Markov Chain Distributed Particle Filters (MCDPF)

Joint 48th IEEE Conference on Decision and Control and ThC07.6 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009

Sun Hwan Lee* and Matthew West†

Abstract— Distributed particle filters (DPF) are known to approaches and consensus-based local information exchange provide robustness for the state estimation problem and can methods. Message passing approaches transfer information reduce the amount of information communication compared along a predetermined route covering an entire network. to centralized approaches. Due to the difficulty of merging multiple distributions represented by particles and associated For example, [7] passes parameters of the parametric model weights, however, most uses of DPF to date tend to approximate of the posterior distribution while [8] transmits the raw the posterior distribution using a parametric model or to use information, particles and weights, or the parameters of a a predetermined message path. In this paper, the Markov GMM approximation of the posterior distribution. Consensus Chain Distributed Particle Filter (MCDPF) algorithm is pro- based methods communicate the information only between posed, based on particles performing random walks across the network. This approach maintains robustness since every the nearest nodes and achieve global consistency by consen- sensor only needs to exchange particles and weights locally and sus filtering. The type of exchanged information can be, for furthermore enables more general representations of posterior example, the parameters of a GMM approximation [2] or the distributions because there are no a priori assumptions on local mean and covariance [9]. distribution form. The paper provides a proof of weak conver- The massage passing approaches [7][8] can have reduced gence of the MCDPF algorithm to the corresponding centralized particle filter and the optimal filtering solution, and concludes robustness because the distributed algorithms cannot them- with a numerical study showing that MCDPF leads to a reliable selves cope with the failure of even one node since the system estimation of the posterior distribution of a nonlinear system. uses fixed message paths. Furthermore, the assumption of synchronization with identical particles at every node can I. INTRODUCTION cause fragility. On the other hand, the consensus based Distributed Particle Filters (DPF) have been emerging as approaches so far proposed [2][9] all use approximations of an efficient tool for state estimation, for instance in target the posterior distribution with GMM because to reduce infor- tracking with a robotic navigation system [1][2]. The general mation flow. The amount of reduced information, however, benefits of distributed estimation include the robustness of is not significant compared with transmitting full particles the estimation, the reduction of the amount of information when the dimension of the system is very large, due to flow, and estimation results comparable to the centralized the covariance matrix of the posterior distribution. For an approach. Much effort has been directed toward the real- n-dimensional system, consensus based DPF with a GMM ization of the decentralized Kalman filtering [3][4][5] but approximation has to transmit O(cn2|E|) data through the decentralized particle filters were thought to be challenging entire network per consensus step, where c is the number of due to the difficulty of merging probability distributions rep- Gaussian mixtures and |E| is the number of network edges. resented by particles and weights [6]. The currently existing If the DPF is realized with exchanging N particles, however, distributed particle filtering methods, however, are not able then the amount of information per Markov chain iteration to gain all of these advantages or turn out to benefit from is O(nmN), where m is the number of nodes. A detailed these properties only with relatively low dimensional systems description of the Markov chain iteration is given below. by introducing an assumption such as Gaussian Mixture For a system with cn|E| ≫ mN, a GMM approximation Model (GMM). The developed methods so far try to avoid no longer benefits from the effect of reduced information exchanging the raw data, namely particles and associated flow. Furthermore, it frequently happens that the posterior weights, mainly due to the large amount of information distribution is not well-approximated by small number of that implies. The communication of such raw data scales combination of Gaussian distribution. better with system dimension, however, than the existing In this paper we propose a distributed particle filter based methods do. The distributed particle filter proposed in this on exchanging particles and associated weights according paper exchanges particles and weights only between nearest to a Markov chain random walk. This approach maintains neighbor nodes and estimates the true state by assimilating the robustness of a distributed system since each node only data with an algorithm based on a Markov chain random needs local information and it scales well in the case of non- walk. Gaussian and high dimensional systems. Past work on distributed particle filters can be broadly The contents of this paper are as follows. In section II, a categorized into two approaches, namely massage passing review of basic properties and theorems for random walks on graphs is provided. Section III contains the centralized *S. H. Lee is with the Department of Aeronautics and Astronautics, particle filter (CPF) algorithm and the proposed decentralized Stanford University, [email protected] †M. West is with the Department of Mechanical Science and Engineering, particle filter algorithm. In addition, the weak convergence University of Illinois at Urbana-Champaign, [email protected] of the posterior distribution of the DPF to that of the CPF

