On Sequential Simulation-Based Methods for Bayesian Filtering

Transcription

1 O Sequetial Simulatio-Based Methods for Bayesia Filterig Araud Doucet Sigal Processig Group, Departmet of Egieerig Uiversity of Cambridge CB PZ Cambridge Techical report CUED/F-INFENG/TR Abstract. I this report, we preset a overview of sequetial simulatiobased methods for Bayesia filterig of oliear ad o-gaussia dyamic models. It icludes i a geeral framewor umerous methods proposed idepedetly i various areas of sciece ad proposes some origial developmets. Keywords: Bayesia estimatio, optimal filterig, oliear o-gaussia state space models, hidde Marov models, sequetial Mote Carlo methods.. Itroductio May problems i statistical sigal processig, automatic cotrol, applied statistics or ecoometrics ca be stated as follows. A trasitio equatio describes the prior distributio of the Marovia hidde sigal of iterest {x ; }, the so-called hidde state process, ad a observatio equatio describes the lielihood of the observatios {y ; }, beig the discrete time idex. The aim is to estimate the hidde state process usig the observatios. I the Bayesia framewor, all relevat iformatio o {x 0, x,..., x } at time is icluded i the posterior distributio p x 0, x,..., x y 0, y,..., y. I may applicatios i sigal processig, we are iterested i estimatig recursively i time this distributio ad especially oe of its margials, the so-called filterig distributio p x y 0, y,..., y. This problem is ow as the Bayesia filterig problem, also called the optimal filterig or stochastic filterig problem. Except for a few cases icludig liear Gaussia state space models Kalma filter ad hidde fiite-state space Marov chais Woham filter, it is impossible to evaluate aalytically these distributios. From the mid 60 s, a huge umber of papers ad boos have bee devoted to obtaiig approximatios of these distributios, see [3] for example. The most popular algorithms, the exteded Kalma filter ad the Gaussia sum filter, rely o aalytical approximatios [5, 6] but early well-ow wor relyig o determiistic umerical itegratio methods was also performed by Bucy ad co-worers, see [3] for example. Other iterestig wor i automatic cotrol was doe durig the 60 s ad 70 s based o sequetial Mote Carlo itegratio methods, see [,, 4, 5, 6, 47, 6]. Most liely because of the primitive computers available at the time, these last algorithms were overlooed ad forgotte. I the late 80 s, the great icrease of computatioal power allowed the rebirth of umerical itegratio methods for Bayesia filterig [33]. Curret research has ow focused o MC Mote Carlo itegratio methods which have the great advatage of ot beig subject to ay liearity or Gaussiaity hypotheses o the model. The mai objective of this report is to iclude i a uified framewor may old ad recet algorithms developed idepedetly i various fields of applied sciece. Some origial developmets are also preseted. The closest wor to this report is the wor of Liu ad Che [4], developed idepedetly, which uderlies similarly the cetral rôle of sequetial importace samplig SIS i sequetial simulatio-based methods for Bayesia filterig. This techical report is a traslatio of chapter 3 of [9] i abbreviated form. To the best of my owledge, these importat wors are cited either i ay stadard article ad boo o optimal estimatio or i ay curret wor o the subject.

2 O Sequetial Simulatio-Based Methods for Bayesia Filterig This report is orgaized as follows. I sectio, we briefly review the Bayesia filterig problem. A classical MC method, Bayesia importace samplig, is proposed to solve it. We the preset a sequetial versio of this method which allows us to obtai a geeral recursive MC filter. This algorithm is based o the itroductio of a probability distributio ow as the importace fuctio. Uder a give criterio, we obtai the optimal importace fuctio. Ufortuately, for umerous models, oe caot use this importace fuctio, which is why we propose several suboptimal distributios of practical iterest ad retrieve as particular cases may algorithms preseted idepedetly i the literature. I Sectio 3, a resamplig scheme is used to limit practically the degeeracy of the algorithm. I Sectio 4, we apply the Rao-Blacwellisatio method to SIS ad obtai efficiet hybrid aalytical/mc filters. I Sectio 5, we show how to use the MC filter to compute the predictio ad fixed-iterval smoothig distributios as well as the lielihood. Fially, a few simulatios are preseted i Sectio 6.. Bayesia Estimatio for Hidde Marov Models usig Importace Samplig The sigal {x ; }, x x, is a uobserved hidde Marov process of iitial distributio p x 0 ad trasitio equatio p x x. The observatios {y ; }, y y, are coditioally idepedet give the process {x ; } of margial distributio p y x. To sum up, the model is a hidde Marov model HMM described by p x 0 ad p x x for p y x for 0 We deote by x {x 0,..., x } ad y {y 0,..., y }, respectively, the sigal ad the observatios up to time. Our aim is to estimate recursively i time the distributio p x y ad its associated features icludig p x y ad the expectatio I f = p x y f x = f x p x y dx 3 for ay p x y -itegrable f : recursive formula for p x y : + x. We obtai straightforwardly a p x + y + = p y + x + p x + p y + = p x y p y + x + p x + x p y + y 4 This recursio is oly academic i the sese that oe caot typically compute the ormalizig costat p y +, the margials of p x + y + i particular p x + y + ad I f + because it requires the ability to evaluate complex high-dimesioal itegrals. A umerical solutio cosists of usig a Mote Carlo itegratio method. Later, we will assume that we ow how to sample accordig to p x x ad that we ca evaluate p x x ad p y x poitwise... Perfect Mote Carlo { samplig. } Let us assume that we are able to simulate N i.i.d. radom samples x i ; i =,..., N accordig to p x y. A empirical estimate of this distributio is give by: P dx y = N i= δ i x dx 5

