Convergence Analysis of Ensemble Filters for Linear Stochastic Systems with Poisson-Sampled Observations

TL;DR

This paper analyzes the convergence of ensemble filters for linear stochastic systems with Poisson-sampled observations, proposing a McKean--Vlasov diffusion approach that simplifies computations while ensuring convergence under certain sampling conditions.

Contribution

It introduces a novel ensemble filtering method for Poisson-sampled observations and proves its convergence to the optimal filter as the number of particles grows.

Findings

Empirical mean and covariance converge to optimal filter values.

The proposed method reduces computational complexity compared to Riccati-based solutions.

Convergence depends on the sampling rate and particle number.

Abstract

For continuous-time linear stochastic dynamical systems driven by Wiener processes, we consider the problem of designing ensemble filters when the observation process is randomly time-sampled. We propose a continuous-discrete McKean--Vlasov type diffusion process with additive Gaussian noise in observation model, which is used to describe the evolution of the individual particles in the ensemble. These particles are coupled through the empirical covariance and require less computations for implementation than the optimal ones based on solving Riccati differential equations. Using appropriate analysis tools, we show that the empirical mean and the sample covariance of the ensemble filter converges to the mean and covariance of the optimal filter if the mean sampling rate of the observation process satisfies certain bounds and as the number of particles tends to infinity.