In this paper, which is a very preliminary version, 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, Positivstellensatze and uniqueness. Based on it, we analyze the error of moments of the density estimates through the filtering process with the proposed density surrogate. An error upper bound in the sense of total variation distance is also given. A discussion on continuous and discrete treatments to the non-Gaussian Bayesian filtering problem is proposed to explain why our proposed filter shall also be a mainstream of the non-Gaussian Bayesian filtering research and motivate the research on continuous parametrization of the system state. Last but not the least, 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:本文是一个非常初步的版本,我们在本文中将我们关于非Gaussian Bayesian 的单象牙过滤器的结果扩大到多变系统,这种系统可以是线性或非线性。 这样做带来了一些具有挑战性的问题, 例如密度代孕的正对称化, 这不仅是一个过滤设计问题, 而且是多种维度汉堡时点问题之一。 我们建议用其存在的证据Positivstellensatze和独特性对密度代谢器进行配比化。 我们在此基础上, 通过拟议密度代孕的过滤程序分析密度估计的误差。 也给出了完全变异差距离感上方的误差。 提议对非Gaussian Bayesian过滤器的连续和离散处理问题进行讨论, 以便解释为什么我们提议的过滤器过滤器只能成为非Gaussian Bayesian 过滤器国家的一种主流研究, 并激励关于系统持续折叠化的研究。 最后, 而不是多变式过滤器功能, 是要对一个不断变异的系统进行最起码的模拟, 。