Sensor localization using nonparametric generalized belief propagation in network with loop

Transcription

1 Sensor localzaton usng nonparametrc generalzed belef propagaton n network wth loop Vladmr Savc and Santago Zazo Post Prnt N.B.: When ctng ths work, cte the orgnal artcle. 009 IEEE. Personal use of ths materal s permtted. However, permsson to reprnt/republsh ths materal for advertsng or promotonal purposes or for creatng new collectve works for resale or redstrbuton to servers or lsts, or to reuse any copyrghted component of ths work n other works must be obtaned from the IEEE. Vladmr Savc and Santago Zazo, Sensor localzaton usng nonparametrc generalzed belef propagaton n network wth loop, 009, IEEE Proc. of Intl. onf. on Informaton Fuson (FUSION. Postprnt avalable at: Lnköpng Unversty Electronc Press

2 Sensor Localzaton usng Nonparametrc Generalzed Belef Propagaton n Network wth Loops Vladmr Savc Sgnal Processng Applcaton Group Polytechnc Unversty of Madrd udad Unverstara S/N, 8040 Madrd, vladmr@gaps.ssr.upm.es Abstract - Belef propagaton (BP s one of the bestknown graphcal model for nference n statstcal physcs, artfcal ntellgence, computer vson, etc. Furthermore, a recent research n dstrbuted sensor network localzaton showed us that BP s an effcent way to obtan sensor locaton as well as approprate uncertanty. However, BP convergence s not guaranteed n a network wth loops. In ths paper, we propose localzaton usng generalzed belef propagaton based on uncton tree method (GBP-JT and nonparametrc (partcle-based appromaton of ths algorthm (NGBP-JT. We llustrate t n a network wth loop where BP shows poor performance. In fact, we compared estmated locatons wth Nonparametrc Belef Propagaton (NBP algorthm. Accordng to our smulaton results, GBP-JT resolved the problems wth loops, but the prce for ths s unacceptable large computatonal cost. Therefore, our appromated verson of ths algorthm, NGBP-JT, reduced sgnfcantly ths cost, wth lttle effect on accuracy. Keywords: Localzaton, generalzed belef propagaton, uncton tree, partcle flters, loops. Introducton The localzaton conssts n obtanng the relatve or absolute poston of a sensor node together wth the uncertanty of ts estmate. Equppng every sensor wth a GPS recever or equvalent technology may be epensve, energy prohbtve and lmted to outdoor applcatons. Therefore, we consder the problem n whch some small number of sensors, called anchor nodes, obtan ther coordnates va GPS or by nstallng them at ponts wth known coordnates, and the rest, unknown nodes, must determne ther own coordnates. If unknown nodes were capable of hgh-power transmsson, they would be able to make measurements wth all anchor nodes (sngle-hop technque. However, we prefer to use energy-conservng devces wthout power amplfer, wth lack the energy necessary for long-range communcaton. In ths, mult-hop, case, each sensor has avalable nosy measurements only to several neghborng sensors. Santago Zazo Sgnal Processng Applcaton Group Polytechnc Unversty of Madrd udad Unverstara S/N, 8040 santago@gaps.ssr.upm.es A recent drecton of research n dstrbuted sensor network localzaton s the use of partcle flters [, ]. In [], Ihler et al. formulated the sensor network localzaton problem as an nference problem on a graphcal model and appled partcle based varant of belef propagaton (BP methods [4], the so-called nonparametrc belef propagaton (NBP algorthm, to obtan an appromate soluton to the sensor locatons. omparng wth determnstc algorthms [5, 6, 7], the man advantages of ths statstcal approach are ts easy mplementaton n a dstrbuted fashon and suffcency of a small number of teratons to converge. Furthermore, NBP s capable of provdng nformaton about locaton estmaton uncertantes and accommodatng non-gaussan dstance measurement errors. However, NBP convergence s not guaranteed n a network wth loops [4, 8], or even f NBP converges, t could provde us less accurate estmates. Therefore, n ths paper, we present a new varant of the NBP method whch