Nonparametric Boxed Belief Propagation for Localization in Wireless Sensor Networks

Transcription

1 Nonparamerc Boxed Belef Propagaon for Localzaon n Wreless Sensor Neworks Vladmr Savc and Sanago Zazo Pos Prn N.B.: When cng hs work, ce he orgnal arcle. 29 IEEE. Personal use of hs maeral s permed. However, permsson o reprn/republsh hs maeral for adversng or promoonal purposes or for creang new collecve works for resale or redsrbuon o servers or lss, or o reuse any copyrghed componen of hs work n oher works mus be obaned from he IEEE. Vladmr Savc and Sanago Zazo, Nonparamerc Boxed Belef Propagaon for Localzaon n Wreless Sensor Neworks, 29, IEEE Proc. of Inl. Conf. on Sensor Technologes and Applcaons, hp://dx.do.org/.9/sensorcomm Posprn avalable a: Lnköpng Unversy Elecronc Press hp://urn.kb.se/resolve?urn=urn:nbn:se:lu:dva-8372

2 Nonparamerc Boxed Belef Propagaon for Localzaon n Wreless Sensor Neworks Vladmr Savc, Sanago Zazo The Sgnal Processng Applcaons Group Polyechnc Unversy of Madrd, Span {vladmr, sanago}@gaps.ssr.upm.es Absrac Obanng esmaes of each sensor s poson as well as accuraely represenng he uncerany of ha esmae s a crcal sep for effecve applcaon of wreless sensor neworks (WSN). Nonparamerc Belef Propagaon (NBP) s a popular localzaon mehod whch uses parcle based approxmaon of belef propagaon. In hs paper, we presen a new varan of NBP mehod whch we call Nonparamerc Boxed Belef Propagaon (NBBP). The man dea s o consran he area from whch he samples are drawn by buldng a box ha covers he regon where anchors rado ranges overlap. These boxes, whch are creaed almos whou any addonal communcaon beween nodes, are also used o fler erroneous samples of he belefs. In order o decrease he compuaonal and he communcaon cos, we also added ncremenal approach usng Kullback-Lebler (KL) dvergence as a convergence parameer. Smulaon resuls show ha accuracy, compuaonal and communcaon cos are sgnfcanly mproved.. Inroducon The localzaon consss n obanng he relave or absolue poson of a sensor node ogeher wh he uncerany of s esmae. Equppng every sensor wh a GPS recever or equvalen echnology may be expensve, energy prohbve and lmed o oudoor applcaons. Therefore, we consder he problem n whch some small number of sensors, called anchor nodes, oban her coordnaes va GPS or by nsallng hem a pons wh known coordnaes, and he res, unknown nodes, mus deermne her own coordnaes. If unknown nodes were capable of hgh-power ransmsson, hey would be able o make measuremens wh all anchor nodes (sngle-hop echnque). However, we would lke o use energyconservng devces whou power amplfer, wh lack he energy necessary for long-range communcaon. In hs case, each sensor has avalable nosy measuremens o several neghborng sensors (mulhop echnque). Typcal measuremens echnques [] are me-of-arrval (TOA), receve-sgnal-srengh (RSS), angle-of-arrval (AOA) or connecvy. There are a large number of prevously proposed localzaon mehods. Range-free or connecvy-based localzaon algorhms [2, 3, 4] rely on connecvy beween he nodes. The prncple of hs algorhm s o deermne wheher or no a sensor s n he ransmsson range of anoher sensor. The mos aracve feaure of he range-free algorhms s her smplcy. However, hey can only provde a coarse graned esmae of each node s locaon, whch means ha hey are only suable for applcaons requrng an approxmae locaon esmae. Range-based or dsance-based localzaon algorhms [5, 6] use he ner-sensor dsance measuremens n a sensor nework o locae he enre nework. Ths ype of algorhms s usually more accurae, bu also sensve o measuremen errors. Based on he approach of processng he ndvdual ner-sensor daa, localzaon algorhms can be also consdered n wo man classes: cenralzed and dsrbued algorhms. Cenralzed algorhms [3, 5] ulze a sngle cenral processor o collec all he ndvdual ner-sensor daa and produce a map of he enre sensor nework, whle dsrbued algorhms [2, 4, 6] rely on self-localzaon of each node n he sensor nework usng he local nformaon collecs from s neghbors. From he perspecve of locaon esmaon accuracy, cenralzed algorhms are lkely o provde more accurae locaon esmaes han dsrbued algorhms. However, cenralzed algorhms suffer from he scalably problem and generally are no feasble o be mplemened for large scale sensor neworks.

