IEEE TRANSACTIONS ON SIGNAL PROCESSING VOL - PDF document

ARULAMPALAMetal.:TUTORIALONPARTICLEFILTERSWebegininSectionIIwithadescriptionofthenonlineartrackingproblemanditsoptimalBayesiansolution.Whencertainconstraintshold,thisoptimalsolutionistractable.TheKalmanfilterandgrid-basedfilter,whichisdescribedinSectionIII,aretwosuchsolutions.Often,theoptimalsolutionisintractable.ThemethodsoutlinedinSectionIVtakeseveraldifferentapproximationstrategiestotheoptimalsolution.TheseapproachesincludetheextendedKalmanfilter,approximategrid-basedfilters,andparticlefilters.Finally,inSectionVI,weuseasimplescalarexampletoillustratesomepointsabouttheapproachesdiscusseduptothispointandthendrawconclusionsinSectionVII.Thispaperisatutorial;therefore,tofacilitateeasyimplementation,the“pseudo-code”foralgorithmshasbeenincludedatrelevantpoints.II.NAYESIANRACKINGTodefinetheproblemoftracking,considertheevolutionofthestatesequence ofatargetgivenby (1)where isapossiblynonlinearfunctionofthestate , isani.i.d.processnoisese- aredimensionsofthestateandprocessnoisevectors,respectively,and isthesetofnaturalnumbers.Theobjectiveoftrackingistorecursivelyestimate frommea- (2)where isapossiblynonlinearfunc- isani.i.d.measurementnoisesequence, aredimensionsofthemeasurementandmeasure-mentnoisevectors,respectively.Inparticular,weseekfilteredestimatesof basedonthesetofallavailablemeasurements , uptotime FromaBayesianperspective,thetrackingproblemistore-cursivelycalculatesomedegreeofbeliefinthestate attime ,takingdifferentvalues,giventhedata uptotime .Thus,itisrequiredtoconstructthepdf .Itisassumedthattheinitialpdf ofthestatevector,whichisalsoknownastheprior,isavailable( beingthesetofnomeasure-ments).Then,inprinciple,thepdf maybeobtained,recursively,intwostages:predictionandupdate.Supposethattherequiredpdf attime isavailable.Thepredictionstageinvolvesusingthesystemmodel(1)toobtainthepriorpdfofthestateattime viatheChapman–Kolmogorovequation Notethatin(3),usehasbeenmadeofthefactthat , as(1)describesaMarkovprocessoforderone.Theprobabilisticmodelofthestateevolution isdefinedbythesystemequation(1)andtheknownstatisticsof Attimestep ,ameasurement becomesavailable,andthismaybeusedtoupdatetheprior(updatestage)viaBayes’rule wherethenormalizingconstant dependsonthelikelihoodfunction definedbythemeasurementmodel(2)andtheknownstatisticsof .Intheupdatestage(4),themeasurement isusedtomodifythepriordensitytoobtaintherequiredposteriordensityofthecurrentstate.Therecurrencerelations(3)and(4)formthebasisfortheoptimalBayesiansolution.Thisrecursivepropagationoftheposteriordensityisonlyaconceptualsolutioninthatingeneral,itcannotbedeterminedanalytically.Solutionsdoexistinare-strictivesetofcases,includingtheKalmanfilterandgrid-basedfiltersdescribedinthenextsection.Wealsodescribehow,whentheanalyticsolutionisintractable,extendedKalmanfilters,ap-proximategrid-basedfilters,andparticlefiltersapproximatetheoptimalBayesiansolution.III.OA.KalmanFilterTheKalmanfilterassumesthattheposteriordensityateverytimestepisGaussianand,hence,parameterizedbyameanandcovariance. isGaussian,itcanbeprovedthat isalsoGaussian,providedthatcertainassumptionshold[21]: and aredrawnfromGaussiandistributionsofknownparameters. isknownandisalinearfunctionof and .• isaknownlinearfunctionof and Thatis,(1)and(2)canberewrittenas (6) (7) and areknownmatricesdefiningthelinearfunctions.Thecovariancesof and are,respectively, and .Here,weconsiderthecasewhen and havezeromeanandarestatisticallyindependent.Notethatthesystemandmeasurementmatrices and ,aswellasnoiseparameters and ,areallowedtobetimevariant.TheKalmanfilteralgorithm,whichwasderivedusing(3)and(4),canthenbeviewedasthefollowingrecursiverelationship: (8) (9) Forclarity,theoptimalBayesiansolutionsolvestheproblemofrecursivelycalculatingtheexactposteriordensity.Anoptimalalgorithmisamethodfordeducingthissolution. IEEETRANSACTIONSONSIGNALPROCESSING,VOL.50,NO.2,FEBRUARY2002 (11) (12) (13) andwhere isaGaussiandensitywithargument ,mean ,andcovariance ,and (15) arethecovarianceoftheinnovationterm ,andtheKalmangain,respectively.Intheaboveequations,thetrans-poseofamatrix isdenotedby Thisistheoptimalsolutiontothetrackingproblem—ifthe(highlyrestrictive)assumptionshold.TheimplicationisthatnoalgorithmcaneverdobetterthanaKalmanfilterinthislinearGaussianenvironment.Itshouldbenotedthatitispossibletoderivethesameresultsusingaleastsquares(LS)argument[22].Allthedistributionsarethendescribedbytheirmeansandco-variances,andthealgorithmremainsunaltered,butarenotcon-strainedtobeGaussian.Assumingthemeansandcovariancestobeunbiasedandconsistent,thefilterthenoptimallyderivesthemeanandcovarianceoftheposterior.However,thisposte-riorisnotnecessarilyGaussian,andtherefore,ifoptimalityistheabilityofanalgorithmtocalculatetheposterior,thefilteristhennotcertaintobeoptimal.Similarly,ifsmoothedestimatesofthestatesarerequired,thatis,estimatesof ,where thentheKalmansmootheristheoptimalestimatorof Thisholdsif isfixed(fixed-lagsmoothing,ifabatchofdataareconsideredand fixed-intervalsmoothing),orifthestateataparticulartimeisofinterest isfixed(fixed-point).Theproblemofcalculatingsmootheddensitiesisofinterestbecausethedensitiesattime arethenconditionalnotonlyonmeasurementsuptoandincludingtimeindex butalsoonfuturemeasurements.Sincethereismoreinformationonwhichtobasetheestimation,thesesmootheddensitiesaretypicallytighterthanthefiltereddensities.Althoughthisistrue,thereisanalgorithmicissuethatshouldbehighlightedhere.Itispossibletoformulateabackward-timeKalmanfilterthatrecursesthroughthedatasequencefromthefinaldatatothefirstandthencombinestheestimatesfromtheforwardandbackwardpassestoobtainoverallsmoothedes-timates[20].Adifferentformulationimplicitlycalculatesthebackward-timestateestimatesandcovariances,recursivelyesti-matingthesmoothedquantities[38].Bothtechniquesarepronetohavingtocalculatematrixinversesthatdonotnecessarilyexist.Instead,itispreferabletopropagatedifferentquantitiesusinganinformationfilterwhencarryingoutthebackward-timerecursion[4].,thentheproblemreducestotheestimationof