Distributed Kalman Filtering over Big Data: Analysis Through Large Deviations of Random Riccati Equations

This paper studies the convergence of the estimation error process and the characterization of the corresponding invariant measure in distributed Kalman filtering for potentially unstable and large linear dynamic systems. A gossip based information exchanging scheme termed Modified Gossip Interactive Kalman Filtering (M-GIKF) is proposed, where sensors exchange their filtered states (estimates and error covariances) and also propagate their observations via inter-sensor communications of rate γ, where γ is defined as the averaged number of intersensor message passages per signal evolution epoch. The filtered states are interpreted as stochastic particles with local interaction and it is shown that the conditional estimation error covariance sequence at each sensor under M-GIKF evolves as a random Riccati equation (RRE) with Markov modulated switching. By formulating the RRE as a random dynamical system, it is shown that the conditional estimation error covariance at each sensor converges weakly (in distribution) to a unique invariant measure from any initial state. Further, it is proved that as γ → ∞ this invariant measure satisfies the Large Deviation (LD) upper and lower bounds, implying that this measure converges exponentially fast (in probability) to the Dirac measure δP∗ , where P ∗ is the stable error covariance of the centralized (Kalman) filtering setup. The LD results answer a fundamental question on how to quantify the rate at which the distributed scheme approaches the centralized performance as the inter-sensor communication rate increases.