A Bayesian algorithm for distributed network localization using distance and direction data

Transcription

1 Notce: Ths work has been submtted to the IEEE for possble publcaton. Copyrght may be transferred wthout notce, after whch ths verson may no longer be accessble. A Bayesan algorthm for dstrbuted network localzaton usng dstance and drecton data Hassan Naser, Student Member, IEEE Vsa Kovunen, Fellow, IEEE arxv:7.98v [cs.it] 8 Aug 7 Abstract A relable, accurate, and affordable postonng servce s hghly requred n wreless networks. In ths paper, the novel Message Passng Hybrd Localzaton (MPHL) algorthm s proposed to solve the problem of cooperatve dstrbuted localzaton usng dstance and drecton estmates. Ths hybrd approach combnes two sensng modaltes to reduce the uncertanty n localzng the network nodes. A statstcal model s formulated for the problem, and approxmate mnmum mean square error (MMSE) estmates of the node locatons are computed. The proposed MPHL s a dstrbuted algorthm based on belef propagaton (BP) and Markov chan Monte Carlo (MCMC) samplng. It mproves the dentfablty of the localzaton problem and reduces ts senstvty to the anchor node geometry, compared to dstance-only or drecton-only localzaton technques. For example, the unknown locaton of a node can be found f t has only a sngle neghbor; and a whole network can be localzed usng only a sngle anchor node. Numercal results are presented showng that the average localzaton error s sgnfcantly reduced n almost every smulaton scenaro, about % n most cases, compared to the competng algorthms. Index Terms Postonng, Cooperatve Localzaton, Message Passng, Hybrd Localzaton I. Introducton In recent years, the servces that utlze the postons of network devces have become a key component of wreless systems []. We consder a wreless network comprsed of anchor nodes wth known locatons, and target nodes wth unknown locatons to be estmated. Cooperatve localzaton s a technque that employs communcaton among all the nodes to fnd the unknown locatons [] []. That s, the measurements among target nodes are also utlzed, n addton to the anchortarget measurements. Utlzng the extra nformaton mproves the dentfablty of the localzaton problem [6]. Hence, cooperatve localzaton can be appled to ad-hoc networks where not every node s connected to all the anchors. Propagaton delays,.e., tme-of-arrvals (TOAs), and drectons-of-arrval (DOAs) of rado sgnals amongst the network nodes can be estmated and employed for cooperatve localzaton. Hghresoluton TOA and DOA estmaton are facltated by the ncreased bandwdth of rado sgnals along wth a wde applcaton of mult-antenna transcevers n most current wreless systems (e.g., G LTE and 8.ac WF) and emergng technologes (e.g., G and evoluton of WF) [7] [9]. Hence, t s sensble to take advantage of the avalable angular doman nformaton for wreless network localzaton n addton to delay data. Dstance and drecton data are two ndependent sources of nformaton from dfferent sensng modaltes. Therefore, they can be combned to reduce the uncertanty n localzaton compared to dstance-only or drecton-only localzaton. Moreover, t helps to solve a localzaton problem wth fewer anchor nodes and fewer connectons among the nodes []. Practcal applcatons nclude locaton based servces (LBS) for G moble networks, locaton-based routng and spectrum sharng for moble devces, postonng of devces n WF networks, localzaton n sensor networks, and navgaton of autonomous vehcles, robots, and frst responders n emergency servces. These applcatons are more mportant n ndoor scenaros (ncludng dense urban envronments and covered paths), where conventonal satellte-based and cellbased localzaton solutons may not be avalable or relable. The man contrbutons of ths paper are the followng: ) A new statstcal model s developed for the problem of cooperatve localzaton usng hybrd dstance and drecton data. A lkelhood functon s derved to combne the jont statstcs of dstance and drecton estmates usng Gaussan and von Mses dstrbutons. Both maxmum lkelhood (ML) and mnmum mean square error (MMSE) estmators are formulated usng the new data model. ) The novel Message Passng Hybrd Localzaton (MPHL) algorthm s proposed to fnd an approxmate soluton for the formulated MMSE estmator. It s the frst algorthm to approxmate an optmal soluton for the cooperatve hybrd localzaton problem usng a statstcal estmaton approach. It s a dstrbuted algorthm based on belef propagaton (BP) and Markov chan Monte Carlo (MCMC) samplng. Numercal results are provded to show that the Message Passng Hybrd Localzaton (MPHL) algorthm sgnfcantly reduces the localzaton error, typcally % compared to the competng algorthms, n almost every scenaro consdered. A theoretcal study of cooperatve localzaton usng hybrd dstance and drecton data was reported n []. However, the results were based on nose-free observatons, and the basc algorthm was only applcable to very specfc network confguratons. The Message Passng Hybrd Localzaton (MPHL) algorthm, proposed n ths paper, employs nosy (erroneous) estmates of dstance and drecton quanttes; and t s applcable to every rgd network confguraton. The concept of rgdty wll be dscussed later, see [] for more detals. Dstance-based cooperatve localzaton has been studed n [], [] []. In a fully connected network, the metrc mult-dmensonal scalng (MDS) [] s an optmal algorthm for dstance-only centralzed localzaton [, chap. 7]. Drecton-only cooperatve localzaton has been studed n [6], [7]. Cooperatve localzaton methods n [], [] do not consder the jont statstcs of dstance and drecton data, unlke the method developed n ths paper. A stochastc search algorthm for cooperatve localzaton,

