Gaussian Processes for Topology Inference of Directed Graphs

In machine learning applications, the data are often high-dimensional and intrinsically related. It is often of interest finding the underlying structure and the causal relationships of the data and representing the findings 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 estimating the topology of the graph that reflects how the nodes of the graph affect each other, if at all. We propose a novel Bayesian method which 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 adjacency matrix as hyperparameters. The method employs an automatic relevance determination (ARD) kernel and allows for learning of the mapping function from selected past data to current data. The resulting adjacency matrix provides the intrinsic structure and answers questions related to causality. Numerical tests show that the proposed method has comparable or better performance than state-of-the-art methods.