Distributed Estimation using Bayesian Consensus Filtering

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1 1 Distributed Estiation using Bayesian Consensus Filtering Saptarshi Bandyopadhyay Student Meber, IEEE, Soon-Jo Chung Senior Meber, IEEE 6 ariv: v3 [ath.oc] 13 Oct 216 Abstract We present the Bayesian consensus filter BCF for tracing a oving target using a networed group of sensing agents and achieving consensus on the best estiate of the probability distributions of the target s states. Our BCF fraewor can incorporate nonlinear target dynaic odels, heterogeneous nonlinear easureent odels, non-gaussian uncertainties, and higher-order oents of the locally estiated posterior probability distribution of the target s states obtained using Bayesian filters. If the agents cobine their estiated posterior probability distributions using a logarithic opinion pool, then the su of Kullbac Leibler divergences between the consensual probability distribution and the local posterior probability distributions is iniized. Rigorous stability and convergence results for the proposed BCF algorith with single or ultiple consensus loops are presented. Counication of probability distributions and coputational ethods for ipleenting the BCF algorith are discussed along with a nuerical exaple. I. INTRODUCTION In this paper, the ter consensus eans reaching an agreeent across the networ, regarding a certain subect of interest called the target dynaics. Distributed and networed groups of agents can sense the target, broadcast the acquired inforation, and reach an agreeent on the gathered inforation using consensus algoriths. Potential applications of distributed estiation tass include environent and pollution onitoring, tracing dust or volcanic ash clouds, tracing obile targets such as flying obects or space debris using distributed sensor networs, etc. Consensus algoriths are extensively studied in controls [1] [6], distributed optiization [7] [1], and distributed estiation probles [11] []. Strictly speaing, consensus is different fro the ter distributed estiation, which refers to finding the best estiate of the target, given a distributed networ of sensing agents. Many existing algoriths for distributed estiation [11] [2] ai to obtain the estiated ean first oent of the estiated probability distribution of the target dynaics across the networ, but cannot incorporate nonlinear target dynaics, heterogeneous nonlinear easureent odels, non-gaussian uncertainties, or higher-order oents of the locally estiated posterior probability distribution of the target s states. It is difficult to recursively cobine local ean and covariance estiates using a linear consensus algorith because the diension of the vector transitted by each agent increases linearly with tie due to correlated process noise [21] and The authors are with the Departent of Aerospace Engineering and Coordinated Science Laboratory, University of Illinois at Urbana-Chapaign, Urbana, IL 6181, USA. eail: bandyop2@illinois.edu; schung@illinois.edu This research was supported by AFOSR grant FA Fig. 1. A heterogeneous sensor networ tracs a target, whose current position is ared by. The sensing region of soe sensors are shown in light blue. The sensors use the Hierarchical BCF algorith to estiate the probability distribution of the target s position and reach a consensus across the networ x the covariance update equation is usually approxiated by a consensus gain [22]. Multi-agent tracing or sensing networs are deployed in a distributed fashion when the target dynaics have coplex teporal and spatial variations. Hence, it is necessary to preserve the coplete inforation captured in the locally estiated posterior probability distribution of the target s states while achieving consensus across the networ. For exaple, while tracing an orbital debris in space, the uncertainty in position along the direction of velocity is uch larger than that orthogonal to the velocity direction, and this extra inforation is lost if a linear consensus algorith is used to cobine the estiated ean fro ultiple tracing stations. As shown in Fig 1, the contour plot represents the consensual probability distribution of the target s final position, where the uncertainty along the velocity direction is relatively larger than that in the orthogonal direction. The ain obective of this paper is to extend the scope of distributed estiation algoriths to trac targets with general nonlinear dynaic odels with stochastic uncertainties, thereby addressing the aforeentioned shortcoings. Bayesian filters [23] [26] recursively calculate the probability density/ass function of the beliefs and update the based on new easureents. The ain advantage of Bayesian filters over Kalan filter based ethods [27], [28] for estiation of nonlinear target dynaic odels is that no approxiation

2 2 is needed during the filtering process. In other words, the coplete inforation about the dynaics and uncertainties of the odel can be incorporated in the filtering algorith. However, Bayesian filtering is coputationally expensive. Advances in coputational capability have facilitated the ipleentation of Bayesian filters for robotic localization and apping [29] [32] as well as planning and control [33] [35]. Practical ipleentation of these algoriths, in their ost general for, is achieved using particle filtering [26], [36] and Bayesian prograing [37], [38]. This paper focuses on developing a consensus fraewor for distributed Bayesian filters. The statistics literature deals with the proble of reaching a consensus aong individuals in a coplete graph, where each individual s opinion is represented as a probability distribution [39], [4]; and under select conditions, it is shown that consensus is achieved within the group [41] [43]. Exchange of beliefs in decentralized systes, under counication constraints, is considered in [44], [45]. Algoriths for cobining probability distributions within the exponential faily, i.e., a liited class of uniodal distributions that can be expressed as an exponential function, are presented in [46], [47]. If the target s states are discrete rando variables, then the local estiates can be cobined using a tree-search algorith [48] or a linear consensus algorith [49]. In contrast, this paper focuses on developing generalized Bayesian consensus algoriths with rigorous convergence analysis for achieving consensus across the networ without any assuption on the shape of local prior or posterior probability distributions. The proposed distributed estiation using Bayesian consensus filtering ais to reach an agreeent across the networ on the best estiate, in the inforation theoretic sense, of the probability distribution of the target s states. A. Paper Contributions and Organization In this paper, we assue that agents generate their local estiate of the posterior probability distribution of the target s states using Bayesian filters with/without easureent exchange with neighbors. Then, we develop algoriths for cobining these local estiates, using the logarithic opinion pool LogOP, to generate the consensual estiate of the probability distribution of the target s states across the networ. Finally, we introduce the Bayesian consensus filter BCF, where the local prior estiates of the target s states are first updated and the local posterior probability distributions are recursively cobined during the consensus stage, so that the agents can estiate the consensual probability distribution of the target s states while siultaneously aintaining consensus across the networ. The flowchart for the algorith is shown in Fig. 2 and its pseudo-code is given in Algorith 1. The first contribution of this paper is the LogOP based consensus algorith for cobining posterior probability distributions during the consensus stage and achieving consensus across the networ. As discussed in Section III-B, this is achieved by each agent recursively counicating its posterior probability distribution of the target s