2 based on a Gaussan model for data and locatons, was proposed n [8]. Ths model s not generally vald for hybrd localzaton, see Secton II for more detals. Although TOA, DOA, and receved sgnal strength (RSS) measurements were suggested as nput data n [8], only RSS-based results were presented. Recent advances n cooperatve hybrd (dstance and drecton) localzaton were presented n [9], []. The CLORIS algorthm, proposed n [9], employed a heurstc cost functon, whch was not derved from a statstcal data model. In [], a Gaussan data model was proposed for hybrd data, The Gaussan dstrbuton s not generally sutable for DOA estmates as t has an nfnte support nstead of a perodc support of π over angular doman []. Although an maxmum lkelhood estmator (MLE) was formulated n [], the proposed algorthm, referred to as SDP_Tomc, was not a drect approxmaton of the MLE. The statstcal nformaton, e.g., the error varances of the data, were dscarded durng the approxmaton. Moreover, the authors n [] combned drecton data wth squared dstance estmates, dvergng more from the assumed model. The dstrbuted versons of the CLORIS and SDP_Tomc algorthms, proposed n [], [], were based on (block) coordnate descent. However, the exact order of the optmzaton (message schedule) were not specfed. In the early stages of dstrbuted cooperatve localzaton, the node locatons may have mult-modal dstrbutons. Hence, a coordnate-descent method s only applcable to certan network confguratons wth suffcent anchor connectvty to avod sufferng from local mnma problem. The proposed MPHL algorthm can be appled to any localzable network confguraton, snce t estmates and propagates probablty dstrbutons rather than node locatons (sngle ponts). The MPHL runs a sum-product message passng algorthm over a loopy factor graph model, a varant of the loopy belef propagaton (LBP) [], []. It s a dstrbuted sequental algorthm,.e., all the nodes update and propagate ther belefs (margnal probablty dstrbutons) n parallel. Dstance-based cooperatve localzaton algorthms stemmng from LBP have been proposed n [] []. A multpathaded hybrd localzaton method based on BP was proposed n [6]. They dd not explctly model dstance and drecton data, but the combned locaton error was modeled as normally-dstrbuted. Ths model s not generally vald for hybrd localzaton, see Secton II for detals. The MPHL algorthm approxmates BP messages usng a set of samples (partcles). It employs MCMC samplng to generate equallyweghted partcles. An approxmate MMSE estmate of a target locaton s obtaned as the sample mean of t posteror dstrbuton. Dstance-based cooperatve localzaton algorthms utlzng BP and mportance samplng (weghted partcles) were proposed [7] [9]. In contrast to MCMC, mportance samplng may suffer from the problem of sample degeneracy due to teratve re-weghtng, and t requres the evaluaton of exact posteror probablty densty functons (PDFs). The partcle belef propagaton (PBP) [] and non-parametrc belef propagaton (NBP) [] are general partcle-base BP algorthms that employ MCMC samplng. The NBP apples kernel smoothng to all the messages (factors) to have welldefned products, whch s not requred by the PBP algorthm. The proposed MPHL algorthm s a varant of the partcle belef propagaton (PBP) []. The man dfferences to the PBP nclude: (a) formulaton of new factors (potental functons) for hybrd localzaton, (b) reduced complexty n communcaton, (c) mproved numercal stablty, and (d) employng a novel message schedulng mechansm desgned for cooperatve localzaton. These dfference wll be dscussed later n more detals. The rest of ths paper s organzed as follows. Secton II states the problem and data model. The Bayesan estmaton framework, the factor graph model, and the proposed algorthm are descrbed n Secton III. Secton IV consders the propertes, requrements and extensons of the algorthm. Smulaton results are presented n Secton V. Fnally, Secton VI concludes the paper. II. Data Model Assume a network comprsed of n nodes x, =,..., n, from whch m are target nodes wth unknown locatons to be estmated, and n m are anchor nodes wth known locatons. The notaton x s used to refer both to a network node and to ts locaton. It s assumed that some parwse dstance estmates r j and drecton estmates α j among the nodes are avalable through TOA and DOA estmaton technques. The goal s to estmate the unknown locatons of the target nodes usng these observatons. In D anchor-based localzaton, a sngle target node could be unambguously localzed usng: (a) three TOAs to three dfferent anchors, (b) two DOAs (wth dfferent values) to two dfferent anchors, (c) a TOA and a DOA to a sngle anchor, (d) a TOA and a DOA to two dfferent anchors that have a certan geometry, and (e) two TOAs and a DOA to three dfferent anchors. The above condtons are suffcent usng error-free observatons wth the excepton of some degenerate cases, e.g., f three anchors le on a sngle lne. In cooperatve hybrd localzaton wth both (error-free) dstance and drecton estmates, all the locatons can be found unambguously usng a sngle anchor node and mnmum connectvty of the network,.e., f the network graph s just sngly connected. In a general cooperatve hybrd localzaton scenaro,.e., dfferent combnatons of dstance and drecton estmates, the topology (connectvty) of the network determnes f the problem s dentfable [], []. That s, the network graph should satsfy certan rgdty requrements n order to unambguously estmate all target locatons, see [] for more detals. These rgdty requrements may be evaluated before startng the localzaton algorthm. The assumpton n ths paper s that, the localzaton problem has a unque soluton wth the gven data. The parwse dstances and azmuth angles between three nodes are shown n Fg.. A parwse dstance d j = x x j s the Eucldean dstance between the nodes, j. The azmuth angle of node j from node, denoted by θ j, s the angle of a vector from node to node j. An observed dstance and drecton at node are modeled as r j = d j + ɛ j, α j = θ j + γ j, ()

