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 of the state to estimate, 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 both the light-tailed and heavy-tailed density surrogates. Error upper bounds of the density estimate 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.
翻译:这个问题的困难在于国家估计参数的参数化。 但是,现有的方法无法很好地处理。 我们提议使用权力时间来获得参数化。 与现有的参数估计方法不同, 我们提议的算法并不要求事先了解国家来估计, 例如模式的数量和可行的功能类别。 此外, 拟议的算法不需要在过滤过程中存储大量参数, 以作为现有非参数性贝叶斯过滤器( 例如粒子过滤器) 的过滤器。 拟议的超光速化参数的参数也可以由带有时间限制的convex优化方案来确定, 解决方案已经证明存在并且是独一无二的。 为密度估计存在和具有限定性的所有能量时刻提供了必要和充分的条件。 对轻尾部和重尾部密度加速器进行了分析; 提出了密度估计的错误上限。 对不同类型的国家密度功能进行了模拟结果, 包括重尾部算法。