Tracking Algorithms Based on Dynamics of Individuals and MultiDimensional Scaling

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1 Trackng Algorthms Based on Dynamcs of Indvduals and MultDmensonal Scalng Jose Mara Cabero, Fernando De la Torre, Galder Unbaso and Artz Sanchez ROBOTIKER-TECNALIA Technology Centre, Telecom Unt, Zamudo, Bzkaa, 4817, Span Emal: Carnege Mellon Unversty, Pttsburgh, PA 1513 Emal: Abstract Accurate locaton of people s a key aspect of many applcatons such as resource management or securty. In ths paper, we explore the use of rado communcaton technologes to track people based on ther dynamcs. The network conssts of two types of rado nodes: statc nodes (anchors) and moble nodes (ndvduals). From a set of sparse dssmlarty matrces wth nformaton about proxmty or estmated dstances between nodes and ndvduals dynamcs at each tme nstant, we nfer ndvduals trajectores. Dependng on the nformaton avalable, two algorthms are proposed: Dynamc Weghted Multdmensonal Scalng wth Bnary Flter (DWMDS-BF) and Dynamc Weghted Multdmensonal Scalng based on Dstance Estmatons (DWMDS-DE). DWMDS-BF s an algorthm that mplements a Bnary Flter functon that obtans very good trackng results when only connectvty nformaton s avalable and DWMDS-DE s desgned for those networks where a good estmaton of dstances between nearby nodes s avalable. Both algorthms mplement a dynamc component that regularzes the obtaned trajectores accordng to ndvduals dynamcs. Extensve smulatons show the effectveness and robustness of the proposed algorthms. I. INTRODUCTION Short and medum range rado communcaton technologes, due to ther cheap cost, are beng ncluded n almost all personal electronc devces, such as moble phones, laptops or PDAs. The wdespread use of these devces makes them deal platforms for locaton-aware applcatons. Unlke specfc trackng technologes such as GPS or those based on ultrasound or mage processng, the man purpose of these technologes s not trackng but communcaton between devces. The most promsng current trend s usng technques that can be appled to almost any rado devce and that are based on features of the rado communcaton technologes, lke the Receved Sgnal Strength (RSS). Most trackng algorthms consder tracked objects as generc denttes, usually called nodes, where no node s dfferent from any other node n the network. They are ether consdered statc nodes or nodes n moton followng synthetcally generated trajectores (usually random), what leads to non-realstc stuatons, such as networks wth thousands or even mllons of nodes wth smlar behavor. One of the man contrbutons of our work s a characterzaton of the moble nodes accordng to ther dynamcs. Although ths characterzaton could be done for almost any knd of network, we are especally nterested n socal networks,.e. networks where the nodes to be tracked are people, wth dfferent dynamcs and patterns. We present a new approach for the trackng procedure as a two-step problem: one dependent on the technology used and the other one dependent on the partcular nature of each moble. We show the effectveness of usng both aspects workng together. The man contrbutons of ths work are: Dynamc Weghted Multdmensonal Scalng wth Bnary Flter (DWMDS-BF), a trackng algorthm to be used on networks where only connectvty nformaton between nodes s avalable. DWMDS based on Dstance Estmatons (DWMDS-DE), a trackng algorthm to be used on networks where dstance estmatons between nodes are avalable. A mathematcal expresson (called Dynamc term along the artcle) that uses the learned moble nodes moton patterns to smooth the trackng soluton accordng to ther partcular dynamcs. Indvdual s dynamcs means ndvdual s speed n ths artcle. A Bnary Flter functon to handle those scenaros where only connectvty nformaton between nearby nodes s avalable. The rest of the paper s organzed as follows: secton II revews prevous work and secton III formulates DWMDS algorthms. The correspondng experments and comparson wth other locaton algorthms are reported n Secton IV. Secton V summarzes the conclusons and dscusses future research trends. II. PREVIOUS WORK Most popular methods to locate people are based on measurements of rado sgnals, such as Tme of Arrval (ToA) [1], Angle of Arrval (AoA) [], Tme Dfference of Arrval (TDoA) [3] and Receved Sgnal Strength (RSS) [4]. The frst three technques need costly customzed hardware whereas RSS s the most attractve one because of the varety of personal rado communcaton devces that cheaply and by default mplement t. The frst ones are better technques to obtan accurate estmaton of the dstances, whereas RSS s prone to errors due to the complexty of the rado channel [5]. In spte of the dffcultes to model the rado channel, some works based on RSS use trlateraton [4], multlateraton [6] or smlar methods [7], [8] to make an estmaton of the dstances between the tracked object and some known /8/$5. 8 IEEE 388 Authorzed lcensed use lmted to: Carnege Mellon Lbrares. Downloaded on September 18, 9 at 17:31 from IEEE Xplore. Restrctons apply.