states with neighboring agents and updating its estiated probability distribution Start th cycle of th agent 1. Copute prior pdf 2. Measure the target 3. Obtain easureent array 4. Copute posterior pdf Exit 6. Set 5. for to. Consensus stage 5.1. Obtain the pdfs 5.2. Cobine using LogOP Fig. 2. Flowchart for BCF LogOP algorith describing the ey steps for a single agent in a single tie step. Steps 1 4 represent the Bayesian filtering stage, while step 5 represents the consensus stage. of the target s states using the LogOP. As shown in Fig. 3, cobining posterior probability distributions using the linear opinion pool LinOP typically results in ultiodal solutions, which are insensitive to the weights [4]. On the other hand, cobining posterior probability distributions using the LogOP typically results in uniodal, less dispersed solutions, thereby indicating a ointly preferred consensual distribution by the networ. Moreover, as discussed in Section III-B, the optial solution does not depend upon the choice of scale of the prior probability distribution and LogOP is is externally Bayesian [4] See Fig. 3 c-d. The KL divergence is the easure of the inforation lost when the consensual estiate is used to approxiate the locally estiated posterior probability distributions. In Theore 6, we show that the LogOP algorith on a strongly connected SC balanced graph iniizes the inforation lost during the consensus stage, i.e., the consensual probability distribution iniizes the su of KL divergences with the locally estiated posterior probability distributions. Methods for counicating probability distributions and the effects of inaccuracies on the consensual probability distribution are discussed in Section III-C. The second contribution of this paper is the BCF algorith presented in Section IV. As illustrated in Fig. 2 and Algorith 1, each agent generates a local estiate of the posterior probability distribution of the target s states using the Bayesian filter. Note that easureent exchanges with neighbors during the Bayesian filtering stage are not andatory and can be oitted. During the consensus stage, the LogOP algorith is executed ultiple ties to reach an agreeent across the networ. The nuber of consensus loops n loop N depends on the second

3 3 largest singular value of the atrix representing a SC balanced counication networ topology. Moreover, the convergence conditions for a given nuber of consensus loops are derived. Note that this consensual probability distribution fro the current tie step is used as the prior probability distribution in the next tie step, as shown in Fig 2. The novel features of the BCF algorith are: The algorith can be used to trac targets with general nonlinear tie-varying target dynaic odels. The algorith can be used by a SC balanced networ of heterogeneous agents with general nonlinear tie-varying easureent odels. The algorith achieves global exponential convergence across the networ to the consensual probability distribution of the target s states. The consensual probability distribution the best estiate, in the inforation theoretic sense because it iniizes the su of KL divergences with the locally estiated posterior probability distributions. If a central agent receives all the local posterior probability distributions and is tased to find the best estiate in the inforation theoretic sense, then it would also yield the sae consensual probability distribution. Hence, we clai to have achieved distributed estiation using the BCF algorith. The Hierarchical BCF algoriths, in Section IV-B, is used when soe of the agents do not observe the target. In Section V, we apply the Hierarchical BCF algorith to the proble of tracing orbital debris in space using the space surveillance networ on Earth. B. Notation The tie index is denoted by a right subscript. For exaple, x represents the true states of the target at the th tie instant. The target is always within the copact state space, i.e., x, N. Also, x 1: represents an array of the true states of the target fro the first to the th tie instant. The agent index is denoted by a lower-case right superscript. For exaple, z represents the easureent taen by the th agent at the th tie instant. The sybol P refers to probability of an event. F represents the estiated probability density function pdf of the target s states over the state space, by the th agent at the th tie instant. The sybol p also refers to pdf or probability ass function pf over the state space. During the consensus stage at the th tie instant, F,ν represents the local pdf of the target s states by the th agent at the ν th consensus step and F represents the consensual pdf to which each F,ν converges. Let be the Borel σ algebra on. The counication networ topology at the th tie instant is represented by the directed tie-varying graph G, where all the agents of the syste for the set of vertices V which does not change with tie and the set of directed edges is denoted by E. The neighbors of the th agent at the th tie instant is the set of agents fro which the th agent receives inforation at the th tie instant and is denoted by N, i.e., l N if and only if l E for all l, V. The set of inclusive neighbors of the th agent is denoted by J := N {}. Let N and R be the sets of natural nubers positive integers and real nubers respectively. The set of all by n atrices over the field of real nubers R is denoted by R n. Let λ and σ represent the eigenvalue and the singular value of a square atrix. Let 1 = [1, 1,..., 1] T, I,, and φ be the ones vector, the identity atrix, the zero atrix of appropriate sizes, and the epty set respectively. The sybols,, and sgn represent the absolute value, ceiling function, and signu function respectively. Let ln and log c represent the natural logarith and the logarith to the base c. Finally, lp represents the l p vector nor. The L p function denotes the set of all functions fx : R nx R with the bounded integral fx p dµx 1/p, where µ is a easure on. II. PRELIMINARIES In this section, we first state four assuptions used throughout this paper and then introduce the proble stateent of BCF. Next, we discuss an extension of the Bayesian filter to sensor fusion over a networ. Assuption 1. In this paper, all the algoriths are presented in discrete tie. Assuption 2. The state space R nx is closed and bounded. Hence, by the Heine Borel theore cf. [5, pp. 86], is copact. Assuption 3. All continuous probability distributions are upper-bounded by soe large value M R. Assuption 4. The inter-agent counication tie scale is uch faster than the tracing/estiation tie scale. Assuptions 1 and 2 are introduced to discretize the tie and bound the state space, so that the algoriths are coputationally tractable. Under these assuptions, particle filters [36], approxiate grid based filters, or histogra filters [32] can be used to execute the algoriths developed in this paper. Assuptions 2 and 3 are introduced to tae advantage of the results in inforation theory and easure theory, which deal with bounded functions on copact support. Under Assuption 4, the agents can execute ultiple consensus loops within each tracing tie step. We envisage that the results in this paper could be extended to continuous tie if the Foer Plan equations are solved efficiently [51] and additional issues due to counication delay and tie scale separation are addressed. Under Assuption 1, we do not discuss continuous tie related issues in this paper. Next, we show that discrete and continuous probability distributions can be handled in a unified anner. Rear 1. Let be the Borel σ algebra for. The probability of a set A ay be written as the Lebesgue Stieltes integral PA = px dµx, where µ is a A easure on. In the continuous case, px is the pdf and µ is the Lebesgue easure. In the discrete case, px is the pf and µ is the the counting easure. Hence, in this paper, we only deal with pdfs over with µ as the Lebesgue easure. Siilar arguents will also wor for pfs or ixed probability distributions.