jz ereduptothispoint.B.Grid-BasedMethodsGrid-basedmethodsprovidetheoptimalrecursionofthefil-tereddensity ifthestatespaceisdiscreteandconsistsofafinitenumberofstates.Supposethestatespaceattime consistsofdiscretestates , .Foreachstate ,lettheconditionalprobabilityofthatstate,givenmea-surementsuptotime bedenotedby ,thatis, .Then,theposteriorpdf canbewrittenas (17)where istheDiracdeltameasure.Substitutionof(17)into(3)and(4)yieldsthepredictionandupdateequations,respec-tively (18) (19)where (20) Theaboveassumesthat and areknownbutdoesnotconstraintheparticularformofthesediscretedensi-ties.Again,thisistheoptimalsolutioniftheassumptionsmadeIV.SInmanysituationsofinterest,theassumptionsmadeabovedonothold.TheKalmanfilterandgrid-basedmethodscannot,therefore,beusedasdescribed—approximationsarenecessary.Inthissection,weconsiderthreeapproximatenonlinearBayesianfilters:a)extendedKalmanfilter(EKF);b)approximategrid-basedmethods;c)particlefilters.A.ExtendedKalmanFilterIf(1)and(2)cannotberewrittenintheformof(6)and(7)becausethefunctionsarenonlinear,thenalocallinearizationoftheequationsmaybeasufficientdescriptionofthenonlinearity.TheEKFisbasedonthisapproximation. isapprox-imatedbyaGaussian (22) (23) (24) ARULAMPALAMetal.:TUTORIALONPARTICLEFILTERS (25) (26) (27) andwherenow, and arenonlinearfunctions,and and arelocallinearizationsofthesenonlinearfunctions(i.e., (29) (30) (31) TheEKFasdescribedaboveutilizesthefirstterminaTaylorexpansionofthenonlinearfunction.AhigherorderEKFthatretainsfurthertermsintheTaylorexpansionexists,butthead-ditionalcomplexityhasprohibiteditswidespreaduse.Recently,theunscentedtransformhasbeenusedinanEKFframework[23],[42],[43].Theresultingfilter,whichisknownasthe“unscentedKalmanfilter,”considersasetofpointsthataredeterministicallyselectedfromtheGaussianapproximation .Thesepointsareallpropagatedthroughthetruenonlinearity,andtheparametersoftheGaussianapproximationarethenre-estimated.Forsomeproblems,thisfilterhasbeenshowntogivebetterperformancethanastandardEKFsinceitbetterapproximatesthenonlinearity;theparametersoftheGaussianapproximationareimproved.However,theEKFalwaysapproximates beGaussian.Ifthetruedensityisnon-Gaussian(e.g.,ifitisbimodalorheavilyskewed),thenaGaussiancanneverdescribeitwell.Insuchcases,approximategrid-basedfiltersandparticlefilterswillyieldanimprovementinperformanceincomparisontothatofanEKF[1].B.ApproximateGrid-BasedMethodsIfthestatespaceiscontinuousbutcanbedecomposedinto “cells,” : ,thenagrid-basedmethodcanbeusedtoapproximatetheposteriordensity.Specifically,supposetheapproximationtotheposteriorpdfat isgiven Then,thepredictionandupdateequationscanbewrittenas (34) (35)where (36) (37)Here, denotesthecenterofthe thcellattimeindex Theintegralsin(36)and(37)ariseduetothefactthatthegrid , ,representregionsofcontinuousstatespace,andthus,theprobabilitiesmustbeintegratedovertheseregions.Inpractice,tosimplifycomputation,afurtherapprox-imationismadeintheevaluationof .Specifically,theseweightsarecomputedatthecenterofthe“cell”corresponding (38) Thegridmustbesufficientlydensetogetagoodapproxi-mationtothecontinuousstatespace.Asthedimensionalityofthestatespaceincreases,thecomputationalcostoftheapproachthereforeincreasesdramatically.Ifthestatespaceisnotfiniteinextent,thenusingagrid-basedapproachnecessitatessometrun-cationofthestatespace.Anotherdisadvantageofgrid-basedmethodsisthatthestatespacemustbepredefinedand,there-fore,cannotbepartitionedunevenlytogivegreaterresolutioninhighprobabilitydensityregions,unlesspriorknowledgeisHiddenMarkovmodel(HMM)filters[30],[35],[36],[39]areanapplicationofsuchapproximategrid-basedmethodsinafixed-intervalsmoothingcontextandhavebeenusedexten-sivelyinspeechprocessing.InHMM-basedtracking,acommonapproachistousetheViterbialgorithm[18]tocalculatetheaposterioriestimateofthepaththroughthetrellis,thatis,thesequenceofdiscretestatesthatmaximizestheprob-abilityofthestatesequencegiventhedata.Anotherapproach,duetoBaum–Welch[35],istocalculatetheprobabilityofeachdiscretestateateachtimeepochgiventheentiredatasequence.V.PARTICLEILTERINGA.SequentialImportanceSampling(SIS)AlgorithmThesequentialimportancesampling(SIS)algorithmisaMonteCarlo(MC)methodthatformsthebasisformostsequentialMCfiltersdevelopedoverthepastdecades;see[13],TheViterbiandBaum–Welchalgorithmsarefrequentlyappliedwhenthestatespaceisapproximatedtobediscrete.Thealgorithmsareoptimalifandonlyiftheunderlyingstatespaceistrulydiscreteinnature. IEEETRANSACTIONSONSIGNALPROCESSING,VOL.50,NO.2,FEBRUARY2002[14].ThissequentialMC(SMC)approachisknownvariouslyasbootstrapfiltering[17],thecondensationalgorithm[29],particlefiltering[6],interactingparticleapproximations[10],[11],andsurvivalofthefittest[24].Itisatechniqueforimple-mentingarecursiveBayesianfilterbyMCsimulations.Thekeyideaistorepresenttherequiredposteriordensityfunctionbyasetofrandomsampleswithassociatedweightsandtocomputeestimatesbasedonthesesamplesandweights.Asthenumberofsamplesbecomesverylarge,thisMCcharacterizationbecomesanequivalentrepresentationtotheusualfunctionaldescriptionoftheposteriorpdf,andtheSISfilterapproachestheoptimalBayesianestimate.Inordertodevelopthedetailsofthealgorithm,let denotearandommeasurethatcharacterizestheposteriorpdf ,where , isasetofsupportpointswithassociatedweights , and , isthesetofallstatesuptotime .Theweightsarenormalizedsuchthat .Then,theposteriordensityat canbeapproximatedas Wethereforehaveadiscreteweightedapproximationtothetrueposterior, .Theweightsarechosenusingtheprincipleofimportancesampling[3],[12].Thisprinciplereliesonthefollowing.Suppose isaprobabilitydensityfromwhichitisdifficulttodrawsamplesbutforwhich beevaluated[aswellas uptoproportionality].Inaddition, , besamplesthatareeasilygener-atedfromaproposal calledanimportancedensity.Then,aweightedapproximationtothedensity isgivenby (41)where isthenormalizedweightofthe thparticle.Therefore,ifthesamples weredrawnfromanimpor-tancedensity ,thentheweightsin(40)aredefinedby(42)tobe Returningtothesequentialcase,ateachiteration,onecouldhavesamplesconstitutinganapproximationto