Two Gaussian Regularization Methods for Time-Varying Networks

We model time-varying network data as realizations from multivariate Gaussian distributions with precision matrices that change over time. To facilitate parameter estimation, we require not only that each precision matrix at any given time point be sparse, but also that precision matrices at neighboring time points be similar. We accomplish this with two different algorithms, by generalizing the elastic net and the fused LASSO, respectively. While similar approaches in the literature for modeling time-varying networks have predominantly extended the graphical LASSO of Friedman et al. (Biostatistics 9(3):432-441, 2008), we extend the regression approach of Meinshausen and Bühlmann (Ann Stat 34(3):1436–1462, 2006) and subsequently of Peng et al. (J Am Stat Assoc 104(486):735–746, 2009). This allows us to explicitly focus on and work with the partial correlation coefficients, which are more directly meaningful and interpretable parameters for the biological sciences. We develop efficient algorithms and convenient degree-of-freedom formulae for choosing tuning parameters. The proposed methods are demonstrated through simulation studies. By applying them to an hourly temperature dataset, we detect interesting time-varying connectivity among thirteen Canadian cities. Supplementary materials accompanying this paper appear online.