On using Likelihood-adjusted Proposals in Particle Filtering: Local Importance Sampling

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1 On using Likelihood-adjused Proposals in Paricle Filering: Local Imporance Sampling Péer Torma Eövös Loránd Universiy, Pázmány Péer séány /c 7 Budapes, Hungary yus@axelero.hu Csaba Szepesvári Compuer and Auomaion Research Insiue of he Hungarian Academy of Sciences, Kende u. 3-7, Budapes, Hungary szcsaba@szaki.hu Absrac of he form An unsaisfacory propery of paricle filers is ha hey may become inefficien when he observaion noise is low. In his paper we consider a simple-o-implemen paricle filer, called LIS-based paricle filer, whose aim is o overcome he above menioned weakness. LIS-based paricle filers sample he paricles in a wo-sage process ha uses informaion of he mos recen observaion, oo. Experimens wih he sandard bearings-only racking problem indicae ha he proposed new paricle filer mehod is indeed a viable alernaive o oher mehods. Inroducion In his paper we consider filering of non-linear sochasic processes. The problem sudied can be formalized as follows. A sequence of values Y, Y, Y 2,... of some Euclidean space, governed by he equaions X + = a(x ; ξ +, X p (, ( Y + = b(x + ; η +, =,,... (2 is observed. Here X is he sae of he sysem a ime sep, p is he iniial disribuion over he possible saes a ime sep zero, and ξ, η are he process and observaion noise processes; hey are assumed o be composed of independen, idenically disribued random variables and also o be independen of each oher. The goal is o deermine he poserior, π (x = p(x = x Y,..., Y, over he saes a any ime-sep. Three major facors can be idenified ha influence he performance of filering algorihms: (i he energy of he process noise; (ii he energy of he observaion noise; and (iii he severiy of percepual aliasing ha makes he recovery of he sae from he sequence of observaions hard in he zero noise limi. Paricle filers (see e.g. [6, 7, 5] and he references herein approximae he poserior by empirical measures π (x = k= w(k k= w(k δ(x X (k where X (k and w (k represen he kh paricle s posiion and weigh, respecively, and δ( is Dirac s dela funcion. Here X (k and w (k def on he sequence of pas observaions Y : are (random quaniies ha depend = (Y,..., Y. Given he empirical measure he esimae of he expecaion of an arbirary funcion h wih respec o he poserior is obained by I,N (h = k= w(k h(x (k k= w(k,. (3 The generic paricle filer works by updaing he paricle s posiions and weighs in a recursive manner. The updae is composed of wo seps: compuaion of he paricle s new posiions is done by sampling from he so-called proposal funcion, followed by he updae of he weighs and an opional resampling sep [6]. In he SIR filer [6] he paricle s posiions are updaed independenly of each oher by sampling he kh paricle s new posiion using a(x (k ; ξ (k, where ξ (k is sampled from he common underlying disribuion of he process noise variables ξ, whils he weigh of he kh paricle is compued by evaluaing he observaion likelihood p(y X (k +. When he level of he observaion noise is low, he observaion likelihood funcion becomes peaky or concenraed around is modes (he modes correspond o he saes ha are locally mos likely o cause he mos recen observaion. If he posiion of a paricle is no sufficienly close o one of hese modes hen he paricle s weigh will become small and hus he paricle will bring in lile informaion ino he esimae of he poserior. If his happens for mos of he paricles hen he qualiy of he approximaion o he poserior degrade seriously. We call his problem he curse of reliable observaions. The curse of reliable observaions is a well-known peculiariy of paricle filers. The general advise o remedy

