Hawkes graphs

This paper introduces the Hawkes skeleton and the Hawkes graph. These objects summarize the branching structure of a multivariate Hawkes point process in a compact, yet meaningful way. We demonstrate how the graph-theoretic vocabulary (ancestor sets, parent sets, connectivity, walks, walk weights, etc.) is very convenient for the discussion of multivariate Hawkes processes. For example, we reformulate the classic eigenvalue-based subcriticality criterion of multitype branching processes in graph terms. Next to these more terminological contributions, we show how the graph view can be used for the specification and estimation of Hawkes models from large, multitype event streams. Based on earlier work, we give a nonparametric statistical procedure to estimate the Hawkes skeleton and the Hawkes graph from data. We show how the graph estimation can then be used for specifying and fitting parametric Hawkes models. Our estimation method avoids the a priori assumptions on the model from a straightforward MLE-approach and is numerically more flexible than the latter. Our method has two tuning parameters: one controlling numerical complexity, and the other controlling the sparseness of the estimated graph. A simulation study confirms that the presented procedure works as desired. We pay special attention to computational issues in the implementation. This makes our results applicable to high-dimensional event-stream data such as dozens of event streams and thousands of events per component.