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

Abylay Zhumekenov, Alexandros Beskos, Dan Crisan, Ajay Jasra, Nikolas Kantas

TL;DR

This work considers the discrete-time filtering problem in scenarios where the observation noise is degenerate or low, and develops new particle filtering algorithms which are robust to both low noise and fine levels of time discretization.

Abstract

We consider the discrete-time filtering problem in scenarios where the observation noise is degenerate or low. We focus on the case where the observation equation is a linear function of the state and that additive noise is low or degenerate, however, we place minimal assumptions on the hidden state process. In this scenario we derive new particle filtering (PF) algorithms and, under assumptions, in such a way that as the noise becomes more degenerate a PF which approximates the low noise filtering problem provably 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.

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

TL;DR

This work considers the discrete-time filtering problem in scenarios where the observation noise is degenerate or low, and develops new particle filtering algorithms which are robust to both low noise and fine levels of time discretization.

Abstract

We consider the discrete-time filtering problem in scenarios where the observation noise is degenerate or low. We focus on the case where the observation equation is a linear function of the state and that additive noise is low or degenerate, however, we place minimal assumptions on the hidden state process. In this scenario we derive new particle filtering (PF) algorithms and, under assumptions, in such a way that as the noise becomes more degenerate a PF which approximates the low noise filtering problem provably 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.
Paper Structure (28 sections, 2 theorems, 77 equations, 12 figures)

This paper contains 28 sections, 2 theorems, 77 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Modeling
  3. State-Space Model
  4. Natural Parametrization
  5. Noise Parametrization and Degenerate Noise
  6. Degenerate Noise
  7. Particular Cases
  8. General Additive Noise
  9. Low Noise Case
  10. Degenerate Noise Case
  11. Mathematical Result
  12. Special Case of Additive Noise
  13. Particle Filter for the Natural Parameterization
  14. Mathematical Result
  15. Partially Observed Diffusions
  16. ...and 13 more sections

Key Result

Proposition 3.1

Assume (Aass:1). Then for any $(n,r,\varphi)\in\mathbb{N}^2\times\mathtt{B}_b(\mathbb{R}^{d_x-d_y})$ we have

Figures (12)

  • Figure 1: Marginals densities of $X_{n,1}$ for $n\in\{1,6,11,16\}$. This is for the linear Gaussian model.
  • Figure 2: ESS (left) and MSE (right) versus $\log_{10}(1/\Delta)$. This is for the linear Gaussian model.
  • Figure 3: Components of $X_n$ for $n\in\{1,\dots,400\}$. This is for the nonlinear model.
  • Figure 4: Marginals densities of $X_{n,8}$ for $n\in\{1,101,201,301\}$. This is for the nonlinear model.
  • Figure 5: ESS versus $n$. This is for the nonlinear model.
  • ...and 7 more figures

Theorems & Definitions (7)

  • Remark 3.1
  • Proposition 3.1
  • Remark 3.2
  • Proposition 3.2
  • Remark 3.3
  • proof
  • proof