Iterative Statistical Linear Regression for Gaussian Smoothing in Continuous-Time Non-linear Stochastic Dynamic Systems

TL;DR

This paper introduces an iterative Gaussian smoothing method for continuous-time non-linear stochastic systems, improving estimation accuracy by re-linearising the system with respect to the current Gaussian approximation.

Contribution

It develops two novel methods for linearising stochastic differential equations, leading to new and generalized Gaussian smoothers for continuous-time systems.

Findings

Better estimation accuracy than existing smoothers.

Effective in challenging tracking scenarios like reentry and radar tracking.

Generalizes discrete-time iterative smoothers to continuous-time setting.

Abstract

This paper considers approximate smoothing for discretely observed non-linear stochastic differential equations. The problem is tackled by developing methods for linearising stochastic differential equations with respect to an arbitrary Gaussian process. Two methods are developed based on 1) taking the limit of statistical linear regression of the discretised process and 2) minimising an upper bound to a cost functional. Their difference is manifested in the diffusion of the approximate processes. This in turn gives novel derivations of pre-existing Gaussian smoothers when Method 1 is used and a new class of Gaussian smoothers when Method 2 is used. Furthermore, based on the aforementioned development the iterative Gaussian smoothers in discrete-time are generalised to the continuous-time setting by iteratively re-linearising the stochastic differential equation with respect to the current Gaussian process approximation to the smoothed process. The method is verified in two challenging tracking problems, a reentry problem and a radar tracked coordinated turn model with state dependent diffusion. The results show that the method has better estimation accuracy than state-of-the-art smoothers.