A Distributed Multiple-Target Identity Management Algorithm in Sensor Networks

Transcription

1 A Disribued Muliple-Targe Ideniy Managemen Algorihm in Sensor Neworks Inseok Hwang, Kaushik Roy, Hamsa Balakrishnan, and Claire Tomlin Dep. of Aeronauics and Asronauics, Sanford Universiy, CA Dep. of Elecrical Engineering, Sanford Universiy, CA ishwang, kroy1, hamsa, Absrac In his paper, we develop a disribued ideniy managemen algorihm for muliple arges in sensor neworks. Each sensor is assumed o have he capabiliy of managing ideniies of muliple arges wihin is surveillance region and of communicaing wih is neighboring sensors. We use he algorihm from our companion paper [1] o incorporae local informaion abou he ideniy of a arge when i is available o a local sensor and a he same ime reduces he uncerainy of he arge s ideniy as measured by enropy. Ideniy informaion fusion is crucial for disribued ideniy managemen o compue he global informaion of he sysem from informaion provided by local sensors. We formulae his problem as an opimizaion problem and presen hree differen cos funcions, namely, Shannon informaion, Chernoff informaion, and he sum of Kullback-Leibler disances, which represen differen performance crieria. Using Bayesian analysis, we derive a daa fusion algorihm ha needs a prior probabiliy of he given daa. Finally, we demonsrae he performance of he disribued ideniy managemen algorihm using scenarios from muliple-aircraf racking in a sensor (radar) nework wih differen fusion crieria. I. INTRODUCTION The las few decades have seen many advances in wireless communicaion echniques and in sensor echnology. These advances, combined wih growing ineres in boh miliary and civilian applicaions in using disribued sensors, have led o he concep of a sensor nework. These applicaions include balefield surveillance and enemy racking in miliary applicaions, and habia monioring, environmen observaion, and raffic surveillance in civilian applicaions ([2], [3] and references herein). A sensor nework is a nework of sensor nodes which have local sensing, processing, and communicaion capabiliies. Many applicaions of sensor neworks, such as arge racking and habia monioring using only local informaion (i.e., informaion obained by each sensor), have a unique problem ha does no arise in a cenralized nework: scalable disribued informaion fusion. This implies ha he global saes of he sysem mus be esimaed and mainained using only local informaion available o local sensors. In his paper, we presen a Disribued Muliple-Targe Ideniy Managemen (DMIM) algorihm which can esimae he ideniies of muliple maneuvering arges in sensor neworks. For DMIM in sensor neworks, informaion abou he ideniy of a arge may become available o a local sensor, and hus we need mehods which can incorporae his new informaion o reduce he uncerainy of he sysem. For he case in which he number of arges is consan, he Sinkhorn algorihm [4] is used in [5], [6], [7]. However, in disribued sensor neworks, he number of arges in he surveillance region of each sensor may change over ime and he Sinkhorn algorihm may no converge for his case. In [1], we have developed an algorihm which can solve his problem in polynomial ime and in his paper, we use his algorihm for local informaion incorporaion. A crucial par of he DMIM algorihm is disribued informaion fusion. In disribued sensor neworks, ideniy informaion of muliple arges is mainained by individual sensors and each sensor can manage only ideniies of he arges wihin is surveillance region. Thus, each sensor has only a knowledge abou is neighborhood, no he global picure of he whole sysem. To ge he global informaion from informaion mainained by individual local sensors, we need an informaion fusion algorihm. To fully exploi he capabiliy of sensor neworks, his algorihm should be scalable, i.e., adding/deleing sensors or arges ino a sensor nework can be handled efficienly, and disribued, i.e., he algorihm can be implemened in individual sensors. We formulae he informaion fusion problem as an opimizaion problem and propose hree differen cos funcions: Shannon informaion, Chernoff informaion, and he sum of Kullback-Leibler disances o represen differen performance crieria. Using Bayesian analysis, we also derive an informaion fusion algorihm ha needs a prior probabiliy of he given daa. Finally, we apply he DMIM algorihm o muliple-aircraf racking problems in sensor neworks and demonsrae he performance of he proposed informaion fusion algorihms under differen scenarios. This paper is organized as follows: In Secion II, he Disribued Muliple-Targe Ideniy Managemen (DMIM) algorihm including local informaion incorporaion and belief informaion fusion is presened. Secion III presens applicaions of he DMIM algorihm o muliple-aircraf racking in sensor neworks. Finally, our conclusions are presened in Secion IV. II. DISTRIBUTED MULTIPLE-TARGET IDENTITY MANAGEMENT (DMIM) In his secion, we consider he problem of managing ideniies of muliple arges in sensor neworks. Each sensor

