Monte Carlo Filter Particle Filter

Transcription

1 205 European Conrol Conference (ECC) July 5-7, 205. Linz, Ausria Mone Carlo Filer Paricle Filer Masaya Muraa, Hidehisa Nagano and Kunio Kashino Absrac We propose a new realizaion mehod of he sequenial imporance sampling (SIS) algorihm o derive a new paricle filer. The new filer consrucs he imporance disribuion by he Mone Carlo filer (MCF) using subparicles, herefore, is non-gaussianiy naure can be adequaely considered while he oher ype of paricle filer such as unscened Kalman filer paricle filer (UKF-PF) assumes a Gaussianiy on he imporance disribuion. Since he sae esimaion accuracy of he SIS algorihm heoreically improves as he esimaed imporance disribuion becomes closer o he rue poserior probabiliy densiy funcion of sae, he new filer is expeced o ouperform he exising, sae-of-he-ar paricle filers. We call he new filer Mone Carlo filer paricle filer (MCF-PF) and confirm is effeciveness hrough he numerical simulaions. A. Problem Definiion I. INTRODUCTION In his paper, we consider he sequenial sae esimaion problem of he following nonlinear non-gaussian sae space model. X f(x ) + Φ, Φ p(φ () Y h(x + Ω, Ω p(ω (2) Here, X is he L-dimensional random vecor called sae a ime and his is he esimaion arge using he pas series of M-dimensional observaions {Y } {Y, Y 2,, Y } or {Y } {Y, Y,, Y }. The former is called he prediced sae esimae : E[X {Y }] and he laer is called he filered sae esimae : E[X {Y }]. Their approximaion errors are calculaed by E[(X (X T {Y }] and E[(X (X T {Y }], respecively. f( ) and h( ) are nonlinear funcions and called sae ransiion and sae observaion, respecively. Φ and Ω are independen whie noises following arbirary probabiliy densiy funcions (PDFs) and hey are called sysem noise and observaion noise, respecively. Equaions () and (2) are referred o as sysem model and observaion model, and hese wo equaions consiue he sae space model of ineres. For his model, since boh sae and noises do no always follow Gaussian disribuions, he well-sudied, sae-of-he-ar Gaussian filers[] such as exended Kalman filer (EKF)[2] and unscened Kalman filer (UKF)[3] can no be adequaely employed. Therefore, designing he sequenial sae esimaor for his model sill remains as a challenging problem. Masaya Muraa, Hidehisa Nagano and Kunio Kashino are research scieniss a NTT Communicaion Science Laboraories, NTT Corporaion, 3-, Morinosao Wakamiya, Asugi-Shi, Kanagawa , Japan. muraa.masaya a lab.n.co.jp Kunio Kashino is also a visiing professor a Naional Insiue of Informaics, 2--2 Hiosubashi, Chiyoda-ku, Tokyo Japan. B. Sequenial Imporance Sampling (SIS) To ackle he problem of ineres, he sequenial imporance sampling (SIS) algorihm[4][5] is ofen used o obain he filered sae esimae. The SIS algorihm approximaes he p(x {Y }), ha is, he poserior PDF of sae a given observaions up o ime, as follows: Sequenial Imporance Sampling (SIS) p(x {Y }) w δ(x where w i p(x X, Y ), i, 2,, N w w w N w(j) p(y X )p(x X p(x X, Y ) ) Here, {, w }, i, 2,, N are weighed se of paricles (paricle values and heir corresponding normalized imporance weighs) a ime and N i w holds. δ( ) is he Dirac s dela funcion and his PDF approximaion is called he Dirac s poin-mass esimae. p(x X, Y ) is he opimal imporance disribuion specified by eqs. () and (2) and i proves o minimizes he variance of he p(x X, Y ), i imporance weighs., 2,, N denoes he random (i.i.d.) sampling from he opimal imporance disribuion. p(y X and p(x X ) are likelihood and ransiion PDFs specified by eq. (2) and (), respecively. The above imporance weigh equaion also indicaes ha he curren weigh is calculaed from he previous weigh and ogeher wih he fac ha he curren paricle value depends on he previous value, his algorihm can be sequenially performed afer he appropriae imporance disribuion selecion. Then he filered sae esimae a ime (denoed as can be obained as follows: i w (j) (3) Noe ha he SIS algorihm is no designed o obain he prediced sae esimae a ime (denoed as. This algorihm proves o easily encouner he degeneracy problem ha almos all he paricles have zero or nearly zero weighs as he ime proceeds and herefore, he paricle resampling procedure is ofen performed afer obaining {, w }, i, 2,, N. The resampling makes he EUCA 2836

