Some statistical approaches to the analysis of matrix-valued data

Abstract

In many modern applications, we encounter data sampled in the form of two-dimensional matrices. Simple vectorization of the matrix-valued observations would destroy the intrinsic row and column information embedded in such data. In this research, we study three statistical problems that are specific to matrix-valued data. The first one concerns dimension reduction for a group of high-dimensional matrix-valued data. We propose a novel dimension reduction approach that has nice approximation property, computes fast for high dimensionality, and also explicitly incorporates the intrinsic two-dimensional structure of the matrices. We discuss the connection of our proposal with existing approaches, and compare them both numerically and theoretically. We also obtain theoretical upper bounds on the approximation error of our method. The second one is a group independent component analysis approach. Motivated by analysis of groups of high-dimensional imaging data, we develop a framework in the frequency domain through Whittle log-likelihood maximization. Our method starts with efficient population value decomposition, and then models each temporally-dependent source signal via parametric linear processes. The superior performance of our approach is demonstrated through simulation studies and the ADHD200 data. The third one addresses the problem of regression with matrix-valued covariates. We consider the bilinear regression model, where two coefficient vectors are used to incorporate matrix covariates. We propose two maximum likelihood based estimators. Both estimators are shown to achieve the information lower bound and hence are theoretically optimal under the classical asymptotic framework. We further propose a bilinear ridge estimator and derive its convergence property. The superior performances of the proposed estimators are demonstrated both theoretically and numerically.Doctor of Philosoph