followingseveralyearsofimplementation,thegeneralconsensusappearstobethatowingtoitssimpleimplementation,lowcomputationcomplexityandlowsamplevariance,thesystematicresamplingalgorithmisthemostwidelyusedresamplingtechnique. Despitethefactthatresamplingalleviatestheproblemofsampledegeneracy,anotherseriousphenomenoncameuptothesurface,calledassampleimpoverishment.Inthisphenomenon,particleswithlowweightsdisappearduringtheresamplingstep,whileparticles 13

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withhigherweightsreplicate.Inanextremecase,asetofNdistinctparticlescanbereducedtoasingleparticlerepeatedNtimes.Sincelossofsamplediversityrepresentslossofstatisticalinformationofthesystemstate,itmaycauseveryharmfuleffectsonstateestimation.Therefore,appropriatestepmustbetakentomaintainsamplediversity.Recently,twomajortechniqueswereproposedinliteraturetoimprovetheparticlediversity: 1. Regularizedparticlelter(RPF)[ 25 ]:RPFresamplesformacontinuousapproximationoftheposterior,asopposedtotheSIRPFwhichdoessofromadiscreteapproximation.Thecontinuousapproximationofposteriordensityisconstructedbytheregularizationtechnique,whichbuildsaGaussiankernelbasedontheexistingparticles. 2. MCMCmovetechnique:ProposedbyGilksetal.in2001[ 8 ]isanotherpotentialsolutiontothesampleimpoverishmentproblem.TheMCMCmovealgorithmusesMarkovchainMonte-Carlo(MCMC)method[ 27 ],torelocate(move)existingparticlestobetterrepresent(i.e.withgreaterdiversity)theposteriordensity.TypicallytheMetropolis-Hastings(MH)algorithm[ 12 24 ]isusedforthispurpose. Abovetwodifferentapproachesaremainlyusedtoimprovesamplediversity.However,wewillnotbeconcernedwiththeRPFinthisthesisbecauseourresearchfocusisanextensionoftheMCMCmove.EventhoughtheMCMCmovestrategywasproposedtogivesamplediversity,itcanbreakdownifprocessnoiseisverysmall[ 14 ]. Inthisthesis,weproposeanewMCMCmovealgorithm,namedadaptiveMCMCmoveparticleltertoaddressthelowprocessnoisescenario.Asmentionedabove,whentheprocessnoiseisverysmall,theMCMCmovePFsometimesfailstotrackamovingobject[ 14 ]duetothesampleimpoverishmentproblem.IntheMetropolis-HastingsMCMCalgorithm[ 12 24 ],samplesstatesarerstproposedfromaproposaldensityfunction,andthenacceptedorrejected.IntherelatedeldofMCMCbasedintegration,ithasbeenseenthatadaptivelyadjustingthesupportoftheproposaldensityfunctionimprovestheperformanceofMCMCintegration[ 28 ].Inthisadaptiveapproach,thesupportadjustmentisperformedsuchthattheacceptancerateismaintained,between0.4and0.5.ThecurrentworkdrawsinspirationfromthisideaandadaptsittothePFframework.Intheproposedmethod,theratioofnumberofparticlesactuallymovedbythestandardMCMCmovemethodtothetotalnumberofparticlesisusedasanindicatorofthequalityofsamplediversityasinMCMCintegration;ifthisratioliesoutsideadesiredwindow 14

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(dependingontheproblem),thecovariancematrixoftheprocessnoiseisadjustedtoproposeanewsetofmovedparticles.Notethattheactualsystemhasnotbeenaltered.Theprocessnoisecovarianceismanipulatedonlyinordertoproposeanewsetofmovedpoints.Thesemaybeacceptedorrejectedbasedontheirlikelihoodofbelongingtotheactualposteriorpdf. Theremainderofthisthesisisorganizedasfollows.Inchapter2,theproblemformulationfortheBayesianstateestimationispresented.Inchapter3,theparticlelter(PF)anditsvariantsarebrieyreviewed,includingtheassociatedalgorithmsforpracticalimplementation.TheabovementionednewadaptiveMCMCmovealgorithmisdevelopedandtodetailedalgorithmisprovided.Inchapter4,wediscussexamplesforpracticalimplementation.Lastly,chapter5containsconclusionandpathsforfuturework. 15

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CHAPTER2NONLINEARBAYESIANTRACKING 2.1ProblemFormulationforStateEstimation Considerthefollowingdiscrete-time,hiddenMarkovmodelwhichiscalledthestatetransitionmodelandthestatemeasurementmodel: Xkj(Xk)]TJ /F9 7.97 Tf 6.59 0 Td[(1=xk)]TJ /F9 7.97 Tf 6.59 0 Td[(1),p(xkjxk)]TJ /F9 7.97 Tf 6.59 0 Td[(1) (2) Zkj(Xk=xk),p(zkjxk) (2) whereXk=fx0;x1;:::;xkgisthesequenceofsystemstatesuptotimek2N,andZk=fz1;z2;:::;zngisthesequenceofmeasurementavailableuptotimek.Also,xk2Rnxdenotesthestateofthesystemattimek,zk2Rnzdenotestheobservationattimestepkandnx,nzdenotethedimensionsofthestatevectorandmeasurementrespectively.WeassumethatXkandZkarestatisticallyindependent.Thestatetransitionandthestatemeasurementmodels(eqn.( 2 )andeqn.( 2 ))canberewritteninfunctionalformasshownbelow: xk=fk)]TJ /F9 7.97 Tf 6.59 0 Td[(1(xk)]TJ /F9 7.97 Tf 6.59 0 Td[(1;vk)]TJ /F9 7.97 Tf 6.59 0 Td[(1) (2)zk=hk(xk;wk) (2) wherevk2Rnvdenotestheprocessnoiseattimek,wk2Rnwdenotesthemeasurementnoiseattimekandnv,nwdenotethedimensionsoftheprocessnoiseandmeasurementnoiserespectively.Notethattheinitialstatedistribution,p(x0),p(x0jz0),isassumedknown.Theobjectiveofthelteringproblemistoobtaintheestimatedstate,xk,givenallavailableinformationuptotimek;themeasurementinformationZkgrowssequentiallylikealtration,meaningthatZkisnotknownattimestepk)]TJ /F7 11.955 Tf 11.96 0 Td[(1.Then,themarginalpdfofthestatep(xkjZk)canbecomputedrecursivelyintwostages;namely,predictionandupdate. 16

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2.1.1PredictionStep Supposethestateestimateattheprevioustimestep(k)]TJ /F7 11.955 Tf 12.56 0 Td[(1)isknown;i.e.p(xk)]TJ /F9 7.97 Tf 6.58 0 Td[(1jZk)]TJ /F9 7.97 Tf 6.59 0 Td[(1).Thenthepriorpdfofthestateisobtainedas: p(xkjZk)]TJ /F9 7.97 Tf 6.58 0 Td[(1)=Zp(xkjxk)]TJ /F9 7.97 Tf 6.59 0 Td[(1)p(xk)]TJ /F9 7.97 Tf 6.58 0 Td[(1jZk)]TJ /F9 7.97 Tf 6.59 0 Td[(1)dxk)]TJ /F9 7.97 Tf 6.58 0 Td[(1(2) TheaboveequationisknownastheChapman-Kolmogorovequation(CKE),whichindifferentialformisnothingbuttheFokker-Plankequation(FPE).NotethattheCKEismoregeneralthanFPEbecauseitholdsforbothcontinuousanddiscretesystems. 2.1.2UpdateStep Atthecurrenttimek,theposteriorpdfcanbeobtainedbyincorporatingthenewmeasurementviatheBayesruleasfollow: p(xkjZk)=p(xkjzk;Zk)]TJ /F9 7.97 Tf 6.59 0 Td[(1)=p(zkjxk;Zk)]TJ /F9 7.97 Tf 6.58 0 Td[(1)p(xkjZk)]TJ /F9 7.97 Tf 6.59 0 Td[(1) p(zkjZk)]TJ /F9 7.97 Tf 6.59 0 Td[(1)=p(zkjxk)p(xkjZk)]TJ /F9 7.97 Tf 6.58 0 Td[(1) p(zkjZk)]TJ /F9 7.97 Tf 6.59 0 Td[(1)(2) wherethenormalizingconstantcanbeobtainedas: p(zkjZk)]TJ /F9 7.97 Tf 6.59 0 Td[(1)=Zp(zkjxk)p(xkjZk)]TJ /F9 7.97 Tf 6.59 0 Td[(1)dxk(2) 2.2ResearchIssues 2.2.1Nonlinearity Iff(;)andh(;)arelinearandvkandwkareGaussianthentherecursivestateestimationcanbesolvedanalytically.Ifnot(thisisthetypicalreal-lifesituation),lteringbecomesaninnitedimensionproblem(equivalenttoestimatingallmomentsofthestatepdf).Thisisanintractableproblemandsub-optimalmethodsmustbedeveloped[ 19 ]. 17

