Performance bounds for cooperative RSS emitter tracking using diffusion particle filters

This paper introduces a methodology for numerical computation of the Posterior Cramér-Rao Lower Bound (PCRLB) for the position estimate mean-square error when a moving emitter is tracked by a network of received-signal-strength (RSS) sensors using a distributed, random exchange diffusion filter. The square root of the PCRLB is compared to the empirical root-mean-square error curve for a particle filter implementation of the diffusion filter, referred to as RndEx-PF, and to the square root of the PCRLB for the optimal centralized filter that assimilates all network measurements at each time instant. In addition, we also compare the proposed RndEx-PF algorithm to three alternative distributed trackers based on Kullback-Leibler fusion using both iterative consensus and non-iterative diffusion strategies.


I. INTRODUCTION
Distributed particle filtering (PF) [1] has become a standard technique for cooperative target tracking in sensor networks where each network node has computational and communication capabilities of its own. A common network cooperation strategy found in other contexts in the literature is the Kullback-Leibler (KL) fusion methodology, see e.g. [2], [3], where, at any given instant n, each node first runs a local particle filter that assimilates only the node's own current measurements. Subsequently, the network nodes run an iterative average consensus protocol [4] which, after multiple iterations, seeks to replace the local state posterior probability density functions (p.d.f's) at each node with a normalized version of the geometric mean of all node posteriors. A non-iterative, online version of the KL fusion approach was also independently introduced by Dedecius et al. in [5].
Alternatively, we introduced in [6] and [7] the non-iterative Random Exchange Diffusion Particle Filter (RndEx-PF), which, building on previous work on distributed Kalman filtering by Kar et al. [8], uses a methodology where network nodes first exchange their local posterior p.d.f's with randomly selected neighbors and then use a particle filter to assimilate all available measurements in their closed neighborhood. As shown in [7], the aforementioned algorithm builds over time, at each node, different state posterior p.d.f's conditioned on different random subsets of all network measurements, with the measurements in each of those random subsets coming from the entire network and not only from the node's immediate vicinity, thus enabling information diffusion.
In this paper, building on previous results in [9], we introduce a methodology for numerical computation of the posterior Cramér-Rao lower bound (PCRLB) for the average network mean-square state estimation error when the proposed random exchange diffusion protocol is used. The computed bound is compared, in a specific application of emitter tracking with received-signal-strength (RSS) sensors, to the empirical performance, assessed via simulations, of the RndEx-PF tracker, and is also compared to the PCRLB for the mean-square state estimation error associated with the optimal centralized filter that assimilates all network measurements at each instant. Finally, we also compare the RndEx diffusion bound to the simulation performance of a standard iterative consensus KL fusion tracker which follows the approach in [2] and of a modified diffusion KL fusion tracker running respectively in non-iterative mode as in [5] and in consensus mode as in [2].
The paper is divided into 7 sections. Section I is this Introduction. In Sec. II, we review briefly the RndEx-PF algorithm and the underlying state and sensor models assumed in the paper. In Sec. III, we introduce the methodology for computation of the PCRLB assuming that the network nodes cooperate using the proposed random exchange protocol. Sec. IV then briefly describes the alternative consensus KL tracker based on [2] and the proposed variation inspired by [5]. Simulation results, are presented in Sec. V. Finally, we offer some conclusions in Sec. VI.

II. RANDOM EXCHANGE DIFFUSION FILTER
In this Section, we review the state and sensor models and the RndEx-PF tracking algorithm from references [6] and [7]. We use lowercase letters to denote both random variables/vectors and realizations of random variables/vectors with the proper interpretation implied in context.

