Sparse multidimensional scaling for blind tracking in sensor networks

Transcription

1 Sparse multdmensonal scalng for blnd trackng n sensor networks R. Rangarajan 1, R. Rach 2, and A. O. Hero III 3 1 Unversty of Mchgan rangaraj@eecs.umch.edu 2 Unversty of Mchgan ravvr@eecs.umch.edu 3 Unversty of Mchgan hero@eecs.umch.edu 1 Introducton In ths chapter, we consder the problem of trackng a movng target usng sensor network measurements. We assume no pror knowledge of the sensor locatons and so we refer to ths trackng as blnd. We use the dstrbuted weghted multdmensonal scalng (dwmds) algorthm to obtan estmates of the sensor postons. Snce dwmds can only fnd sensor poston estmates up to rotaton and translaton, there s a need for algnment of sensor postons from one tme frame to another. We ntroduce a sparsty constrant to dwmds to algn current tme sensor postons estmates wth those of the prevous tme frame. In the presence of a target, locaton estmates of sensors n the vcnty of the target wll vary from ther ntal values. We use ths phenomenon to perform lnk level trackng relatve to the ntally estmated sensor locatons. Wreless sensor networks have been deployed for a number of montorng and control applcatons such as target trackng [29], envronmental montorng [30], manufacturng logstcs [27], geographc routng, and precson agrculture [45]. For many target trackng applcatons such as anomaly detecton [21, 46], speces dstrbuton and taxonomy [19], and survellance [4], the man purpose of the sensor network s to locate and track changes n remote envronments. For example, speces dstrbuton and classfcaton are currently documented usng sghtngs, captures, and trap locatons, whch nvolve consderable manpower, tme, and effort. Deployng moble sensors wth cameras can mprove remote counts of the speces as they move around n the envronment. For survellance applcatons, the sensors must be able to locate where the ntruders or the vehcles are movng n the network. Another example s the problem of locatng equpment n a warehouse. The sensors that tag the equpment must regster ther physcal locatons and actvate an alarm f they are about to ext the buldng. As another example, n secure protocol and network routng t s crtcal to track anomales such as worm actvty, flash crowds, outages, and denal of servce attacks n the network.

2 2 R. Rangarajan, R. Rach, and A. O. Hero III Automatc self-confguraton and self-montorng of sensor networks s the key enablng technology for these trackng applcatons. To respond to changes n the sensor network, t s crtcal to know where the changes are occurrng. Data measurements from the sensors must be regstered to ther physcal locatons n the network n order to make optmal decsons. For dense sensor networks, the large sze makes t mpractcal for humans to manually enter the physcal locaton of the sensors and t s too expensve to attach the GPS to every devce n the network. The sensors must have the capabltes to automatcally estmate ther relatve postons and detect changes n the network at low cost, e.g., wth mnmum battery power. Self-localzaton algorthms can be broadly classfed nto two categores, centralzed strateges and decentralzed strateges. In a centralzed approach, all the data collected by the sensors must be communcated to the fuson center whch then makes a decson based on ths nformaton. Algorthms that use multdmensonal scalng (MDS) [41], maxmum lkelhood estmaton [31], and convex optmzaton [14] have been proposed for centralzed estmaton and have shown to perform well. However, ths may be mpractcal when the sensors operate wth lmted power and bandwdth. For networks wth thousands of sensors, transmsson of sensor data to a fuson center overwhelm the low-bandwdth capacty of sensor networks. Furthermore, remote sensors are frequently battery operated and battery replacement may be nfeasble or expensve. The need to conserve power and bandwdth has set the stage for more effcent decentralzed strateges for localzaton. Among the popular approaches are adaptve trlateraton [33, 40] and successve refnement [9, 23] algorthms. In trlateraton, each sensor gathers nformaton about ts locaton wth respect to anchor nodes, also referred to as seeds [32], through a shortest path. Usng the range estmates from the seeds, a sensor uses trlateraton to estmate ts locaton n the network. In successve refnement algorthms, each sensor localzes ts poston n ts own coordnate system based on the nformaton communcated from only ts neghbors. Sensors refne ther locaton estmates teratvely usng updates from neghborng sensors and fnally merge ther local coordnates systems, effectvely fndng the soluton to the localzaton problem. Recently, there has been research emphass on localzaton based on a movng target, called a moble n [6, 35, 43]. The moble moves randomly n the network whle transmttng sgnals thereby allowng the sensors to calbrate ther range to the moble. Ths provdes a large number of measurements wth greater dversty whch helps overcome envronmental obstacles and enables mproved estmaton of the sensor node locatons. Most localzaton algorthms assume the presence of anchor nodes,.e., certan sensors whch have knowledge of ther postons n the network. In the absence of anchor nodes, the sensor locaton estmates are only accurate up to a rotaton and translaton. The ntuton behnd ths result s the followng: consder an ad-hoc network of N sensors. The objectve s to fnd the N sensor locatons gven the N(N 1)/2 nter-sensor dstance measurements.