4 4 A. Proble Stateent Let R nx be the n x -diensional state space of the target. The dynaics of the target in discrete tie {x, N, x } is given by: x = f x 1, v 1, 1 where f : R nx R nv R nx is a possibly nonlinear tievarying function of the state x 1 and an independent and identically distributed i.i.d. process noise v 1, where n v is the diension of the process noise vector. Let heterogeneous agents siultaneously trac this target and estiate the pdf of the target s states where does not change with tie. The easureent odel of the th agent is given by: z = h x, w, {1,..., }, 2 where h : Rnx R nw R nz is a possibly nonlinear tievarying function of the state x and an i.i.d. easureent noise w, where n z, n w are diensions of the easureent and easureent noise vectors respectively. Note that the easureent odel of agents is quite general since it accoodates heterogeneous sensors with various bandwidths, ranges, and noise characteristics and partial state observation. The obective of the BCF is to estiate the target s states and aintain consensus across the networ. This obective is achieved in two steps: i each agent locally estiates the pdf of the target s states using a Bayesian filter, and ii each agent s local estiate converges to a global estiate during the consensus stage see Fig. 1. The obective of Bayesian filtering with/without easureent exchange, discussed in Section II-B, is to estiate the posterior pdf of the target s states at the th tie instant, which is denoted by F, {1,..., }, using the estiated prior pdf of the target s states F 1 fro the 1th tie instant and the new easureent array obtained at the th tie instant. The obective of the consensus stage, discussed in Section III, is to guarantee pointwise convergence of each estiated pdf F,ν to the consensual pdf F. B. Bayesian Filter with Measureent Exchange A Bayesian filter consist of two steps: i the prior pdf of the target s states is obtained during the prediction stage, and ii the posterior pdf of the target s states is updated using the new easureent array during the update stage [23] [26]. The Bayesian filter gives the exact posterior probability distribution, hence it is the best possible estiate of the target fro the available inforation. Exchange of easureents is optional since heterogeneous agents, with different priors, fields of view, resolutions, tolerances, etc., ay not be able to cobine easureents fro other agents. For exaple, if a satellite in space and a low flying quadrotor are observing the sae target, then they cannot exchange easureents due to their different fields of view. Furtherore, a centralized estiator ay not be able to cobine easureents fro all heterogeneous agents in the networ to estiate p x z {1,...,}, because it would have to use a coon prior for all the agents. Hence, in this paper, we let the individual agents generate their own posterior pdfs of the target s states and then cobine the to get the best estiated pdf fro the networ. If an agent can cobine easureents fro another neighboring agent during its update stage, then we call the easureent neighbors. In this section, we extend the Bayesian filter by assuing that each agent transits its easureents to other agents in the networ, and receives the easureents fro its easureent neighbors. Let z S := {z l, l S } denote the array of easureents taen at the th tie instant by the easureent neighbors of the th agent, where S J denotes the set of easureent neighbors aong the inclusive neighbors of the th agent. Next, we assue that the prior is available at the initial tie. Assuption 5. For each agent, the initial prior of the states F = p x, is assued to be available. In case no nowledge about x is available, F is assued to be uniforly distributed over. In Bayesian Filtering with Measureent Exchanges, the th agent estiates the posterior pdf of the target s states F = p x z S at the th tie instant using the estiated consensual pdf of the target s states F 1 = p 1 x 1 fro the 1 th tie instant and the new easureent array z S obtained at the th tie instant. The prediction stage involves using the target dynaics odel 1 to obtain the estiated pdf of the target s states at the th tie instant via the Chapan Kologorov equation: ˆ p x = p x x 1 p 1 x 1 dµx 1 3 The probabilistic odel of the state evolution p x x 1 is defined by the target dynaics odel 1 and the nown statistics of the i.i.d. process noise v 1. Proposition 1. The new easureent array z S is used to copute the posterior pdf of the target s states F = p x z S p x z S = during the update stage using Bayes rule 4: l S p l zl x p x, 4 l S p l zl x p x dµx The lielihood function p l zl x, l S is defined by the easureent odel 2, and the corresponding nown statistics of the i.i.d. easureent noise w l. Proof: We need to show that the ter pz S x in the Bayesian filter [26] siplifies to l S p l zl x. Let the agents r, r + 1,... be easureent neighbors of the th agent at the th tie instant i.e., r, r + } 1,... N. Let us define z S \{r} := {z l, l S \ {r} as the easureent array obtained by the th agent at the th tie instant, which does not contain the easureent fro the r th agent. Since the easureent noise is i.i.d., 2 describes a Marov process of order one, we get:

5 5 pz S, x px = pzs, x pz S \{r}, x =pz r z S \{r}, x pz r+1 pz S \{r}, x pz S \{r,r+1}, x z S \{r,r+1}... pz, x px, x... pz x Thus, we obtain pz S x = pz S, x /px = p l zl x. l S If the estiates are represented by pfs, then we can copare these estiates using entropy [52, pp. 13], which is a easure of the uncertainty associated with a rando variable or its inforation content. Rear 2. Let Y be a rando variable with the pf p x z given by a stand-alone Bayesian filter [26], Y S is a rando variable with the pf p x z S given by Bayesian filter with easureent exchange, and H refers to the entropy of the rando variable. Since Y S is obtained by conditioning Y with zs \{} because p x z S = p x z, zs \{}, the clai follows fro the theore on conditioning reduces entropy cf. [52, pp. 27]: IY ; zs \{} = HY HY zs \{}, where I ; refers to the nonnegative utual inforation between two rando variables. Since HY S = HY zs \{} HY, the estiate given by Bayesian filter with easureent exchange is ore accurate than that obtained by a stand-alone Bayesian filter. Note that 4 is siilar to the epirical equation for Independent Lielihood Pool given in [35] and a generalization of the Distributed Sequential Bayesian Estiation Algorith given in [53]. The structure of 4 ensures that an arbitrary part of the prior distribution does not doinate the easureents. There is no consensus protocol across the networ because each agent receives inforation only fro its neighboring agents and never receives easureents even indirectly fro any other agent in the networ. III. COMBINING PROBABILITY DISTRIBUTIONS In this section, we present the algoriths for achieving consensus in probability distributions across the networ. As discussed before, the obective of the consensus stage in Algorith 1 is to guarantee pointwise convergence of each F to a consensual pdf F, which is independent of. This is achieved by each agent recursively transitting its estiated pdf of the target s states to other agents, receiving estiates of its neighboring agents, and updating its estiate of the target. Let F, = F represent the local