Graph Laplacian Distributed Particle Filtering

Transcription

1 26 24h European Signal Processing Conference (EUSIPCO) Graph Laplacian Disribued Paricle Filering Michael Rabba and Mark Coaes Dep. Elecrical and Compuer Engineering McGill Universiy, Monréal, QC, Canada Sephane Blouin DRDC Alanic Research Cenre Darmouh, NS, Canada Absrac We address he problem of designing a disribued paricle filer for racking one or more arges using a sensor nework. We propose a novel approach for reducing he communicaion overhead involved in he daa fusion sep. The approach uses graph-based signal processing o consruc a ransform of he join log likelihood values of he paricles. This ransform is adapive o paricle locaions and in many cases leads o a parsimonious represenaion, so ha he he join likelihood values of all paricles can be accuraely approximaed using only a few ransform coefficiens. The proposed paricle filer uses gossip o perform disribued, approximae compuaion of he ransform coefficiens. Numerical experimens highligh he poenial of he proposed approach o provide accurae racks wih reduced communicaion overhead. I. INTRODUCTION Paricle filers have proven o be highly effecive for racking scenarios wih non-linear and/or non-gaussian dynamic and/or measuremen models. In disribued racking sysems, where muliple nodes obain measuremens and collaborae o achieve improved racking accuracy, developing efficien and accurae disribued paricle filers is an imporan challenge. An imporan design goal is o reduce he communicaion overhead by ransmiing fewer and/or smaller messages. This paper focuses on disribued paricle filering algorihms, wherein a nework of nodes collaborae o rack one or more arges. Wihin he class of disribued paricle filering algorihms, we concenrae on gossip-based mehods, where each node mainains is own local paricle filer and nodes communicae o share informaion obained hrough heir local measuremens. There is a rade-off beween he communicaion overhead and he racking accuracy. Relaed Work: Several disribued paricle filers have been proposed over he pas decade; [] provides a valuable survey of he mehods. The approaches involve disribued, approximae compuaion of he paricle weighs. This is achieved by gossiping on he paricles wih he highes join likelihood [2], [3], or forming approximaions of he join likelihood funcion or poserior and gossiping on sufficien saisics [4] [7]. Alhough hese filers can perform very effecively in some seings, he former family of filers can ofen incur a high communicaion overhead, and he laer can sruggle if he assumpions underpinning he approximaions are no jusified. Conribuion: We propose a novel approach for reducing he communicaion overhead involved in he daa fusion sep of a disribued paricle filer. The reducion is obained by approximaing he join log likelihood values. The approach uses graph-based signal processing o consruc a ransform of he join log likelihood values for each paricle which is adaped o he paricle locaions. In many cases, he ransform is such ha he join likelihood values a all paricle can be accuraely approximaed using only a few ransform coefficiens. Numerical experimens highligh he poenial of he proposed approach o provide accurae racks wih reduced communicaion overhead. In comparison o oher sae-of-hear disribued paricle filering approaches, we find ha he proposed approach can significanly reduce he communicaion overhead while suffering only a modes decrease in racking accuracy. Among he specrum of disribued racking algorihms, his makes i possible o achieve performance in a differen regime han was previously possible. II. PROBLEM FORMULATION AND BACKGROUND A. Targe Dynamics and Measuremen Model In his paper we focus on he problem of racking he sae x 2 R nx of a arge over imes. The sae is assumed o evolve according o he discree-ime dynamics x = f(x )+, () where f( ) :R nx! R nx is a possibly non-linear mapping, and 2 R nx is