978-1-4244-3872-3/09/$25.00 ©2009 IEEE 5496 ThC07.6 and the optimal ﬁlter is proved in this section. A numerical kernel and conditional probability distribution of Y attain example using a bearing only tracking problem with four Lebesgue measures. sensors is given in section IV. Conclusion and future works are discussed in section V. Pr(Xt ∈ A|Xt−1 = xt−1) = K(xt|xt−1)dxt (5) ZA

II. RANDOM WALKS ON GRAPHS Pr(Yt ∈ B|Xt = xt) = ρ(yt|xt)dyt (6) ZB

A sensor network system can modeled as a graph, G = where ρ(yt|xt) is the transition probability density of a (V, E), with a normalized adjacency matrix A. The vertices measurement yt given the state xt. V = {1,..., m} correspond to nodes or sensors in the The filtering problem is to estimate the true state xt network and edges E represent the connections between at time t given the time series of observations y1:t. The sensors. The neighbors of node i are given by Ni = {j ∈ prediction and updating of the optimal filtering based on V : Aij 6= 0}. Matrix A is a Markov transition probability Bayes’ recursion are given as follows. matrix defined on the graph because it satisfies A ≥ 0 and A1 = 1. Consequently a random walk on the network can be p(xt|y1:t−1) = p(xt−1|y1:t−1)K(xt|xt−1) dxt−1 (7) defined according to matrix A and we assume no self-loops ZRn in the chain. Here we review several properties of random ρ(yt|xt)p(xt|y1:t−1) p(xt|y1:t) = . (8) walks on graphs which are useful for the development of Rn ρ(yt|xt)p(xt|y1:t−1) dxt DPF. Analytic solutionsR for the posterior distribution in (8) do Proposition 2.1: If A is the normalized adjacency matrix not generally exist except in special cases, such as linear of an undirected, connected graph G then Markov chain dynamical systems with Gaussian noise. In the particle defined by A has a unique stationary distribution Π and filtering setting, the posterior distribution is represented by a Πi > 0 for all i ∈ V . Moreover, for any initial distribution, group of particles and associated weights so that the integral in (8) is approximated by the sum of discrete values. M(i, k) lim = Πi, (1) k→∞ k A. Centralized Particle Filters where M(i, k) is the number of visits to state i during k Particle filtering is a recursive method to estimate the true steps. state, given the time series of measurements [10][11]. Sup- pose the posterior distribution at time t−1, π (dx ), Theorem 2.2: If A is the normalized adjacency matrix t−1|t−1 t−1 is approximated by N particles xi N . Then we have of an undirected connected graph G then the stationary { t−1}i=1 , distribution of the Markov chain defined by A is given by p(xt−1|y1:t−1) πt−1|t−1(dxt−1) (9) d(i) Π = (Π1, Π2,..., Πm), where Πi = 2|E| . Here d(i) is the N degree of node i and E is the number of edges in the graph. N 1 | | ≈ π (dxt−1) = δxi (dxt−1). t−1|t−1 N t−1 Proof: We compute Xi=1