3 x d θ θ x d θ d θ x θ Fg. : Parwse dstances and drectons where ɛ j and γ j are random error terms. These quanttes are assumed to be obtaned usng hgh-resoluton TOA and DOA estmaton technques. The estmaton of these para are not n the scope of ths paper; and the estmated quanttes are referred to as observatons. See [] [6] for examples of hgh resoluton channel estmaton technques n wreless networks. It s assumed that the varances of dstance and drecton estmaton errors are known or estmated relably. These varances mght be obtaned from the measurements or approxmated usng performance bounds. It s also assumed that the dstance and drecton estmates are absolute quanttes wth respect to a general frame of reference. In order to get absolute dstance estmates, the clock offsets of the nodes and other delays n the system should be compensated for,.e., by tme synchronzaton; see [], [6] for more detals. In order to have absolute drecton estmates,.e., wth respect to a common coordnate axs, the orentatons of all the nodes should be known. Let us defne three sets H, A, X that nclude all dstance and drecton estmates, and node locatons respectvely. The observatons (obtaned by a same node or by dfferent nodes) are assumed to be condtonally ndependent, only dependng on the locatons of the two nodes nvolved. That s, the condtonal PDFs of the observatons,.e., lkelhoods, may be wrtten as f r j (r j H \ r j, A, X) = f r j (r j x, x j ), f α j (α j H, A \ α j, X) = f α j (α j x, x j ), where \ denotes a set dfference. Gaussan dstrbuton s commonly used n the lterature to model short-range dstance estmaton error [], [7]. Hence, the condtonal PDF of a dstance observaton wth mean d j = x x j and varance σj s gven by f r j (r j x, x j ) = exp ( ) rj πσj σj x x j. () Drectonal data s usually modeled usng von Mses dstrbuton, whch s a drectonal doman counterpart for Gaussan dstrbuton [], [8]. It provdes an approprate support for drectonal data,.e., satsfes f (θ) = f (θ+πk) for any nteger k. The von Mses PDF fully defnes the dstrbuton of a DOA estmate usng only two para, gven by f α j (α j θ j ) = () πi (κ j ) exp { κ j cos(α j θ j ) }, () where I (.) s the modfed Bessel functon of the frst knd of order zero. The two para are the mean drecton θ j (symmetry center) and the concentraton parameter κ j [, ) (nverse scale), whch are analogous to mean and varance n the normal dstrbuton. The mean drecton θ j corresponds to the true value of a drecton parameter,.e., the value of an error-free DOA estmate. If the concentraton parameter s large, e.g., κ j >, t can be approxmated as κ j = /ζ j, where ζj s the DOA estmaton varance. Let us defne two unt vectors u j = [cos α j, sn α j ] T as an observed drecton vector, and v j = [cos θ j, sn θ j ] T = x j x x x j, () as a true drecton vector from node to j. The cosne of the angle between these two vectors s gven by ther nner product,.e., cos(α j θ j ) = u T j v j. Hence, the PDF of a drecton observaton condtoned on the node locatons may be wrtten as f α j (α j x, x j ) = { πi (κ j ) exp κ j u T j (x j x ) x x j }. (6) Usng () and (6) and ndependence propertes of the observatons (), the log-lkelhood functon for locaton para X s gven by L ( X; H, G ) = δ (, j) U j (, j) V κ j u T j ( hj x x j ) x j x x x j, where x are unknown para for =,..., m,, and known anchor locatons for = m +,..., n,. Index sets U and V contan ndex tuples (, j) for every observed dstance and drecton, respectvely, e.g., f the dstance between nodes, s observed at node then (, ) U. Assumng that unknown locatons X are determnstc quanttes, a ML estmate of the locaton para may be found by maxmzaton the loglkelhood functon (7). Ths s a hghly non-convex problem that cannot be solved usng conventonal convex optmzaton technques. Hence, a Bayesan formulaton for the localzaton problem s proposed n the next secton, that can be solved usng approxmate technques. The approxmaton nvolves factorzng the jont estmaton problem nto several sngle-target localzaton problems, that can be solved usng a sequental algorthm. III. Bayesan estmaton In the framework of Bayesan estmaton, an unknown target locaton s treated as a random varable. In ths secton, an MMSE estmator s formulated for the hybrd localzaton problem; and the MPHL algorthm s proposed to solve t. It s an teratve Bayesan estmaton algorthm that approxmates the posteror dstrbutons of target locatons. The MPHL works by updatng and propagatng the margnal dstrbutons of target locatons over a factor graph model. The man reasons behnd usng ths framework are as follows. (7)

4 An MMSE estmator can optmally combne the statstcs of the dstance and drecton data. It can also handle mult-modal dstrbutons when an ML or maxmum a posteror probablty (MAP) estmator may suffer from local mnma. Solvng the MLE (7), or MMSE of hybrd localzaton, analytcally or usng convex optmzaton methods, s not a tractable problem. The factor graph approach breaks ths problem nto several small problems of sngle-node localzaton, that can be solved usng a message passng algorthm. A. Mnmum mean square error estmaton The MMSE estmate of a locaton varable x s the expected value of ts margnal posteror dstrbuton, gven by " # x = f (H, A X) f () (X) X \ x, (8) where f (H, A X) s the jont condtonal PDF of the observatons (lkelhood functon), f () (X) s a pror dstrbuton on locatons, and X \ x = {dx,..., dx, dx+,..., dxn } denotes ntegraton wth respect to all locaton varables except x. The logarthm of the lkelhood functon f (H, A X) was gven n (7). x x x Fg. : An example of a posteror PDF for node gven the postons of nodes and, and two observatons. The drecton α and the dstance r are observed. Combnng dstance and drecton nformaton reduces uncertanty about the locaton of node. An example of a posteror dstrbuton for the locaton of a sngle node x wth unform pror s llustrated n Fg.. There s one drecton observaton between x and x, and one dstance observaton between x and x. The locatons of x and x are gven. The locatons of x may be found as the mean or maxmum of ths posteror dstrbuton. The PDF n Fg. has a complex shape wth two modes. Mult-modal dstrbutons arse n the context of cooperatve localzaton due to: (a) nsuffcent number of anchor connectons for a node, (b) large errors n TOA and DOA estmates, e.g., caused by multpath propagaton. Many of the locaton para may have mult-modal dstrbutons n the early stages of a dstrbuted localzaton algorthm. By fusng more measurements and trough cooperaton the uncertanty can be further reduced, and the PDF may become unmodal. B. Factor graph model Factor graph s a graphcal tool to represent the factorzaton of a jont probablty dstrbuton by explotng the condtonal ndependence propertes of the varables. Ths factorzaton makes t possble to apply message passng algorthms to compute the margnals of an otherwse ntractable jont PDF. Theoretcal studes have proven the convergence and effcency of such algorthms [9]. Moreover, the factor graph model for the cooperatve localzaton problem drectly maps to the communcaton network topology (nodes and lnks). Hence, t s a natural choce for ths problem. A message passng algorthm