estiated posterior pdf of the target s states, by the th agent at the start of the consensus stage, obtained using Bayesian filters with/without easureent exchange. During each of the n loop iterations within the consensus stage in Algorith 1, this estiate is updated as follows: F,ν =T l J {F,ν 1} l, {1,..., }, ν N, 5 where T is the linear or logarithic opinion pool for cobining the pdf estiates. Note that the proble of easureent neighbors does not arise here since all pdfs are expressed over the coplete state space. We introduce Lea 2 to show that pointwise convergence of pdfs is the sufficient condition for convergence of their induced easures in total variation TV. Let F 1,..., li n F n, F be real-valued easurable functions on, be the Borel σ-algebra of, and A be any set in. If µ Fn A = A F nxdµx for any set A, then µ Fn is defined as the easure induced by the function F n on. Let µ Fn, µ F denote the respective induced easures of F n, F on. Definition 3. Convergence in TV If µ Fn µ F TV := sup A µ Fn A µ F A tends to zero as n, then the easure µ Fn converges to the easure µ F in TV, i.e., li n µ Fn µ F. Lea 2. Pointwise convergence iplies convergence in TV If F n converges to F pointwise, i.e., li n F n = F pointwise; then the easure µ Fn converges in TV to the easure µ F, i.e., li n µ Fn µ F. Proof: Siilar to the proof of Scheffé s theore [54, pp. 84], under Assuption 3, using the doinated convergence theore cf. [54, Theore 1.5.6, pp. 23] for any set A gives: li n ˆ A ˆ F n xdµx= A ˆ li F nxdµx= F xdµx. n This relation between easures iplies that li n µ Fn µ F TV = and li n µ Fn µ F. A. Consensus using the Linear Opinion Pool The first ethod of cobining the estiates is otivated by the linear consensus algoriths widely studied in the literature [5] [7]. The pdfs are cobined using the Linear Opinion Pool LinOP of probability easures [39], [4]: F,ν = a l,ν 1 F,ν 1, l {1,..., }, ν N, 6 l J a l where l J,ν 1 = 1 and the updated pdf F,ν after the ν th consensus loop is a weighted average of the pdfs of the inclusive neighbors F,ν 1 l, l J fro the ν 1 th consensus loop, at the th tie instant. Let W,ν := T F,ν,ν 1,..., F denote an array of pdf estiates of all the A agents after the ν th consensus loop, then the LinOP 6 can be expressed concisely as: W,ν = P,ν 1 W,ν 1, ν N, 7 where P,ν 1 is a atrix with entries P,ν 1 [, l] = a l,ν 1. Assuption 6. The counication networ topology of the ulti-agent syste G is strongly connected SC. The weights a l,ν 1,, l {1,..., } and the atrix P,ν 1 have the following properties: i the weights are the sae for all consensus loops within each tie instant, i.e., a l,ν 1 = al

6 6 and P,ν 1 = P, ν N; ii the atrix P confors with the graph G, i.e., a l > if and only if l J, else al = ; and iii the atrix P is row stochastic, i.e., l=1 al = 1. Theore 3. Consensus using the LinOP on SC Digraphs Under Assuption 6, using the LinOP 6, each F,ν asyptotically converges pointwise to the pdf F = π if, i where π = [π 1,... π ] T is the unique stationary distribution of P. Furtherore, their induced easures converge in total variation, i.e., li ν µ F µ F,ν, {1,..., }. Proof: See Appendix A. Theore 3 is a generalization of the linear consensus algorith for cobining oint easureent probabilities [55]. Moreover, if π = 1 and each F, is a L 2 function, then F = 1 F, i globally iniizes the su of the squares of L 2 distances with the locally estiated posterior pdfs. As shown in Fig. 3 a-b, the ain difficulty with the LinOP is that the resulting solution is typically ultiodal, so no clear choice for ointly preferred estiate eerges fro it [4]. Moreover, the LinOP algorith critically depends on the assuption that the sae -1 scale is used by every agent as shown in Fig. 3 c-d. Hence, better schees for cobining probability distributions are needed for the proposed BCF algorith. B. Consensus using the Logarithic Opinion Pool Note that F,ν = p,ν x, x represents the pdf of the estiated target s states by the th agent during the ν th consensus loop at the th tie instant. The LogOP is given as [41]: F,ν =p,ν x = a l l J p l,ν 1 x,ν 1 l J p l,ν 1 x a l,ν 1 dµx {1,..., }, ν N, 8 where l J a l,ν 1 = 1 and the integral in the denoinator of 8 is finite. Thus the updated pdf F,ν,ν 1 after the ν th consensus loop is the weighted geoetric average of the pdfs of the inclusive neighbors F,ν 1 l, l J fro the ν 1 th consensus loop, at the th tie instant. As shown in Fig. 3 a-b, the LogOP solution is typically uniodal and less dispersed, indicating a consensual estiate ointly preferred by the networ [4]. As shown in Fig. 3 c-d, the LogOP solution is invariant under under rescaling of individual degrees of belief, hence it preserves an iportant credo of uni Bayesian decision theory; i.e., the optial decision should not depend upon the choice of scale for the utility function or prior probability distribution [56]. When the paraeter space is finite and a -1 probability scale is adopted, the LogOP is equivalent to the Nash product [57]. Note that if the local probability distribution of the target s states is inherently ultiodal, as shown in Fig. 3 e-f, then LogOP preserves this ultiodal nature while cobining these local estiates. The ost copelling reason for using LogOP is that it is, f 1x f 2x x a f 1x f 2x 1 x c f3x f4x x e f LinOP x f LogOP x α 1 = x b f LinOP x f LogOP x α 1 = x.1 d f LinOP x f LogOP x α1 = x Fig. 3. In a, two uniodal pdfs f 1 x and f 2 x are shown. In b, these pdfs are cobined using the LinOP and LogOP using the weight α 1 =.5, i.e., f LinOP x = α 1 f 1 x + 1 α 1 f 2 x and f LogOP x = f α 1 1 f 1 α 1 2 / f α 1 1 f 1 α 1 2 dµx. Note that the LinOP solution is ultiodal while the LogOP solution is uniodal, indicating a consensual pdf. In c, the scale of the function f 2 x is changed to -1 fro the standard -1 scale. In d, the noralized LinOP solution changes drastically but the LogOP solution reains unaffected. In e, the pdfs f 3 x and f 4 x have biodal nature. In f, the LogOP solution preserves this biodal nature. externally Bayesian; i.e., finding the consensus distribution coutes with the process of revising distributions using a coonly agreed lielihood distribution. Thus [4]: { lxp l T x } l J lxpl xdµx lxt = lxt l J l J f {p l x}, 9 {p l x} dµx where T refers to the LogOP 8, p l x, l J are pdfs on and lx is an arbitrary lielihood pdf on. Due to these advantages, LogOP is used for cobining prior distributions [58] and conditional rando fields for natural language processing tass [59]. Next, we present consensus theores using the LogOP. Assuption 7. The local estiated pdf at the start of the consensus stage is positive everywhere, i.e., F, = p, x >, x, {1,..., }. Assuption 7 is introduced to avoid regions with zero probability, since they would constitute vetoes and unduly great ephasis would get placed on the. Moreover, the LogOP

