A Scalable Strategy for the Identification of Latent-Variable Graphical Models

In this article, we propose an identification method for latent-variable graphical models associated with autoregressive (AR) Gaussian stationary processes. The identification procedure exploits the approximation of AR processes through stationary reciprocal processes thus benefiting of the numerical advantages of dealing with block-circulant matrices. These advantages become more and more significant as the order of the process gets large. We show how the identification can be cast in a regularized convex program and we present numerical examples that compares the performances of the proposed method with the existing ones.

I. Introduction

Graphical models for time-series have been introduced in [1] and [2] and have become an important tool for the analysis and the identification of stochastic processes: among the first papers dealing with these issues, we mention [3], [4]; in [5], complexity criteria for selecting the best graphical model have been introduced; a Bayesian point of view has been discussed in [6]. These models provide a graphical representation of the conditional independence relations among the process’ components, thus highlighting the interdependence among such components. As a consequence, graphical models may be used to uncover the topological structure of the system generating the observed data. Observe that the smaller is the number of edges of a graphical model the richer is the information provided by the model on the structure of the system (in the extreme case when the graph is full, it does not provide any information and does not uncover any structure). In other words, sparse graphs correspond to parsimonious models. It may happen, however, that most of the components of the graph are genuinely interconnected thus corresponding to a graph that is almost full and hence essentially uninformative. This may be due to the fact that the data structure is based on the presence of a few latent (i.e., unobserved) variables that explain most of the interdependence among the process components. Thus, by taking into account the presence of latent variables we may unveil a whole hidden structure in the system. A simple example is that of the electrical power consumption of each of the houses in a certain city. It may happen that the strong correlation among all the components is mostly explained by a common correlation with a single unobserved variable such as the external temperature. Graphical models containing also nodes accounting for the effect of the latent variables are called latent-variable graphical models [7], [8].