A Gauss Implementation of Particle Filters The PF library

Transcription

1 A Gauss Implementaton of Partcle Flters The PF lbrary Therry Roncall Unversty of Evry Gullaume Wesang Bentley Unversty Ths verson: December 24, 2008

2 Contents 1 Introducton Installaton Gettng started readme.tt fle Setup What s PF? Usng Onlne Help An ntroducton to partcle flters Framewor Defnton of the tracng problem Bayesan flters Partcle flters Numercal Algorthms Command Reference 13 4 Some eamples 21 Bblography 25

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4 Chapter 1 Introducton 1.1 Installaton 1. The fle pf.zp s a zpped archve fle. Copy ths fle under the root drectory of Gauss, for eample D:\GAUSS Unzp the fle. Drectores wll then be created and fles wll be coped over them: target path target path\dlb target path\lb target path\pf\eamples target path\pf\src target path\src readme.tt DLLs lbrary fle eample and tutoral fles source code fles source code fles 3. If your root of Gauss s D:\GAUSS60, the nstallaton s fnshed, otherwse you have to modfy the paths of the lbrary usng notepad or the LbTool. Another way to update the lbrary s to run Gauss, log on to the pf\src drectory, delete the path wth the command lb pf -n and add the path to the lbrary wth the command lb pf -a. 1.2 Gettng started Gauss for Wndows s requred to use the PF routnes readme.tt fle The fle readme.tt contans last mnute nformaton on the PF procedures. Please read t before usng them Setup In order to use these procedures, the PF lbrary must be actve. Ths s done by ncludng PF n the LIBRARY statement at the top of your program: lbrary pf; 3

5 4 CHAPTER 1. INTRODUCTION To reset global varables n subsequent eecutons of the program and n order to load DLLs, the followng nstructon should be used: pfset; 1.3 What s PF? PF s a Gauss lbrary for computng partcle flters. PF contans the procedures whose lst s gven below: Partcle Flter Set Generc Partcle Flter Partcle Smoother Regularzed Partcle Flter Smulate Tracng Problem SIR Partcle Flter SIS Partcle Flter 1.4 Usng Onlne Help PF lbrary supports Wndows Onlne Help. Before usng the browser, you have to verfy that the PF lbrary s actvated by the lbrary command.

6 Chapter 2 An ntroducton to partcle flters 2.1 Framewor We have developed ths lbrary n the contet of tracng problems [9]. In the net paragraphs, we recall the defnton of tracng problems and we present Bayesan flters whch are generally used to solve them Defnton of the tracng problem We follow [1] and [8] n ther defnton of the general tracng problem. We note R n the vector of states and z R n z the measurement vector at tme nde. In our settng, we assume that the evoluton of s gven by a frst-order Marov model: = f (t, 1, ν ) (2.1) where f s a non-lnear functon and ν a nose process. In general, the state s not observed drectly, but partally through the measurement vector z. Thus, t s further assumed that the measurement vector s lned to the target state vector through the followng measurement equaton: z = h (t,, η ) (2.2) where h s a non-lnear functon, and η s a second nose process ndependent from ν. Our goal s thus to estmate from the set of all avalable measurements z 1: = {z, = 1,..., }. The goal n a tracng problem s to estmate the state varable, the current state of the system at tme t, usng all avalable measurement z 1: = {z l } l=1: Bayesan flters The pror densty of the state vector at tme s gven by the Chapman-Kolmogorov equaton: p ( z 1: 1 ) = p ( 1: 1 ) p ( 1 z 1: 1 ) d 1 (2.3) where we used the fact that our model s a frst-order Marov model to wrte p ( 1: 1, z 1: 1 ) = p ( 1: 1 ). Ths equaton s nown as the Bayes predcton step. It gves an estmate of the probablty densty functon of gven all avalable nformaton untl 5