7 7 guarantees that F,ν consensus loop. will reain positive for all subsequent Definition 4. H,ν vector for LogOP For the purpose of analysis, let us choose ψ such that p,ν ψ [ >, ] {1,..., }, ν N. Let us define H,ν := ln p,ν x. p,ν ψ Under Assuption 7, H,ν is a well-defined function, but need not be a L 1 function. Then, by siple algebraic anipulation of 8, we get [6]: l J p l,ν 1 x al,ν 1 p,ν x p,ν ψ = l J p l,ν 1 x al,ν 1 dµx, H,ν = l J l J l J p l,ν 1 ψ al,ν 1 p l,ν 1 x al,ν 1 dµx a l,ν 1 Hl,ν 1, {1,..., }, ν N. 1 Note that 1 is siilar to the LinOP 6. Let U,ν := H 1,ν,..., H,ν T be an array of the estiates of all the agents during the ν th consensus loop at the th tie instant, then the equation 1 can be expressed concisely as: U,ν = P,ν 1 U,ν 1, ν N, 11 where P,ν 1 is a atrix with entries a l,ν 1. Thus we are able to use the highly nonlinear LogOP for cobining the pdf estiates, but we have reduced the coplexity of the proble to that of consensus using the LinOP. Theore 4. Consensus using the LogOP on SC Digraphs Under Assuptions 6 and 7, using the LogOP 8, each F,ν asyptotically converges pointwise to the pdf F given by: F = p p i, x πi x = p i, x πi, 12 dµx where π is the unique stationary distribution of P. Furtherore, their induced easures converge in total variation, i.e., li ν µ F µ F,ν, {1,..., }. Proof: Siilar to the proof of Theore 3, each H,ν converges pointwise to H = π T U, = π ih, i asyptotically. We additionally need to show that convergence of H,ν to H iplies pointwise convergence of F,ν to F. We have x : li ln p ν,ν x ln p,ν ψ =ln p x ln p ψ. 13 We clai ψ such that li ν p,ν ψ = p ψ. If this clai is untrue, then < li ν p,ν x < p x, x or vice versa. Hence li ν p,ν x dµx = 1 < p x dµx, which results in contradiction since p x is also a pdf. Hence, substituting ψ into equation 13 gives li ν p,ν x = p x, x. Thus each F,ν converges pointwise to the consensual pdf F given by 12. By Lea 2, the easure induced by F,ν on converges in total variation to the easure induced by F on, i.e., li ν µ F µ F,ν. Since Perron Frobenius theore only yields asyptotic convergence, we next discuss the algorith for achieving global exponential convergence using balanced graphs. Assuption 8. In addition to Assuption 6, the weights a l are such that the digraph G is balanced. Hence for every vertex, the in-degree equals the out-degree, i.e., l J a r, where, l, r {1,..., }. r J r a l = Theore 5. Consensus using the LogOP on SC Balanced Digraphs Under Assuption 7 and 8, using the LogOP 8, each F,ν globally exponentially converges pointwise to the pdf F given by: 1 F = p p i, x x = 1 14 p i, x dµx at a rate faster or equal to λ 1 P T P = σ 1 P. Furtherore, their induced easures globally exponentially converge in total variation, i.e., li ν µ F µ F,ν, {1,..., }. Proof: Since Assuption 8 is stronger than Assuption 6, we get li ν P ν = 1πT. Moreover, since P is also a colun stochastic atrix, therefore π = 1 1 is its left eigenvector corresponding to the eigenvalue 1, i.e., P T 1 1 = and satisfying the noralizing condition. Hence, we get li ν P ν = 1 11T and each H,ν converges pointwise to H = 1 1T U, = 1 Hi,. Fro the proof of Theore 4, we get that each F,ν converges pointwise to the consensual pdf F given by 14. Note that F,ν are L [ 1 functions ] but H,ν need not be L 1 1 functions. Let V tr = 1, V s be the orthonoral atrix of eigenvectors of the syetric priitive atrix P T P. By spectral decoposition [[61], we get: ] Vtr T P T 1 P V tr = Vs T P T P, V s where 1 1T P T P 1 1 = 1, 1 T P T P V s = 1 1, and Vs T P T P 1 1 = 1 1 are used. Since the eigenvectors are orthonoral, V s Vs T T = I. The rate at which 1 U,ν synchronizes to is equal to the rate at which Vs T substituting V T 1 or U U,ν 1 1. Pre-ultiplying 11 by Vs T s 1 = results in: Vs T U,ν = Vs T P V s Vs T T U,ν 1 = Vs T P V s Vs T U,ν 1. and Let z,ν = Vs T U,ν. The corresponding virtual dynaics is represented by z,ν = Vs T P V s z,ν 1, which has both Vs T U,ν and as particular solutions. Let Φ,ν = z T,ν z,ν be a candidate Lyapunov function for this dynaics. Expanding this gives:

8 8 Φ,ν =z T,ν 1Vs T P T P V s z,ν 1 λ ax Vs T P T P V s Φ,ν 1. Note that Vs T P T P V s contains all the eigenvalues of P T P other than 1. Hence λ ax Vs T P T P V s = λ 1 P T P < 1 and Φ,ν globally exponentially vanishes with a rate faster or equal to λ 1 P T P. Hence each H,ν globally exponen- converges pointwise to H with a rate faster or equal to tially λ 1 P T P = σ 1 P. Next, we need to find the rate of convergence of F,ν to F. Fro the exponential convergence of H,ν, we get: [ p ln,ν x ] p ψ p x p,ν ψ [ p σ 1 P ln,ν 1 x ] p ψ p x p,ν 1 ψ. Let us define the function α,ν x such that α,ν x = [ ] p,ν x p p x ψ p,ν ψ [ α,ν x = p x p,ν x if p,ν x p ψ p x p,ν ψ and ] p,ν ψ p ψ otherwise. Note that α,ν x is a continuous function since it is a product of continuous functions. Since α,ν x 1 and ln α,ν x, x, siplifies to: ln α,ν x σ 1 P ln α,ν 1 x. σ 1P α,ν x α, x ν. 16 Since p,ν x tends to p x, i.e., li ν α,ν x = 1, we can write 16 as: σ 1P α,ν x 1 α, x ν 1 σ 1P ν. 17 Using the ean value theore cf. [5], the right hand side of 17 can be siplified to 18, for soe c [1, α, x ]. σ 1P α, x ν 1 σ 1P ν = σ 1 P ν c σ 1P ν 1 α, x As σ 1 P < 1, the axiu value of c σ 1P ν 1 is 1. Substituting this result into 17 gives: α,ν x 1 σ 1 P ν α, x Hence α,ν x exponentially converges to 1 with a rate faster or equal to σ 1 P. Irrespective of the orientation of α,ν x and α, x, 19 can be written as 2 by 1 ultiplying with or 1, and then with α,ν x α, x p x. p ψ p,ν ψ p,ν x p x σ 1 P ν p ψ p, ψ p, x p x. 2 As shown in the proof of Theore 4, we can choose ψ such that p, ψ = p ψ. Now we discuss two cases to reduce the left hand side of 2 to p,ν x p x. p ψ p,ν ψ p,ν x p x p,ν x p x p + ψ p,ν ψ 1 p,ν x p if ψ = p,ν ψ 1 p x p,ν x + 1 p ψ p,ν ψ p,ν x p if ψ p,ν ψ < 1 p x p,ν x. Hence, for both the cases, we are able to siplify 2 to: p,ν x p x σ 1 P ν p, x p x. Thus each F,ν = p,ν x globally exponentially converges to F = p x with a rate faster or equal to σ 1 P. The KL divergence is a easure of the inforation lost when the consensual pdf is used to approxiate the locally estiated posterior pdfs. We now show that the consensual pdf F obtained using Theore 5, which is the weighted geoetric average of the locally estiated posterior pdfs F,, {1,..., }, iniizes the inforation lost during the consensus stage because it iniizes the su of KL divergences with those pdfs. Theore 6. The consensual pdf F given by 14 globally iniizes the su of Kullbac Leibler KL divergences with the locally estiated posterior pdfs at the start of the consensus stage F,, {1,..., }, i.e., F = arg in D KL ρ F i,, 21 ρ L 1 where L 1 is the set of all pdfs over the state space satisfying Assuption 7. Proof: The su of the KL divergences of a pdf ρ L 1 with the locally estiated posterior pdfs is given by: D KL ρ F i, = ˆ ρx lnρx ρx lnp i,x dµx. 22 Under Assuption 7, D KL ρ F, i is well defined for all agents. Differentiating 22 with respect to ρ using Leibniz integral rule [54, Theore A.5.1, pp. 372], and equating it to zero gives: ˆ lnρx + 1 lnp i,x dµx =, 23 where ρ x = 1 e pi, x 1/ is the solution to 23. The proection of ρ on the set L 1, obtained by