3 O Sequetial Simulatio-Based Methods for Bayesia Filterig 3 ad oe obtais the followig estimate: I N f = f x P dx y = N From the strog law of large umbers SLLN i= f x i 6 I N f a.s. N + I f 7 where a.s. deotes almost sure covergece. If the posterior variace of f x satisfies σ f var p y [f x ] 8 [ f x ] p y [f x ] < + = p y the a cetral limit theorem holds: N[IN f I f ] = N + N 0, σ f where = deotes covergece i distributio. { The advatage of this perfect MC method is clear. From the set of radom samples x i }, ; i =,..., N oe ca easily estimate ay quatity p y [f x ] ad the speed of covergece of this estimate either depeds o x or o f x but oly o N. Ufortuately, it is usually impossible to sample efficietly from the posterior distributio p x y at ay time, p x y beig multivariate, o stadard ad oly ow up to a proportioality costat... Bayesia Importace Samplig. A alterative solutio cosists of usig the importace samplig IS method. The basic idea of this method is the followig. We choose a so-called importace fuctio, that is a probability distributio π x y which depeds here o the observatios util time from which oe ca easily sample. The IS method is based o the followig simple remar. If p x y > 0 implies π x y > 0 the oe ca write: I f = f x p x y π x y π x y dx 0 = π y [f x w x ] where w x = p x y /π x y Thus if oe ca simulate N i.i.d. samples {x, i =,..., N} accordig to π x y, a possible estimate of I f is: { where the importace weights w i Î N f N w i = w x i = i= 9 f x i w i 3 }, i =,..., N are equal to: px i π x i y = py i x pxi y py π x i 4 y The estimate 3 is ubiased ad coverges a.s. accordig to the SLLN toward I f whe N +.

4 O Sequetial Simulatio-Based Methods for Bayesia Filterig 4 I a Bayesia framewor, this estimate caot geerally be used as it requires the owledge of the ormalizig costat p y : p y = p y x p x dx 5 Typically p y caot be expressed i closed form. However, oe ca observe that I f = π y [f x w x ] π y [w x ] 6 where w x = p y x p x /π x y 7 Thus a estimate of I f is give by the ratio of the estimates of the umerator ad deomiator obtaied usig the classical importace samplig method: x i w i Î N f = N N i= f N N j= wj where the uormalised importace weights = i= { } w i, i =,..., N f x i w i 8 are equal to w i = w x i = p y x i p x i /π x i y 9 w i 0 meas proportioal to ad the ormalised importace weights are equal to w i w i = N j= wj The true importace weights w i have bee replaced by the followig estimate: ŵ i = N w i This method is well-ow i the statistical literature as Bayesia IS, see for example [, 5]. We recall here some classical results o this MC method. Assumptio { } x i ; i =,..., N is a set of i.i.d. vectors distributed accordig to π x y. The support π = { x x + /π x y > 0 } of π x y icludes the support p = { x x + /p x y > 0 } of p x y. I f exists ad is fiite. Assumptio p. y [w x ] < + ad p. y [f x w x ] < +. A sufficiet coditio to verify assumptio is []: var p. y [f x ] < + ad w x < C < + for ay x π 3

5 O Sequetial Simulatio-Based Methods for Bayesia Filterig 5 Propositio. For N fiite, ÎN f is biased but asymptotically, uder assumptio, the SLLN yields: a.s. Î N f I f 4 N + Uder assumptio, the previous propositio implies a covergece of the empirical distributio N i= wi δ i x dx towards P dx y i the sese of a.s. covergece of ÎN f for ay fuctio f such that I f exists ad is fiite. This result is importat as it meas that we ca iterpret the IS method as a simulatio method to sample from P dx y rather tha as a itegratio method, see [] for a similar iterpretatio. Usig the delta method, we also obtai the followig propositio. Propositio. Gewee 989 [] Uder assumptios ad, N[ Î N f I f ] = N + N 0, σ f 5 where f σ f = p. y x p. y f x w x 6 We show i the followig subsectio how it is possible to obtai easily a recursive MC filter usig Bayesia IS..3. Mote Carlo filter usig sequetial importace samplig. Oe ca always rewrite the importace fuctio as follows: π x y = π x 0 y π x x 0:, y 7 where π x x 0:, y is the probability desity fuctio of x coditioal upo x 0: ad y. Our aim is to obtai at time a estimate of the distributio p x 0: y 0: ad to be able to propagate { this estimate i time without modifyig subsequetly the past simulated trajectories x i 0: }. ; i =,..., N This meas that the importace fuctio at time + admits as a margial distributio at time the importace fuctio π x 0: y 0:. This is possible if we restrict ourselves to importace fuctios of the followig form: Iteratig, it yields: = π x y = π x y π x x, y 8 π x y = π x 0 y 0 π x x 0:, y 0: 9 This importace fuctio allows to evaluate recursively i time the importace weights 9 ad. Remar. This assumptio could be weaeed. For example, oe ca cosider the case where oe is iterested i a estimate of the fixed-lag distributio p x 0: y 0:+p, p beig fixed. I this case, oe ca choose: π x y +p = π x 0 y 0:p = π x x 0:, y 0:+p =

6 O Sequetial Simulatio-Based Methods for Bayesia Filterig 6 Uder the assumptio 9, we obtai straightforwardly the followig MC filter. Algorithm : Sequetial Importace Samplig SIS. At time = 0, For i =,..., N, sample x i 0 π x 0 y 0. For i =,..., N, evaluate the importace weights up to a ormalizig costat: p w i 0 = y 0 x i 0 p x i 0 30 π y 0 x i 0 For i =,..., N, ormalise the importace weights: w i w i 0 0 = N j= wj 0 3. For times, For i =,..., N, sample x i π x x i 0:, y 0: ad x i 0: x i 0:, xi. For i =,..., N, evaluate the importace weights up to a ormalizig costat: p y x i w i = w i p x i x i πx i 3 x i 0:, y 0: For i =,..., N, ormalise the importace weights: w i w i = N j= wj 33 Numerous algorithms proposed i the literature are special cases of this geeral ad simple algorithm. A particular case of this algorithm was itroduced i 969 by Hadschi ad Maye [5, 6]! The umerical complexity of this algorithm is O N. This is importat as we tae N i practice but it has the great advatage of beig parallelizable. I the geeral case, the memory requiremets are O + N as it is ecessary to eep all the N simulated trajectories from time 0 to time. However, if π x x i 0:, y 0: = π x x i, y ad if oe is oly iterested i the filterig distributio p x y 0:, the memory requiremets are O N. I the geeral case, oe obtais at time the followig estimate of the joit posterior distributio: P dx 0: y 0: = w i δ dx x i 0: 34 0: ad a estimate of I f : Î N f = i= f x 0: P dx 0: y 0: 35 = i= w i f x i 0:

