Regularized Covariance Analysis For Multiple Response Longitudinal Data

Multiple response longitudinal data analysis is becoming popular in a number of contemporary quantitative research fronts, of which the covariance estimation is much more complicated than univariate response longitudinal data. In this paper, we develop block-wise methods for estimating the sparse precision matrix. We first ex-pand the Cholesky Decomposition to blocked precision matrix and obtain the positive-definiteness of the estimate. The Cholesky factors can be related to the regression coefficients and the prediction error covariance of a certain linear auto-regression model. Then we establish a biconvex optimization problem, which is generated from the penalized negative log-likelihood minimization, and we develop Alternate Convex Search Algorithm to solve it. We use the Graphical Lasso and block-wise coordinate descent methods alternately to minimize the biconvex objective function and prove the convergence of the algorithm. Simulation evidence shows that our method is ef-fective and efficient compared with other approaches, especially in models of sparse correlation.