2 anchors. However most RSS-based methods do not try to estmate dstances drectly from RSS, but consst of a prevous measurement phase where a RSS map of the scene s bult [4], [8], [9], [], [11]. Ths methodology s extremely dependent on the envronment and any sgnfcant change to the topology mples a costly new re-calbraton. In the context of sensor networks, classcal MDS [1] has been used to locate statc sensor nodes n dense wreless sensor networks [13], [14], [15], [16]. When the networks are more sparse, the accuracy quckly decreases. These prevous works rely on hop counts and shortest path measurements to estmate the dstances between nodes needed for classcal MDS, whch leads to poor results n non-unform networks [15]. In order to solve ths drawback, both [14] and [16] have a prevous phase where the network s splt n subnetworks that locally apply shortest path measurements and classcal MDS (the error due to non-unform networks s reduced but not removed), and that are fnally merged to get the resultant network. These methods assume contnuous communcaton between the nodes n the network to transmt nformaton of the state of the network, whch mply a hgh communcaton cost n terms of bandwdth and energy consumpton. Our approach only needs to dentfy the nearby nodes, whch s provded by default by most rado communcaton technologes, so the communcaton cost s neglgble. The work reported by [17] stresses the mportance of reducng the communcaton cost, tryng to reduce t wth a technque whch chooses adaptvely a neghborhood of nodes, applyng MDS locally and transmttng the updates to the neghbors. The best results n terms of accuracy are reported by the MDS-MAP(P,R) algorthm [16] usng classcal MDS and shortest path measurements as a startng pont followed by a subsequent optmzaton phase based on least squares mnmzaton. III. TRACKING AS A LOW DIMENSIONAL EMBEDDING PROBLEM A. Problem Formulaton We approach the trackng problem from two dfferent perspectves dependng on whch nformaton s avalable: connectvty nformaton or estmated dstances between connected nodes. Two nodes and j are consdered connected or neghbors f and only f node s nsde the coverage radus of node j and vce versa. In both scenaros, the network conssts of moble and statc nodes called anchors used as reference to obtan the trajectores of the moble nodes. The network can be represented as a graph wth vertces V and edges E (G=(V,E)), where the vertces are ndvduals postons at each tme nstant, and the edges jon connected nodes at that tme nstant. The value of the edge s 1 n the connectvty scenaro and the correspondng estmated dstance n the dstance scenaro. From now on the term dssmlarty wll be also used to address the value of the edges, no matter the scenaro descrbed. Once the dssmlartes are gathered through tme, they are used together wth ndvduals dynamcs as nput of DWMDS algorthm to obtan ther trajectores. Dssmlarty at one tme nstant between two nodes n the network s recorded accordng to the followng procedure: f a node s connected to a node j, thej and j terms of a dssmlarty matrx are set to 1 (connectvty scenaro) or to the estmated dstance between them (dstance scenaro). If both nodes are not connected, they are set to no matter the scenaro (see fgure 1). Nodes and j can be anchors or ndvduals. As secton IV wll show, not only the anchors work actvely n the trackng process but the nodes n moton help too. Consderng fgure 1: f node A were an anchor and node B and C nodes n moton such that at tme t are at those postons, then even although there s not a drect connecton between nodes A and C, the node B acts as a brdge between them, and our trackng system wll take advantage of t. Fg. 1. Connectvty scenaro: node B detects A and C. C and A are out of range, so they do not detect each other. Colored crcles represent the correspondng coverage areas. DWMDS-BF s the algorthm for the scenaros wth connectvty data, the mnmum nformaton avalable n almost all communcaton networks. DWMDS-DE can be appled whenever dstance estmatons are avalable, no matter the technque used to get them. Its accuracy depends on the accuracy of the technque used to estmate the dstances. Both approaches are