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-series. Traditionally, graphical models are estimated under the assumption that data are drawn identically from a generating distribution. Introducing sparsity and sparse-difference 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 dataset. Supplementary material for this article is available online.

KEYWORDS:

• Changepoint
• High-dimensional
• M-estimator
• Sparsity
• Time-series

Supplementary Materials

An Appendix containing technical details and further information can be found in the online supplemental material for this article. In addition to the Appendix, one can also find a compressed folder with example code to implement the GFGL and IFGL estimators as discussed in the article.

Acknowledgments

In the bright memory of Dr. James D. B. Nelson who sadly passed away Sep. 2016. The authors gratefully acknowledge support from the UK Defence Science & Technology Laboratory (DSTL).