can be easly dstrbuted over a wreless network to run wthout a central control mechansm. From (), and assumng that the pror probabltes are all ndependent, the jont posteror probablty may be factorzed as a parwse Markov random feld (MRF). Ths factorzaton (up to a normalzaton constant) s gven by Y Y f (X H, A) φ (x ) φ j (x, x j ). (9) (, j) N The set N = U V contans ndex tuples (, j) for every connected par of nodes. A parwse factor φ j (x, x j ) s (proportonal to) the lkelhood of the locatons x, x j gven the observatons. A local factor φ (x ) s the evdence for node. It s proportonal to the pror probablty of x multpled by the probablty of any local observaton. If both dstance and drecton observatons r j, α j between the nodes, j are avalable, then we have φ j (x, x j ) f r j (r j x, x j ) f α j (α j x, x j ). () Snce anchor locatons are known, the observatons between target nodes and anchor nodes can be ncorporated nto the defntons of local factors φ (x ). The Bethe cluster graph s a smple graph model to represent a parwse MRF [9, chap. ]. For every factor or varable n (9) we create a vertex. Each varable s connected to all the factors sharng that varable. The result s a bpartte graph wth the frst layer for factors (clusters of varables) and the second layers for ndvdual varables. Ths smple model can be constructed automatcally. Fg. shows an example of a φ X φ φ φ X X φ φ φ φ X φ φ Fg. : Factor graph model for the jont probablty dstrbuton of the observatons and varables. It s a bpartte graph and each parwse factor s only connected to two varables. factor graph correspondng to a network of four nodes wth full connectvty. Ths s an undrected model assumng that the observatons are recprocal. That s, f r j s observed then r j = r j s also gven. Smlarly, f α j s observed then α j = α j. The model and the rest of the results n ths paper can be easly extended to the case of nonrecprocal observatons. The margnal posteror dstrbutons of all varables n a factor graph can be computed by graph calbraton usng a sum-product message passng algorthm, also known as belef propagaton [9, chap. -]. In a Bethe cluster graph, due

5 to the exstence of loops, these margnals are computed usng the loopy belef propagaton (LBP) algorthm [], []. The class of LBP algorthms perform approxmate nference over a loopy graph by: (a) computng local belefs (margnal posteror dstrbutons) at each cluster/node usng the messages receved from ts neghbors, (b) propagatng the updated belefs (messages) over the graph. Ths procedure s contnued teratvely followng a certan schedule untl convergence,.e., when the adjacent factors approxmately agree on the dstrbutons of the para. The convergence crtera of the MPHL algorthm wll be dscussed n the next secton. An LBP algorthms may run n parallel or seres over the clusters/nodes dependng on the schedulng mechansm; and the parallel mode may requre a centralzed message synchronzaton. The MPHL algorthm employs an asynchronous schedulng mechansm,.e., t runs n parallel over the nodes wth no centralzed control. The proposed schedulng mechansm wll be descrbed later. In the standard sum-product algorthm,.e., BP, the messages are defned on a contnuous state space. There are two types of messages correspondng to the graph model n Fg. : (a) a message from parwse factor to a varable µ (t) x φ j (x ), and (b) a message from a varable to a parwse factor µ (t) x φ j (x ), see [] for detals on the BP messages. At each teraton, the belef at node, whch s proportonal to the margnal posteror probablty of x, may be obtaned by multplyng all ncomng messages at node wth ts local evdence. C. The proposed MPHL algorthm The margnalzaton ntegrals of the jont posteror PDF n (9) cannot be computed usng analytcal methods. Hence, mplementng an exact message passng algorthm on a contnuous parameter space s not tractable. Several non-parametrc message passng algorthms have been proposed n the lterature to approxmate LBP messages for contnuous random varables [], [], [9, chap. ]. The MPHL s manly based on the partcle belef propagaton (PBP) algorthm [], whch obtans consstent estmates of the LBP messages. The dfferences to the PBP are as follows. ) New factor functons are formulated for the MPHL, snce the PBP has not been appled before to the problem of hybrd localzaton. ) To reduce the communcaton cost: (a) each node broadcasts ts current partcle set nstead of PBP messages; (b) these outgong messages are sub-sampled. ) All the factors n MPHL are computed n log-doman for numercal stablty. ) The convergence and results of LBP depend on the message order. Whle the PBP does not specfy an order, the MPHL has a schedulng mechansm desgned for cooperatve localzaton. The fast convergence of ths schedule s demonstrated by varous numercal tests. In the MPHL algorthm, the probablty dstrbutons of contnuous varables are represented usng fnte sets of unweghted random samples. It avods any bases assocated wth densty estmaton methods by usng the partcles to drectly compute the messages. The MPHL algorthm s summarzed n Table, and descrbed n the followng. Table : Message passng hybrd localzaton (MPHL) algorthm Data: Set of factors Φ and network graph G Result: Samplng dstrbutons for each node locaton // Create schedulng sets S (t) createschedule(g), t =,..., N ter // Start message passng teratons Anchor nodes transmt ther locatons M j = {(x, )}. for t = to N ter do // Recevng messages and samplng All the nodes lsten to ncomng messages. for every node recevng messages do 6 Construct a factor for each message receved by node, gven by µ (t) φ j x (x ) (ˆx j,w j ) M (t ) φ j (x, ˆx j )/w j, j 7 Draw a new sample yx (t) of sze N from the current belef at node, gven by (x ) φ (x ) j N µ (t) φ j x (x ). b (t) 8 end // Transmttng messages 9 for every node n the schedulng set S (t) do Select a random subset Y (t) of sze M from yx (t). for every node j n the neghborhood of do Construct a message M (t) j and transmt t to node j, gven by M (t) j = ) } {(ˆx, w j ˆx Y (t), w j = µ (t) φ j x (ˆx ). end end end A message from a varable to a parwse factor s an standard BP message, gven by µ (t) x φ j (x ) = φ (x ) µ (t) φ k x (x ), () k N \j where N s the neghborhood of node,.e., the ndces of ts neghbor, and µ (t) φ k x (x ) s a message from a parwse factor to a varable. Ths multplcaton stage combnes all the nformaton about the locaton of node receved from ts neghbors except the message from node j. The belef at node at teraton t s obtaned by multplyng all ncomng messages to node wth ts local evdence, as b (t) (x ) = φ (x ) µ (t) φ j x (x ). () j N A local belef of a node s a fnte-sample approxmaton of ts margnal posteror PDF, up to a normalzaton constant. Usng local belefs, the equaton () may be expressed alternatvely as µ (t) x φ j (x ) b (t) (x )/µ (t) φ j x (x ). () Usng ths alternatve form, a message from a parwse factor to a varable s approxmated by a sum over a set of samples