2 is assumed o have is own surveillance region, and o communicae wih is neighboring sensors. A wo-sensor example is shown in Figure 1 in which he circles represen he surveillance regions of he sensors. We assume ha each sensor has he capabiliy o compue he posiion esimaes and manage he ideniy of arges wihin is surveillance region. For disribued ideniy managemen, we have o consider he possibiliy ha he number of arges wihin he surveillance region of a sensor could change over ime. For example, a arge migh leave or ener he surveillance region of a sensor. Anoher imporan problem ha has o be addressed is scalable and disribued informaion fusion o ge he global esimae of he sysem from informaion compued by individual local sensors. These problems are unique for disribued sensor nework applicaions. Therefore, we propose a scalable Disribued Muliple-Targe Ideniy Managemen (DMIM) algorihm ha can manage muliple-arge ideniies efficienly in a disribued sensor nework environmen. The srucure of DMIM is shown in Figure 2: le us sar wih he ideniy managemen algorihm. ac1 sensor 1 ac2 sensor 2 Fig. 1. A disribued muliple-arge ideniy managemen scenario for a wo-sensor nework. Informaion Fusion Ideniy Managemen ac3 ac4 even-driven, query-based communicaion communicaion communicaion belief vecor sensor 1 Daa Associaion belief vecor Ideniy Managemen sensor 2 Local Informaion Incorporaion Belief Marix Updae Ideniy managemen Informaion Fusion Fig. 2. The srucure of he Disribued Muliple-Targe Ideniy Managemen (DMIM) algorihm for a wo-sensor example. A. Daa Associaion Suppose here are T arges and T ideniies, for example, T aircraf wih ideniies {piper, cherokee, cessna, }, in he surveillance region of sensor i. Then, he problem of managing ideniies of muliple arges is o mach each arge o is ideniy over ime. For his, we use he idea of he Ideniy-Mass-Flow in [5]. The idea of he Ideniy- Mass-Flow is ha an ideniy is reaed as a uni mass assigned o a arge. These masses canno be desroyed or creaed, and flow from a arge ino anoher hrough he mixing marix, M(k) a ime k. The mixing marix is an T T marix whose elemen M ij (k) represens he probabiliy ha arge i a ime k 1 has become arge j a ime k. Thus, he mixing marix is a doubly sochasic marix; ha is, is column sums and row sums are equal o 1. The oupu of he Daa Associaion block is he mixing marix. B. Belief Marix Updae We use a belief vecor o represen he ideniy of a arge probabilisically. For muliple arges, we have a belief marix B(k) whose columns are belief vecors of he arges. Thus, enry B ij (k) represens he probabiliy ha arge j can be idenified as label (or name) i a ime k. The Belief Marix Updae block mainains ideniy informaion sored in a T T belief marix B(k) over ime. The evoluion of his belief marix is governed by he equaion [5]: B(k) = B(k 1)M(k) (1) We can show ha (1) keeps row and column sums of he belief marix consan when he numbers of arges and ideniies are he same. However, his is no he case for disribued ideniy managemen since he number of he arges wihin he surveillance region of individual sensors may change over ime. There are wo possible cases: a arge leaves or eners he surveillance region of a sensor. When a arge leaves, we delee he corresponding column in he belief marix managed by he sensor. When a arge eners he surveillance region of a sensor, here are wo possible cases: (i) he arge comes from he surveillance region of anoher sensor, which may be queried, or (ii) he arge comes from he ouside of he surveillance region of a sensor nework. For hese cases, we propose a scalable, even-driven, query-based belief marix updae algorihm: Algorihm 1: Even-driven, query-based Belief Marix Updae For sensor i and arge if arge leaves he surveillance region of sensor i. hen delee he corresponding column in he belief marix. if a arge eners he surveillance region of sensor i. hen send a query abou he ideniy of arge. if here is an answer yes and receive he belief vecor of arge, hen augmen he belief marix wih he belief vecor received. else augmen he belief marix wih a belief vecor wih a new ideniy assigned o he arge.

