  In many applications of finance, biology and sociology, complex systems
involve entities interacting with each other. These processes have the
peculiarity of evolving over time and of comprising latent factors, which
influence the system without being explicitly measured. In this work we present
latent variable time-varying graphical lasso (LTGL), a method for multivariate
time-series graphical modelling that considers the influence of hidden or
unmeasurable factors. The estimation of the contribution of the latent factors
is embedded in the model which produces both sparse and low-rank components for
each time point. In particular, the first component represents the connectivity
structure of observable variables of the system, while the second represents
the influence of hidden factors, assumed to be few with respect to the observed
variables. Our model includes temporal consistency on both components,
providing an accurate evolutionary pattern of the system. We derive a tractable
optimisation algorithm based on alternating direction method of multipliers,
and develop a scalable and efficient implementation which exploits proximity
operators in closed form. LTGL is extensively validated on synthetic data,
achieving optimal performance in terms of accuracy, structure learning and
scalability with respect to ground truth and state-of-the-art methods for
graphical inference. We conclude with the application of LTGL to real case
studies, from biology and finance, to illustrate how our method can be
successfully employed to gain insights on multivariate time-series data.