7 6 CHAPTER 2. AN INTRODUCTION TO PARTICLE FILTERS 1. At tme, as a new measurement value z becomes avalable, one can update the probablty densty of : p ( z 1: ) p (z ) p ( z 1: 1 ) (2.4) Ths equaton s nown as the Bayes update step. The Bayesan flter corresponds to the system of the two recursve equatons (2.3) and (2.4). In order to ntalze the recurrence algorthm, we assume the probablty dstrbuton of the ntal state vector p ( 0 ) to be nown. Usng Bayesan flters, we do not only derve the probablty dstrbutons p ( z 1: 1 ) and p ( z 1: ), but we may also compute the best estmates ˆ 1 and ˆ whch are gven by: ˆ 1 = E [ z 1: 1 ] = p ( z 1: 1 ) d and: ˆ = E [ z 1: ] = p ( z 1: ) d When loong at Bayesan flters, the frst dstncton should be between the type of state varables. On the one hand, n the case of a state varable wth a fnte number of dscrete states, one can use Grd-based methods to get an optmal soluton to the Bayesan flter, ndependently of the form of the densty functons. On the other hand, f the state varable s contnuous, then there ests no method n general provdng an optmal soluton, ecept for the normal case. Snce the Gaussan famly s ts own conjugate, models wth Gaussan denstes have a partcular attracton. If, furthermore, the functons f and h n (2.1) and (2.2) are lnear, then the optmal soluton of the Bayesan flter s gven by the Kalman flter. Moreover, n the case where the nose denstes are Gaussan but the functons f and h are nonlnear, one can use an appromate method called Etended Kalman flter (EKF) where the functons f and h are replaced by local lnear appromaton usng ther frst dervatves at each recurson. In the more general case of non Gaussan denstes, one has to resort to sub-optmal algorthms, called partcle flters, to appromate the soluton to the Bayesan flter. The dea behnd partcle flters s rather smple. Snce no closed-form soluton to the tracng problem can be found n general, one smply smulate at each step a sample of partcles whch wll be used to provde a dscrete estmaton of the densty functon, the flterng densty, p( z 1: ). 2.2 Partcle flters Partcle flterng methods are technques to mplement recursve Bayesan flters usng Monte- Carlo smulatons. The ey dea s to represent the posteror densty functon by a set of random samples wth assocated weghts and to compute estmates based on these samples and weghts [1, 3, 5, 6, 7, 8]. As the samples become very large N s 1, ths Monte-Carlo appromaton becomes an equvalent representaton on the functonal descrpton of the posteror pdf. To clarfy deas 1, let {, } Ns w denotes a set of support ponts { =1, = 1,..., N } s and ther assocated weghts { w, = 1,..., N s} characterzng the posteror densty p ( z 0: ). The posteror densty at tme can then be appromated as: 1 Note that the succnct presentaton gven here of partcle flters s adapted to our frst-order Marovan framewor. N s p ( z ) wδ ( ) =1 (2.5)