3 For disribued ideniy managemen, a belief marix managed by each sensor may no be a square marix bu migh more likely be a skinny marix which has more rows han columns. The belief marix may no be a doubly sochasic marix, bu i should be a sochasic marix wih column sums equal o one. Is row sums remain consan because an ideniy mass canno be desroyed or creaed. I also has he propery ha he sum of column sums is equal o he sum of row sums; ha is, even hough he number of arges in he surveillance region of each sensor changes, he ideniy mass is conserved in he surveillance region. Since he evoluion of he belief marix is governed by (1), hese characerisics of he belief marix are preserved over ime. C. Local Informaion Incorporaion In his secion, we consider he case in which local informaion abou he ideniy of a arge is available o a local sensor. The local informaion has he form of a belief vecor and when available, we use he local informaion o decrease he uncerainy of he belief marix measured by enropy. The enropy of a L T belief marix is defined as H(B(k)) L T i=1 j=1 B ij(k) log B ij (k). Then, he problem is how o incorporae his informaion o he belief marix. From he idea of he Ideniy-Mass- Flow and he characerisics of (1), we know ha he belief marix should have he following properies: is column sums are equal o one, is row sums remain consan, and he sum of row sums and he sum of column sums are equal. However, if we replace he column in he belief marix wih he local informaion, i is no guaraneed ha he new belief marix has he above properies. Thus, we have developed a polynomial-ime algorihm ha can check wheher he available local informaion can be incorporaed (i.e., he new belief marix is scalable or almos scalable) and if so, can make he new belief marix have he above properies. We refer he reader o our companion paper [1] for he deails. The local informaion incorporaed may no necessarily decrease he uncerainy (enropy) of he belief marix. Therefore, local informaion is incorporaed only when i reduces he uncerainy of he belief marix. The local informaion incorporaion algorihm can be described as follows: Algorihm 2: Local Informaion Incorporaion Given: local informaion (belief vecor) of a arge and a belief marix B(k). Make a marix B (k) by replacing he column corresponding o he arge in B(k) wih he local informaion. if B (k) is almos scalable hen B new (k) := S(B (k)) if H(B new (k)) H(B(k)) hen B(k) := B new (k) else B(k) := B(k) else B(k) := B(k) where is he operaor S represens he marix scaling process in [1]. D. Belief Informaion Fusion In his secion, we consider he problem of combining wo belief vecors of he same arge from wo differen sensors. Informaion fusion can be formulaed as an opimizaion problem such ha he fused informaion is he one ha minimizes a cos funcion which represens a performance crierion. For opimizaion, we propose hree differen cos funcions: Shannon informaion, Chernoff informaion, and he sum of Kullback-Leibler disances. Shannon informaion: The Shannon informaion is defined as n H(b ) = b (i) log b (i) (2) i=1 where b [0, 1] n wih i b (i) = 1. The Shannon informaion (also known as enropy) is a measure of he uncerainy of a sysem. Thus, he minimizaion of he Shannon informaion selecs a belief vecor ha is mos informaive in he sense of minimum enropy. Suppose b 1 and b 2 are belief vecors of arge compued by sensor 1 and sensor 2 respecively. Since he mos common daa fusion algorihms compue a linear combinaion of wo daa, we propose he following fusion sraegy: b = ωb 1 + (1 ω)b 2 (3) where ω [0, 1], b i [0, 1] n wih n j=1 b i(j) = 1 for i {1, 2}, and n j=1 b (j) = 1. Then, he problem of compuing he fused belief vecor becomes a problem o find a weigh, ω, which minimizes he cos funcion in (2). If we use he fusion sraegy in (3), he Shannon informaion of he new fused informaion is H(b ) = H(ωb 1 +(1 ω)b 2 ) ωh(b 1 )+(1 ω)h(b 2 ) (4) From (4), we can see ha he