ime-varying process noise. Our aim is o rack x from noisy observaions made a a collecion of n v sensor nodes. We denoe he se of nodes by V = {,...,n v }. Le z,v 2 R nz,v denoe he measuremen aken a node v 2V a ime. We assume ha he measuremens follow he model z,v = h,v (x )+,v, (2) where h,v ( ) :R nx! R nz,v is a node-dependen, possibly non-linear mapping, and,v is he measuremen noise a sensor v a ime. We allow he measuremen funcion h,v ( ) o vary wih ime. Le z = {z,v : v 2V}denoe he collecion of all measuremens made a all nodes a ime, and z : = {z,...,z } denoe he collecion of all measuremens from imes up o. We assume ha he measuremens aken a differen nodes a ime are condiionally independen given he sae x. Under his assumpion, he join likelihood of he measuremens z facorizes as p(z x )= Y v2v p(z,v x ). (3) /6/$3. 26 IEEE 493

2 26 24h European Signal Processing Conference (EUSIPCO) We assume ha he model f( ) for he arge dynamics and he disribuion of he process noise are known o all nodes. In addiion, we assume ha node v knows he local measuremen funcions {h,v ( )}, as well as he disribuion of he local measuremen noise,v. Alhough each node could individually rack x using is own observaions, in general more accurae esimaes of x are obained by fusing measuremens from muliple nodes. To his end, nodes may share informaion wih each oher by communicaing over a nework. We assume ha if wo nodes can communicae in he nework, hen bidirecional communicaion is achievable, and ha hese undireced links form a conneced opology. We focus on algorihms where each node mainains is own esimae of he sae x, and nodes communicae using gossip proocols (simple message-passing mehods) in order o keep heir sae esimaes synchronized. The use of gossip proocols is desirable because hey do no require cenral coordinaion and hey are robus o ypical challenges arising in wireless disribued sysems, such as ime-varying nework opologies and dropped packes [8], [9]. The paricle filering algorihms makes use of wo ypes of gossip. The firs reaches consensus, asympoically, on he average of a se of scalar values a differen nodes []; he second reaches consensus on he maximum of he scalar values []. The fusion in he paricle filering algorihm involves averaging, so we need o employ average gossip. The convergence of average gossip is asympoic, bu we need all nodes o have exacly he same esimaes, so we subsequenly apply he maximum gossip procedure, which can be shown o converge almos surely o he sae where all nodes know he maximum value afer a finie number of ieraions []. B. Paricle Filers Paricle filers approximae he poserior densiy p(x z : ) using a se of N weighed paricles, X = {bx, } N i=, where bx 2 R nx and 2 R; specifically, he approximaion is given by bp(x z : )= NX i= (x bx ), (4) where ( ) is he Dirac dela funcion. Given an iniial arge deecion a locaion bx and ime =, he paricle locaions bx are iniialized by sampling from a densiy p ( bx ) and he weighs are iniialized o =/N for all i. Each paricle is updaed from one ime sep o he nex by firs sampling a new locaion bx from a proposal densiy q(x bx, z :). Then he weigh associaed wih bx is / p(z bx w q(bx )p(bx bx ) bx, z :), (5) where he proporionaliy consan is such ha P N i= w Finally, a new se of paricles {bx by sampling N {bx =., } N i= is obained paricles wih replacemen from he se }N i=, where bx is sampled wih probabiliy w, and seing he corresponding weighs o =/N. This las sep is referred o as resampling in he lieraure. For more background on paricle filers see, e.g., [2]. In he boosrap paricle filer, q( bx, z :) is aken o be p( bx ), he densiy of x given x ined by he dynamic model () wih x = bx. For his choice, he propagaion of bx only depends on bx ; in paricular, i does no depend on he measuremen z. C. Gossip-Based Disribued Paricle Filering We focus on a disribued framework where each node mainains and updaes is own copy of paricles X,v, and