Modeling and applications of discrete-time Hawkes processes: flexible and scalable methods

Abstract

Many count time series show short-term clustering: after one event, more tend to follow soon after. This thesis develops fast and flexible methods for modelling such self-exciting behaviour in data observed on a regular time grid using discrete-time Hawkes models. Our contributions are twofold: first, we derive linear-time algorithms for the log-likelihood and its gradient, which enable efficient estimation for long multivariate sequences. We also build a marked, multivariate model for operational risk in a forensic psychiatric hospital, with an alarm mark for rare severe incidents that prompt a hospital-wide response; in that setting the model improves prediction and yields management-relevant risk signals. Second, we propose the Gaussian Process Discrete Hawkes Process (GP-DHP), a Bayesian nonparametric model that places independent Gaussian process priors on the baseline intensity and the excitation kernel. A collapsed representation over the latent additive intensity supports scalable maximum a posteriori estimation with complexity proportional to the number of observations, and a post hoc decomposition recovers smooth, data-adaptive baseline and excitation functions that separate exogenous background from endogenous feedback. We evaluate the methods on synthetic data and on two applications: weekly Cryptosporidiosis counts and U.S. terrorism incidents. Across these studies GP-DHP outperforms parametric discrete Hawkes baselines in predictive accuracy while revealing a flexible excitation function and seasonal structure.

The overall result is a practical toolkit for discrete-time self-excitation that is both scalable and interpretable.