m m where particle i is at position xi in state space. Now, parti- d(i) |E | t−1 (ΠA) = Π A = ij = Π , (2) cles go through the prediction and measurement update steps j i ij 2|E| d(i) j Xi=1 Xi=1 to approximate the posterior distribution at time t. Given N particles, new particles are sampled from the transition kernel where |Eij| is the number of edges connecting nodes i and j i N 1 N i m density, x˜t ∼ πt−1|t−1K(dxt) = N i=1 K(xt|xt−1). This and i=1|Eij| = d(j). Since Π satisfies Π = ΠA, Π is the set of particles is the approximationPof πt|t−1, stationaryP distribution of the Markov chain defined by A. , p(xt|y1:t−1) πt|t−1(dxt) (10) III. PARTICLE FILTERS N N 1 ≈ π˜ (dxt) = δ i (dx˜t). (11) Suppose we have the general state space model, t|t−1 N x˜t Xi=1 If the empirical distribution in (11) is substituted in (8), we xt+1 = f(xt, wt) (3) have the following distribution approximating the posterior yt = g(xt, vt), (4) distribution p(xt|y1:t). n p N where xt ∈ R , yt ∈ R , and wt, vt are process and ρ(y |x )π˜ (dx ) N , t t t|t−1 t measurement noises respectively. We define two stochastic π˜t|t(dxt) N (12) Rn ρ(yt|xt)π˜t|t−1(dxt) dxt processes, X = {Xt, t ∈ N} and Y = {Yt, t ∈ N}, where R N i N ρ(y |x˜ )δ i (dx˜ ) X and Y are a signal process and an observation process i=1 t t x˜t t i = = w δ i (dx˜t), respectively. The signal process X is a Markovian pro- P N i t x˜t i=1 ρ(yt|x˜t) Xi=1 cess with an initial distribution µ(x0) and transition kernel N i P i K(dxt|xt−1) and the observation process Y is conditionally where i=1 wt = 1 and wt are called the importance independent given X. For simplicity, we assume that the weights.PTo avoid the degeneracy problem, particles are

