In this paper, we explore the class of the Hidden Semi-Markov Model (HSMM), a flexible extension of the popular Hidden Markov Model (HMM) that allows the underlying stochastic process to be a semi-Markov chain. HSMMs are typically used less frequently than their basic HMM counterpart due to the increased computational challenges when evaluating the likelihood function. Moreover, while both models are sequential in nature, parameter estimation is mainly conducted via batch estimation methods. Thus, a major motivation of this paper is to provide methods to estimate HSMMs (1) in a computationally feasible time, (2) in an exact manner, i.e. only subject to Monte Carlo error, and (3) in a sequential setting. We provide and verify an efficient computational scheme for Bayesian parameter estimation on HSMMs. Additionally, we explore the performance of HSMMs on the VIX time series using Autoregressive (AR) models with hidden semi-Markov states and demonstrate how this algorithm can be used for regime switching, model selection and clustering purposes.
翻译:在本文中,我们探讨了隐藏半马尔科夫模型(HSMM)的类别,这是流行的隐藏马可夫模型(HMM)的灵活扩展,使基本随机过程成为半马尔科夫链。由于在评估概率函数时计算难度增加,HSMM模型的使用通常比基本HMM模型使用频率低。此外,虽然这两种模型都是相继性的,但参数估计主要通过批量估计方法进行。因此,本文的一个主要动机是提供方法,在计算可行的时间内估算HSMM(1)(1), 精确地说,(2) 仅受蒙特卡洛错误的影响,(3) 依次设定。我们提供并核实了巴伊西亚人对HSMMS参数估计的有效计算方法。此外,我们探索了HSMMS在使用隐藏半马尔科夫状态的自动递进模型的VIX时间序列上的性能,并演示如何将这一算法用于系统转换、模式选择和组合目的。