Distributed Particle Filters for Sensor Networks

Transcription

1 Disribued Paricle Filers for Sensor Neworks Mark Coaes Deparmen of Elecrical and Compuer Engineering, McGill Universiy 3480 Universiy S, Monreal, Quebec, Canada H3A 2A7 WWW home page: hp:// coaes Absrac. This paper describes wo mehodologies for performing disribued paricle filering in a sensor nework. I considers he scenario in which a se of sensor nodes make muliple, noisy measuremens of an underlying, ime-varying sae ha describes he moniored sysem. The goal of he proposed algorihms is o perform on-line, disribued esimaion of he curren sae a muliple sensor nodes, whils aemping o minimize communicaion overhead. The firs algorihm relies on likelihood facorizaion and he raining of parameric models o approximae he likelihood facors. The second algorihm adds a predicive scalar quanizer raining sep ino he more sandard paricle filering framework, allowing adapive encoding of he measuremens. As is primary example, he paper describes he applicaion of he quanizaion-based algorihm o racking a manoeuvring objec. The paper concludes wih a discussion of he limiaions of he presened echnique and an indicaion of fuure avenues for enhancemen. 1 Inroducion The goal of many sensor neworks is o deec and rack changes in he moniored environmen. Such scenarios arise in arge racking [1,9], in ime-varying densiy esimaion, and in he ask of robo navigaion [3, 4]. In hese siuaions, he class of signal (or informaion) processing ask ha mus be collaboraively performed by he sensor nodes migraes from saic esimaion or deecion o on-line esimaion, filering or change-poin deecion. I becomes imporan o design sequenial algorihms ha can dynamically fuse he informaion recorded by he sensors wihou requiring excessive exchange of eiher daa or algorihmic informaion. In his paper, we consider he siuaion where he naure of he moniored environmen can be capured by a Markovian sae-space model ha involves poenially nonlinear dynamics, nonlinear observaions, and non-gaussian innovaion and observaion noises. Our goal is o perform sequenial esimaion of he curren sysem sae a muliple sensor nodes in he nework. In nonlinear and non-gaussian scenarios, he decenralized Kalman filer [2], which admis an aracive, relaively low-dimensional parameric informaion exchange beween

2 2 sensor nodes, becomes inapplicable. Exended Kalman filers, grid-based mehods and Gaussian-sum filers are possible alernaives [5, 6], bu hese all have limiaions, and informaion exchange is no as simple. The class of sequenial Mone Carlo (or paricle filering) mehods [7] is aracive because of is power and flexibiliy. These mehods keep rack of a se of paricles, or candidae sae descripions. The mehods evaluae how well each paricle conforms o he dynamic model and explains he observaions, using his assessmen o generae a weighed pariculae approximaion o he filering disribuion, and hence form sae esimaes. There are wo causes for concern in he adopion of sequenial Mone Carlo algorihms in sensor neworks. Firs, he algorihms are subsanially more compuaionally demanding han more parameric alernaives. Second, he developmen of decenralized or disribued paricle filers has been limied, so i is somewha unclear wha informaion mus be exchanged in order o implemen a collaboraive algorihm. The second concern is more pressing han he firs; a sensor node wih reasonable processing power and memory is likely o be able o cope wih he compuaional demands of a paricle filer unless sae changes are very rapid, in which case sensing and nework communicaion also begin o be very difficul exercises. Idenifying he informaion ha needs o be exchanged beween nodes is more difficul. I is cerainly undesirable o ransmi he raw (or finely quanized) daa o a se of processing nework nodes. The communicaion cos is high and subsanial sensor node energy is consumed. On he oher hand, he naural informaion represenaion wihin a paricle filer is a se of paricles and associaed weighs. The exchange of hese in raw form almos cerainly involves he ransmission of many more bis han he exchange of he raw daa. In some cases, i is possible o develop a parameric approximaion of he paricle-based filering disribuion (a mehod adoped in [8, 9]). We explore his idea furher in his paper, examining some of he resricions in model srucure ha his approach implies. In his paper, we propose wo disribued paricle filering algorihms for sensor neworks. The firs approach is based on facorizing he likelihood, and forming parameric approximaions o producs of likelihood facors (using he paricles and heir associaed likelihoods as raining daa). The model parameers are hen exchanged beween sensor nodes, insead of he daa or exac paricle informaion. This approach places resricions on he srucure of he problem, because i mus be possible o boh facorize he likelihood and develop reasonably accurae, low-dimensional parameric models o describe he facors. The approach resuls in subsanial communicaion savings when he daa dimension is much higher han he dimension of he parameer space of he models approximaing he likelihood-facors. The second disribued algorihm uses an adapive daa-encoding approach. I involves he raining of predicive linear quanizers a every ime-sep based on a common paricle filer mainained a all nodes. Sensor nodes ransmi he quanized daa o one anoher; he compression can be subsanial because he paricle filer can provide a very good indicaion of where a sensor measuremen