minimum is always achieved a eiher ω = 0 or ω = 1. This means ha a fused belief vecor ha has he minimum Shannon informaion is eiher of he wo given belief vecors, which is a hard choice. For some applicaions such as ideniy managemen in his paper, he hard choice may no be desirable since i ignores one possibiliy compleely and hus migh quickly lead o a wrong answer over ime if no immediaely. Thus, we propose a sof choice mehod which has ω (0, 1) for almos all cases. Moivaed by he fac ha Shannon informaion minimizaion chooses a belief vecor which has he minimum enropy, we propose o use he inverse of he Shannon informaion of a belief vecor as a weigh. Thus, we pu large confidence on a belief vecor which has small Shannon informaion. Then, a new belief vecor b = [b (i)] is b H(b 1 ) 1 b 1 (i) (i) = H(b 1 ) 1 + H(b 2 ) 1 + H(b 2 ) 1 b 2 (i) H(b 1 ) 1 (5) + H(b 2 ) 1

4 From (3) and (5), we ge ω = H(b 1 ) 1 H(b 1 ) 1 + H(b 2 ) 1 = H(b 2 ) H(b 1 ) + H(b 2 ) When H(b 1 ) = H(b 2 ) = 0, we se ω = 1 2. ω = 0 if H(b 2 ) = 0 (no uncerainy in b 2 ) and ω = 1 when H(b 1 ) = 0 (no uncerainy in b 1 ). In hese cases, he fused belief vecor compued by he proposed fusion algorihm is a belief vecor which has no uncerainy. This fusion algorihm is a sof choice mehod since he fused daa is a convex combinaion of he wo given daa wih a larger weigh on he daa which has smaller enropy han he oher. From (4) and (6), he Shannon informaion of he new belief H(b ) has he propery ha: H(b ) 2H(b 1)H(b 1 ) H(b 1 ) + H(b 2 ) or 2H(b ) 1 H(b 1 ) 1 +H(b 2 ) 1 (7) Inequaliy (7) ells us ha he achievable minimum uncerainy of he fused belief vecor wih he fusion sraegy in (3) wih (6) as a weigh is under-bounded by uncerainies of he given informaion. In oher words, he maximum achievable cerainy (inverse of he Shannon informaion) is upper-bounded by he arihmeic mean of he inverse of he Shannon informaion of he given belief vecors. If we use he fusion sraegy in (3), we can also derive he upper bound of he Shannon informaion of he new belief vecor: H(b ) ω 2 H(b 1) + (1 ω) 2 H(b 2) + ω(1 ω)(h(b 1) +H(b 2) + D(b 1 b 2) + D(b 2 b 1)) (8) where D(p q) p(i) i p(i) log( q(i) ) is he Kullback-Leibler disance [8]. If we use ω in (6), hen The new belief vecor in (11) saisfies ([8], [9]) (6) D(b b 1 ) = D(b b 2 ) (12) This fusion sraegy is differen from ha in (3) which is a convex combinaion of he wo daa. From (12), he minimizaion of he Chernoff informaion is equivalen o finding a funcion ha is in he middle of he wo original funcions, where he middle is defined in erms of he Kullback-Leibler disance. In oher words, Chernoff informaion minimizaion could be inerpreed as selecing a probabiliy vecor which is equally close in erms of he Kullback-Leibler disance o he original probabiliy vecors. This fusion algorihm does no pu more confidence on one han he oher. Thus, his cos funcion could be useful when we do no know he qualiy of informaion obained from individual sensors; by choosing he middle poin of he wo pieces of informaion, we could minimize he bias over ime. However, he fused belief vecor compued by he Chernoff informaion minimizaion algorihm may have larger enropy han ha compued by he algorihm in (3) wih (6). Sum of he Kullback-Leibler disances: Since he Kullback-Leibler disance is no symmeric, we consider wo possible opimizaion problems: minimize D(b b 1 ) + D(b b 2 ) n subjec o j=1 b (j) = 1 b (j) 0 minimize D(b 1 b ) + D(b 2 b ) n subjec o j=1 b (j) = 1 b (j) 0 (13) (14) where b (j) is he jh elemen of a vecor b. Le us firs consider he opimizaion problem in (13). The Lagrangian is given by H(b ) 2H(b1)H(b1) H(b1)H(b2)[D(b1 b2) + D(b2 b1)] n + H(b 1 ) + H(b 2 ) (H(b 1 ) + H(b 2 )) 2 L(b, λ) = D(b b 1 )+D(b b 2 )+λ( b (j) 1) (15) (9) j=1 Thus, we can analyically compue he upper and lower To ge an opimal soluion, we se he derivaives of L wih bounds of he Shannon informaion of he new belief vecor respec o b (i) and o λ o be