9 9 noralizing ρ to 1, is the consensual pdf F L 1 given by 14. The KL divergence is a convex function of pdf pairs [52, Theore 2.7.2, pp. 3], hence the su of KL divergences 22 is a convex function of ρ. If ρ 1, ρ 2,..., ρ n L 1 and η 1, η 2,..., η n ρ n [, 1] such that η i = 1, then = n η iρ i L 1 ; because i since ρ i x >, x, i {1,..., n} therefore ρ x >, x ; and ii since ρ ix dµx = 1, i {1,..., n} therefore ρ x dµx = 1. Moreover, since is a copact set, therefore L 1 is a closed set. Hence L 1 is a closed convex set. Hence 21 is a convex optiization proble. The gradient of D KL constant, i.e., d dρ D KL ρ F i, ρ=f = ln ρ F i, evaluated at F is a e p i, x 1. dµx This indicates that for further iniizing the convex cost function, we have to change the noralizing constant of F, which will result in exiting the set L 1. Hence F is the global iniu of the convex cost function 21 in the convex set L 1. This is illustrated using a siple exaple in Fig. 4. Another proof approach involves taing the logarith, in the KL divergence forula, to the base 1 c := p i, x dµx. Then differentiating D KL ρ F, i with respect to ρ gives: ˆ logc ρx + 1 log c p i,x dµx =, which is iniized by F. Hence F is indeed the global iniu of the convex optiization proble 21. Note that if a central agent receives all the locally estiated posterior pdfs F,, {1,..., } and is tased to find the best estiate in the inforation theoretic sense, then it would also yield the sae consensual pdf F given by 14. Hence we clai to have achieved distributed estiation using this algorith. In Rear 5, we state that the ethods for recursively cobining probability distributions to reach a consensual distribution are liited to LinOP, LogOP, and their affine cobinations. Rear 5. The LinOP and LogOP ethods for cobining probability distributions can be generalized by the g Quasi Linear Opinion Pool g QLOP, which is described by the following equation: F,ν = g 1 l J α l,ν 1 gf,ν 1 l g 1 l J α l,ν 1 gf,ν 1 l dµx, {1,..., }, ν N, 24 where g is a continuous, strictly onotone function. It is shown in [6] that, other than the linear cobination of LinOP and LogOP, there is no function g for which the final consensus can be expressed by the following equation: Probx L 1 ρ F.5 1 Probx 1 a DKL ρ F i F 1 F 2 F 3 F.5 1 Probx 1 Fig. 4. Let the discrete state space have only two states x 1 and x 2. All valid pfs ust lie on the set L 1 where Px 1 + Px 2 = 1. Given three initial pfs F i, i = {1, 2, 3}, the obective is to find the pf that globally iniizes the convex cost function 3 D KL ρ F i. In a, ρ = 1 3 e F i1/ globally iniizes the cost function, but it does not lie on L 1. In b, F L 1, which is the proection of ρ on the set L 1 obtained by noralizing ρ to 1, indeed globally iniizes the cost function on the set L 1. li F ν,ν =F g 1 =1 = π gf, l g 1 =1 π gf, l dµx, {1,..., }, ν N, 25 where π is the unique stationary solution. Moreover, the function g is said to be -Marovian if the schee for cobining probability distribution 24 yields the consensus 25 for every regular counication networ topology and for all initial positive densities. It is also shown that g is -Marovian if and only if the g QLOP is either LinOP or LogOP [6]. C. Counicating Probability Distributions The consensus algoriths using either LinOP or LogOP need the estiated pdfs to be counicated to other agents in the networ. We propose to adopt the following ethods for counicating pdfs. The first approach involves approxiating the pdf by a weighted su of Gaussians and then transitting this approxiate distribution. Let N x i, B i denote the Gaussian density function, where the ean is the n x -vector i and the covariance is the positive-definite syetric atrix B i. The Gaussian su approxiations lea of [62, pp. 213] states that any pdf F = px can be approxiated as closely as desired in the L 1 R nx space by a pdf of the for ˆF = ˆpx = n g α in x i, B i, for soe integer n g and positive scalars α i with n g α i = 1. For an acceptable counication error ε co >, there exists n g, α i, i and B i such that F ˆF L1 ε co. Several techniques for estiating the paraeters are discussed in the Gaussian ixture odel literature, lie axiu lielihood ML and axiu a posteriori MAP paraeter estiation [63] [65]. Hence, in order to counicate the pdf ˆF, the agent needs to transit 1 2 n gn x n x + 3 real nubers. Let us study the effect of this counication error ε co on the LinOP consensual distribution. Let F,ν be the LinOP solution after cobining local pdfs corrupted by counication error, i.e., F,ν := T l J { ˆF,ν 1 l } where T is b

10 1 LinOP 6. We prove by induction that F,ν F,ν L 1 νε co, ν N, where F,ν is the true solution obtained fro uncorrupted local pdfs. As the basis of induction holds, the inductive step for the ν th consensus step is as follows: F,ν F,ν L 1 F,ν 1 l F,ν 1 l L1 l J a l,ν 1 + F l,ν 1 ˆF l,ν 1 L1 ν 1ε co + ε co. 26 Siilarly, it follows fro the proof of Theore 5 that the LogOP solution after n loop iterations F,n loop, under counication inaccuracies, is always within a ball of n loop ε co radius fro the true solution using LogOP F,n loop in the L 1 space, i.e., F,n loop F,n loop L1 n loop ε co. If particle filters are used to evaluate the Bayesian filter and cobine the pdfs [36], [65], then the resapled particles represent the agent s estiated pdf of the target. Hence counicating pdfs is equivalent to transitting these resapled particles. The inforation theoretic approach for counicating pdfs is discussed in [66]. Let the local pdf F,ν be transitted over a counication channel using a finite sequence and the pdf ˆF,ν is reconstructed by the other agent. For a given error threshold, the iniu rate such that the variational distortion between F,ν and ˆF,ν is bounded by the error threshold, is given by the utual inforation between transitted and received finite sequences. Now that we have established that counication of pdfs is possible, let us discuss the coplete BCF algorith. Algorith 1 BCF LogOP on SC Balanced Digraphs 1: one cycle of th agent during th tie instant 2: Given the pdf fro previous tie step F 1 = p 1 x 1 3: Set n loop, the weights a l } Theores 5, 7 4: while tracing do 5: Copute the prior pdf p x using 3 6: Copute the posterior pdf F = p x z S using 4 and easureent array z S 7: for ν = 1 to n loop 8: if ν = 1 then Set F, = F end if 9: Obtain the counicated pdfs F,ν 1, l l J 1: Copute the new pdf F,ν using the LogOP 8 end for 11: Set F = F,n loop end while Bayesian Filtering Stage Sec. II-B LogOP based Consensus Stage Sec. III-B IV. MAIN ALGORITHMS: BAYESIAN CONSENSUS FILTERING In this section, we finally solve the coplete proble stateent for BCF discussed in Section II-A and Algorith 1. We also introduce an hierarchical algorith that can be used when soe agents in the networ fail to observe the target. A. Bayesian Consensus Filtering The BCF is perfored in two steps: i each agent locally estiates the pdf of the target s states using a Bayesian filter with/without easureents fro neighboring agents, as discussed in Section II-B, and ii during the consensus stage, each agent recursively transits its pdf estiate of the target s states to other agents, receives estiates of its neighboring agents, and cobines the using the LogOP as discussed in Section III-B. According to [67], this strategy of first updating the local estiate and then cobining these local estiates to achieve a consensus is stable and gives the best perforance in coparison with other update cobine strategies. In this section, we copute the nuber of consensus loops n loop in Algorith 1 needed to reach a satisfactory consensus estiate across the networ and discuss the convergence of this algorith. Definition 6. Disagreeent vector θ,ν Let us define T θ,ν := θ,ν 1,..., θ,ν, where θ,ν := F,ν F L 1. Since the L 1 distance between pdfs is upper bounded by 2, the l 2 nor of the disagreeent vector θ,ν l2 is upper bounded by 2. This