andwanttoapproximate withanewsetofsamples.Iftheimportancedensityischosentofactorizesuchthat thenonecanobtainsamples byaugmentingeachoftheexistingsamples thenewstate , .Toderivetheweightupdateequation, isfirstexpressedintermsof , ,and .Notethat(4)canbederivedbyintegrating(45) (45) Bysubstituting(44)and(46)into(43),theweightupdateequationcanthenbeshowntobe Furthermore,if , , ,thentheimportancedensitybecomesonlydependenton and .Thisisparticularlyusefulinthecommoncasewhenonlyafilteredestimateof isrequiredateachtimestep.Fromthispointon,wewillassumesuchacase,exceptwhenexplicitlystatedotherwise.Insuchscenarios,only needbestored;therefore,onecandiscardthepath andhistoryofobservations .Themodifiedweightisthen andtheposteriorfiltereddensity canbeapproxi-matedas wheretheweightsaredefinedin(48).Itcanbeshownthatas ,theapproximation(49)approachesthetrueposterior TheSISalgorithmthusconsistsofrecursivepropagationoftheweightsandsupportpointsaseachmeasurementisreceivedsequentially.Apseudo-codedescriptionofthisalgorithmisgivenbyalgorithm1. Algorithm1:SISParticleFilter SIS FOR —Draw , —Assigntheparticleaweight, accordingto(48) ENDFOR 1)DegeneracyProblem:AcommonproblemwiththeSISparticlefilteristhedegeneracyphenomenon,whereafterafewiterations,allbutoneparticlewillhavenegligibleweight.Ithas ARULAMPALAMetal.:TUTORIALONPARTICLEFILTERSbeenshown[12]thatthevarianceoftheimportanceweightscanonlyincreaseovertime,andthus,itisimpossibletoavoidthedegeneracyphenomenon.Thisdegeneracyimpliesthatalargecomputationaleffortisdevotedtoupdatingparticleswhosecon-tributiontotheapproximationto isalmostzero.Asuitablemeasureofdegeneracyofthealgorithmistheeffectivesamplesize introducedin[3]and[28]anddefinedas Var where , isreferredtoasthe“trueweight.”Thiscannotbeevaluatedexactly,butanestimate of canbeobtainedby (51)where isthenormalizedweightobtainedusing(47).Notice ,andsmall indicatesseveredegeneracy.Clearly,thedegeneracyproblemisanundesirableeffectinpar-ticlefilters.Thebruteforceapproachtoreducingitseffectistouseaverylarge .Thisisoftenimpractical;therefore,werelyontwoothermethods:a)goodchoiceofimportancedensityandb)useofresampling.Thesearedescribednext.2)GoodChoiceofImportanceDensity:Thefirstmethodin-volveschoosingtheimportancedensity tomin-imizeVar sothat ismaximized.Theoptimalimpor-tancedensityfunctionthatminimizesthevarianceofthetrue conditionedon and hasbeenshown[12]tobe Substitutionof(52)into(48)yields Thischoiceofimportancedensityisoptimalsinceforagiven , takesthesamevalue,whateversampleisdrawnfrom , .Hence,conditionalon ,Var Thisisthevarianceofthedifferent resultingfromdifferent Thisoptimalimportancedensitysuffersfromtwomajordrawbacks.Itrequirestheabilitytosamplefrom andtoevaluatetheintegraloverthenewstate.Inthegeneralcase,itmaynotbestraightforwardtodoeitherofthesethings.Therearetwocaseswhenuseoftheoptimalimportancedensityispossible.Thefirstcaseiswhen isamemberofafiniteset.Insuchcases,theintegralin(53)becomesasum,andsampling ispossible.Anexampleofanapplication isamemberofafinitesetisaJump–Markovlinearsystemfortrackingmaneuveringtargets[15],wherebythedis-cretemodalstate(definingthemaneuverindex)istrackedusingaparticlefilter,and(conditionedonthemaneuverindex)thecontinuousbasestateistrackedusingaKalmanfilter.Analyticevaluationispossibleforasecondclassofmodelsforwhich isGaussian[12],[9].Thiscanoccurifthedynamicsarenonlinearandthemeasurementslinear.Suchasystemisgivenby (54) (55)where (56) (57)and : isanonlinearfunction, isanobservationmatrix,and and aremutuallyindependenti.i.d.Gaussiansequenceswith and .Defining (58) oneobtains (60)and Formanyothermodels,suchanalyticevaluationsarenotpossible.However,itispossibletoconstructsuboptimalapproximationstotheoptimalimportancedensitybyusinglocallinearizationtechniques[12].SuchlinearizationsuseanimportancedensitythatisaGaussianapproximationto .AnotherapproachistoestimateaGaussianap-proximationto usingtheunscentedtransform[40].Theauthors’opinionisthattheadditionalcomputationalcostofusingsuchanimportancedensityisoftenmorethanoffsetbyareductioninthenumberofsamplesrequiredtoachieveacertainlevelofperformance.Finally,itisoftenconvenienttochoosetheimportanceden-sitytobetheprior Substitutionof(62)into(48)thenyields Thiswouldseemtobethemostcommonchoiceofimpor-tancedensitysinceitisintuitiveandsimpletoimplement.How-ever,thereareaplethoraofotherdensitiesthatcanbeused,andasillustratedbySectionVI,thechoiceisthecrucialdesignstepinthedesignofaparticlefilter.3)Resampling:Thesecondmethodbywhichtheeffectsofdegeneracycanbereducedistouseresamplingwheneverasig-nificantdegeneracyisobserved(i.e.,when fallsbelowsomethreshold ).Thebasicideaofresamplingistoelimi-nateparticlesthathavesmallweightsandtoconcentrateonpar-ticleswithlargeweights.Theresamplingstepinvolvesgener-atinganewset byresampling(withreplacement) timesfromanapproximatediscreterepresentationof givenby sothat .Theresultingsampleisinfactani.i.d.samplefromthediscretedensity(64);therefore,the IEEETRANSACTIONSONSIGNALPROCESSING,VOL.50,NO.2,FEBRUARY2002weightsarenowresetto .Itispossibletoimple-mentthisresamplingprocedurein operationsbysam- ordereduniformsusinganalgorithmbasedonorderstatistics[6],[37].Notethatotherefficient(intermsofreducedMCvariation)resamplingschemes,suchasstratifiedsamplingandresidualsampling[28],maybeappliedasalternativestothisalgorithm.Systematicresampling[25]istheschemepreferredbytheauthors[sinceitissimpletoimplement,takes time,andminimizestheMCvariation],anditsoperationisde-scribedinAlgorithm2,where istheuniformdistributionontheinterval (inclusiveofthelimits).Foreachresam-pledparticle ,thisresamplingalgorithmalsostorestheindexofitsparent,whichisdenotedby .Thismayappearunneces-saryhere(andis),butitprovesusefulinSectionV-B2.AgenericparticlefilteristhenasdescribedbyAlgorithm3.Althoughtheresamplingstepreducestheeffectsofthede-generacyproblem,itintroducesotherpracticalproblems.First,itlimitstheopportunitytoparallelizesincealltheparticlesmustbecombined.Second,theparticlesthathavehighweights