equal o zero. Then, we ge using he fusion sraegy in (3) wih (6). Thus, he Shannon a new belief vecor: informaion cos funcion would be useful when we have good knowledge abou he performance and/or fideliy of b b1 (i)b 2 (i) (i) = n (16) each sensor, since we can ge a soluion which has lower j=1 b1 (j)b 2 (j) enropy by weighing informaion ha has smaller enropy From (16), we see ha he fused daa is a geomeric mean more han he oher. However, if we do no have such knowledge, we may ge a biased soluion by consisenly puing for Chernoff informaion minimizaion when ω = 1 2 of he given daa. The fused daa is he same as ha in (11) more confidence on one piece of informaion (possibly he. Thus, his daa fusion sraegy can be inerpreed as a special case wrong one) han he oher. of he Chernoff informaion minimizaion mehod. Chernoff informaion: The Chernoff informaion is defined as n Now, le us consider he opimizaion problem in (14). C(b 1, b 2 ) = min log( The Lagrangian is given by b 1 (i) ω b 2 (i) 1 ω ) (10) n 0 ω 1 i=1 L(b, λ) = D(b 1 b )+D(b 2 b )+λ( b (j) 1) (17) If ω minimizes (10), he new belief vecor b (= [b j=1 (i)] for i = {1, 2,, n}) is Similarly, we ge an opimal soluion: b b 1 (i) ω b 2 (i) 1 ω (i) = n j=1 b (11) 1(j) ω b 2 (j) b b 1 (i) + b 2 (i) (i) = 1 ω n j=1 [b 1(j) + b 2 (j)] = b 1(i) + b 2 (i) (18) 2 In his case, he fused daa is he arihmeic mean of he given daa. This fusion sraegy is he same as ha in (3)

5 when ω = 1 2. Thus, from (4) and (8), we ge he lower and upper bounds of Shannon informaion of he new belief vecor: H(b ) H(b1)+H(b2) 2 H(b ) H(b 1)+H(b 2 ) 2 + D(b 1 b 2 )+D(b 2 b 1 ) (19) 4 Therefore, he fusion algorihms obained by solving he opimizaion problems in (13) or (14) are o average he given daa eiher geomerically or arihmeically. This is similar o Chernoff informaion minimizaion and hus hese fusion sraegies would be useful when we wan o ge unbiased fused daa in siuaions where we do no have good a prior informaion abou a sysem. An example would be a case in which informaion from one sensor is wrong due o failure of he sensor or he malicious inen of he sensor ha is unknown a priori. These informaion fusion sraegies would be robus o his wrong informaion since hey do no pu more confidence on one (possibly incorrec informaion) han he oher, bu average hem o compue a fused belief vecor. Bayesian approach: In his secion, we derive a fused belief vecor using a Bayesian approach. Suppose he arge s ideniy θ {1, 2,, N} and wihou loss of generaliy, suppose here are wo sensors. Denoe evens X 1 and X 2 o be observaions a sensor 1 and sensor 2 respecively. We are assumed o be given informaion b 1 (θ) P (θ X 1 ) from sensor 1 and b 2 (θ) P (θ X 1 ) from sensor 2 where P ( ) is a condiional probabiliy. Then, he problem of informaion fusion is o find he a poseriori probabiliy P (θ X 1, X 2 ). We assume P (X 1, X 2 θ) = P (X 1 θ)p (X 2 θ) since given he ideniy of a arge, he evens ha i is observed by sensor 1 or sensor 2 are independen in disribued ideniy managemen. Using he Bayes rule, we ge Since P (θ X i ) = P (θ X 1, X 2 ) = P (X 1 θ)p (X 2 θ)p (θ) P (X 1, X 2 ) P (Xi θ)p (θ) P (X i ) for i {1, 2}, we obain P (θ X 1, X 2 ) = b 1(θ)b 2 (θ) P (X 1 )P (X 2 ) P (θ) P (X 1, X 2 ) Therefore, a fused belief vecor is b (ˆθ) = arg max P (θ X 1, X 2 ) = b 1(θ)b 2 (θ) 1 θ P (θ) c (20) (21) (22) where c is a normalizaion consan. This is an ineresing resul because he fused daa does depend only on he given daa (b 1 (θ), b 2 (θ)) and he a priori probabiliy P (θ). Thus, if we knew a priori informaion, we could compue he a poseriori probabiliy (i.e., he fused daa). However, since we may no know he a priori probabiliy for some applicaions such as disribued ideniy managemen in his paper, we canno compue he fused daa from (22). In order o compue he fused daa for his case, we have o assume P (θ) eiher from he characerisics of he sysems or from ha of applicaions. For example, due o he lack