5497 ThC07.6 selected according to a resampling step that samples N adjacency matrix A to compute the importance weights. The N particles from the empirical distribution, π˜t|t(dxt), and resets main idea is that each particle gains ρi(yi,t|xt) exponentially 1 proportional to the expected number of visit to node i. the weights to N . We then have the empirical distribution approximating the posterior at time t given by Suppose we have the graph G = (V, E) based on the N sensor network and the normalized adjacency matrix A. In N 1 the MCDPF setting, the Markov chain is run k steps on π (dxt) = δ i (dxt). (13) t|t N xt every particle after the prediction step and the number of Xi=1 visits to the i-th node is defined by M(i, k). Considering B. The Markov Chain Distributed Particle Filter (MCDPF) the number of visits to each node, each particle multiplies The main difference between the CPF and DPF is that the 2|E(G)| ρ (y |x ) kd(i) to its previous weight every time it visits CPF has a central unit to collect the entire measurements i i,t t the i-th node. If we have N particles after k Markov chain from all nodes and update particles using all whole measure- steps at a node, then the posterior distribution of the MCDPF ments simultaneously. During the process of data collection, is given as follows. the CPF might suffer from the bottlenecks in information 2|E(G)| N m i ×M(j,k) flow. On the other hand, a DPF can overcome this problem by ρ (y |x˜ ) kd(j) δ i (dx˜ ) N i=1 j=1 j j,t t x˜t t passing information only locally between connected nodes. π˜t|t,k(dxt) = 2|E(G)| P QN m i kd(j) ×M(j,k) If we have m nodes measuring the partial observations i=1 j=1 ρj(yj,t|x˜t) independently, then we can decompose the general state N P Q i space model (4) as follows. = w δ i (dx˜ ). (17) t,k x˜t t Xi=1 xt+1 = f(xt, wt) (14) The MCDPF is defined in algorithm 1 below. We use the y1,t g1(xt, v1,t) i notation xj,t for the i-th particle of node j at time t and  y2,t   g2(xt, v2,t)  . = . . (15) N(j) for the number of particle at node j. Also Ii→j is the  .   .  indices of particles moving from node i to j in the current     y  g (x , v ) Markov chain step and we recall that A is the adjacency  m,t  m t m,t  m matrix of the network. Rn Rpi Here xt ∈ , yi,t ∈ with i=1 pi = p and subscript i represents node i. In addition,P the measurement noise Algorithm 1 Markov Chain Distributed Particle Filter E T at each nodes is assumed to be uncorrelated, [vtvt ] = (MCDPF) diag(R1, R2,..., Rm). Uncorrelated noise structure enables Initialization: us to have conditionally independent measurements at each i N i N 1 {xj,0}i=1 ∼ p(x0), {wj,0}i=1 = N for j = 1,..., m node, yi,t, given the true state xt. As a consequence of this Importance Sampling: For j = 1,..., m assumption, the function ρ(y |xi) in (8) can be factorized by t t {x˜i }N(j) ∼ p x |{xi }N(j) , {w˜i }N(j) = 1 a product of ρ (y |xi) at each node, j,t i=1 t j,t−1 i=1 j,t i=1 j j,t t for k iterationsdo m Move {x˜i }N(j), {w˜i }N(j) according to matrix A ρ(y |xi) = ρ (y |xi). (16) ·,t i=1 ·,t i=1 t t j j,t t for j = 1 to m do jY=1 i N(j) i {x˜j,t}i=1 = l∈N {x˜l,t}i∈Il→j We propose a distributed particle filtering method using a j {w˜i }N(j) = S {w˜i } j,t i=1 l∈Nj l,t i∈Il→j random walk on the graph defined by the network topol- 2|E(G)| i N(j) S i N(j) i N(j) kd(i) ogy. In the sensor network, node i measures the partial {w˜j,t}i=1 ← {w˜j,t}i=1 ×ρj(yj,t|{x˜j,t}i=1 ) observation yi,t at time t and data at every node has to end for be fused to reach the global estimation of the true state. end for While achieving a global estimate by exchanging data, it Resample: For j = 1,..., m i N(j) i N(j) is desirable to maintain a robustness with respect to the Resample {xj,t}i=1 according to {w˜j,t}i=1 and set unexpected changes of global properties such as losing a i N(j) 1 weights {wj,t}i=1 = N(j) node. The DPF proposed in this paper is robust since the information, consisting of particles and weights, is trans- ferred only to the connected neighborhood of each node. In other words, every node only needs local information. C. Convergence to CPF and Algorithm Transferring particle data is inefficient for low-dimensional We will show that the empirical posterior distribution of systems, but scales well (only linearly) with dimension size, the MCDPF converges weakly to that of the CPF as the as opposed to existing methods using GMM approximations Markov chain steps k per measurement goes to infinity. of posterior distribution [9][2][8]. As briefly explained in The notation that will be used throughout the proof mainly section I, communicating raw data is more efficient in terms follows that of [10]. In the stochastic filtering problem, of bandwidth capacity for relatively high-order systems. functions at and bt are defined on a metric space (E, d) to MCDPF moves particles around the network according to itself and are considered as continuous maps from πt|t−1 → k the Markov chain on the network defined by the normalized πt|t and πt−1|t−1 → πt|t−1, respectively. Additionally, at is