solves problem wth loops. We propose localzaton usng generalzed belef propagaton based on uncton tree (GBP-JT and nonparametrc (partcle based appromaton of ths algorthm (NGBP- JT. Juncton tree model s a generalzaton of belef propagaton (BP that s correct for arbtrary graphs. Jordan proved t usng elmnaton procedure [9]. omparng wth Ihler s Nonparametrc Belef Propagaton (NBP algorthm, GBP-JT converges well n network wth loops, but the prce for ths s unacceptable large computatonal cost. Therefore, we mplemented appromated verson of ths algorthm, NGBP-JT, by drawng hgh-dmensonal partcles from approprate clques n the network. Moreover, n order to draw samples n hgh-probablstc area, we used mproved samplng procedure whch utlzes nformaton from frst phase of the algorthm as an a pror. Ths verson reduces sgnfcantly computatonal cost (over a 00 tmes, wth lttle effect on accuracy. The remander of ths paper s organzed as follows. In Secton, we revew standard BP and condton for ts convergence. In Sectons and 4, we propose GBP-JT and NGBP-JT algorthms, respectvely. Smulaton results are presented n Secton 5. Fnally, Secton 6 provdes some conclusons and future work perspectve.

3 onvergence of Belef Propagaton In the standard BP algorthm, the belef at a node s proportonal to the product of the local evdence at that node ( ψ (, and all the messages comng nto node : M ( = kψ ( m ( ( N( where k s a normalzaton constant and N( denotes the neghbors of node. The messages are determned by the message update rules: m ( ψ ( ψ (, m ( = ( k k N( \ where ψ (, s parwse potental between nodes and. On the rght-hand sde, there s a product over all messages gong nto node ecept for the one comng from node. In practcal computaton, one starts wth nodes at the edge of the graph, and only computes a message when one has avalable all the messages requred. It s easy to see [4] that each message needs to be computed only once for sngle connected graphs. That means that whole computaton takes a tme proportonal to the number of lnks n the graph, whch s dramatcally less that the eponentally large tme that would be requred to compute margnal probabltes navely. In other words, BP s a way of organzng the "global" computaton of margnal belefs n terms of smaller local computatons. The BP algorthm, defned by equatons ( and (, does not make a reference to the topology of the graph that t s runnng on. Thus, there s nothng to stop us from mplementng t on a graph that has loops. One starts wth some ntal set of messages, and smply terates the message-update rules ( untl they eventually converge, and then can read off the appromate belefs from the belef equatons (. But f we gnore the estence of loops and permt the nodes to contnue communcatng wth each other, messages may crculate ndefntely around these loops, and the process may not converge to a stable equlbrum. One can ndeed fnd eamples of graphcal models wth loops, where, for certan parameter values, the BP algorthm fals to converge or predcts belefs that are naccurate. On the other hand, the BP algorthm could be successful n graphs wth loops, e.g. error-correctng codes defned on Tanner graphs that have loops [0]. Ths can be proved usng Bethe appromaton to the "free energy" [4, 8]. The fed ponts of the BP algorthm correspond to the statonary ponts of the Bethe "free energy". To make ths more clear, let's defne for one graphcal model, a ont probablty functon p({ }. If we have some other appromate ont probablty functon b({ }, we can defne a "dstance" between p({ } and b({ }, called Kullback-Lebler (KL dstance, by: b({ } Db (({} p({} = b({}ln p ({ ( } { } The KL dstance s useful because t s always nonnegatve and s zero f and only f the two probablty functons p({ } and b({ } are equal. Statstcal physcsts generally assume that Boltzmann's law s true: E({ }/ T p({ } = e (4 Z where Z s a normalzaton constant, and the "temperature" T s ust a parameter that defnes a scale of unts for the "energy" E. For smplcty, we can choose T =. Usng