Regularized Estimation of Piecewise Constant Gaussian Graphical Models: The Group-Fused Graphical Lasso

Abstract

The time-evolving precision matrix of a piecewise-constant Gaussian graphical model encodes the dynamic conditional dependency structure of a multivariate time- is drawn identically from a generating distribution. Introducing sparsity and sparsedifference inducing priors we relax these assumptions and propose a novel regularized M-estimator to jointly estimate both the graph and changepoint structure. The resulting estimator possesses the ability to therefore favor sparse dependency structures and/or smoothly evolving graph structures, as required. Moreover, our approach extends current methods to allow estimation of changepoints that are grouped across multiple dependencies in a system. An efficient algorithm for estimating structure is proposed. We study the empirical recovery properties in a synthetic setting. The qualitative effect of grouped changepoint estimation is then demonstrated by applying the method on a genetic time-course data-set.

A Tutorial on Estimating Time-Varying Vector Autoregressive Models

“…Third, all methods discussed in this paper are based on the assumption that the true parameters are smooth functions of time. However, in some situations it might be more appropriate to assume different kinds of local stationarity, for example piece-wise constant functions (e.g., Bringmann & Albers, 2019;Gibberd & Nelson, 2017). It would be useful to make those alternative estimation methods available to applied researchers, and possibly combine them with the methods presented here.…”

“…Another interesting way to modeling time-varying parameters is by using the fused lasso (Hastie et al, 2015). However, to our best knowledge there are currently only implementations available to estimate time-varying Gaussian Graphical Models with this type of method: a Python implementation (R. Monti, 2014) of the SINGLE algorithm (R. P. Monti et al, 2014) and a Python implementation (Gibbert, 2017) of the (group) fused-lasso based method as presented in Gibberd and Nelson (2017).…”

Algorithms for Estimating Trends in Global Temperature Volatility

“…Our methodology is similar in flavor to [12] or related work in [10,19], but with several fundamental differences. These papers aim to discover the time-varying structure of a network.…”

The Time-Varying Dependency Patterns of NetFlow Statistics

“…In this work we utilise an Alternating Directed Method of Multipliers (ADMM) algorithm which has computational complexity of order O(p 3 T log(T )). In the interests of space we refer the reader to Gibberd et al [5] for details of this implementation, code is availiable on request.…”