Non-Gaussian Bayesian Filtering by Density Parametrization Using Power Moments

TL;DR

This paper introduces a novel non-Gaussian Bayesian filtering method using power moments for density parametrization, which does not require prior knowledge or extensive storage, and is validated through simulations.

Contribution

The paper proposes a new density parametrization approach using power moments that avoids prior assumptions and large storage, with a convex optimization scheme ensuring unique solutions.

Findings

The method accurately estimates non-Gaussian densities including heavy-tailed ones.

The proposed algorithm outperforms existing methods in terms of storage and prior knowledge requirements.

Simulation results confirm the effectiveness of the density estimation across various density types.

Abstract

Non-Gaussian Bayesian filtering is a core problem in stochastic filtering. The difficulty of the problem lies in parameterizing the state estimates. However the existing methods are not able to treat it well. We propose to use power moments to obtain a parameterization. Unlike the existing parametric estimation methods, our proposed algorithm does not require prior knowledge about the state to be estimated, e.g. the number of modes and the feasible classes of function. Moreover, the proposed algorithm is not required to store massive parameters during filtering as the existing nonparametric Bayesian filters, e.g. the particle filter. The parameters of the proposed parametrization can also be determined by a convex optimization scheme with moments constraints, to which the solution is proved to exist and be unique. A necessary and sufficient condition for all the power moments of the density estimate to exist and be finite is provided. The errors of power moments are analyzed for the density estimate being either light-tailed or heavy-tailed. Error upper bounds of the density estimate for the one-step prediction are proposed. Simulation results on different types of density functions of the state are given, including the heavy-tailed densities, to validate the proposed algorithm.