Estimating Covariance Matrices at Different Levels in Repeated Measurements

Repeated measurements are common in many fields, where random variables are observed repeatedly across different subjects.Such data have an underlying hierarchical structure, and it is of interest to learn covariance/correlation at different levels.Most existing methods for sparse covariance/correlation matrix estimation assume independent samples.Ignoring the underlying hierarchical structure and correlation within the subject may lead to erroneous scientific conclusions.In this paper, we propose to distinguish between the between-subject covariance structure and the within-subject covariance structure.In the presence of repeated measurement, this leads to the problem of sparse and positive-definite estimation of between-subject and within-subject covariance matrices.Our estimators are solutions to convex optimization problems that can be solved efficiently.We establish estimation error rates for the proposed estimators and demonstrate their favorable performance through theoretical analysis and comprehensive simulation studies.We further apply our methods to construct between-subject and within-subject covariance graphs of clinical variables from