3 Sparse multdmensonal scalng for blnd trackng n sensor networks 3 The dstance nformaton depends only on the dfferences n the sensor locatons so that the postons of the N sensors n the network can be rotated and translated wthout changng these dstances. In ths chapter, we present a sparsty constraned dwmds algorthm, whch can localze the relatve postons of the sensor nodes even n the absence of anchor nodes. The dwmds algorthm proposed n [9] s a successve refnement method, where a global cost functon s dvded nto multple local cost functons at each sensor locaton and the computatonal load nvolved n fndng the sensor locaton estmates s dvded among the sensors n a dstrbuted fashon. The allocaton of non-negatve contnuous weghts to the measured data overcomes the problem of combnng local maps to one global map, a problem that s common to other decentralzed methods [23]. We call our new algorthm sparsty penalzed dwmds. More mportantly, we explan how the anchorless sparse dwmds algorthm can effcently track changes n the network. Sensor localzaton s frequently vewed as an essental prelude to the montorng and trackng of actve phenomena. Target trackng and detecton has been one such motvatng applcaton of sensor networks [24, 44, 1, 26]. Most target trackng applcatons assume known sensor locatons or estmate the locaton of sensor nodes separately before employng the trackng algorthm. The standard model used for descrbng the state dynamcs of a movng target s the lnear Gaussan model [38]. When the measurement model s also Gaussan, the optmal tracker s gven by the Kalman flter. For nonlnear state space and measurement models, other technques such as Extended Kalman Flter (EKF) [24], unscented Kalman Flter (UKF) [44], and Gaussan sum approxmaton [1] have been proposed. Partcle flterng algorthms were then formulated for target trackng, where the probablty densty of the state s approxmated by a pont mass functon on a set of dscrete ponts [13]. The dscrete ponts are chosen through mportance samplng. The advantage of partcle flterng s ts applcablty to a large range of denstes, nose processes, and measurement models. More recently, researchers have looked at the smpler problem of trackng n a bnary sensng modalty [2, 25]. The sensor outputs a hgh, when the target s wthn a sensng range and outputs a low, when the target falls outsde ts range. Based on the fuson of the sensor outputs, an approxmate lnk level trajectory can be realzed to track the target. Such a bnary sensng modalty has lmted accuracy but requres mnmal power consumpton and has the advantage of analytcal tractablty [42]. Ths procedure can also be nterpreted as a target detecton problem mplemented for multple tme steps. Dstrbuted target detecton methods have been proposed n the lterature [34] n the context of desgnng an optmal decson statstcs at the sensor fuson center. The detecton problem has also been addressed for under communcaton constrants, where the sensor transmttng the nformaton needs to send an optmal summary of the gathered nformaton to the fuson center [7]. In the context of anomaly detecton n nternet data, approxmate densty of ncomng traffc s constructed for each locaton. Dstance between denstes

4 4 R. Rangarajan, R. Rach, and A. O. Hero III s then used as a smlarty measure n the MDS algorthm to form a map of the nternet network. By performng MDS over tme, t s shown that anomales such as network scans, worm attacks, and denal of servce attacks can be dentfed and classfed [37, 16]. For wormhole detecton n ad hoc sensor networks, most research efforts requre moble nodes equpped wth specal hardware or GPS devces [22, 5] to localze the wormhole. In contrast to the methods proposed n the lterature, we present the sparsty penalzed dwmds algorthm whch localzes the sensor nodes n the absence of anchors and tracks multple targets amongst the sensor lnks. The prncple behnd our proposed algorthm s the followng: n the acquston phase or ntalzaton, an ntal estmate of sensor locatons s acqured. Once the sensors have been ntally localzed, t s only the network topology that s crtcal to the problem of trackng. Hence, durng the trackng phase, we ntroduce a sparsty constrant to the dwmds problem formulaton, whch attempts to fx the algnment of the sensor network wth respect to the algnment of the localzed network at the prevous tme nstance. By dong so, we keep montorng the network wth respect to a fxed geometry obtaned by the localzaton algorthm at the frst tme nstance (t = 1). The sparsty constrant only reassgns a small fracton of the sensor locatons, whle mantanng the locatons of remanng sensors close to ther prevous estmates. When the sensor network s then used for trackng, only the sensors affected by the presence of a target are perturbed, whle the rest of the locaton estmates reman unchanged. Based on the dfferences n the sensor locaton estmates between two tme-frames, we propose a novel perturbaton based lnk level trackng algorthm, whch accurately localzes a target to wthn a small set of sensor lnks. Fgure 1 shows the localzaton process n the absence of targets. The actual sensor locatons are marked as crcles and the anchor nodes are hghlghted usng squares. The sensors communcate among themselves and the anchor nodes to obtan locaton estmates ndcated as crossed crcles. Fgure 2 shows the localzaton process n the presence of a target. The measurements of the sensor nodes closest to the target are affected and the sensors appear further apart than they are n realty. Ths change n the sensor locaton estmates can be used to perform lnk level trackng. Lnk level trackng has many attractve features, the most mportant of whch s that t does not requre a physcal model for the target, whch s fundamental to most trackng algorthms n the lterature [3]. Moreover, the goal of certan sensor networks s to obtan an estmate of the locaton of the targets, or detect changes n the network. For example, n mltary applcatons, the sensors can locate a target relatve to the network and the network can actvate the approprate sensors to dentfy the target. For anmal trackng n bologcal research, t s suffcent to have a low resoluton trackng algorthm to montor anmal behavor and nteractons wth ther own clan and wth other speces. We ntroduce the sparsty constraned dwmds algorthm for smultaneous sensor localzaton and lnk level trackng n ths chapter. We gve a flavor of

5 Sparse multdmensonal scalng for blnd trackng n sensor networks 5 xxx xx x xxx xxx xxx xx xx xxx xx xx Fg. 1. Localzaton n the absence of target. Anchor nodes (square), true sensor locatons (crcle), estmated sensor coordnates (crossed crcle). xxx xxx x xx xx xx xx Fg. 2. Lnk level trackng based on localzaton n the presence of target. how the algorthm can be extended to estmate actual target coordnates usng standard trackng algorthms. Furthermore, the algorthm we present here can be used to desgn optmal sensor schedulng strateges for trackng to lmt power consumpton n sensor networks. We ncorporate the sparsty constrant such that the localzaton algorthm s stll dstrbuted n ts mplementaton to mnmze communcaton and computatonal costs. Ths chapter s organzed as follows: Secton 2 formally ntroduces the problem of sensor localzaton. Secton 3 ntroduces the classcal MDS al-