II-A. State and Sensor Model
We track the sequence of hidden state vectors {x n x nẋn y nẏn T } that collect the positions and velocities of a moving emitter's centroid respectively in dimensions x and y at each time instant n ≥ 0. The sequence {x n } evolves in time according to the linear, white-noise acceleration model x n+1 = Fx n +u n , described in [10], with {u n } a sequence of independent, identically distributed (i.i.d) Gaussian vectors with zero mean and covariance matrix Q. Moreover, let T and σ accel denote respectivelly the sampling period and the acceleration standard deviation, matrices F and Q are given by F = diag(F,F) and Q = diag(Q,Q) A network of R RSS sensors records at each instant n the measurements z n,r in dBm at each network location r ∈ V {1, 2, . . . , R} such that [11] where ν n,r is a zero-mean i.i.d. Gaussian noise process with known variance σ 2 r , x r is the r-th sensor position, ||.|| is the Euclidean norm, and the parameters (P 0 , d 0 , ζ r ) are known model parameters (see [11] for details) and H is a 2 × 4 projection matrix such that H(1, 1) = H(2, 3) = 1 and H(i, j) = 0 otherwise. In the remainder of the paper, the closed neighborhood of node r, denoted N(r), is defined as the union of {r} and the set of all indices { } such that node is a neighbor of node r according to the topology of the network graph. The measurement noise ν n,r is also assumed independent from node to node.

II-B. Random Exchange Diffusion Particle Filter
Assume first that a given node s ∈ V has at instant n − 1 a parametric approximationp s (x n−1 ) of its local posterior state p.d.f. built from a weighted set of samples {w n−1,s }, j = 1, . . . , N p , that represents the true posterior p.d.f. p(x n−1 |Z 0:n−1,s ) where Z 0:n−1,s is the set of network measurements assimilated by s up to instant n − 1. At instant n, after all network nodes have executed the random exchange protocol originally introduced in [6] and [7] and summarized in Sec. III-A, node s and another random network node r exchange their parametric representations of the state posterior such thatp s (.) is now available at node r. Upon receivingp s (.), node r at instant n, for j = 1, . . . , N p , where, in Line 3, the proportionality constant is such that j w (j) n,r = 1. Barring the parametric approximation step prior to the random exchange protocol, it can be shown, see [7], that {w n,r } is now a properly weighted set to represent the posterior p(x n |Z 0:n,r ) at node r at instant n, where Z 0:n,r Z 0:n−1,s ∪Z n,r and Z n,r {z n, , ∈ N(r)}. Finally, we approximate the integral in E{x n |Z 0:n,r } by the Monte Carlo approximation

III. PCRLB FOR THE RNDEX DIFFUSION FILTER
We introduce in this section a methodology to compute the posterior Cramér-Rao lower bound (PCRLB) for the distributed emitter tracking error considering that network nodes diffuse information using the random exchange protocol described in the sequel.

III-A. Random Exchange Protocol
More specifically, at instant n − 1, each network node s randomly chooses an available neighboring node and sends then a single request to exchange the parametric representation of its local posterior with node . Upon receiving an acknowledgement from node , node s thus sends the set of parameters that representp s (.) to node and, likewise, receives from node the corresponding set of parameters that representp (.). The random exchange procedure terminates when all network nodes have their requests processed by an available node in their vicinity. Finally, after performing the random exchange step, each node r broadcasts its own measurement at instant n to its neighbors and then assimilates the set of measurements Z n,r using the procedure in Line 3 of Section II-B.
As the initial posterior p.d.f. p(x 0 |Z 0,r0 ) on a node r 0 ∈ V at time 0 follows a path P n,rn {r 0 , . . . , r k , . . . , r n } along the network according to the random exchange protocol, it assimilates, at each time instant k ∈ {0, . . . , n}, the available measurements Z k,r k at each visited node r k ∈ P n,rn . Mathematically, at a given time n, the posterior p(x n |Z n,rn , Z 0:n−1,rn−1 ) at a node r n is equivalent then (again barring parametric approximations) to the posterior provided by a fusion center that, at each instant k ∈ {0, . . . , n}, randomly chooses a closed neighborhood N(r k ) and assimilates all observations contained in Z k,r k . That interpretation plays a crucial role in the derivation of the PCRLB for the state estimation error as discussed in the sequel.