eqs. ( and (4, we fnd the KL dstance: Db (({} p({} = b({ } E({ } + b({ }ln b({ } + ln Z (5 { } { } So we see that ths KL dstance wll be zero when appromate probablty functon b({ } wll equal to the eact probablty functon p({ }.The Bethe appromaton s the case when ont belef b({ } s functon of sngle-node belefs b ( and two-node belefs b (,. Yedda et al. proved [4] that for a sngleconnected graph, values of these belefs that mnmze the Bethe free energy, wll correspond to the eact margnal probabltes. For graph wth loops, these belefs wll only be appromatons, although a lot of them are qute good. Localzaton usng Generalzed Belef Propagaton Our goal n ths secton s to develop new localzaton algorthm usng generalzed belef propagaton based on uncton tree method (GBP-JT. Juncton tree algorthm s a standard method for eact nference n graphcal model [9]. The graph s frst trangulated (added vrtual edges so that every loop of length more than has a chord. Gven a trangulated graph, wth clques and potentals ψ (, and gven correspondng uncton tree whch defnes lnks between the clques, we send the followng message from clque to clque by the message update rule: m ( ψ ( m ( = (6 S k S \ S k N(\ k where S =, and where N( are the neghbors of clque n the uncton tree. The belef at clque s proportonal to the product of the local evdence at that clque and all the messages comng nto clque :

4 Fgure. Eample of 0-node network wth loop M ( = kψ ( m ( (7 S N( Belefs for sngle nodes can be obtaned va further margnalzaton: M ( = M ( for (8 \ The equatons (6, (7, and (8 represent generalzed belef propagaton algorthm whch s vald for arbtrary graphs. The BP algorthm defned wth ( and ( s a specal case of GBP-JT, obtanng by notng that the orgnal tree s already trangulated, and has only pars of nodes as clques. In ths case, sets S are sngle nodes, and margnalzaton usng eq. (8 s unnecessary. Let s show how t works n our eample n Fgure. The network has 0 nodes, 5 anchors (nodes 6-0 and 5 unknowns (nodes -5. There s a loop , so we have to trangulate t by addng two more edges (- and - 4. Then we can defne 8 clques n the graph: = {,, }, = {,, 4}, = {, 4, 5}, 4 = { 4, 9}, 5 = { 5, 0}, 6 = {, 6}, 7 = {, 7}, 8 = {, 8}. The approprate potentals of -node clques are gven by: ψ (,, = ψ(, ψ(, ψ (,, 4 = ψ4(, 4 ψ (,, = ψ (, ψ (, Note that vrtual edges do not appear n these equatons snce they are used only to defne clques. Other clques, defned over pars of nodes, are nothng else than potental functons between two nodes already known from standard BP: 0 (9 ψ ( 4 4, 9 = ψ 49( 4, 9, ψ ( 5 5, 0 = ψ 50 ( 5, 0, ψ ( 6, 6 = ψ6(, 6, ψ ( 7, 7 = ψ 7(, 7, (0 ψ (, = ψ (, The uncton tree correspondng to the network n Fgure s shown n Fgure. As we can see, anchor clques ( 4 8 do not receve messages, so ths graph does not contan loops. Actually, these anchor clques also nclude one unknown node so we can send them messages, but ths node also could be located margnalzng the belef of some other clque. In the net step, we can compute all messages usng equaton (6. The complete set of messages s gven by: m ( = ψ (,, m ( = ψ (, * * m ( = m ( =ψ (, * m ( = m ( = ψ (, * m ( = m ( = m ( = ψ (, * m (, = ψ (, ψ (, ψ (, ψ (,, 7 * * * m (, = ψ (, ψ (, ψ (, ψ (,, * * * ( (, = * * * 7 (, 7 8(, 8 49 ( 4, 9 (,, 4 m (, 4 m ψ ψ ψ ψ 4 m (, = ψ (, ψ (, ψ (, ψ (,, (, * * * m where astersk denotes the known locaton of the anchor node and the messages from "anchor clques" are drectly replaced by approprate potental functon. The belefs of clques are computed usng equaton (7: M(,, = ψ * * * (,, ψ6(, 6 ψ7 (, 7 ψ8(, 8 m(, M(,, 4 = ( ψ * * * (,, 4 ψ7 (, 7 ψ8(, 8 ψ49 ( 4, 9 m(, m (, 4 M(, 4, 5 = ψ (,, ψ (, * ψ (, * ψ (, * m (, Now t s easy to compute belefs of sngle nodes by margnalzng belefs of clques usng eq. (8. Obvously, t s suffcent to know belefs of and snce these clques nclude all unknown nodes. Margnalzaton of provdes degree of freedom and could be used to check the estmated postons of some nodes (n our case, for nodes, and 4.

