Generalised Particle Filters with Gaussian Mixtures

TL;DR

This paper provides a rigorous theoretical analysis of Gaussian mixture particle filters for stochastic filtering, establishing convergence rates and demonstrating the effectiveness of the new method through numerical examples.

Contribution

It introduces a theoretically justified Gaussian mixture approximation for filtering, with proven convergence rates, filling a gap in the existing literature.

Findings

Established L^2-convergence rate for the Gaussian mixture filter

Provided numerical examples validating the new algorithm

Filled a theoretical gap in Gaussian mixture filtering methods

Abstract

Stochastic filtering is defined as the estimation of a partially observed dynamical system. A massive scientific and computational effort is dedicated to the development of numerical methods for approximating the solution of the filtering problem. Approximating the solution of the filtering problem with Gaussian mixtures has been a very popular method since the 1970s (see [1],[2],[46],[49]). Despite nearly fifty years of development, the existing work is based on the success of the numerical implementation and is not theoretically justified. This paper fills this gap and contains a rigorous analysis of a new Gaussian mixture approximation to the solution of the filtering problem. We deduce the L^2-convergence rate for the approximating system and show some numerical example to test the new algorithm.