7 O Sequetial Simulatio-Based Methods for Bayesia Filterig 7 Assumptio which esures the asymptotic covergece of these estimates is quite wea. I practice, oe obtais however poor performace of these estimates whe the importace fuctio is ot well-chose. The choice of the importace fuctio is the topic of the followig sectios..4. Degeeracy of the algorithm. Whe iterpretig IS as a MC samplig method rather tha as a MC itegratio method, the best possible choice would cosist of selectig the posterior distributio of iterest p x 0: y 0: as importace fuctio π x 0: y 0:. The we would obtai for the importace weights π y 0: w x 0: = ad var π y0: w x 0: = 0. We would lie to be close to this case. But for importace fuctios of the form 9, the variace of the importace weights ca oly icrease stochastically over time. Propositio 3. The ucoditioal variace of the importace weights, i.e. observatios y 0: beig iterpreted as radom variables, icreases over time. with the The proof of this propositio is a straightforward extesio of a Kog-Liu-Wog [37, p. 85] theorem to the case of a importace fuctio of the form 9. Thus, it is impossible to avoid a degeeracy pheomeo. Practically, after a few iteratios of the algorithm, all but oe of the ormalised importace weights are very close to zero, a large computatioal burde is devoted to updatig trajectories whose cotributio to the fial estimate is almost zero..5. Selectio of the importace fuctio. Practically, at time, the importace weights w i, i =,..., N are fixed. To limit degeeracy of the algorithm, a atural strategy cosists of selectig the importace fuctio which miimizes the variace of the importace weights coditioal upo the simulated trajectory x i 0: ad the observatios y 0:. Propositio 4. p x x i, y is the importace fuctio which miimizes the variace of the importace weight w i The proof is straightforward [9]. importace fuctio p x x i, y. Optimal importace fuctio. coditioal upo x i 0: ad y 0:. First we preset how to implemet the optimal The optimal importace fuctio p x x i, y has bee itroduced by Zaritsii et al. [6] the by Aashi et al. for a particular case [4]. More recetly, this importace fuctio has bee used i [4, 5, 6, 30, 37, 38, 40]. For this distributio, we obtai usig 3 the followig expressio for the importace weight: w i = w i p y x i = w i p y x i p x i x i /p x i x i, y Remar. I this case, the importace weight w i does ot deped { o x i. This is } iterestig i practice as it allows parallelizatio of the simulatio of x i { ; i =,..., N ad the evaluatio of w i }. ; i =,..., N 36 Remar 3. To verify Propositio, a sufficiet coditio which esures that the importace weights are bouded cosists of assumig that the lielihood is bouded. Ufortuately, this boud is time-depedet.

8 O Sequetial Simulatio-Based Methods for Bayesia Filterig 8 The optimal importace fuctio suffers from two major drawbacs. It requires the ability to sample from p x x i, y ad to evaluate, up to a proportioality costat, p y x i where p y x i = p y x p x x i dx 37 It requires the evaluatio of a itegral which does ot admit a aalytical expressio i the geeral case. Nevertheless, this evaluatio is possible for the importat class of models preseted below. Example 5. Partial Gaussia State Space Models. Let us cosider the followig model: x = f x + v, v N 0, Σ v 38 y = Cx + w, w N 0, Σ w 39 where f : x x, C is a real y x matrix, v ad w are two mutually idepedet i.i.d. Gaussia sequeces with Σ v > 0 et Σ w > 0. Deotig oe obtais ad p y x exp Σ = Σ v + C t Σ w C 40 m = Σ Σ v f x + C t Σ 4 w y x x, y N m, Σ 4 y Cf x t Σ v + CΣ w C t y Cf x For may other models, such evaluatios are impossible. We ow preset suboptimal methods which allow approximatio of the optimal importace fuctio. The first proposed method is based o a secod MC step. MC approximatio of the optimal importace fuctio. We assume here that p y x i caot be evaluated aalytically ad that it is ot possible to sample from p x x i, y. If the lielihood p y x < M is bouded the the ratio p x x i, y /p x x i is bouded. It is possible to sample from p x x i, y usig the accept/reject procedure. Accept/Reject procedure. Sample x p x x i ad u U [0,].. Accept x i = x if u p y x /M ; otherwise retur to. Ufortuately, this procedure requires a radom umber of iteratios before obtaiig a radom sample distributed accordig to p x x i, y. I the framewor of olie applicatios, this strategy must be avoided. Aother more severe problem is that p y x i is ot evaluated. 43

9 O Sequetial Simulatio-Based Methods for Bayesia Filterig 9 A aive approach cosists of usig a secod MC step based o Bayesia IS to sample from p x x i, y ad/or to evaluate p y x i. For each x i i =,..., N, oe { } ca sample N i.i.d. radom variables x i,j ; j =,..., N distributed accordig to x i,j where p x x i. We obtai the followig approximatio of p P dx x i, y = p p j i = N N j= p p y x i beig a estimate of p y x : x x i, y : p j i δ dx x i,j 44 y x i,j 45 y x i p y x i = N N j= p y x i,j 46 This approximatio is theoretically valid oly if N +. Moreover, this solutio, although simple, is computatioally very expesive. Other MC methods based o MCMC methods have bee proposed to simulate approximately from p x x i, y ad/or to evaluate p y x i, see [, 8, 4]. These iterative algorithms appear to be of limited iterest i a o-lie framewor ad there is a lac of theoretical covergece results. I fact, the geeral framewor of SIS allows us to cosider other importace fuctios built so as to approximate aalytically the optimal importace fuctio. The advatages of this alterative approach are that it is computatioally less expesive that MC methods ad that the previous give covergece results o Bayesia IS are still valid. There is o geeral method to build suboptimal importace fuctios ad it is ecessary to build these o a case by case basis, depedet o the model studied. To this ed, it is possible to base these developmets o previous wor o stadard suboptimal filterig methods [6, 60]. Importace distributio obtaied by local liearisatio. A simple choice cosists of selectig as the importace fuctio π x x, y a parametric distributio π x θ x, y, of fiite-dimesioal parameter θ θ Θ determied by x ad y, θ : x y Θ beig a determiistic mappig. May strategies are possible. To illustrate such methods, we preset here two origial methods that result i a Gaussia importace fuctio whose parameters are evaluated usig local liearisatios, i.e. which are depedet o the simulated trajectory i =,..., N. Local liearisatio of the Marov state space model. We propose to liearise the model locally as i the Exteded Kalma Filter. However, i our case, this liearisatio is performed with the aim of obtaiig a importace fuctio ad the algorithm obtaied still coverges asymptotically towards the optimal solutio uder the assumptios give previously. Example 6. Let us cosider the followig model x = f x + v, v N 0 v, Σ v 47 y = g x + w, w N 0 w, Σ w 48