5 8 6 m6( m(, m(, 4 5 m5( 5, 6,, m ( 8 m(, 8, 8 m ( m (,, 4, 4, 5 m (, 4 8, 5 0 m ( 7 m7( m ( 4 4, 7 4, m ( 4 4 Fgure. The uncton tree correspondng to the network n Fgure Fnally, n order to use ths method for localzaton, we have to defne potental functons. In our case, we assumed that we ddn't obtan a pror nformaton about node poston, so sngle-node potentals are equal to (n opposte case, belefs computed usng eq. (7 have to be multpled by ther own potentals. The parwse potental between nodes t and u s gven by []: Pd ( t, u pv ( dtu t u, f otu =, ψ tu ( t, u = Pd( t, u, otherwse. ( where P d s the probablty of detectng nearby sensors; n our case we used mproved model whch assumes that the probablty of detectng nearby sensors falls off eponentally wth squared dstance: Pd( t, u ep = t u / R (4 where R s the transmsson radus. The bnary varable o tu ndcates whether ths observaton s avalable ( o tu = or not ( o tu = 0. And the last remanng parameter s measured dstance. The unknown node t obtans a nosy measurement d of ts dstance from detected node u: tu d = + v, v p (, (5 tu t u tu tu v t u In our case, we used Gaussan dstrbuton for p v, but, as we can see, t's very easy to change t to any desred dstrbuton (e.g. obtaned by runnng tranng eperment n the deployment area. The proposed GBP-JT algorthm s not unque. There are a lot of varatons of ths method; the best known s cluster varaton method [8]. However, t can be shown that they are qute smlar. For eample, n [8] s descrbed the relatonshp between dfferent regon-based appromatons. The man goal s acheved n all of them: estmated belefs are correct n network wth loops. However, the prce for ths s unacceptable large computatonal cost, so we are gong to mplement appromated verson of GBP-JT algorthm. 4 Nonparametrc Generalzed Belef Propagaton In order to obtan acceptable spatal resoluton for unknown nodes, number of dscrete ponts n the deployment area (e.g. N Ny for D grd must be too large for GBP to be computatonally feasble []. Besdes, the presence of nonlnear relatonshps and potentally hghly non-gaussan uncertantes n sensor localzaton makes GBP undesrable. However, usng partcle-based representatons va nonparametrc generalzed belef propagaton (NGBP, enables the applcaton of GBP to nference n sensor networks. In ths secton we propose NGBP-JT, partcle-based appromaton of GBP-JT method for the same eample of network from prevous secton (Fgures and. 4. Drawng ntal partcles Let s draw : N weghted partcles from clques and { W, X} = { W,[ X,, X,, X,]} { W, X } = { W,[ X, X, X ]},,4,5 (6 where Wm represents weght of 6-dmensonal partcle X m from clque m whch conssts from three -dmensonal partcles from node t ( X mt,. For now, we don t need partcle from clque. There s a lot of ways to draw these partcles. In general, we can draw all partcles unformly wthn the deployment area, but t requres

