A Nonparametric Discrete Hawkes Model with a Collapsed Gaussian-Process Prior
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| Format: | Preprint |
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2025
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| author | Brisley, Trinnhallen Ross, Gordon Paulin, Daniel |
| author_facet | Brisley, Trinnhallen Ross, Gordon Paulin, Daniel |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_21996 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Nonparametric Discrete Hawkes Model with a Collapsed Gaussian-Process Prior Brisley, Trinnhallen Ross, Gordon Paulin, Daniel Machine Learning 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. |
| title | A Nonparametric Discrete Hawkes Model with a Collapsed Gaussian-Process Prior |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2509.21996 |