2 imporance weighs for all of he new paricles become /N. The new paricles are randomly drawn from he discree se { }, i, 2,, N wih probabiliies {w }, i, 2,, N. Noe ha afer he resampling procedure, he sequenial weigh equaion becomes w p(y X p(x X p(x X, Y ) ) since he resampling makes {w }, i, 2,, N become /N. C. Imporance Disribuion Selecion The selecion of he imporance disribuion affecs he enire sae esimaion performance of he SIS algorihm and generally speaking, he smaller he difference beween he assumed imporance disribuion and rue poserior PDF of sae, he beer he sae esimaion accuracy becomes. The simples choice is p(x X, Y ) p(x X ) and i derives he boosrap filer (BF)[6]. Here, he p(x X ) is he sae ransiion PDF specified by eq. () and he BF is one of he mos popular paricle filers when ackling he problem of ineres. However, one obvious problem is ha he possible large deviaion beween he assumed imporance disribuion and he rue one is likely o arise. To alleviae his gap, we may need he large number of paricles and i someimes resuls in he severe compuaional burden. The oher ofen-used selecion is o assume p(x X, Y ) p(x,, Y where, is supposed o follow eqs. () and (2) and E[, {Y }] is se o. We call, ih paricle sae. Moreover he p(x,, Y is assumed as Gaussian disribued. Then, using N paricles which specify he expecaions of he N paricle saes, N differen p(x,, Y are esimaed by using he Gaussian filer such as EKF or UKF. The former filer is called exended Kalman filer paricle filer (EKF-PF)[7] and he laer one is called unscened Kalman filer paricle filer (UKF-PF)[7][8]. Since he laes observaion Y is incorporaed when consrucing he imporance disribuion, hese filers heoreically are expeced o be more accurae han he BF ha does no ake Y ino accoun. The UKF-PF is also heoreically beer han he EKF-PF since he unscened ransformaion (UT)[3] in he UKF algorihm handles he nonlinear sae ransformaion wih higher accuracy han he runcaed Taylor series based linearizaion approach in he EKF algorihm. D. Conribuion In his paper, we propose a mehod o consruc he p(x X, Y ) by using Mone Carlo filer (MCF)[9] wih sub-paricles. Therefore he non-gaussianiy of he imporance disribuion is aken ino accoun, conribuing o he improvemen in he sae esimaion accuracy. Alhough he compuaional burden increases due o he MCF execuion for all of he paricles under consideraion, he new (4) filer is expeced o be superior over he exising filers. We call he new filer Mone Carlo filer paricle filer (MCF-PF) and confirm is effeciveness using he numerical examples. The res of his paper is organized as follow. Secion I I describes he MCF algorihm and is relaionship o he BF algorihm. In secion III, we explain he proposed filer formulaion and summarize he sequenial sae esimaion algorihm. Numerical examples are shown in secion IV and secion V summarizes he basic findings and conclusion of his paper. A. Algorihm II. MONTE CARLO FILTER (MCF) As well as he SIS algorihm, he Mone Carlo filer (MCF) is also a promising soluion for he aforemenioned sequenial sae esimaion problem. Unlike he SIS, he MCF approximaes he Bayesian filer algorihm using randomly drawn samples called paricles. And, he MCF no only provides he filered sae esimae, bu also provides he prediced sae esimae. The algorihm of MCF can be summarized as follows: Mone Carlo Filer (MCF) ˆ 0 N ( 0, P 0 ), ŵ 0 /N, i, 2,, N For each, 2, Prediced sae esimae Φ w p(φ, i, 2,, N f( ) + Φ ŵ w (j) (j) Filered sae esimae ˆ ŵ w p(y X ŵ ŵ N ŵ(j) w (j) (j) Here, 0 p(x 0 ) and Φ p(φ are he ih paricles randomly drawn from he iniial sae PDF and he sysem noise PDF, respecively. p(y X ) is he likelihood PDF specified by eq.(2). and are he prediced and he filered sae esimaes, and hey are calculaed by he weighed sum of he paricle values as shown in he algorihm. As well as he SIS algorihm, he paricle resampling procedure is performed afer he filered sae esimae o randomly drawn paricles from he esimaed poserior sae }, i, 2,, N all become /N and he weigh equaion in he filered sae esimae can be simply expressed as follows: PDF. Then {w ŵ p(y X (5) 2837