6 6 R. Rangarajan, R. Rach, and A. O. Hero III gorthm and ts varatons. We then present our sparsty penalzed dwmds algorthm n Secton 4. In Secton 5, we explan how ths algorthm can be appled for lnk level trackng. Fnally, Secton 6 concludes ths chapter by dscussng the extensons of ths formulaton for model-based multple target trackng and sensor management strateges. 2 Problem formulaton We begn by ntroducng the nomenclature used n ths chapter. We denote vectors n Ê M by boldface lowercase letters and matrces n Ê M N by boldface uppercase letters. The dentty matrx s denoted by I. We use ( ) T to denote the transpose operator. We denote the l 2 -norm of a vector by,.e., x = xt x. A Gaussan random vector wth mean µ and covarance matrx C s denoted as N(µ,C). The purpose of the sparsty constraned MDS algorthm s to smultaneously localze and track targets. We frst formally state the sensor localzaton problem. Consder a network of N = n+m nodes n d dmensonal space. The localzaton algorthms can be appled to arbtrary d (d < N) dmensonal spaces. Snce applcatons for localzaton typcally occur n physcal space, we wll restrct our attenton to d = 2, 3 dmensons. Let {x } N =1,x Ê d be the true locaton of the n sensors. The m sensor nodes {x } n+m =n+1 are anchor nodes,.e., whose locatons are known. We ntroduce the anchor nodes to keep the formulaton as general as possble. Later, we set m = 0 for anchor free localzaton. Denote X = [x 1,x 2,...,x N ] as the d N matrx of actual sensor locatons. Let D = (d,j ) N,j=1 be the matrx of the true nter-sensor dstances, where d,j denotes the dstance between sensor and sensor j. It s common that some wreless sensor networks may have mperfect a pror knowledge about the locatons of certan sensor nodes. Ths nformaton s encoded through parameters r and x, where x s the sensor locaton and r s the correspondng confdence weght. If x s unavalable, then we set r = 0. The problem settng s explaned through an llustraton of a sensor network n Fg. 3. In ths sensor network, each sensor communcates to ts three nearest neghbors and hence, the weghts correspondng to lnks between non neghborng sensors are zero. Sensor localzaton s the process of estmatng the locaton of the n sensor nodes {x } n =1 gven {x } n+m =n+1, {r }, { x } and parwse range measurements {δ t,j } taken over tme t = 1, 2,...,K. The ndces (, j) run over a subset of {1, 2,..., N} {1, 2,..., N}. The range measurements can be obtaned by sensng modaltes such as tme-of-arrval (TOA), receved sgnal strength (RSS), or proxmty.

7 Sparse multdmensonal scalng for blnd trackng n sensor networks 7 y x k w,k = 0 x r x δ,j x j r j x j Fg. 3. Sensor localzaton setup: Anchor nodes (square), sensor nodes (crcle), a pror sensor locatons (blocked crcle). The communcatng sensors are connected usng sold lnes. The non neghborng sensor lnks have zero weght. x 3 Classcal MDS and varatons Multdmensonal scalng (MDS) s a methodology for recoverng underlyng low dmensonal structure n hgh dmensonal data. The measured data can come from confuson matrces, group data, or any other (ds)smlarty measures. MDS has found numerous applcatons n cogntve scence, marketng, ecology, nformaton scence, and manfold learnng [11, 12]. In the context of sensor localzaton, the goal n MDS s to dscover the sensor locatons (lower dmensonal embeddng) from the nter-sensor dstances obtaned by a gven sensng method (hgh dmensonal data). Classcal MDS [18] provdes a closed-form soluton to the sensor locatons when the nter-sensor measurements are the nter-sensor Eucldean dstances,.e., n the absence of nose or nonlnear effects. We assume all parwse range measurements are avalable, and so we can compute the complete matrx of dstances: d,j = x x j = (x x j ) T (x x j ). (1) Denote by D (2) the matrx of squared dstances,.e., D (2) = (d 2,j )N,j=1. Then D (2) can be rewrtten as D (2) = ψ1 T 2X T X + 1ψ T, (2)

8 8 R. Rangarajan, R. Rach, and A. O. Hero III where 1 s an N-element vector of ones and ψ = [x T 1 x 1,x T 2 x 2,...,x T N x N] T. Let H = I (1/N)11 T. Multplyng on the left of D (2) by 1/2H and the rght by H, we obtan A = 1 2 HD(2) H = HX T XH. (3) Gven A, one can dscover the matrx X to a rotaton and translaton by solvng the followng varatonal problem mn Y A YT Y 2 F, (4) where F ndcates the Frobenus norm and the search space s over all full rank d N matrces. The soluton to X s then gven by X = dag(λ 1/2 1,...,λ 1/2 d )V T 1, (5) where the sngular value decomposton (SVD) of A s gven by A = [V 1 V 2 ] dag(λ 1,...,λ d, λ d+1,..., λ N ) [V 1 V 2 ] T. (6) The matrx V 1 conssts of the egenvectors of the frst d egenvalues λ 1,...,λ d, whle the rest of the N d egenvectors are represented as V 2. The term dag(λ 1,..., λ N ) refers to a N N dagonal matrx wth λ as ts th dagonal element. Though the soluton to the classcal MDS s obtaned n closed-form, the algorthm has the followng defcences: 1. MDS requres knowledge of all nter-sensor dstances. Obtanng all parwse range measurements s prohbtve due to the sze of the sensor network and the lmted power of the sensors. In our problem formulaton, ths mples that w,j 0,, j, whch makes MDS fall under the category of a centralzed approach,.e., all the nformaton needs to be transmtted to the fuson center whch then performs the MDS algorthm. Due to power and bandwdth lmtatons n the sensor network, ths process s nfeasble. 2. The nter-sensor range measurements δ,j are corrupted by envronment and recever nose whch further degrades the qualty of the measurements,.e., δ,j s only an estmate of the nter-sensor dstance d,j. 3. MDS uses the squared dstance matrx whch tends to amplfy the measurement nose, resultng n poor performance. As mentoned n Secton 1, there has been sgnfcant effort drected towards desgnng decentralzed strateges for sensor localzaton. However, consstent reconstructon of the sensor locatons s attanable only n the presence of anchor nodes. If the current localzaton algorthms are mplemented for anchor free localzaton, the geometry of the sensor network assumes dfferent algnments as localzaton s performed over varous tme nstants. Ths makes t

9 Sparse multdmensonal scalng for blnd trackng n sensor networks 9 mpossble to locate where the changes are occurrng n the network. To llustrate ths phenomenon, we mplement the dwmds algorthm for sensor localzaton n the absence of anchor nodes and n the absence of target. We provde snapshots of the sensor locaton estmates (cross) along wth ther actual locatons (crcle) n Fg. 4 as a functon of tme. Observe that the geometry of the network s mantaned, whle the true locatons are subject to rotaton and translaton. Now consder a target movng through ths network. In ths scenaro, the localzaton process s affected by two factors: the lack of anchor nodes and some naccurate nter-sensor measurements n the vcnty of the target. Wth anchor free localzaton, the process of trackng a target becomes extremely dffcult. To overcome ths problem, we propose our sparsty constraned dwmds algorthm that algns the current sensor locaton estmates to those of prevous tme frames. (a) t=1 (b) t=2 (c) t=3 (d) t=4 Fg. 4. Anchor free sensor localzaton by dwmds. True sensor locatons (crcle), estmated sensor locatons (cross).

