Topology Inference of Directed Graphs by Gaussian Processes with Sparsity Constraints

In machine learning applications, data are often high-dimensional and intricately related. It is often of interest to find the underlying structure and causal relationships among the data and represent these relationships with directed graphs. In this paper, we study multivariate time series, where each series is associated with a node of a graph, and where the objective is to estimate the topology of a sparse graph that reflects how the nodes of the graph affect each other, if at all. We propose a novel fully Bayesian approach that employs a sparsity-encouraging prior on the hyperparameters. The proposed method allows for nonlinear and multiple lag relationships among the time series. The method is based on Gaussian processes, and it treats the entries of the graph adjacency matrix as hyperparameters. It utilizes a modified automatic relevance determination (ARD) kernel and allows for learning the mapping function from selected past data to current data as edges of a graph. We show that the resulting adjacency matrix provides the intrinsic structure of the graph and answers causality-related questions. Numerical tests show that the proposed method has comparable or better performance than state-of-the-art methods.