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.

翻译：本文将在单变量非高斯贝叶斯滤波中使用幂矩的研究成果扩展至线性或非线性的多元系统。这引出了若干具有挑战性的问题，例如密度替代函数的正定参数化——这不仅涉及滤波器设计问题，还与多维汉堡矩问题相关。我们提出了一种密度替代函数的参数化方法，并证明了其存在性、正定性定理（Positivstellensatz）及唯一性。基于此参数化方法，我们分析了其密度估计矩的误差。通过讨论非高斯贝叶斯滤波问题的连续与离散处理方法，我们论证了系统状态连续参数化研究的必要性。针对不同类型多元密度函数估计的仿真结果验证了所提滤波器的有效性。据我们所知，本文提出的滤波器是首个将系统状态参数化为连续函数并仅要求真实状态为勒贝格可积的多元贝叶斯滤波器实现。