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2.2.2Dimensionality Inanylteringapplication,thecurseofdimensionalityreferstovariousnegativephenomenaoccurringduetoanalysisofhigh-dimensionalspaces.Oneofthecommonmanifestationsofthisproblemisthat,whennx(i.e.statedimensionality)increases,thenumericalimplementationofbothprediction(CKE)andupdate(Bayes'rule)becomesincreasingly(exponentially)difcult[ 19 ]. 2.2.3Real-time Stateestimationisespeciallydifcultbecauseitmustbeperformedinrealtime;i.e.theprediction(CKE)andupdate(Bayes'rule)stagesmustbecompletedbeforethenextsetofmeasurementsbecomeavailable.Thisplacesspecialefciencydemandsonthedevelopedalgorithms[ 19 ]. 18

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CHAPTER3PARTICLEFILTERINGMETHODS ThesequentialMonteCarlo(SMC)methodsareaclassofsub-optimallters.Theyareknownvariouslyasthebootstraplter[ 9 ],thecondensationalgorithm[ 21 ],theparticlelter(PF)[ 3 ]etc.InordertodescribeSMC,wewillbeginwithadiscussionoftheimportancesamplingbasedMonteCarlointegrationtechnique.Then,wewillreviewthedetailsoftheconventionalPFmethods,namelythesequentialimportancesampling(SIS,oneoftherstversionsofPF)andthesequentialimportanceresampling(SIR)particlelter.TheMarkovchainMonteCarlo(MCMC)basedPFwillthenbeintroducedtoovercometheshortcomingsoftheSIRPF.Finally,wewilldescribethenewadaptiveMCMCmovemethodtoobtainimprovedsamplediversity. 3.1ImportanceSamplingMonteCarloIntegration Supposewewanttointegrateafunctiong(x), I=Zg(x)dx(3) wherex2Rnx.Weconsiderthespecialcaseinwhichtheintegrandg(x)canbefactoredasfollows: g(x)=f(x)(x)(3) where(x)isaprobabilitydensityfunction(pdf)satisfyingtheproperties,(x)0andR(x)dx=1.If(x)isreadilyavailableandeasytosamplefrom;i.e.wecanndasetofpoints(particles),fxi;i=1;:::;Ng(x),N1;wecandecomposethefunctionusingIasfollows: IN=1 NNXi=1f(xi)(3) where,Nrepresentsthenumberofsamples.ThisisthewellknownMakovchainMonteCarlo(MCMC)methodofnumericalintegration.Moreover,ifthesamples,xi(x);fori=1;:::;N 19

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areindependentandidenticallydistributed(iid)randomvariables,thenE[IN)]TJ /F5 11.955 Tf 12.86 0 Td[(I]=0,asN!1.INwillconvergealmostsurelytoIaccordingtothelawoflargenumbers.Thevarianceoff(x)canbegivenas: 2=Z(f(x))]TJ /F5 11.955 Tf 11.95 0 Td[(I)2(x)dx;(3) Ifthevarianceisnite,theestimationerrorconvergestonormaldistributionwithmean0and2, limN!1p N(IN)]TJ /F5 11.955 Tf 11.95 0 Td[(I)N(0;2):(3) Unfortunately,(x)isinvariablynotknown,ornoteasytosamplefrom.Inthissituation,animportancedensity,q(x)isintroduced,andisalsoknownastheproposaldensity.Wecanrewriteeqn.( 3 )asfollows: I=Rf(x)(x)dx=Rf(x)(x) q(x)q(x)dx(3) Theproposalq(x)isspecicallychosentobeeasytosamplefromandmustsatisfysomebasicrequirements[ 4 ],e.g.: q(x)c(x):(3) NowwecanestimateIusingsamplesdrawnfromq(x),i.e.fxi;i=1;:::;Ngq(x)asfollows: IN=1 NNXi=1f(xi)~w(xi)(3) where~w(xi)aretheunnormalizedimportanceweightsandgivenby: ~w(xi)=(xi) q(xi);(fori=1;:::;N):(3) 20

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Notethatthesediscreteweightsmaynotautomaticallybenormalized:i.e.PNi=1~w(xi)6=1.Tomaintaintheunbiasednatureoftheapproximation,theweightsmustbenormalized,resultinginthenalapproximationasfollows: IN=1 NNXi=1f(xi)~w(xi) 1 NNXj=1~w(xj)=NXi=1f(xi)w(xi);(3) where,thenormalizedweightcanbecomputedas: w(xi)=~w(xi) PNj=1~w(xj):(3) Iftheimportancedensity(proposaldensity)isnotchosenproperly[ 4 ],theaboveapproachcanresultinparticleswithverylow(negligiblylow)weights.ThisaffectsthequalityofapproximationandwillmanifestintheformofsampledegeneracyintheimportancesamplingbasedPFdescribednext. 3.2SequentialImportanceSampling(SIS) 3.2.1SequentialImportanceSampling Essentially,SISisasequenceofthebasicimportancesamplingMonteCarlointegrationstepdescribedabove.Theunknownpdf(x)istheunderlyingstatepdfwhichmustbeapproximatedusingparticles.Sinceitcannotbedirectlysampledfrom;asequenceofimportancedensityisusedtogenerateparticleswithaboveweights.Therefore,letfXik;wikgNi=1denotearandommeasurethatcharacterizesp(XkjZk),wherep(XkjZk)representsthejointposteriordensityattimek;fXikgNi=1isthesetofsupportpointswithassociatedweightsfwikgNi=1whichsatisfyPNi=1wik=1(thenormalizedweights).Thejointposteriordensityattimekcannowbedepictedasfollows[ 26 ]: 21

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p(XkjZk)NXi=1wik(Xk)]TJ /F5 11.955 Tf 11.95 0 Td[(Xik)(3) where()istheDiracdeltafunctionand(Xk)]TJ /F5 11.955 Tf 12.47 0 Td[(Xik)denotestheDiracdeltamasslocatedatsupportXik.Theaboveisadiscreteweighted.Thenormalizedweights,wikareselectedaccordingtoeqn.( 3 )as: wik/p(XikjZk) q(XikjZk):(3) Sincelteringisarecursivestateestimationtechnique,assumethatwehavesamplesdescribingthepreviousjointposterior,p(Xk)]TJ /F9 7.97 Tf 6.59 0 Td[(1jZk)]TJ /F9 7.97 Tf 6.59 0 Td[(1).Now,wewishtoapproximatethejointposteriorp(XkjZk)byincorporatingthenewmeasurementsobtainedattimek.Iftheimportancedensityisdenedas: q(XkjZk),q(xkjXk)]TJ /F9 7.97 Tf 6.59 0 Td[(1;Zk)q(Xk)]TJ /F9 7.97 Tf 6.58 0 Td[(1jZk)]TJ /F9 7.97 Tf 6.59 0 Td[(1);(3) thenwecanobtainimportancesamplesatcurrenttimek(i.e.Xikq(XkjZk))bycombiningexistingsamplessuchasXik)]TJ /F9 7.97 Tf 6.58 0 Td[(1q(Xk)]TJ /F9 7.97 Tf 6.59 0 Td[(1jZk)]TJ /F9 7.97 Tf 6.58 0 Td[(1)andxikq(xkjXk)]TJ /F9 7.97 Tf 6.59 0 Td[(1;Zk).Beforeweobtaintherecursiveformoftheimportanceweight(weightupdating),therecursiveformofthejointposteriorp(XkjZk)mustbeobtainedasbelow: p(XkjZk)=p(Xk;Zk) p(Zk)=p(Xk;zk;Zk)]TJ /F9 7.97 Tf 6.59 0 Td[(1) p(Xk;Zk)]TJ /F9 7.97 Tf 6.58 0 Td[(1)p(Xk;Zk)]TJ /F9 7.97 Tf 6.59 0 Td[(1) p(Zk)]TJ /F9 7.97 Tf 6.59 0 Td[(1) p(Zk) p(Zk)]TJ /F9 7.97 Tf 6.59 0 Td[(1)=p(zkjXk;Zk)]TJ /F9 7.97 Tf 6.59 0 Td[(1)p(XkjZk)]TJ /F9 7.97 Tf 6.58 0 Td[(1) p(zkjZk)]TJ /F9 7.97 Tf 6.59 0 Td[(1)=p(zkjXk;Zk)]TJ /F9 7.97 Tf 6.59 0 Td[(1)p(xkjXk)]TJ /F9 7.97 Tf 6.58 0 Td[(1;Zk)]TJ /F9 7.97 Tf 6.59 0 Td[(1)p(Xk)]TJ /F9 7.97 Tf 6.59 0 Td[(1jZk)]TJ /F9 7.97 Tf 6.59 0 Td[(1) p(zkjZk)]TJ /F9 7.97 Tf 6.58 0 Td[(1): (3) Intheeqn.( 3 ),wecanregard: 22

