Sparse Positive-Definite Estimation for Large Covariance Matrices with Repeated Measurements

In many fields of biomedical sciences, it is common that random variables are measured repeatedly across different subjects. In such a repeated measurement setting, dependence structures among random variables that are between subjects and within a subject may be different, and should be estimated differently. Ignoring this fact may lead to questionable or even erroneous scientific conclusions. In this paper, we study the problem of sparse and positive-definite estimation of between-subject and within-subject covariance matrices for high-dimensional repeated measurements. Our estimators are defined as solutions to convex optimization problems, which can be solved efficiently. We establish estimation error rate for our proposed estimators of the two target matrices, and demonstrate their favorable performance through theoretical analysis and comprehensive simulation studies. We further apply our methods to recover two covariance graphs of clinical variables from hemodialysis patients.