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：在本文中，我们将基于幂矩的单变量非高斯贝叶斯滤波器结果扩展至多变量系统，此类系统既可为线性也可为非线性。这一扩展引入了若干具有挑战性的问题，例如密度替代函数的正参数化，这不仅涉及滤波器设计问题，同时也是多维哈默尔矩问题。我们提出了一种密度替代函数的参数化方法，并给出了其存在性、正性定理及唯一性的证明。基于此，我们分析了利用所提密度替代函数进行密度估计时的矩误差。通过讨论连续与离散两种处理方式对非高斯贝叶斯滤波问题的不同影响，我们论证了对系统状态进行连续参数化研究的必要性。通过估计不同类型多变量密度函数的仿真结果，验证了所提滤波器的有效性。据我们所知，所提滤波器是首个以连续函数参数化系统状态实现多变量贝叶斯滤波的方法，且仅要求真实状态满足勒贝格可积性。