The well-established methodology for the estimation of hidden semi-Markov models (HSMMs) as hidden Markov models (HMMs) with extended state spaces is further developed to incorporate covariate influences across all aspects of the state process model, in particular, regarding the distributions governing the state dwell time. The special case of periodically varying covariate effects on the state dwell-time distributions - and possibly the conditional transition probabilities - is examined in detail to derive important properties of such models, namely the periodically varying unconditional state distribution as well as the overall state dwell-time distribution. Through simulation studies, we ascertain key properties of these models and develop recommendations for hyperparameter settings. Furthermore, we provide a case study involving an HSMM with periodically varying dwell-time distributions to analyse the movement trajectory of an arctic muskox, demonstrating the practical relevance of the developed methodology.

翻译：针对将隐半马尔可夫模型（HSMM）作为具有扩展状态空间的隐马尔可夫模型（HMM）进行估计的成熟方法，本研究进一步拓展以纳入协变量对状态过程模型所有方面的影响，特别是关于控制状态驻留时间的分布。本文详细考察了协变量对状态驻留时间分布（以及可能的条件转移概率）产生周期性影响的特殊情形，推导出此类模型的重要性质，即周期性变化的无条件状态分布以及整体状态驻留时间分布。通过模拟研究，我们确定了这些模型的关键特性，并就超参数设置提出了建议。此外，我们通过一个包含周期性变化驻留时间分布的HSMM案例研究，分析了北极麝牛的运动轨迹，证明了所开发方法的实际应用价值。