Estimation and application of Bayesian Hawkes process models

Abstract

In this thesis, we examine various facets of Bayesian approaches to Hawkes Processes. Hawkes Processes are a flexible class of point processes that are used to model events that occur in clusters or bursts, as classic Hawkes processes capture the self-exciting behaviour where one event makes future events more likely. While they are popular in the earthquake literature, they are also successfully used in other applications, such as crime, email or Twitter messaging patterns, or tradings on the stock market.

A variety of estimation procedures, both in the frequentist and Bayesian domains, exist to estimate the parameters of the Hawkes process. The goal of this thesis is to enable and improve parameter estimation for different scenarios, such as missing data and inhibition. We use these findings to apply Hawkes processes to product sales analysis, specifically to identify product cannibalisation, and to model data from a group chat setting.

We address issues in parameter estimation in the excitation-only case when data from a Hawkes process is missing. This can severely bias the learning of the Hawkes process parameters. As such, we develop a novel estimation approach based on Approximate Bayesian Computation.

We then consider an extension of the Hawkes process which incorporates inhibition, where events can decrease the intensity function. This leads to additional complexities in the estimation procedure. We resolve challenges regarding the integration of the intensity function and introduce a new, less restrictive condition for stability as existing conditions are unnecessarily strict under inhibition.

Based on these findings, we use the multivariate Hawkes process to model product sales. In particular, we are interested in product cannibalisation, which refers to the decrease in the sales of one product due to competition from another product. We examine this phenomenon in a wholesale data set provided by an international company using a multivariate Hawkes process with inhibition. For this, we design a dimension-independent prior for inhibition based on a reparametrisation.

Finally, we propose an extension to the classic multivariate Hawkes model, which permits different influences for immigrant and triggered events subject to the latent branching structure. We showcase this extended model on data from a group chat.