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p(xkjXk)]TJ /F9 7.97 Tf 6.58 0 Td[(1;Zk)]TJ /F9 7.97 Tf 6.59 0 Td[(1)=p(xkjxk)]TJ /F9 7.97 Tf 6.59 0 Td[(1) (3) p(zkjXk;Zk)]TJ /F9 7.97 Tf 6.59 0 Td[(1)=p(zkjxk); (3) sincep(xkjXk)]TJ /F9 7.97 Tf 6.59 0 Td[(1;Zt)]TJ /F9 7.97 Tf 6.58 0 Td[(1)andp(zkjXk;Zk)]TJ /F9 7.97 Tf 6.58 0 Td[(1)areMarkovprocess.Pluggingtheeqn.( 3 )andeqn.( 3 )intotheeqn.( 3 )yieldstherecursivejointposterior,p(XkjZk)asfollows: p(XkjZk)=p(zkjxk)p(xkjxk)]TJ /F9 7.97 Tf 6.58 0 Td[(1) p(zkjZk)]TJ /F9 7.97 Tf 6.58 0 Td[(1)p(Xk)]TJ /F9 7.97 Tf 6.58 0 Td[(1jZk)]TJ /F9 7.97 Tf 6.59 0 Td[(1) (3) /p(zkjxk)p(xkjxk)]TJ /F9 7.97 Tf 6.59 0 Td[(1)p(Xk)]TJ /F9 7.97 Tf 6.58 0 Td[(1jZk)]TJ /F9 7.97 Tf 6.59 0 Td[(1): (3) Substitutingtheimportancedensityfunction(eqn.( 3 ))andtherecursivejointposterior(eqn.( 3 ))intotheeqn.( 3 )yieldsthedesiredweightupdatingequationas: wik/p(XikjZk) q(XikjZk)=p(zkjxik)p(xikjxik)]TJ /F9 7.97 Tf 6.59 0 Td[(1) q(xikjXik)]TJ /F9 7.97 Tf 6.59 0 Td[(1;Zk)p(Xik)]TJ /F9 7.97 Tf 6.59 0 Td[(1jZk)]TJ /F9 7.97 Tf 6.59 0 Td[(1) q(Xik)]TJ /F9 7.97 Tf 6.59 0 Td[(1jZk)]TJ /F9 7.97 Tf 6.58 0 Td[(1)| {z }=wik)]TJ /F15 5.978 Tf 5.75 0 Td[(1=wik)]TJ /F9 7.97 Tf 6.59 0 Td[(1p(zkjxik)p(xikjxik)]TJ /F9 7.97 Tf 6.59 0 Td[(1) q(xikjXik)]TJ /F9 7.97 Tf 6.59 0 Td[(1;Zk): (3) Moreover,intheeqn.( 3 ),ifq(xkjXk)]TJ /F9 7.97 Tf 6.59 0 Td[(1;Zk)=q(xkjxk)]TJ /F9 7.97 Tf 6.59 0 Td[(1;zk),thentheimportanceweightisonlyrelativetoxk)]TJ /F9 7.97 Tf 6.59 0 Td[(1andzkandcanberepresentedasasfollows: wik/wik)]TJ /F9 7.97 Tf 6.59 0 Td[(1p(zkjxik)p(xikjxik)]TJ /F9 7.97 Tf 6.58 0 Td[(1) q(xikjxik)]TJ /F9 7.97 Tf 6.59 0 Td[(1;zk):(3) Fromtheeqn.( 3 ),thediscreteapproximationofthemarginalposteriordensity,p(xkjZk),canthenbeobtainedasbelow: 23

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p(xkjZk)NXi=1wik(xk)]TJ /F5 11.955 Tf 11.95 0 Td[(xik):(3) SISparticlelteringalgorithmrecursivelypropagateNsupportpointsandassociatedimportanceweights(i.e.fxik;wikgNi=1)usingeqn.( 3 )andeqn.( 3 ).TheaboveSISPFprocesslaysthegroundworkofmostPFlters. NotethataselectionoftheimportancedensityfunctionisacriticaldesignissueofanyPFsystem.Theoptimalimportancedensityfunctionisonlyavailableinlimitedcases[ 5 6 ].Therefore,wewillusethemostcommonlyusedsuboptimalimportancedensity,thetransitionalprior,asfollows: q(xkjxik)]TJ /F9 7.97 Tf 6.59 0 Td[(1;zk),p(xkjxik)]TJ /F9 7.97 Tf 6.59 0 Td[(1):(3) Pluggingthetransionalpriorintotheweightupdatingeqn.( 3 )yields: wik/wik)]TJ /F9 7.97 Tf 6.59 0 Td[(1p(zkjxik)p(xikjxik)]TJ /F9 7.97 Tf 6.59 0 Td[(1) q(xikjxik)]TJ /F9 7.97 Tf 6.59 0 Td[(1;zk)=wik)]TJ /F9 7.97 Tf 6.59 0 Td[(1p(zkjxik): (3) Seetheg.( 3-1 )describingonecycleofaboveSISPFprocessgraphically.Also,seethedetailsoftheSISPFalgorithmforparcticalimplementationispresentedintheg.( 3-2 )[ 26 ] 3.2.2ProblemofSIS:SampleDegeneracy Eventhoughtheidealimportancedensityfunctionistheposteriordensityfunctionitself,wearenotnormallyabletodrawimportanceweightsdirectlyfromtheposterior.Iftheimportancedensityq(x)innotchosenproperly(i.e.eqn.( 3 )),thevarianceoftheimportanceweightsonlyincreasesoverthetime;forexample,seeRef[ 5 ].ThevarianceincrementleadstoapotentiallyriskyphenomenonoftheSISPF,calledthesampledegeneracyproblem.Afteracertainnumberofrecursivecycles,onlyoneparticleoccupiesthemostoftheweightandtherestoftheparticles 24

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Figure3-1. Graphicalrepresentationofthesequentialimportancesampling(SIS)particlelter(PF)(1cycle) Algorithm [fxik;wikgNi=1]=SIS[fxik)]TJ /F9 7.97 Tf 6.59 0 Td[(1;wik)]TJ /F9 7.97 Tf 6.58 0 Td[(1gNi=1;zk] 1. FORi=1:N (a) Draw xik=q(xkjxik)]TJ /F9 7.97 Tf 6.59 0 Td[(1;zk) (b) Evaluatetheunnormalizedimportanceweightsviaeqn.( 3 )oreqn.( 3 ) wik/wik)]TJ /F9 7.97 Tf 6.59 0 Td[(1p(zkjxik)p(xikjxik)]TJ /F9 7.97 Tf 6.59 0 Td[(1) q(xikjxik)]TJ /F9 7.97 Tf 6.59 0 Td[(1;zk) 2. ENDFOR 3. Normalizetheimportanceweight:fwikgNi=1=f~wikgNi=1 PNj=1~wjk Figure3-2. SISPFalgorithm becomenegligible(seetheg.( 3-3 )).IntheSISPFframework,thesampledegeneracyproblemisinevitable.Therefore,aresamplingtechniquewasintroducedtohandlethisproblem. 25