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m R j=1 ρj(yj,t|xt)e(xt) ρ(yt|xt)e(xt) N = − The perturbation c is defined as a function that maps from a Q m Rn j=1 ρj(yj,t|xt)e(dxt) Rn ρ(yt|xt)e(dxt) measure ν to a random sample size of size N of the measure, R Q R so that = 0. (27) 1 N The first equality is due to proposition 2.1 and the second cN,w(ν) = δ , (19) N {Vj (w)} equality comes from conditional independence of the mea- Xj=1 surements at each node. Theorem 3.3: Consider a connected sensor network with Rn where Vj : Ω → is an IID random variable with the measurements at different nodes conditionally independent distribution ν and w ∈ Ω. For notational simplicity, let given the true state. Then the estimated distribution of the N N N ht , h1:t be defined as the composition of functions at, bt, c MCDPF in Algorithm 1 converges weakly to the estimated as follows. distribution of the CPF as the number of Markov chain steps k per measurement goes to infinity. That is, N , N N N , N N ht c ◦ at ◦ c ◦ bt, h1:t ht ◦ ··· ◦ h1 (20) N N lim πt|t,k = πt|t. (28) N , N k N N , N N k→∞ ht,k c ◦ at ◦ c ◦ bt, h1:t,k ht,k ◦ ··· ◦ h1,k. (21) pointwise. Thus the posterior distribution of CPF and MCDPF at time Proof: Combining lemmas 3.1 and 3.2 with the optimal k t can then be expressed as Bayesian filtering functions at, bt and at gives (28). D. Convergence to Optimal Filtering N N N N πt|t = ht (πt−1|t−1) = h1:t(π0) (22) So far we showed that the MCDPF converges weakly N N N N to the CPF. The next step is to prove the convergence of πt|t,k = ht,k(πt−1|t−1,k) = h1:t,k(π0). (23) MCDPF to the optimal filtering distribution as N → ∞ N N as well as k → ∞. The convergence of the classical To prove limk→∞ πt|t,k = πt|t, several lemmas are reviewed here. particle filter to optimal filtering distribution is shown in [10][12]. The only difference between the MCDPF and the Lemma 3.1: Let (E, d) be a metric space with functions standard particle filter used in the proof is the function ak, ak, a , b : E → E such that lim ak = a pointwise for t t t t k→∞ t t which needs to satisfy the following condition to ensure each t. Then the convergence of MCDPF to optimal filtering. For all N N sequences e → e ∈ E we have lim h1:t,k = h1:t. (24) N k→∞ k lim lim at (eN ) = at(e). (29) N→∞ k→∞ pointwise for each t and N. k N This property is not obvious because the function at con- Proof: For e ∈ E and arbitrary t, we have c (bt(e)) ∈ E. Since we assumed pointwise convergence of ak to a , for verges only pointwise to a function at. Fortunately, however, t t a is continuous, which is sufficient to give (29), as we see all ǫ > 0 there exists k(e, ε) such that for k > k(e, ε), t from the following lemma. k k N N Lemma 3.4: Suppose (E, d) is a metric space and a , a : kat (c (bt(e))) − at(c (bt(e)))k < ε, (25) E → E are continuous functions so that ak converges pointwise to a as k → ∞. For a convergent sequence where k·k is the supremum norm on functions from (E, d) limN→∞ eN = e ∈ E, we have to itself. Equivalently, hN hN pointwise for all limk→∞ t,k = t k k lim lim a (eN ) = lim lim a (eN ) = a(e). (30) t. By induction over t we have (24). N→∞ k→∞ k→∞ N→∞ k Lemma 3.2: For the MCDPF and CPF as defined above, Proof: For functions a → a pointwise and eN → e ∈ E, we have k k lim at = at (26) lim a (eN ) = a(eN ) for all N (31) k→∞ k→∞ k ⇒ lim lim a (eN ) = lim a(eN ) = a(e), (32) pointwise for all t. N→∞ k→∞ N→∞

