On the Mathematical Theory of Ensemble (Linear-Gaussian) Kalman-Bucy Filtering

Abstract

The purpose of this review is to present a comprehensive overview of the theory of ensemble Kalman-Bucy filtering for continuous-time, linear-Gaussian signal and observation models. We present a system of equations that describe the flow of individual particles and the flow of the sample covariance and the sample mean in continuous-time ensemble filtering. We consider these equations and their characteristics in a number of popular ensemble Kalman filtering variants. Given these equations, we study their asymptotic convergence to the optimal Bayesian filter. We also study in detail some non-asymptotic time-uniform fluctuation, stability, and contraction results on the sample covariance and sample mean (or sample error track). We focus on testable signal/observation model conditions, and we accommodate fully unstable (latent) signal models. We discuss the relevance and importance of these results in characterising the filter's behaviour, e.g. it's signal tracking performance, and we contrast these results with those in classical studies of stability in Kalman-Bucy filtering.We also provide a novel (and negative) result proving that the bootstrap particle filter cannot track even the most basic unstable latent signal, in contrast with the ensemble Kalman filter (and the optimal filter). We provide intuition for how the main results extend to nonlinear signal models and comment on their consequence on some typical filter behaviours seen in practice, e.g. catastrophic divergence.

References in corpus (10)

• The Ensemble Kalman Filter for Inverse Problems
• Data Assimilation: A Mathematical Introduction
• Convergence of the Square Root Ensemble Kalman Filter in the Large Ensemble Limit
• Uniform Time Average Consistency of Monte Carlo Particle Filters
• Observability and nonlinear filtering
• The stability of conditional Markov processes and Markov chains in random environments
• McKean-Vlasov SDEs in nonlinear filtering
• Mean field limit of Ensemble Square Root Filters -- discrete and continuous time
• On the continuous time limit of Ensemble Square Root Filters
• Multi-index ensemble Kalman filtering

Cited by in corpus (2)

• Non-Asymptotic Analysis of Ensemble Kalman Updates: Effective Dimension and Localization
• Rough McKean-Vlasov dynamics for robust ensemble Kalman filtering