Finite Dimensional Hidden State Estimation Schemes for Non-Gaussian Dynamics with Stochastic Parameters

April 7, 2011
Time: 3:30pm-4:30pm
Speaker: W. P. Malcolm, Joint Operation Division, Defence Science and Technology Organisation of Australia

Abstract

In this seminar we develop several finite dimensional estimation schemes for a class of jump Markov systems.

Jump Markov systems arise quite naturally in Electrical Engineering, where, for example, no single set of dynamics can fully describe the time evolution of a given system. In such cases, one may wish to consider approximate dynamics formed by a finite collection of candidate individual dynamics. That is, a collection of separate dynamical systems each of which are in effect, or not, according to the state of a Markov chain.

In some cases exact solutions to these estimation problems can be obtained, however, no such solution will be finite dimensional.

Two main difficulties emerge in computing estimators for Jump Markov systems. The first concerns evaluating conditional expectations for triply stochastic systems. The second and more challenging difficulty, concerns managing the time-dependent dimensional growth of those schemes which compute the conditional expectations of interest.

To evaluate conditional expectations, we appeal to the techniques of reference probability measures, resulting in un-normalized dynamics for filters, smoothers, detectors and parameter estimation schemes.

The issue of dimensional growth is managed through a novel probability-based selection scheme. This scheme can be made be made adaptive and driven by alternative criteria such as measures of entropy in the mixture approximations. Initially we consider first order Markov chains as a law for the switching between parameter sets, however, it is shown that Markov chains or any order can be considered in our context. Computer-based simulation examples illustrating numerical performance are provided.


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