A Multivariate Non-Gaussian Bayesian Filter Using Power Moments

Abstract: In this paper, we extend our results on the univariate non-Gaussian Bayesian filter using power moments to the multivariate systems, which can be either linear or nonlinear. Doing this introduces several challenging problems, for example a positive parametrization of the density surrogate, which is not only a problem of filter design, but also one of the multiple dimensional Hamburger moment problem. We propose a parametrization of the density surrogate with the proofs to its existence, Positivstellensatz and uniqueness. Based on it, we analyze the errors of moments of the density estimates by the proposed density surrogate. A discussion on continuous and discrete treatments to the non-Gaussian Bayesian filtering problem is proposed to motivate the research on continuous parametrization of the system state. Simulation results on estimating different types of multivariate density functions are given to validate our proposed filter. To the best of our knowledge, the proposed filter is the first one implementing the multivariate Bayesian filter with the system state parameterized as a continuous function, which only requires the true states being Lebesgue integrable.