3 3 is likely o be. This approach places no resricions on he naure of he likelihood funcion, bu he raining of opimal linear quanizers is compuaionally expensive, so he amoun of daa ha can be processed a each ime-sep is limied by he compuaional power of he sensor nodes. Wih his limiaion in mind, we describe a hierarchical sensor nework framework involving wo classes of nodes, A and B. Class A nodes are more compuaionally powerful, and have more energy resources. In he framework we describe, all compuaion is performed by class A nodes, and all sensing by class B nodes. The class A nodes manage he class B nodes, selecing only a small se o make measuremens a any ime sep. The adapive daa-encoding approach is hen feasible. The paper is organized as follows. Secion 2 saes he problem and reviews he core seps of cenralized paricle filering algorihms. Secion 3 describes he wo disribued paricle filering algorihms. Secion 3 describes he wo disribued paricle filering algorihms. Secion 4 oulines a hierarchical sensor nework framework ha uses he adapive-encoding based filering algorihm. Secion 5 repors on simulaions in which he hierarchical sensor nework was used o rack an objec manouevring hrough a sensor field. Finally, Secion 6 discusses he proposed algorihms, indicaing limiaions and proposing avenues for improvemen and furher research. 1.1 Relaed Work There have been some effors o design disribued Bayes (or paricle) filers in he sensor neworks [8, 9] and in he arificial inelligence communiy [4]. The work in [8] is argeed a racking an objec or several objecs. In he single objec scenario, one leader node is acive a any ime insan. The leader node mainains a belief sae, effecively a filering disribuion (represened eiher paramerically or hrough a paricle se). Based on is belief sae, he leader node evaluaes he expeced uiliy of neighbouring sensors and chooses he node wih highes uiliy o become he new leader node for he nex ime sep. I hen propagaes is belief sae o he new leader node, eiher by exchanging parameers or by ransmiing paricle locaions and weighs. In follow-up work, he auhors have invesigaed echniques for approximaing he paricle disribuion and ransmiing his approximaion. The work in [4] designs disribued paricle filers for decenralized daa fusion. The aim is o use local paricle filers o deermine which measuremens are worh sharing. The scheme works using a query-response sysem. Each sensor node mainains a local paricle filer. Neighbouring nodes query one anoher for useful sensor measuremens; a query is comprised of a small se of randomlyseleced paricles wih enire sae rajecories. Based on his se of paricles, he queried node examines is se of unshared measuremens (is own or hose received from oher sensors), and ransmis only he mos informaive measure, as evaluaed by some form of divergence measure. The disribued paricle filers we describe in his paper differ in purpose and implemenaion. The key difference is ha we srive o mainain a common

4 4 paricle represenaion of he poserior disribuion a muliple nodes in he nework a every ime insan. This means ha he manner in which he measured daa a any ime insan is uilised mus be consisen across he nework. 2 Generalized Problem Saemen and Cenralized Approaches In his paper, we are concerned wih he problem of performing on-line sae esimaion for muli-dimensional signals ha can be modelled using Markovian sae-space models ha are (poenially) nonlinear and non-gaussian. The unobserved global sae {x ; N} is modelled as a Markov process wih iniial disribuion p(x 0 ) and ransiion probabiliy p(x x 1 ). The observaions {y ; N } are assumed o be condiionally independen (in ime) given he process x and of marginal disribuion p(y x ). We denoe by x 0: {x 0,...,x } and by y 1: {y 1,...,y }, respecively, he sysem sae and he observaions up o ime. The measuremens y are recorded by K sensors, and we use y k o denoe he subse of observaions made by he k-h sensor. The aim is o esimae on-line he poserior disribuion p(x 0: y 1: ), and funcions derived from i, such as he filering disribuion p(x y 1: or expecaions of he form I(f )=E p(x0: y 1:)[f (x 0: )]. 2.1 Cenralized Paricle Filering Approach In he case where he sensor measuremens are available a a cenral locaion, several approaches have been proposed o perform he ask oulined in he previous secion. In he linear, Gaussian sae-space case, he Kalman filer provides analyical updae expressions for racking he evoluion of he poserior disribuions. In he case of nonlinear and/or non-gaussian models, approximaion approaches such as he exended Kalman filer [5] or grid-based mehods [6] can be adoped. An alernaive approach is o employ one of he algorihms belonging o he class of sequenial Mone Carlo (or paricle filering) mehods [7]. This secion now briefly oulines sequenial Mone Carlo mehods, drawing on descripions from [7] and references herein. Sequenial Mone Carlo mehods have been dubbed paricle filers because hey mainain a se of sae rajecories (or paricles) ha are candidae represenaions of he sysem sae. There is an imporance weigh associaed wih each paricle; a a given ime insan, his weigh is represenaive of how well he sae rajecory conforms o model dynamics and describes he se of observaions, relaive o he oher paricles. Whenever here is a ransiion beween ime insans and a new observaion becomes available, each rajecory is exended, and is associaed weigh adjused according o how well i explains he new observaion. There are hree generic seps ha appear, in one form or anoher, in he majoriy of sequenial Mone Carlo mehods. Individual algorihms have variaions

