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.

KEYWORD:

• Filtering
• Poisson observation
• Closed-form expression
• Reference probability
• Autoregression model

Additional information

Funding

This work is supported by National Natural Science Foundation of China (70971109), Doctoral Scientific Research Foundation of Northwest University (2017.Math.Liujr), Natural Science Foundation of Education Department of Shaanxi Provincial Government (2018NWU-LIU).