analyzed n the followng sectons, although we especally focus on the scenaro wth connectvty data, whch s more wdely used and realstc n terms of assumptons. B. Dynamc Weghted MultDmensonal Scalng (DWMDS) Multdmensonal scalng (MDS) [1] s a powerful statstcal dmensonalty reducton technque for data analyss whch s extensvely used n socal scences, engneerng and marketng. The startng pont of MDS s a matrx consstng of parwse dssmlartes between data samples n the orgnal space. MDS attempts to fnd an embeddng n a metrc space, so that the dstances n a low-dmensonal space correspond to the gven dssmlartes between samples n the orgnal space. Let 1 y 1,..., y n be the samples n the orgnal space, and δ j the correspondng dssmlartes between sample and j n that space. Let x 1,..., x n be the coordnates of the samples 1 Bold non-captal letters are used to denote vectors. Bold captal letters are used to denote matrces. All non-bold letters wll represent varables of scalar nature. d j denotes the scalar n the row and column j of the matrx D. The number of moble devces s p, the number of statc devces s q, T s the number of tme nstants. The superndexes s and d correspond to the statc and moble nodes respectvely. sd, dd and ss represent the terms statc vs moble nodes, and moble vs moble nodes and statc vs statc nodes respectvely. Authorzed lcensed use lmted to: Carnege Mellon Lbrares. Downloaded on September 18, 9 at 17:31 from IEEE Xplore. Restrctons apply. 389

3 n the embedded space and d j the correspondng dstance between sample and j n that space. The man goal of MDS s to fnd an embeddng (.e. x 1,..., x n ) such that d j n the low-dmensonal space s as close as possble to the orgnal dssmlarty n the orgnal space δ j n the least square sense. It s not usually possble that d j = δ j, j, and t s common to fnd a unque soluton by averagng the least square error usng dfferent normalzaton errors such as the ones n equaton 1. A local mnmum of the these error functons w.r.t X =[x 1,..., x n ] s usually found by usng standard gradent technques [18]. The general expresson of the error functon of the DWMDS algorthms conssts of two terms, the Statc one that uses dssmlartes between nodes and the Dynamc one that comprses the dynamcs of the moble nodes n the network (frst and second terms respectvely n equaton ). The nput to DWMDS algorthms wll be a set of matrces wth dssmlarty nformaton δj t between nodes at each tme nstant and the prevously learned ndvduals dynamcs. The fnal goal s to obtan the coordnates of the nodes through tme that mnmze the error functon. Ω 1 (X) = <j (d j δ j ) <j ; Ω (X) = (d j δ j ) ; δ j <j δ j Ω 3 (X) = ( ) dj δ j. (1) δ <j j Let us denote the coordnates of the nodes (ndvduals and anchors) n the network as: 1 x s X d,t 1 =, X. s x s =., x s r q = { 1,xd,t } x s = {xs 1,xs } R 1 x s R1 X d,t R r X s R q, where X d,t corresponds to the coordnates of the moble nodes at tme nstant t. X s are the tme nvarant coordnates of the statc nodes. x 1 and x are the x and y coordnates for the devce n the bdmensonal plane. Assumng the algorthm gathers data over T tme nstants: X d = ( X d,1 X d,... X d,t ) R r T, Ω DWMDS (X ALL )= + T t=1 T n <j j= p t= =1 α m t j (f t j δt j ) () ( ) 1 F, where at any tme nstant: + <j <j m j (f j δ j ) = <j m ss ss j (fj δss j ) m dd j (d dd j δj dd ) + m sd j (fj sd δj sd ). <j X ALL comprses the coordnates of the trajectores X d and the poston of the anchors X s. Insde the Dynamc term, α s a tradeoff parameter dependent on the moble nodes velocty to equlbrate ( the contrbuton ) of the dynamcs of each moble node. 1 s the squared Frobenus norm of F the dfference between the poston of a moble node n two consecutve tme nstants,.e. the squared dstance covered between consecutve tmes nstants (l ).TheDynamcterm s the same for all DWMDS algorthms, however accordng to the nature of the dssmlartes, the Statc term changes, generatng two dfferent expressons of DWMDS that are explaned below (See table I). 