(556d) Parameter Estimation with a Moving Window Particle Filtering for Nonlinear Dynamic Models

Particle Filtering has successful applications for state estimation of nonlinear dynamic models. However, parameter estimation with particle filtering remains as a major problem so far. Bayesian inference under particle filtering considers unknown parameters as random variables with some prior distribution, and augment the original state space by the parameters, thus making the parameters estimation problem into optimal filtering to fit into the framework of particle filtering. To this end, marginal posterior distribution of parameters needs to be known analytically, or can be easily sampled from, such that new values of parameters can be generated at each time point. Otherwise, fixing parameters at their initial values (often from priors) would inevitably lead to particle degeneracy. One can also impose an artificial dynamic behavior on the parameters, like random walk, to diversify parameters.

We propose to use the Expectation-Maximization (EM) method, under particle filtering, to get Maximum Likelihood Estimate (MLE) of the parameters. MLE is an appealing method for its consistency and asymptotic normality properties under certain regularity conditions. Recursive optimization through EM is needed since direct usage of MLE in nonlinear dynamic models is nearly impossible. The states estimated in particle filtering provide Monte Carlo approximation for the expectation step of EM. To shorten convergence time of EM and make particle filtering less sensitive to possibly unreliable initial parameter values, a moving window is used to limit the data set to which particle filtering is performed.