A class of discrete‐time random processes arising in engineering and econometrics applications consists of a linear state‐space model whose parameters are modulated by the state of a finite‐state Markov chain. Typical filtering approaches are collapsing methods, which approximate filtered distributions by mixtures of Gaussians, each Gaussian corresponding to one possibility of the recent history of the Markov chain, and particle methods. This article presents an alternative approach to filtering these processes based on keeping track of the values of the underlying filtered density and its characteristic function on grids. We prove that it has favorable convergence properties under certain assumptions. On the other hand, as a grid method, it suffers from the curse of dimensionality, and so is only suitable for low‐dimensional systems. We compare our method to collapsing filters and a particle filter with examples, and find that it can outperform them on 1‐ and 2‐dimensional problems, but loses its speed advantage on 3‐dimensional systems. Meanwhile, our method has a proven theoretical convergence rate that is probably not achieved by collapsing and particle methods.

中文翻译：

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一种带马尔可夫切换的线性状态空间模型的数值滤波方法

工程和计量经济学应用中出现的一类离散时间随机过程由线性状态空间模型组成，其参数由有限状态马尔可夫链的状态调制。典型的滤波方法是折叠方法，它通过混合高斯分布近似滤波分布，每个高斯分布对应于马尔可夫链最近历史的一种可能性，以及粒子方法。本文提出了一种基于跟踪基础过滤密度的值及其在网格上的特征函数来过滤这些过程的替代方法。我们证明它在某些假设下具有良好的收敛特性。另一方面，作为一种网格方法，它受到维数灾难的影响，因此只适用于低维系统。我们将我们的方法与折叠过滤器和粒子过滤器与示例进行比较，发现它可以在一维和二维问题上胜过它们，但在 3 维系统上失去了速度优势。同时，我们的方法具有经过验证的理论收敛速度，这可能是折叠和粒子方法无法实现的。