10 10 R. Rangarajan, R. Rach, and A. O. Hero III 4 Sparsty penalzed MDS Consder usng the MDS algorthm ndependently to obtan the sensor locaton estmates at tme t and at tme t 1. Algnment between these two sets of ponts can be performed n varous ways. For example, n Procrustes analyss [17] algnment s performed by fndng the optmal affne transformaton of one set of nodes that yelds the set closest to the second set of ponts n the least squares sense. However, ths procedure cannot guarantee that many sensor locatons estmates wll reman unchanged from ther prevously estmated values. The errors n the sensor locaton estmates between two tme steps may accumulate over tme resultng n algnment errors. In contrast, we ntroduce a sparseness penalty on the dstances between the sensor locaton estmates at tme t (x ) and at tme t 1 (x (t 1) ) drectly to the sensor localzaton algorthm. Construct a vector of Eucldean dstances between the locaton estmates at tme t and at tme t 1 [ T g (t) = x 1 x (t 1) 1,..., x n x (t 1) n ]. (7) Defne the l 0 -measure of a vector v = [v 1, v 2,..., v n ] as the number of nonzero elements gven by n v 0 I(v 0), (8) =1 where I( ) s the ndcator functon. Usng an l 0 -constrant on the dstance vector g (t) of the form g (t) 0 q, we guarantee that no more than q of the locaton estmates wll vary from ther prevous tme frame values. Mnmzng a cost functon under the l 0 -constrant requres a combnatoral search whch s computatonally nfeasble. Defne the l p -measure of a vector v as v p ( n v p) 1/p. (9) =1 For a quadratc cost functon, an l p -constrant (0 < p 1) nduces a sparse soluton. Among all l p sparsfyng constrants, only p = 1 offers a convex relaxaton to the l 0 -constrant [15]. To promote sparsty, we next advocate the use of the l p -constrant as a penalty term va the Lagrange multpler n the dwmds algorthm to solve for the sensor locaton estmates. Hence the term sparsty penalzed MDS. The cost functon of the dwmds algorthm [9] s motvated by the varatonal formulaton of the classcal MDS, whch attempts to fnd sensor locaton estmates that mnmze the nter-sensor dstance errors. Keepng n mnd that t s the geometry of the sensor network whch s crucal for trackng, we present a novel extenson of the dwmds algorthm through the addton of the sparseness nducng l p -constrant. At any tme t, we seek to mnmze the overall cost functon C (t) gven by

11 Sparse multdmensonal scalng for blnd trackng n sensor networks 11 C (t) = 1 n j n+m 1 l M ( ) 2 n w (t),l,j δ (t),l,j d,j (X) + r x x 2 =1 +λ g (t) p p. (10) The Eucldean dstance d,j (X) s defned n (1). For each tme t, there are M range measurements δ (t),l,j for each sensor lnk, j. As n [9], the weghts w (t),l,j can be chosen to quantfy the accuracy of the predcted dstances. When no measurement s made between sensor and sensor j, w (t),l,j = 0. Furthermore, the weghts are symmetrc,.e., w (t),l,j = w (t),l j,, and w (t),l, = 0. If avalable, the a pror nformaton of sensor locatons s encoded through the penalty terms {r x x 2 }. Fnally, we ntroduce an l p -constrant (0 p 1) on the dstances between the sensor locatons at tme t and the estmated sensor locatons at tme t 1. The Lagrange multpler of the sparseness penalty s denoted as λ. We can tune the value of λ to yeld the desred sparsty level n g (t). Later, when we apply the algorthm for trackng, the sparseness wll be advantageous as only those sensors whch are hghly affected by the target wll vary from ther ntal postons, thereby allowng for a detecton of the target through the process of relatve sensor localzaton. To solve ths optmzaton problem, we propose to use the successve refnement technque, where each sensor node updates ts locaton estmate by mnmzng the global cost functon C (t), after observng range measurements at node and recevng poston estmates from ts neghborng nodes. 4.1 Mnmzng cost functon by optmzaton transfer Unlke classcal MDS for whch we could obtan a closed-form expresson for the estmates, there s no closed-form soluton to mnmzng C (t). Therefore, we solve the local nonlnear least squares problem teratvely usng a quadratc majorzaton functon smlar to SMACOF (Scalng by MAjorzng a COmplcated Functon [20]). Ths procedure can be vewed as a specal case of optmzaton transfer algorthms through surrogate objectve functons [28], e.g., the popular EM algorthm. A majorzng functon T(x,y) of C(x) s a functon T : Ê d Ê d Ê, whch satsfes the followng propertes: T(x,y) C(x), y and T(x,x) = C(x). In other words, the majorzng functon upper bounds the orgnal cost functon. Usng ths property, we can formulate an teratve mnmzaton procedure as follows: denote the ntal condton as x 0. Startng from n = 1, obtan x n by solvng x n = argmn x T(x,x n 1 ), untl a convergence crteron for C(x) s met. We can easly observe that ths teratve scheme always produces a non decreasng sequence of cost functons,.e., C(x n+1 ) T(x n+1,x n ) T(x n,x n ) = C(x n ).