6 6 (partcles) from ts local belef yx (t) j, gven by φ j (x, ˆx j ) µ (t) φ j x (x ) = ˆx j yx (t ) j µ (t ) φ j x j (ˆx j ). () where t denotes the tme nstance,.e., teraton number. Ths approxmate margnalzaton computes the lkelhood of a node locaton x gven observatons between nodes, j and the latest message from x j. The partcles are generated by drect samplng from the local belefs of the nodes. The samples are drawn usng the Metropols-Hastngs random walk MCMC algorthm [9, chap. ] [, chap. 6]. Snce the samples are equally-weghted, the s no need for sample re-weghtng. The Metropols-Hastngs algorthm can produce samples from a dstrbuton by evaluatng any functon proportonal to ts PDF,.e., node belefs b (t) (x ). Hence, t does not requre an extremely dffcult computaton of normalzaton terms. The algorthm works by drawng proposal samples from Gaussan dstrbuton, whch are then ether accepted or rejected after evaluatng the local belefs b (t) (x ) analytcally. The varance of the Gaussan proposal s tuned to acheve the desred acceptance rate for the samples. It has been shown that the deal acceptance rate for a mult-dmensonal target dstrbuton s about / []. The lax requrements of the Metropols-Hastngs algorthm, makes t possble to mplemented all the computaton n log-doman for mproved numercal stablty. From (), a logarthmc parwse factor s gven by φ j (x, x j ) = δ j ( hj x x j ) + κj u T j x j x x x j. () The factor graph model n Fg. drectly maps to the structure of the underlyng communcaton network. The clusters of parameter and measurement factors (vertces) are mapped to the network nodes, and the edges between them are mapped to the actual lnks of the wreless network. At every teraton, each network node collects messages form ts neghbors to construct ts local belef functon b (t) (x ). Then t draws a new sample yx (t) of sze N from ts local belef. If the node s propagatng ths turn (dependng on the schedule), then t selects a random subset of sze M < N from ts current sample and broadcasts t to ts neghbors along wth the values of the ncomng messages evaluated at samplng ponts. Thus, a transmt message M (t) j s a set of tuples (ˆx, µ (t) φ j x (ˆx ) ). Ths teratve procedure s contnued followng a certan message passng schedule. The graphcal model for the localzaton problem s a cyclc graph, see Fg.. Message passng n cyclc graphs, also known as loopy belef propagaton, s an approxmate nference method []. The fnal result and the convergence propertes of the algorthm depend on the order of the messages. A dynamc schedulng mechansm s employed n the MPHL algorthm to guarantee the convergence. It starts by only anchor nodes transmttng ther locatons. The other nodes jon the propagaton schedule f they have already receved a certan number of messages. The detals of the schedulng and convergence propertes of the algorthm wll be dscussed n next secton. A fnal approxmate MMSE estmate of a node locaton x s gven by the the mean of yx (end),.e., the sample mean of t local belef at last teraton. A. Convergence IV. Propertes of the algorthm In the standard BP algorthm, a cluster graph s calbrated (the algorthm s converged) f every par of adjacent clusters (measurements factors) agree on the dstrbuton of ther shared nodes [9, chap. -]. Snce an exact agreement can not be acheved n the LBP algorthm, a loopy graph may be consdered calbrated f the factors approxmately agree on the dstrbutons of the para []. A partcle-based BP algorthm can be stopped f the samplng dstrbutons of the para are approxmately constant, e.g., by checkng the varaton of the sample mean. That s, the localzaton algorthm can be stopped f the estmated locatons of all the nodes reman approxmately unchanged (wthng a gven error tolerance) n consecutve teratons. That s max { ˆx (t) ˆx (t ) =,, m } ɛ, (6) where ɛ s a gven error tolerance for localzaton. However, evaluatng such a crtera s not easy n the context of dstrbuted localzaton. It requres communcaton between all the nodes before every teraton or a centralzed control mechansm. The stoppng crteron for the proposed MPHL algorthm s not based on the graph calbraton, but the maxmum number of teratons determned by the message passng schedule. The convergence rate of the algorthm depends on the network sze, the connectvty and geometry of the network, and the qualty of the observatons. The MPHL algorthm employs an automatc schedulng mechansm. The algorthm s stopped when the message passng schedule s fnshed. The proposed message passng schedule s descrbed n the next subsecton. The experments show that, for unquely localzable confguratons consdered n ths paper, the MPHL algorthm converges very fast (e.g. n teratons) and fnds accurate estmates of the target locatons. The convergence results are presented n the next secton. B. Message schedulng The MPHL algorthm runs teratvely untl the message passng schedule s completed. There s no general message passng schedule to guarantee the convergence to exact margnals n a cyclc factor graph. However, a carefully desgned schedule can ensure the convergence for an specfc problem. In the problem of cooperatve localzaton, an schedule can be desgned n advance by analyzng the network connectvty, or t can be done dynamcally by each node [] []. A dynamc schedulng mechansm employed by the MPHL, as descrbed n the followng. ) Frst, only anchor nodes transmt ther locatons. ) Later, target nodes also transmt ther locatons f they have receved a certan number (γ) of messages n last teraton. ) The schedule ends when last nodes n the schedule transmtted a certan number (ν) of messages.