1) Estmated dstances as dssmlartes. DWMDS-DE algorthm: n those scenaros where estmated dstances are avalable, δj t and fj t are the estmated dstance and the unknown Eucldean dstance (dependent on the coordnates of the and j nodes) respectvely between nodes and j at tme t. m t j = wt j where w t δj t j s a weght that stresses the dfference between fj t and δt j. The more accurate the estmated dstances are the more accurate DWMDS-DE s. So far, the scenaros where only RSS nformaton s avalable are not relable enough to accurately estmate dstances between nodes, so a connectvty-based approach s more adequate for these networks. ) Connectvtes as dssmlartes. DWMDS-BF algorthm: n scenaros when only connectvty nformaton s avalable, δj t and f j t corresponds to the bnary connectvty nformaton (1 f both nodes are connected and otherwse) and a Bnary Flter functon respectvely, between the node and j at tme t. Herem t j = wt j. DWMDS-BF reles only on connectvty data, nformaton perfectly avalable from most rado technologes, whch makes ths DWMDS varant very attractve. Whle the expresson of f t j n DWMDS-DE s the Eucldean dstance between two nodes at tme t, the expresson used n DWMDS-BF corresponds to a Bnary Flter functon dependent on three parameters: the Eucldean dstance between nodes, the coverage radus, and a control parameter β that s used to change the slope of the Bnary Flter functon. Fgure shows the Bnary Flter functon versus the dstance between nodes (d j ) for a coverage radus R =for dfferent values of β. The Statc term penalzes the dfference between the connectvty value (1 f connected, otherwse) and the Bnary Flter functon whch deally works n the followng way: f the coverage area of any node were a crcle of radus the nomnal coverage radus (R) specfed by the manufacturer, the output of ths flter would be 1 when the dstance between nodes and j s less than R (connected nodes),.5 f t s equal to R and otherwse (dsconnected nodes). However, the Authorzed lcensed use lmted to: Carnege Mellon Lbrares. Downloaded on September 18, 9 at 17:31 from IEEE Xplore. Restrctons apply. 39

4 Bnary Flter functon Fg DWMDS-DE DWMDS-BF m t j wj t δ t j w t j f t j d t j e β(dt j R) e β(dt j R) +1 TABLE I STATIC TERM PARAMETERS R = β =.1 β =. β =.3 β =.4 β =.5 β =.6 β =.7 β =.8 β =.9 β = 1 β = 3 β = β = Dstance between nodes Bnary Flter functon for dfferent values of β parameter. electromagnetc envronment s far from beng deal [19], so the queston s when to consder two nodes connected. In a real scenaro, the complex nature of the electromagnetc feld can generate stuatons where two nodes separated by a dstance d j < R were not connected or even f they were located farther than R, they could detect each other. The trckest zone s the one around the nomnal coverage radus ( n fgure ) where the value of the functon changes abruptly. Far from the nomnal radus and for a fxed value of β, the value of the functon grows when the dstance between nodes tends to (1 s the lmt) and tends to when the dstance between nodes tends to nfnty. The slope of the Bnary Flter functon and consequently ts output s controlled wth the β parameter. Hgh values of β mean steep slopes,.e. almost deal scenaros, whereas low values of β have the contrary effect, wth flatter slopes, for characterzng very unsteady scenaros. Secton IV-A shows the mpact of the β parameter n medum scenaros, nether especally nosy nor steady. C. Optmzaton phase Once DWMDS expressons are defned, the algorthm enters an optmzaton phase based on a gradent descent technque to fnd the optmal soluton of equaton w.r.t. X ALL n,whchs X ALL at teraton n. In ths paper, we assume that X s s known and we do not update t. The gradent updates are gven by: where Ω DW MDS X ALL n X ALL n+1 = X ALL n η Ω DW MDS X ALL n, (3) s a unt vector n the drecton of the gradent. One major problem wth the update of equaton 3 s to determne the optmal η. In our case η s determned usng a lne search strategy [18]. D. Weghts, Dynamc Term, local mnma, ntalzaton, computatonal cost and connectvty degree 1) Adjustment of weghts and the mportance of the Dynamc term: weghts wj t and α have to fulfll two requrements: one regardng the term they are workng on and the other as a tradeoff parameter between the Dynamc and Statc terms. Focusng on the Statc term, wj t wll have a hgher value when the correspondng j term s a term of hgh confdence, otherwse ts