6 sgnfcant large number of partcles (e.g. 00 partcles drawn for each node, corresponds to = 06 partcles of ts clque. Therefore, we wll mmedately nclude all nformaton avalable wthn the clque: potental functons gven by (9 and ( whch represent our nformaton about dstance between nodes wthn the clque. Frst, we draw partcles of node t unformly wthn the deployment area. To draw partcle of any neghborng node u, we shft partcle of node t n the random drecton for an amount whch represents the observed dstance between these two nodes: Xmu, = Xmt, + ( dtu+ v[sn( θ cos( θ ], θ ~ Unf[0, π, =,..., N We wll use smplfed notaton of the above equaton: mu, mt, (7 X = shft( X, d (8 Assumng that we have already drawn samples, e.g. from nodes and 5, we can compute partcles of other nodes: tu 4. omputng messages Havng drawn all partcles, we can now compute all messages. Messages m and m are functon of m and m respectvely (see eqs. (, so they wll be computed later. Also, messages from the anchor clques wll be drectly replaced wth approprate potental functon. So we start wth messages m and m whch depends on ψ and ψ from whch we already have drawn partcles. Let s represent these two messages n slghtly dfferent form: m (, = M (,, ( M (,, = ψ (, ψ (, ψ (, ψ (,, * * * m (, = M (,, ( M (,, = ψ (, ψ (, ψ (, ψ (,, * * * Defned factors, M and M, are some knd of unmargnalzed messages, so we ll call them ont messages. Now t s very easy to compute weghted partcles from these ont messages ({ W, X }: mn mn X = shft( X, d, X = shft( X, d,,,, X = shft( X, d, X = shft( X, d,4,5 45,,5 5 (9 X = X = [ X, X, X ],,,, W = ψ ( X, ψ ( X, ψ ( X, W ( * * * 7, 7 8, 8 6, 6 Snce these partcles are drawn from ψ and ψ respectvely, and that we already ncluded all nformaton whch place these partcles n hgh-probablstc regons wth respect to X, and X,5 (see eqs. ( and (9, all clque s weghts can be appromated wth the same value: W = W =, =,..., N (0 N Note that all partcles of nodes wthn the clque have one common weght, e.g. { W,[ X,, X,, X,]}. Our ntal set of partcles for clque s llustrated n Fgure. X = X = [ X, X, X ] W X X X W,,4,5 * * * = ψ49 (,4, 9 ψ8(,, 8 ψ50(,5, 0 (4 Before computng fnal messages, we notced usual problem, sample depleton [], the problem when one, or few, of the weghts are much larger than the rest. Ths means that any sample-based estmate wll be unduly domnated by the nfluence of ust few partcles. In our case, t s epected because we are workng n 6- dmensonal space where t s very hard to draw good sample (clque wth poston and shape smlar to the rght one see Fgure. Therefore, we resample wth replacement [, ], whch wll produce N equal-weght partcles ( Wmn = / N. In our case, we have to resample from clques, thus the easest way s to resample from sngle nodes usng standard resamplng procedure [], and then to synchronze ndees n order to keep orgnal shapes of the partcles. Ths procedure s llustrated for M, by the followng pseudocode: d X + v, d + v X, X, Fgure. Intal set of partcles for clque [ W, X,, nde] = resample( W, X, ; for =: N nde( nde( X, = X, ; X, = X, ; end Note the dfference between X and X,! (5

7 where { W, X,} s the vector of N partcles from node (part of ont message, and nde s the vector of old (preresampled ndees of new partcles. Now we are ready to compute partcles from messages ({ wmn, mn}. The margnalzaton of ont messages s very easy snce we already have weghted partcles from them. So we ust need to dscard one data, and keep the same weghts. Thus, they are gven by:,,,,4 = X ( : = [ X, X ], w = W = / N = X (: = [ X, X ], w = W = / N (6 for =: N end X = (; X = (;,, whle( k < k ma X = shft( X, d ;,, f abs( X, X ( d ε, d + ε break; end k = k + ; end,, W = ψ ( X, ψ ( X, ψ ( X, w ; * * * 6, 6 7, 7 8, 8 (9 Fnally, we can compute partcles