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Figure3-3. Graphicalrepresentationofthesampledegeneracyproblem 3.3GenericParticleFilter(PF) 3.3.1Resampling Basically,theresamplingprocessinvolvesmappingtherandommeasuresfxik;wikgNi=1intoanewuniformlyweightedrandommeasuresfxjk;1 NgNj=1.Inotherwords,theresamplingprocessuniformlydrawsasamplexikwithprobabilityofwikfromthediscretesetfxik;wikgNi=1.Afterresamplingprocess,particleswithlowweightsdisappearwhileparticleswithhigherweightsreplicate.Moreover,theweightsetfwikgNi=1isnormalizedasfwik=1 NgNi=1. Figure.( 3-4 )[ 23 ]depictsthemultinomialresamplingmethodwhichispresentedinthepaperofGordon,SalmondandSmith[ 9 ].Inthemultinomialresamplingprocess,thecumulativedistributionfunction(cdf)isconstructedwiththeimportanceweights,fwikgNi=1.Initially,uniformlydrawnithsample(i.e.uiU(0;1),fori=1;:::;N)isprojectedontothedistributionrange.Theintersectionofuiisthenprojectedagainontothedistributiondomaintogetthenewsampleindexj.ThisdepictshowthenewparticlesetfxjkgNj=1andtheindexofparentssetfijgNj=1areobtained.AftertheresamplingstepwasrstimplementedtotheSISlter,manyotherresamplingideaswereproposedtoimproveSMCmethods.Forexample,multinomial 26

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Figure3-4. Graphicalrepresentationofthemultinomialresamplingprocess resampling[ 9 ],residualresampling[ 20 ],strariedresampling[ 17 ],andsystematicresampling[ 17 ]algorithmshavebeenproposedbymanyauthors.Followingseveralyearsofimplementation,thegeneralconsensusappearstobethatowingtoitssimpleimplementation,lowcomputationcomplexityandlowsamplevariance,thesystematicresamplingalgorithmisthemostwidelyusedresamplingtechnique.Inthisthesis,wewillonlyreviewthesystematicresamplingalgorithmwhichispresentedbelowing.( 3-5 )[ 26 ].Generallyinanyparticleltersystems,theindexofthesampleparent,ij,isnotrequiredexceptforauxiliaryPFandtheMCMCmovebasedPF. 3.3.2SequentialImportanceResampling(SIR) WenowpresentonecycleofthegenericPFprocess(SIRPF);Seetheg.( 3-6 ).Inthisprocess,weconsideredthetransitionpriorisconsideredastheimportancedensity(i.e.q(xkjxik)]TJ /F9 7.97 Tf 6.58 0 Td[(1;zk),p(xkjxik)]TJ /F9 7.97 Tf 6.59 0 Td[(1)).Atthetopofthegure,westartwiththeposteriorapproximationattimek)]TJ /F7 11.955 Tf 12.79 0 Td[(1,i.e.fxik)]TJ /F9 7.97 Tf 6.58 0 Td[(1;1 NgNi=1,whichisuniformlyweighted.Inthepredictionstep,aboveparticlesarepropagatedthroughthesystemdynamics(eqn.( 2 ))andthen,theparticles 27

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Algorithm [fxjk;wjk;ijgNj=1]=RESAMPLE[fxik;wikgNi=1] 1. ConstructthecdffcigNi=1withparticles'importanceweightfwikgNi=1 2. Setthecdfindex:i=1 3. Drawarandomsample:u1U(0;1 N) 4. FORj=1:N (a) Movealongthecdf:uj=ui+j)]TJ /F9 7.97 Tf 6.59 0 Td[(1 N (b) WHILEuj>ci i. i=i+1 (c) ENDWHILE (d) Assignsample:xjk=xik (e) Assignweight:wjk=1 N (f) Assignparent:ij=i 5. ENDFOR Figure3-5. Systematicresamplingalgorithm approximatethepriordensity,fxik;1 NgNi=1p(xkjZk)]TJ /F9 7.97 Tf 6.59 0 Td[(1).Whenanewobservationzkisavailable,weupdatetheimportanceweightofeachparticlessuchthattheparticlesapproximatethemarginalposterior,fxik;wikgNi=1p(xkjZk).Thenweexcutetheresamplingprocesstocircumventthesampledegeneracyproblem.Duringtheresamplingprocess,theparticleswithlowweightsareremovedwhileparticleswithhigherweightsareduplicated.Theresampledparticlesstillapproximatethemarginalposterioratthecurrenttimekasfxi0k;1 NgNi0=1p(xkjZk).ThedetailsoftheSIRPFalgorithmforpracticalimplementationispresentedintheg.( 3-7 )[ 26 ] 3.3.3AdaptiveSIR TheonlydifferencebetweentheSIRlterandtheadaptiveSIRlteristheexistanceofcomputingtheeffectivesamplesize(dNeff)step.InthestandardSIRlterprocess,theresamplingprocessisperformedeverytimestep(i.e.time=1;:::;k)aftereveryweightupdating 28

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Figure3-6. Graphicalrepresentationofthesequentialimportanceresampling(SIR)PF(1cycle) Algorithm [fxikgNi=1]=SIR[fxik)]TJ /F9 7.97 Tf 6.59 0 Td[(1gNi=1;zk] 1. FORi=1:N (a) Drawxikp(xkjxik)]TJ /F9 7.97 Tf 6.58 0 Td[(1) (b) Computetheunnormalizedimportanceweight:~wik=p(zkjxik) 2. ENDFOR 3. Normalizetheimportanceweights:fwikgNi=1=f~wikgNi=1 PNj=1~wjk 4. Resampleusingtheresamplingalgorithming.( 3-5 ) (a) [fxik;)]TJ /F5 11.955 Tf 9.3 0 Td[(;gNi=1]=RESAMPLE[fxik;wikgNi=1] Figure3-7. SIRPFalgorithm step.However,intheadaptiveSIRlterprocess,wecomputetheeffectivesamplesizedNeffasanindicatorofthesampledegeneracy.IfdNefffallsbelowthethresholdofparticlepopulationNthr(i.e.dNeffNthr),theadaptiveSIRlterperformstheresamplingstepsoastocircumventthesampledegeneracy.IntheadaptiveSIRPF,Nthrisanusers'choiceofdesignparameter.Theeffectivesamplesizeisintroducedinthepaper[ 18 ]andcomputedasfollows: 29

