A Lagged Particle Filter for Stable Filtering of certain High-Dimensional State-Space Models

TL;DR

This paper introduces a lagged particle filter for high-dimensional state-space models, providing a biased but stable approximation method with controlled error, suitable for complex models like shallow-water systems.

Contribution

The paper proposes a novel lagged approximation for filtering in high-dimensional SSMs, with theoretical guarantees on bias and computational cost, extending applicability to complex models.

Findings

Bias of the approximation is uniformly controlled in dimension.

Cost to achieve stable estimation scales as O(Nd^2) per unit time.

Method successfully applied to high-dimensional examples like shallow-water models.

Abstract

We consider the problem of high-dimensional filtering of state-space models (SSMs) at discrete times. This problem is particularly challenging as analytical solutions are typically not available and many numerical approximation methods can have a cost that scales exponentially with the dimension of the hidden state. Inspired by lag-approximation methods for the smoothing problem, we introduce a lagged approximation of the smoothing distribution that is necessarily biased. For certain classes of SSMs, particularly those that forget the initial condition exponentially fast in time, the bias of our approximation is shown to be uniformly controlled in the dimension and exponentially small in time. We develop a sequential Monte Carlo (SMC) method to recursively estimate expectations with respect to our biased filtering distributions. Moreover, we prove for a class of class of SSMs that can contain dependencies amongst coordinates that as the dimension $d\rightarrow\infty$ the cost to achieve a stable mean square error in estimation, for classes of expectations, is of $\mathcal{O}(Nd^2)$ per-unit time, where $N$ is the number of simulated samples in the SMC algorithm. Our methodology is implemented on several challenging high-dimensional examples including the conservative shallow-water model.