Parsimonious hidden Markov models for matrix-variate longitudinal data

Abstract

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.

Modeling matrix variate time series via hidden Markov models with skewed emissions

“…In Asilkalkan and Zhu [20], the authors consider modeling matrix variate time series by means of HMMs with matrix variate normal emissions. Furthermore, Tomarchio et al [21] implement parsimonious HMMs with matrix variate normal emissions. Although the matrix‐variate normal distribution is well‐known and frequently used in the literature due to its mathematical properties, the use of a matrix‐variate normal distribution may become problematic when the data is either skewed or contains outliers.…”

Finite mixture model of hidden Markov regression with covariate dependence

“…We refer the readers to the works of Rabiner and Juang (1986) and Rabiner (1989) for methodological developments and interesting applications. Some notable developments in recent years include contaminated Gaussian HMM (Punzo & Maruotti, 2016), HMM for matrix time series (Asilkalkan & Zhu, 2021) and parsimonious HMM for matrix‐variate data (Tomarchio et al, 2021).…”