2 his problem is o use a proposal ha depends on he mos recen observaion [5]. However, finding a proposal ha is boh racable and sill yields good performance can be nooriously hard. A sraighforward alernaive is o increase he number of paricles unil i is ensured ha many paricles are close o he peaks of he observaion likelihood funcion. In high dimensional sae-spaces his approach may require an enormous number of paricles, which in urn slows down he filering process. Hence mehods ha make i possible o keep he number of paricles low are of considerable ineres. Since he inefficiency sems from he paricles posiions no being close enough o he modes of he observaion likelihood, i is a naural idea o le he modes of he likelihood arac he paricles. In his paper we consider algorihms ha subscribe o his idea. The algorihms considered in his paper generae he paricles posiions in a wo-sage sampling process, where he firs sep uses he prior saes, whils he second uses he mos recen observaion. In his paper we consider wo mehods, he firs mehod, he local likelihood sampling (LLS based paricle filer inroduced in [3], is used o moivae he second. In he LLSfiler he firs sampling sep is he same as in SIR, whils in he second sep of he LLS-filer a localized version of he observaion likelihood funcion is used o adjus he paricles posiions. Weighs are calculaed so ha he process remains (locally unbiased. In he case of he second algorihm proposed here, he firs sampling sep remains he same, whils in he second sep he observaion likelihood is replaced by a user-chosen proposal densiy funcion ha should be designed o be close o he likelihood. The weigh updae equaions are modified so ha unbiasedness is reained. We call his second algorihm he local imporance sampling based paricle filer. A paricular varian ha employs mixure of Gaussian proposals is sudied in greaer deail. Experimenal resuls wih he sandard bearings only racking problem indicae he superioriy of he proposed mehod o some of is alernaives. 2 Noaion Le us denoe he ransiion kernel corresponding o he dynamics a (cf. ( by K = K(u v, i.e., K(u v du = U P (a(v, ξ U, where U is any measurable subse of he sae-space. Furher, le us denoe he observaion likelihood densiy by r = r(y x, i.e., Y r(y x dy = P (b(x, η Y, where Y is any measurable subse of he observaion space. 3 Algorihms 3. LLS-filers The basic idea of LLS-based paricle filers [3], is o draw a sample from he predicion densiy as in SIR, bu hen allow he observaion densiy o perurb he posiion of he paricles. A window-funcion (g is used o localize he observaion densiy s effec on he sample, hence he name of he procedure. The role of localizaion is o preven paricles jump around in he sae-space, i.e. o keep he informaion in he previous esimae of he poserior. The procedure is shown in Figure. Iniialize {(Z (k, /N}N k= from he prior p (. For =, 2,... For k =, 2,..., N Resample S = {(Z (j, w(j }N j= o obain a new sample (Z (k, /N Predic X (k by drawing a sample from K( Z k Perurb X (k by drawing Z (k from r(y g(x (k, where α (k α (k = Updae weigh w (k ŵ (k r(y xg(x (k x dx. = α (k using K(Z (k Z (k K(X (k Normalize weighs using w (k Z (k. = ŵ (k / j= ŵ(j. Figure. LLS-based Paricle Filer The following proposiion was shown o hold in [3]: Proposiion Le (X, Y evolve according o Equaions (-(2. Assume ha g is a non-negaive, inegrable funcion saisfying I(g =. Le {(w (k, Z (k } be he paricle sep obained a ime sep of he LLS-based paricle filer. Then E[w (k h(z (k Y : ] = E[h(X Y : ]p(y Y :. In words, he saemen of he proposiion means ha he weighed sample obained a ime sep represens he poserior properly, up o a consan facor (dependen only on he observaions. As a consequence of he proposiion, we ge ha E[h(X Y : ] can be approximaed using weighed averages of he form (3 and convergence resuls (almos sure convergence, convergence in disribuion, mean-square, finie sample performance bounds can be derived along he lines of previous proofs (see [3] and references herein. Wha is more ineresing is ha in [3] a resul was proven where i was shown ha he LLS-filer can be more efficien han SIR (see Proposiion 2, [3] provided ha he cross-correlaion beween he observaion densiy and he observaion densiy muliplied by he predicion densiy is