5 5 on he oulined seps or have addiional seps o improve esimaion performance, bu he seps form he foundaion for he mehodology. The firs sep is he iniialisaion of N paricles, denoed by {x (i) 0: ; i =1,...,N}. In his iniialisaion phase, each paricle is sampled from he iniial disribuion: x (i) 0 p(x 0 ), and every imporance weigh is iniialised o w (i) 0 = 1/N. Afer iniialisaion, he remaining wo seps, he imporance sampling sep and he selecion sep, are repeaed a every ime insan. Firs, in he imporance sampling sep, for each i =1,...,N, x (i) is sampled from an imporance disribuion π(x x (i) 0: 1, y 1:), which may be any disribuion ha has he same suppor as he poserior. A rajecory proposal is hen( formed by) appending his sample o he exising i- h paricle o form x (i) 0: = x (i) 0: 1, x(i). The imporance weighs are evaluaed according o w (i) = p(y x(i) )p( x (i) x (i) 1 ) π( x (i) x (i) 0: 1,y1:). The se of imporance weighs is normalized o sum o one. In he selecion sep, N paricles {x (i) 0: formed by sampling wih replacemen from he se { x (i) 0: ; i =1,...,N} are ; i =1,...,N} where ; i =1,...,N} and he associaed, normalized weighs w (i). The poserior disribuion is esimaed by he weighed empirical disribuion P N (dx 0: y 1: ) = (dx 0: ). he probabiliy of sampling he i-h rajecory is w (i). Esimaes of he poserior disribuion, he filering disribuion, or of expecaions are formed based on he N paricles { x (i) 0: 1 N N i=1 w(i) δ x (i) 0: Expecaions I(f ) are esimaed as ÎN (f )= N i=1 f (x (i) 0: ). In he cenralized paricle filering case, wih K sensors, he communicaion required per ime insan (neglecing overhead bis) is K k=1 D km k bis. For each sensor k, M k is he number of communicaion hops o he fusion cenre, and D k is he number of bis necessary for an adequae represenaion of is measured daa. D k is dependen on he required accuracy of he racking funcion and he encoding mechanism. In he nex secion, we focus on developing a mehod ha can dramaically reduce he value of D k (as compared o a naive implemenaion). 3 Disribued Paricle Filers This secion develops wo disribued algorihms designed o rack he sae of a non-linear dynamic sysem based on noisy measuremens made by a se of physically isolaed sensors. The algorihms consiss of K paricle filers, wih one filer running a every sensor in he nework. I is he naure of he communicaion beween sensors ha differs beween he wo algorihms. A his poin, i is useful o define synchronized paricle filers; in his paper, hese are paricle filers whose random number generaors have been iniialized a he same poin and which generae he same number of random numbers per ime sep. This means ha heir random draws are always he same).

6 6 3.1 Disseminaion of Raw Daa In order o updae is paricle approximaions o he poserior disribuions when new daa becomes available a ime, each filer needs o calculae, for each paricle i =1,...N, he likelihood funcion p(y x (i) ), which deermines he new weigh of he paricle. If he imporance sampling disribuion is daa-dependen, he daa is also required for he sampling procedure. The disseminaion of quanized daa hrough he nework generaes a similar communicaion overhead as he cenralized approach, approximaely K k=1 D km k bis, where D k is, as before, he number of bis required for adequae daa represenaion, bu M k is now he number of communicaion hops required o disseminae he daa of sensor k hroughou he nework. In he wors case, M k should be linear in K, and he communicaion cos is O(KD) where D = K k=1 D k. 3.2 Facorizable Likelihoods: A Parameric Modelling Approach I is possible o adop an alernaive approach o daa disseminaion in he special case where he likelihood funcion is facorizable, p(y x )= K k=1 p(yk x ), and each facor p(y k x ) can be described (or approximaed) by a parameric model ( F k (x ; θ k ). The) model parameers θ k are esimaed from raining daa pairs {,p(y k x (i) ; i = 1,...,N}, and he parameers disseminaed insead x (i) ) of he daa iself. Unforunaely, unless he daa represened by each sensor a each ime insan has high dimension, i is likely ha an adequae parameric represenaion of he likelihood funcion will require more bis han he daa iself. The poenial advanage of he scheme is ha here is he possibiliy of performing model aggregaion a each sensor node, hereby avoiding he communicaion expense engendered by he need o disseminae he enire parameer se {θ k ; k =1,...,K} hroughou he sensor nework. We now describe an algorihm ha performs a form of model aggregaion across sensor nodes. Figure 1 presens he algorihm in high-level pseudo-code forma. In his algorihm, we need o make a furher assumpion on he naure of he likelihood. No only do we need o be able o approximae individual likelihood facors by a parameric model, we need models G k (x ; φ k )hacan approximae producs of such likelihood facors, j S(k) p(yj x ). Here S(k) is he se of likelihood facors whose produc is modelled a node k. Below, we consider he scenario where here is a single communicaion chain from node 1 o node K, wih any node k in he inerior of he chain communicaing only wih nodes k 1 andk +1. The algorihm is very similar if here is a ree srucure for communicaion; parameers are simply exchanged beween parens and children insead of neighbours in he chain. The paricle filer a each node is iniialized as in he sandard sequenial Mone Carlo framework, by sampling N paricles from p(x 0 ). A ime insan, Node 1 samples from is imporance disribuion π 1 (x x (i) 0: 1, y1 1:) o generae N paricles { x (i) ; i =1,...,N}. The imporance disribuion may depend on y1:, 1 bu i canno depend on measuremens a oher sensors, which are unavailable