12 12 R. Rangarajan, R. Rach, and A. O. Hero III The frst and last relatons follows from the propertes of majorzng functons whle the mddle nequalty follows from the fact that x n+1 mnmzes T(x,x n ). Now the trck s to choose a majorzng functon that can be mnmzed analytcally, e.g., a lnear or quadratc functon. We propose a quadratc majorzng functon T (t) (X,Y) for the global cost C (t) (X). Mnmzng C (t) (X) through the majorzaton algorthm s the smple task of mnmzng the quadratc functon T (t) (X,Y),.e., T (t) (X,Y) x = 0, = 1, 2,..., n. (11) If we denote the estmates of the sensor nodes at teraton k as X k, the recurson for the update of locaton estmates for node from (11) s gven by x k = 1 ( c + X k 1 b k 1 ), (12) a where b k 1, a, and c are defned n (30)-(33) respectvely. The detals of the dervaton of the sparsty penalzed MDS algorthm can be found n Secton 8. For each sensor, the j th element of the vector b k 1 depends on the weght w,j. Snce the weghts of the nodes not n the neghborhood of the sensor are zero, the correspondng elements n the vector b k 1 are also zero; therefore the update rule for node n (12) wll depend only the locaton of ts nearest neghbors and not on the entre matrx X k 1. Ths facltates the dstrbuted mplementaton of the algorthm. The proposed algorthm s summarzed n Fg. 5. We llustrate the majorzaton procedure n Fg. 6. The orgnal cost functon (sold) and the correspondng surrogate (dotted) s presented for every teraton, along wth the track of the estmates at teraton k (crcle). Inputs: { w (t) (t),j }, { δ,j }, {r}, { x}, {x(t 1) }, ǫ, X 0 (ntal condton for teratons). Set k = 0, compute cost functon C (t),0 and a from equatons (10) and (32) respectvely repeat k=k+1 for = 1 to n compute ( b k 1 from equaton ) (30) x k = 1 c a + X k 1 b k 1 compute C (t),k update C (t),k to C (t),k C (t),k 1 + C (t),k communcate x k to neghbors of sensor (nodes for whch w,j > 0) communcate C (t),k to next node (( + 1) mod n) end for untl C (t),k C (t),k 1 < ǫ Fg. 5. Descrpton of the sparsty constraned MDS algorthm.

13 Sparse multdmensonal scalng for blnd trackng n sensor networks 13 Cost functon k=3 k=2 k=1 Surrogate functon Cost functon Optmal value x coordnate of sensor Fg. 6. Majorzaton procedure: cost functon (sold curve), surrogate functon (dotted curve), optmal locaton estmate at each teraton (crcle). Only a sngle coordnate s updated n ths pcture. Our proposed algorthm ntroduces a sparseness penalty on the dstance between estmate at tme t 1 and the current estmate. If the sparsty regularzaton parameter λ s not chosen properly, many sensor postons estmates mght slowly vary wth tme, thereby creatng cumulatve error n the sensor localzaton. An nterestng way to counteract ths problem would be to penalze the dstance between the current estmate and the ntal estmate at t = 1. Usng such a constrant would mean that the sensors are always compared to the fxed ntal frame and errors do not accumulate over tme. The mplementaton of ths algorthm would be straghtforward as t would smply nvolve changng the ndex t 1 to 1 n the orgnal algorthm presented n Fg. 5. However, usng the estmate from tme t 1 has the property that t s easly adapted to the case of moble sensors. 4.2 Implementaton Weghts: When RSS measurements are used to compute dstance estmates, the weghts are set usng the locally weghted regresson methods (LOESS) scheme [8] smlar to one used n the dwmds algorthm [9]. The weght assgnment s gven by { exp( δ 2 w,j =,j /h 2,j ), f and j are neghbors 0, otherwse, where h,j s the maxmum dstance measured by ether sensor or j. A nave equal weght assgnment to all measurements s also shown to work well wth our algorthm. Intalzaton: For the successve refnement procedure, the sensor locatons estmates must be ntalzed for every tme frame. Though several ntalzaton algorthms have been proposed n the lterature, we use a nave random

14 14 R. Rangarajan, R. Rach, and A. O. Hero III ntalzaton. We would lke to pont out that the ntalzaton s not a crtcal component to our algorthm, as we are solely nterested n the algnment of sensors n the network and not on the exact locatons. Irrespectve of the ntal estmates, the sparseness penalty wll ensure that the estmated sensor locatons are relatvely close to those of prevous tme frames. Our algorthm s found to be robust wth respect to the ntal estmates. Neghborhood selecton: Tradtonally, the neghbors are chosen based on the dstance measure obtaned from the RSS measurements,.e., select all sensors wthn a dstance R as your neghbors. When the RSS measurements are nosy, there s a sgnfcant bas n the neghborhood selecton rule. Ths method has a tendency to select sensors whch are, on average, less than the actual dstances x x j. A smple two-stage adaptve neghborhood selecton rule s proposed n [9] to overcome the effect of ths bas. We use ths selecton rule n our algorthm. Range measurement models: The nter-sensor measurements can be obtaned by RSS, TOA, or proxmty. Any one of these approaches can be used n our algorthm. Our sparsty constraned MDS algorthm s farly robust to ether of these measurement models. For the smulatons n ths chapter, we use the RSS to obtan a range measurement between two sensors. It can be shown through the central lmt theorem (CLT) that the RSS s log-normal n ts dstrbuton [10],.e., f P,j s the measured power by sensor transmtted by sensor j n mllwatts, then 10 log 10 (P,j ) s Gaussan. Thus P,j n dbm s typcally modeled as P,j = N( P,j, σ 2 0 ) P,j = P 0 10n p log ( d,j d 0 where P,j s the mean receved power at dstance d,j, σ 0 s the standard devaton of the receved power n dbm, and P 0 s receved power n dbm at a reference dstance d 0. n p s referred to as the path-loss exponent that depends on the multpath n the envronment. Gven the receved power, we use maxmum lkelhood estmaton to compute the range,.e., dstance between the sensor nodes and j. The maxmum lkelhood estmator of d,j s gven by δ,j = 10 ((P0 P,j)/10np). (13) ), Smulaton of tracker wthout a target The smulaton parameters are chosen as follows: we deploy a unform grd of sensors n a network. We consder anchor free localzaton,.e., m = 0 and we assume we make a sngle nter-sensor measurement (M = 1). We set the sparseness parameter λ to produce a change n the locaton estmates for

15 Sparse multdmensonal scalng for blnd trackng n sensor networks 15 only a small porton of the sensors. The value of λ wll depend on the sze of the network and the nose n the measurements. If the RSS measurements are very nosy, then range estmates become naccurate whch tend to vary the sensor locaton estmates. Hence λ s selected to ensure that sensor locaton estmates reman algned wth the prevous tme frame estmates. In ths smulaton, we set λ = 0.1 and the nose varance σ 0 = Each sensor communcates wth ts 15 nearest neghbors. The weghts of the RSS measurements were chosen based on the LOESS scheme descrbed earler. The weghts of lnks for non communcatng sensors were set to zero. We demonstrate the performance of the sparsty constraned MDS algorthm on ths sensor network as a functon of tme n Fg. 7. The true locatons are denoted as crcles and the estmated locatons as crosses. (a) t=1 (b) t=2 (c) t=3 (d) t=4 Fg. 7. Anchor free sensor localzaton by sparsty penalzed MDS. True sensor locatons (crcle), sensor poston estmates (cross).