3 A recen drecon of research n dsrbued sensor nework localzaon s he use of parcle flers [7, 8]. In [9], Ihler e al. formulaed he sensor nework localzaon problem as an nference problem on a graphcal model and appled one varan of belef propagaon (BP) mehods [], he so-called nonparamerc belef propagaon (NBP) algorhm, o oban an approxmae soluon o he sensor locaons. Comparng wh deermnsc approaches [2-6], he man advanages of hs sascal approach are s easy mplemenaon n a dsrbued fashon and suffcency of a small number of eraons o converge. Furhermore s capable of provdng nformaon abou locaon esmaon unceranes and accommodang non-gaussan dsance measuremen errors. In hs paper, we presen a new varan of hs mehod. The remander of hs paper s organzed as follows. In Secon 2, we revew Ihler s graphcal model for he localzaon problem. In Secon 3, we propose nonparamerc boxed belef propagaon mehod (NBBP). Smulaon resuls are presened n Secon 4. Fnally, Secon 5 provdes some conclusons and fuure work perspecve. 2. Graphcal model We begn by descrbng a graphcal model [9] for he sensor nework localzaon problem. We suppose ha all sensors wh unknown posons oban nosy dsance measuremens of nearby subse of he oher sensors n he nework. Typcally, hs measuremen procedure can be accomplshed usng a broadcas ransmsson (acousc or wreless) from each sensor as all oher sensors lsen. Le us assume ha we have N s sensors ( N a anchors and N u unknowns) scaered randomly n a planar regon, and denoe he wo-dmensonal locaon of sensor by x. The unknown node obans a nosy measuremen du of s dsance from node u wh some probably Pd( x, x u) : d = x x + v, v p ( x, x ) () u u u u v u The bnary varable o u wll ndcae wheher hs observaon s avalable ( o u = ) or no ( o u = ). Each sensor has some pror dsrbuon denoed p( x ). Ths pror could be an unnformave one for he unknowns and he dela mpulse for he anchors. Then, he on dsrbuon s gven by: px (,..., xnu { ou},{ du}) = po ( x, x) pd ( x, x) p( x) = u u u u (, u) (, u) (2) The fnal goal of hs localzaon problem s o esmae he maxmum a poseror (MAP) sensor locaon x gven a se of observaons { d u}. There are wo dfferen ways o do hs, o esmae MAP of each x, or o esmae MAP of all x only. We selec he frs one because hs s he only way for our ncremenal approach where unknowns are locaed one by one a dfferen me pons. Ihler also choose he same way n order o mnmze he b-error rae [9]. The measured dsances d u and d u may be dfferen, and s even possble o have ou ou, ndcang ha only one of he sensors u and can observe he oher. However, we can assume ha boh sensors oban he same sngle observaon, so d u = d u and o u = o u, oherwse here s no observed dsance. For large-scale sensor neworks, s reasonable o assume ha only a subse of parwse dsances wll be avalable, prmarly beween sensors whch are locaed whn he some radus R. We use mproved model [9] whch assumes ha he probably of deecng nearby sensors falls off exponenally wh squared dsance: 2 2 Pd( x, xu) exp = x xu / R 2 (3) We also need o exchange nformaon beween he nodes whch are no drecly conneced. Le s defne a par of nodes s and o be a -sep neghbors of one anoher f hey observe her parwse dsance d s. Then, we defne 2-sep neghbors of node s o be all nodes such ha we do no observe he d s, bu do observe d su and du for some node u. We can follow he same paern for he 3-sep neghbors, and so forh. These n-sep neghbors ( n > ) conan some nformaon abou he dsance beween hem. Therefore, f wo nodes do no observe he dsance beween hem, hey should be far away from each oher. In our case, we wll nclude all -sep and 2-sep neghbors, ohers could be negleced whou losng accuracy of he resuls. The relaonshp beween he graph of he nodes and on dsrbuon may be quanfed n erms of poenal funcons ψ [9, ] whch are defned over each of he graph s clques: px (,..., xnu ) ψ C({ x : C}) (4) clquesc