gossip proocols are used o homogenize he paricles across nodes over ime. We assume ha nodes have a shared source of randomness which can be used o sample from he disribuions p ( bx ) and q( bx, z :). Suppose ha he weighed paricle ses a all nodes are exacly he same a ime, and also suppose ha he nework implemens he boosrap paricle filer. Then he proposal disribuion q( ) only depends on he curren paricle locaions. Based on he assumpion of shared randomness, he newly sampled paricle locaions bx will be idenical a all nodes. For he boosrap paricle filer, he weigh updae (5) hus simplifies o / w p(z bx ), and so new weigh values are a funcion of he measuremens aken a all nodes, z = {z,v : v 2V}. Under he condiional independence assumpion (3), he join likelihood can be expressed as ( ) X p(z x )=exp log p(z,v x ).,v v2v Le = log p(z,v bx ) denoe he log-likelihood of he measuremen a node v and ime. Then he weigh updaes can be expressed as where = exp{ } P N j= w(j) exp{ (j) },,v. Because = P v2v is a linear funcion of values,v a each node, an approximaion of he values } N i= can be compued a all nodes in a disribued manner using gossip proocols. { In pracice, sampling is implemened using a pseudo-random number generaor, and shared randomness can be achieved by having he nodes iniially agree on he seed. If he nodes are iniially no synchronized, each node can draw a random number. Then hey can use he max-gossip proocol o deermine he maximum value and use i as he common seed o he pseudorandom number generaor. 494

3 26 24h European Signal Processing Conference (EUSIPCO) III. GRAPH LAPLACIAN APPROXIMATION FOR DISTRIBUTED PARTICLE FILTERING This secion proposes an approach o updaing paricle weighs in a disribued manner. The echnique consrucs a ransform ha is adaped o he paricle locaions. Since paricles near each oher generally have more similar join likelihood values han paricles ha are far apar, we build a graph over he paricles by placing edges beween pairs of paricles ha are close o each oher. Then he Laplacian marix of his graph is used o consruc a ransform such ha he join log-likelihood values a all paricles can be succincly approximaed using only a few coefficiens in he ransform domain. Paricles wih similar locaions in he sae space ypically have similar weighs; i.e., kbx bx (j) k being small ypically implies ha w (j) is also small. This suggess ha some form of ransform coding may be effecive. We propose o use a ransform adaped o he propagaed locaions {bx }N i= in order o obain a reduced-order approximaion o he weighs. Specifically, we propose o use he eigenvecors of { }N i= a graph consruced from he propagaed locaions {bx }N i= as a ransform basis. Le A denoe he N N adjacency marix of a graph wih one verex for each paricle in he se {bx }N i=. Fix a posiive ineger apple N. We place an edge beween paricles i and j (equivalenly, we se A i,j = A j,i =) if paricle i is one of he apple neares neighbours of j, or vice versa, where he disance beween wo paricles is kbx bx (j) k 2 ; his is someimes referred o as he symmerized apple-neares neighbour graph. Le D denoe a diagonal N N marix wih diagonal enries D i,i = P N j= A i,j conaining he number of neighbours of node i. Noe ha A is symmeric and ha no all paricles necessarily have he same number of neighbours since he appleneares neighbour relaionship is no necessarily symmeric. Le 2 R N denoe a vecor obained by sacking he paricle log-likelihoods { } N i= using he same indexing used when forming he graph corresponding o A. We view as a signal suppored on he graph and seek a ransform ha concenraes he energy of ino a small number of coefficiens. The Laplacian marix L of he graph wih adjacency marix A is L = D A. Because L is a symmeric real marix i has an eigendecomposiion of he form L = F F T, where F is an N N orhonormal marix of eigenvecors and is a diagonal marix of eigenvalues. Le apple 2 apple apple N denoe he eigenvalues (he diagonal elemens of ) sored in ascending order, and le f j denoe he j-h column of F. The eigenvalues and eigenvecors of he graph Laplacian have a number of ineresing properies for processing signals suppored on he graph [3] [5]. In paricular, he columns of F can be inerpreed as a Fourier basis for signals suppored on he graph. Eigenvecors corresponding o larger eigenvalues have he inerpreaion of being higher frequency basis vecors. Typical log-likelihood funcions log p(z bx ) vary smoohly as a funcion of bx, and so one may hope ha mos of he energy of concenraes in a few low frequency componens. Given {bx,w }N i=, execue a all nodes v 2Vin parallel: : for i =,...,N do 2: Sample bx q( bx, z:) 3: end for 4: Form he apple neares neighbour graph for paricles {bx } N i=. 5: Calculae he Laplacian L and perform he eigendecomposiion L = F F T. 6: Calculae Laplacian ransform coefficiens,v = Fm T for F m ined in (6). 7: {b (j) 8: Se b = F m b. 9: for i =,...,N do : = } m j= = GOSSIP v({n v (j),v} m j=). w exp{b } P Nj= w (j) exp{b(j) } : end for 2: for i =,...,N do 3: Sample bx Pr(bx 4: end for = bx (j) from {bx (j) )=w(j) }N j= wih replacemen, wih. Se w =/N. Fig.. Pseudo-code for one sep of he graph Laplacian approximaion disribued paricle filer. Line 7 involves an applicaion of average and maxconsensus. Given a signal 2 R N on he graph, we refer o = F T as he vecor of Laplacian ransform coefficiens. Inuiively, smooh signals on he graph hose for which he values a neighbouring nodes are similar should have mos of heir energy concenraed in he coefficiens corresponding o lower frequency basis vecors. For a posiive ineger m apple N, an m-erm approximaion o can be obained by firs ransforming ino he Laplacian eigenvalue domain, hresholding he magniudes corresponding o he larges eigenvalues o zero, and aking he inverse ransform. Equivalenly, we ake mx b = (fj T )f j, j= where {f j } m j= are he graph Laplacian eigenvecors corresponding o he m smalles Laplacian eigenvalues. Since F is an orhonormal marix, when m = N here is no approximaion error and b is idenical o. In he seing of disribued paricle filering, le us again assume ha he value m is fixed in advance and known o all nodes. All nodes also know he paricle locaions {bx } N i=, and so each node can locally compue he adjacency marix A, he corresponding Laplacian marix L, and a marix F m =[f,...,f m ] (6) composed of he m eigenvecors corresponding o he m smalles eigenvalues of L. Node v hen locally compues he m-dimensional vecor of coefficiens,v = F T m,v, 495

4 26 24h European Signal Processing Conference (EUSIPCO) α(j ) w ( i ) ŵ( i ) N i = j m (a) (b) (c) Fig. 2. (a) An example paricle cloud shown wih a apple-neares neighbour graph (apple =7). (b) Magniude of he coefficiens (j) of he join log-likelihood expressed in erms of he graph Laplacian eigenvecors. (c) Typical paricle weigh approximaion error for his example using only he coefficiens corresponding o he m firs Laplacian eigenvecors for differen values of m. where,v is he vecor of paricle log-likelihood values compued using only he measuremens a node v. The nodes hen perform K ieraions of disribued averaging using he gossip algorihm, followed by he applicaion of maximum gossip. Le b denoe he resuling m-dimensional vecor of coefficiens. To obain he log-likelihood esimaes, each node ses b = F m b and proceeds wih he weigh updae. Figure provides pseudocode ha summarizes he proposed approach. Figure 2 illusraes he graph Laplacian approach o reducing communicaions when compuing he weighs associaed wih each paricle. We ake he paricle cloud for he example shown in Fig. 2(a) and consruc a neares neighbour graph wih apple =7. Fig. 2(b) depics he magniude of he elemens of he vecor ; i.e., (j) = fj T, where f j is he j h column of F. Observe ha he bulk of he energy is concenraed in he firs few coefficiens. Finally, Fig. 2(c) shows he absolue error in paricle weighs when