7 7 o node 1. Node 1 calculaes he value of is likelihood facor for each one of hese paricles for he curren observaion, p(y k x (i) ), and hen rains he model G 1 (x ; φ 1 ) using daa pairs from he raining se {( x (i),p(y 1 x (i) )); i =1,...,N}. The values φ 1 are hen appropriaely quanized and ransmied o node 2 in he chain. Node 2 also exends he rajecories by sampling from is imporance disribuion π 2 (x x (i) 0: 1, y2 1:, φ 1 ), generaing values x (i). Noe ha he imporance disribuion can depend on he ransmied parameers φ 1, and he samples may differ from hose generaed a node 1. Node 2 rains a model G 2 (x ; φ 2 )o fi he produc of likelihood facors p(y 1 x )p(y( 2 x ). The raining is performed ) according o daa pairs from he raining se { x (i), G 1 ( x (i) ; θ 1 )p(y 2 x (i) ) ; i = 1,...,N}. The process coninues, wih node k raining a model G k (x ; φ k )ofi ) k j=1 p(yj x ) using he raining daa {( x (i), G k 1 ( x (i) ; φ k 1 )p(y k x (i) ) ; i = 1,...,N}. AheK-h sensor node, he parameers of a model G K (x ; φ K ) have been rained. This model aemps o fi he global likelihood p(y x )= K k=1 p(yk x ). In he nex phase of he algorihm, he esimaed parameers φ K are propagaed back along he communicaion chain. Node k uses is knowledge of he model G K o calculae esimaes of likelihood for is samples x (i). These esimaes are hen used o deermine imporance weighs and o perform he resampling sep. The algorihm, as described hus far, resuls in a differen se of paricles and differen esimaes a each sensor node. If synchronized paricle filers are used, hen his effec can be eliminaed. Insead of performing resampling based on is iniial rajecory exensions, each sensor node generaes a new se of rajecory exensions x (i) according o a common imporance disribuion π(x x (i) 0: 1, φk ). Due o filer synchronizaion, hese paricle exensions are common o all nodes, as are he calculaed imporance weighs and he resampling sep. The se of paricles remains common across all nodes, as do he esimaes. This procedure allows imporance sampling o ake place from a disribuion ha is dependen on all he daa in he nework. The originally sampled paricle exensions could have been generaed in areas where he poserior is insubsanial, due o consideraion of only he local daa. The communicaion cos of his algorihm per ime sep, neglecing overhead bis, is KP K + K k=1 P k, where P k is he number of bis required o represen he parameer se φ k. Bounding P k by P, he wors-case for any node k, he cos becomes O(KP). Comparing his o he cos of disseminaing he raw daa, which was O(KD), wih D being he number of bis required o represen all he measured daa, we see ha here is subsanial saving if P D. This will be he case when here are many sensors, so ha he daa dimension is high, and when he likelihood facors can be represened by simple models. Each G k is a funcion on he sae-space, whose dimension should be subsanially smaller han ha of he daa. Care mus be aken in model selecion o ensure ha he dimension of he parameer space is reasonable and ha he parameers can be

8 8 Disribued Parameric Approximaion Paricle Filer 1. Iniialisaion, =0. For each sensor k =1,...,K For i =1,...,N, sample x (i) 0 p(x 0)andse =1. 2. Firs Imporance sampling sep and forward parameer exchange Define φ 0 =, he empy se, and G 0(x ; φ 0 )=1 x. For k =1,...,K For i =1,...,N, sample x (i) π k (x x (i) 0: 1, yk 1:, φ k 1 ). Esimae he parameers φ k of he model G k (x ; φ k ), designed o approximae k j=1 p(yj x ), using daa pairs from he raining se ( ) { x (i), G k 1 ( x (i) ; φ k 1 )p(y k x (i) ) ; i =1,...,N}. If k<k,quanizeφ k andsendonodek Backward parameer exchange and weigh calculaion For k = K, K 1,...,1 For i = 1,...,N, sample x (i) π(x x (i) 0: 1, φk ) and se x (i) ( ) 0: = x (i) 0: 1, x(i). For i = 1,...,N, evaluae he (approximae) imporance weighs w (i) = GK( x(i) π( x (i) ; φ K ) p( x (i) x (i) 0: 1, φk ) x (i) 1 ) Normalize he imporance weighs. If k>1, ransmi he quanized parameers φ K o node k Selecion sep For k = 1,...,K, resample wih replacemen N paricles {x (i) 0: ; i = 1,...,N} from he se { x (i) 0: ; i =1,...,N} according o he imporance weighs. Se +1andgoosep2.. Fig. 1. High-level algorihm descripion of he disribued paricle filer ha uses parameric models o approximae likelihood facors (see Secion 3.2).