conservative bound is used to obtain the iniu nuber of consensus loops for achieving ε-consensus across the networ, while tracing a oving target. Let us now quantify the divergence of the local pdfs during the Bayesian filtering stage. Definition 7. Error propagation dynaics Γ Let us assue that the dynaics of the l 2 nor of the disagreeent vector during the Bayesian filtering stage can be obtained fro the target dynaics and easureent odels 1 and 2. The error propagation dynaics Γ estiates the axiu divergence of the local pdfs during the Bayesian filtering stage, i.e., θ, l2 Γ θ 1,nloop l2, where θ 1,nloop l2 is the disagreeent vector with respect to F 1 at the end of the consensus stage during the 1 th tie instant; and θ, l2 is the disagreeent vector with respect to F after the update stage during the th tie instant. Next we obtain the iniu nuber of consensus loops for achieving ε-consensus across the networ and also derive conditions on the counication networ topology for a given nuber of consensus loops. Theore 7. BCF LogOP on SC Balanced Digraphs Under Assuptions 5, 7, 8, and an acceptable counication error ε co >, each agent tracs the target using the BCF algorith. For soe acceptable consensus error ε cons > and γ = in Γ θ 1,nloop l2, 2 : i for a given P, if the nuber of consensus loops n loop satisfies σ 1 P n loop γ + 2n loop ε co εcons ; 27 or ii for a given n loop, if the counication networ topology P during the th tie instant is such that

11 11 1 εcons 2n loop ε co n loop σ 1 P ; 28 then the l 2 nor of the disagreeent vector at the end of the consensus stage is less than ε cons, i.e., θ,nloop l2 ε cons. Proof: In the absence of counication inaccuracies, Theore 5 states that the local estiated pdfs F, globally exponentially converges pointwise to a consensual pdf F given by 14 with a rate of σ 1 P, i.e. F,ν F L 1 σ 1 P ν F, F L 1. If θ, is the initial disagreeent vector at the start of the consensus stage, then θ,nloop l2 σ 1 P nloop θ, l2 σ 1 P nloop γ. In the presence of counication error, cobining 26 with the previous result gives F,ν F L 1 σ 1 P ν F, F L 1 + νε co. Since θ,ν F,ν F L 1 + νε co, the disagreeent vector after n loop iterations is given by θ,nloop l2 σ 1 P nloop θ, l2 + 2n loop ε co. Thus, we get the conditions on nloop or σ 1 P fro the inequality σ 1 P nloop γ + 2n loop ε co εcons. Note that in the absence of counication inaccuracies, γ lnεcons/γ ln σ 1P 27 siplifies to n loop and 28 siplifies 1 n ε to σ 1 P cons loop γ. In the particular case where n loop = 1 and counication errors are present, 28 siplifies to σ 1 P and the necessary εcons 2εco γ condition for a valid solution is 2ε co < εcons. In the genral case, it is desireable that 2ε co εcons for a valid solution to Theore 7. B. Hierarchical Bayesian Consensus Filtering In this section, we odify the original proble stateent such that only 1 out of agents are able to observe the target at the th tie instant. In this scenario, the other 2 = 1 agents are not able to observe the target. Without loss of generality, we assue that the first 1 agents, i.e., {1, 2,..., 1 }, are tracing the target. During the Bayesian filtering stage, each tracing agent i.e., agent tracing the target estiates the posterior pdf of the target s states at the th tie instant F = p x z S {1,...1}, {1,..., 1 } using the estiated prior pdf of the target s states F 1 and the new easureent array zs {1,...1} { } := z l, l S {1,... 1} obtained fro the neighboring tracing agents. Each non-tracing agent i.e., agent not tracing the target only propagates its prior pdf during this stage to obtain p x, { 1 + 1,..., }. The obective of hierarchical consensus algorith is to guarantee pointwise convergence of each F,ν, {1,..., } to a pdf F and only the local estiates of the agents tracing the target contribute to the consensual pdf. This is achieved by each tracing agent recursively transitting its estiate of the target s states to other agents, only receiving estiates fro its neighboring tracing agents and updating its estiate of the target. On the other hand, each non-tracing agent recursively transits its estiate of the target s states to other agents, receives estiates fro all its neighboring agents and updates its estiate of the target. This is illustrated using the pseudocode in Algorith 2 and the following equations: F,ν = T l J {1,...,1}{F,ν 1} l, {1,..., 1 }, ν N, 29 F,ν = T l J {F,ν 1} l, { 1 + 1,..., }, ν N, 3 where, T refers to the LogOP 8 for cobining pdfs. Let D represent the counication networ topology of only the tracing agents. Algorith 2 Hierarchical BCF LogOP on SC Balanced Digraphs 1: one cycle of th agent during th tie instant 2: Given the pdf fro previous tie step F 1 = p 1 x 1 3: Set n loop, the weights a l } Theores 7, 8 4: while tracing do 5: Copute prior pdf p x using 3 6: if 1 then 7: Copute the posterior pdf F using 4 and z S {1,..., 1} end if 8: for ν = 1 to n loop 9: if ν = 1 then 1: if 1 then Set F, = F 11: else Set F, = p x end if end if 12: if 1 then 13: Obtain the pdfs F,ν 1, l l J {1,..., 1} fro tracing neighbors 14: else Obtain the pdfs F,ν 1, l l J fro neighbors end if : Copute the new pdf F,ν using the LogOP 8 end for 16: Set F = F,n loop end while l J Bayesian Filtering Stage Sec. II-B Hierarchical LogOP based Consensus Stage Sec. IV-B Assuption 9. The counication networ topologies G and D are SC and the weights a l are such that the digraph D is balanced. The weights a l,ν 1,, l {1,..., } and the atrix P,ν 1 have the following properties: i The weights are the sae for all consensus loops within each tie instants, i.e., a l,ν 1 = al and P,ν 1 = P, ν N. Moreover, P can be decoposed into four parts P = [ P 1 P 2 ] P 3 P 4, where P 1 R 1 1, P 2 = R 1 2, P 3 R 2 1, and P 4 R 2 2. ii If {1,..., 1 }, then a l > if and only if l J {1,..., 1}, else a l = ; hence P 2 = 1 2. Moreover, P 1 is balanced, i.e., a l = r J a r r, where, l, r {1,..., 1}; iii

12 12 If { 1 + 1,..., }, then a l > if and only if l J, else a l = ; iv The atrix P is row stochastic, i.e., l=1 al = 1. Theore 8. Hierarchical Consensus using the LogOP on SC Balanced Digraphs Under Assuptions 7 and 9, using the LogOP 8, each F,ν globally exponentially converges pointwise to the pdf F given by: 1 1 F = p p i, x 1 x = 1 31 p i, x 1 dµx 1 at a rate faster or equal to λ 1 1P1 T P 1 = σ 1 1P 1. Only the initial estiates of the tracing agents contribute to the consensual pdf F. Furtherore, their induced easures converge in total variation, i.e., li ν µ F,ν {1,..., }. µ F, Proof: The atrix P 1 confors to the balanced digraph D. Let 1 1 = [1, 1,..., 1] T, with 1 eleents. Siilar to the proof of Theore 5, we get P 1 is a priitive atrix and li ν P1 ν = T 1. Next, we decopose U,ν fro equation 11 into two parts such that U,ν = [Y,ν ; Z,ν ], where Y,ν = T T H,ν,ν 1,..., and H1 Z,ν = H 1+1,ν,..., H,ν. Since P 2 is a zero atrix, 11 generalizes and hierarchically decoposes to: Y,ν+1 = P1Y ν,, ν N 32 Z,ν+1 = P 3 Y,ν + P 4 Z,ν, ν N 33 Cobining equation 32 with the previous result gives li ν Y,ν = T 1 Y,. Thus li ν H,ν = H = T 1 Y, = Hi,, {1,..., 1}. Fro the proof of Theore 5, we get F,ν, {1,..., 1} globally exponentially converges pointwise to F given by 31 with a rate faster or equal to σ 1 1P 1. Since G is strongly connected, inforation fro the tracing agents reach the non-tracing agents. Taing