4 Ths only requres poenal funcons defned over varables assocaed wh sngle nodes and pars of nodes. Sngle-node poenal a each node, and he parwse poenal beween nodes and u, are respecvely gven by: ψ ( x ) = p ( x ), (5) Pd ( x, xu ) pv ( du x xu ), f ou =, ψ u ( x, xu ) = (6) Pd( x, xu), oherwse. Fnally, he on poseror dsrbuon for each node s gven by: px (,..., x { o, d }) ψ ( x) ψ ( x, x) (7) Nu u u u u, u Havng defned a graphcal model for sensor localzaon, we can now esmae he sensor locaons by applyng he belef propagaon (BP) algorhm. The form of BP as an erave, local message passng algorhm makes hs procedure rval o dsrbue among he wreless sensor nodes. We apply BP o esmae each sensor s poseror margnal, and use he mean value of hs margnal and s assocaed uncerany o characerze sensor posons. Each node compues s belef M ( x ) (he poseror margnal dsrbuon of wo-dmensonal poson x ( a, b x x ) a eraon ) by akng a produc of s local poenal ψ wh he messages from s se of neghbors G : M ( x ) ψ ( x ) m ( x ) (8) u u G The messages m u, from node u o node, are compued by: M ( x ) m ( x ) ( x, x ) dx (9) u u u ψ u u u mu ( xu ) In he frs eraon of hs algorhm s necessary o nalze m u = and M = p for all u,, hen repea compuaon usng (8) and (9) unl suffcenly converge. In fac, he number of eraon should be a leas he lengh of he longes pah n he graph. 3. Nonparamerc Boxed Belef Propagaon (NBBP) The presence of nonlnear relaonshps and poenally hghly non-gaussan unceranes n sensor localzaon makes descrbed BP algorhm unaccepable. Besdes, o oban accepable spaal resoluon for he sensors, he dscree sae space mus be made oo large for BP o be compuaonally feasble. However, usng parcle-based represenaons va nonparamerc belef propagaon (NBP) enables he applcaon of BP o nference n sensor neworks. The belef and message updae equaons, (8) and (9), are performed usng sochasc approxmaons, n wo sages: frs, drawng samples from he belef M ( x ), hen usng hese samples o approxmae each ougong message m u. In hs secon we propose nonparamerc boxed belef propagaon (NBBP) whch ncludes hree modfcaons: ) Consran he area from whch he samples are drawn by buldng a box ha covers he regon where anchors rado ranges overlap (Fgure ). 2) In each eraon, fler erroneous samples of he belefs (all he samples whch are ou of he approprae box). 3) Locae nodes n he ncremenal way: As soon as he belef suffcenly converges, characerze sensor posons wh mean value and uncerany, and from ha pon consder hs node as an anchor. 3.. Compung messages,, Gven N weghed samples { W, X } from he belef M ( x ) obaned a eraon, we can compue a Gaussan mxure esmae of he ougong BP message m u. We frs consder he case of observed edges (- sep) beween unknown nodes. The dsance measuremen d u provdes nformaon abou how far sensor u s from sensor, bu no nformaon abou s relave drecon. To draw a sample of he message, ( x u + ), gven he sample X whch represens he, poson of sensor, we smply selec a drecon θ a random ( = ), unformly n he nerval [,2 π ). However, sarng from he second eraon, we nclude nformaon from he prevous eraon usng already compued belefs. Therefore, for >, a,,, drecon s calculaed by θ = arcan( Xu X ). We hen shf X n he drecon of θ, by an amoun whch represens he esmaed dsance beween nodes u and ( d + v ):,,,, u u u x + = X + ( d + v )[sn( θ ) cos( θ )], () The samples are hen weghed by he remnder of (9):,,, W wu + = Pd( X, xu) (), m ( X ) u