only he firs m graph Laplacian coefficiens are used o obain an approximaion ˆ. IV. NUMERICAL EVALUATION We evaluae he performance of he proposed approach and compare i wih sae-of-he-ar mehods using a simulaed daa se. The scenario is a bearings-only racking problem wih wo arges and n v = 25 sensor nodes arranged in a perurbed grid over a km by km area. The wo arges make a counerclockwise maneuver over a period of T = 5 ime seps. A each ime sep, he second arge senses he firs and moves in is direcion wih a consan velociy. Because he dynamics of he wo arges are coupled, we ake he sae x o be an 8- dimensional vecor, wih he firs four coordinaes represening he posiion and velociy of he firs arge, and he second four coordinaes represening he posiion and velociy of he second arge. A each ime sep, a random subse of he nodes gaher bearings measuremens. On average five measuremens are made across he nework per arge per ime sep. Each measuremen is of he bearings from he arge o he sensor making he measuremen, and he measuremens are corruped by addiive zero-mean Gaussian noise wih a sandard deviaion of 5 degrees. To make a fair comparison among he algorihms, and o avoid issues relaed o daa associaion, we assume ha each measuremen is associaed wih he corresponding arge by an oracle and hese same associaions are used by all of he algorihms. The sensors communicae over a grid opology, wih each node able o communicae wih up o four neighbours. Below we evaluae he performance of he proposed algorihms. To quanify accuracy, we reporqhe ime-averaged roo P T mean square posiion error (ARMSE), T = kx bx k 2. All resuls repored are drawn from Mone Carlo rials. To sudy he performance of he proposed algorihm, deailed in he pseudocode of Figure, we fix he number of paricles o N = 2 and examine he algorihm performance as we vary m and he number of gossip ieraions. A each sep, a graph is formed over he paricle cloud using apple = 2 neighbours. Figure 3 shows he oal ARMSE as a funcion of he average number of scalars ransmied per node per sep for differen values of m. Each curve corresponds o fixing a value of m and varying he number of gossip ieraions from 5 o 2 in incremens of 5. The figure illusraes ha a filer using only or 5 coefficiens can achieve accuracy approaching ha of a cenralized filer if here are sufficien gossip ieraions (a leas in his case). Nex, we compare he performance of op-m selecive gossip and he graph Laplacian approximaion approach o oher disribued paricle filering algorihms from he lieraure. Specifically, we compare wih he disribued boosrap filer running gossip ieraions direcly on he paricle weighs, wih he Gaussian approximaion disribued paricle filer [5], wih he se-membership consrained (SMC) disribued paricle filer [2], and wih he op-m selecive gossip paricle filer [3]. All filers use 2 paricles. The communicaion overhead of he disribued boosrap, Gaussian approximaion, and SMC paricle filers can be adjused by varying he number of gossip ieraions used by each filer. The radeoff beween oal mean ARMSE (for boh arges) 496

5 26 24h European Signal Processing Conference (EUSIPCO) Toal ARMSE (Targe + Targe 2) m = m = 5 m = m = 5 m = 9 Toal ARMSE (Targe + Targe 2) Disribued Boosrap Gaussian Approximaion Se Membership Consrained Top m Selecive Gossip Graph Laplacian Approx Avg. Scalars per Node per Sep x 5 Fig. 3. The sum of he mean ARMSE for boh arges as a funcion of he average number of scalar values ransmied per node per ime sep of he simulaed scenario, running he disribued paricle filer wih graph Laplacian approximaion. The poins on each curve are obained by fixing he value of m and varying he number of gossip ieraions per racking ime sep from 5 o 25 in incremens of 5. and he communicaion overhead, as measured in erms of he average number of scalars ransmied per sensor per sep of he disribued filer, is shown in Figure 4. The Gaussian approximaion filer has much less