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Flexible and Scalable Bayesian Modelling of Spatio-Temporal Hawkes Processes

Wenqing Liu, Xenia Miscouridou, Déborah Sulem

Abstract

Existing spatio-temporal Hawkes process models typically rely on either parametric or semiparametric assumptions, limiting the model's ability to capture complex endogenous and exogenous event dynamics. We propose a fully Bayesian nonparametric framework for spatio-temporal Hawkes processes using additive Gaussian processes for the prior distributions on the background rate and the triggering kernel. This additive structure enhances interpretability by decoupling temporal and spatial effects while maintaining high modelling flexibility across the entire spatio-temporal domain. To address scalability, we develop a sparse variational inference scheme based on the Gaussian variational family. Synthetic experiments demonstrate that the proposed method accurately recovers background and triggering structures, achieving superior performance compared to existing alternatives. When applied to real-world datasets, it achieves higher held-out log-likelihoods and reveals interpretable spatio-temporal structures of the self-excitation mechanism. Overall, the framework provides a flexible, scalable, interpretable, and uncertainty-aware approach for modelling complex excitation patterns in spatio-temporal event data.

Flexible and Scalable Bayesian Modelling of Spatio-Temporal Hawkes Processes

Abstract

Existing spatio-temporal Hawkes process models typically rely on either parametric or semiparametric assumptions, limiting the model's ability to capture complex endogenous and exogenous event dynamics. We propose a fully Bayesian nonparametric framework for spatio-temporal Hawkes processes using additive Gaussian processes for the prior distributions on the background rate and the triggering kernel. This additive structure enhances interpretability by decoupling temporal and spatial effects while maintaining high modelling flexibility across the entire spatio-temporal domain. To address scalability, we develop a sparse variational inference scheme based on the Gaussian variational family. Synthetic experiments demonstrate that the proposed method accurately recovers background and triggering structures, achieving superior performance compared to existing alternatives. When applied to real-world datasets, it achieves higher held-out log-likelihoods and reveals interpretable spatio-temporal structures of the self-excitation mechanism. Overall, the framework provides a flexible, scalable, interpretable, and uncertainty-aware approach for modelling complex excitation patterns in spatio-temporal event data.

Paper Structure

This paper contains 46 sections, 61 equations, 15 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Related methods
  3. Methodology
  4. Problem Setup
  5. Gaussian Process Prior
  6. Model Likelihood
  7. Inference
  8. Sparse GP Approximation
  9. Experiments
  10. Baselines and Metrics
  11. Parametric Hawkes Process bernabeu2025spatio:
  12. Cox--Hawkes Process miscouridou2023cox:
  13. Posterior Mean Error ($PM_{\mathrm{mse}}$):
  14. Posterior Expected Error ($PE_{\mathrm{mse}}$):
  15. Expected Log-Likelihood ($ELL$):
  16. ...and 31 more sections

Figures (15)

  • Figure 1: Estimated Hawkes process components for Scenario 1. The figure is divided into two main panels: the background rate $\mu(t,x,y)$ (left) and the triggering kernel $\phi(\Delta t, \Delta x, \Delta y)$ (right), each shown at two different time snapshots. For each component, the ground truth is on the left and the posterior mean estimate is on the right.
  • Figure 2: Three-dimensional visualisation of the estimated Hawkes process components for Scenario 1, including posterior uncertainty. Left: background rate $\mu(t,x,y)$ at $t=5$. Right: triggering kernel $\phi(\Delta t,\Delta x,\Delta y)$ at $\Delta t=0.25$. The plots show the variational posterior mean (blue), ground truth (green), and the 95% CIs (grey shaded area).
  • Figure 3: Spatial averages of the Hawkes process components for Scenario 1. Left: spatial average of the background rate over its domain. Right: spatial average of the triggering kernel over its domain. The plots show the variational posterior mean (blue), ground truth (green), and 95% CIs.
  • Figure 4: Estimated Hawkes process components for Scenario 2. The background rate $\mu$ (left) and triggering kernel $\phi$ (right) are shown at different time snapshots. For each component, the ground truth is on the left and the variational posterior mean estimate is on the right.
  • Figure 5: Three-dimensional visualisation of the estimated Hawkes process components for Scenario 2, demonstrating captured posterior uncertainty. Left: background rate $\mu(t,x,y)$ at $t=5$. Right: triggering kernel $\phi(\Delta t,\Delta x,\Delta y)$ at $\Delta t=0.25$. The plots display the variational posterior mean (blue), ground truth (green), and the 95% CIs (grey shaded area).
  • ...and 10 more figures