TVEG: Model Selection of the Time-Varying Exponential Family Distributions Graphical Models

The undirected graphical model, a popular class of statistical model, offers a way to describe and explain the relationships among a set of variables. However, it remains a challenge to choose a certain graphical model to explain the relationships of variables adequately, especially when the relationships of variables are rewiring over time. This paper proposes the Time-Varying Exponential Family Distributions Graphical (TVEG) models, with time-varying structures and exponential family node-wise conditional distributions. TVEG models extend the scope of available graph models and can be applied to time-varying and exponential family distribution observation data in reality. We propose the Temporally Smoothed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L_{1}$</tex-math></inline-formula>-regularized exponential family graphical estimator (TSLEG), an estimator to infer the structure of TVEG from observations. We derive sufficient conditions for the TSLEG to recover the block partition and sparse pattern with high probability. We derive a message-passing optimization method to solve the TSLEG for time-varying Ising, Gaussian, exponential, and Poisson graphs based on the ADMM. The synthetic network simulations corroborate the theoretical analysis. Analysing of real data of stocks and the US Senate by the time-varying exponential model and Poisson model indicates the effectiveness and practicality of TVEG models.