16 16 R. Rangarajan, R. Rach, and A. O. Hero III 5 Trackng usng sparse MDS Here we present an algorthm for performng lnk level trackng usng the sparsty constraned MDS algorthm. By lnk level trackng, we refer to localzaton of targets to wthn a set of nter-sensor lnks. Lnk level trackng s attractve n the sense that there s no need to assume a physcal model for a target. However, t s mportant to know the effect of the target on the nter-sensor measurements. Researchers have proposed varous models for the sgnal strength measurements rangng from the tradtonal lnear Gaussan model to the bnary sensng model. These are approxmate statstcal models and the dstrbuton of the measurements n the presence of a target remans an open queston. To model the statstcs under the settng of vehcle trackng, we conducted experments usng RF sensors hardware n the presence of a target. We constructed a fne grd of locatons, where the target was placed and RSS measurements were recorded between two statc sensors for postons on the grd. Upon gatherng the data, we ft the followng statstcal model n the presence of target. The RSS measurements at sensor lnk, j are dstrbuted as P,j k ˆP,j N( ˆP,j, σ0 2 ),..d, ˆP,j N( P,j, σ1), 2 k = 1, 2,...,M where P,j k s the kth nter-sensor measurement when the target s n the neghborhood of the sensors. The M sensor lnk measurements are correlated through the random varable ˆP,j. The values obtaned from our actual experments were σ dBm and σ 1 1.5dBm. The nose varance n the measurements σ 1 was roughly an order of 10 tmes larger than σ 0. In other words, RSS measurements tend to have a larger varance due to scatterng and attenuaton of the sgnals n the presence of a target. A confdence measure for such a log-normal dstrbuton of the RSS data s obtaned usng the Kolmogorov-Smrnov (KS) test n [36] and the model s shown to work well for sensor localzaton. We assume ths statstcal model for the RSS measurements, when the target s wthn a specfed dstance R of the sensor lnk, j. The dstance R depends on the reflectvty of the object. If the object s hghly reflectve, then the varaton n the RSS measurements s detected by more lnks. Gven the measurement model, we formulate the optmal decson statstc to detect a presence of a target n a partcular sensor lnk usng the lkelhood rato test (LRT). For a fxed false alarm level α, the LRT for each lnk, j s gven by 1 M M l=1 P (t),l,j P,j H 1 H 0 γ, (14) where γ = (σ 0 / M)Q 1 (α/2) and P,j s the mean receved power n the sensor lnk estmated usng an ntal set of range measurements. {P (t),l,j } M l=1

17 Sparse multdmensonal scalng for blnd trackng n sensor networks 17 s the set of nter-sensor measurements made by lnk, j at tme t. We assume that the sensor network s n ts steady state operaton mode. We do not consder the transent effects n the measured data when t s obtaned n the absence of any target. Ths most powerful test of level α yelds the probablty of correct detecton ( ) β = 2Q Q 1 (α/2) σ 2 0 σ Mσ2 1. (15) A dervaton of the decson rule and ts performance s gven n Secton 9. The performance of the optmal detector s clearly dependent on the number of samples avalable for the nter-sensor measurements. As the number of measurements M becomes very large, β n (15) tends to 1. However, f only few samples are avalable, β may not approach 1 and msdetect type errors may become non neglgble. In such a case, nstead of usng LRT, we can use a test on the varaton of the sensor locaton estmates at tme t from ther estmates at a prevous tme. In other words, we can perform a smple hypothess test for each lnk of the form, H 1 d t,j d T,j γ,j, (16) H 0 where T = 1 or T = t 1 dependng on whether the sensors are statc or moble. Our objectve s to approxmately locate the target relatve to the locaton of the sensors. There are a number of ways n whch ths lnk level estmate can be translated nto actual target coordnates. For example, take the mdpont of the convex hull generated by the postons of those sensors that yeld a hgh n the optmal LRT. Another estmate can be found by the ntersecton of convex regons correspondng to the sensor lnks that show the presence of the target through the optmal decson rule. We have made no assumptons on the physcal model of the target trajectory n our analyss. However, gven a target moton model, standard flterng technques such as the Kalman flter, EKF, or partcle flterng can be used to obtan accurate target locatons based on the bnary decsons for each sensor lnk. Smulaton of tracker n the presence of target We present our results by smulatng movng targets n a unform grd of sensors. We set m = 0,.e., no anchor nodes. We assume no a pror knowledge of the sensor coordnates,.e., r = 0. Each sensor communcates only to ts 15 nearest neghbors and the weghts for those lnks were chosen by the LOESS strategy. The rest of the weghts were set to zero. We obtan M = 50 data measurements for each communcatng sensor lnk n the network. We set the

18 18 R. Rangarajan, R. Rach, and A. O. Hero III sparseness parameter λ to produce a change n the locaton estmates for only a small porton of the sensors. We allow any number of targets to appear n a sensor network wth probablty 0.4. Though our algorthm s robust to randomly movng targets n the network, we consder a state-space model for the purposes of ths smulaton to produce a vsually pleasng target trajectory. We apply the sparsty constraned MDS algorthm as multple targets move through the sensor network and the results are shown n Fg. 8. The true sensor locatons are shown as crcles and the estmated sensor locatons are ndcated usng crosses. The sensors correspondng to those sensor lnks that yelded a hgh under the optmal LRT are shown n blocked crcles. The target trajectores are shown as nverted trangles. We observe that as the targets move, the sparsty constraned MDS algorthm localzes most sensors close to the estmates from the prevous tme frame whle changng the locaton of very few sensors that are n the vcnty of the target. Furthermore, the LRT s able to localze the targets to wthn a small set of sensor lnks. Numercal comparsons We perform a numercal comparson of trackng performance usng the LRT and dstance based decson rules descrbed n (14) and (16), respectvely. We form the convex hull of the sensor coordnates correspondng to the sensor lnks that fall under the H 1 hypothess and consder the mdpont of the convex hull as our estmate of the target coordnates. We apply the sparsty penalzed MDS algorthm and compute the coordnates of the target based on each of the decson rules. An example of our mdpont trackng algorthm s shown n Fg. 9. The true sensor locatons are marked as crcles. The trajectory of the target s ndcated usng nverted trangles. The estmate of the target coordnates s denoted usng a plus symbol. The mean-squared error (MSE) n trackng s computed as the average error n the estmated target coordnates over tme. We evaluate trackng performance n terms of the MSE as a functon of the number of nter-sensor measurements M. In the smulatons, we set σ 0 = 0.15, σ 1 = 3.75, λ = 0.1, α = 0.01, and P 0 = 1, where P 0 s the power transmtted by each sensor. Threshold values γ,j n (16) are chosen based on a numercal search to acheve false alarm rate α. The MSE n trackng s evaluated over 20 randomly generated runs of the target trajectory n the sensor network. The results are presented n Fg. 10. We observe that as M ncreases, the performance of both methods mprove. For the LRT based trackng, the optmal performance n terms of probablty of correct decson s gven by (15). As M ncreases, β tends to 1. As a result, lnks assocated wth the movng target are correctly dentfed to acheve an mprovement n trackng performance. However, note that β 1 does not mply a zero trackng error as we only construct an approxmate estmate of the targets locaton usng the mdpont strategy. For the dstance based trackng, as M ncreases, accurate nter-sensor dstances are avalable for all sensors except for