3 Such paricle resampling is also effecive o avoid he paricle impoverishmen problem ha almos all of he paricles have he same values afer he ime proceeds o some exen. B. Relaionship o he Boosrap Filer (BF) The difference beween he MCF and BF menioned in he previous secion is ha while he MCF is based on he Bayesian filer, he BF is formulaed from he SIS algorihm. The MCF uses f( ) + Φ o approximae he poserior PDF of sae, while he BF uses p(x X ). Then, since heir weigh equaions are he same (eq. (4) in secion I-B becomes eq. (5) in secion II-A by selecing he ransiion prior as he imporance disribuion), he MCF is very similar o he BF. Therefore he same problem occurring in he BF algorihm also exiss in he MCF algorihm such ha he large number of paricles are required o guaranee he sae esimaion accuracy due o he deviaion beween he simple imporance disribuion and he rue poserior PDF of sae. To address his issue, we propose he new paricle filer formulaion ha employs he MCF algorihm o accuraely esimae he imporance disribuion. The resuling esimaed imporance disribuion is expeced o be much closer o he rue poserior PDF of sae han ha for he BF. III. MONTE CARLO FILTER PARTICLE FILTER (MCF-PF) A. Imporance Disribuion Esimaion using Sub-paricles The new filer called Mone Carlo filer paricle filer (MCF-PF) consrucs he imporance disribuion in he SIS algorihm by using MCF algorihm. The imporance disribuion becomes as follows: p(x X, Y ) p(y X p(x X ) p(y X p(x X )dx C p(y X p(x X ) (6) Here, C p(y X p(x X )dx. By random sampling from p(x X ) specified by eq. (), he following Dirac s poin-mass approximaion holds: p(x X ) M δ(x,(j) (7) Here,,(j) p(x X ), j, 2,, M and he sub-paricle number M is no necessarily equal o N (paricle number in he SIS algorihm). We can hen obain he following equaions: p(y X p(x X ) M p(y X C M p(y X k δ(x,(j) (8),(k) (9) Subsiuing eqs. (8) and (9) ino eq. (6) yields, p(x X, Y ) p(y X M k p(y X ŵ,(j) δ(x where ŵ,(j),(j),(k) δ(x,(j),(j) (0) p(y X,(j) ŵ,(j) ŵ,(j) M k ŵ,(k),(j) Equaion (0) means ha he imporance disribuion for he ih paricle is approximaed by he MCF algorihm using he M sub-paricles. Therefore, random sampling from he esimaed imporance disribuion can be replaced wih he,(j) uniform sampling from he discree se { }, i, 2,, M wih probabiliies {ŵ,(j) }, i, 2,, M. Such sampling corresponds o he paricle resampling procedure menioned in he previous secion. In shor, we consruc he imporance disribuion for each paricle in conjuncion wih he MCF algorihm using he sub-paricles and obain he new paricle by performing he resampling procedure o he consruced imporance disribuion. B. Imporance Weigh Calculaion As explained in he previous secion, by using eq. (0), he new paricle sampled from he esimaed imporance disribuion can be expressed as follows: ˆ p(x X Y ) ŵ,(j),(j) δ(x,() as,(l),(m), Since he is eiher,,(2),, or we can also denoe, where l is eiher, 2,, or M. Therefore, he sequenial imporance weigh calculaion and he normalized weigh become ŵ ŵ ŵ p(y X ŵ p(y X ŵ N k ŵ(k) p(x p(x X ) X, Y ),(l) p(x ŵ,(l) δ(0),(l) X ) () (2) Alhough δ(0) appears in eq. (), i will be omied when calculaing he normalized weigh ŵ in eq. (2). Tha is, he weigh calculaed wihou he δ(0) (righ side of he 2838