value wll be low. In DWMDS-DE, f all the estmated dstances δj t have the same level of confdence (all of them have been estmated n the same way, whch s usual), they wll have the same weght, except for those tme nstants when nodes are not connected, when there are not estmated dstances, and consequently the correspondng weghts are. Unlke DWMDS-DE, DWMDS-BF takes nto account those postons out of the coverage radus ( output n the flter), so weghts wll have the same value n and out of the coverage radus. The terms n the dagonal = j are always n both algorthms. The α weghts work on the Dynamc term n such a way that the gradent descent technque respects each moble node s dynamcs. Every moble node wll have a specfc speed v, that lets cover a specfc dstance l between two consecutve tme nstants (the dfference between two consecutve tme nstants s the tme step Δt). Assumng that every moble node has a constant velocty n moton (whch s pretty accurate when the moble nodes are people walkng), the relaton between the dstance covered by two dfferent moble nodes and j n Δt s l j = vj v l. If we consder α equal for all and apply gradent descent method (see secton III-C) to mnmze the error expresson (equaton ), then the fastest nodes (nodes wth larger values of l ), would have more mportance n ths mnmzaton process. In order to compensate for that effect and treat all the nodes n a democratc way we consder α such that α = α v k where v k = max{v } ɛp. (4) v We show an example wth two nodes durng T tme nstants, where node 1 s 3 tmes faster than node (α =3α 1 ): T ( l = ( α t= =1 ) 1 F t,, ) 1 = F α 1(T 1)l1 + α (T 1)l = α (T 1)(l 1 +9l ), where l1 =9l, so no matter the speed of the moble nodes, the gradent descent technque wll treat them smlarly, and respect ther dfferent dynamcs. Fgure 3 shows the dstorton n the obtaned trajectores when there are two anchors and two lnear trajectores wth dfferent speeds that last T tme nstants, but whose dynamcs are consdered the same (α 1 = α ). α parameter n equaton 4 s a tradeoff parameter between the Dynamc and Statc term, that equalzes the value of both terms so that both of them have a smlar weght n the general stress expresson n equaton. In ths artcle we deal wth synthetc trajectores, so we assume that the dynamcs of Authorzed lcensed use lmted to: Carnege Mellon Lbrares. Downloaded on September 18, 9 at 17:31 from IEEE Xplore. Restrctons apply. 391

5 Fg. 3. Left: ground truth. Rght: obtaned trajectores when α 1 = α. each moble node s known. In real testbeds, there wll be a short learnng stage where the dynamcs of every person n the network s modeled. The procedure s straghtforward and conssts of measurng the average tme t takes for every ndvdual to cover a known dstance between two anchors. Ths can be measured automatcally as the tme between the connecton wth one anchor (startng tme) and the connecton wth the other (stop tme). The Dynamc term s especally necessary n those networks that are very sparse, where there are gaps wth solated nodes (nodes not connected to any other nodes). Ths term acts nterpolatng the poston of the nodes n those out-of-coverage parts of the network. When the network becomes more connected, the mportance of the Dynamc term vares dependng on the scenaro: When only connectvty data s avalable (DWMDS-BF), the Dynamc term s always useful no matter the topology of the network. The reason s that n addton to the connectvty gap fller functon, ths term s necessary to counteract and smooth the otherwse abrupt resultant trajectores from the Statc term. When estmated dstances are avalable (DWMDS-DE) and the network becomes more connected (fewer connectvty gaps), the obtaned trajectores converge gradually wth and wthout the Dynamc term. A dfferent approach s to consder the dssmlarty matrces as the degrees of freedom of the network, so the more connected the network s (the matrces are less sparse because more estmated dstances are avalable), the smaller the ambguty s, and consequently the less necessary the Dynamc term s. The extreme stuaton s when the coverage radus of the nodes covers the entre network (dssmlarty matrces totally full), then the DWMDS-DE algorthm has smlar