3 much larger han he cross-correlaion beween he convoluion of he window funcion and he observaion densiy, and he convoluion of he window funcion and he produc of he observaion and predicion densiies. Dropping condiioning on pas observaions and ime indexes, le us denoe by f he observaion densiy, and by p he predicion densiy. Then he condiion for improved performance has he form ɛ f, fp f g, (fp g. Here ɛ is a consan ha depends on p and g (we omi he definiion of ɛ due o he lack of space; he ineresed reader can find i in [3] and u v denoes he convoluion of u and v. Since for ypical choices of he window funcion convoluion wih i cus high frequencies, he condiion can be expeced o hold for a wide class of problems, especially when he predicion densiy, p, is broad as compared o he observaion densiy, f. Noe ha he algorihm can be generalized easily o use window funcions ha change depending on X (k. One jus needs o replace g(x (k x in he sample-perurbaion sep by g(x (k x; X (k and redefine α (k accordingly: α (k = p(y xg(x (k x; X (k dx. The simples applicaion of his is o fi g o mach he energy disribuion of he observaion noise. Similarly g can be made dependen on he observaion Y. 3.2 Local Imporance Sampling One problem wih LLS-filers is ha hey depend on wheher sampling sampling from r(y g(x (k /α (k can be implemened efficienly. One possible remedy for his problem is o inroduce a proposal densiy o replace r(y. The corresponding algorihm, called Local Imporance Sampling, is given in Figure 2. The following proposiion can be shown o hold (he proof is omied due o he lack of space: Proposiion 2 Le (X, Y evolve according o Equaions (-(2. Assume ha g is a non-negaive, inegrable funcion saisfying I(g = and le q x,y ( > be a bounded, inegrable funcion for all x, y. Le {(w (k, Z (k } be he paricle se obained a ime sep of he LIS-based paricle filer. Then E[w (k h(z (k Y : ] = E[h(X Y : ]p(y Y :. As a consequence of his resul, he LIS-based paricle filer enjoys similar heoreical properies as SIR. Building on he previous argumen ha shows ha LLS is more efficien han he naive algorihm, one expecs ha under similar condiions LIS will also be more efficien provided ha he proposal funcion q x,y fis r(y around x for any x, y. The efficiency of he new algorihm will be demonsraed in he nex secion on he sandard bearings-only racking problem in he nex secion. However, firs le us consider an imporan pracical varian of his algorihm. Iniialize a sample se {(Z (k, /N}N k= according o he prior p (. For =, 2,... For k =, 2,..., N Resample S = {(Z (j, w(j }N j= o obain a new sample (Z (k, /N Predic X (k by drawing a sample from K( Z k Perurb X (k by drawing Z (k from q α (k (k X,Y ( g(x (k, where α (k = Updae weigh w (k ŵ (k = α (k q (k X,Y (xg(x (k x dx. using r(y Z (k q (k X,Y (Z (k K(Z (k Z (k K(X (k Z (k. Normalize weighs using w (k = ŵ (k / j= ŵ(j Figure 2. LIS-based Paricle Filer 3.3 Using Gaussian Mixure Proposal in LISbased paricle filers A paricularly aracive, easy o implemen LIS-based paricle filer is obained when he proposal funcion is chosen o be a mixure of Gaussians and he window funcion is chosen o be a Gaussian, oo. The purpose of his secion is o give he deails of he resuling procedure. Le u denoe he sae-observaion pair (x, y and choose q x,y = q u o be a mixure of Gaussians wih n componens, having priors p,..., p n, means m u,,..., m u,n and variances σ 2 u,,..., σ 2 u,n. Furher, choose he window funcion o be a zero-mean Gaussian wih variance σ 2 g. Mixure of Gaussians are aracive due o heir universal approximaion properies for coninuous densiies [, ] and also due o heir analyic racabiliy which we build heavily on here: For implemening he LIS-based paricle filer, one needs o be able o draw samples from q u ( g(x, as well as o evaluae (q u g(x. In our case, as i is well known, q u ( g(x is a mixure of Gaussians wih means variances and m u,i = xσ2 u,i + m u,iσg 2 σg 2 + σu,i 2, (4 σ u,i = σ gσ u,i σ 2 g + σ 2 u,i (5

