Hidden Markov models (HMMs) have been extensively used in the univariate and multivariate literature. However, there has been an increased interest in the analysis of matrix-variate data over the recent years. In this manuscript we introduce HMMs for matrix-variate balanced longitudinal data, by assuming a matrix normal distribution in each hidden state. Such data are arranged in a four-way array. To address for possible overparameterization issues, we consider the eigen decomposition of the covariance matrices, leading to a total of 98 HMMs. An expectation-conditional maximization algorithm is discussed for parameter estimation. The proposed models are firstly investigated on simulated data, in terms of parameter recovery, computational times and model selection. Then, they are fitted to a four-way real data set concerning the unemployment rates of the Italian provinces, evaluated by gender and age classes, over the last 16 years.

隐马尔可夫模型 (HMM) 已广泛用于单变量和多变量文献中。然而，近年来，人们对矩阵变量数据的分析越来越感兴趣。在这篇手稿中，我们通过假设每个隐藏状态中的矩阵正态分布来介绍矩阵变量平衡纵向数据的 HMM。这些数据以四路阵列排列。为了解决可能的过度参数化问题，我们考虑了协方差矩阵的特征分解，总共产生了 98 个 HMM。讨论了用于参数估计的期望条件最大化算法。所提出的模型首先在模拟数据上进行了参数恢复、计算时间和模型选择方面的研究。然后，