Regularized estimation of precision matrix for high-dimensional multivariate longitudinal data

Abstract

Estimating covariance matrix is one of the most important analytical tasks in analyzing multivariate longitudinal data, which provides a unique opportunity in studying the joint evolution of multiple response variables over time. This paper introduces a BiConvex Blockwise Regularization (BCBR) sparse estimator for the precision matrix (inverse of the covariance matrix) of high dimensional multivariate longitudinal responses. Using the modified Cholesky block decomposition, we impose a block banded structure on the Cholesky factor and sparsity on the innovation variance matrices via a novel convex hierarchical penalty and lasso penalty, respectively. The blockwise banding structure is a generalization of the existing banding structure for univariate longitudinal data. An efficient alternative convex optimization algorithm is developed by using ADMM algorithm. The resulting estimators are shown to converge in an optimal rate of Frobenius norm, and the exact bandwidth recovery is established for the precision matrix. Simulations and real-life data analysis show that the proposed estimator outperforms its competitors.