of other two messages, m and m. Accordng to eqs. (, they are functon of ψ m and ψ m, respectvely, so we wll draw partcles from these products and then re-weght by remander of eq. (. Actually, two sngle-node partcles of messages m and m are already computed (see eqs. (6, so we have ust to draw mssng partcle usng nformaton from ψ, the observed dstance between nodes and 4. The result of ths procedure are partcles of ont messages M and M. Margnalzng them, we obtan fnal messages m and m. The complete procedure s gven as follows: X = [shft( (, d, (, (] W X X X w resample and synchronze = X (: = [ X (, X (], w = W = / N 4 * * * = ψ49 ( (, 9 ψ8( (, 8 ψ7 ( (, 7 X = [ (, (, shft( (, d ] W X X X w resample and synchronze = X ( : = [ X (, X (], w = W = / N 4 * * * = ψ49 ( (, 9 ψ8( (, 8 ψ7 ( (, 7 4. omputng fnal belefs (7 (8 To estmate belefs of unknown nodes, we are gong to compute belefs of clques usng already computed partcles of messages. Accordng to eqs. (, belefs M, M and M are functon of ψ m, ψ m m and ψ m, respectvely, so we wll draw partcles from these products and then re-weght by remander of eqs. (. Let s start wth M and ts correspondng product ψ m. As we can see n eqs. (9, ψ ncludes nformaton about dstance between nodes and, as well as between nodes and. Besdes, message m ncludes nformaton about postons of nodes and. So we ust need to locate node, usng avalable postons and dstances. It could be done geometrcally by ntersectng crcles, but we prefer statstcal approach whch s qute faster. It s done by followng pseudocode: where,, abs( X, X s estmated dstance between these two partcles, and ε s predefned tolerance. If we do not obtan good partcle after k ma teratons, that s mean that these two crcles cannot ntersect, so our partcle s the poston shfted for d n random drecton. Ths s not a problem because ths wrong partcle wll obvously have later a very small weght (fltered by potental functons from anchors. The other problem s bmodalty, f the crcles ntersect n two ponts; but the wrong partcle wll be also fltered n same way. The same procedure s done for M snce we have dstance between nodes 5 and 4, as well as between nodes 5 and, and message m whch ncludes nformaton about postons of nodes and 4. As we already mentoned, the belef M s not necessary snce other two clques nclude postons of all unknown nodes. Anyway, we wll show the procedure because t s slghtly dfferent. We have to draw partcles from product of two 4-dmensonal messages and as result we epect 6-dmensonal message. So, f we want to avod to draw randomly mssng partcles (e.g. for message m(,, we would have to draw sngle-node partcles from 4, we wll drectly draw partcles from the product ψ (,, 4 m(, m(, 4 and then re-weght by remander of eq. (. The followng procedure shows t:, X = [shft( (, d4, (, (];, X = [ (, (, shft( (, d4];,, X = choose( X X, N; W = / N (0 m ( X,, X, m ( X,, X,4 W = W m ( X,, X, + m ( X,, X,4 W = ψ (, * (, * (, * X ψ X ψ X W 7, 7 8, 8 49,4 9,, where functon choose( X X, N chooses randomly N partcles from N. Ths procedure s known as mportance samplng [], the appromaton of orgnal dstrbuton ( mm wth proposal one ( m + m from,, whch s easy to draw samples ( X X, and then reweghtng ( m m /( m + m to compensate the error. For the smplcty, updated and old partcles are denoted by same symbols.