5 anchor Approxmaed dsance beween anchor and unknown sep anchors, he bounds of he box for node x ( xa, x b) are gven by: u u u d d 3 N a Na a a* a a* u,mn = u u,max = + u = = x max( x d ), x mn( x d ), N a Na b b* b b* u,mn = u u,max = + u = = x max( x d ), x mn( x d ) (6) d 2 Samples drawn whn he box Fgure. Buldng he box and samplng The opmal value for bandwdh h u + could be obaned n a number of possble echnques. The smples way s o apply he rule of umb esmae []: + + u u h = N 3 Var({ x }) (2) We have also o defne messages from anchor * nodes, usng (9) and he belef of he anchor node x * ( M ( x ) = δ ( x x )): + m ( x ) ψ ( x, x ) (3) * u u u u Messages along unobserved edges (2-sep, ) mus be represened as analyc funcons snce her poenals have he form Pd( x, xu) whch s ypcally no normalzable. Usng he probably of deecon P d and samples from he belef M, an esmae of ougong message o node u s gven by: + m x = W P X x (4),, u ( u ) d (, u ) Fnally, he messages along unobserved edges from, anchor nodes ( W = / N ) are gven by: + m ( x ) = P ( x, x ) (5) * u u d u 3.2. Compung belefs In he nalzaon phase for each unknown, we consruc he box (Fgure ) usng only -sep and 2- sep edges from he anchor nodes. -sep edges are already measured wh some nose v u, and 2-sep edges are approxmaed wh he wors case scenaro: sum of he wo measured dsances. Usng hs * dsances ( d u ) and he posons ( x ) of -sep and 2- where N a s he number of -sep and 2-sep edges beween hs unknown and anchors. Of course, hese bounds are lmed o he bounds of he deploymen area. To nalze belef M u( x u), we draw N samples wh he unform dsrbuon whn hs box. + To esmae he belef M u ( xu) usng (8), we draw samples from he produc of several Gaussan mxure and analyc messages. In our case s very dffcul o draw samples from hs produc, so we use proposal dsrbuon, sum of he Gaussan mxures, and hen re-wegh all samples. Ths procedure s well-known as mxure mporance samplng [2]. Denoe he se of neghbors of u, havng observed edges o u and no ncludng anchors, by G u, and he se of of all neghbors by G u. In order o draw N samples, we creae a collecon of kn weghed samples (where k s a parameer of he samplng algorhm) by drawng kn / G u samples from each message m u wh G u and assgnng each sample a wegh equal o he rao: W = m / m (7), u vu vu v Gu v Gu + If he sample of he belef M u ( xu) s ou of s, box, we fler by assgnng W u + =. Some of hese calculaed weghs are much larger hen he res, especally afer more eraons. Ths means ha any sample-based esmae wll be unduly domnaed by he nfluence of a few of he parcles, and he esmae could be erroneous. To avod hs, we hen draw N values ndependenly from collecon, +, + { Wu, Xu } wh probably proporonal o her wegh, usng resamplng wh replacemen [8]. Ths means ha we creae N equal-wegh samples drawn from he produc of all ncomng messages Convergence A node s locaed when a convergence crera s me. We use Kullback-Lebler (KL) dvergence [3], a common measure of error beween wo dsrbuons. For he parcle based belefs n our algorhm, KLdvergence beween belefs n wo consecuve eraons, s gven by:

6 (a) (b) Fgure 2. Comparson of he resuls for a 5-node nework (a) NBP, (b) NBBP KL = W log[ M ( X ) / M ( X )] (8) +, + +, +, u u u u u u When KL + u drops below he predefned hreshold, he node u s locaed and sars o behave as an anchor. In hs way, we can locae all nodes ncremenally. The execuon s over when KL drops below he hreshold for all nodes, or when he maxmum number of eraon s reached. In any case, esmaed posons of all unknowns and her unceranes wll be avalable. Average error [% dmax] NBBP - parcles NBBP - 5 parcles NBP - parcles NBP - 5 parcles 4. Smulaon resuls In he smulaon, we placed 5 nodes randomly n 2m 2m area, 4 of hem are unknowns. The values of parameers are se as follow: sandard devaon of he Gaussan nose ( sgma =.m ), ransmsson radus ( R = 3% of dagonal lengh of he deploymen area - d max = 2 2m), number of parcles ( N = ), and he KL hreshold ( KL mn =.2 ). We run he smulaon for boh algorhm (NBP and NBBP), and obaned resuls shown n Fgure 2. Obvously, he locaon esmaes for he NBBP are more accurae snce all esmaes are placed whn s box. In Fgure 3 and 4, we show comparson of he average error and coverage (percenage of locaed nodes) wh respec o ransmsson radus. Fnally, n Fgure 5 and 6, we show comparson of he compuaonal and communcaon cos wh respec o he number of parcles. The man concluson s ha NBBP algorhm performs beer han NBP. Ths resul s expeced because consruced boxes ncrease accuracy, and ncremenal approach decreases compuaonal and communcaon cos. Average coverage (error<5%) Transmsson radus [% dmax] Fgure 3. Comparson of accuracy 3 NBBP - parcles NBBP - 5 parcles 2 NBP - parcles NBP - 5 parcles Transmsson radus [% dmax] Fgure 4. Comparson of coverage (percenage of locaed nodes wh error less hen 5%)