the liit of equation 33 and substituting the above result gives: li ν Z,ν+1 = 1 1 P T 1 Y, + P 4 li ν Z,ν 34 Let 1 2 = [1, 1,..., 1] T, with 2 eleents. Since P is row stochastic, we get P = [I P 4 ]1 2. Hence, fro equation 34, we get li ν Z,ν = T 1 Y,. Moreover, the inessential states die out geoetrically fast [68, pp. 12]. Hence li ν H,ν = H = T 1 Y,, { 1 + 1,..., }. Hence, the estiates of the non-tracing agents F,ν, { 1 + 1,..., } also converge pointwise geoetrically fast to the sae consensual pdf F given by 31. By Lea 2 we get li ν µ F µ F,ν, {1,..., }. Note that Theore 7 can be directly applied fro Section IV-A to find the iniu nuber of consensus loops n loop for achieving ɛ-convergence in a given counication networ topology or for designing the P 1 atrix for a given nuber of consensus loops. A siulation exaple of Hierarchical 9in 3 7in 6in 5in 97in 22 1in 1 3in 4in Fig. 5. The SSN locations are shown along with their static SC balanced counication networ topology. The orbit of the Iridiu 33 debris is shown in red, where ars its actual position during particular tie instants. BCF LogOP algorith for tracing orbital debris in space is discussed in the next section. V. NUMERICAL EAMPLE Currently, there are over ten thousand obects in Earth orbit, of size.5 c or greater, and alost 95% of the are nonfunctional space debris. These debris pose a significant threat to functional spacecraft and satellites in orbit. The US has established the Space Surveillance Networ SSN for ground based observations of the orbital debris using radars and optical telescopes [69], [7]. In February 29, the Iridiu 33 satellite collided with the Kosos 2251 satellite and a large nuber of debris fragents were created. In this section, we use the Hierarchical BCF LogOP Algorith to trac one of the Iridiu 33 debris created in this collision. The orbit of this debris around Earth and the location of SSN sensors are shown in Fig. 5. The actual two-line eleent set TLE of the Iridiu 33 debris was accessed fro North Aerican Aerospace Defense Coand NORAD on 4 th Dec 213. The nonlinear Siplified General Perturbations SGP4 odel, which uses an extensive gravitational odel and accounts for the drag effect on ean otion [71], [72], is used as the target dynaics odel. The counication networ topology of the SSN is assued to be a static SC balanced graph, as shown in Figure 5. If the debris is visible above the sensor s horizon, then it is assued to create a single easureent during each tie step of one inute. The heterogeneous easureent odel of the th sensor is given by: z = x + w, where w = N, I, where x R 3 is the actual location of the debris and the additive Gaussian easureent noise depends on the sensor nuber. Since it is not possible to ipleent the SGP4 target dynaics on distributed estiation algoriths discussed in the literature [11] [2], we copare the perforance of our Hierarchical BCF LogOP algorith Algorith 2 against the

13 13 18 All SSN 1th SSN 1 rd Particles of 3 SSN 22nd SSN n [Revs per day] 3rd SSN Tie [ins] 8 1 a 18 1 b 18 th Particles of SSN 17 n [Revs per day] n [Revs per day] 17 a = nd Particles of 1 SSN n [Revs per day] 5 Tie [ins] Tie [ins] c 1 5 Tie [ins] 1 d 25 5 b 75 = n [Revs per day] Nuber of Sensors c d Fig. 6. a Nuber of SSN sensors observing debris. Traectories of particles for stand-alone Bayesian filters for b 3rd, c 1th, and d 22nd SSN sensor. Fig. 7. Traectories of particles of all sensors for a Hierarchical BCF LinOP and b Hierarchical BCF LogOP. The color-bar on the right denotes the 33 SSN sensors. Evolution of the consensual probability distribution for c Hierarchical BCF LinOP and d Hierarchical BCF LogOP. Hierarchical BCF LinOP algorith, where the LinOP is used during the consensus stage. VI. C ONCLUSION In this siulation exaple, we siplify the debris tracing proble by assuing only the ean otion n of the debris is unnown. The obective of this siulation exaple is to estiate n of the Iridiu 33 debris within 1 inutes. Hence, each sensor nows the other TLE paraeters of the debris and an unifor prior distribution F is assued. Note that at any tie instant, only a few of the SSN sensors can observe the debris, as shown in Fig 6a. The results of three stand-alone Bayesian filters, ipleented using particle filters with resapling [36], are shown in Fig 6b-d. Note that the estiates of the 22nd and 1th sensors initially do not converge due to large easureent error, in spite of observing the debris for soe tie. The estiates of the 3rd sensor does converge when it is able to observe the debris after 7 inutes. Hence we propose to use the Hierarchical BCF LogOP algorith where the consensual distribution is updated as and when sensors observe the debris. Particle filters with resapling are used to evaluate the Bayesian filters and counicate pdfs in the Hierarchical BCF algoriths. 1 particles are used by each sensor and 1 consensus loops are executed during each tie step of one inute. The traectories of all the particles of the sensors in the Hierarchical BCF algorith using LinOP and LogOP and their respective consensual probability distributions at different tie instants are shown in Figure 6a-d. As expected, all the sensors converge on the correct value of n of 14.6 revs per day. The Hierarchical BCF LinOP estiates are ultiodal for the first 9 inutes. On the other hand, the Hierarchical BCF LogOP estiates converges to the correct value within the first 1 inutes because the LogOP algorith efficiently counicates the best consensual estiate to other sensors during each tie step and achieves consensus across the networ. In this paper, we extended the scope of distributed estiation algoriths in a Bayesian filtering fraewor in order to siultaneously trac targets, with general nonlinear tievarying target dynaic odels, using a strongly connected networ of heterogeneous agents, with general nonlinear tievarying easureent odels. We introduced the Bayesian filter with/without easureent exchange to generate local estiated pdfs of the target s states. We copared the LinOP and LogOP ethods of cobining local posterior pdfs and deterined that LogOP is the superior schee. The LogOP algorith on SC balanced digraph converges globally exponentially, and the consensual pdf iniizes the inforation lost during the consensus stage because it iniizes the su of KL divergences to each locally estiated probability distribution. We also explored several ethods of counicating pdfs across the sensor networ. We introduced the BCF algorith, where the local estiated posterior pdfs of the target s states are first updated using the Bayesian filter and then recursively cobined during the consensus stage using LogOP, so that the agents can trac a oving target and also aintain consensus across the networ. Conditions for exponential convergence of the BCF algorith and constraints on the counication networ topology have been studied. The Hierarchical BCF algorith, where soe of the agents do not observe the target, has also been investigated. Siulation results deonstrate the effectiveness of the BCF algoriths for nonlinear distributed estiation probles. ACKNOWLEDGMENT The authors would lie to than F. Hadaegh, D. Bayard, S. Hutchinson, P. Voulgaris, M. Egerstedt, A. Gupta, A. Dani, D. Morgan, S. Sengupta, and A. Olshevsy for stiulating discussions about this paper.