10 O Sequetial Simulatio-Based Methods for Bayesia Filterig 0 where f : x x, g : x y is differetiable, v ad w are two mutually idepedet i.i.d. sequeces with Σ v > 0 ad Σ w > 0. Performig a approximatio up to the first order of the observatio equatio [6], we get y = g x + w g f x + g x x x =fx x f x + w 49 We have ow defied a ew model with a similar evolutio equatio to 47 but with a liear Gaussia observatio equatio 49, obtaied by liearisig g x i f x. This model is ot Marovia as 49 depeds o x. However, it is of the form ad oe ca perform similar calculatios to obtai a Gaussia importace fuctio π x x, y N m, Σ with mea m ad covariace Σ evaluated for each trajectory i =,..., N usig the followig formula: Σ m = Σ [ = Σ v + g x x Σ v f x + x =fx [ g x x ] t Σ y g f x + g x x w g x x x =fx The associated importace weight is evaluated usig 3. x =fx 50 ] t Σ w 5 f x x =fx Local liearisatio of the optimal importace fuctio. We assume here that l x l p x x, y is twice differetiable wrt x o x. We defie: l x l x 53 x l x l x x x t x =x Usig a secod order Taylor expasio i x, we get : x =x 5 54 l x l x + [l x] t x x + x x t l x x x 55 The poit x where we perform the expasio is arbitrary but determied by a determiistic mappig of x ad y. Uder the additioal assumptio that l x is egative defiite, which is true if l x is cocave, the settig yields Σ x = l x 56 m x = Σ xl x 57 [l x] t x x + x x t l x x x = C x x m x t Σ x x x m x 58

11 O Sequetial Simulatio-Based Methods for Bayesia Filterig This suggests to adopt as importace fuctio: π x x, y = N m x + x, Σ x 59 If p x x, y is uimodal, it is judicious to adopt x as the mode of p x x, y, thus m x = 0 x. The associated importace weight is evaluated usig 3. Example 7. Liear Gaussia Dyamic/Observatios accordig to a distributio from the expoetial family. We assume that the evolutio equatio satisfies: x = Ax + v where v N 0 v, Σ v 60 where Σ v > 0 ad the observatios are distributed accordig to a distributio from the expoetial family, i.e. p y x = exp y t Cx b Cx + c y 6 where C is a real y x matrix, b : y ad c : y. These models have umerous applicatios ad allow cosideratio of Poisso or biomial observatios, see for example [60]. We have This yields l x = C + y t Cx b Cx x Ax t Σ v x Ax 6 l x = b Cx x x t x =x Σ v = b x Σ v 63 but b x is the covariace matrix of y for x = x, thus l x is defiite egative. Oe ca determie the mode x = x of this distributio by applyig a iterative Newto- Raphso method iitialised with x 0 = x, which satisfies at iteratio j: x j+ = x j [ l x j ] l x j 64 Remar 4. This last method is close to the oe developed idepedetly by Shephard ad Pitt [48] i a differet framewor. They propose a MCMC algorithm for off-lie estimatio of o-gaussia measuremets time series based o the Metropolis-Hastigs algorithms. The proposal distributio of this algorithm is build i the case where l x is cocave usig a similar method 3. We ow preset two simpler methods. Prior importace fuctio. A simple choice cosists of selectig as importace fuctio the prior distributio of the hidde Marov model. This is the choice made by Hadschi et Maye [5, 6] i their semial wor. This distributio has bee recetly adopted by Taizai et al. [56, 57]. I this case, we have ad π x x 0:, y 0: = p x x 65 w i = w i p y x i This method is ofte iefficiet i simulatios as the state space is explored without ay owledge of the observatios. It is especially sesitive to outliers. 3 I fact, all the methods developed i the literature to build clever proposal distributios for the Metropolis-Hastigs M-H algorithms ca be applied i a sequetial framewor ad vice versa. But, while covergece of the M-H algorithm is esured uder wea assumptios whe the umber of iteratios of the simulated Marov chai teds towards ifiity, i the sequetial framewor, covergece of the algorithm is esured uder wea assumptios whe the umber N of simulated trajectories teds towards ifiity. 66