communicaion overhead han he oher approaches, bu he oal ARMSE is much higher, probably because he Gaussian approximaion does no adequaely capure he shape of he poserior disribuion for he racking scenario considered. The disribued boosrap, semembership consrained, and op-m selecive gossip filers all exhibi a similar radeoff. The graph Laplacian approximaion offers ineresing performance for communicaion overheads in he range 5 3 o 5 4 scalars ransmied per node per filering sep. In his range, he communicaion overhead is reduced by nearly one order of magniude, as compared o he se-membership consrained filer or op-m selecive gossip filers, and he oal ARMSE only increases by a lile over. km. V. CONCLUSION We have presened a novel disribued paricle filer for racking one or more arges using a sensor nework. A graph signal processing approach is used o approximae he loglikelihoood values of he paricles. Numerical experimens highligh he poenial of he proposed approach o provide accurae racks wih reduced communicaion overhead. In fuure work, we will explore how o reduce he compuaional overhead of consrucing a neares neighbour graph and exracing he eigenvecors from he Laplacian. REFERENCES [] O. Hlinka, F. Hlawasch, and P. Djurić, Disribued paricle filering in agen neworks: A survey, classificaion, and comparison, IEEE Signal Processing Mag., vol. 3, no., pp. 6 8, Jan Avg. Scalars per Node per Sep Fig. 4. The sum of he mean ARMSE (km) for boh arges in he simulaed scenario as a funcion of he average number of scalar values ransmied per node per ime sep. [2] S. Farahmand, S. Roumeliois, and G. Giannakis, Se-membership consrained paricle filer: Disribued adapaion for sensor neworks, IEEE Trans. Signal Processing, vol. 59, no. 9, pp , Sep. 2. [3] D. Üsebay, M. J. Coaes, and M. G. Rabba, Disribued auxiliary paricle filers using selecive gossip, in Proc. IEEE In. Conf. Acousics, Speech, and Signal Proc. (ICASSP), Prague, Czech Republic, May 2, pp [4] D. Gu, J. Sun, Z. Hu, and H. Li, Consensus based disribued paricle filer in sensor neworks, in Proc. IEEE ICIA, Zhangjiajie, China, Jun 28, pp [5] B. Oreshkin and M. Coaes, Asynchronous disribued paricle filer via decenralized evaluaion of Gaussian producs, in Proc. ISIF In. Conf. Informaion Fusion, Edinburgh,UK, July 2. [6] O. Hlinka, O. Slu ciak, F. Hlawasch, P. Djurić, and M. Rupp, Likelihood consensus and is applicaion o disribued paricle filering, IEEE Trans. on Signal Processing, vol. 6, no. 8, pp , Aug 22. [7] A. Mohammadi and A. Asif, A consrain sufficien saisics based disribued paricle filer for bearing only racking, in IEEE Inl. Conf. on Communicaions, June 22, pp [8] R. Olfai-Saber, J. Fax, and R. Murray, Consensus and cooperaion in neworked muli-agen sysems, Proceedings of he IEEE, vol. 95, no., pp , Jan. 27. [9] A. G. Dimakis, S. Kar, J. M. Moura, M. G. Rabba, and A. Scaglione, Gossip algorihms for disribued signal processing, Proceedings of he IEEE, vol. 98, no., pp , November 2. [] S. Boyd, A. Ghosh, B. Prabhakar, and D. Shah, Randomized gossip algorihms, IEEE Trans. Informaion Theory, vol. 52, no. 6, pp , Jun 26. [] F. Iuzeler, P. Cibla, and J. Jakubowicz, Analysis of max-consensus algorihms in wireless channels, IEEE Trans. on Signal Processing, vol. 6, no., pp , Nov. 22. [2] A. Douce and A. M. Johansen, A uorial on paricle filering and smoohing: Fifeen years laer, in Handbook of Nonlinear Filering. Oxford, UK: Oxford Universiy Press, 29, ch. 2, pp [3] F. Chung, Specral Graph Theory. American Mahemaical Sociey, 997. [4] X. Zhu and M. Rabba, Approximaing signals suppored on graphs, in Proc. IEEE Inl. Conf. on Acousics, Speech, and Signal Processing, Kyoo, Japan, March 22. [5] D. Shuman, S. Narang, P. Frossard, A. Orega, and P. Vandergheyns, The emerging field of signal processing on graphs: Exending highdimensional daa analysis o neworks and oher irregular domains, IEEE Signal Processing Mag., vol. 3, no. 3, pp , May