4 and (un-normalized weighs: L u,i = p i e (x m u,i 2 2 (σ u,i 2 +σ2 g. (6 2π(σg 2 + σu,i 2 Hence, sampling from q u ( g(x can be implemened by firs drawing an index from he normalized weighs (L u, /L u,..., L u,n /L u, where L u = n i= L u,i, and hen drawing a sample from he appropriae Gaussian. Furher, (q u g(x is jus equal o L u. Hence, LIS-based paricle filers are paricularly easy o implemen when used wih mixure of Gaussian proposals and a Gaussian window funcion. The calculaions are furher elaboraed on in Figure 3 for a single paricle and a single ime-sep, assuming ha he predicion densiy is p and he observaion densiy is f (ime indexes and condiioning on previous samples/observaions are dropped. Before pre- Inpus: predicion densiy (p, observaion densiy (f, observaion (Y Draw he jh paricle X j from p Calculae he Gauss-Mixure parameers (m Xj,Y,i, σ Xj,Y,i, L Xj,Y,i using Equaions (4 (6. Draw an index k from { LXj },Y,i n L Xj,Y i= Draw Z j a Gaussian wih mean m Xj,Y,k and variance σ Xj,Y,k. Calculae he weigh ( n w j = i= L Xj,Y,i f(z j p(z j q Xj,Y (Z j p(x j. Figure 3. Local Imporance Sampling wih mixure-of-gaussian proposals and Gaussian window funcion sening our experimenal resuls we discuss some relevan relaed algorihms. In he experimens we will compare he performance of he proposed new mehod o ha of one of hese algorihms (AVM. 4 Relaed Work Due o space resricions, we give only a few key references. One of he bes known paricle filer whose aim is o overcome he curse of reliabiliy is he Auxiliary Variable Mehod (AVM inroduced by Pi and Shephard [9]. AVM uses a proposal densiy of he form q(x X (,..., X(, Y = k= r(y X (k K(x X (k, where X(k is e.g. he expeced nex sae for paricle k (i.e. X (k = E[a(X (k, ξ X (k ]. Sampling is implemened by firs doing a weighed resampling sep using he weighs r(y X (k and hen drawing he nex saes using he ransiion densiy kernel K. AVM can be more efficien han SIR when he process noise variance is low and he observaion likelihood is no oo peaky. AVM is similar o LLS/LIS-filers in ha i is also a wo-sage scheme. However, in AVM he paricles posiion is sill sampled from he predicion densiy, whils in LLS/LIS-filers he observaion direcly influences he paricles s posiions. One issue wih AVMs is ha when he observaion likelihood is peaky and he number of paricles is no high enough hen resampling he paricle se using {r(y X (k } migh no be successful a picking righ paricles. A similar problem occurs when he predicion densiy is muli-modal (hough in such a case X could be replaced by random samples from he predicion densiy. Also, when he predicion densiy has a large variance hen even if he firs sage is successful, sampling he paricle s nex sae from K migh yield o a se of paricles ha are spread ou oo much in he sae-space. The advanage of AVM o LIS-filers is ha in LIS-filers he user has o design he proposals, hough his choice can be made auomaic by employing sandard densiy esimaion procedures. Anoher recen mehod is likelihood sampling considered e.g. in deails in [4]. In his approach i is he likelihood funcion p(y ha is used as he proposal, whils he predicion densiy is used o calculae he weighs. Thus he success of his mehod depends on wheher he likelihood is a good predicor of he rue sae. For muli-modal likelihoods (when aliasing effecs are severe a large number of paricles can be generaed away from he likely nex posiions of he rue sae. These paricles will ge low weighs in he weighing process and hus will have no significan effec on he esimaed poserior. Hence he effecive samplesize would be small in his case. Our mehod overcomes his problem by firs sampling from he predicion densiy and hence concenraing he samples in he viciniy of he correc peaks of he likelihood funcion and using a localized version of he likelihood. The LS-N-IPS algorihm, inroduced in [2], uses he predicion densiy o derive he new paricle se which is hen locally modified by climbing he observaion likelihood. This algorihm inroduces some bias and relies on he availabiliy of a mehod o climb he observaion likelihood. Alhough, according o he well-known biasvariance dilemma, inroducing bias is no necessarily bad, LLS/LIS-filers may achieve roughly he same variance reducion ha is possible o ge using LS-N-IPS, bu wih no addiional bias (weighed imporance sampling iself yields biased esimae of he poserior. Furher, LLS/LIS-filers do no require a hill-climbing algorihm. Ye anoher recen mehod is he Boosed Paricle Filer [8]. Using our noaion, his mehod uses he following