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Algorithm [fxik;wikgNi=1]=ASIR[fxik)]TJ /F9 7.97 Tf 6.59 0 Td[(1;wik)]TJ /F9 7.97 Tf 6.58 0 Td[(1gNi=1;zk] 1. FilteringviaSIS(g.( 3-2 )) [fxik;wikgNi=1]=SIS[fxik)]TJ /F9 7.97 Tf 6.59 0 Td[(1;wik)]TJ /F9 7.97 Tf 6.59 0 Td[(1gNi=1;zk] 2. CalculatedNeffusingeqn.( 3 ) dNeff=1 PNi=1(wik)2 3. IFdNeff
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multipleMCMCcyclemeansthatmoreprocessingtimeisspentbutguranteesthatparticlesareasymptoticallyconvergedtothedesiredposterior.Thederivation[ 26 ]oftheMCMCmovestepispresentedinthenextsubsection.Inaddition,whentheprocessnoiseinthesystemdynamicmodeliscompletelyzero,thenanyPFmethodswillperformpoorlybecausePFmethodsareproperintheestimationofstochasticdynamicsystems[ 26 ].However,theRPF[ 25 ]andtheMCMCmovePF[ 8 ]showsbetterperformancewhentheproblemofsampleimpoverishmentissevereorwhentheprocessnoiseinthedynamicsystemissmall. 3.4.1DerivationofMetropolis-Hastings(MH)AlgorithmBasedMCMCMove RememberthatXk=fxj;j=0;:::;kgrepresentsthesequenceofallstatesuptotimek;p(XkjZk)representsthejointposteriordensityattimestepk;p(xkjZk)representsthemarginaldensityattimek;Fromtherecursiverelationshipofthejointposterioreqn.( 3 ),wehavethefollowingequation: p(XkjZk)=p(zkjxk)p(xkjxk)]TJ /F9 7.97 Tf 6.59 0 Td[(1) p(zkjZk)]TJ /F9 7.97 Tf 6.59 0 Td[(1)p(Xk)]TJ /F9 7.97 Tf 6.58 0 Td[(1jZk)]TJ /F9 7.97 Tf 6.59 0 Td[(1):(3) Aftertheresamplingstepattimestepk,wehavetheindexofparentsijandtheresampledparticlesxik.Usingtheindexofparentsij,wecanobtaintheresampledparentsasfollows: xijk)]TJ /F9 7.97 Tf 6.58 0 Td[(1,xik)]TJ /F9 7.97 Tf 6.58 0 Td[(1:(3) Themostimportantideaisthatwecomparearesampledparticlexikandaproposalsamplefromatransitionalpriorx?ikp(xkjxik)]TJ /F9 7.97 Tf 6.59 0 Td[(1).Thenwedecideparticlemovementwiththeacceptanceprobabilitywhenu(uU(0;1))as: Weacceptmovementxik=8>><>>:x?ikwithprobabilityxikotherwise(3) Wecanobtainparticlesamplesfromtheposteriordensityusingtheeqn.( 3 )as: 31

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p(X?ikjZk)=p(zkjx?ik)p(x?ikjxik)]TJ /F9 7.97 Tf 6.59 0 Td[(1) p(zkjZk)]TJ /F9 7.97 Tf 6.58 0 Td[(1)p(Xk)]TJ /F9 7.97 Tf 6.58 0 Td[(1jZk)]TJ /F9 7.97 Tf 6.59 0 Td[(1)p(XikjZk)=p(zkjxik)p(xikjxik)]TJ /F9 7.97 Tf 6.59 0 Td[(1) p(zkjZk)]TJ /F9 7.97 Tf 6.59 0 Td[(1)p(Xk)]TJ /F9 7.97 Tf 6.59 0 Td[(1jZk)]TJ /F9 7.97 Tf 6.58 0 Td[(1):(3) AssumethatMCMCproposaldensitya(jxik)issymmetric.ThenwecancanceltheMCMCproposalswiththerelationshipa(x?ikjxik)=a(xikjx?ik).Thentheacceptanceprobabilitycanbecomputedasfollows: =min1;p(X?ikjZk)a(xikjx?ik) p(XikjZk)a(x?ikjxik)=min1;p(X?ikjZk) p(XikjZk)=min0@1;p(zkjx?ik)p(x?ikjxik)]TJ /F15 5.978 Tf 5.76 0 Td[(1) (((((p(zkjZk)]TJ /F15 5.978 Tf 5.75 0 Td[(1)((((((((p(Xk)]TJ /F9 7.97 Tf 6.59 0 Td[(1jZk)]TJ /F9 7.97 Tf 6.59 0 Td[(1) p(zkjxik)p(xikjxik)]TJ /F15 5.978 Tf 5.76 0 Td[(1) (((((p(zkjZk)]TJ /F15 5.978 Tf 5.75 0 Td[(1)((((((((p(Xk)]TJ /F9 7.97 Tf 6.59 0 Td[(1jZk)]TJ /F9 7.97 Tf 6.58 0 Td[(1)1A=min1;p(zkjx?ik)p(x?ikjxik)]TJ /F9 7.97 Tf 6.59 0 Td[(1) p(zkjxik)p(xikjxik)]TJ /F9 7.97 Tf 6.58 0 Td[(1)(3) 3.4.2MCMCMoveParticleFilter WenowpresentonecycleofthestandardMCMCmovePFprocess(timek)]TJ /F7 11.955 Tf 12.78 0 Td[(1!k,N=10):Seetheg.( 3-9 ).Notethatthetransitionalpriorisusedastheimportancedensity(i.e.q(xkjxik)]TJ /F9 7.97 Tf 6.58 0 Td[(1;zk),p(xkjxik)]TJ /F9 7.97 Tf 6.58 0 Td[(1)).Beforethepredictionstep,westartwiththeuniformlyweightedparticlesfxik)]TJ /F9 7.97 Tf 6.59 0 Td[(1;1 NgNi=1whichisthediscreteapproximationoftheposteriorattimek)]TJ /F7 11.955 Tf 12.72 0 Td[(1.Inthepredictionstep,theparticleswillbepropagatedviathesystemdynamicsf()(eqn.( 2 ))andweobtaintheparticlesapproximatingthepriordensityfxik;1 NgNi=1p(xkjZk)]TJ /F9 7.97 Tf 6.59 0 Td[(1).Afteranewobservationzkisobtained,weupdatetheimportanceweightofeachparticlesuchthattheparticlesapproximatetheposteriordensity,fxik;wikgNi=1p(xkjZk).Duringtheresamplingstep,weobtaintheresampledparticlesxi0kandtheresampledparentsxijk)]TJ /F9 7.97 Tf 6.58 0 Td[(1viaeqn.( 3 ).ThentheMCMCmovestepcomparestheresampledparticlesxikandtheproposalsamples(x?ikp(xkjxik)]TJ /F9 7.97 Tf 6.58 0 Td[(1)),toselectbetterparticlesviaeqn.( 3 ).Therelocatedparticlesstillapproximatetheposteriorfxik;1 NgNi=1p(xkjZk)buthavinganimprovedsamplediversity.ThedetailsoftheMCMCmovealgorithmandtheMCMCmovePFalgorithmforpractical 32

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Figure3-9. GraphicalrepresentationoftheMarkovchainMonteCarlo(MCMC)movePF(1cycle) implementationarepresentedintheg.( 3-10 )andtheg.( 3-11 )respectively.NotethatmultiplecyclesoftheMCMCmovecanberepeated(e.g.s=Sn(Sn>1))sothattheapproximationoftheposteriorMCMCcomverges(e.g.SnSburn)]TJ /F6 7.97 Tf 6.59 0 Td[(in). 3.5AdaptiveMCMCMove Thesampleimpoverishmentproblemisknowntobeseverewhentheprocessnoiseofthesystemdynamicsissmall[ 26 ].Insuchcases,wenormallychooseeithertheregularizedPFortheMCMCmovePF.EventhoughtheMCMCmovePFperformsbetterthantheSIRPF,theMCMCmovePFsometimesbreaksdownduetothesampleimpoverishmentproblem[ 14 ].Thisisbecausetheproposalsamplesarenotgeneratedfrombroadstateduetothesmallprocessnoise.IftheMCMCmovePFcanadjusttheprocessnoisecovariance(orstandarddeviation)purelyforthepurposeofgeneratingbetterproposalstate,wecanhavebetterproposalsamplesviatheMCMCmovestep.Therefore,weproposeanewMCMCmovestepwithadaptiveprocessnoise 33

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Algorithm [fxikgNi=1]=MCMC[fxikgNi=1;fxijk)]TJ /F9 7.97 Tf 6.59 0 Td[(1gNij=1] 1. FORs=1:Sn (a) FORi=1:N i. SampleuU(0;1) ii. Sampletheproposalcandidatex?ikfromthetransitionalprior iii. IFu=min1;p(zkjx?ik)p(x?ikjxik)]TJ /F15 5.978 Tf 5.76 0 Td[(1) p(zkjxik)p(xikjxik)]TJ /F15 5.978 Tf 5.76 0 Td[(1),acceptmove: xik=x?ik iv. ELSErejectmove: xik=xik v. ENDIF (b) ENDFOR 2. ENDFOR 3. Xik=fXik)]TJ /F9 7.97 Tf 6.59 0 Td[(1;xikg Figure3-10. MCMCmovealgorithm covariance(orstandarddeviation)selectionintheMCMCmovesteptogiveimprovedsamplediversity. IntheadaptiveMCMCmovePF,theacceptancerate,AR,isusedasanindicatorofthesampleimpoverishmentproblem.Thatis,ifthelterachieveslowerARthanthedesiredAR(i.e.ARth),weassumethatparticlesarediversiedenoughtorepresenttheuncertainty.Ontheotherhand,ifthelterachiveshigherARthanthedesiredAR,thelterneedtoadjusttheproposalnoisecovariance(orstandarddeviation)soastoobtainbetterparticlesamples.ARisdenedastheproportionoftheacceptednumberofparticlesovertheentireparticlepopulationN.ARiscomputedasfollows: AR=Naccepted N:(3) 34