12 O Sequetial Simulatio-Based Methods for Bayesia Filterig Fixed importace fuctio. A simpler choice cosists of fixig a importace fuctio idepedetly of the simulated trajectories ad from the observatios. I this case, we have π x x 0:, y 0: = π x 67 ad 3 : w i = w i p y x i p x i x i /π This is the choice adopted by Taizai et al. [54, 55] who presets this method as a stochastic alterative to the umerical itegratio method of Kitagawa [33]. The results obtaied are rather poor as either the dyamic of the model or the observatios are tae ito accout. It leads i most cases to ubouded importace weights. 3. Resamplig As it has bee previously illustrated, the degeeracy of the algorithm based o SIS ca ot be avoided. I [5], a forgettig factor o the weights associated with the optimal importace fuctio is itroduced ad, uder stability ad regularity assumptios o the Marov model, a iterestig time-uiform covergece result is obtaied as N +. Practically, N < + ad this regularizatio slows dow but does ot avoid degeeracy of the algorithm [6]. It is ecessary to itroduce aother procedures. The basic idea of resamplig methods cosists of elimiatig the trajectories which have wea ormalised importace weights ad to multiply trajectories with strog importace weights. We adopt as a measure of degeeracy of the algorithm the effective sample size. This criterio, itroduced by Liu [37, 39], is defied usig the variaces of the estimates of I f respectively obtaied usig imagiary i.i.d. samples accordig to π x 0: y 0: ad a importace samplig method based o i.i.d. samples distributed accordig to p x 0: y 0:. For fuctios f x 0: which vary slowly with x 0:, Liu shows that: ] var π y0: [ÎN f [ var p y0: IN f ] [ + var π y0: w x 0: ] 69 The effective sample size N eff is thus defied as: N eff = = N + var π y0: w x 0: N [w x 0: ] N N N i= π y 0: Oe ca ot evaluate exactly N eff but, owig to, a estimate N eff of N eff is give by: N N eff = = ŵ i 7 N i= w i Whe N eff is below a fixed threshold N thres, we use a resamplig procedure. The most popular resamplig scheme is the SIR algorithm Samplig Importace Resamplig itroduced by Rubi [46, 50]. This scheme is based o two steps: a first step is a IS step, the secod step is a samplig step based o the obtaied discrete distributio. 3.. SIS/Resamplig Mote Carlo filter. At time, we have the followig approximatio 34 : P dx 0: y 0: = i= x i w i δ dx x i 0: 7 0:

13 O Sequetial Simulatio-Based Methods for Bayesia Filterig 3 At time, the modified Mote Carlo filter proceeds as follows. Algorithm : SIS/Resamplig Mote Carlo filter. Importace samplig For i =,..., N, sample x i πx x i 0:, y 0: ad x i 0: x i 0:, xi. For i =,..., N, evaluate the importace weights up to a ormalizig costat: p y x i w i = w i p x i x i π x i 73 x i 0:, y 0: For i =,..., N, ormalise the importace weights: w i w i = N j= wj 74 Evaluate N eff usig 7.. Resamplig If N eff N thres x i 0: = xi 0: otherwise for i =,..., N. For i =,..., N, sample a idex j i distributed accordig to the discrete distributio with N elemets satisfyig Pr{j i = l} = w l for l =,..., N. For i =,..., N, x i 0: = xji 0: ad wi = N. If N eff N thres, the algorithm preseted i sectio is thus ot modified. If N eff < N thres the SIR algorithm is applied ad we obtai the followig approximatio of the joit distributio: P dx 0: y 0: = N i= δ i x dx 0: 75 0: Remar 5. I [4], other more iterestig resamplig schemes are preseted which reduce the MC variatio of the SIR. 3.. Implemetatio of the resamplig procedure. If N eff < N thres, it is ecessary to implemet the algorithm to sample N radom variates accordig to a discrete distributio with N elemets. A straightforward applicatio of the SIR procedure has a complexity i O N l N [3]. This complexity is very importat ad, so as to reduce it, Beadle et al. [0] have recetly proposed several ad hoc methods. I fact, it is possible to implemet exactly the SIR procedure i O N operatios by oticig that it is possible to sample i O N operatios N i.i.d. variables distributed accordig to U [0,] ad ordered, i.e. u u u N, usig a classical algorithm [45, pp. 96]. Algorithm [45, pp. 96] For i =,..., N, sample ũ i U [0,].

14 O Sequetial Simulatio-Based Methods for Bayesia Filterig 4 u N = [ũ N ] /N. For i = N,...,, u i = [ũ i ] /i u i+. We deduce straightforwardly the algorithm to sample N i.i.d. samples accordig to the discrete distributio i O N operatios. Remar 6. This algorithm is also preseted i [49] which attributed the idea of usig this algorithm to Carpeter, Clifford ad Fearhead Limitatios of the resamplig scheme. The resamplig procedure decreases algorithmically the degeeracy problem but itroduces practical ad theoretical problems. From a practical poit of view, the resamplig scheme seriously limits the parallelisability of the algorithm. From a theoretical poit of view, after oe resamplig step, the simulated trajectories are o loger statistically idepedet ad { so we lose the} simple covergece results give previously. Moreover the trajectories x i 0:, i =,..., N which are statistically selected may times. I 75, umer- have high importace weights w i ous trajectories x i 0: ad xi 0: are i fact equal for i i [,..., N]. There is a loss of diversity. Recetly, Berzuii et al. [] have however established a cetral limit theorem for the estimate of I f which is obtaied whe the SIR procedure is applied at each iteratio. Despite its drawbacs, the SIR algorithm is the basis of umerous wors. The popular bootstrap filter of Gordo, Salmod et Smith [9, 0,, 3, 3], simultaeously developed by Kitagawa [9, 34, 35, 36], applies at each iteratio a resamplig step usig π x x i, y = p x x i, see also [5] for a similar method developed i the closely related field of Bayesia etwors. To limit the loss of diversity, may ad hoc procedures have bee proposed. I [3], the trajectories are artificially perturbed after the resamplig step. Aother simple solutio cosists of buildig a semi-parametric approximatio of P dx = N i= p ik x x i before resamplig [8, 4] but the choice of a good erel K is difficult. Higuchi [7, 8] proposes various heuristic procedures tae from the geetic algorithms literature to itroduce such a diversity amog samples. Oe ca otice that, i fact, the SIR procedure has a similar mathematical structure to the selectio step of geetic algorithms. Iterestig extesios of the SIR algorithm have bee recetly developed by Shephard ad Pitt [49]. 4. Rao-Blacwellisatio for Sequetial Importace Samplig We propose here to improve SIS usig variace reductio methods desiged to mae the most of the model studied. Numerous methods have bee developed so as to reduce the variace of MC estimates icludig atithetic samplig [5, 6] ad cotrol variates [, 6]. We apply here the Rao-Blacwellisatio method [4]. We show how it is possible to apply this method successfully to a importat class of HMM ad obtai hybrid filters where a part of the calculatios is realized aalytically ad the other part usig MC methods. Let us assume that we ca partitio the state x x j 0,..., xj. We have: as x, x ad deote x j I f = f x p y x p x dx p y x p x dx 76 = [ f x, x p y x, ] x p x x dx p x dx [ p y x, x p x x ] dx p x dx = gx p x dx p y x p x dx

