Learning Spatio-Temporal Graphical Models From Incomplete Observations

This paper investigates the problem of learning a graphical model from incomplete spatio-temporal measurements. Our purpose is to analyze a time-varying graph signal represented by an incomplete data matrix, the rows and columns of which correspond to spatial and temporal features/measurements of the signal, respectively. In contrast to the conventional approaches which utilize either a directed or an undirected graphical model for data analysis, we propose a compound multi-relational model exploiting both directed and undirected structures. Our approach is based on statistical inference in which a spatio-temporal signal is considered as a random graph process to which we can apply maximum-a-posteriori estimation methods for model identification. We incorporate the Gaussian-Markov random field (GMRF) and the vector auto-regressive (VAR) models to capture both the (undirected) spatial correlations and the (directed) temporal dependencies. We propose an algorithm for joint estimation of the signal and the graphical models, from incomplete measurements. For this purpose, we formulate an optimization problem that we solve using the block successive upperbound minimization (BSUM) method. Our simulation results confirm the efficiency of the proposed method for signal recovery and graph learning.