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Algorithm [fxikgNi=1]=MCMCPF/AMCMCPF[fxik)]TJ /F9 7.97 Tf 6.59 0 Td[(1gNi=1;zk] 1. FORi=1:N (a) Drawxikp(xkjxik)]TJ /F9 7.97 Tf 6.58 0 Td[(1) (b) Calculatetheunnormalizedimportanceweight:~wik=p(zkjxik) 2. ENDFOR 3. Normalizetheimportanceweights:fwikgNi=1=f~wikgNi=1 PNj=1~wjk 4. Resampleusingthealgorithming.( 3-5 ) (a) [fxjk;)]TJ /F5 11.955 Tf 9.3 0 Td[(;ijgNj=1]=RESAMPLE[fxik;wikgNi=1] 5. FORj=1:N (a) Assignparents:xjk)]TJ /F9 7.97 Tf 6.58 0 Td[(1=xijk)]TJ /F9 7.97 Tf 6.58 0 Td[(1 6. ENDFOR 7. [fxikgNi=1]=MCMC/AdaptiveMCMC[fxikgNi=1;fxjk)]TJ /F9 7.97 Tf 6.58 0 Td[(1gNj=1] Figure3-11. MCMCmovePFalgorithm TheadaptiveMCMCmovePFmethodrepeatsSn(Sn>1)timesofthepropagationphaseuntilthedesiredARisachieved.IfwegetdesiredAR,weassumetheapproximationoftheposteriorisMCMCconverged(e.g.SnSburn)]TJ /F6 7.97 Tf 6.59 0 Td[(in).Itisintuitivethattherepeatingthepropagationphaseharmtheprocessingtimebutbenetthelteraccuracy(uncertaintyinformation).Insomecases(seeRef[ 14 ]),ithasseenthatrepeatingthestandardMCMCmove[ 8 ]isveryslowtoconvergetothedesiredtargetdensity.However,theadaptiveMCMCmovePFacceleratesMCMCconvergenceusingadaptiveadjustmentoftheproposalnoisecovariance(orstandarddeviation)step.Asaresult,theadaptiveMCMCmovePFmethodcanachieveabetterstateestimationbyimprovingsamplediversity. Theadaptiveadjustmentoftheproposalnoisecovariance(orstandarddeviation)isperformedbymultiplyingextensionmagnitude,(1),totheproposalnoisecovariance 35

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(orstandarddeviation).ThechoiceofisveryimportantdesignissueinthedesignoftheadaptiveMCMCmovePF.Itisrecommendedtodesignatemorethan2levelsfortheacceptancerate.Forexample: AR>AR1!Qk(ork)=1Q(or); (3) AR1>AR>AR2!Qk(ork)=2Q(or); (3) Otherwise(ARARth)!Qk(ork)=Q(or): (3) AgraphicalrepresentationoftheadaptiveMCMCmovePFprocess(timek)]TJ /F7 11.955 Tf 12.74 0 Td[(1!k,N=10andonecycleadaptiveMCMCmovephase)ispresentedintheg.( 3-12 ).ThepreviousstepsbeforetheadaptiveMCMCmoveisidenticaltothestandardMCMCmove;seetheg.( 3-10 ).AfteronephaseofthestandardMCMCmove,theadaptiveMCMCmovePFdetermines(basedontheAR)iftheproposalnoisecovariance(orstandarddeviation)adjustmentisrequiredornot.IfahighAR(i.e.higerthanthedesiredAR)isobtained,extensionparameter,,willbemultipliedtothenoisecovariance(orstandarddeviation).Wethenre-propagatetheparentsofresampledparticles(xijk)]TJ /F9 7.97 Tf 6.59 0 Td[(1,xik)]TJ /F9 7.97 Tf 6.59 0 Td[(1)toobtainbetterproposalsamples(x?ikp(xkjxik)]TJ /F9 7.97 Tf 6.59 0 Td[(1)).Notethattheparentsofresampledparticlesareobtainedviatheindexofparents.Inaddition,theyrepresenttheparticlesattheprevioustimestep(k-1)whichhavesurvivedoffspringafterresamplingstep.Next,theadaptiveMCMCmovePFwillcomparetheresampledparticles(xik)tothenewproposalsamples(x?ik)andchoosebetterparticlesamplesviaMHalgorithmeqn.( 3 ).AftertheadaptiveMCMCmove,particlesapproximatetheposteriorfxik;1 NgNi=1p(xkjZk)withmoreimprovedsamplediversitythanthestandardMCMCmovePF.ThedetailsoftheadaptiveMCMCmovealgorithmforpracticalimplementationarepresentedintheg.( 3-13 ).ThemainPFalgorithmisidenticaltothestandardMCMCmovePF(seetheg.( 3-11 )). 36

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Figure3-12. GraphicalrepresentationoftheadaptiveMCMCmovePF(1cycle) 37

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Algorithm [fxikgNi=1]=AdaptiveMCMC[fxikgNi=1;fxik)]TJ /F9 7.97 Tf 6.59 0 Td[(1gNi=1] 1. Settheinitialparameters (a) SETthethresholdacceptancerateARth (b) SETtheextensionlevelsusingeqn.( 3 )througheqn.( 3 ) 2. FORs=1:Sn (a) ComparetheARtothelevels,adjustproposalnoisecovariance(orstandarddeviation) i. IFARAR1;THENQk=1Q(seetheeqn.( 3.5 )fordesign) ii. ELSEIFARARth,THENbreak iii. ELSEQk=Q iv. ENDIF (b) SETNaccepted=0 (c) FORi=1:N i. SampleuU(0;1) ii. Sampletheproposalcandidatex?ikfromthetransitionalpriorwiththeproposalnoisecovarianceQk iii. IFu=min1;p(zkjx?ik)p(x?ikjxik)]TJ /F15 5.978 Tf 5.76 0 Td[(1) p(zkjxik)p(xikjxik)]TJ /F15 5.978 Tf 5.76 0 Td[(1),acceptmove: A. xik=x?ik B. Naccepted=Naccepted+1 iv. ELSErejectmove: A. xik=xik v. ENDIF (d) ENDFOR (e) Calculatetheacceptancerate:AR=Naccepted N 3. ENDFOR 4. Xik=fXik)]TJ /F9 7.97 Tf 6.59 0 Td[(1;xikg Figure3-13. AdaptiveMCMCmovealgorithm 38

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CHAPTER4RESULTS TheadaptiveMCMCmovePFwillbecomparedtothestandardMCMCmovePFandtheSIRPF,inthefollowingtwoexamples:asyntheticexperimentandthebearingonlytrackingmodel(BOTM). 4.1SyntheticExperiment Thesyntheticexperiment[ 23 ]istherstexampleoftheadaptiveMCMCPF. 4.1.1Model Fortherstexperiment,thesystemdynamicmodelisgivenas: xk+1=1+sin(!k)+xk+vk(4) where!andaremodelparametersandgivenby!=0:04and=0:5,respectively.TheprocessnoisevkisdistributedwithGammarandomdistributionwithparameterssuchthat(3;2).Andthenon-stationaryobservationmodelisgivenasfollows: zk=x2k+wk;(k30) (4) =xk)]TJ /F7 11.955 Tf 11.96 0 Td[(2+wk;(k>30) (4) whereandaretheobservationmodelparameters,=0:2,=0:5.TheobservationnoisewkisdistributedwiththeGaussianrandomdistributionsuchthatwk=N(0;0:00001).Notethatnewobservationsareavailableeveryonesecond. 4.1.2ComparativeResults Whenasequenceofobservationdatafzk;k=1;:::;60gisgiven,thestateestimationf^xk;k=1;:::;60gisperformedwithvariouslteringmethods.Notethattheexperimentdatawereobtainedbasedontheaverageof50runs.Ref[ 23 ]providesanextendedcomparison,althoughtheprocessingtimeforeachmethodhasnotbeenprovided.Inthisexperiment,thesecondSIRPFhadequivalentamountofparticlescomparedtotheprocessingtimeofthe 39