arestatisticallyselectedmanytimes.Thisleadstoalossofdi-versityamongtheparticlesastheresultantsamplewillcontainmanyrepeatedpoints.Thisproblem,whichisknownasimpoverishment,issevereinthecaseofsmallprocessnoise.Infact,forthecaseofverysmallprocessnoise,allparticleswillcollapsetoasinglepointwithinafewiterations.sincethediversityofthepathsoftheparticlesisreduced,anysmoothedestimatesbasedontheparticles’pathsdegenerate.Schemesexisttocounteractthiseffect.Oneapproachconsidersthestatesfortheparticlestobepredeterminedbytheforwardfilterandthenobtainsthesmoothedestimatesbyrecalculatingtheparticles’weightsviaarecursionfromthefinaltothefirsttimestep[16].AnotherapproachistouseMCMC[5]. Algorithm2:ResamplingAlgorithm , RESAMPLE InitializetheCDF: FOR —ConstructCDF: ENDFOR StartatthebottomoftheCDF: Drawastartingpoint: FOR —MovealongtheCDF: —WHILE —ENDWHILE—Assignsample: —Assignweight: —Assignparent: ENDFOR Iftheprocessnoiseiszero,thenusingaparticlefilterisnotentirelyap-propriate.Particlefilteringisamethodwellsuitedtotheestimationofdynamicstates.Ifstaticstates,whichcanberegardedasparameters,needtobeestimatedthenalternativeapproachesarenecessary[7],[27].Sincetheparticlesactuallyrepresentpathsthroughthestatespace,bystoringthetrajectorytakenbyeachparticle,fixed-lagandfixed-pointsmoothedesti-matesofthestatecanbeobtained[4]. Algorithm3:GenericParticleFilter PF , FOR —Draw —Assigntheparticleaweight, accordingto(48) ENDFOR Calculatetotalweight: SUM FOR —Normalize: ENDFOR Calculate using(51) IF —Resampleusingalgorithm2: RESAMPLE ENDIF Therehavebeensomesystematictechniquesproposedrecentlytosolvetheproblemofsampleimpoverishment.Onesuchtechniqueistheresample-movealgorithm[19],whichisnotbedescribedindetailinthispaper.Althoughthistechniquedrawsconceptuallyonthesametechnologiesofimportancesampling-resamplingandMCMCsampling,itavoidssampleimpoverishment.Itdoesthisinarigorousmannerthatensurestheparticlesasymtoticallyapproximatesamplesfromtheposteriorand,therefore,isthemethodofchoiceoftheauthors.Analternativesolutiontothesameproblemisregularization[31],whichisdiscussedinSectionV-B3.Thisapproachisfrequentlyfoundtoimproveperformance,despitealessrigorousderivationandisincludedhereinpreferencetotheresample-movealgorithmsinceitsuseissowidespread.4)TechniquesforCircumventingtheUseofaSuboptimalIm-portanceDensity:Itisoftenthecasethatagoodimportancedensityisnotavailable.Forexample,iftheprior usedastheimportancedensityandisamuchbroaderdistribu-tionthanthelikelihood ,thenonlyafewparticleswillhaveahighweight.Methodsexistforencouragingtheparticlestobeintherightplace;theuseofbridgingdensities[8]andprogressivecorrection[33]bothintroduceintermediatedistri-butionsbetweenthepriorandlikelihood.Theparticlesarethenreweightedaccordingtotheseintermediatedistributionsandre-sampled.This“herds”theparticlesintotherightpartofthestateAnotherapproachknownaspartitionedsampling[29]isusefulifthelikelihoodisverypeakedbutcanbefactorizedintoanumberofbroaderdistributions.Typically,thisoccursbecauseeachofthepartitioneddistributionsarefunctionsofsome(notall)ofthestates.Bytreatingeachofthesepartitioneddistributionsinturnandresamplingonthebasisofeachsuchpartitioneddistribution,theparticlesareagainherdedtowardthepeakedlikelihood.B.OtherRelatedParticleFiltersThesequentialimportancesamplingalgorithmpresentedinSectionV-Aformsthebasisformostparticlefiltersthathave ARULAMPALAMetal.:TUTORIALONPARTICLEFILTERSbeendevelopedsofar.ThevariousversionsofparticlefiltersproposedintheliteraturecanberegardedasspecialcasesofthisgeneralSISalgorithm.ThesespecialcasescanbederivedfromtheSISalgorithmbyanappropriatechoiceofimportancesamplingdensityand/ormodificationoftheresamplingstep.Below,wepresentthreeparticlefiltersproposedintheliteratureandshowhowthesemaybederivedfromtheSISalgorithm.Theparticlefiltersconsideredarei)samplingimportanceresampling(SIR)filter;ii)auxiliarysamplingimportanceresampling(ASIR)filter;iii)regularizedparticlefilter(RPF).1)SamplingImportanceResamplingFilter:TheSIRfilterproposedin[17]isanMCmethodthatcanbeappliedtorecur-siveBayesianfilteringproblems.TheassumptionsrequiredtousetheSIRfilterareveryweak.Thestatedynamicsandmea-surementfunctions and in(1)and(2),respec-tively,needtobeknown,anditisrequiredtobeabletosamplerealizationsfromtheprocessnoisedistributionof fromtheprior.Finally,thelikelihoodfunction tobeavailableforpointwiseevaluation(atleastuptopropor-tionality).TheSIRalgorithmcanbeeasilyderivedfromtheSISalgorithmbyanappropriatechoiceofi)theimportanceden-sity,where ischosentobethepriordensity ,andii)theresamplingstep,whichistobeappliedateverytimeindex.Theabovechoiceofimportancedensityimpliesthatweneedsamplesfrom .Asample canbegeneratedbyfirstgeneratingaprocessnoisesample andsetting , ,where isthepdfof .Forthisparticularchoiceofimportancedensity,itisevidentthattheweightsaregivenby However,notingthatresamplingisappliedateverytimeindex,wehave ;therefore Theweightsgivenbytheproportionalityin(66)arenormalizedbeforetheresamplingstage.AniterationofthealgorithmisthendescribedbyAlgorithm4.AstheimportancesamplingdensityfortheSIRfilterisinde-pendentofmeasurement ,thestatespaceisexploredwithoutanyknowledgeoftheobservations.Therefore,thisfiltercanbeinefficientandissensitivetooutliers.Furthermore,asresam-plingisappliedateveryiteration,thiscanresultinrapidlossofdiversityinparticles.However,theSIRmethoddoeshavetheadvantagethattheimportanceweightsareeasilyevaluatedandthattheimportancedensitycanbeeasilysampled. Algorithm4:SIRParticleFilter SIR FOR —Draw —Calculate ENDFOR Calculatetotalweight: SUM FOR —Normalize: ENDFOR Resampleusingalgorithm2: RESAMPLE , 2)AuxiliarySamplingImportanceResamplingFilter:ASIRfilterwasintroducedbyPittandShephard[34]asavariantofthestandardSIRfilter.ThisfiltercanbederivedfromtheSISframeworkbyintroducinganimportancedensity whichsamplesthepair ,where referstotheindexoftheparticleat ByapplyingBayes’rule,aproportionalitycanbederivedfor as TheASIRfilteroperatesbyobtainingasamplefromthejoint andthenomittingtheindices inthepair toproduceasample fromthemarginalized .Theimportancedensityusedtodrawthe isdefinedtosatisfytheproportionality (68)where issomecharacterizationof ,given .Thiscouldbethemean,inwhichcase, orasample .Bywriting anddefining itfollowsfrom(68)that Thesample isthenassignedaweightpropor-tionaltotheratiooftheright-handsideof(67)to(68) ThealgorithmthenbecomesthatdescribedbyAlgorithm5. Algorithm5:AuxiliaryParticleFilter APF FOR —Calculate —Calculate . ENDFOR Calculatetotalweight: SUM FOR —Normalize: ENDFOR Resampleusingalgorithm2: RESAMPLE FOR —Draw asintheSIRfilter.