of informaion abou he sysem, we assume ha he a priori probabiliy is a geomeric mean of he given daa (P (θ) = b1 (θ)b 2 (θ)). Then, we can ge exacly he same resul as ha in (16) which minimizes he sum of Kullback-Leibler disances o he original daa in (13). Thus, he a poseriori probabiliy is he same as he a priori probabiliy; ha is, we canno exrac any informaion from he given daa. From Bayesian analysis, we can see ha he daa fusion sraegies such as Chernoff informaion minimizaion and he minimizaion of he sum of Kullback-Leibler disances in (13) compue he soluion in a similar form o he soluion produced by he Bayesian approach. III. SIMULATIONS Several simulaions are presened in his secion o highligh he performance and capabiliies of he DMIM algorihm. Specifically, he scenarios consis of saionary sensors (e.g., air raffic conrol radars) racking muliple aircraf hrough wo-dimensional space. Individual sensors are assumed o have he capabiliy o compue he posiion esimaes of aircraf. Measuremens are available o sensors when inside a sensing radius, se o 10 km, while wo sensors can communicae if inside he communicaion radius, se o 20 km. P 1 (A) Fig. 3. y posiion [km] x 10 4 L x posiion [km] x 10 4 Aircraf rajecories for wo-aircraf, wo-sensor scenario (a) K L disance (bis) R Fig. 4. A cooperaive wo-sensor scenario. (a) Belief of aircraf 1 having ideniy A. (b) Efficiency of belief esimaes as measured in bis. (Solid lines are local esimaes and dashed lines are global esimaes. Symbols x and + are used for sensor L and R respecively. Symbols,, and are used for he ideniy fusion algorihms using he sum of Kullback-Leibler disances beween he local esimaes and he global esimae (corresponding o an arihmeic mean), he sum of Kullback-Leibler disances beween he global esimae and he local esimaes (corresponding o a geomeric mean), and Shannon informaion, respecively.) The firs wo simulaions involve a sysem of wo sensors, denoed sensor L for lef and R for righ, observing wo (b)

6 aircraf, denoed 1 and 2. Aircraf 1 sars in sensor L s surveillance region, while aircraf 2 sars in R s surveillance region, as shown in Figure 3. This figure shows he rue posiions of he aircraf (solid lines and x s) and he global posiion esimaes made by he sensors (dashed lines and o s). Boh aircraf ravel souheas wih velociy 200 m/s. The rue ideniy of aircraf 1 and 2 are A and B respecively, bu simulaion is iniialized wih belief vecors [0.8 ] T for aircraf 1 and [ 0.8] T for aircraf 2 respecively. [0.8 ] T means ha aircraf 1 is hough o have ideniy A wih 80% probabiliy and B wih 20% probabiliy. Three imes correspond o imporan evens in he simulaion. A ime 7, aircraf 1 eners R s surveillance region, allowing global belief esimaes based on fusion echniques discussed in he previous secion and sensor R receives he ideniy informaion abou aircraf 1 from sensor L using Algorihm 1. A ime 11, local informaion is obained by sensor R. Aircraf 2 is now hough o have belief [ ] T. Since his local informaion makes a new belief marix scalable and decreases enropy (uncerainy) of he belief marix, we replace he belief vecor of aircraf 2 in he belief marix wih his local informaion. Since row sums of he belief marix should remain consan, he belief vecor for aircraf 1 in he new belief marix becomes [ ] T. This local informaion incorporaion is performed using Algorihm 2. Finally, a ime 18, aircraf 1 leaves sensor L s surveillance region. Because he wo aircraf mus share ideniy A and B, an esimae of he probabiliy of aircraf 1 having ideniy A deermines fully he belief marix for he sysem. Thus, only P 1 (A) (probabiliy ha aircraf 1 has ideniy A) is ploed in Figure 4-(a), which presens belief informaion according o differen esimaors. Solid lines indicae P 1 (A) for each observer, while he various dashed lines indicae P 1 (A) for hree global esimae mehods discussed in he previous secion. The hree mehods are using Shannon informaion, minimizing