9 9 esimaed using an algorihm ha is no exremely compuaionally demanding. The sensor nodes mus run a raining algorihm every ime-sep, so he raining mus be a reasonable exercise given processing power and memory consrains. I should also be noed ha each sensor mus be aware of he funcional srucure of he models of is neighbours and also ha of he global model G K. 3.3 Disribued Paricle Filer using Adapive Encoding In his secion, we describe a disribued paricle filering algorihm ha uses he predicive capabiliies of he paricle filer locaed a each sensor node o perform efficien, adapive encoding of he measured daa. In conras o he parameric algorihm, which is effecive when he daa dimension is high, sensor compuaional limiaions combine wih he srucure of he encoding-based paricle filer o impose a resricion on he feasible daa dimension. In he descripion of he algorihm ha follows, we again consider he scenario where here is a single communicaion chain from node 1 o node K, wih any node k in he inerior of he chain communicaing only wih nodes k 1and k + 1. As in he parameric-approximaion filer, he adapive-encoding paricle filer is equally applicable if here is a ree srucure for communicaion. Sensor k records a vecor of measuremens y k a each ime sep. Denoe he dimension of his vecor d k. The vecor of daa colleced a all sensors, y,has dimension d K k=1 d k. The paricle filers a all nodes are synchronized wih one anoher, and each paricle filer is iniialized as in he sandard sequenial Mone Carlo framework, by sampling {x (i) 0 ; i =1,...,N} from p(x 0). Each node also mainains a se of d ime-varying, scalar quanizers q, consising of a codebook and a pariion funcion. We will index he se using a pair of inegers (k, j), so ha a ime, q k,j represens he quanizer corresponding o he j h elemen of he measuremen vecor y k. A ime, sensor node k samples from he prior disribuion p(x x (i) 0: 1 ) o generae N paricles { x (i) ; i =1,...,N}. (As an alernaive o hese seps, one can follow he idea of auxiliary variable-based paricles filers [10], and have he sensor generae a se of values {µ (i) ; i =1,...,N}, where µ (i) is he mean, mode, or some oher likely value associaed wih p(x x (i) 0: 1 )). Based on he se of paricles x (i) (or he µ (i) ), he sensor node hen generaes a se of predicive daa values {ỹ (i) ; i =1,...,N} by drawing from he likelihood funcions p(y x (i) (or p(y µ (i) ). Noe ha his sep requires ha each sensor have knowledge of he global likelihood funcion. The paricle filers are synchronized, so every node generaes he same se of predicive daa values. The nex sep of he algorihm is he quanizer raining phase. In designing and raining he quanizers q, a nauaral choice is o aemp o minimize he error funcion [ ] p(y x 0: 1 ) (p(y x ) p(q (y ) x )) 2 dx dy, where q (y ) is he quanized value of y. The paricle approximaion o his error funcion ( is N N 2. i=1 j=1 )) Vecor quanizaion can be explored as a possible pah for minimizing his funcion, bu in any even, sensor p(ỹ (i) x (j) ) p(q (ỹ (i) ) x (j)

10 10 Disribued Adapive Encoding-based Paricle Filer 1. Iniialisaion, =0. Synchronize he paricle filers by equaing he random seeds. For each sensor k =1,...,K For i =1,...,N, sample x (i) 0 p(x 0)andse =1. 2. Quanizer raining sep For each sensor k =1,...,K For i =1,...,N, sample x (i) p(x x (i) 0: 1 )andỹ(i) p(y x (i) ). For each elemen y k,j of he daa y,rainad k,j elemen quanizer q k,j using he Lloyd-Max algorihm applied o he daa se ; i =1,...,N} using he error funcion {ỹ (i) N ( i=1 p(ỹ (i) ( x (i) ) p ỹ k,j,(i),q k,j (ỹ k,j,(i) )) 2 ) x (i). (1) 3. Quanizaion For each sensor k, quanizey k using he se of quanizers q k = {q k,j } o form quanized daa ( ). y k Disribue he quanized daa y = { ( ) y k ; k =1,...,K} o all sensors. 4. Imporance sampling For each sensor k =1,...,K For i = 1,...,N, sample x (i) ( ) π(x x (i) 0: 1, y 0:), and se x (i) 0: = x (i) 0: 1, x(i). For i = 1,...,N, evaluae he (approximae) imporance weighs w (i) = p(y x (i) Normalize he imporance weighs. ) p( x (i) x (i) 1 ) π( x (i) x (i) 0: 1, y 0: ). 5. Selecion sep For k = 1,...,K, resample wih replacemen N paricles {x (i) 0: ; i = 1,...,N} from he se { x (i) 0: ; i =1,...,N} according o he imporance weighs. Se +1andgoosep2. Fig. 2. High-level algorihm descripion of he disribued paricle filer ha performs adapive encoding of sensor daa (see Secion 3.3).