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adaptiveMCMCmovePFtaken.TherstSIRPFwithN=20,thesecondSIRPFwithN=500,thestandardMCMCmovePFwithN=20andrepeatedmultiplepropagationphaseSn=35andtheadaptiveMCMCmoveparticlelterwithbelowparametersweretested: N=20;Sn=35;(AR>70)Qk!1(=3)Q;(70AR>25)Qk!2(=2)Q;Otherwise(ARth25)Qk!Q: Intheparticlelteringmethods,thesystematicresamplingmethodwasused.Seethetable.( 4-1 )whichsummarizestheperformanceofeachparticleltermethod.Forthecomparisonpurposes,meansquarederror(MSE)ofthestateestimates,varianceoftheMSE,andprocessingtimearepresented. Table4-1. Averageofperformancesfromeachlterkindsover50runs FiltertypeAvg.MSEAvg.varianceAvg.time PF(N=20)0.82290.02700.3369PF(N=500)0.25670.02154.6724MCMCPF(Sn=35;N=20)0.56570.03445.4291AdaptiveMCMCPF(Sn=35;N=20)0.17360.00105.5802 Seetheg.( 4-1 )tocomparethetrajectoriesofeachlteringmethod.Dot-dashedlineshowsthetrajectoryofSIRPFwithN=100,dot-dottedlineshowsthetrajectoryofgenericPF(SIR)withN=500,stitchedlineshowsthetrajectoryoftheMCMCmovePFwithN=20and35cycles,andsolidlineshowsthetrajectoryofadaptiveMCMCmovePFmethodwithN=20,35cycles.EventhoughtheadaptiveMCMCmovemethodusesonly20particles,itshowsthelowestMSEamongtheparticipatinglters. 40

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Figure4-1. Trajectoriesofthesyntheticexperimentmodel. 4.2BearingOnlyTracking Forthesecondexample,weexperimentedaconventionallteringapplicationknownasthebearingonlytracking[ 2 ].Duringtheapplication,wetrackamovingobjectwithacoupleofanglesensors.Thosesensorsareonlyallowedtomeasurebearings(orangles)ofthemovingtargetwithrespecttothesensorlocation.ThefollowingliteraturecontainstheperformancecomparisonoftheclassiclteringmethodssuchastheextendedKalmanlters(EKFs)andtheunscentedKalmanlters(UKFs,seethepaperbyJ.Hartikainenetal.[ 11 ]).Thestateofthemovingobjectattimestepkcompriseofthecartesiancoordinatesofthetargetlocationxkandykandthevelocityoftheeachxandydirectionssuchas_xkand_yk.Thenthestatecanbeexpressedasfollows: 41

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xk=xkyk_xk_yk(4) 4.2.1Model Thesystemdynamicmodelisgivenas xk=10k0010k00100001xk)]TJ /F9 7.97 Tf 6.58 0 Td[(1yk)]TJ /F9 7.97 Tf 6.59 0 Td[(1_xk)]TJ /F9 7.97 Tf 6.58 0 Td[(1_yk)]TJ /F9 7.97 Tf 6.59 0 Td[(1+vk)]TJ /F9 7.97 Tf 6.59 0 Td[(1(4) wherevk)]TJ /F9 7.97 Tf 6.58 0 Td[(1istheprocessnoise,whichisdistributedasGaussianwithzeromean.Theprocessnoisecovariancecanbecomputedas E[vk)]TJ /F9 7.97 Tf 6.58 0 Td[(1vTk)]TJ /F9 7.97 Tf 6.59 0 Td[(1]=1 3k301 2k2001 3k301 2k21 2k20k001 2k20kv;(4) wherevrepresentsthenoisespectraldensitywhichissettov=0:2.Thebearingsobservationmodeloftheithsensorisdenedas ik=arctan(yk)]TJ /F5 11.955 Tf 11.95 0 Td[(siy xk)]TJ /F5 11.955 Tf 11.96 0 Td[(six)+wik(4) andS=s1s2isthelocationsofthesensorsmatrix,wheretherstsensorislocatedats1=s1xs1y=)]TJ /F7 11.955 Tf 9.29 0 Td[(1:5)]TJ /F7 11.955 Tf 9.3 0 Td[(0:5andthesecondsensorisats2=s2xs2y=11.Eachobservationerroroftheithsensor(wik)isindependentanddistributedasGaussianwithzero 42

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meanandvariance,2(i.e.wik=N(0;2)).Thenoisedistributionparameterisgivenby=0:01radians.Notethatnewobservationsareavailableevery0.1second.Theinitialconditionsforthelteringaregivenasfollows: x0=0010T(4) P0=0:100000:1000010000010(4) Theinitialconditionscanbereadthatwehavesomeinformationaboutthemovingobject.Thatis,weareconvincedoftheobjectlocation,butweareuncertainaboutthevelocityofthemovingobject. 4.2.2ComparativeResults Forthesecondexperiment,SIRPF(N=100),SIRPF(N=2000),MCMCmovePF(N=100;Sn=15)andadaptiveMCMCmovePF(N=100)weretested.BecausetheperformanceoftheSIRPFislargelydependentonthenumberofparticles,alargepopulationSIRPF(N=2000)wasselectedforthesecondSIRPF.Duetothetimelimitation(60seconds),anySIRPFwithgreaterthanN=2000wasnotconsidered.TheadaptiveMCMCmoveparametersweresetasfollows.OnlywhenAR190,extensionmagnitude1=5wasmultipliedtotheproposalnoisestandarddeviation.Notethattheexperimentdatawereobtainedbasedontheaverageof100runs. Seetheg.( 4-2 )fordetailcomparisonofthesamplediversitybetweentheMCMCmovePFandtheadaptiveMCMCmovePF.ThelefthandsideshowstheparticledistributionoftheregularMCMCmovePFandtherighthandsideshowstheparticledistributionoftheadaptiveMCMCmovePFmethod.Theintersectionofstitchedlinesarewherethemesurementwasmade.Fromtherightgure,theadaptiveMCMCmovePFisseentoworkgreaterbasedonthefactthat 43

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Figure4-2. Samplediversitycomparisonofthebearingonlytrackingmodel. particlesaredistributedideallyoverthestate.Comparetheresultwiththeleftgure(theregularMCMCmovePFhaspoorersamplediversity).Seethetable.( 4-2 )whichissummarizingtheperformanceofeachdifferentparticlelteringmethods.Forthecomparisonpurpose,rootmeansquarederror(RMSE)ofthestateestimates,andprocessingtimearepresentedineachcolumn. Table4-2. Averageofperformancesfromeachlterkindsover100runs FiltertypeAvg.RMSEAvg.varianceAvg.time PF(N=100)1.10560.11312.4203PF(N=2000)0.45510.001252.4790MCMCPFw/multiplecycles(Sn=15;N=100)0.75770.040250.5498AdaptiveMCMCPF(Sn=15;N=100)0.29640.000955.5341 Finally,theestimatedstatetrajectoriesoftheparticipatingltersareseeninandg.( 4-3 ).Dot-dotedlineshowsthetrajectoryofgenericPFwithN=100,stitchedlineshowsthetrajectoryofgenericPFwithN=2000,dot-dashedlineshowthetrajectoryofPFwithMCMCmovewithN=100,15cycles,andsolidlineshowsthetrajectoryofPFwithAdaptiveMCMCmethodwithN=100,15cycles.Fromthegure,wecanconrmthatthetrajectoryoftheadaptiveMCMCmovePFwassuperiortotheothersinspiteofthesmallnumberofparticles(N=100).First,theSIRlter(N=100)couldnottracktheobjectforsometimeatthebeginning.EventhoughthesecondSIRPF(N=2000)hassmoothertrajectorybutstillhassomedifcultiestotrackthemovingobjectcompletely.Next,theMCMCmovePFhasseenapoorertrajectorythanoneof 44

