  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 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
spectral 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.
