April 2023 A theoretical analysis of one-dimensional discrete generation ensemble Kalman particle filters
Pierre Del Moral, Emma Horton
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Ann. Appl. Probab. 33(2): 1327-1372 (April 2023). DOI: 10.1214/22-AAP1843

Abstract

Despite the widespread usage of discrete generation ensemble Kalman particle filtering methodology to solve nonlinear and high-dimensional filtering and inverse problems, little is known about their mathematical foundations. As genetic-type particle filters (a.k.a. sequential Monte Carlo), this ensemble-type methodology can also be interpreted as mean-field particle approximations of the Kalman–Bucy filtering equation. In contrast with conventional mean-field type interacting particle methods equipped with a globally Lipschitz interacting drift-type function, Ensemble Kalman filters depend on a nonlinear and quadratic-type interaction function defined in terms of the sample covariance of the particles.

Most of the literature in applied mathematics and computer science on these sophisticated interacting particle methods amounts to designing different classes of useable observer-type particle methods. These methods are based on a variety of inconsistent but judicious ensemble auxiliary transformations or include additional inflation/localisation-type algorithmic innovations, in order to avoid the inherent time-degeneracy of an insufficient particle ensemble size when solving a filtering problem with an unstable signal.

To the best of our knowledge, the first and the only rigorous mathematical analysis of these sophisticated discrete generation particle filters is developed in the pioneering articles by Le Gland–Monbet–Tran and by Mandel–Cobb–Beezley, which were published in the early 2010s. Nevertheless, besides the fact that these studies prove the asymptotic consistency of the ensemble Kalman filter, they provide exceedingly pessimistic mean-error estimates that grow exponentially fast with respect to the time horizon, even for linear Gaussian filtering problems with stable one-dimensional signals.

In the present article we develop a novel self-contained and complete stochastic perturbation analysis of the fluctuations, the stability, and the long-time performance of these discrete generation ensemble Kalman particle filters, including time-uniform and nonasymptotic mean-error estimates that apply to possibly unstable signals. To the best of our knowledge, these are the first results of this type in the literature on discrete generation particle filters, including the class of genetic-type particle filters and discrete generation ensemble Kalman filters. The stochastic Riccati difference equations considered in this work are also of interest in their own right, as a prototype of a new class of stochastic rational difference equation.

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Pierre Del Moral. Emma Horton. "A theoretical analysis of one-dimensional discrete generation ensemble Kalman particle filters." Ann. Appl. Probab. 33 (2) 1327 - 1372, April 2023. https://doi.org/10.1214/22-AAP1843

Information

Received: 1 July 2021; Published: April 2023
First available in Project Euclid: 21 March 2023

zbMATH: 07692291
MathSciNet: MR4564428
Digital Object Identifier: 10.1214/22-AAP1843

Subjects:
Primary: 60F05 , 60G35 , 60K35 , 65C35
Secondary: 39A30 , 60J10 , 65L07 , 93D23

Keywords: Ensemble Kalman filter , Exponential stability , Feynman–Kac formulae , noncentral χ-square , rational difference equations , stochastic perturbation theorems , stochastic Riccati difference equation , uniform estimates with respect to the time horizon

Rights: Copyright © 2023 Institute of Mathematical Statistics

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Vol.33 • No. 2 • April 2023
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