5 proposal: q( X, Y = α(x, Y r(y g(x + ( α(x, Y K( X, where X is he expeced nex sae given he curren sae X and { A, if r < max x α(x, y = :d(x,x λ r(y x ; B, oherwise. Here < B A <, and r, λ are parameers o be chosen by he user. Hence, when he observaion likelihood is sufficienly large in a neighborhood of he expeced nex sae hen he observaion likelihood is used o draw he nex sample; whils if he observaion likelihood is no sufficienly large hen he predicion densiy is used o draw he nex posiion. Noe ha he version of his algorihm presened in [8] uses heurisically derived approximaions o he observaion likelihood and i is slighly more complicaed han he one presened here. In any case, he above proposal depends on he mos recen observaion and drawing samples from i can be done in an efficien manner. However, jus like in he case of AVM, when he predicion densiy is muli-modal or when he predicion densiy has a large variance hen since he expeced value of he nex sae is a bad predicor of where he nex sae migh be, he algorihm degrades o SIR. 5 Experimens In his secion we compare he performance of he LISfiler o ha of SIR and AVM on he sandard bearingsonly racking problem ha has been considered previously by several auhors [6, 9, 2, ]. In his problem he aim is o rack he (horizonal moion of a ship, while observing only angles o i. Wihou he loss of generaliy,le us assume ha he coordinae sysem is fixed o he observer. The ship s sae is assumed o follow a second order AR process, wih is acceleraion driven by whie noise: X + = X + σ η 2 2 ξ, where X, ξ R 2, ξ, ξ 2 N (,, and X, X 3 represen he ship s verical and horizonal posiions, respecively, whils X 2, X 4 represen he ship s verical and horizonal velociies. The iniial sae is sampled from a 4- dimensional Gaussian wih a diagonal covariance marix. Wha makes his problem paricularly challenging is ha he observaions depend on he sae only hrough he angle, θ = an (X 3 /X, a which he ship is observed. The observaion noise is defined using a wrapped Cauchy densiy: r(y θ = ρ 2 2π + ρ 2 2ρ cos(y θ. This densiy (when ρ is close o one is hough o reflec well a sonar s behaviour: angle measuremens are ypically very reliable, whils possible ouliers are well modelled by he heavy ails of he wrapped Cauchy disribuion. The parameers of he model used in he experimens are as follows: σ η =., ρ =.5 2, and he iniial sae is sampled from a Gaussian wih means (.5,.,.2,.55 and wih a diagonal covariance marix wih diagonal enries given by. (.5 2,.5 2,.3 2,. 2. Figure 4 gives an example of he ship s moion and he observed angles Figure 4. An example of he ship s moion ogeher wih he observaion direcions. 5. Resuls The performance of he LIS-based paricle filer was compared o ha of AVM and SIR. All paricle filers in hese experimens use N = 3 paricles. For he implemenaion of he LIS-based paricle filer he imporance funcion q x,y is bes described using polar coordinaes: Basically, q x,y is a Gaussian in he angle coordinaes wih variance δ 2 x,y = ρ. The window funcion g is defined in he angular space wih dispersion δ g =.5. Figure 5 shows paricle clouds generaed by he hree algorihms for an arbirary seleced ime-sep. I should be clear from he figure ha he paricle se generaed by LIS (shown as dos on he figure is much beer concenraed around he rue sae han he ses generaed by boh AVM ( + and SIR (. In paricular, he paricles are more concenraed along he lines poining owards he rue sae. Also, he paricle ses generaed by AVM are more concenraed around he rue sae han hose generaed by SIR. We noe ha a sraighforwardly implemened likelihood sampling algorihm would perform much weaker han any of hese algorihms as i would have no clue abou he disance of he ship, and hus i would need o disribue samples evenly along he measuremen lines. In order o ge a