he sum of Kullback-Leibler disances beween he local esimaes and he global esimae (corresponding o an arihmeic mean of he local beliefs), and minimizing he sum of Kullback-Leibler disances beween he global esimae and he local esimaes (corresponding o a geomeric mean of he local beliefs). The Bayesian approach and he Chernoff approach have a similar form o he geomeric mean and are hus no ploed. Figure 4-(b) presens he efficiency of each mehod of esimaing he belief of aircraf 1. This figure only covers imes 7 o 17, since hese are he imes when he informaion fusion algorihms are used. We use he Kullback-Leibler disance beween he correc belief vecor and he esimaed belief vecor (i.e., D(b rue b es )) as a performance measure since his Kullback-Leibler disance measures he inefficiency of esimaors [8], [10], [11]. As shown in he figure, he inefficiency in sensor L s belief of aircraf 1 is consan, since i always has belief P 1 (A) = 0.8. Sensor R s inefficiency drops a ime 11 when local informaion is incorporaed. The various global esimaes lie beween he wo local belief esimaes. For his scenario, global belief esimaes are bes obained hrough Shannon informaion mixing in (3) wih (6) as expeced. P 1 (A) (a) K L disance (bis) Fig. 5. A malicious wo-sensor scenario. (a) Belief of aircraf 1 having ideniy A. (b) Efficiency of belief esimaes as measured in bis. (Solid lines are local esimaes and dashed lines are global esimaes. Symbols x and + are used for sensor L and R respecively. Symbols,, and are as in Figure 4.) The second scenario presens he same aircraf and rajecory, bu sensor R, hrough error or malice, reverses he local informaion received a ime 11. Tha is, i believes P 1 (A) = 0.1, raher han P 1 (A) = 0.9. The rajecory plo is exacly he same as in Figure 3, bu belief esimaes P 1 (A) are now hose shown in Figure 5-(a). The inefficiency of he belief esimae is shown in Figure 5-(b). As in he previous scenario, sensor L s belief of aircraf 1 and hus is inefficiency in ha belief remain consan. However, sensor R receives incorrec local informaion, hereby increasing is inefficiency o 3.32 bis, above he scale of Figure 5-(b). The global belief esimaes again fall somewhere in beween. However, because sensor R conribues incorrec informaion, he global belief esimae wih he leas inefficiency is now he arihmeic average of he local esimaes in (18), while he mehod based on he Shannon informaion resuls in he highes inefficiency. Tha is, he arihmeic average mehod is mos robus o incorrec informaion. To analyze he performance of he various Cooperaive efficiency (b) Malevolen efficiency Fig. 6. Cooperaive efficiency versus malicious efficiency for each global belief esimae, for he wo scenarios presened above. Symbols,, and are as in Figure 4.) belief fusion mehods, we presen Figure 6 ha plos he ordered pairs (D coop,d mal ) for each mehod of global belief fusion. The quaniies D coop and D mal refer respecively o he efficiency of he global esimaes for he cooperaive

7 and malicious scenarios and are aken from Figures 4-(b) and 5-(b). A poin closer o he origin in boh axes is more efficien overall han anoher poin. From Figure 6, one can argue ha using Shannon informaion is more efficien for scenarios in which all sensors are known o be cooperaive, while arihmeic combinaion is more efficien for scenarios in which here is a high probabiliy of malicious sensors. However, his conclusion depends on he configuraion of sensor neworks: he number of malicious sensors and heir locaions. y Prob(Targe 1=Aircraf A) A B = x Arih. mean Geom. mean Shannon info Fig. 7. A general sensor nework scenario. (Top) Aircraf rajecories for wo-aircraf, 41-sensor scenario. (Boom) Global belief esimaes of aircraf 1 having ideniy A. Figure 7 shows a complex scenario involving many sensors. I is assumed ha sensors ha can see he arges are all iniialized o he same belief vecors. Targe 1 is iniialized o [0.8 ] T and arge 2 o [ 