—Assignweight using(72) IEEETRANSACTIONSONSIGNALPROCESSING,VOL.50,NO.2,FEBRUARY2002 ENDFOR Calculatetotalweight: SUM FOR —Normalize: ENDFOR Althoughunnecessary,theoriginalASIRfilterasproposedin[34]consistedofonemorestep,namely,aresamplingstage,toproduceani.i.d.sample withequalweights.ComparedwiththeSIRfilter,theadvantageoftheASIRfilteristhatitnaturallygeneratespointsfromthesampleat which,conditionedonthecurrentmeasurement,aremostlikelytobeclosetothetruestate.ASIRcanbeviewedasresamplingattheprevioustimestep,basedonsomepointestimates thatcharacterize .Iftheprocessnoiseissmallsothat iswellcharacterizedby ,thenASIRisoftennotsosensitivetooutliersasSIR,andtheweights aremoreeven.However,iftheprocessnoiseislarge,asinglepointdoesnot well,andASIRresamplesbasedonapoorapproximationof .Insuchscenarios,theuseofASIRthendegradesperformance.3)RegularizedParticleFilter:RecallthatresamplingwassuggestedinSectionV-B1asamethodtoreducethedegen-eracyproblem,whichisprevalentinparticlefilters.However,itwaspointedoutthatresamplinginturnintroducedotherprob-lemsand,inparticular,theproblemoflossofdiversityamongtheparticles.Thisarisesduetothefactthatintheresamplingstage,samplesaredrawnfromadiscretedistributionratherthanacontinuousone.Ifthisproblemisnotaddressedproperly,itmayleadto“particlecollapse,”whichisaseverecaseofsampleimpoverishmentwhereall particlesoccupythesamepointinthestatespace,givingapoorrepresentationoftheposteriordensity.Amodifiedparticlefilterknownastheregularizedpar-ticlefilter(RPF)wasproposed[31]asapotentialsolutiontotheaboveproblem.TheRPFisidenticaltotheSIRfilter,exceptfortheresam-plingstage.TheRPFresamplesfromacontinuousapproxima-tionoftheposteriordensity ,whereastheSIRresam-plesfromthediscreteapproximation(64).Specifically,intheRPF,samplesaredrawnfromtheapproximation (73)where istherescaledKerneldensity , istheKernelband-width(ascalarparameter), isthedimensionofthestatevector ,and , arenormalizedweights.TheKerneldensityisasymmetricprobabilitydensityfunctionsuch TheKernel andbandwidth arechosentominimizethemeanintegratedsquareerror(MISE)betweenthetrueposteriordensityandthecorrespondingregularizedempiricalrepresenta-tionin(73),whichisdefinedas (75)where denotestheapproximationto givenbytheright-handsideof(73).Inthespecialcaseofallthesampleshavingthesameweight,theoptimalchoiceofthekernelistheEpanechnikovkernel[31] if otherwise(76)where isthevolumeoftheunithyperspherein .Fur-thermore,whentheunderlyingdensityisGaussianwithaunitcovariancematrix,theoptimalchoiceforthebandwidthis[31] (77) (78) Algorithm6:RegularizedParticleFilter RPF FOR —Draw —Assigntheparticleaweight, accordingto(48) ENDFOR Calculatetotalweight: SUM FOR —Normalize: ENDFOR Calculate using(51) IF —Calculatetheempiricalcovariance of , —Compute suchthat —Resampleusingalgorithm2: RESAMPLE , —FOR Draw fromtheEpanechnikov —ENDFOR ENDIF Alhoughtheresultsof(76)and(77)and(78)areoptimalonlyinthespecialcaseofequallyweightedparticlesandunderlyingGaussiandensity,theseresultscanstillbeusedinthegeneralcasetoobtainasuboptimalfilter.OneiterationoftheRPFisde-scribedbyAlgorithm6.TheRPFonlydiffersfromthegenericparticlefilterdescribedbyAlgorithm3asaresultoftheaddi-tionoftheregularizationstepswhenconductingtheresampling.NotealsothatthecalculationoftheempiricalcovariancematrixAsobservedbyoneoftheanonymousreviewers,itisworthnotingthattheuseoftheKernelapproximationbecomeincreasinglylessappropriateasthedimensionalityofthestateincreases. ARULAMPALAMetal.:TUTORIALONPARTICLEFILTERSTABLEIABLEOFTHEECTIONSOFTHERTICLEIGURESTHATELATETOTHERMSEVVERAGED100MCR iscarriedoutpriortotheresamplingandisthereforeafunc-tionofboththe and .Thisisdonesincetheaccuracyofanyestimateofafunctionofthedistributioncanonlydecreaseasaresultoftheresampling.Ifquantitiessuchasthemeanandcovarianceofthesamplesaretobeoutput,thentheseshouldbecalculatedpriortoresampling.Byfollowingtheaboveprocedure,wegenerateani.i.d.randomsample drawnfrom(73).Intermsofcomplexity,theRPFiscomparablewithSIRsinceitonlyrequires additionalgenerationsfromthekernel ateachtimestep.TheRPFhasthetheoreticdisadvantagethatthesamplesarenolongerguaranteedtoasymtoticallyapprox-imatethosefromtheposterior.Inpracticalscenarios,theRPFperformanceisbetterthantheSIRincaseswheresampleim-poverishmentissevere,forexample,whentheprocessnoiseisVI.EHere,weconsiderthefollowingsetofequationsasanillus-trativeexample: (79) orequivalently (81) (82)where andwhere and arezeromeanGaussianrandomvariableswithvariances and ,respectively.Weuse and .Thisexamplehasbeenanalyzedbeforeinmanypublications[5],[17],[25].WeconsidertheperformanceofthealgorithmsdetailedinTableI.Inordertoqualitativelygaugeperformanceanddis-cussresultingissues,weconsideroneexemplarrun.Inordertoquantifyperformance,weusethetraditionalmeasureofper-formance:theRootMeanSquaredError(RMSE).Itshouldbenotedthatthismeasureofperformanceisnotexceptionallymeaningfulforthismultimodalproblem.However,ithasbeenusedextensivelyintheliteratureandisincludedhereforthatreasonandbecauseitfacilitatesquantitativecomparison.Forreference,thetruestatesfortheexemplarrunareshowninFig.1andthemeasurementsinFig.2. Fig.1.Figureofthetruevaluesofthestate asafunctionoffortheexemplarrun. Fig.2.Figureofthemeasurements ofthestates showninFig.1forthesameexemplarrun.Theapproximategrid-basedmethoduses50stateswithcen-tersequallyspacedon .Alltheparticlefiltershave50particlesandemployresamplingateverytimestep( Theauxiliaryparticlefilteruses .Theregu-larizedparticlefilterusesthekernelandbandwidthdescribedinSectionV-B3.Tovisualizethedensitiesinferredbytheapproximategrid-basedandparticlefilters,thetotalprobabilitymassatanytimeineachof50equallyspacedregionson isshownasimagesinFigs.5–9.Atanygiventime(andinanyverticalslicethroughtheimage),darkerregionsrepresenthigherprobabilitythanlighterregions.Agraduatedscalerelatingintensitytoprob-abilitymassinapixelisshownnexttoeachimage.A.EKFTheEKFslocallinearizationandGaussianapproxima-tionarenotasufficientdescriptionofthenonlinearandnon-Gaussiannatureoftheexample.OncetheEKFcannotadequatelyapproximatethebimodalnatureoftheunderlying IEEETRANSACTIONSONSIGNALPROCESSING,VOL.50,NO.2,FEBRUARY2002 Fig.3.EvolutionoftheEKFsmeanestimateofthestate. Fig.4.EvolutionoftheupperandlowerpositionsofthestateasestimatedbytheEKF(dotted)withthetruestatealsoshown(solid).posterior,theGaussianapproximationfails—theEKFispronetoeitherchoosingthe“wrong”modeorjustsittingontheaveragebetweenthemodes.Asaresultofthisinabilitytoadequatelyapproximatethedensity,thelinearizationapproxi-mationbecomespoor.ThiscanbeseenfromFig.3.Themeanofthefilterisrarelyclosetothetruestate.WerethedensitytobeGaussian,onewouldexpectthestatetobewithintwostandarddeviationsofthemeanapproximately95%ofthetime.FromFig.4,itisev-identthattherearetimeswhenthedistributionissufficientlybroadtocapturethetruestateinthisregionbutthattherearealsotimeswhenthefilterbecomeshighlyoverconfidentofabi-asedestimateofthestate.TheimplicationofthisisthatitisverydifficulttodetectinconsistentEKFerrorsautomaticallyonline.TheRMSEmeasureindicatesthattheEKFistheleastaccu-rateofthealgorithmsatapproximatingtheposterior.Theap-proximationsmadebytheEKFareinappropriateinthisex- Fig.5.Imagerepresentingevolutionofprobabilitydensityforapproximategrid-basedfilter.B.ApproximateGrid-BasedFilterThisexampleislowdimensional,andtherefore,onewouldexpectthatanapproximategrid-basedapproachwouldperformwell.Fig.5showsthisisindeedthecase.Thegrid-basedap-proximationisabletomodelthemultimodalityoftheproblem.Usingtheapproximategrid-basedfilterratherthananEKFyieldsamarkedreductioninRMSerrors.Aparticlefilterwith particlesconducts operationsperiteration,whereasanapproximategrid-basedfiltercarriesout operationswith cells.ItisthereforesurprisingthattheRMSerrorsfortheapproximategridarelargerthanthoseoftheparticlefilter.Theauthorssuspectthatthisisanartifactofthegridbeingfixed;theresolutionofthealgorithmispredefined,andthefixedposi-tionofthegridpointsmeansthatthegridpointsnear 25con-tributesignificantlytotheerrorwhenthetruestateisfarfromthesevalues.C.SIRParticleFilterUsingthepriordistributionastheimportancedensityisinsomesenseregardedasastandardSIRparticlefilterand,there-fore,isanappropriateparticlefilteralgorithmwithwhichtobegin.AscanbeseenfromFig.6,theSIRparticlefiltergivesdisappointingresultswiththelownumberofparticlesusedhere.Thespeckledappearanceofthefigureisaresultofsamplingalownumberofparticlesfromthe(broad)prior.Itisanartifactresultingfromtheinadequateamountofsampling.TheRMSEmetricshowsamarginalimprovementovertheapproximategrid-basedfilter.Toachievesmallererrors,onecouldsimplyincreasethenumberofparticles,buthere,wewillnowinvestigatetheeffectofusingthealternativeparticlefilteralgorithmsdescribeduptothispoint.D.AuxiliaryParticleFilterOnewaytoreduceerrorsmightbethattheproposedpar-ticlepositionsarechosenbadly.Onemightthereforethinkthatchoosingtheproposedparticlesinamoreintelligentmannerwouldyieldbetterresults.Anauxiliaryparticlefilterwouldthen ARULAMPALAMetal.:TUTORIALONPARTICLEFILTERS Fig.6.ImagerepresentingevolutionofprobabilitydensityforSIRparticlefilter. Fig.7.Imagerepresentingevolutionofprobabilitydensityforauxiliaryparticlefilter.seemtobeanappropriatecandidatereplacementalgorithmforSIR.Here,wehave asasamplefrom AsshownbyFig.7,forthisexample,theauxiliaryparticlefilterperformswell.ThereisarguablylessspeckleinFig.7thaninFig.6,andtheprobabilitymassappearstobebetterconcentratedaroundthetruestate.However,onemightthinkthisproblemisnotverywellsuitedtoanauxiliaryparticlefiltersincethepriorisoftenmuchbroaderthanthelikelihood.Whenthepriorisbroad,thoseparticleswithanoiserealizationthathappenstohaveahighlikelihoodareresampledmanytimes.ThereisnoguaranteethatothersamplesfromthepriorwillalsolieinthesameregionofthestatespacesinceonlyasinglepointisbeingusedtocharacterizethefiltereddensityforeachTheRMSerrorsareslightlyreducedfromthoseforSIR. Fig.8.Imagerepresentingevolutionofprobabilitydensityforregularizedparticlefilter.E.RegularizedParticleFilterUsingtheregularizedparticlefilterresultsinasmoothingoftheapproximationtotheposterior.ThisisapparentfromFig.8.Thespeckleisreducedandthepeaksbroadenedwhencomparedwiththepreviousparticlefilters’images.TheregularizedparticlefiltergivesverysimilarRMSerrorstotheSIRparticlefilter.Theregularizationdoesnotresultinasignificantreductioninerrorsforthisdataset.F.“Likelihood”ParticleFilterAlltheaforementionedparticlefilterssharethepriorasapro-posaldensity.Forthisexample,muchofthetime,thelikelihoodisfartighterthantheprior.Asaresult,theposterioriscloserinsimilaritytothelikelihoodthantotheprior.Theimportancedensityisanapproximationtotheposterior.Therefore,usingabetterapproximationbasedonthelikelihood,ratherthantheprior,canbeexpectedtoimproveperformance.Fig.9showsthattheuseofsuchanimportancedensity(seetheAppendixfordetails)yieldsareductioninspeckleandthatthepeaksofthedensityarecloseronaveragetothetruestatethanforanyoftheotherparticlefilters.TheRMSerrorsaresimilartothosefortheAuxiliaryparticlefilter.G.CrucialStepintheApplicationofaParticleFilterTheRMSerrorsindicatethatinhighlynonlinearenviron-ments,anonlinearfiltersuchasanapproximategrid-basedfilterorparticlefilteroffersanimprovementinperformanceoveranEKF.Thisimprovementresultsfromapproximatingthedensityratherthanthemodels.Whenusingaparticlefilter,onecanoftenexpectandfre-quentlyachieveanimprovementinperformancebyusingfarmoreparticlesoralternativelybyemployingregularizationorusinganauxiliaryparticlefilter.Forthisexample,aslightim-provementinRMSerrorsispossiblebyusinganimportancedensityotherthan .Theauthorsassertthatanim-portancedensitytunedtoaparticularproblemwillyieldanap-propriatetradeoffbetweenthenumberofparticlesandthecom- IEEETRANSACTIONSONSIGNALPROCESSING,VOL.50,NO.2,FEBRUARY2002 Fig.9.Imagerepresentingevolutionofprobabilitydensityfor“likelihood”particlefilter.putationalexpensenecessaryforeachparticle,givingthebestqualitativeperformancewithaffordablecomputationaleffort.Thecrucialpointtoconveyisthatalltherefinementsoftheparticlefilterassumethatthechoiceofimportancedensityhasalreadybeenmade.Choosingtheimportancedensitytobewellsuitedtoagivenapplicationrequirescarefulthought.Thechoicemadeiscrucial.VII.CForaparticularproblem,iftheassumptionsoftheKalmanfilterorgrid-basedfiltershold,thennootheralgorithmcanout-performthem.However,inavarietyofrealscenarios,theas-sumptionsdonothold,andapproximatetechniquesmustbeem-ployed.TheEKFapproximatesthemodelsusedforthedynamicsandmeasurementprocessinordertobeabletoapproximatetheprobabilitydensitybyaGaussian.Approximategrid-basedfil-tersapproximatethecontinuousstatespaceasasetofdiscreteregions.Thisnecessitatesthepredefinitionoftheseregionsandbecomesprohibitivelycomputationallyexpensivewhendealingwithhigh-dimensionalstatespaces[3].Particlefilteringapprox-imatesthedensitydirectlyasafinitenumberofsamples.Anumberofdifferenttypesofparticlefilterexist,andsomehavebeenshowntooutperformotherswhenusedforparticularap-plications.However,whendesigningaparticlefilterforapar-ticularapplication,itisthechoiceofimportancedensitythatisMPORTANCEENSITYFORARTICLEILTERThisAppendixdescribestheimportancedensityforthe“like-lihood”particlefilter,whichisintendedtoillustratethecrucialnatureofthechoiceofimportancedensityinaparticlefilter.Thisimportancedensityisnotintendedtobegenericallyappli-cablebuttobeonechosentoworkwellforthespecificproblemandparametersdescribedinSectionVI.Tokeepthenotationsimple,throughoutthisAppendix, .Forauniformprioron ,thedensity canbewrittenbyBayes’ruleas Wecanthensample [samples arerepeat-edlydrawnfrom untiloneisdrawnsuch ,i.e.,onesuchthat ].Then, canbechosentobeapairofdeltafunctions Thiscanthenbeusedtoforma“Likelihood”basedimpor-tancedensitythatsamples conditionalon andindepen-dentlyfrom Theweightofthesamplecanbecalculatedaccordingto(47) (87) (88) Now, , and areconstant;there-fore,theydisappear,leaving Now,theratioof to needscarefulconsid-eration.Althoughthevaluesof and beinitiallythoughttobeproportional,theyareprobabilityden-sitiesdefinedwithrespecttoadifferentmeasure(i.e.,adif-ferentparameterizationofthespace).Since integratestounityover while integratestounityover theratiooftheprobabilitydensitiesisthenproportionaltotheinverseoftheratioofthelengths, and .Theratioof to isthedeterminantoftheJacobianofthetransformationfrom to Anexpressionfortheweightisthenforthcoming: TheparticlefilterthatresultsfromthissamplingprocedureisgiveninAlgorithm7.Therefore,ratherthandrawsamplesfromthestateevolu-tiondistributionandthenweightthemaccordingtotheirlikeli-hood,samplesaredrawnfromthelikelihoodandthenassignedweightsonthebasisofthestateevolutiondistribution. ARULAMPALAMetal.:TUTORIALONPARTICLEFILTERS Algorithm7:“Likelihood”ParticleFilter LPF FOR —REPEAT Draw —UNTIL —IF —ELSE —ENDIF ENDFOR Calculatetotalweight: SUM FOR —Normalize: ENDFOR Calculate using(51) IF —Resampleusingalgorithm2: RESAMPLE ENDIF CKNOWLEDGMENTTheauthorswouldliketothanktheanonymousreviewersandtheeditorsofthisSpecialIssuefortheirmanyhelpfulsug-gestions,whichhavegreatlyimprovedthepresentationofthispaper.Theauthorswouldalsoliketothankvariousfundingsourceswhohavecontributedtothisresearch.N.GordonwouldliketothankQinetiQLtd.[1]S.ArulampalamandB.Ristic,“ComparisonoftheparticlefilterwithrangeparameterizedandmodifiedpolarEKF’sforangle-onlytracking,”Proc.SPIE,vol.4048,pp.288–299,2000.[2]Y.Bar-ShalomandX.R.Li,Multitarget–MultisensorTracking:Princi-plesandTechniques.Urbana,IL:YBS,1995.[3]N.Bergman,“RecursiveBayesianestimation:Navigationandtrackingapplications,”Ph.D.dissertation,LinköpingUniv.,Linköping,Sweden,[4]N.Bergman,A.Doucet,andN.Gordon,“OptimalestimationandCramer–Raoboundsforpartialnon-Gaussianstatespacemodels,”Inst.Statist.Math.,vol.53,no.1,pp.97–112,2001.[5]B.P.Carlin,N.G.Polson,andD.S.Stoffer,“AMonteCarloapproachtononnormalandnonlinearstate-spacemodeling,”J.Amer.Statist.Assoc.vol.87,no.418,pp.493–500,1992.[6]J.Carpenter,P.Clifford,andP.Fearnhead,“Improvedparticlefilterfornonlinearproblems,”Proc.Inst.Elect.Eng.,Radar,Sonar,Navig.,1999.[7]T.Clapp,“Statisticalmethodsfortheprocessingofcommunicationsdata,”Ph.D.dissertation,Dept.Eng.,Univ.Cambridge,Cambridge,U.K.,2000.[8]T.ClappandS.Godsill,“ImprovementstrategiesforMonteCarlopar-ticlefilters,”inSequentialMonteCarloMethodsinPractice,A.Doucet,J.F.G.deFreitas,andN.J.Gordon,Eds.NewYork:Springer-Verlag,[9]P.DelMoral,“Measurevaluedprocessesandinteractingparticlesys-tems.Applicationtononlinearfilteringproblems,”Ann.Appl.Probab.vol.8,no.2,pp.438–495,1998.[10]D.Crisan,P.DelMoral,andT.J.Lyons,“Non-linearfilteringusingbranchingandinteractingparticlesystems,”MarkovProcessesRelatedFields,vol.5,no.3,pp.293–319,1999.[11]P.DelMoral,“Non-linearfiltering:Interactingparticlesolution,”MarkovProcessesRelatedFields,vol.2,no.4,pp.555–580.[12]A.Doucet,“OnsequentialMonteCarlomethodsforBayesianfiltering,”Dept.Eng.,Univ.Cambridge,UK,Tech.Rep.,1998.[13]A.Doucet,J.F.G.deFreitas,andN.J.Gordon,“AnintroductiontosequentialMonteCarlomethods,”inSequentialMonteCarloMethodsinPractice,A.Doucet,J.F.G.deFreitas,andN.J.Gordon,Eds.NewYork:Springer-Verlag,2001.[14]A.Doucet,S.Godsill,andC.Andrieu,“OnsequentialMonteCarlosam-plingmethodsforBayesianfiltering,”Statist.Comput.,vol.10,no.3,pp.[15]A.Doucet,N.Gordon,andV.Krishnamurthy,“ParticlefiltersforstateestimationofjumpMarkovlinearsystems,”IEEETrans.SignalPro-,vol.49,pp.613–624,Mar.2001.[16]S.Godsill,A.Doucet,andM.West,“MethodologyforMonteCarlosmoothingwithapplicationtotime-varyingautoregressions,”inProc.Int.Symp.FrontiersTimeSeriesModeling,2000.[17]N.Gordon,D.Salmond,andA.F.M.Smith,“Novelapproachtonon-linearandnon-GaussianBayesianstateestimation,”Proc.Inst.Elect.Eng.,F,vol.140,pp.107–113,1993.