11 11 k does no have access o y. Sensor k requires a quanizer q k ha maps y k o a quanized value. In he algorihm proposed here, we ake his one sep furher, and consider scalar quanizaions, raining q k,j a each ime sep o quanize individual measuremens y k,j. In raining q k,j, we use he error funcion: [ ( ) ] 2 p(y x 0: 1 ) p(y x ) p(y k,j,q k,j (y k,j ) x ) dx dy, where y k,j denoes all elemens of by excep y j,k. The paricle approximaion o his error funcion is: N N ( i=1 j=1 p(ỹ (i) x (j) ) p(ỹ k,j,(i),q k,j 2 (ỹ k,j,(i) ) x (j) )). (2) In he simulaions in his paper we have used he Lloyd-Max approach o rain he scalar quanizers [11, 12] (i is also possible o consider compuaionally simpler mehods or enropy-coded quanizaion approaches [13]). Noe ha using he error funcion (2) involves he evaluaion of N 2 likelihoods upon every ieraion of he algorihm. We have observed in simulaions ha reducing his o N evaluaions by only including he erms i = j resuls in similar filer performance. Our goal is o deermine a good quanizer (no necessarily he opimal quanizer), so we can also limi he number of raining ieraions. However, even wih hese modificaions, he raining of several Lloyd-Max quanizer is a compuaionally demanding exercise. This resuls in limiaions on he viable daa dimension per ime sep for his filering approach. Once he quanizers have been rained a all sensor nodes, he sensor daa are quanized o y q (y ) and disribued hroughou he nework. The sensors hen perform he sandard paricle filering seps (imporance sampling, weigh evaluaion, and paricle selecion) based on he quanized daa y. Figure 2 presens he algorihm in high-level pseudo-code forma. The only reducion in communicaion cos compared o he disseminaion of he raw daa lies in he achievable compression in he quanizaion. The communicaion cos in bis per ime sep, neglecing overhead bis, is K k=1 D km k, where D k is he number of bis required afer quanizaion and M k is he number of communicaion hops required o send sensor k s quanized daa hroughou he sensor nework. As illusraed in Secion 5, subsanial compression can be achieved whils mainaining esimaion accuracy. The algorihm s compuaional complexiy grows linearly in daa dimension (he quanizer raining is he dominan compuaional expense), and his resrics he feasible size of he sensor nework for real-ime operaion. 4 A Hierarchical Sensor Nework For he remainder of he paper, we focus on he disribued paricle filering algorihm based on adapive daa encoding. I is clear from he preceding secion ha he algorihm becomes impracical for a large number of sensors. Every sensor mus mainain and rain a codebook for each scalar measuremen in he

12 12 nework, and he compuaional expense incurred in his exercise soon becomes overwhelming. Moreover, every sensor mus say awake and paricipae in he algorihm every ime sep, so ha filers do no lose synchronizaion. In his secion, we describe he high-level srucure of a hierarchical sensor nework ha addresses hese issues. In he hierarchical nework, we consider ha here are wo classes of sensor nodes, which we denoe classes A and B for convenience. Class A nodes have subsanially more energy and compuaional power han he more numerous class B nodes. Class B nodes are responsible for sensing he moniored environmen and reporing heir measuremens o a single paren class A node. We assume ha he densiy of class A nodes is sufficien ha each class B node can direcly communicae wih a leas one class A node. Class A nodes are responsible for performing all compuaion and managing he class B sensor nodes. Class A nodes are always acive; a class B node is acivaed for measuremen and communicaion by is paren node. Due o he limiaions of he adapive encoding paricle filering algorihm, he class A nodes, numbered 1,...,K only acivae a small number of class B nodes (for measuremen) per ime sep. We denoe his se of sensors by V. Oher class B nodes may be acive for he purpose of relaying messages beween he se of class A nodes. The class A nodes implemen he disribued paricle filering algorihm described in Secion 3.3. A ime = 0, he algorihm is iniialized by sampling paricles a each class A node from p(x 0 ), and a random se of sensors V 1 is seleced for he firs measuremen. Each sensor v V 1 makes is se of measuremens y1 v and ransmis he daa o is paren class A node. The ransmission a his sep involves fine quanizaion (of he order of 16 or 32 bis per measuremen). The K class A nodes hen perform he disribued paricle filering algorihm exacly as described in Figure 2. The nodes only generae a small number of linear quanizers, one for each daa measuremen made by he acive class B sensor nodes, so he compuaional requiremens are manageable. The exchange beween he class A nodes involves highly compressed daa (2-5 bis per measuremen). A a subsequen ime sep, insead of a random se of sensors being acivaed for measuremen, he class A nodes decide upon a se based on a predicion of which sensors will provide he mos informaive measuremens. Algorihms for performing his ype of sensor managemen exercise have been described in [1, 9, 14]. As he decision is made on he basis of a common paricle filer, every class A node is aware of he se of acive sensor nodes ha will perform measuremen one sep ahead in ime. This is imporan because i means ha he number of bis required for daa labelling is minimal, and relaed o he cardinaliy of he acive se raher han he oal number of sensors. The number of informaion bis ha need o be ransferred hroughou he nework is K k=1 D k, where D k is he number of bis required by class A node k. This expression excludes he iniial finely-quanized communicaions beween he acive class B sensors and heir parens.