0.8] T. Figure 7-(Top) shows he rack of each arge and he coverage areas of he 41 sensors. Sensors wih solid lines are malicious, meaning heir local esimaes are [β α] T when [α β] T is sensed. Sensors wih hick solid lines gaher local informaion; his informaion is always ha arge 1 is aircraf A wih probabiliy 0.9. Targe 2 exiss o creae confusion a he sar of he scenario; however, ideniy informaion is no presened for his arge. For arge 1, local esimaes are made a each ime sep by hose sensors ha can observe he arge. The global esimaes are made a each ime sep by combining he local ideniy esimaes. In Figure 7-(Boom), he global esimaes, using each of he hree mehods described in he previous secion, are ploed over ime. Iniially, all sensors have local belief [0.8 ] T, yielding he same global esimae. A imes 6, 26, and 31, sensors wih local informaion sar observing arge 1, leading o improved global esimaes; hese evens are noed by dashed lines in Figure 7-(Boom). A imes 16 and 24, malicious sensors begin observing arge 1, leading o degraded global esimaion, as noed by doed lines in he figure. This scenario exhibis he scalabiliy of he DMIM algorihm applied o a large se of sensors racking he ideniy of a maneuvering arge. IV. CONCLUSIONS We have developed a scalable Disribued Muliple-Targe Ideniy Managemen (DMIM) algorihm which can manage ideniies of muliple maneuvering arges in sensor neworks and efficienly incorporae local informaion abou he ideniy of a arge, when available, o reduce he uncerainy of he sysem. For ideniy fusion o obain global informaion using local sensor informaion, we have formulaed an opimizaion problem and have presened hree differen cos funcions: Shannon informaion, Chernoff informaion, and he sum of Kullback-Leibler disances, which represen differen performance crieria ha could be useful for differen applicaions. Using Bayesian analysis, we have also derived an informaion fusion algorihm ha needs a priori probabiliy of he given daa. Finally, we have applied he DMIM algorihm o he problem of managing ideniies of muliple aircraf in sensor neworks and demonsraed he performance of he proposed fusion algorihms. REFERENCES [1] H. Balakrishnan, I. Hwang, and C. Tomlin. Polynomial approximaion algorihms for belief marix mainenance in ideniy managemen. In Proceedings of he 43nd IEEE Conference on Decision and Conrol, Alanis, Paradise Island, Bahamas, December [2] I.F. Akyidliz, W. Su, Y. Sankarasubramaniam, and E. Cayirci. A survey on sensor neworks. IEEE Communicaions Magazine, 40(8): , Augus [3] Ning Xu. A survey of sensor nework applicaions. hp://enl.usc.edu/ ningxu/papers, [4] R. Sinkhorn. Diagonal equivalence o marices wih prescribed row and column sums. American Mahemaical Monhly, 74: , [5] J. Shin, L.J. Guibas, and F. Zhao. A disribued algorihm for managing muli-arge ideniies in wireless ad-hoc sensor neworks. In F. Zhao and L. Guibas, ediors, Informaion Processing in Sensor Neworks, Lecure Noes in Compuer Science 2654, pages , Palo Alo, CA, April [6] I. Hwang, H. Balakrishnan, K. Roy, J. Shin, L. Guibas, and C. Tomlin. Muliple-arge Tracking and Ideniy Managemen algorihm for Air Traffic Conrol. In Proceedings of he Second IEEE Inernaional Conference on Sensors, Torono, Canada, Ocober [7] I. Hwang, H. Balakrishnan, K. Roy, and C. Tomlin. Muliple-arge Tracking and Ideniy Managemen algorihm in cluer, wih applicaion o aircraf racking. In Proceedings of he AACC American Conrol Conference, Boson, MA, June [8] T.M. Cover and J.A. Thomas. Elemens of Informaion Theory. Wiley-Inerscience, New York, [9] J.N. Kapur and H.K. Kesavan. Enropy Opimizaion Principles wih Applicaion. Academic Press, Inc., [10] D. Fox. Adaping he sample size in paricle filers hrough KLDsampling. Inernaional Journal of Roboics Research, 22, Ocober [11] J. Kasuri, R. Acharya, and M. Ramanahan. An informaion heoreic approach for analyzing emporal paern of gene expression. Bioinformaics, 19(4): , 2003.