8 2.2. PARTICLE FILTERS 7 We have thus a dscrete weghted appromaton to the true posteror dstrbuton. One common way of choosng the weghts s by way of mportance samplng see for eample [1, 3, 5, 8]. Ths prncple reles on the followng dea. In the general case, the probablty densty p ( z ) s such that t s dffcult to draw samples from t. Assume for a moment that p () π () s a probablty densty from whch t s dffcult to draw sample from, but for whch π () s easy to evaluate. Hence, up to proportonalty, so s p (). Also, let s q () be samples that are easly drawn from a proposal q ( ), called an mportance densty. Then, smlarly to 2.5, a weghted appromaton of the densty p ( ) can be obtaned by usng: where: N s p () w δ ( ) =1 w π ( ) q ( ) s the normalzed weght of the -th partcle. Thus, f the samples { } were drawn from a proposal densty q ( z ), then the weghts n (2.5) are defned to be: w p ( z ) q ( z ) (2.6) The PF sequental algorthm can thus be subsumed n the followng steps. At each teraton, one has samples consttutng an appromaton of p ( 1 z ) ( ) 1 and wants to appromate p z wth a new set of samples. If the mportance densty can be chosen so as to factorze n the followng way: q ( z ) = q ( 1, z ) q ( 1 z 1 ) (2.7) then one can obtan samples { } by drawng samples from q ( z ). To derve the weght update equaton: p ( z ) = p (z, z 1 ) p ( z 1 ) p (z z 1 ) = p (z, z 1 ) p ( 1, z 1 ) p ( 1 z 1 ) p (z z 1 ) = p (z ) p ( 1 ) p ( 1 z 1 ) p (z z 1 ) p (z ) p ( 1 ) p ( 1 z 1 ) (2.8) By substtutng (2.7) and (2.8) nto (2.6), the weght equaton can be derved to be: w w 1 p ( ( ) z ) p 1 q ( 1, z ) (2.9) and the posteror densty p ( z ) can be appromated usng (2.5). We refer the reader to [1] for a more detaled but concse eposé of the dfferences between the dfferent PF algorthms: sequental mportance samplng (SIS), generc partcle flter, samplng mportance resamplng (SIR), aulary partcle flter (APF), and regularzed partcle flter (RPF). We provde a succnct eposé of the SIS, SIR algorthms as well as the generc partcle flter s and the regularzed partcle flter s n the net secton. One mportant feature of PF s that not one mplementaton s better than all the others. In dfferent contets, dfferent PFs may have wldly dfferent performances.

9 8 CHAPTER 2. AN INTRODUCTION TO PARTICLE FILTERS 2.3 Numercal Algorthms In the prevous secton, we presented the algorthm, nown under the name Sequental Importance Samplng (SIS), whch forms the bass for most sequental Monte Carlo flters developed over the past decade [1]. We start by provdng ts pseudo code n Algorthm 1, before eposng the more advanced algorthms we used: a generc Partcle Flter (GPF), a Samplng Importance Resamplng (SIR) algorthm, and a regularzed Partcle Flter (RPF). Algorthm 1 SIS Partcle Flter procedure { SIS Partcle Flter(z 1:T,N s ) Runs a SIS Partcle Flter 0, w0} =1:N s p 0 (.) Intalzaton 1 whle { < T do, w} =1:N s SIS Step( 1, w 1, z ) + 1 end whle return { 1:T, } w 1:T =1:N s end procedure procedure SIS step( 1, w 1, z ) Propagates the sample from state 1 to state for = 1 : N s do Draw q ( 1, z ) Assgn the partcle a weght, w, accordng to 2.9 end for return {, } w =1:N s end procedure The SIS algorthm s thus a very smple algorthm, easy to mplement. However, t commonly suffers from a degeneracy phenomenon, where after only a few teratons, all but one partcle wll have neglgble weghts. Ths degeneracy problems mples that a large computatonal effort wll be devoted to updatng partcles whose contrbuton to the appromaton of the flterng densty p ( z 1: ) s quas null. In order to allevate ths problem, more advanced algorthm have been devsed. One way to deal wth degeneracy s to carefully choose the mportance densty functon q ( 1, z ). We leave to the reader to consult [1] for a dscusson of the mportance of the choce of the mportance densty. Another smple dea s to resample the partcles when a certan measure of degeneracy becomes too large (or too small). For eample, one could calculate the effectve sample sze N eff defned as: N eff = N s 1 + σ ( w where w = p ( z 1:) /q ( 1, z ) s referred to as the true weght. As ths cannot be valued eactly, ths quantty can be estmated usng: ˆN eff = Ns =1 ) 2 1 ( ) w 2 (2.10) We provde n Algorthm 2 and n Algorthm 3 respectvely the resamplng algorthm we used and the generc Partcle Flter whch s deduced from the SIS algorthm by addng ths resamplng step to avod degeneracy.

