Undirected graphs are often used to describe high dimensional distributions. Under sparsity conditions, the graph can be estimated using ℓ1 penalization methods. However, current methods assume that the data are independent and identically distributed. If the distribution, and hence the graph, evolves over time then the data are not longer identically distributed. In this paper we develop a nonparametric method for estimating time varying graphical structure for multivariate Gaussian distributions using an ℓ1 regularization method, and show that, as long as the covariances change smoothly over time, we can estimate the covariance matrix well (in predictive risk) even when p is large.
Graph selection ℓ1 regularization High dimensional asymptotics Risk consistency