7 Average compuaon me [s] NBBP - 2 unknowns NBBP - unknowns NBP - 2 unknowns NBP - unknowns Number of parcles Fgure 5. Comparson of compuaonal cos Messages per node x 4 NBBP - 2 unknowns NBBP - unknowns NBP - 2 unknowns NBP - unknowns Number of parcles Fgure 6. Comparson of communcaon cos 5. Concluson and fuure work In hs paper, we presened nonparamerc boxed belef propagaon (NBBP), a new varan of NBP. Our man goal was o ncrease he performance of he algorhm and we acheved by addng boxes whch consran he area from whch he samples are drawn. These boxes, whch are creaed almos whou any addonal communcaon beween nodes, are also used o fler erroneous samples of he belefs. We also added ncremenal approach n order o decrease he compuaonal and he communcaon cos. There reman few open drecons for mprovng hs algorhm. Accuracy could be mproved by cluserng he nodes and passng messages beween he groups usng some verson of general belef propagaon (GBP) []. Moreover, communcaon cos could be decreased usng some specfc message passng proocol, e.g. groupng smlar samples and sendng hem lke one. Ths wll be a par of our fuure work. 6. Acknowledgmen Ths work has been performed n he framework of he ICT proec ICT-2733 WHERE, whch s parly funded by he European Unon and parly by he Spansh Educaon and Scence Mnsry under Gran TEC C2-/2/TCM. Furhermore, we hank paral suppor by program CONSOLIDER- INGENIO 2 CSD28- COMONSENS. 7. References [] N. Pawar, J.N. Ash, S. Kyperounas, A.O. Hero III, R.L. Moses and N.S. Correal, Locang he nodes, IEEE Sgnal Processng Magazne, vol. 22, ssue 4, pp , July 25. [2] D. Nculescu and B. Nah, Ad hoc posonng sysem (APS), n IEEE GLOBECOM, vol. 5, pp , November 2. [3] Y. Shang, W. Ruml, Y. Zhang, and M. Fromherz, Localzaon from Connecvy n Sensor Neworks, IEEE Transacons on Parallel and Dsrbued Sysems, vol. 5, no., pp , November 24. [4] V. Vvekanandan and V.W.S. Wong, Concenrc Anchor Beacon Localzaon Algorhm for Wreless Sensor Neworks, IEEE Transacons on Vehcular Technology, vol. 56, ssue 5, pp , Sepember 27. [5] A. Savvdes, H. Park, and M. B. Srvasava, The Bs and Flops of he N-hop Mullaeraon Prmve for Node Localzaon Problems, n Inernaonal Workshop on Sensor Neworks Applcaon, pp. 2 2, Sepember 22. [6] N.B. Pryanha, H. Balakrshnan, E. Demane and S. Teller, Anchor-Free Dsrbued Localzaon n Sensor Neworks, MIT Laboraory for Compuer Scence, Tech Repor, Aprl 23. [7] M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, A Tuoral on Parcle Flers for Onlne Nonlnear/Non-Gaussan Bayesan Trackng, IEEE Transacons on Sgnal Processng, vol. 5, ssue 2, pp , February 22. [8] P.M. Durc, J.H. Koecha, J. Zhang, Y. Huang, T. Ghrma, M.F. Bugallo, J. Mguez, Parcle Flerng, IEEE Sgnal Processng Magazne, vol. 2, ssue 5, pp. 9-38, Sepember 23. [9] A. T. Ihler, J. W. Fsher III, R. L. Moses, and A. S. Wllsky, Nonparamerc Belef Propagaon for Self- Localzaon of Sensor Neworks, IEEE Journal On Seleced Areas In Communcaons, vol. 23, ssue 4, pp , Aprl 25. [] J.S. Yedda, W.T. Freeman, and Y. Wess, Undersandng belef propagaon and s generalzaons, Explorng arfcal nellgence n he new mllennum, ACM, pp , 23. [] B.W. Slverman, Densy Esmaon for Sascs and Daa Analyss, Chapman and Hall, New York, 986. [2] D. MacKay, Inroducon o mone carlo mehods, In M. I. Jordan, edor, Learnng n Graphcal Models. MIT Press, 999. [3] A. T. Ihler, J. W. Fsher, III, and A. S. Wllsky, Communcaon-Consraned Inference, Techncal Repor TR-26, Laboraory for Informaon and Decson Sysems, MIT, 24.