10 2.3. NUMERICAL ALGORITHMS 9 Algorthm 2 Resamplng Algorthm procedure Resample( {, } w c 1 0 for = 2 : N s do c c 1 + w end for =1:N s ) Intalse the CDF Construct the CDF 1 Start at the bottom of the CDF u 1 U [ ] 0, Ns 1 Draw a startng pont for j = 1 : N s do u j u 1 + Ns 1 (j 1) Move along the CDF whle u j > c do + 1 end whle j = Assgn sample w j = N s 1 Assgn weght parent j Assgn parent end for{ } return j, wj, parent j j=1:n s end procedure Algorthm 3 Generc Partcle Flter procedure { Generc Partcle Flter(z 1:T,N s ) Runs a Generc Partcle Flter 0, w0} p =1:N 0 (.) Intalzaton s 1 whle { < T do, w} PF =1:N s Step( 1, w 1, z ) + 1 end whle return { 1:T, } w 1:T =1:N s end procedure procedure PF Step( 1, w 1, z ) for = 1 : N s do Draw q ( 1, z ) Assgn the partcle a weght, w, accordng to 2.9 end for t N s =1 w Calculate total weght for = 1 : N s do w t 1 w end for Calculate N eff usng 2.10 f N { eff < N s then, w, } =1:N s Resample( {, } w =1:N s ) end f end procedure

11 10 CHAPTER 2. AN INTRODUCTION TO PARTICLE FILTERS Algorthm 4 SIR Partcle Flter procedure { SIR Partcle Flter(z 1:T,N s ) Runs a SIR Partcle Flter 0, w0} =1:N s p 0 (.) Intalzaton 1 whle { < T do, w} =1:N s SIR Step( 1, w 1, z ) + 1 end whle return { 1:T, } w 1:T =1:N s end procedure procedure SIR Step( 1, w 1, z ) for = 1 : N s do Draw p( 1 ) w p(z ) end for t N s =1 w Calculate total weght for = 1 : N s do w t 1 w end for {, w, } =1:N s Resample( {, w } end procedure =1:N s ) Systematc resamplng In many partcle flters mplementatons, one uses the pror densty p ( 1) as the mportance densty q ( 1, z ) for even though t s often suboptmal, t smplfes the weghts update equaton 2.9 nto: w w 1 p ( z ) Furthermore, f resamplng s appled at every step ths partcular mplementaton s called the Samplng Importance Resamplng (SIR) of whch we gve the algorthm n pseudo code n Algorthm 4 then we have w 1 = 1/N s, and so: w p ( z ) The weghts gven n 2.11 are normalzed before the resamplng stage. =1 (2.11) The regularzed Partcle Flter s based on the same dea as the Generc Partcle Flter, wth the same resamplng condton, but the resamplng step provdes an entrely new sample based on a contnuous appromaton of the posteror flterng densty p ( z ), such that we have the followng appromaton: N s ˆp ( z ) = wk ( h ) (2.12) where: K h () = 1 ( ) h n K h s the re-scaled Kernel densty K ( ), h > 0 s the Kernel bandwdth, n s the dmenson of the state vector, and w, = 1,..., N s are normalzed weghts. The Kernel K ( ) and bandwdth

12 2.3. NUMERICAL ALGORITHMS 11 h should be chosen to mnmze the Mean Integrated Square Error (MISE), between the true posteror densty and the correspondng regularzed emprcal representaton n 2.12, defned as: [ ] MISE (ˆp) = E [ˆp ( z ) p ( z )] 2 d One can show that n the case where all the samples have the same weght, the optmal choce of the Kernel s the Epanechnov Kernel: K opt = { n +2 2c n (1 2) f < 1, 0 otherwse where c n s the volume of the unt hypersphere n R n. Furthermore, when the underlyng densty s Gaussan wth a unt covarance matr, the optmal choce for the bandwdth s: h opt = AN 1 n+4 s A = [ 8c 1 n (n + 4) ( 2 π ) n ] 1 n+4 We can now provde the algorthm for the regularzed Partcle Flter n Algorthm 5.

