Time series of matrix-valued data are increasingly available in various areas including economics, finance, social science, etc. These data may shed light on the inter-dynamical relationships between two sets of attributes, for instance countries and economic indices. The matrix autoregressive (MAR) model provides a parsimonious approach for analyzing such data. However, the MAR model, being a linear model with parametric constraints, cannot capture the nonlinear patterns in the data, such as regime shifts in the dynamics. We propose a mixture matrix autoregressive (MMAR) model for analyzing potential regime shifts in the dynamics between two attributes, for instance, due to recession vs. blooming, or quiet period vs. pandemic. We propose an EM algorithm for maximum likelihood estimation. We derive some theoretical properties of the proposed method including consistency and asymptotic distribution, and illustrate its performance via simulations and real applications.

翻译：矩阵值数据的时间序列在经济学、金融学、社会科学等多个领域中越来越常见。这类数据可能揭示两组属性（例如国家与经济指标）之间的相互动态关系。矩阵自回归（MAR）模型为分析此类数据提供了一种简洁的方法。然而，作为带有参数约束的线性模型，MAR模型无法捕捉数据中的非线性模式，例如动态过程中的机制转换。我们提出一种混合矩阵自回归（MMAR）模型，用于分析两组属性之间可能存在的机制转换，例如由经济衰退与繁荣、或平静期与疫情等因素引起的动态变化。我们提出一种用于最大似然估计的EM算法。我们推导了所提方法的一些理论性质，包括一致性和渐近分布，并通过模拟和实际应用展示了其性能。