15 O Sequetial Simulatio-Based Methods for Bayesia Filterig 5 where gx f x, x p y x, x p x x dx 77 Uder the assumptio that, coditioal upo a realizatio of x, gx ad p y x ca be evaluated aalytically, two estimates of I f based o IS are possible. The first classical oe is obtaied usig as importace distributio π x, x y : Î N f = N N N f D N f = i= f x,i, x,i w x,i, x,i 78 N i= x w,i, x,i where w x,i, x,i = p π x,i, x,i x,i, x,i y 79 y The secod Rao-Blacwellised estimate Ĩ N f is obtaied by itegratig out aalytically x ad usig as importace distributio The estimate is give by: where π x y = N Ñ N f i= Ĩ p x y,x,i N f = = D N f w x,i = π x, x y dx 80 f x,i, x w N i= w x,i p π x,i x,i x,i 8 y 8 y The followig propositio shows that if oe ca itegrate aalytically oe of the compoets the the variace of the obtaied estimate is weaer tha the oe of the crude estimate. Propositio 8. The variaces of the importace weights, the umerator ad the deomiator, obtaied by Rao-Blacwellisatio, are smaller tha those obtaied usig a crude Mote Carlo method: [ var π x y w x ] [ varπ x,x y w x, ] x 83 ad var π x y ÑN f var π x y DN f var π x,x y NN f var π x,x y DN f The proof is straightforward [9]. We ca use this simple result to estimate the margial distributio p x y but also:

16 O Sequetial Simulatio-Based Methods for Bayesia Filterig 6 If f x, x = f x the gx = f x p y x ad N i= Ĩ f x,i w x,i N f = 86 N i= x w,i If f x, x = f x the gx = ad Ĩ N f = N i= p x y,x,i p x y,x f x p y x N i= w x,i f x w x,i 87 I all cases, it is possible to use the MC methods developed i the previous sectios to x. Nevertheless, eve if the observatios y are idepedet coditioal upo x, x, they are geerally o loger idepedet coditioal upo the sigle process x. The modificatios are straightforward. We obtai for the optimal importace fuctio p x y0:, x 0: ad its associated importace weight p y y 0:, x 0:. We ow preset two importat applicatios of this geeral method. Example 9. Coditioally liear Gaussia state space model Let us cosider the followig model p x x 88 x = A x x + B x v 89 y = C x x + D x w 90 where x is a Marov process, v N 0 v, I v ad w N 0 w, I w. Oe wats to estimate p x y, f x y, x y ad x x t y. It is possible to use a MC filter based o Rao-Blacwellisatio. Ideed, coditioal upo x, x is a liear Gaussia state space model ad the itegratios required by the Rao-Blacwellisatio method ca be realized usig the Kalma filter. Aashi ad Kumamoto [, 4, 58] itroduced this algorithm uder the ame of RSA Radom Samplig Algorithm i the particular case where x is a homogeeous fiite state-space Marov chai 4. I this case, they adopted the optimal importace fuctio p x y0:, x0:. Ideed, it is possible to sample from this discrete distributio ad to evaluate the importace weight p y y 0:, x0: usig the Kalma filter [4]. Similar developmets have bee proposed by Sveti et al. [53]. The algorithm for blid decovolutio recetly proposed by Liu et al. [38] is also a particular case of this method where x = h is a time-ivariat chael of Gaussia prior distributio 5. Usig the Rao-Blacwellisatio method i this framewor is particularly attractive as, while x has some cotiuous compoets, we restrict ourselves to the exploratio of a discrete state space. Example 0. Fiite State-Space HMM Let us cosider the followig model p x x p x x, x p y x, x 4 Aashi ad Kumamoto made the coectios with the wor of Hadschi ad Maye i []. 5 I this framewor, the extesio to a time-varyig chael h modeled by a liear Gaussia statespace model is straightforward.

17 O Sequetial Simulatio-Based Methods for Bayesia Filterig 7 where x is a Marov process ad x is a fiite state-space Marov chai whose parameters at time deped o x. We wat to estimate p x y, f x y ad f x y. It is possible to use a Rao-Blacwellised MC filter. Ideed, coditioal upo x, x is a fiite state-space Marov chai of ow parameters ad thus the itegratios require by the Rao-Blacwellisatio method ca be doe aalytically [6]. 5. Predictio, smoothig ad lielihood The estimate of the joit distributio p x 0: y 0: based o SIS, i practice coupled with a resamplig procedure to limit the degeeracy, is at ay time of the followig form: P dx 0: y 0: = i= w i δ dx x i 0: 9 0: We show here how it is possible to obtai based o this distributio some approximatios of the predictio ad smoothig distributios as well as the lielihood. 5.. Predictio. Based o the approximatio of the filterig distributio P dx y 0:, we wat to estimate the p step-ahead predictio distributio, p, give by: +p p x +p y 0: = p x y 0: p x j x j dx :+p 9 j=+ Replacig p x y 0: i 9 by its approximatio obtaied from 9, we obtai: i= w i +p p x + x i p x j x j dx +:+p 93 j=+ To evaluate these itegrals, it is sufficiet to exted the trajectories x i 0: usig the evolutio equatio. Algorithm. p step-ahead predictio For j = to p For i =,..., N, sample x i +j p x +j x i +j We obtai radom samples give by Thus ad x i 0:+j x i 0:+j, xi +j. { x i 0:+p ; i =,..., N }. A estimate of P dx 0:+p y 0: is P dx 0:+p y 0: = P dx +p y 0: = i= i= w i δ dx x i 0:+p 0:+p w i δ dx x i +p 94 +p 5.. Fixed-Lag smoothig. We wat to estimate the fixed-lag smoothig distributio p x y 0:+p, p beig the legth of the lag. At time + p, the MC filter yields the followig approximatio of p x 0:+p y 0:+p : P dx 0:+p y 0:+p = i= w i +p δ dx x i 0:+p 95 0:+p

