Separable covariance arrays via the Tucker product, with applications to multivariate relational data

Modern datasets are often in the form of matrices or arrays, potentially having correlations along each set of data indices. For example, data involving repeated measurements of several variables over time may exhibit temporal correlation as well as correlation among the variables. A possible model for matrix-valued data is the class of matrix normal distributions, which is parametrized by two covariance matrices, one for each index set of the data. In this article we describe an extension of the matrix normal model to accommodate multidimensional data arrays, or tensors. We generate a class of array normal distributions by applying a group of multilinear transformations to an array of independent standard normal random variables. The covariance structures of the resulting class take the form of outer products of dimension-specific covariance matrices. We derive some properties of these covariance structures and the corresponding array normal distributions, discuss maximum likelihood and Bayesian estimation of covariance parameters and illustrate the model in an analysis of multivariate longitudinal network data. Some key words: Gaussian, matrix normal, multiway data, network, tensor, Tucker decomposition.