19 Sparse multdmensonal scalng for blnd trackng n sensor networks 19 (a) t=1 (b) t=2 (c) t=3 (d) t=4 (e) t=5 (f) t=6 (g) t=7 (h) t=8 Fg. 8. Anchor free sensor localzaton by sparsty constraned MDS n the presence of targets. True sensor locatons (crcle), estmated sensor locatons (cross), sensors localzng the target (blocked crcle), target trajectory (nverted trangle).

20 20 R. Rangarajan, R. Rach, and A. O. Hero III Fg. 9. A smple trackng algorthm based on lnk level trackng. True sensor locatons (crcle), true trajectory of the target (nverted trangle), estmated trajectory (plus). those n the vcnty of the target. The sparseness penalty algns most sensors to ther prevous locaton estmates, whch were obtaned n the absence of the target. Only those sensors affected by the target may vary ther locaton estmates. The dstance based rule can then select the correct set of perturbed sensors resultng n mproved trackng performance. For small M, we note that the lkelhood based approach has a larger trackng error than the dstance based approach. Ths s because for small M, LRT decson rule results n more msdetect type errors whle the sparsty constrant s able to counteract these errors n the dstance based approach. Moreover, the performance of the LRT based trackng yelds better performance than the dstance based approach as M becomes large snce the LRT approaches perfect detecton (β 1) whle no such property can be clamed for the dstance approach. Next, we evaluate the MSE for varous values of SNR = Mσ 1 /σ 0 keepng M constant. We set M = 1, σ 0 = 0.15, P 0 = 1, and λ = 0.1. The performance of the two methods s shown n Fg. 11. Snce we observe only one measurement per lnk, the performance of the lkelhood based algorthm s nferor to the dstance based trackng for small SNR due to a hgh false alarm rate,.e., lnks falsely detect the presence of the target and the mdpont of the sensor postons falls further away from the actual coordnate of the target. However, the dstance based algorthm performs better as t makes use of past sensor locaton estmates through the sparsty constrant. For large SNR, the LRT yelds β close to 1 and the LRT based trackng algorthm acheves better performance. A seemngly counter ntutve observaton from Fg. 10 and Fg. 11 s that the dwmds based trackng offers better performance than the LRT based al-

21 Sparse multdmensonal scalng for blnd trackng n sensor networks Lkelhood based Dstance based 10 log 10 MSE M Fg. 10. Performance of trackng algorthms n terms of MSE versus M Lkelhood based Dstance based 10 log 10 MSE SNR (db) Fg. 11. Performance of trackng algorthms n terms of MSE versus SNR. gorthm n the low SNR regme. The reasonng s as follows: n the presence of a movng target, the RSS measurements of the sensor lnks are correlated spatally and temporally. The presence of a target n a gven lnk mples that wth hgh probablty the target s present n neghborng sensor lnks. Furthermore, gven a set of sensors that detected a target at the prevous tme-frame, there s a hgh probablty that the target mght be detected n the vcnty of these sensors at the next tme-frame. The LRT s lmted n ts performance as we fnd the optmal decson statstc on each sensor lnk, j ndependent of other sensor lnk measurements and ndependent of past observatons. However, the dstance based approach uses the sparsty penalzed dwmds to track the tar-

22 22 R. Rangarajan, R. Rach, and A. O. Hero III get. Snce the nter-sensor dstances are computed after each sensor obtans nformaton from ts nearest neghbors, ths method makes use of the spatal nformaton n ts decson statstc. Moreover, the temporal correlaton of the RSS measurements s captured through the sparsty constrant used for algnng the sensors locatons estmates. At hgh SNR, the LRT s able to outperform the dstance based approach as t asymptotcally acheves optmal detecton performance. Future work Gven the set of tagged sensors,.e., sensor lnks wth hgh output n the LRT, we have reduced the problem to that of bnary sensng, where the knowledge of the presence of the target s stored as the decsons made on each of the lnks. For accurate estmaton of targets postons, we can now use the popular partcle flterng technques proposed on bnary sensng models [26] to perform mult-target trackng gven a small set of anchor nodes. Moreover, most sensor networks are remotely operated and lmted n power. We can pose a power constrant by lmtng the number of nter-sensor measurements to a small s of the n(n 1) (s n(n 1)) total sensor lnks at each tme step. The problem of choosng s from n(n 1) lnks s a combnatorally hard problem. So we propose a convex relaxaton to the problem, whch chooses the set of actve lnks by mnmzng the predcated mean square error of the state of the target. Ths approach has been shown to acheve near optmal performance n our earler work [39]. 6 Conclusons In ths chapter, we presented the sparsty penalzed MDS algorthm for smultaneous localzaton and trackng. We are nterested n trackng a target relatve to the sensor coordnates. The subset selecton capablty of our proposed sparsty constrant allows the algorthm to fnd only those whch have changed ther locaton estmate due to the presence of a target. We use these sensors to perform lnk level trackng. We formulate a model for the nter-sensor RSS measurements n the presence and absence of a target by conductng actual experments n free space. Usng ths model, we llustrated the performance of our algorthm for lnk level target trackng. Currently, we are n pursut of optmal sensor schedulng strateges for physcal level trackng. 7 Acknowledgements We would lke to thank Mr. Xng Zhou for hs enthusastc assstance n conductng the experments dscussed n ths chapter. Ths work was partally supported by a grant from the Natonal Scence Foundaton CCR