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Figure4-3. Trajectoriesofthebearingonlytrackingmodelwithvariousltermethods. thelargepopulationSIRPF(N=2000).Sincetheactualnumberofparticlesisonly100withsmallprocessnoise,theMCMCmovePFisnobetterthantherstSIRPFinthesenseofsamplediversity. Remark.Allofthesimulationexamplesareexecutedunderthefollowingenvironment,MATLAB7.9,AMDAthlon64DualCore1.7GHzandWindows7ProfessionalOS. 45

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CHAPTER5CONCLUSION 5.1ClosingStatement Theconventionalparticlelteringalgorithmsarebroadlyusedinvariousapplications.EventhoughtheMCMCmovePFisapopularchoicetoavoidthesampleimpoverishmentproblem,therestillexistpotentialdangersoffailureduetosampleimpoverishment.Inthisthesis,weproposedadaptiveMCMCmovealgorithmtoavoidthesampleimpoverishmentproblemeffectivelybygivingimprovedsamplediversity.Thisnewmethodadjuststheproposalnoisecovariance(orstandarddeviation)adaptivelybythescaleofacceptancerate(AR),whichplaysanimportantrole:apoorsamplediversityindicator.Theseriesofexperiment(giveninchapter4)showedthatthisadaptiveMCMCmovePF(withfewerparticles)improvessamplediversitywhilstincreasingthelteraccuracy.Inaddition,inthesenseofgreatercondence,theadaptiveMCMCmovePFshowedthelesserrorvariationamongalloftheparticipatedparticlelteringmethods.Consequently,wecanusethisalgorithmwhereanyharshenvironmentisrequiredtargettrackingapplications(describedearlierinthechapter1atthebeginningofthemotivationsection)bytheaccuracyandreliability. 5.2FutureWork Whilepromisehasbeenshown,thereexistsnumerousavenuesforfutureresearch;e.g. 1. IntherecentstudyofMCMCintegration[ 28 ]method,ithasbeenseenthatadaptivelyadjustingthesupportoftheproposaldensityfunctionimprovestheperformanceofMCMCintegration.Insuchcases,thesupportoftheproposaldensityfunctionisadjustedintwoways:extensionandshrinkage.Inthisthesis,wehaveshownonlythehigh-ARcases(i.e.extensioncase).Therefore,follow-upstudyofthereversedcases(i.e.thelow-ARcases)andtheireffectsonthelteraccuracywillprovideanextensionofthisresearch. 2. DesignationofthemultipleARlevelswassuggestedintheadaptiveMCMCmovePFprocess.Inthisstep,newextensionparameterwasobtainedviaempericalmethod.However,iftheoptimalrelationshipbetweenandARexists(i.e.afunctionalform),itwouldbebenecialtomaintainingtheappropriatesamplediversityinminimaltime. 46

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REFERENCES [1] M.S.Arulampalam,S.Maskell,N.Gordon,T.Clapp,Atutorialonparticleltersforonlinenonlinear/non-gaussianbayesiantracking,IEEETRANSACTIONSONSIGNALPROCESSING50(2002). [2] Y.Bar-Shalom,X.R.Li,T.Kirubarajan,EstimationwithApplicationstoTrackingandNavigation,WileyInterscience,2001. [3] J.Carpenter,P.Clifford,P.Fearnhead,Improvedparticlelterfornon-linearproblems,IEEProc.PartF:RadarandSonarNavigation146(1999)2. [4] A.Doucet,N.deFreitas,N.J.Gordon,eds.,SequentialMonteCarloMethodsinPractice,NewYorkSpringer,2001. [5] A.Doucet,S.Godsill,C.Andrieu,Onsequentialmontecarlosamplingmethodsforbayesianltering,Statisticsandcomputing10(2000)197. [6] A.Doucet,N.Gordon,V.Krishnamurthy,Particleltersforstateestimationofjumpmarkovlinearsystems,IEEETrans.SignalProcessing49(2001)613. [7] A.Doucet,A.M.Johansen,Atutorialonparticlelteringandsmoothing:fteenyearslater(2008). [8] W.R.Gilks,C.Berzuini,Followingamovingtarget-montecarloinferencefordynamicbayesianmodels,J.roy.statist.Soc.Ser.B63(2001)127. [9] N.J.Gordon,D.J.Salmond,A.F.M.Smith,Novelapproachtononlinear/non-gaussianbayesianstateestimation,IEEProc.-F140(1993)107. [10] J.M.Hammersley,K.W.Morton,Poorman'smontecarlo,JournaloftheRoyalStatisticalSociety16(1954)23. [11] J.Hartikainen,S.Sarkka,Optimallteringwithkalmanltersandsmoothers(2008). [12] W.K.Hastings,Montecarlosamplingmethodsusingmarkovchainsandtheirapplications,Biometrika57(1970)97. [13] A.H.Jazwinski,StochasticProcessesandFilteringTheory,SanDiego,CA:Academic,1970. [14] L.Jing,P.Vadakkepat,Interactingmcmcparticlelterfortrackingmaneuveringtarget,DigitalSignalProcessing20(2010)561. [15] S.J.Julier,J.K.Uhlmann,Anewextensionofthekalmanltertononlinearsystems,Proc.ofAeroSense:The11thInternationalSymposiumonAerospace/DefenceSensing,simulationandcontrols,Orlando,FloridaMultiSensorFusion,TrackingandResourceManagementII(1997). 47

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[16] R.E.Kalman,Anewapproachtolinearlteringandpredictionproblems,TransactionsoftheASME-JournalofBasicEngineering82(1960)35. [17] G.Kitagawa,Montecarlolterandsmootherfornon-gaussiannon-linearstatespacemodels,JournalOfComputationalandGraphicalStatistics5(1996)1. [18] A.Kong,J.S.Liu,W.H.Wong,Sequentialimputationsandbayesianmissingdataproblems,JournaloftheAmericanStatisticalAssociation89(1994)278. [19] M.Kumar,DesignandanalysisofstochasticdynamicalsystemswithFokker-Planckquation,Ph.D.thesis,TexasA&MUniversity,2009. [20] J.Liu,R.Chen,Sequentialmontecarlomethodsfordynamicsystems,JournalofAmericanStatisticalAssociation93(1998)1032. [21] J.MacCormick,A.Blake,Aprobabilisticexclusionprinciplefortrackingmultipleobjects,ProcInt.Conf.ComputerVision(1999)572. [22] P.S.Maybeck,Stochasticmodels,estimationandcontrol,AcademicPress2(1982). [23] R.vanderMerwe,A.Doucet,N.deFeritas,E.Wan,Theunscentedparticlelter,Tech.Rep.CUED/F-INFENG/TR380,CambridgeUniversityEngineeringDepartment(2000). [24] N.Metropolis,A.W.Rosenbluth,M.N.Rosenbluth,E.Teller,Equationsofstatecalculationsbyfastcomputingmachines,,JournalofChemicalPhysics21(1953)1087. [25] C.Musso,N.Oudjane,F.LeGland,Improvingregularisedparticlelters,SequentialMonteCarloMethodsinPractice(A.Doucet,N.deFeritas,andN.J.Gordon),NewYork:Springer,2001. [26] B.Ristic,S.Arulampalam,N.Gordon,BeyondtheKalmanFilter,ArtechHousePublishers,2004. [27] C.P.Robert,G.Casella,MonteCarloStatisticalMethods,NewYork:Springer,1999. [28] Z.Zhao,M.Kumar,Acomparativestudyofrandomizedalgorithmsformultidimensionalintegrationincomputationalphysics(2012). 48

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BIOGRAPHICALSKETCH JaeMyungYoonwasborninSeoul,RepublicofKorea,in1980.HereceivedhisBachelorofEngineeringdegreeincontrolengineeringfromKwangwoonUniversity,Seoul,RepublicofKorea,in2006.HethenworkedasanequipmentdevelopmentengineerattheMemorybusinessofSamsungElectronicsCo.Ltdfrom2006through2008.HejoinedtheUniversityofFloridatopursuehismaster'sdegreeinmechanicalengineeringin2010.HethenjoinedtheStochasticSystemsLaboratorytoresearchundertheadvisementofDr.MrinalKumar.HecompletedMasterofSciencefromtheUniversityofFloridainMay2012. 49