Sequential Monte Carlo Methods for General State-Space Models英文文献资料

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1、IntroductionFilteringSmoothingandPredictionParameterEstimationTheoreticalAspectsSimulationsSequentialMonteCarloMethodsforGeneralState-SpaceModelsJanCNeddermeyerUniversitätHeidelbergHeidelberg,20061/70IntroductionFilteringSmoothingandPredictionParameterEstimationTheoreticalAspectsSim

2、ulationsContents1Introduction2Filtering3SmoothingandPrediction4ParameterEstimation5TheoreticalAspects6Simulations2/70IntroductionFilteringSmoothingandPredictionParameterEstimationTheoreticalAspectsSimulationsOutline1Introduction2Filtering3SmoothingandPrediction4ParameterEstimation5T

3、heoreticalAspects6Simulations3/70IntroductionFilteringSmoothingandPredictionParameterEstimationTheoreticalAspectsSimulationsGeneralState-SpaceModelRandomvariables:1Hiddenstatessequence{Xt;t≥0}2Observationssequence{Yt;t≥1}Xt=ft(Xt−1,Ut−1)(processmodel)Yt=ht(Xt,Vt)(observationmodel)•f

4、t:Rnx×Rnu→Rnx,ht:Rnx×Rnv→Rnyarbitraryfunctions•Ut,Vti.i.d.noisesequences(notnecessarilyGaussian)Aim:EstimatedistributionofthehiddenstatesusingknownobservationsYt=yt.4/70IntroductionFilteringSmoothingandPredictionParameterEstimationTheoreticalAspectsSimulationsDependenciesinaGSSMTheG

5、SSMasadirectedacyclicgraphY1···Yt−1YtYt+1···↑↑↑↑X0→X1→···→Xt−1→Xt→Xt+1→···⇒Given{Xt;t≥0},theobservations{Yt;t≥1}areindependent.Furthermore,p(yt

6、y1:t−1,x0:t)=p(yt

7、xt)p(xt

8、y1:t−1,x0:t−1)=p(xt

9、xt−1)5/70IntroductionFilteringSmoothingandPredictionParameterEstimationTheoreticalAspectsSimu

10、lationsFirstExampleofaGSSMThelinearGaussiancaseXt=FtXt−1+Ut−1Yt=HtXt+Vt•Ft,Htmatrices•Ut,Vti.i.d.Gaussiannoisesequencesoptimalfilterexists⇒TheKalmanfilter6/70IntroductionFilteringSmoothingandPredictionParameterEstimationTheoreticalAspectsSimulationsSecondExampleofaGSSMAStochasticvolat

11、ilitymodel[Stroud2003]•Ytreturnsofsomefinancialunderlying(i.e.stockindex)•XtvolatilityofreturnsXt=α+βXt−1+γUtXtYt=expVt2whereUt,Vt∼N(0,1).Parametervectorθ=(αβγ)oftenunknown.⇒Additionalproblem:Parameterestimation7/70IntroductionFilteringSmoothingandPredictionParameterEstimationTheor

12、eticalAspectsSimula