Particle Filtering for a Class of State-Space Models with Low and Degenerate Observational Noise

TL;DR

This paper introduces new particle filtering methods that remain effective even when observational noise is very low or degenerate, extending to diffusion processes and demonstrating robustness through theoretical analysis and numerical examples.

Contribution

The authors develop particle filters that are robust to low and degenerate observation noise, with theoretical guarantees and applicability to diffusion processes.

Findings

Particle filters maintain performance with low or degenerate noise.

The methods are theoretically proven to inherit properties from degenerate noise cases.

Numerical examples demonstrate effectiveness across scenarios.

Abstract

We consider the discrete-time filtering problem in scenarios where the observation noise is low or degenerate. We focus on the case where the observation equation is a linear function of the state and the data involve additive noise. However, we place minimal assumptions on the hidden state process. For such a class of models we derive new particle filters (PFs) with the key property that their performance is robust to the size of the observation noise. As a consequence, the developed PFs are well-defined in the limiting case of degenerate observation noise. Indicatively, we prove (under assumptions) that the PF applied in this low noise setting inherits the properties of the PF used in the degenerate case. We extend our framework to the case where the hidden states are drawn from a diffusion process. In this scenario we develop new PFs which are robust to both low noise and fine levels of time discretization. We illustrate our algorithms numerically on several examples.