Model-based statistical depth for matrix data

The field of matrix data learning has witnessed significant advancements in recent years, encompassing diverse datasets such as medical images, social networks, and personalized recommendation systems. These advancements have found widespread application in various domains, including medicine, biology, public health, engineering, finance, economics, sports analytics, and environmental sciences. While extensive research has been conducted on estimation, inference, prediction, and computation for matrix data, the ranking problem has not received adequate attention. Statistical depth, a measure providing a center-outward rank for different data types, has been introduced in the past few decades. However, its exploration has been limited due to the complexity of the second and higher order-statistics. In this paper, we propose an approach to rank matrix data by employing a model-based depth framework. Our methodology involves estimating the eigen-decomposition of a 4th-order covariance tensor. To enable this process using conventional matrix operations, we specify the tensor product operator between matrices and 4th-order tensors. Furthermore, we introduce a Kronecker product form on the covariance to enhance the robustness and efficiency of the estimation process, effectively reducing the number of parameters in the model. Based on this new framework, we develop an efficient algorithm to estimate the model-based statistical depth. To validate the effectiveness of our proposed method, we conduct simulations and apply it to two real-world applications: field goal attempts of NBA players and global temperature anomalies.