7 7 Settng γ = guarantees that a target node propagates ts locaton only f t can be unquely determned. However, fewer connectons, e.g., γ = or γ =, mght be suffcent for some nodes f both dstance and drecton data are avalable, as dscussed n Secton II. If a target node never receves γ messages n a sngle teraton, t may stll transmt f the total number of receved messages equal γ tot. Ths s necessary for cooperatve localzaton n sparsely connected networks. In ths paper, the value γ tot = γ s used to make sure that nodes wth nsuffcent connectvty are adequately delayed n jonng the schedule. The stoppng number ν s proportonal to the dameter (δ) of the network graph [6],.e., the greatest hop dstance between any par of nodes. In a cycle-free graph (tree), a dstrbuted LBP algorthm converges after δ teratons [6]. In our experments for loopy graphs, we set ν = δ. Hence, the length of the schedule (and the total number of teratons) depend on the network sze and connectvty. An mportant advantage of the proposed schedulng mechansm s that, t does not requre the message propagaton and belef update stages over the network to be synchronzed and centrally controlled. Hence, t allows for a fully autonomous and parallel executon of the localzaton algorthm over all the nodes. Although there s no theoretcal guarantee for convergence of a loopy belef propagaton algorthm, the numercal results show that the proposed schedulng mechansm provdes good convergence propertes n all the scenaros consdered. C. Numercal stablty Samplng from the posteror dstrbuton usng a fnteprecson arthmetcs n a computer can lead to numercal stablty ssues. Evaluatng a belef functon for samplng, whch ncludes products of many Gaussan and von-mses denstes, can easly overflow or underflow double precson arthmetc f the varances are very small. Underflow s a more severe problem, snce t can prevent Markov chan from movng towards the modes of the dstrbuton, unless ntal state s n a regon of hgh probablty. To mprove numercal stablty the Metropols-Hastngs algorthm s mplemented n logarthmc doman. Logarthmc transformaton turns factor products to summatons, helpng to avod underflows due to multplcaton of very small probabltes. D. Computaton complexty The most computatonally expensve part of the MPHL algorthm s samplng from posteror dstrbutons, whch runs at every node at every teraton. Hence, the overall computatonal complexty of the algorthm s determned by the complexty of the samplng mechansm. The followng para mpact the cost of samplng. ) Sample sze N: at each teraton every node draws a sample of sze N to construct ts outgong messages. Note that, constructng messages does not requre extra computaton snce the correspondng weghts are already calculated for samplng. ) Neghborhood sze N : Drawng a sngle sample requres evaluatng the product of ncomng messages. The number of messages depends on the neghbors of each node. N can be the largest neghborhood sze n the network. ) Message sze M: Each ncomng message s a sum of M functons (observaton factors) correspondng to M samples receved from a neghbor. Hence, drawng a sngle sample requres evaluatng O( N M) lkelhood functons. The complexty of samplng per teraton at a sngle node s O( N M N). Snce the algorthm can be dstrbuted over the network, each node can draws ts samples locally. The total computaton cost of the algorthm also depends on the number of message passng teratons requred untl convergence. The number of teratons n the schedule depends on the network confguraton and connectvty. The schedule length does not drectly grow by the network sze, but s proportonal to the network dameter (longest shortest path),.e., the maxmum hop dstance between any two nodes n the network. The experments show that, for the studed cases, the algorthm always converge n fewer than teratons. E. Communcaton complexty The complexty n communcaton,.e., the total amount of data transfered among the nodes, depends on the ) Message sze M: at each teraton a node transmts M samples to ts neghbors. ) Neghborhood sze N : messages from one node nclude dfferent factor values (weghts) for each of ts neghbors. Hence, the message sze depends on the number of ts neghbors. ) Neghborhood sze N : snce the nodes n a sngle neghborhood transmt messages over a shared medum (n wreless networks), the amount of resources used for communcaton depend on the neghborhood sze. Hence, the total complexty n communcaton per teraton s O( N M). Smlar to the computaton complexty, the total communcaton cost of the algorthm also depends on the number of message passng teratons requred untl convergence. V. Results Numercal results for dfferent stages of the algorthm are presented n ths secton. The results are produced usng a smulated network of nodes, from whch are anchor nodes. The network s partally connected,.e., only some dstance and drecton observatons are avalable. It s assumed that the observatons are recprocal. Fg. shows an example of a smple network confguraton. All dstances are n but they can be easly scaled, as long as the measurement model s vald. In ths example, there are 7 lnks between the nodes wth both dstance and drecton observatons. The standard devaton s. for dstance observatons and degrees for drecton observaton. These error levels can be acheved usng state of the art network technology, e.g., a mult-antenna WF system, and hgh resoluton estmaton algorthms [8], [9]. The results of the MPHL algorthm for nodes and 9 are shown n Fg. usng only dstance observatons, and n Fg. 6 usng both dstance and drecton observatons (hybrd). These results are for, 7, and