4 following equaion) yields he same normalized weigh as follows: ŵ ŵ p(y X,(l),(l) p(x X ) ŵ,(l) (3) Therefore, when execuing he MCF-PF, eq. (3) can be used o obain he one-sep-ahead imporance weigh. However, alhough an approximaion error is included, we can also subsiue eq. (7) ino eq. () o cancel he δ(0) as follows: ŵ ŵ p(y X ŵ,(l) ŵ p(y X,(l) M δ(0) δ(0),(l) Mŵ,(l) (4) This formulaion may be more appropriae ha he eq. (3) since he eq () is equal o he division by infiniy. Since he imporance weighs all become /N afer he paricle resampling procedure and he sub-paricle number M in eq. (4) do no change he resuling normalized weigh, above sequenial weigh equaions can be expressed as follows: ŵ ŵ p(y X p(y X ŵ,(l),(l),(l) p(x X ) ŵ,(l) (5),(l) (6) In eiher case, he compuaional cos for performing he SIS algorihm is increased compared o he oher SIS realizaion such as EKF-PF or UKF-PF due o he sub-paricle calculaion. However, since no assumpions are imposed when consrucing he imporance disribuion such as he Gaussianiy, he MCF-PF algorihm is expeced o ouperform hese PFs. We summarize he MCF-PF algorihm for obaining he prediced and he filered sae esimaes in he nex secion. C. Algorihm The algorihm of MCF-PF is shown as follows: Mone Carlo Filer Paricle Filer (MCF PF) ˆ 0 p(x 0 ), ŵ 0 /N, i, 2,, N For each, 2, Prediced sae esimae Φ w p(φ, i, 2,, N f( ) + Φ ŵ k w (j) (j) Filered sae esimae Imporance disribuion esimaion,(j) p(x X ), (j, 2,, M) ŵ,(j) p(y X,(j) ŵ,(j) ŵ,(j) M k ŵ,(k) Sequenial imporance sampling ˆ ˆ,(l) ŵ p(y X ŵ ŵ,(j) δ(x,(l) p(x or p(y X ŵ,(l) ŵ ŵ N k ŵ(k) k ŵ (k) (k) ŵ,(l),(l),(j), (i, 2,, N),(l) X ) The algorihm for obaining he prediced sae esimae is he same as ha for he MCF. We call he MCF-PF using he weigh equaion in eq. (5) and ha using he weigh equaion in eq. (6) as MCF-PF- and MCF-PF- 2, respecively. Their performance difference is numerically invesigaed in he nex secion in which we compare he performance of he MCF-PF wih hose of he oher saeof-he-ar filers. IV. NUMERICAL SIMULATIONS We firs invesigae he effeciveness of he MCF-PF on he scalar nonlinear sae-space models[0] corruped by Gaussian sysem and observaion noises [Prob.]. We hen examine he performance of he MCF-PF on non-gaussian sysem and observaion noises such as he Laplace sysem noise and Cauchy observaion noise [Prob.2]. The comparison filers are UKF (unscened Kalman filer), MCF (Mone Carlo filer), GPF (Gaussian paricle filer[]), and UKF-PF (unscened Kalman filer paricle filer). Here, he GPF assumes he Gaussian-disribued poserior PDF of sae in he MCF algorihm and a every ime sep, new paricles are randomly drawn from ha PDF. Therefore he paricle resampling procedure required in he MCF algorihm is no necessary for he GPF and he so-called paricle impoverishmen problem ha he almos all of he paricles have he same values can be circumvened. The GPF is expeced o work wih relaively small number of paricles, leading o he less compuaional burden compared o he 2839

5 MCF. Therefore, we also compare he performance of he MCF-PF wih ha of he GPF in his secion. The [Prob.] is on he sequenial sae esimaion problem for he following sae-space model. [Prob.] 25x x 2 x + + (x ) 2 + 8cos(.2) + φ, ( ) where φ N(φ ; 0, ) exp φ2 (2π) 2 y (x ) ω, where ω N(ω ; 0, ) ( ) exp ω2 (2π) 2 We generaed he observaion daa from o 000 based on he iniial sae x 0 0. The iniial condiions for he filers were all se o N(0, 0 2 ). Table I shows average 000 x ˆx ) of absolue sae esimaion errors ( 000 UKF, MCF, GPF, UKF-PF, MCF-PF- and MCF-PF-2 over 00 Mone Calro simulaions. Here, he x is he rue sae a ime. Paricle numbers for he MCF, GPF, UKF-PF and MCF-PF were all N 00 and he number of sub-paricles used in he MCF-PF algorihms was M 00. In Table I, he bold number indicaes he smalles sae esimaion error. TABLE I AVERAGE ABSOLUTE STATE ESTIMATION ERRORS OVER 00 MONTE CARLO SIMULATIONS WITH N 00 AND M 00 SETTING. UKF MCF GPF UKF-PF MCF-PF- MCF-PF From he resul in Table I, we firs confirmed ha he PFs are much beer han he Gaussian filer represened by he UKF