13 13 5 A Simulaion Example: Tracking a Manoeuvring Objec In his secion, we invesigae an example of he applicaion of he adapive encoding disribued paricle filer using he hierarchical sensor nework framework described in Secion 4. We consider he exercise of racking an objec manoeuvring in a 2-D plane. The dynamic sysem (a jump-sae Markov model) is described by an iniial disribuion p(u 0,θ 0, x 0 ) and he updae equaions u p(u u 1 ), (3) θ = θ 1 + c(u )+v, (4) x = x 1 + m[cos θ, sin θ ]. (5) Here u 0, 1, 2 is a ime varying sae, indicaing no urn (c(0) = 0), lef-urn (c(1) = 0.1 radians), and righ-urn (c(2) = 0.1 radians), respecively. The angle of moion is deermined by θ, v is he innovaion noise (Gaussian), and x is he posiion. The velociy is consan and specified by m Fig. 3. An example realizaion of he sensor nework for he simulaion in Secion 5. Solid squares indicae class A nodes a fixed locaions, equally spaced in he plane. Circles are class B angle-measuremen nodes; sars are class B disance-measuremen nodes. Class B nodes are uniformally disribued. The hin lines indicae he class A paren of each class B node. The hick line indicaes an example rajecory of he objec over 500 ime seps, saring a he solid circle and moving o he riangle. In our sysem, here are wo ypes of class B nodes: hose capable of measuring he angle of he objec s posiion relaive o he node φ, and hose capable

14 14 of measuring disance o he objec r. The observaion equaions for a node v wih posiion g v are: φ v = arcan(x g v )+n (6) r v = max( x g v + s, 0) (7) The noise erms, n and s are modelled as zero-mean Gaussian wih variances σn 2 and σs, 2 respecively. We performed simulaions wih 128 class B sensors, 64 of each ype, and 16 class A nodes. The posiions of he class B nodes in he plane (covering [ 64, 64] [ 64, 64]) were random, drawn according o a uniform disribuion. The class A nodes were equally spaced across he plane. Figure 3 depics an example realizaion of he sensor field. In our sudies, he objec was racked for 500 ime seps. The objec s iniial posiion was deermined by a Gaussian disribuion cenred a [2,2] and diagonal covariance enries se o 1. The iniial angle of moion of he objec was deermined by a Gaussian, cenred a π/4, wih variance 0.01, and u 0 was se o 0. The objec moved wih a consan velociy m =0.5, and he innovaion noise had variance The observaion noise in our simulaions was generaed by seing σ n =0.02 and σ s =0.02. The sae ransiion probabiliy marix p(u u 1 )wasseas: A any ime sep, he class A nodes acivae eigh class B nodes for measuremen, including four angle-measuremen sensors and four disance-measuremen sensors. These are chosen as he closes (of heir kind) o he prediced posiion of he objec, he predicion being made one sep-ahead. We conduced a Mone Carlo sudy of he performance of he adapive encoding disribued paricle filer. The sudy involved S = 20 realizaions of he sensor field and he objec pah. For each sudy, he paricle filer was applied wih REP = 10 differen random seeds for iniializaion. In our simulaions, he paricle filers knew he model parameers, and N, he number of paricles per filer, was se o 500. We examined he performance for hree fixed seings of D v, he number of bis used for daa represenaion by each acive class B sensor (and per measuremen in his case). We compared he esimaion performance of D v = {2, 3, 4} o he raw daa case, where we use 16 bis o represen each measuremen via sraighforward quanizaion over he feasible range. For each seing of D v, we calculaed he mean-squared error beween he rue objec posiion and he paricle filer-based mean-square esimae of objecposiion. This was derived for each realizaion s by averaging over he REP insances of he algorihm. Based on his we obained a log mean squared-error measure: LMSE = log 1 S [ S s=1 1 REP REP r=1 x,s x,r,s 2 ] (8)