8 (a (b Fgure 4. omparson of the results for a 0-node network (a NBP, (b NGBP-JT The fnal estmated postons of unknown nodes are gven by mean values of partcles from clque : N N est u m m, u / m = = W X W = ( 4.4 Improved samplng procedure There s one mportant modfcaton to ths algorthm that can reduce sgnfcantly the ntal number of partcles. As we already mentoned, f we draw N partcles from one node, generally t corresponds to N = N partcles of a - node clque. However, we ncluded nformaton about dstance, so our new number for the same clque s N NN θ angles. But ths number s stll very large, so we would lke to nclude addtonal nformaton. We assumed that there s no a pror nformaton about node poston. However, after very frst phase of algorthm, = where N θ represents the number of possble we computed ont messages M and M whch nclude current nformaton about postons of clques and. At ths pont, partcles are concentrated n a smaller regon (ecept for very few of them, so we can draw new set of partcles around sngle-node partcles of ont messages. For, t s done by followng procedure: d = Unf (0, r, X, = shft( X,, d X, = shft( X,, d, X, = shft( X,, d W = / N compute messages agan m ( where r s the radus of deployment area of new partcles. omputng messages agan s very mportant snce we draw new set of partcles, whch means that we have to run algorthm from the begnnng. Of course, for clque, we use analog procedure. Ths mproved procedure allows us to decrease ntal number of samples to N = NNθ / n where n s the reducng factor that could be found epermentally. Theoretcally, t s proportonal to the rato of the new to the old deployment area. 5 Smulaton Results We smulated the network from Fgure usng NBP, GBP- JT and NGBP-JT algorthms. We placed 0 nodes n m m area, 5 anchors and 5 unknowns. We set the values of transmsson radus ( R = 5% of dagonal length of the deployment area and standard devaton of measured dstance ( sgma = 0.m = 4% R. Number of teraton for NBP s set to the length of the longest path n the graph ( N ter = 5, and for GBP-JT/NGBP-JT there s obvously, n our eample, ust one teraton. Number of partcle for NBP s set to N = 400, so correspondng number of grd ponts for GBP-JT s N G = 0 0. For NGBP-JT we used mproved samplng procedure wth radus r = 0. and epermentally we found out reducng factor whch do not change the accuracy ( n 4. Assumng that mnmum number of possble angles could be appromated wth N θ = 0, we set the number of clque s partcles to N = NNθ / n = We ran the smulaton for NBP and NGBP-JT, and obtaned results shown n Fgure 4. Obvously, the locaton estmates for the NGBP-JT are more accurate snce ths algorthm s correct for network wth loops. NBP algorthm does not converge well for a few nodes, but for some other values of parameters, or wth dfferent postons of some nodes, t provdes estmates wth almost same accuracy as NGBP-JT. However, comparng uncertantes for NBP and NGBP-JT (contours n Fgures 4a and 4b, we can see that NBP provdes us better guarantees of ts estmate.