6 Figure 5. Paricle clouds generaed by AVM ( +, SIR ( and LIS ( for he bearings only racking problem. more precise picure of he performance of hese algorihms, we have measured he racking performance by compuing he Euclidean disance beween he prediced and he acual ship posiions as a funcion of ime. The errors were measured wih 2 independenly generaed racking (measuremen sequences, and by running each algorihms imes on each of he 2 measuremen sequences. Figure 6 shows he resuling racking error of SIR, AVM and LIS as a funcion of he number of ime seps. As expeced, AVM performs beer han baseline SIR, bu LIS improves upon he performance of boh SIR and AVM by a considerable margin. The observed performance differences were found o be significan. Error SIR AVM LIS Tracking Error Time Figure 6. Tracking errors of SIR, AVM and LIS as a funcion of ime 6 Conclusions We have proposed a family of algorihms o enhance paricle filers wih he aim o overcome he curse of reliable observaions. The proposed algorihms, he LLS/LISfilers of which LIS-filers were inroduced here, are modificaions of he sandard SIR algorihm whereas afer he predicion sep he posiion of he paricles are randomly re-sampled from a localized version of he observaion densiy or a localized imporance funcion. We argued ha using he new mehod higher effecive sample sizes can be achieved when he observaions are reliable and when he design parameers of he new algorihm are chosen appropriaely. One expecs his increase o be refleced by an improved racking performance. Experimens wih he sandard bearings-only racking problem indicae ha he proposed algorihm is indeed capable of improving he racking performance as compared wih he performance of SIR and AVM. References [] N. Bergman. Recursive Bayesian Esimaion: Navigaion and Tracking Applicaions. PhD hesis, Deparmen of Elecrical Engineering, Linkoping, 2. [2] J. Carpener, P. Clifford, and P. Fearnhead. An improved paricle filer for nonlinear problems. IEEE Proceedings-Radar Sonar and Navigaion, 46(:2 7, 999. [3] D. Crisan and A. Douce. A survey of convergence resuls on paricle filering for praciioners. IEEE Trans. Signal Processing, 5(3: , 22. [4] W. B. D. Fox, S. Thrun and F. Dellaer. Paricle filers for mobile robo localizaion. In A. Douce, N. de Freias, and N. Gordon, ediors, Sequenial Mone Carlo Mehods in Pracice, New York, 2. Springer. [5] A. Douce. On sequenial simulaion based mehods for Bayesian filering. Saisics and Compuing, (3:97 28, 998. [6] N. J. Gordon, D. J. Salmond, and A. F. M. Smih. Novel approach o nonlinear/non-gaussian bayesian sae esimaion. Proc. Ins. Elec. Eng. F, 4(2:7 3, 993. [7] P. D. Moral and G. Salu. Non-linear filering using mone carlo paricle mehods. C.R. Acad. Sci. Paris, pages 47 52, 995. [8] K. Okuma, A. Taleghani, N. de Freias, J. J. Lile, and D. G. Lowe. A boosed paricle filer: Muliarge deecion and racking. In European Conference on Compuer Vision, volume, pages 28 39, 24. [9] M. Pi and N. Shephard. Filering via simulaion: Auxiliary paricle filer. Journal of he American Saisical Associaion, 94:59 599, 999. [] D. Sco. Mulivariae Densiy Esimaion. Theory, Pracice, and Visualizaion. J. Wiley & Sons, New York, London, Sydney, 992. [] D. Tieringon, A. Smih, and U. Makov. Saisical Analysis of Finie Mixure Disribuions. J. Wiley & Sons, New York, London, Sydney, 985. [2] P. Torma and C. Szepesvári. LS-N-IPS: an improvemen of paricle filers by means of local search. Proc. Non-Linear Conrol Sysems(NOLCOS S. Peersburg, Russia, 2. [3] P. Torma and C. Szepesvári. Enhancing paricle filers using local likelihood sampling. ECCV24, Prague. Lecure Noes in Compuer Science, pages 6 28, 24.