5499 ThC07.6 since a is continuous. Conversely, continuity of ak gives giving the desired result. k k lim a (eN ) = a (e) for all k (33) IV. NUMERICAL EXAMPLE N→∞ k k We consider a bearing-only tracking example in this ⇒ lim lim a (eN ) = lim a (e) = a(e). (34) k→∞ N→∞ k→∞ section for the purpose of demonstrating the performance of the MCDPF. The dynamic model was a time-dependent k Lemma 3.5: Consider at, bt from (8), at from (18), and linear system but the measurement model was nonlinear, N k c from (8). Assume that the sequence at satisfies the with one moving target tracked by 4 bearing sensors linked property (29) for each t. Then we have in a network. There were two modes for the movement N of the target, straight and turning mode. The target moved lim lim h1:t,k = h1:t. (35) N→∞ k→∞ with linear dynamics, turning to the right by 90 degrees Moreover for all sequences eN → e ∈ E we have between 0.5 and 1, 2 and 2.5, and 3.5 and 4 seconds. The N state vector is [xt yt x˙ t y˙t] and the state-space system and lim lim h1:t,k(eN ) = h1:t(e). (36) N→∞ k→∞ measurements at each sensor were given by N Proof: From [10, Lemma 2], for all eN → e ∈ E, c Ft∆t satisfies xt+1 = e xt + qt, (46) N i lim c (eN ) = e. (37) yt − s (y) N→∞ θi ri, (47) t = arctan i + t xt − s (x) Thus, for all eN → e ∈ E and any continuous function bt, N where lim c (bt(eN )) = bt(e). (38) N→∞ 0 0 1 0 From the property (29), 0 0 0 1  0 Straight mode Ft = , at = π N 0 0 0 at Turning mode lim c (bt(eN )) = bt(e) (39)   2×51∆t N→∞ 0 0 −at 0  k N   ⇒ lim lim at (c (bt(eN ))) = at(bt(e)). (40) 3 2 N→∞ k→∞ ∆t ∆t 3 0 2 0 And again from the property (37) of cN ,   ∆t3 ∆t2  i 2 0 3 0 2 rt ∼ N (0, 0.05 ), qt ∼ N 0, 2 . k N ∆t 0 ∆t 0 lim lim a (c (bt(eN ))) = at(bt(e)) (41)   2  t   2  N→∞ k→∞   0 ∆t 0 ∆t  N k N   2  ⇒ lim lim c (at (c (bt(eN )))) = at(bt(e)). (42) N→∞ k→∞ Here ∆t = 0.01, (si(x), si(y)) was the position of ith N i Thus we have limN→∞ limk→∞ ht,k(eN ) = ht(e) sensor, and rt was the measurement noise. Each sensor was and from induction over t we can conclude connected to its nearest two neighboring sensors. The true N limN→∞ limk→∞ h1:t,k(eN ) = h1:t(e). trajectory of the moving target was estimated by the CPF Recall that a kernel K is said to have the Feller property and MCDPF. The centralized particle filter tracked the true if Kϕ is a continuous bounded function whenever ϕ is a trajectory with N = 400 particles and bearing information continuous bounded function. For such kernels we have the gathered from all four different sensors. The trajectory was following result. also estimated by the MCDPF with N = 400 particles at Lemma 3.6: Suppose at and bt are functions defined each node. in (8). Then at is continuous provided the function ρ(yt|·) Figures 1 and 2 show the estimation results of the CPF and is bounded, continuous and strictly positive. Furthermore, bt MCDPF with k = 4 Markov chain steps per measurement for is a continuous function if the transition kernel K is Feller. the MCDPF. Even with such a small number of Markov chain Proof: See [10, Section IV.B.]. steps per measurement, the MCDPF at each sensors obtained Putting all the above lemmas together gives the following a reasonable estimate of the true trajectory by exchanging main result. information only with connected neighbor according to a Theorem 3.7: Assume that the kernel K is Feller and the random walk of particles and weights on the sensor network. function ρ is bounded, continuous and strictly positive. Then the estimated distribution of the MCDPF in Algorithm 1 To numerically study the convergence of the posterior converges to the optimal filtering distribution as the number distribution of the MCDPF at each node to the posterior of particles N and the number of Markov chain steps k per distribution of the CPF (as proved in theorem 3.3), we define measurement go to infinity: the root mean square error (RMSE) to be N 1/2 lim lim πt|t,k = πt|t. (43) 2 N→∞ k→∞ RMSE(xˆDPF) = E kxˆDPF − xôptk (48) Proof: For the initial probability measure µ0, we know 2 1/2 N ≈ E kxˆDPF − xˆCPFk . (49) that limN→∞ µ0 = µ0. From lemma 3.5, N N N Figure 3 shows the RMSE versus the number of Markov lim lim πt|t,k = lim lim h1:t,k(µ0 ) (44) N→∞ k→∞ N→∞ k→∞ Chain steps k per measurement, computed by averaging 2 × = h1:t(µ0) = πt|t, (45) 105 executions of the CPF and MCDPF at time t = 0.1 in the

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2.5 True trajectory 2 CPF estimate Position of sensors 1.5 Particles

−3 0.5 10

0 2 x

−0.5 RMSE

−1 1 −1.5 2 −2

−4 −2.5 10

−3 2 3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 10 10 x 1 Markov chain step, k

Fig. 1. The trajectory estimation by CPF. Fig. 3. Convergence of RMSE in number of Markov chain steps k.