results wth and wthout the Dynamc term and the accuracy of the obtaned trajectores s % dependent on the error n the estmated dstances. ) Local mnma: whle DWMDS-BF algorthm s pretty robust, DWMDS-DE can converge to local mnma due to errors n the estmated dstances. Such errors make the gradents of the Statc and Dynamc terms of equaton try to reduce the error n opposte ways, whch makes the algorthm unpredctable. Fgure 4 s an example of such a stuaton, where a moble node, gong from anchor A 1 to A and consequently at t +1 s farther from anchor A 1 than at tme t, seems to be nearer due to the error n the estmated dstance. The Dynamc term gradent would try to follow the trajectory accordng to the learned dynamcs from node A 1 to node A, whle the Statc term would try to take the moble node backwards. Fg. 4. Up: dotted crcles represent estmated dstances. Blue crcles represent coverage areas. Down: cross drecton of the gradent of the Statc term over the real postons because of the error n the estmated dstances. The DWMDS-DE performs well when the errors n the estmated dstances wth respect to the real ones are smaller than the dstance covered by the correspondng moble nodes durng a tme step Δt, what makes the Dynamc and Statc term gradent agree. In very well or full connected networks (nodes connected to all the rest of moble nodes), the Statc term predomnates over the Dynamc one, and the contradctory effect dsappears. 3) Connectvty degree, ntalzaton and computatonal cost: connectvty degree s the metrc used to measure the average number of connectons per node offerng a quck vew of the densty of the network. As we ntroduce a dynamc term that lnks dfferent tme nstants to do the trackng procedure, the connectvty degree n ths paper does not measure only the average connectvty at a tme nstant, but throughout the whole set of tme nstants n the survey. Equaton 5 shows how to calculate the connectvty degree (CD Total) of a network, where γj t s 1 f nodes and j are connected at tme nstant t, otherwse. CD Total = T n t=1 =1 n j=1 j = γ t j nt The ntalzaton of the algorthm s random, although any nformaton a pror regardng moble nodes postons could be used to make the algorthm converge faster. The computatonal cost of the algorthm s O(n 3 T ) for the Statc term and O(p T ) for the Dynamc term, where n s the total number of nodes, p s the number of moble nodes and T s the trackng tme (number of tme nstants). IV. PERFORMANCE EVALUATION AND COMPARISON In ths secton, we report extensve smulaton results of the performance of DWMDS-BF and DWMDS-DE n the followng scenaro: a square of r x r (r s the reference unt used durng the experments) where there are a varable number of anchors and moble nodes wth random trajectores and speeds. Every moble node moves accordng to a constant speed durng the trackng tme. The velocty range s: (5) Authorzed lcensed use lmted to: Carnege Mellon Lbrares. Downloaded on September 18, 9 at 17:31 from IEEE Xplore. Restrctons apply. 39

6 [.1r, r] v ɛ ɛp Δt Random trajectores have a duraton of T = tme nstants. The postons of the anchors are known and the goal s to obtan the trajectores of the moble nodes. The reconstructon error n the followng sectons s the mean error of the dfference between the obtaned trajectores from the algorthm and the ground truth, unless otherwse stated. The coverage radus (R) s the same for all the nodes, unless otherwse stated. All the smulatons are done wth MATLAB 7.1 (R14SP3). A. Scenaro wth connectvty data: DWMDS-BF algorthm Secton III-B remarks the mportance of the β parameter n the performance of the Bnary Flter functon. The goal of ths parameter s to adapt the flter to the stuaton expected n a real scenaro where the coverage radus s not the same for every node, even dfferent through tme accordng to changes n the surroundngs. To analyze the mpact of the β parameter n the performance of the flter and consequently n the whole trackng algorthm, ths secton analyzes the error of the obtaned trajectores n a medum scenaro (nor very nosy nether very steady) when the β parameter changes. The nose n the scenaro s ntroduced by modelng the coverage radus of every node as a normal dstrbuton whose