Robust Kalman Filters Based on the Sub-Gaussian $\alpha$-stable Distribution

TL;DR

This paper introduces a novel robust Kalman filter framework that models heavy-tailed measurement noise using the sub-Gaussian alpha-stable distribution, employing variational Bayesian methods and multiple estimators to improve filtering accuracy under non-Gaussian noise.

Contribution

It proposes the RKF-SGαS framework for heavy-tailed noise, utilizing variational Bayesian approximation and novel estimators, advancing robust filtering techniques for non-Gaussian noise scenarios.

Findings

RKF-SGαS outperforms existing RKFs in heavy-tailed noise conditions

The proposed estimators improve the accuracy of the scale function estimation

Simulation results confirm the efficiency and robustness of the proposed method

Abstract

Motivated by filtering tasks under a linear system with non-Gaussian heavy-tailed noise, various robust Kalman filters (RKFs) based on different heavy-tailed distributions have been proposed. Although the sub-Gaussian $\alpha$-stable (SG$\alpha$S) distribution captures heavy tails well and is applicable in various scenarios, its potential has not yet been explored in RKFs. The main hindrance is that there is no closed-form expression of its mixing density. This paper proposes a novel RKF framework, RKF-SG$\alpha$S, where the signal noise is assumed to be Gaussian and the heavy-tailed measurement noise is modelled by the SG$\alpha$S distribution. The corresponding joint posterior distribution of the state vector and auxiliary random variables is approximated by the Variational Bayesian (VB) approach. Also, four different minimum mean square error (MMSE) estimators of the scale function are presented. The first two methods are based on the Importance Sampling (IS) and Gauss-Laguerre quadrature (GLQ), respectively. In contrast, the last two estimators combine a proposed Gamma series (GS) based method with the IS and GLQ estimators and hence are called GSIS and GSGL. Besides, the RKF-SG$\alpha$S is compared with the state-of-the-art RKFs under three kinds of heavy-tailed measurement noises, and the simulation results demonstrate its estimation accuracy and efficiency. All the code needed to reproduce the results presented in this work are available at: https://github.com/PengchengH/Robust-Kalman-Filters-Based-on-the-Sub-Gaussian-alpha-stable-Distribution.