8 Anchor node Target node Fg. : Network model wth four anchors and two target nodes, three dstance observatons and two drecton observatons. teratons. The sample sze s, and the message sze s. That s, n each teraton every node draws a sample of sze, and propagates of them to ts neghbors. It s evdent from the network topology n Fg. that none of the target nodes can be relably localzed usng only anchor-target dstance observatons. However, as seen n Fg., cooperatve localzaton can provde accurate locaton estmates for all the nodes after few teratons. The samplng dstrbutons of the locatons change from spread-out mult-modal functons at teraton (Fg. a, Fg. d) to sharp sngle-mode denstes at teraton (Fg. c, Fg. f). Fg. 6a shows that all the nodes n the consdered scenaro may be unquely localzed usng only anchor-target dstance and drecton observatons (combned),.e., the belefs are unmodal. However, as seen n Fg. 6b the uncertanty n locaton estmates can be sgnfcantly reduced through cooperaton among the target nodes. The uncertanty regons sgnfcantly shrnk after only 7 teratons. Comparng Fg. and Fg. 6, shows that dstance and drecton data can be effcently combned to reduce the uncertanty n localzaton. 6 8 (a) Node, Iteraton 6 8 (d) Node 9, Iteraton 6 8 (b) Node, Iteraton (e) Node 9, Iteraton 7 Anchor node Target node 6 8 (c) Node, Iteraton Anchor node Target node 6 8 (f) Node 9, Iteraton Fg. : Samplng dstrbutons for target nodes and 9 at teratons, 7, and usng the MPHL algorthm wth only dstance observatons for the network confguraton n Fg.. The samplng dstrbutons at teraton are computed usng anchor-target observatons, as the target nodes are not transmttng any message yet. The have crcular shape because each of these target nodes (,9) s drectly connected to only one anchor. The samplng dstrbutons gradually concentrate around true target postons after each teraton,.e., the varance of the sample decreases. The samplng dstrbutons at teraton has sngle modes at approxmately true target locatons. The mprovement n locaton estmates after few teratons, s obtaned by cooperaton between the target nodes. 6 8 (a) Iteraton Anchor node Target node 6 8 (b) Iteraton 7 Fg. 6: Samplng dstrbutons for target nodes and 9 at teratons and 7 usng the MPHL algorthm wth hybrd dstance and drecton data. The samplng dstrbutons at teraton are computed usng anchor-target observatons, as the target nodes are not transmttng any message yet. These ntal dstrbutons are spread out (hgh uncertanty) because each of these target nodes s drectly connected to only one anchor. Each samplng dstrbuton concentrates around a true target poston after few teratons,.e., the varance of the sample decreases. The samplng dstrbutons at teraton 7 has sngle modes at approxmately true target locatons. The mprovement n locaton estmates after few teratons, s obtaned by cooperaton between the target nodes. or non-hybrd methods such as satellte-based localzaton. For example the localzaton performance degrades f (a) three or more nodes (especally anchors) le on a sngle lne, (b) many nodes are outsde the convex hull of the anchors, (c) the spatal dstrbuton of the nodes s very nonunform. The mpact of network geometry on localzaton performance can be partally llustrated usng geometrc dluton of precson (GDOP) plots [7]. In order to control for the mpact of network geometry, multple random confguratons are studed. In each confguraton, nodes are placed randomly n a m m area, wth a constrant of m mnmum separaton between nodes, and four anchor nodes are randomly selected. Both dstance and drecton observatons are avalable recprocally for every connected par of nodes. Wth nodes, n a fully connected network there are connecton. However, the followng results are produced for partally connected networks. A par of nodes are connected f they are closer than 6 m. The resultng average connectvty s about 8% n the smulated networks,.e., 6 connectons n average. Multple realzatons of observaton error s used to study each network confguraton. Fg. 7 shows an example of a random network realzaton. The performance crtera for evaluatng the results Anchor node Target node Fg. 7: Network model wth four anchors and two target nodes, three dstance observatons and two drecton observatons. 9 The qualty of localzaton results always depends on the network geometry, although the proposed approach makes t less dependent on the geometry compared to non-cooperatve s the average localzaton error versus the standard devatons (STDs) of dstance or drecton estmates (observaton error). The average localzaton error s the Eucldean dstance be-

9 9 tween the estmated and actual target locatons averaged over all the target nodes n dfferent network confguratons wth multple realzatons of observaton error. The STDs of dstance and drecton estmates depend on dfferent para of the measurement system and the envronment, ncludng the receved sgnal-to-nose rato (SNR), sgnal bandwdth, the sze of a measurement packet (symbols), the number of antenna elements, the severty of multpath propagaton, etc. The relatonshp between each term, e.g., SNR, and the observaton error may be establshed usng TOA and DOA estmaton technques or performance bounds [], [], [6]. Fg. 8a shows the average localzaton error n the network for the MPHL algorthm at dfferent teratons usng dstanceonly, drecton-only and hybrd observatons. The algorthm converges very fast and the localzaton error decreases monotoncally. The algorthm converges n after only teratons. In ths scenaro, the average localzaton error s reduced more than % usng hybrd observatons compared to the dstance-only and drecton-only cases. The standard devaton of samplng dstrbuton (sum for all varables) s plotted n Fg. 8b. It shows that the uncertanty n locaton s decreasng over tme and the hybrd localzaton provdes more relable results. Avg. localzaton