23 Sparse multdmensonal scalng for blnd trackng n sensor networks 23 8 Appendx: Dervaton of sparsty penalzed dwmds To smplfy our dervaton, we dvde the global cost functon nto multple local cost functons as follows: n C (t) = C (t) + c (t), (17) =1 where c (t) s a constant ndependent of the sensor locatons X and the local cost functon at each sensor node s C (t) = n j=1,j w (t) (t),j ( δ,j d,j(x)) n+m j=n+1 w (t) (t),j ( δ,j d,j(x)) 2 +r x x 2 + λ x x (t 1) p, (18) where w (t),j = M l=1 w(t),l (t),j and δ,j = M l=1 w(t),l,j δ(t),l,j / w(t),j. The cost functon depends only the measurements made by sensor node and the postons C (t) of the neghborng nodes,.e., nodes for whch w (t),l,j > 0; C (t) s local to node [9]. The local cost functon n (18) can be rewrtten as where c (t) 1 = C (t) (X) = c (t) n j=1,j c (t) 2 (X) = n j=1,j c (t) 3 (X) = 2 n j=1,j 1 + c(t) 2 w (t) (t),j ( δ,j )2 + 2 (X) c(t) (X) + c(t) (X), (19) n+m j=n+1 n+m w (t),j d2,j (X) + 2 j=n+1 w (t) (t),j δ,j d,j(x) w (t) (t),j ( δ,j )2 4 w (t),j d2,j (X) + r x x 2 n+m j=n+1 w (t) (t),j δ,j d,j(x) c (t) 4 (X) = λ x x (t 1) p. (20) The term c (t) 1 s ndependent of x. The term c (t) 2 s quadratc n x. Terms c (t) 3 and c (t) 4 are nether affne nor quadratc functons of x. A majorzng functon for the term c (t) 3 s motvated by the followng Cauchy-Schwarz nequalty, d,j (X) = x x j (x x j ) T (y y j ), Y, (21) d,j (Y) where Y = [y 1,...,y n ]. For c (t) 4, we present a quadratc majorzng functon, whch can be obtaned from the followng relaton

24 24 R. Rangarajan, R. Rach, and A. O. Hero III α p/2 α p/2 0 + p 2 (α α 0)(α 0 ) ( p 2 1), α, α 0 > 0. (22) The above nequalty follows from a lnear approxmaton to the concave functon f(α) = α p/2 va Taylor seres expanson. Choosng α = x x t 1 2 and α 0 = y x t 1 2 yelds x x t 1 p y x t 1 p + p 2 x x t 1 2 y x t 1 2 y x t 1, (23) 2 p the majorzng functon for the c (t) 4 term. Substtutng the nequaltes from (21) and (23) n (19), we obtan the majorzng functon for the local cost functon as T (t) (X,Y) = c (t) n j=1,j n j=1,j +4 n+m j=n+1 n+m w (t),j d2,j (X) + 2 w (t),j w (t),j + λ y x (t 1) j=n+1 (t) (x x j ) δ T (y y j ),j d,j (Y) (t) (x x j ) δ T (y y j ),j d,j (Y) p + λp 2 w (t),j d2,j (X) + r x x 2 x x (t 1) 2 y x (t 1) 2. y x (t 1) 2 p (24) Snce T (t) (X,Y) s a majorzng functon to C (t) (X), t s easy to verfy that the functon T (t) (X,Y) = n =1 T (t) (X,Y) s a majorzng functon to the global cost functon C (t) (X). The partal dervatve of T (t) (X,Y) wth respect to x s straghtforward as all the expressons n (24) are lnear or quadratc n x. The partal dervatve of T (t) (X,Y) wth respect to x s gven by T (t) (X,Y) x = (t) T (X, Y) + x k T (t) k (X,Y) x, (25) where T (t) (X, Y) x = 2 n j=1,j +4 ( w (t) ),j (x x j ) w (t) (t) (y y j ),j δ,j y y j w (t),j (x x j ) w (t) (t) (y y j ),j δ,j y y j n+m j=n+1 +2r (x x ) + λp (x x (t 1) ) y x (t 1) 2 p (26)

25 and Sparse multdmensonal scalng for blnd trackng n sensor networks 25 T (t) k (X,Y) ( = 2 w (t),k x (x x k ) w (t),k δ (t),k Substtutng (26) and (27) n (25) yelds, T (t) n+m (X,Y) = 4 w (t),j x (x x j ) w (t),j j=1,j (y y k ) y y k δ (t),j +2r (x x ) + λp (x x (t 1) y x (t 1) (y y j ) y y j ). (27) ) (28) 2 p. Settng the dervatves to zero yelds the followng recursve update rule x k = 1 [ ] ) (c + x (k 1) 1,...,x (k 1) N b (k 1), (29) a where x k denotes the locaton of node at teraton k. Furthermore, b k = [b k 1, bk 2,..., bk N ] and n+m b k = 4 w (t) (t),j δ,j x k j=1,j xk j, (30) ( ) b k j = 4 w (t),j w(t) (t),j δ,j x k xk j, j, (31) a = 4 n+m j=1,j c = 2r x + w (t),j + 2r + x k 1 λpx (t 1) λp x k xt 1 2 p, (32) x (t 1). (33) The dwmds algorthm n [9] obtans a recursve update for locaton x by settng the dervatves of the surrogate to the th local cost functon (T (t) (X,Y)) to zero. Ths s equvalent to mnmzng the global cost functon only under anchor free localzaton (m = 0) and no a pror nformaton (r = 0). However, n our algorthm, we use the local cost functons only to derve a majorzng functon for the global cost functon and not n the mnmzaton. Moreover, the algorthm s stll decentralzed n ts mplementaton though we mnmze the global cost functon wth respect to the sensor locatons X. 9 Appendx: Optmal lkelhood rato test To test the presence of a target on a sensor lnk, j, we pose the followng hypotheses testng problem