9 Average error [%R] NBP (.8 MFlops NGBP-JT (7.89 MFlops GBP-JT ( MFlops Dstance devaton [%R] Fgure 5. omparson of accuracy Fnally, we checked the averaged accuracy wth respect to the devaton of measured dstance for all three methods (Fgure 5. The accuracy of GBP-JT s always hgher than accuracy of NBP and NGBP-JT. NGBP-JT provdes us better accuracy than NBP for some usual values of dstance devaton (e.g. for measurements usng tme of arrval, the error s 5-0 %R [], and unepectedly worse accuracy for hgher values of mentoned devaton. Anyway, ths accuracy could be ncreased, usng larger number of partcles (e.g. ncreasng N θ, untl the bottom lne defned by the accuracy of GBP-JT. omparng wth NBP/NGBP-JT, the computatonal cost of GBP-JT s, of course, very large ( MFlops and absolutely unacceptable. Nonparametrc appromaton of ths algorthm decreased t around 5 tmes, and mproved samplng addtonal 4 tmes. So the fnal computatonal cost of NGBP-JT n smulated eample s 7.89 MFlops, ust double as many comparng wth NBP (.8 MFlops. 6 onclusons As presented n ths artcle, uncton tree model s a generalzaton of belef propagaton that s correct for arbtrary graphs. We proposed localzaton usng generalzed belef propagaton based on uncton tree method and nonparametrc appromaton of ths algorthm (NGBP-JT. Our man goal was to solve the problem wth loops wth some acceptable computatonal cost and we acheved t usng NGBP-JT approach. We can conclude that ths algorthm could provde hgher accuracy wth acceptable computatonal cost. The man open drecton for future work s generalzng ths algorthm for an ad-hoc network. Ths would probably requre addtonal computaton necessary for constructon of uncton tree clques wthn the network. Moreover, communcaton cost has to be consdered snce t s obvous that we can not echange thousands of partcles wthout any compresson. Ths wll be a part of our future research. Acknowledgment Ths work has been performed n the framework of the IT proect IT-70 WHERE, whch s partly funded by the European Unon and partly by the Spansh Educaton and Scence Mnstry under Grant TE /0/TM. Furthermore, we thank partal support by program ONSOLIDER-INGENIO 00 SD OMONSENS. References [] M. S. Arulampalam, S. Maskell, N. Gordon, and T. lapp, A Tutoral on Partcle Flters for Onlne Nonlnear/Non-Gaussan Bayesan Trackng, IEEE Transactons on Sgnal Processng, vol. 50, ssue, pp , February 00. [] P.M. Durc, J.H. Kotecha, J. Zhang, Y. Huang, T. Ghrma, M.F. Bugallo, J. Mguez, Partcle Flterng, IEEE Sgnal Processng Magazne, vol. 0, ssue 5, pp. 9-8, September 00. [] A. T. Ihler, J. W. Fsher III, R. L. Moses, and A. S. Wllsky, Nonparametrc Belef Propagaton for Self- Localzaton of Sensor Networks, IEEE Journal On Selected Areas In ommuncatons, vol., ssue 4, pp , Aprl 005. [4] J.S. Yedda, W.T. Freeman, and Y. Wess, Understandng belef propagaton and ts generalzatons, Eplorng artfcal ntellgence n the new mllennum, AM, pp. 9-69, 00. [5] D. Nculescu and B. Nath, Ad hoc postonng system (APS, n IEEE GLOBEOM, vol. 5, pp. 96 9, November 00. [6] Y. Shang, W. Ruml, Y. Zhang, and M. Fromherz, Localzaton from onnectvty n Sensor Networks, IEEE Transactons on Parallel and Dstrbuted Systems, vol. 5, no., pp , November 004. [7] A. Savvdes, H. Park, and M. B. Srvastava, The Bts and Flops of the N-hop Multlateraton Prmtve for Node Localzaton Problems, n Internatonal Workshop on Sensor Networks Applcaton, pp., Sept. 00. [8] J.S. Yedda, W.T. Freeman, and Y. Wess, onstructng Free Energy Appromatons and Generalzed Belef Propagaton Algorthms, Informaton theory, IEEE Transacton On Informaton Theory, vol. 5, ssue 7, pp. 8-, July 005. [9] M.I. Jordan and Y. Wess, Graphcal model: Probablstc nference, The Handbook of Bran Theory and Neural Networks, nd edton. ambrdge, MA: MIT Press, 00. [0] B.J. Frey, A revoluton: Belef propagaton n graphs wth cycles, Adv. n Neural Informaton Processng Systems, vol. 0, MIT Press, 998. [] N. Patwar, J.N. Ash, S. Kyperountas, A.O. Hero III, R.L. Moses and N.S. orreal, Locatng the nodes, IEEE Sgnal Processng Magazne, vol., ssue 4, pp , July 005.