[18]G.D.Forney,“TheViterbialgorithm,”Proc.IEEE,vol.61,pp.268–278,Mar.1973.[19]W.R.GilksandC.Berzuini,“Followingamovingtarget—MonteCarloinferencefordynamicBayesianmodels,”J.R.Statist.Soc.B,vol.63,pp.127–146,2001.[20]R.E.Helmick,D.Blair,andS.A.Hoffman,“Fixed-intervalsmoothingforMarkovianswitchingsystems,”IEEETrans.Inform.Theory,vol.41,pp.1845–1855,Nov.1995.[21]Y.C.HoandR.C.K.Lee,“ABayesianapproachtoproblemsinsto-chasticestimationandcontrol,”IEEETrans.Automat.Contr.,vol.AC-9,pp.333–339,1964.[22]A.H.Jazwinski,StochasticProcessesandFilteringTheory.NewYork:Academic,1970.[23]S.Julier,“Askewedapproachtofiltering,”Proc.SPIE,vol.3373,pp.271–282,1998.[24]K.Kanazawa,D.Koller,andS.J.Russell,“Stochasticsimulational-gorithmsfordynamicprobabilisticnetworks,”inProc.EleventhAnnu.Conf.UncertaintyAI,1995,pp.346–351.[25]G.Kitagawa,“MonteCarlofilterandsmootherfornon-Gaussiannon-linearstatespacemodels,”J.Comput.Graph.Statist.,vol.5,no.1,pp.1–25,1996.[26]G.KitagawaandW.Gersch,SmoothnessPriorsAnalysisofTimeSe-.NewYork:Springer-Verlag,1996.[27]J.LiuandM.West,“Combinedparameterandstateestimationinsim-ulation-basedfiltering,”inSequentialMonteCarloMethodsinPrac-,A.Doucet,J.F.G.deFreitas,andN.J.Gordon,Eds.NewYork:Springer-Verlag,2001.[28]J.S.LiuandR.Chen,“SequentialMonteCarlomethodsfordynamicalJ.Amer.Statist.Assoc.,vol.93,pp.1032–1044,1998.[29]J.MacCormickandA.Blake,“Aprobabilisticexclusionprinciplefortrackingmultipleobjects,”inProc.Int.Conf.Comput.Vision,1999,pp.[30]F.MartinerieandP.Forster,“DataassociationandtrackingusinghiddenMarkovmodelsanddynamicprogramming,”inProc.Conf.ICASSP[31]C.Musso,N.Oudjane,andF.LeGland,“Improvingregularisedparticlefilters,”inSequentialMonteCarloMethodsinPractice,A.Doucet,J.F.G.deFreitas,andN.J.Gordon,Eds.NewYork:Springer-Verlag,[32]M.OrtonandA.Marrs,“Particlefiltersfortrackingwithout-of-se-quencemeasurements,”IEEETrans.Aerosp.Electron.Syst.,submittedforpublication.[33]N.OudjaneandC.Musso,“Progressivecorrectionforregularizedpar-ticlefilters,”inProc.3rdInt.Conf.Inform.Fusion,2000.[34]M.PittandN.Shephard,“Filteringviasimulation:Auxiliaryparticlefilters,”J.Amer.Statist.Assoc.,vol.94,no.446,pp.590–599,1999.[35]L.R.Rabiner,“AtutorialonhiddenMarkovmodelsandselectedappli-cationsinspeechrecognition,”Proc.IE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IEEETRANSACTIONSONSIGNALPROCESSING,VOL.50,NO.2,FEBRUARY2002[40]R.vanderMerwe,A.Doucet,J.F.G.deFreitas,andE.Wan,“Theunscentedparticlefilter,”Adv.NeuralInform.Process.Syst.,Dec.2000.[41]M.WestandJ.Harrison,“Bayesianforecastinganddynamicmodels,”SpringerSeriesinStatistics,2nded.NewYork:Springer-Verlag,[42]E.A.WanandR.VanderMerwe,“TheunscentedKalmanfilterfornonlinearestimation,”inProc.Symp.AdaptiveSyst.SignalProcess.,Commun.Contr.,LakeLouise,AB,Canada,Oct.2000. ,“TheunscentedKalmanfilter,”inKalmanFilteringandNeuralNetworks.NewYork:Wiley,2001,ch.7,tobepublished. M.SanjeevArulampalamreceivedtheB.Sc.degreeinmathematicalsciencesandtheB.E.degreewithfirst-classhonorsinelectricalandelectronicengineeringfromtheUniversityofAdelaide,Adelaide,Australia,in1991and1992,respectively.In1993,hewonaTelstraPostgraduateFellowshipawardandreceivedthePh.D.degreeinelectricalandelectronicengineeringattheUniversityofMelbourne,Parkville,Australia,in1997.Hisdoctoraldissertationwas“PerformanceanalysisofhiddenMarkovmodelbasedtrackingalgorithms.”In1992,hejoinedthestaffofComputerSciencesofAustralia(CSA),whereheworkedasaSoftwareEngineerintheSafetyCriticalSoftwareSystemsGroup.In1998,hejoinedtheDefenceScienceandTechnologyOrganization(DSTO),Canberra,Australia,asaResearchScientistintheSurveillanceSystemsDivision,wherehecarriedoutresearchinmanyaspectsofairbornetargettrackingwithaparticularemphasisontrackinginthepresenceofdeceptionjamming.Hisresearchinterestsincludeestimationtheory,targettracking,andsequentialMonteCarlomethods.Dr.ArulampalamwontheAnglo–Australianpostdoctoralfellowship,awardedbytheRoyalAcademyofEngineering,London,in1998. SimonMaskellreceivedtheB.A.degreeinengi-neeringandtheM.Eng.degreeinelectronicandinformationsciencesfromCambridgeUniversityEngineeringDepartment(CUED),Cambridge,U.K.,bothin1999.HecurrentlypursuingthePh.D.degreeatCUED.HeiswiththePatternandInformationProcessingGroup,QinetiQLtd.,Malvern,U.K.HisresearchinterestedincludeBayesianinference,signalprocessing,tracking,anddatafusion,withparticularemphasisontheapplicationofparticlefilters.Mr.MaskellwasawardedaRoyalCommissionfortheExhibitionof1851IndustrialFellowshipin2001. NeilGordonreceivedtheB.Sc.degreeinmath-ematicsandphysicsfromNottinghamUniversity,Nottingham,U.K.,in1988andthePh.D.degreeinstatisticsfromImperialCollege,UniversityofLondon,London,U.K.,in1993.HeiscurrentlywiththePatternandInformationProcessingGroup,QinetiQLtd.,Malvern,U.K.HisresearchinterestsincludeBayesianestimationandsequentialMonteCarlomethods(a.k.a.particlefilters)withaparticularemphasisontargettrackingandmissileguidance.Hehasco-edited,withA.DoucetandJ.F.G.deFreitas,SequentialMonteCarloMethodsinPractice(NewYork:Springer-Verlag). TimClappreceivedtheB.A.,M.Eng.,andPh.D.de-greesfromtheSignalProcessingandCommunica-tionsGroup,CambridgeUniversityEngineeringDe-partment,Cambridge,U.K.Hisresearchinterestsincludeblindequalization,MarkovchainMonteCarlotechniques,andparticlefilters.Heiscurrentlyinvolvedwithtelecommuni-cationssatellitesystemdesignforthePayloadPro-cessorGroup,AstriumLtd.,Stevenage,U.K. IEEETRANSACTIONSONSIGNALPROCESSING,VOL.50,NO.2,FEBRUARY2002ATutorialonParticleFiltersforOnlineNonlinear/Non-GaussianBayesianTrackingM.SanjeevArulampalam,SimonMaskell,NeilGordon,andTimClappIncreasingly,formanyapplicationareas,itisbecomingimportanttoincludeelementsofnonlinearityandnon-Gaussianityinordertomodelaccuratelytheunderlyingdynamicsofaphysicalsystem.Moreover,itistypicallycrucialtoprocessdataon-lineasitarrives,bothfromthepointofviewofstoragecostsaswellasforrapidadaptationtochangingsignalcharacteristics.Inthispaper,wereviewbothoptimalandsuboptimalBayesianalgorithmsfornonlinear/non-Gaussian