III-B. Computation of the PCRLB
Let J n,rn denote the Fisher information matrix at instant n for node r n conditioned on a particular path realization P n,rn along the network. In precise mathematical terms, we write J n,rn E −∆ xn xn log p(x 0:n , Z 0:n,rn )|P n,rn , where ∆ x denote respectively the Laplacian and the Jacobian operators. Thus, conditioned on P n,rn , the posterior Cramér-Rao lower bounds (PCRLB) P n,rn (J n,rn ) −1 for the estimatex n,rn computed by node r n satisfies the inequality E ||x n,rn − x n || 2 | P n,rn ≥ tr[P n,rn ], where tr [.] denotes the trace of a matrix. Additionally, provided the usual independence assumptions for 0 ≤ k ≤ n, the joint probability of x 0:n (x 0 , . . . , x n ) and Z 0:n,rn Z 0,r0 ∪ . . . ∪ Z n,rn can be factorized as We can apply therefore the method introduced in [9] to recursively compute the Fisher matrix J n+1,rn+1 at instant n + 1 given the Fisher matrix J n,rn originated from a node r n and the set of measurements Z n+1,rn+1 available at node r n+1 .
We extend first the recursive equations derived in [9] to consider multiple observers as in (1) Furthermore, since the measurements in Z n+1,rn+1 are independent, we can further write Next, we rewrite (5), (6) and (7) for the particular filtering problem introduced in Sec. II-A with linear dynamics and log-normal observation model (1) where with D x, (1, 1) = (x − x ) 2 , D x, (3, 3) = (y − y ) 2 , D x, (1, 3) = D x, (3, 1) = (x−x )(y −y ) and D x, (i, j) = 0 otherwise. Note that, as opposed to (9) and (10), Eq. (11) does not have an exact closed form solution. Thus, we simulate several emitter trajectories x (i) 0:n+1 , i ∈ {1, . . . , N T }, according to the emitter dynamic model and use the following Monte Carlo approximation to compute the expectation in the right-hand side of (11) (13) Now, let P n,rn denote the unconditioned PCRLB for each node r n at instant n such that E ||x n,rn − x n || 2 ≥ tr[P n,rn ] tr[(J n,rn ) −1 ], (14) where J n,rn is the unconditioned Fisher information matrix at instant n. From the law of total expectation and Eq. (2), we can write J n,rn E −∆ xn xn log p(x 0:n , Z 0:n,rn ) = E E −∆ xn xn log p(x 0:n , Z 0:n,rn )|P n,rn = E {J n,rn } with the expectation in (15) taken over all possible paths P n,rn along the network that end at node r n at instant n.

IV. KULLBACK-LEIBLER FUSION
Assume, as before, that a given node r has at instant n − 1 a weighted set of particles {w (j) n−1,r , x (j) n−1,r }, j = 1, . . . , N p , that represents its current posterior belief about the unknown target state. The application of the consensus KL fusion procedure detailed in [2] and briefly described in Sec. I would entail the following steps at instant n at each node r ∈ V: n−1,r ) and update the weights asw r (x n ) ∝ [ ∈Vp (x n )] 1/R , ∀r, which, as argued in [2], is, after normalization, the p.d.f. p * that minimizes (1/R) ∈V D(p * ||p ), where D(.||.) denotes the Kullback-Leibler divergence.

Modified Version
We modify the KL fusion algorithm replacing the weight update rule in Line 1 withw n,r ) to allow network nodes to assimilate the available measurements in their vicinity before the K max consensus iterations. In addition, following the lead in [5], we can optionally make the algorithm non-iterative by setting K max = 1 in Line 3. As shown in [5], the choice of K max = 1 leads to β

