Exact and approximate hidden Markov chain filters based on discrete observations

IF 1.3 Q2 STATISTICS & PROBABILITY

Statistics & Risk Modeling Pub Date : 2014-11-04 DOI:10.1515/strm-2015-0004

N. Bäuerle, Igor Gilitschenski, U. Hanebeck

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引用次数: 1

Abstract

Abstract We consider a Hidden Markov Model (HMM) where the integrated continuous-time Markov chain can be observed at discrete time points perturbed by a Brownian motion. The aim is to derive a filter for the underlying continuous-time Markov chain. The recursion formula for the discrete-time filter is easy to derive, however involves densities which are very hard to obtain. In this paper we derive exact formulas for the necessary densities in the case the state space of the HMM consists of two elements only. This is done by relating the underlying integrated continuous-time Markov chain to the so-called asymmetric telegraph process and by using recent results on this process. In case the state space consists of more than two elements we present three different ways to approximate the densities for the filter. The first approach is based on the continuous filter problem. The second approach is to derive a PDE for the densities and solve it numerically. The third approach is a crude discrete time approximation of the Markov chain. All three approaches are compared in a numerical study.

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基于离散观测的精确和近似隐马尔可夫链滤波器

摘要考虑一个隐马尔可夫模型(HMM)，该模型在受布朗运动摄动的离散时间点上可以观测到积分连续马尔可夫链。目的是为底层的连续时间马尔可夫链导出一个滤波器。离散时间滤波器的递推公式很容易推导，但涉及的密度很难得到。本文导出了隐马尔可夫模型的状态空间只有两个元素时所需密度的精确公式。这是通过将潜在的集成连续时间马尔可夫链与所谓的不对称电报过程联系起来，并使用该过程的最新结果来完成的。如果状态空间包含两个以上的元素，我们提出了三种不同的方法来近似滤波器的密度。第一种方法是基于连续滤波问题。第二种方法是推导密度的偏微分方程并进行数值求解。第三种方法是马尔可夫链的离散时间近似。在数值研究中对这三种方法进行了比较。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Statistics & Risk Modeling STATISTICS & PROBABILITY-

CiteScore

1.80

自引率

6.70%

发文量

期刊介绍： Statistics & Risk Modeling (STRM) aims at covering modern methods of statistics and probabilistic modeling, and their applications to risk management in finance, insurance and related areas. The journal also welcomes articles related to nonparametric statistical methods and stochastic processes. Papers on innovative applications of statistical modeling and inference in risk management are also encouraged. Topics Statistical analysis for models in finance and insurance Credit-, market- and operational risk models Models for systemic risk Risk management Nonparametric statistical inference Statistical analysis of stochastic processes Stochastics in finance and insurance Decision making under uncertainty.