18 O Sequetial Simulatio-Based Methods for Bayesia Filterig 8 By margialisig, we obtai a estimate of the fixed-lag smoothig distributio: P dx y 0:+p = i= w i +p δ dx x i 96 Whe p is high, such a approximatio will geerally perform poorly. Remar 7. To estimate p x y 0:+p, it would be better to use a importace fuctio of the form π x x 0:, y 0:+p, see Remar. Uder a straightforward modificatio of the criterio proposed previously, the optimal importace fuctio ad the associated importace weight are respectively equal to p x x 0:, y 0:+p ad p y +p x 0:, y :+p. Usually, it is difficult to sample from p x x 0:, y 0:+p ad impossible to evaluate aalytically p y +p x 0:, y :+p. It is possible to build suboptimal importace fuctios based for example o exteded Kalma smoother techiques but it remais to evaluate the term p y +p x 0:, y :+p which occurs i the expressio of the importace weight. It is possible to evaluate this term usig MC itegratio Fixed-iterval smoothig. Give y, we wat to estimate p x y for ay = 0,...,. At time, the filterig algorithm yields the followig approximatio of p x y : P dx y = i= w i δ dx x i 97 Thus oe ca theoretically obtai p x y for ay by margialisig this distributio. Practically, this method caot be used as soo as is sigificat as the degeeracy problem { requires } use of a resamplig algorithm. At time, the simulated trajectories x i ; i =,..., N have bee usually resampled may times: there are thus oly a few distict trajectories at times for ad the above approximatio of p x y is bad. This problem is eve more severe for the bootstrap filter where oe resamples at each time istat. It is ecessary to develop a alterative algorithm. We propose a origial algorithm to solve this problem. This algorithm is based o the followig formula [8, 33]: p x+ y p x + x p x y = p x y 0: dx + 98 p x + y 0: We see here a approximatio of the fixed-iterval smoothig distributio with the followig form: i.e. P dx y P dx y has the same support N i= w i δ dx x i 99 { x i ; i =,..., N } as the filterig distribu- { tio P dx y 0: but } the weights are differet. A algorithm to obtai these weights w i ; i =,..., N is the followig. Algorithm. Fixed-iterval smoothig.. Iitialisatio at time =. For i =,..., N, w i = wi.. For =,..., 0.

19 O Sequetial Simulatio-Based Methods for Bayesia Filterig 9 For i =,..., N, evaluate the importace weight N w i = w i w j p x j + x i [ + N ] j= l= wl p x j 00 + x l This algorithm is obtaied by the followig argumet. Replacig p x + y by its approximatio 99 yields p x+ y p x + x p x i dx + w i + x p x + y 0: + p x i 0 y 0: where, owig to 9, p x i p x i + + y 0: i= y 0: ca be approximated by = j= p x i + w j p x i A approximatio P dx y of p x y is thus = = + x p x y 0: dx 0 + x j P dx y 03 [ N ] w i δ N p x j dx x i w j + x [ + N ] i= j= l= wl p x j + x l p x j w i w j + x i [ + N ] i= j= l= wl p x j δ + x l i x dx w i δ dx x i i= The algorithm follows. This algorithm requires storage of the margial distributios P dx y 0: weights ad supports for ay = 0,...,. The memory requiremet is O N. Its complexity is O N, which is quite importat as N. However this complexity is a little lower tha the oe of the previous developed algorithms of Kitagawa [35, 36] ad Taizai [56, 57] as it does ot require ay ew simulatio step Lielihood. I some applicatios, i particular for model choice [33, 36], we may wish to estimate the lielihood of the data p y. A simple estimate of the lielihood is give, usig to 5 ad 8, by p y = N j= w j 04 I practice, the itroductio of resamplig steps maes this approach impossible. We will use a alterative decompositio of the lielihood: p y = p y 0 p y y 0: 05 =

20 O Sequetial Simulatio-Based Methods for Bayesia Filterig 0 where: p y y 0: = = p y x p x y 0: dx 06 p y x p x y 0: dx 07 Usig 06, a estimate of this quatity is give by where the samples p y y 0: = i= p y x i w i 08 { x i ; i =,..., N } are obtaied usig a oe-step ahead predictio based o the approximatio P dx y 0: of p x y 0:. Usig expressio 07, it is possible to avoid a MC itegratio if we ow aalytically p y x i : p y y 0: = i= p y x i w i Simulatios I this sectio, we apply the methods developed previously to a liear Gaussia state space model ad to a classical oliear model. We mae for these two models M = 00 simulatios of legth = 500 ad we evaluate the empirical stadard deviatio for the filterig estimates x = [ x y 0: ] obtaied by the MC methods: where: V AR x l = M = M j= / x j l xj x j is the simulated state for the jth simulatio, j =,..., M. x j l x j,i N i= wi l xj,i is the MC estimate of [ x y 0:l ] for the j th test sigal ad is the i th simulated trajectory, i =,..., N, associated with the sigal j. We deote w i w i. These calculatios have bee realized for N = 00, 50, 500, 000, 500 ad The implemeted filterig algorithms are the bootstrap filter, the SIS with the prior importace fuctio ad the SIS with the optimal or a suboptimal importace fuctio. The fixed-iterval smoothers associated with these SIS filters are the computed. For the SIS-based algorithms, the SIR procedure has bee used whe N eff < N thres = N/3. We state the percetage of iteratios where the SIR step is used for each importace fuctio. 6.. Liear Gaussia model. Let us cosider the followig model x = x + v 0 y = x + w where x 0 N 0,, v ad w are white Gaussia oises mutually idepedet, v N 0, σv ad w N 0, σw with σ v = σ w =. For this model, the optimal filter is the Kalma filter [6].

