Maximum likelihood recursive state estimation in state-space models: A new approach based on statistical analysis of incomplete data

TL;DR

This paper introduces a new recursive maximum likelihood particle filtering method for state-space models, utilizing statistical analysis of incomplete data, and extends classical results to nonlinear and linear models, including Kalman filtering.

Contribution

It develops a novel EM-gradient particle filtering approach based on incomplete data analysis, extending existing methods for nonlinear state estimation.

Findings

The method provides a recursive EM-gradient particle filter for nonlinear models.

In linear cases, the approach confirms Kalman filter's efficiency and optimality.

Explicit covariance matrices are derived, matching Cramer-Rao bounds.

Abstract

This paper revisits the work of Rauch et al. (1965) and develops a novel method for recursive maximum likelihood particle filtering for general state-space models. The new method is based on statistical analysis of incomplete observations of the systems. Score function and conditional observed information of the incomplete observations/data are introduced and their distributional properties are discussed. Some identities concerning the score function and information matrices of the incomplete data are derived. Maximum likelihood estimation of state-vector is presented in terms of the score function and observed information matrices. In particular, to deal with nonlinear state-space, a sequential Monte Carlo method is developed. It is given recursively by an EM-gradient-particle filtering which extends the work of Lange (1995) for state estimation. To derive covariance matrix of state-estimation errors, an explicit form of observed information matrix is proposed. It extends Louis (1982) general formula for the same matrix to state-vector estimation. Under (Neumann) boundary conditions of state transition probability distribution, the inverse of this matrix coincides with the Cramer-Rao lower bound on the covariance matrix of estimation errors of unbiased state-estimator. In the case of linear models, the method shows that the Kalman filter is a fully efficient state estimator whose covariance matrix of estimation error coincides with the Cramer-Rao lower bound. Some numerical examples are discussed to exemplify the main results.