mean s the nomnal coverage radus (usually provded by the manufacturer of the rado devce and consdered.5r n ths experment) and whose standard devaton s % of ths nomnal coverage radus. The coverage radus changes also through tme accordng to ths normal dstrbuton. Fgure 5 shows the best results (and very smlar) for β values that make the flter nether very flat nor very steep (β =.3,1). Values of the β parameter that make the slope very steep (β = ) or flat (β =.1) become n larger reconstructon errors. Although ths s the general behavor, there are dfferences dependng on the network confguraton: For a constant number of anchors: Hgh number of anchors (upper left corner graph): no matter the number of moble nodes, the dfference n the error due to β changes remans pretty constant. Low number of anchors (lower rght corner graph): the reconstructon errors obtaned wth dfferent β values are very smlar n sparse networks (they almost converge for very sparse networks) and the dfferences between them reman pretty constant when the network becomes more connected. From now on the smulatons are carred out wth β =1. Fgure 6 shows the reconstructon error n DWMDS-BF when the connectvty degree changes due to changes n the coverage radus and the number of nodes. For nstance, the lower left corner shows and scenaro wth anchors and a varable number of moble nodes when the coverage radus changes from 1.5r to.5r n.5r steps. Gven a constant number of moble nodes, the colored area shows the reconstructon error when the connectvty degree changes 8 6 β =.1 β =.3 β =1 β = Anchors 3 Moble Nodes Anchors 3 Moble Nodes Anchors 3 Moble Nodes 8 6 Anchors 3 Moble Nodes Fg. 5. Reconstructon error when β =.1,.3, 1, due to changes n the coverage radus. The results show that whenever the coverage radus ncreases, the accuracy s hgher. When the number of nodes ncreases (statc or moble ones), the accuracy also ncreases, wth a hgher mprovement when the nodes are anchors, whch was expected because they are known statc references. B. Scenaro wth estmaton of dstances: DWMDS-DE algorthm In Ths secton we focus on the performance of DWMDS- DE when the avalable estmated dstance between nodes are not accurate enough and the trackng algorthm suffers from the perncous effect detaled n secton III-D.The upper graph of fgure 7 compares the performance of DWMDS-BF and DWMDS-DE n networks where the estmaton of the dstances follows a normal dstrbuton whose mean s the real dstance between nearby nodes wth standard devaton 5% of ths real dstance, and the coverage radus used to calculate the connectvtes n DWMDS-BF changes accordng to a normal dstrbuton of mean.5r and standard devaton 5% of ths radus. When the coverage radus s.5r, the errors n the estmated dstances are larger than some nodes speed, happenng the effect explaned n secton III-D, and consequently the results obtaned wth DWMDS-BF are even better than the ones obtaned wth DWMDS-DE. When the rado coverage s enough to cover almost the whole network (very well connected network), the Statc term becomes more mportant than the Dynamc one and takes over the optmzaton phase. Then the contradctory effect dsappears and the results wth and wthout Dynamc term converge (pcture at the bottom of fgure 7). C. Comparson wth other trackng algorthms Most proposed trackng algorthms [13], [14], [15], [16], formulate the trackng problem as a sequence of ndependent tme nstants, takng advantage of the hgh densty of nodes n the network to nfer the poston of the nodes at each tme nstant. When the network s not very dense, or t does not have unformty, these methods lose accuracy exponentally. Unlke Authorzed lcensed use lmted to: Carnege Mellon Lbrares. Downloaded on September 18, 9 at 17:31 from IEEE Xplore. Restrctons apply. 393