2.5 True trajectory 2 Position of sensors number of Markov chain steps k per measurement tends to Particles inﬁnity, and we presented a numerical example to verify this. 1.5 DPF estimates

1 REFERENCES

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2 [2] D. Gu, ”Distributed Particle Filter for Target Tracking”, Robotics and x −0.5 Automation, 2007 IEEE International Conference on, 2007, pp. 3856- 3861. −1 [3] B.S. Rao and H.F. Durrant-Whyte, Fully decentralised algorithm for −1.5 multisensor Kalman filtering, Control Theory and Applications, IEEE Proceedings D, 1999, pp 413-420. −2 [4] Olfati-Saber, R., ”Distributed Kalman Filter with Embedded Consen- −2.5 sus Filters”, Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC ’05. 44th IEEE Conference on, 2005, pp. 8179- −3 8184. −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 x [5] Olfati-Saber, R., ”Distributed Kalman filtering for sensor networks”, 1 Decision and Control, 2007 46th IEEE Conference on, 2007, pp. 5492- 5498. Fig. 2. The trajectory estimation by DPF with Markov chain steps k = 4. [6] L. Ong and B. Upcroft and M. Ridley and T. Bailey and S. Sukkarieh and H. Durrant-Whyte, ”Consistent methods for Decentralised Data Fusion using Particle Filters”, Multisensor Fusion and Integration for Intelligent Systems, 2006 IEEE International Conference on, 2006, pp. simulation. This figure confirms numerically the convergence 85-91. proved in Theorem 3.3. Based on Figure 3 it appears that the [7] M. Coates, ”Distributed particle filters for sensor networks”, In Proc. RMSE converges at a rate of k1/2 (see [13] for a proof). of 3nd workshop on Information Processing in Sensor Networks, 2004, pp. 99-107. [8] X. Sheng and Y. Hu and P. Ramanathan, ”Distributed particle fil- V. CONCLUSIONS AND FUTURE WORK ter with GMM approximation for multiple targets localization and In this paper, we introduced the new Markov Chain Dis- tracking in wireless sensor network”, IPSN ’05: Proceedings of the 4th international symposium on Information processing in sensor tributed Particle Filter (MCDPF) which exchanges particles networks, LA, CA, 2005, pp. 24. between distributed sensor nodes, thus providing robust state [9] D. Gu and J. Sun and Z. Hu and H. Li, ”Consensus based distributed estimation and good scaling in the dimension of the system particle filter in sensor networks”, Information and Automation, 2008. ICIA 2008. International Conference on, 2008, pp. 302-307. being estimated. The robustness is maintained by the fact that [10] Crisan, D. and Doucet, A., A survey of convergence results on the proposed MCDPF algorithm only exchanges information particle filtering methods for practitioners, IEEE Transactions on locally, while good scaling is due to the use of only particles Signal Processing, vol. 50, 2002, pp 736-746. [11] Doucet, A. and De Freitas, N. and Gordon, N., Sequential Monte Carlo to store state. This does mean, however, that the MCDPF methods in practice, Springer-Verlag, 2001. will be less efficient for lower-dimensional systems that can [12] Hu, Xiao-Li and Schon, T. B. and Ljung, L., A Basic Convergence be efficiently represented by Gaussian Mixture Models or Result for Particle Filtering, IEEE Transactions on Signal Processing, vol. 56, 2008, pp 1337-1348. other similar approximations. We proved that the estimated [13] S. Lee and M. West, Convergence of Markov Chain Distributed distribution of the MCDPF algorithm converges weakly to Particle Filters, in preparation. that of the classical Centralized Particle Filter (CPF) as the

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