13 12 CHAPTER 2. AN INTRODUCTION TO PARTICLE FILTERS Algorthm 5 Regularzed Partcle Flter procedure { Regularzed Partcle Flter(z 1:T,N s ) Runs a Regularzed Partcle Flter 0, w0} =1:N s p 0 (.) Intalzaton 1 whle { < T do, w} =1:N s RPF Step( 1, w 1, z ) + 1 end whle return { 1:T, } w 1:T =1:N s end procedure procedure RPF Step( 1, w 1, z ) for = 1 : N s do Draw q ( 1, z ) Assgn the partcle a weght, w, accordng to 2.9 end for t N s =1 w Calculate total weght for = 1 : N s do w t 1 w end for Calculate N eff usng 2.10 f N eff < N s then Compute the emprcal covarance matr S of {, } w =1:N s Compute { D Chol(S ) Cholesy decomposton of S : D D = S, w, } Resample( { =1:N s, } w ) =1:N s for = 1 : N s do Draw ɛ K opt from the Epanechnov Kernel + h optd ɛ end for return {, } w else return {, } w end f end procedure =1:N s =1:N s

14 Chapter 3 Command Reference The followng global varables and procedures are defned n PF. They are the reserved words of PF. pf Ft, pf Ht, pf mportance densty, pf mportance samplng, pf ntal densty, pf ntal samplng, pf lelhood densty, pf Partal Gaussan, pf pror densty, pf pror samplng, pf Qt, pf Resamplng Threshold, pf Rt, pf Save Partcles, rpf Bounds; rpf Kernel N. The default global control varables are pf Resamplng Threshold pf Save Partcles 1 rpf Bounds 0 pf Partal Gaussan 0 rpf Kernel N 201

15 14 CHAPTER 3. COMMAND REFERENCE Purpose Set all the nformaton to run a PF. Partcle Flter Set Format call Partcle Flter Set(F,Q,H,R, mportance pdf,pror pdf,lelhood pdf,ntal pdf, mportance samplng,pror samplng,ntal samplng); Input F scalar, ponter to a procedure that computes F (t, ) Q scalar, ponter to a procedure that computes Q (t, ) H scalar, ponter to a procedure that computes H (t, ) R scalar, ponter to a procedure that computes R (t, ) mportance pdf scalar, ponter to a procedure that computes the mportance densty functon q ( 1, z ) pror pdf scalar, ponter to a procedure that computes the pror densty functon p ( 1 ) lelhood pdf scalar, ponter to a procedure that computes the lelhood densty functon p (z ) ntal pdf scalar, ponter to a procedure that computes the ntal densty functon p ( 0 ) mportance samplng scalar, ponter to a procedure that smulates the state vector accordng to the mportance densty pror samplng scalar, ponter to a procedure that smulates the state vector accordng to the pror densty ntal samplng scalar, ponter to a procedure that smulates the ntal state vector 0 Output Globals Remars The functon F, Q, H and R are only need f you want to perform Monte Carlo runs. In ths case, the model consdered by the procedure Smulate Partcle Flter s the followng: { = F (t, 1 ) + ν z = H (t, ) + η wth ν N (0, Q (t, 1 )) and η N (0, R (t, )). If you don t need to use ths procedure, you may set the ponters to these functons equal to 0. The procedure ntal pdf computes the ntal weghts w 0. If ts ponter s set to zero, the ntal weghts are unform. Source pf.src

16 Purpose Run a generc partcle flter. Generc Partcle Flter 15 Format {,w,m,cov} = Generc Partcle Flter(z,Ns); Input z Ns Output w m cov Globals pf Save Partcles matr N G, observed values z scalar, number of partcles array N N s P, sampled partcles at each step matr N N s, weghts assocated wth the samples matr N P, mean vector of the sample array N P P, covarance matr of the sample scalar, 1 to save the partcles (default), 0 f not Remars Combnng the outputs provdes the emprcal posteror densty at each step whch can be appromated by: N s p ( = z ) = wδ ( ) Source pf.src =1

