Stochastic Average Consensus Filter for Distributed HMM Filtering: Almost Sure Convergence
Nader Ghasemi, Subhrakanti Dey and J. S. Baras
Proceedings of the 2nd IFAC Workshop on Distributed Estimation and Control in Networked Systems (NecSys'10), pp. 335-340, Annecy, France, September 13-14 , 2010.
This paper studies almost sure convergence of a dynamic average consensus algorithm which allows distributed computation of the product of n time-varying conditional probability density functions. These conditional probability density functions (often called as “belief functions”) correspond to the conditional probability of observations given the state of an underlying Markov chain, which is observed by n different nodes within a sensor network. The network topology is modeled as an undirected graph. The average consensus algorithm is used to obtain a distributed state estimation scheme for a hidden Markov model (HMM), where each sensor node computes a conditional probability estimate of the state of the Markov chain based on its own observations and the messages received from its immediate neighbors. We use the ordinary differential equation (ODE) technique to analyze the convergence of a stochastic approximation type algorithm for achieving average consensus with a constant step size. This allows each node to track the time varying average of the logarithm of conditional observation probabilities available at the individual nodes in the network. It is shown that, for a connected graph, under mild assumptions on the first and second moments of the observation probability densities and a geometric ergodicity condition on an extended Markov chain, the consensus filter state of each individual sensor converges P–a.s. to the true average of the logarithm of the conditional observation probability density functions of all the sensors. Convergence is proved by using a perturbed stochastic Lyapunov function technique. Numerical results suggest that the distributed Markov chain state estimates obtained at the individual sensor nodes based on this consensus algorithm track the centralized state estimate (computed on the basis of having access to observations of all the nodes) quite well, while more formal convergence results for the distributed HMM filter to the centralized one are currently under investigation.