error. MPHL (range) MPHL (angle) MPHL (hybrd) 6 Number of teraton (a) Est. STD of locatons.7. MPHL (range) MPHL (angle) MPHL (hybrd) 6 Number of teraton Fg. 8: Average localzaton error (a) and standard devaton of samplng dstrbuton (b) versus number of teratons. STD of observatons are. for dstances and degrees for drectons. Fg. 9a shows the average localzaton error versus STD of dstance estmates. The STD of drecton estmaton s degrees n ths smulaton. The results of the proposed MPHL algorthm s compared wth two recent hybrd localzaton algorthms, SDP_Tomc [] and CLORIS [], [8]. These are the only two methods n the lterature appled to hybrd data (dstance and drecton) for cooperatve localzaton. Both the competng algorthms combne TOA and DOA observatons usng convex relaxaton,.e., the sem-defnte programmng (SDP) relaxaton (SDP_Tomc) and second-order cone programmng (SOCP) relaxaton (CLORIS). Message schedules for the dstrbuted versons of SDP_Tomc and CLORIS methods are not descrbed n the correspondng artcles. Hence, both the algorthms are mplemented as centralzed convex optmzaton methods usng the CVX package [9]. The proposed MPHL algorthm outperform both SDP_Tomc and CLORIS methods at a wde range of observaton error. At hgher levels of rangng error (STD of ), the average (b) localzaton error of the proposed MPHL algorthm s less than % of the competng methods. Fg. 9b shows the average localzaton error of MPHL, SDP_Tomc and CLORIS algorthms versus STD of drecton estmates. The STD of dstance estmates s. m n ths smulaton. The SDP_Tomc performs better at hgher levels of dstance estmaton error, whle the CLORIS has ts best performance at lower levels of angle estmaton error. The results show that the MPHL algorthm outperforms both SDP_Tomc and CLORIS at almost every error level. In these scenaros, the MPHL provdes up to % of reducton n localzaton error compared to the competng methods. The better performance of SDP_Tomc method compared to the MPHL at drecton estmaton STD of m s due to the centralzed processng n SDP_Tomc compared to the dstrbuted fashon of MPHL. Moreover, the results of the MPHL may be mproved by ncreasng the number of partcles. Fg. shows the average localzaton error Avg. localzaton error MPHL (hybrd) SDP Tomc CLORIS STD of range observaton (a) Avg. localzaton error. MPHL (hybrd) SDP Tomc CLORIS STD of angle observaton Fg. 9: Average localzaton error (a) versus dstance estmaton error and (b) versus drecton estmaton error, n a partally-connected network. Number of teratons s. STD of angle observatons s degrees n (a). STD of dstance estmaton s. n (b). The proposed MPHL algorthm outperforms the competng hybrd localzaton methods at a wde range of observaton error. versus STD of dstance estmates for the proposed MPHL algorthm, and metrc MDS [], SDP_Tomc [], and CLORIS [], [8] algorthms. The STD of drecton estmates s degrees. These results are produced usng the same randomly generated node confguratons as above, but wth full network connectvty. The metrc MDS requres all parwse dstance observatons to be avalable. It s shown that n ths scenaro, the MDS algorthm fnds the optmal soluton for dstance-only localzaton [, chap. 7]. The results show that the MPHL algorthm outperforms the competng hybrd localzaton methods and the metrc MDS algorthm at a wde range of observaton error. The average localzaton error of the MPHL s below the dstance estmaton error; and t s % % smaller compared to the competng algorthms. The mprovement over the dstance-only MDS comes from usng hybrd observatons. If dstance estmates are suffcently accurate, e.g., at STD of., the MDS performs better than the SDP_Tomc and CLORIS methods. However, the MPHL algorthm outperforms the MDS at every scenaro by effcently combnng dstance and drecton data. (b)

10 Avg. localzaton error MPHL (hybrd) SDP Tomc MDS (range) CLORIS STD of range observaton Fg. : Average localzaton error versus dstance estmaton error, n a fullyconnected network. STD of drecton estmates s degrees. The proposed MPHL algorthm outperforms the competng hybrd localzaton methods and the metrc MDS algorthm at a wde range of observaton error. VI. Conclusons In wreless networks, ncludng WF and cellular networks, a relable, affordable, and accurate postonng servce s crucal. Due to the avalablty of both dstance and drecton nformaton n modern wreless systems, t s essental to combne these two sensng modaltes for wreless localzaton. The man contrbutons of ths paper were: (a) the problem of cooperatve network localzaton usng hybrd dstance and drecton data was statstcally modeled; and (b) the novel Message Passng Hybrd Localzaton (MPHL) algorthm was proposed to solve t. It s a fully dstrbuted algorthm employng a novel schedulng mechansm. It effcently combnes the dstance and drecton data to fnd approxmate MMSE estmates of target locatons. Numercal results were provded to show the mprovement n localzaton performance compared to exstng dstance-only and hybrd localzaton methods. For example, n the studed fully-connected networks of nodes ( anchors) wth error varances of m for dstances and for drectons, the average localzaton error of the MPHL s about cm compared to 7 9 cm for the competng algorthm. Possble drectons for future work are to extend the algorthm for: (a) jont synchronzaton and localzaton, (b) jont estmaton of locatons and orentatons of the nodes, and (c) to study and mprove the schedulng mechansm for message passng. References [] S. Dhar and U. Varshney, Challenges and busness models for moble locaton-based servces and advertsng, Communcatons of the ACM, vol., no., pp. 8,. [] N. Patwar, J. Ash, S. Kyperountas, I. Hero, A.O., R. Moses, and N. 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