15 LMSE Time Fig. 4. Plo of mean squared error performances of he adapive encoding disribued paricle filer a various quanizaion levels. Dashed line corresponds o a 2-bi quanizaion; solid line o a 3-bi quanizaion; doed line wih * markers o a 4-bi quanizaion; solid line wih diamond markers o a non-adapive 16-bi quanizaion. Figure 4 plos he LMSE for he four quanizaion levels over he firs 60 ime insans. The algorihm performance is as expeced, wih esimaion performance improving (exponenially) as he number of bis increases. The loss in accuracy hrough he use of four-bi encoding as opposed o non-adapive 16-bi quanizaion of he raw daa is no paricularly subsanial, paricularly considering he reducion in communicaion cos. The oal number of informaion bis ransmied by he sensor nework is D v M, where M is he number of hops needed o send a measuremen o all class A nodes. In moving from he nonadapive 16-bi o he adapive 4-bi quanizaion, we achieve (approximaely) a four-fold saving in communicaion. 6 Discussion and Conclusions We have presened wo disribued paricle filering algorihms for racking poserior disribuions in Markovian sae-space models using sensor neworks. The firs algorihm is applicable o sae-space models for which i is possible o facorize he likelihood funcion and approximae he facors using low-dimensional parameric models. I resuls in a subsanial communicaion saving in siuaions where he daa dimension per ime sep is large. The second algorihm uses disribued paricle filers o adap linear quanizers for individual sensor measuremens. Whils he laer algorihm is applicable o more general models, i canno be applied when he daa dimension is high, because subsanial

16 16 compuaion is needed o rain efficien quanizers. In ligh of his, we described a hierarchical sensor nework ha suppors implemenaion of he algorihm. Boh algorihms have several limiaions and he reamen in his paper has glossed over some issues. The algorihms require ha each sensor has a knowledge of he global likelihood funcion. Neiher algorihm provides a mechanism for handling local sae parameers (such as sensor posiion) separaely from he global sae. For he parameric approximaion-based algorihm, mehods for idenifying and raining appropriae parameric models for he likelihood facors remain o be developed. In fuure research, we hope o provide a more refined characerizaion of he sae-space models for which he parameric approach is applicable. In he case of he adapive encoding algorihm, he resricion o a small daa dimension is frusraing. We are currenly exploring daa aggregaion and vecor quanizaion sraegies in our aemps o alleviae he resricion. References 1. Liu, J., Liu, J., Reich, J., Cheung, P., Zhao, F.: Disribued group managemen for rack iniiaion and mainenance in arge localizaion applicaions. In: Proc. IEEE Conf. Informaion Processing in Sensor Neworks, Palo Alo, CA (2003) 2. Muambara, A.: Decenralized esimaion and conrol for mulisensor sysems. CRC Press, Boca Raon, FL (1998) 3. Kam, M., Zhu, X., Kalaa, P.: Sensor fusion for mobile robo navigaion. Proc. IEEE 85 (1997) Rosencranz, M., Gordon, G., Thrun, S.: Decenralized sensor fusion wih disribued paricle filers. In: Proc. Conf. Uncerainy in Arificial Inelligence, Acapulco, Mexico (2003) 5. Anderson, B., Moore, J.: Opimal Filering. Prenice-Hall, New Jersey (1979) 6. Bucy, R., Senne, K.: Digial synhesis of nonlinear filers. Auomaica 7 (1971) Douce, A., de Freias, N., Gordon, N., eds.: Sequenial Mone Carlo Mehods in Pracice. Series: Saisics for Engineering and Informaion Science. Springer- Verlag, New York (2001) 8. Zhao, F., Shin, J., Reich, J.: Informaion-driven dynamic sensor collaboraion for racking applicaions. IEEE Signal Processing Magazine 19 (2002) Shin, J., Guibas, L., Zhao, F.: A disribued algorihm for managing muli-arge ideniies in wireless ad-hoc sensor neworks. In: Proc. IEEE Conf. Informaion Processing in Sensor Neworks, Palo Alo, CA (2003) 10. Pi, M., Shephard, N.: Filering via simulaion: auxiliary paricle filers. J. Amer. Sa. Assoc. 94 (1999) Lloyd, S.: Leas squares quanizaion in PCM. IEEE Trans. Info. Theory 28 (1982) Max, J.: Quanizing for minimum disorion. IRE Trans. Info. Theory 6 (1960) Gersho, A., Gray, R.M.: Vecor Quanizaion and Signal Compression. Kluwer Academic Press, Boson MA (1992) 14. Kreucher, C., Casella, K., Hero, A.O.: Muliarge sensor managemen using alpha divergence measures. In: Proc. IEEE Conf. Informaion Processing in Sensor Neworks, Palo Alo, CA (2003)