Graphical Models for Multivariate Time-Series

Gaussian graphical models have received much attention in the last years, due to their flexibility and expression power. In particular, lots of interests have been devoted to graphical models for temporal data, or dynamical graphical models, to understand the relation of variables evolving in time. While powerful in modelling complex systems, such models suffer from computational issues both in terms of convergence rates and memory requirements, and may fail to detect temporal patterns in case the information on the system is partial. This thesis comprises two main contributions in the context of dynamical graphical models, tackling these two aspects: the need of reliable and fast optimisation methods and an increasing modelling power, which are able to retrieve the model in practical applications. The first contribution consists in a forward-backward splitting (FBS) procedure for Gaussian graphical modelling of multivariate time-series which relies on recent theoretical studies ensuring global convergence under mild assumptions. Indeed, such FBS-based implementation achieves, with fast convergence rates, optimal results with respect to ground truth and standard methods for dynamical network inference. The second main contribution focuses on the problem of latent factors, that influence the system while hidden or unobservable. This thesis proposes the novel latent variable time-varying graphical lasso method, which is able to take into account both temporal dynamics in the data and latent factors influencing the system. This is fundamental for the practical use of graphical models, where the information on the data is partial. Indeed, extensive validation of the method on both synthetic and real applications shows the effectiveness of considering latent factors to deal with incomplete information.