V. SIMULATION RESULTS
We assessed the performance of the RndEx-PF tracker using 100 independent Monte Carlo runs in a simulated scenario consisting of R = 25 RSS sensors over a simulation period of 100 s. We used the same simulation parameters described in [6] to simulate the emitter dynamics and generate sensor measurements. We assumed a Gaussian initial state distribution p(x 0 ) with mean x 0 = 50 m 0 m/s 50 m 0 m/s T and covariance matrix P 0 = diag( (5 m) 2 (0.05 m/s) 2 (5 m) 2 (0.05 m/s) 2 ) . At any given node r 0 , we initialized the conditioned Fisher matrix as J 0,r0 = P −1 0 . All filters employed N p = 1000 particles. The particles at each node were initialized at instant 0 according to the distribution p(x 0 ). At instant n + 1, we used a distinct set of N T = 10000 trajectories in (13) to recursively compute the conditioned Fisher matrix J n+1,rn+1 at node r n+1 given the Fisher matrix J n,rn at node r n . Finally, at any instant n, we employed N f = 1000 different Fisher matrices {J (m) n,rn , 1 ≤ m ≤ N f } conditioned on distinct network paths P (m) n,rn to compute the unconditioned Fisher matrix J n,rn = (P n,rn ) −1 at node r n .
Note that, in the RndEx filter, each node has a different position estimate and a different associated mean-square estimation error (MSEE). Fig. 1 shows the RndEx-PF empirical root-mean-square (RMS) position estimate error averaged over all network nodes from instant zero up to instant 100 s and compares it to the average PCRLB for the square-root of the RndEx filter position MSEE obtained by averaging P n,rn (1, 1) + P n,rn (3, 3) over all possible 25 node indices r n at each instant n. For comparison, we also show in Fig. 1 the PCRLB for the square-root of the centralized filter position MSEE, and the average RMS position estimate error curves for the Consensus KL Fusion filter [2] using K max = 10 consensus iterations and for the modified KL fusion algorithm proposed in Sec. IV using either K max = 1 and K max = 10 consensus iterations. We refer to the modified trackers with K max = 1 and K max = 10 as the Non-iterative and the Iterative KL Fusion filters, respectively. The KL fusion trackers used Metropolis weights a r, [12], [13] and the bars shown in Fig. 1 along the curves represent the standard deviation of the error norm across all nodes. An alternative algorithm for distributed computation of the optimal centralized PCRLB is discussed in [14]. As expected, the RndEx-PF RMS error is above its corresponding PCRLB. The region between the empirical RndEx-PF RMS error curve and its PCRLB in Fig. 1 is filled in light red to highlight the gap between the two curves. Furthermore, despite being non-iterative, the RndEx-PF tracker actually showed better error performance in our simulations than the standard Consensus KL Fusion algorithm, probably because the latter does not include local cooperation before fusion. However, RndEx-PF had a higher RMS error than the Non-iterative KL Fusion algorithm with K max = 1, which suggests that local cooperation plus KL fusion allows nodes to incorporate more information at each time than the random exchange protocol. The performance of the modified KL fusion scheme was further improved by increasing K max from 1 to 10, in which case the corresponding RMS error curve further approaches the PCRLB for the centralized filter error. Note also that the performance of the modified KL fusion tracker is not limited by the theoretical RndEx-PF PCRLB as KL fusion is a different algorithm. Table I summarizes the performance metrics for each evaluated algorithm. They were computed according to the methodology previously introduced in [6]. The RndEx-PF tracker has half of the average reception (RX) communication cost per node of the modified Non-Iterative KL Fusion algorithm with a similar processing cost and at the expense of a slight degradation in error performance. In turn, in addition to having a worse error performance, the Consensus KL Fusion tracker has an average RX communication cost one order of magnitude higher than that of RndEx-PF employing however roughly one-third of the processing cost. Finally, the Iterative KL Fusion tracker has better error performance than the Consensus KL Fusion algorithm at the expense of a slight increase in the average transmission (TX) communication cost and a huge increase of 170 % in processing cost since the nodes assimilate all measurements in their vicinity.

VI. CONCLUSIONS
We computed the PCRLB for the MSEE of a distributed, random exchange (RndEx) diffusion filter. The RndEx PCRLB was compared to the centralized filter PCRLB and to the empirical error of a Gaussian PF implementation of the RndEx filter, denoted RndEx-PF. In addition, we also compared the RndEx empirical error and the centralized PCRLB's to the empirical error of three KL fusion filters. The error curve of the RndEx-PF tracker, as expected, was above its corresponding PCRLB. On the other hand, the Consensus KL Filter without local cooperation before fusion performed worse than RndEx-PF with a communication cost one order of magnitude higher than that of RndEx-PF, which suggests that, in the evaluated scenario, local cooperation is preferable to standard consensus cooperation. However, the Non-iterative KL Filter with local cooperation before fusion was close to the RndEx PCRLB with a communication cost only twice as high as that of RndEx-PF. Finally, by enabling local cooperation before consensus KL fusion, the Iterative KL Filter, although still worse than the centralized PCRLB, outperformed the RndEx PCRLB.