Estimating Time-Varying Graphical Models

In this article, we study time-varying graphical models based on data measured over a temporal grid. Such models are motivated by the needs to describe and understand evolving interacting relationships among a set of random variables in many real applications, for instance, the study of how stock prices interact with each other and how such interactions change over time. We propose a new model, <i>LOcal Group Graphical Lasso Estimation</i> (loggle), under the assumption that the graph topology changes gradually over time. Specifically, loggle uses a novel local group-lasso type penalty to efficiently incorporate information from neighboring time points and to impose structural smoothness of the graphs. We implement an ADMM-based algorithm to fit the loggle model. This algorithm utilizes blockwise fast computation and pseudo-likelihood approximation to improve computational efficiency. An R package loggle has also been developed and is available at https://cran.r-project.org/. We evaluate the performance of loggle by simulation experiments. We also apply loggle to S&amp;P 500 stock price data and demonstrate that loggle is able to reveal the interacting relationships among stock prices and among industrial sectors in a time period that covers the recent global financial crisis. The supplemental materials for this article are available online.