in erms of sae esimaion accuracy. However, since he UKF was much faser han he oher PFs, wheher using he UKF or PFs depends on he problem we solve. For example, when he case ha he filering (processing) ime is raher imporan, he UKF sill remains as one of he candidae filering algorihms. Alhough he GPF was also superior in erms of calculaion cos over he oher PFs, is sae esimaion accuracy was very slighly worse han ha for he MCF. This was due o he Gaussian assumpion on he poserior PDF of sae. The UKF-PF scored he second bes esimaion accuracy. The MCF-PF-2 was beer han he MCF-PF-. This resul indicaed ha he dela funcion in he weigh equaion () was beer o be eliminaed. Alhough he saisical significance difference was no observed beween he resuls of UKF-PF and MCF-PF-2, he MCF-PF-2 scored he bes sae esimaion accuracy among he comparison filers. We also execued he same 00 Mone Carlo simulaions wih less number of paricles for all of he PFs o invesigae he performance dependency on he paricle number. We se N 0 for all of he PFs and M 0 in he MCF- PF algorihm. The resuls are shown in Table II below and again, he bold number indicaes he smalles sae esimaion error. From Table II, we can see ha when he number of TABLE II AVERAGE ABSOLUTE STATE ESTIMATION ERRORS OVER 00 MONTE CARLO SIMULATIONS WITH N 0 AND M 0 SETTING. MCF GPF UKF-PF MCF-PF- MCF-PF paricles became small, he GPF became superior over he MCF. This was due o he random sampling effec from he Gaussian-assumed poserior sae PDF and he GPF seemed o work for he paricle impoverishmen problem ha was inheren in paricle resampling procedure for he MCF algorihm. We also found ha he UKF-PF scored he lowes esimaion error and his observaion leads o he conclusion ha when he sysem and observaion noises follow Gaussian disribuions, he UKF-PF is he bes choice for addressing he filering problem. The sae-space model for he [Prob.2] is described as follows. A his ime, he sysem noise follows a Laplace disribuion and he observaion noise follows a Cauchy disribuion. [Prob.2] x 2 x + 25x + (x ) 2 + 8cos(.2) + φ, where φ Laplace(φ ; 0, ) 2 exp( φ ) y (x ) ω, where ω Cauchy(ω ; 0, ) π ( + (ω 2 ) As well as he previous problem, we again generaed he observaion daa from o 000 based on x 0 0. The iniial condiions for he filers were N(0, 0 2 ). Table III shows he average absolue sae esimaion errors ( N 0 N(000) x ˆx ) of MCF, GPF, UKF-PF, MCF- PF- and MCF-PF-2 over 00 Mone Calro simulaions. Here, when employing UKF and UKF-PF, we assumed ha he Laplace sysem noise was approximaed by N(0, 2) and also assumed ha he Cauchy observaion noise was approximaed by N(0, 0), respecively. These Gaussianassumed PDF seings were required for he UKF and he UKF-PF algorihms. As well as in he previous simulaions, we se N 00 for all of he PFs and M 00 for he wo MCF-PF algorihms. From Table III, we can see ha he UKF was no capable of filering observaions corruped by he non-gaussian noises. MCF was beer han he GPF while sacrificing he filering speed. The UKF-PF someimes suffered from he problem ha all of he imporance weighs became zero, which implied ha sampled paricles from he imporance disribuions were no locaed wihin he suppor domain of 2840

6 he likelihood PDF or he sae ransiion PDF. Because of he large noise realizaion of he Cauchy observaion noise, he observaion updae for each paricle sae someimes produced he deviaed imporance disribuions for each paricle, and ha made he sampled paricles being locaed ou of he aforemenioned suppor domains. This resul indicaed ha for filering problems of noisy observaions, he UKF- PF migh be no suiable. The seing of large observaion noise variance in he UKF calculaion is one remedy for his problem, however, i leads o degrade he enire sae esimaion accuracy. TABLE III AVERAGE ABSOLUTE STATE ESTIMATION ERRORS OVER 00 MONTE CARLO SIMULATIONS WITH N 00 AND M 00 SETTING. UKF MCF GPF UKF-PF MCF-PF- MCF-PF Figure shows