7 Moble Nodes Moble Nodes Anchors 3 7 Anchors Anchors 3 6 Anchors Moble Nodes Moble Nodes 1 1 Anchors Anchors Anchors Anchors Moble Nodes Moble Nodes 1 1 Anchors Anchors Total Connectvty Degree 6 Mean Error (%r) Anchors Anchors Total Connectvty Degree Mean Error (%r) Fg. 6. Mean reconstructon error for dfferent network confguratons and coverage radus. Each fgure represents networks wth a varable number of moble nodes and the same number of anchors. prevous works, the localzaton system proposed on ths paper, takes advantage of ndvduals dynamcs and approaches the trackng problem as a whole pcture, correlated on tme accordng to each ndvdual s dynamcs. Unlke locaton methods such as MDS-MAP, MDS-MAP(P) and MDS-MAP(P,R), DWMDS algorthms are not based on shortest path measurements, so there s not communcaton cost between nodes (wth the correspondng savngs on battery and bandwdth), and ther performance s the same no matter the dstrbuton of the nodes n the network (unform or non-unform networks). The computatonal cost domnant term s O(n 3 T ) n DWMDS and MDS-MAP(P,R), the method that obtans the best results of the MDS-MAP famly. DWMDS algorthms obtan better results when the number of anchors ncreases, whle the other methods do not mprove the performance above anchors. DWMDS algorthms mprove ther performance when the rado coverage ncreases, whle the MDS-MAP algorthms do not get better performance from a crtcal pont. Fgure 8 shows the medan reconstructon error of four dfferent trackng methods, MDS-MAP, MDS-MAP(P), MDS- MAP(P,R) and DWMDS-BF, n two scenaros where only connectvty data s avalable: a random-unform network of nodes and a random-non-unform network that conssts of 16 1 Radus.5r Moble Nodes 14 1 Radus r Anchors DWMDS DE 5 Anchors DWMDS DE Anchors DWMDS CI 5 Anchors DWMDS CI Anchors DWMDS DE wth Dynamc Term 5 Anchors DWMDS DE wth Dynamc Term Anchors DWMDS DE wthout Dynamc Term 5 Anchors DWMDS DE wthout Dynamc Term Moble Nodes Fg. 7. Up: scenaros wth nosy estmated dstances. Down: performance of nosy DWMDS-DE wth and wthout Dynamc term. Medan Reconstructon Error (%r) DWMDS BF anchors 19 people DWMDS BF 15 anchors 185 people DWMDS BF anchors 18 people MDS MAP(P,R) anchors 19 nodes MDS MAP(P) anchors 19 nodes MDS MAP anchors 19 nodes Coverage Radus (%r) Medan Reconstructon Error (%r) 1 DWMDS BF anchors 1 people DWMDS BF 15 anchors 145 people DWMDS BF anchors 13 people MDS MAP(P,R) anchors 1 nodes MDS MAP(P) anchors 1 nodes MDS MAP anchors 1 nodes Coverage Radus (%r) Fg. 8. Left: error n random-unform networks. Rght: error n random-nonunform networks. Authorzed lcensed use lmted to: Carnege Mellon Lbrares. Downloaded on September 18, 9 at 17:31 from IEEE Xplore. Restrctons apply. 394

8 nodes (n both scenaros, some of the nodes are anchors and all the rest are ether moble (DWMDS-BF) or statc (MDS-MAP famly) nodes). The coverage radus ncreases from 1.5r to.5r n.5r steps. DWMDS-BF gets better accuracy than MDS-MAP algorthms when the coverage radus ncreases (MDS-MAP famly algorthms reach an almost steady level) and more n random-non-unform networks than n randomunform ones. The accuracy of DWMDS-BF ncreases when the number of anchors ncreases. MDS-MAP algorthms do not mprove the performance over anchors [16]. V. CONCLUSIONS AND FUTURE RESEARCH In ths paper we have proposed two effcent and numercally stable trackng algorthms to nfer ndvduals trajectores from a set of dssmlarty matrces through tme. DWMDS- BF s an algorthm that obtans very good trackng results for any network when only connectvty nformaton s avalable and DWMDS-DE s the algorthm desgned for those networks when an estmaton of the dstances between neghbors s avalable. The man contrbutons of the algorthms of DWMDS famly are a Dynamc term that effectvely lnks the dssmlarty data through tme regularzng the trackng soluton accordng to the dynamcs of the ndvduals tracked and a novel Bnary Flter functon n the Statc term of the DWMDS- BF algorthm. Compared to other locaton algorthms such as the well known MDS-MAP famly, DWMDS algorthms do not need communcaton between nodes, what saves energy and network bandwdth, and work equally well n unform or non-unform networks. Currently, we are extendng ths work n several ways: Settng up a testbed based on Bluetooth technology n an offce scenaro (headquarters of ROBOTIKER- TECNALIA Technology Centre, Span) wth more than people carryng specal desgned Bluetooh devces (Bluetooth Medallons). Ths testbed wll be used as a real envronment where DWMDS-BF wll be tested and refned wth real data. 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