UC Santa Barbara

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 dependence structures at different levels. Most existing methods for sparse estimation of dependence and conditional dependence structures assume independent samples. Ignoring the underlying hierarchical structure within the subject may lead to erroneous scientific conclusion.

In Part I, we study the problem of sparse and positive-definite estimation of between-subject and within-subject covariance matrices for repeated measurements. 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 hemodialysis patients.

Part II shifts the focus towards learning temporal, contemporaneous and between-subjects conditional dependence graphs with a graphical vector autoregression model. We propose a two-stage procedure for the simultaneous estimation of these three graphs. Furthermore, Bayesian information criteria are formulated for tuning parameters selection in our two-stage method. The performance of the proposed method is evaluated through extensive simulation studies and one real data application.