he sae filering resuls for he MCF-PF- 2. We can observe wo large esimaion errors a he noisy Fig.. Plos of observaion examples, rue saes and sae esimaes of MCF-PF-2. The horizonal axis is he ime sep from 400 and 450. observaions corruped by he realizaions of he addiive Cauchy noise. The MCF-PF-2 esimaed he oher rue saes well from he nonlinear, bimodal observaions. The MCF-PF-2 was also beer han he MCF-PF- for his problem. The MCF-PF-2 scored he bes sae esimaion accuracy among he comparison filers. As well as in he previous problem, we also execued he same 00 Mone Carlo simulaions wih N 0 seing for all of he PFs and M 0 seing in he MCF-PF algorihm. The resuls are shown in Table IV. From Table IV, when he limied number TABLE IV AVERAGE ABSOLUTE STATE ESTIMATION ERRORS OVER 00 MONTE CARLO SIMULATIONS WITH N 0 AND M 0 SETTING. MCF GPF UKF-PF MCF-PF- MCF-PF of paricles was allowed, we found ha he GPF became superior over he MCF. The UKF-PF again suffered from he problem of imporance weighs becoming zero afer he ime proceeded. For his problem, he GPF also ouperformed he MCF-PF-2 and i was he fases filering algorihm among he five comparison filers. The MCF-PF-2 scored he second bes esimaion accuracy, showing he effeciveness of he proposed filer. We found ha he low sae esimaion accuracy of he UKF were implying ha he prior sae PDF was no beer o be assumed as Gaussian. On he oher hand, since he GPF was successful, he poserior sae PDF could be assumed as Gaussian for resampling new paricles. To summarize he overall resuls, when he relaively large number of paricles was allowed in he filer execuion, we confirmed ha he MCF-PF-2 was he bes among he oher PFs. However, for a siuaion ha he small number of paricles was desired due o he compuaional concern, he GPF provided good sae esimaion resuls. To he conrary, when he sufficienly large number of paricles is allowed, he accuracy difference beween MCF and MCF-PF becomes negligible. In ha case, since he MCF is much faser han he MCF-PF, he choice of he MCF will be preferred. V. CONCLUSION We propose he Mone Carlo filer paricle filer (MCF-PF) in which he imporance disribuion in he SIS algorihm is approximaed by he MCF using sub-paricles. Therefore, he non-gaussianiy naure of he imporance disribuion can be incorporaed and i leads o he beer sae esimaion accuracy han he oher PFs such as GPF and UKF-PF. The numerical simulaions show he effeciveness of he MCF-PF and also reveal ha when he limied number of paricles is allowed, he GPF oupus good sae esimaion resuls. REFERENCES [] K. Io and K. Xiong. Gaussian Filers for Nonlinear Filering Problems, IEEE Trans. on Auomaic Conrol, Vol. 45, No. 5, pp , [2] A. Jazwinski. Adapive Filering, Auomaica, 5(4), pp , 969. [3] S. J. Julier, J. K. Uhlmann and H. F. Durran-Whye. A New Approach for Filering Nonlinear Sysems, In Proc. American Conrol Conference, pp , 995. [4] A. Douce. On Sequenial Simulaion-based Mehods for Bayesian Filering, Tech. Rep. CUED/FINFENG/TR 30, Deparmen of Engineering, Cambridge Universiy, 998. [5] S. Sarkka. Bayesian Filering and Smoohing, Cambridge Universiy Press, Cambridge CB2 8BS, 203. [6] N. Gordon, D. J. Salmond and A. F. M. Smih. Novel Approach o Nonlinear/Non-Gaussian Bayesian Sae Esimaion, IEEE Proceedings-F 40(2); pp. 07-3, 993. [7] R. Van der Merwe and N. De Freias, A. Douce and E. Wan. The Unscened Paricle Filer, Advances in Neural Informaion Processing Sysems 3, pp , 200. [8] A. Douce, S. J. Godsill and C. Andrieu. On Sequenial Mone Carlo Sampling Mehods for Bayesian Filering, Saisics and Compuing, 0(3), pp , [9] G. Kiagawa. Mone Carlo Filer and Smooher for Non- Gaussian Nonlinear Sae Space Models, Journal of Compuaional and Graphical Saisics, 5(), pp. -25, 996. [0] T. Kaayama. Nonlinear Kalman Filer, Asakura-Shoen, 20 (in Japanese). [] J. H. Koecha and P. M. Djuric. Gaussian Paricle Filering, IEEE Trans. Signal Processing, Vol.5, No.0, pp ,