Abstract Hidden Markov models (HMMs) describe the relationship between two stochastic processes, namely, an observed outcome process and an unobservable finite-state transition process. Given their ability to model dynamic heterogeneity, HMMs are extensively used to analyze heterogeneous longitudinal data. A majority of early developments in HMMs assume that observation times are discrete and regular. This assumption is often unrealistic in substantive research settings where subjects are intermittently seen and the observation times are continuous or not predetermined. However, available works in this direction restricted only to certain special cases with a homogeneous generating matrix for the Markov process. Moreover, early developments have mainly assumed that the number of hidden states of an HMM is fixed and predetermined based on the knowledge of the subjects or a certain criterion. In this article, we consider a general continuous-time HMM with a covariate specific generating matrix and an unknown number of hidden states. The proposed model is highly flexible, thereby enabling it to accommodate different types of longitudinal data that are regularly, irregularly, or continuously collected. We develop a maximum likelihood approach along with an efficient computer algorithm for parameter estimation. We propose a new penalized procedure to select the number of hidden states. The asymptotic properties of the estimators of the parameters and number of hidden states are established. Various satisfactory features, including the finite sample performance of the proposed methodology, are demonstrated through simulation studies. The application of the proposed model to a dataset of bladder tumors is presented.

中文翻译：

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纵向数据的连续时间隐马尔可夫模型

摘要 隐马尔可夫模型 (HMM) 描述了两个随机过程之间的关系，即观察到的结果过程和不可观察的有限状态转换过程。鉴于其对动态异质性建模的能力，HMM 被广泛用于分析异质纵向数据。HMM 的大多数早期发展都假设观察时间是离散的和规则的。这种假设在实质性研究环境中通常是不切实际的，在这些环境中，受试者被间歇性地观察，观察时间是连续的或不是预先确定的。然而，在这个方向上的可用工作仅限于某些特殊情况，具有用于马尔可夫过程的齐次生成矩阵。而且，早期的发展主要假设 HMM 的隐藏状态的数量是固定的，并且是基于主体的知识或某个标准预先确定的。在本文中，我们考虑具有协变量特定生成矩阵和未知数量的隐藏状态的一般连续时间 HMM。所提出的模型高度灵活，从而使其能够适应定期、不规则或连续收集的不同类型的纵向数据。我们开发了一种最大似然方法以及一种用于参数估计的有效计算机算法。我们提出了一个新的惩罚程序来选择隐藏状态的数量。建立了参数估计量和隐藏状态数的渐近性质。各种令人满意的功能，包括所提出方法的有限样本性能，通过模拟研究得到证明。提出了所提出的模型在膀胱肿瘤数据集上的应用。