21 O Sequetial Simulatio-Based Methods for Bayesia Filterig Optimal importace fuctio. The optimal importace fuctio is x x, y N m, σ where [σ ] = σ w + σ v 3 m = σ x σ + y v σ 4 w ad the associated importace weight is equal to: p y x exp y x σ v + σ w 5 Results. For the Kalma filter, we obtai V AR x = For the differet MC filters, the results are preseted i Tab. ad Tab.. V AR x bootstrap prior dist. optimal dist. N = N = N = N = N = N = Table : MC filters: liear Gaussia model Percetage SIR prior dist. optimal dist. N = N = N = N = N = N = Table : Percetage of SIR steps: liear Gaussia model With N = 500 trajectories, the estimates obtaied usig MC methods are similar to those obtaied by Kalma. The SIS algorithms have similar performaces to the bootstrap filter for a smaller computatioal cost. The most iterestig algorithm is based o the optimal importace fuctio which limits seriously the umber of resamplig steps. 6.. Noliear series. We cosider here the followig oliear referece model [7, 3, 35, 56]: x = f x + v 6 x = x x + 8 cos. + v y = g x + w 7 = x 0 + w

22 O Sequetial Simulatio-Based Methods for Bayesia Filterig where x 0 N 0, 5, v ad w are mutually idepedet white Gaussia oises, v N 0, σ v ad w N 0, σ w with σ v = 0 ad σ w =. I this case, it is ot possible to evaluate aalytically p y x or to sample simply from p x x, y. We propose to apply the method described i.5 which cosists of liearisig locally the observatio equatio. Importace fuctio obtaied by local liearisatio. We get y g f x + g x x f x + w x = f x 0 = f x 0 + f x 0 x =fx x f x + w + f x x + w 8 0 x ; m, σ The we obtai the liearised importace fuctio π x x, y = N where σ = σ v + σ f x w 00 ad m = σ [ σ v f x + σ w f x 0 y + f ] x Results. I this case, it is ot possible to estimate the optimal filter. For the MC filters, the results are displayed i Tab. 3. The average percetages of SIR steps are preseted i Tab. 4. V AR x bootstrap prior dist. liearised dist. N = N = N = N = N = N = Table 3: MC filters: oliear time series Percetage SIR prior dist. liearised dist. N = N = N = N = N = N = Table 4: Percetage of SIR steps: oliear time series This model requires simulatio of more samples tha the precedig oe. I fact, the variace of the dyamic oise is more importat ad more trajectories are ecessary to explore the space. The most iterestig algorithm is the SIS with a suboptimal importace fuctio which greatly limits the umber of resamplig steps over the prior importace fuctio while avoidig a MC itegratio step eeded to evaluate the optimal importace

23 O Sequetial Simulatio-Based Methods for Bayesia Filterig 3 fuctio. This ca be roughly explaied by the fact that the observatio oise is rather small so that y is highly iformative ad allows a limitatio of the regios explored. 7. Coclusio I this report, we have preseted a overview of sequetial simulatio-based methods for Bayesia filterig of geeral hidde Marov models. This overview icludes i the geeral framewor of SIS umerous approaches that have bee previously proposed idepedetly i the literature for early 30 years. Several origial extesios have also bee preseted. I this re-emergig area, there are umerous ways of improvemet icludig amog may others ew variace reductio methods [0, ] or efficiet hybrid IS/MCMC methods. 8. Acowledgmets I acowledge Dr. T. Higuchi, Pr. G. Kitagawa ad Dr. H. Taizai who set me their preprits o this subject. I also acowledge C. Adrieu, T. Clapp ad Dr. S. Godsill for various commets that helped me to improve this report. Refereces [] H. Aashi ad H. Kumamoto, State Estimatio for Systems uder Measuremets Noise with Marov Depedet Statistical Property - a Algorithm based o Radom Samplig, i Proc. 6 th Cof. IFAC, 975. [] H. Aashi ad H. Kumamoto, Costructio of Discrete-time Noliear Filter by Mote Carlo Methods with Variace-reducig Techiques, Sys. Cot., vol. 9, o. 4, 975, pp. - i Japaese. [3] H. Aashi, H. Kumamoto ad K. Nose, Applicatio of Mote Carlo Method to Optimal Cotrol for Liear Systems uder Measuremet Noise with Marov Depedet Statistical Property, It. J. Cot., vol., o. 6, 975, pp [4] H. Aashi ad H. Kumamoto, Radom Samplig Approach to State Estimatio i Switchig Eviromets, Automatica, vol. 3, 977, pp [5] D.L. Aspach ad H.W. Soreso, Noliear Bayesia Estimatio usig Gaussia Sum Approximatio, IEEE Tras. Auto. Cot., vol. 7, o. 4, pp , 97. [6] B.D.O. Aderso ad J.B. Moore, Optimal Filterig, Eglewood Cliffs, 979. [7] M.L. Adrade, L. Gimeo ad M.J. Medes, O the Optimal ad Suboptimal Noliear Filterig Problem for Discrete Time Systems, IEEE Tras. Auto. Cot., vol. 7, 978, pp [8] M. Asar ad H. Deri, A Recursive Algorithm for the Bayes Solutio of the Smoothig Problem, IEEE Tras. Auto. Cot., vol. 6, o., 98, pp [9] D. Avitzour, A Stochastic Simulatio Bayesia Approach to Multitarget Tracig, IEE Proc. o Radar, Soar ad Navigatio, vol. 4, o., 995, pp [0] E.R. Beadle ad P.M. Djuric, A Fast Weighted Bayesia Bootstrap Filter for Noliear Model State Estimatio, IEEE Tras. Aeros. Elec. Sys., vol. 33, o., 997, pp [] P.R. Beyo, Mote Carlo ad Other Methods for Noliear No-Gaussia Estimatio, Math. Comp. Simul., o. 3, 990, pp [] C. Berzuii, N. Best, W. Gils ad C. Larizza, Dyamic Coditioal Idepedece Models ad Marov Chai Mote Carlo Methods, forthcomig J. Am. Stat. Assoc., 997.

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