A theoretical analysis of one-dimensional discrete generation ensemble Kalman particle filters

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/localisationtype 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 meanerror 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 non-asymptotic 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.

A note on Riccati matrix difference equations

“…Using the Lyapunov fixed point equation (15), for any n ě 0 we readily check that the sequence of Gramian matrices G n solving the time varying Lyapunov recursion (12) can alternatively be defined in terms of solution H of the algebraic Lyapunov equation by the formula…”

“…Thus, in this article, we provide a self-contained study of the latter. We also note that the theory developed in this article is crucial to the stability analysis of discrete time Kalman Ensemble filter and thus the results obtained in this article will allow one to develop the theory of such filters, as discussed in [12], to higher dimensions.…”

“…In Theorem 1.3 we provide a Floquet-type representation analogous to the continuous time version given in [4]. In section 2 we provide some preliminary results concerning the Riccati maps and the Gramian matrices introduced in (12). We also state and prove several useful properties of the Riccati maps that are often used in the literature but for which we have been unable to find a proof.…”

On the Stability of Positive Semigroups

“…General asymptotic stability results are also provided in [175]. Stabilising changes of Feynman-Kac measures and related importance-sampling-type and h-process techniques that apply to possibly unstable killed processes on unbounded domains are also discussed in [12,100,163], see also [71] and well as section B.1 in [16] and in section 7.1 in [72] in the context of discrete time models. The stability of the nonlinear filtering equations with deterministic hyperbolic signals is also developed in the recent article [155].…”

“…Thus, whenever ( 72) is met with V s " V t " r 0 " 1, choosing ǫ " 0 in (72) we have, for any r ą 1, lim δÑ0 β V 0,δ s ,V 0,δ t pP s,t q " βpP s,t q ď 1 ´αprq.…”