17 16 CHAPTER 3. COMMAND REFERENCE Partcle Smoother Purpose Run a properly defned partcle smoother, by drawng N s realzatons of p ( z 1:K ). Format {ps,ps w} = Partcle Smoother(pf,pf w,ns); Input pf pf w Ns Output ps ps w array N N s P, stored partcles from the run of a partcle flter matr N N s, stored weghts assocated to the samples from the PF run scalar, number of smulatons array N N s P, smoothed partcles at each step matr N N s, weghts assocated wth the samples ps Globals Remars Runnng a partcle smoother requres to run a partcle flter frst, and feedng the smoothng procedure wth the output of the PF run. Ths algorthm assumes that the procedures used are based on Importance Resamplng. Fnally, the sze of the samples of smoothed partcles s equal to the sze of the samples from the partcle flter. Source pf.src

18 Purpose Run a regularzed partcle flter. Regularzed Partcle Flter 17 Format {,w,m,cov} = Regularzed Partcle Flter(z,Ns); Input z Ns Output w m cov matr N G, observed values z scalar, number of partcles array N N s P, sampled partcles at each step matr N N s, weghts assocated wth the samples matr N P, mean vector of the sample array N P P, covarance matr of the sample Globals pf ernel scalar, defnes the ernel for the regularzaton step (default = 1) 1 for the Epanechnov ernel 2 for the Gaussan ernel pf Save Partcles scalar, 1 to save the partcles (default), 0 f not Remars Combnng the outputs provdes the emprcal posteror densty at each step whch can be appromated by: N s p ( = z ) = wδ ( ) Source pf.src =1

19 18 CHAPTER 3. COMMAND REFERENCE Smulate Tracng Problem Purpose Smulate a tracng problem for Monte Carlo analyss. Format {t,,z} = Smulate Tracng Problem(0,N,Ns); Input 0 vector P 1, ntal values 0 N scalar, number of tme perods Ns scalar, number of smulatons Output t z vector N 1, tme nde aray N N s P, sample of aray N N s G, sample of z Globals Remars The model consdered by the procedure Smulate Tracng Problem s the followng: { = F (t, 1 ) + ν z = H (t, ) + η wth ν N (0, Q (t, 1 )) and η N (0, R (t, 1 )). The functons F, Q, H and R are ntalzed usng the procedure Partcle Flter Set. Source pf.src

20 Purpose Run a SIR partcle flter. SIR Partcle Flter 19 Format {,w,m,cov} = SIR Partcle Flter(z,Ns); Input z Ns Output w m cov Globals pf Save Partcles matr N G, observed values z scalar, number of partcles array N N s P, sampled partcles at each step matr N N s, weghts assocated wth the samples matr N P, mean vector of the sample array N P P, covarance matr of the sample scalar, 1 to save the partcles (default), 0 f not Remars Combnng the outputs provdes the emprcal posteror densty at each step whch can be appromated by: N s p ( = z ) = wδ ( ) Source pf.src =1

21 20 CHAPTER 3. COMMAND REFERENCE Purpose Run a SIS partcle flter. Format {,w,m,cov} = SIS Partcle Flter(z,Ns); SIS Partcle Flter Input z Ns Output w m cov Globals pf Save Partcles matr N G, observed values z scalar, number of partcles array N N s P, sampled partcles at each step matr N N s, weghts assocated wth the samples matr N P, mean vector of the sample array N P P, covarance matr of the sample scalar, 1 to save the partcles (default), 0 f not Remars Combnng the outputs provdes the emprcal posteror densty at each step whch can be appromated by: N s p ( = z ) = wδ ( ) Source pf.src =1

