A Nonparametric Discrete Hawkes Model with a Collapsed Gaussian-Process Prior

• URL: http://arxiv.org/abs/2509.21996v1
• Date: Fri, 26 Sep 2025 07:23:57 GMT
• Title: A Nonparametric Discrete Hawkes Model with a Collapsed Gaussian-Process Prior
• Authors: Trinnhallen Brisley, Gordon Ross, Daniel Paulin,
• Abstract summary: We propose a nonparametric framework that places Gaussian process priors on both the baseline and the excitation.<n>This yields smooth, data-adaptive structure without prespecifying trends, periodicities, or decay shapes.<n>In simulations, GP-DHP recovers diverse excitation shapes and evolving baselines.
• Score: 0.5352699766206809
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Hawkes process models are used in settings where past events increase the likelihood of future events occurring. Many applications record events as counts on a regular grid, yet discrete-time Hawkes models remain comparatively underused and are often constrained by fixed-form baselines and excitation kernels. In particular, there is a lack of flexible, nonparametric treatments of both the baseline and the excitation in discrete time. To this end, we propose the Gaussian Process Discrete Hawkes Process (GP-DHP), a nonparametric framework that places Gaussian process priors on both the baseline and the excitation and performs inference through a collapsed latent representation. This yields smooth, data-adaptive structure without prespecifying trends, periodicities, or decay shapes, and enables maximum a posteriori (MAP) estimation with near-linear-time \(O(T\log T)\) complexity. A closed-form projection recovers interpretable baseline and excitation functions from the optimized latent trajectory. In simulations, GP-DHP recovers diverse excitation shapes and evolving baselines. In case studies on U.S. terrorism incidents and weekly Cryptosporidiosis counts, it improves test predictive log-likelihood over standard parametric discrete Hawkes baselines while capturing bursts, delays, and seasonal background variation. The results indicate that flexible discrete-time self-excitation can be achieved without sacrificing scalability or interpretability.

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