Filtering and smoothing formulas of AR(p)-modulated Poisson processes

Abstract

Recursive formulas are presented to compute smoothed estimates and online filtered estimates for a hidden AR(p) processes, using modulated Poisson observations. We utilize techniques of a transformation of probability, a duality between a forward equation and a backward equation, and develop a new partial fraction decomposition of rational polynomials with several variables. The established polynomial time algorithm computes efficiently closed-form expressions of estimates for the hidden AR(p) process. With a truncating technique, complexity of the algorithm can be reduced further. Finally, we discuss some computational issues related to the proposed algorithm and compare it with other numerical methods.