22 Chapter 4 Some eamples 1. eample1.prg We consder the eample 1 of Arulampalam et al. [1]: { p ( 1 ) = N ( (F () 1 ), Q ) 2 p (z ) = N 20, R or equvalently: { = F ( 1) + ν where: F ( 1 ) = 1 2 z = η cos (1.2) 1 We have ν N (0, Q ) and η N (0, R ). We use Q = 1 and R = 10. Usng the procedure Partcle Flter Set, we buld the correspondng tracng problem. We consder 1000 partcles and perform a Monte Carlo analyss n order to compare the RMSE between SIS, GPF, SIR and RPF algorthms (Fgure 4.1). In Fgure 4.2, we report the results of one MC tral. 2. eample2.prg We use the prevous tracng problem n order to llustrate the nfluence of the number of partcles n the convergence of the SIS algorthm. Results are reported n Fgure eample3.prg Same eample than the eample2.prg program, but wth the SIR algorthm. Results are reported n Fgure eample4.prg We estmate the probablty densty of the state varables usng the RPF algorthm and represent t n Fgures 4.5 and eample5.prg In ths program, we reproduce the eample 2 of Roncall and Wesang [9]. Results are reported n Fgure Ths eample has been already studed by Carln et al. [2] and Ktagawa [4]. 2 Append C, Fgure 28, page

23 22 CHAPTER 4. SOME EXAMPLES Fgure 4.1: Densty of the RMSE statstc Fgure 4.2: An eample of a MC run

24 23 Fgure 4.3: Densty of the RMSE statstc for the SIS algorthm Fgure 4.4: Densty of the RMSE statstc for the SIR algorthm

25 24 CHAPTER 4. SOME EXAMPLES Fgure 4.5: Probablty densty evoluton (partcles representaton) Fgure 4.6: Probablty densty evoluton (mass probablty representaton)

26 25 6. eample6.prg An eample to llustrate the procedure Partcle Smoother. Fgure 4.7: Solvng a GTAA tracng problem wth partcle flters

27 26 CHAPTER 4. SOME EXAMPLES

28 Bblography [1] Sanjeev Arulampalam, Smon Masell, Nel J. Gordon, and Tm Clapp. A tutoral on partcle flters for onlne nonlnear/non-gaussan Bayesan tracng. IEEE Transacton on Sgnal Processng, 50(2): , February [2] Bradley P. Carln, Ncholas G. Polson, and Davd S. Stoffer. A monte carlo approach to nonnormal and nonlnear state space modelng. Journal of the Amercan Statstcal Assocaton, 87(418): , [3] Mchael S. Johannes and Nc Polson. Partcle flterng and parameter learnng. Unversty of Chcago, Worng Paper, March [4] Genshro Ktagawa. Monte carlo flter and smoother for non-gaussan nonlnear state space models. Journal of Computatonal and Graphcal Statstcs, 5(1):1 25, [5] Mchael K. Ptt and Nel Shephard. Flterng va smulaton: Aulary partcle flters. Journal of the Amercan Statstcal Assocaton, 94(446): , June [6] George Poyadjs, Arnaud Doucet, and Sumeetpal S. Sngh. Mamum lelhood parameter estmaton n general state-space models usng partcle methods. In Proceedngs of the Amercan Statstcal Assocaton, JSM 05, August [7] George Poyadjs, Arnaud Doucet, and Sumeetpal S. Sngh. Partcle methods for optmal flter dervatve: applcaton to parameter estmaton. In Proceedngs IEEE Internatonal Conference on Acoustcs, Speech, and Sgnal Processng, March [8] Brano Rstc, Sanjeev Arulampalam, and Nel J. Gordon. Beyond the Kalman Flter: Partcle Flters for Tracng Applcatons. Artech House, Boston, 1st edton, [9] Therry Roncall and Gullaume Wesang. Tracng problems, hedge fund replcaton and alternatve beta. Worng Paper, Avalable at SSRN: 27