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Flexible Parametric Inference for Space-Time Hawkes Processes

Emilia Siviero, Guillaume Staerman, Stephan Clémençon, Thomas Moreau

TL;DR

This work develops a fast, flexible parametric inference framework for space-time Hawkes processes by combining finite-support kernels, space-time discretization, and precomputation with an $\ell_2$ gradient-based solver. By recasting the intensity as a convolution and employing a discretized 3D grid, the method achieves linear scalability in the number of events and supports arbitrary differentiable kernels. The authors provide theoretical guarantees on discretization bias, introduce a computable approximation for a bottleneck precomputation term, and validate the approach with synthetic experiments and seismic data, showing improved efficiency and accuracy. The framework enables richer space-time interactions to be modeled and has practical impact for applications like earthquake aftershock modeling and other spatio-temporal clustering problems.

Abstract

Many modern spatio-temporal data sets, in sociology, epidemiology or seismology, for example, exhibit self-exciting characteristics, triggering and clustering behaviors both at the same time, that a suitable Hawkes space-time process can accurately capture. This paper aims to develop a fast and flexible parametric inference technique to recover the parameters of the kernel functions involved in the intensity function of a space-time Hawkes process based on such data. Our statistical approach combines three key ingredients: 1) kernels with finite support are considered, 2) the space-time domain is appropriately discretized, and 3) (approximate) precomputations are used. The inference technique we propose then consists of a $\ell_2$ gradient-based solver that is fast and statistically accurate. In addition to describing the algorithmic aspects, numerical experiments have been carried out on synthetic and real spatio-temporal data, providing solid empirical evidence of the relevance of the proposed methodology.

Flexible Parametric Inference for Space-Time Hawkes Processes

TL;DR

This work develops a fast, flexible parametric inference framework for space-time Hawkes processes by combining finite-support kernels, space-time discretization, and precomputation with an gradient-based solver. By recasting the intensity as a convolution and employing a discretized 3D grid, the method achieves linear scalability in the number of events and supports arbitrary differentiable kernels. The authors provide theoretical guarantees on discretization bias, introduce a computable approximation for a bottleneck precomputation term, and validate the approach with synthetic experiments and seismic data, showing improved efficiency and accuracy. The framework enables richer space-time interactions to be modeled and has practical impact for applications like earthquake aftershock modeling and other spatio-temporal clustering problems.

Abstract

Many modern spatio-temporal data sets, in sociology, epidemiology or seismology, for example, exhibit self-exciting characteristics, triggering and clustering behaviors both at the same time, that a suitable Hawkes space-time process can accurately capture. This paper aims to develop a fast and flexible parametric inference technique to recover the parameters of the kernel functions involved in the intensity function of a space-time Hawkes process based on such data. Our statistical approach combines three key ingredients: 1) kernels with finite support are considered, 2) the space-time domain is appropriately discretized, and 3) (approximate) precomputations are used. The inference technique we propose then consists of a gradient-based solver that is fast and statistically accurate. In addition to describing the algorithmic aspects, numerical experiments have been carried out on synthetic and real spatio-temporal data, providing solid empirical evidence of the relevance of the proposed methodology.
Paper Structure (28 sections, 2 theorems, 31 equations, 6 figures, 2 tables)

This paper contains 28 sections, 2 theorems, 31 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Space-time Hawkes processes - background and framework
  3. A fast method for parametric inference in space-time Hawkes models
  4. The inference approach - Key components
  5. Convolutional writing.
  6. Finite-support kernels.
  7. Discretization.
  8. Efficient inference with Empirical Risk Minimization
  9. Approximation of $\Psi$.
  10. Gradient-based optimization.
  11. On the bias of spatio-temporal discretization - theoretical guarantees
  12. Numerical experiments
  13. On the bias of spatio-temporal discretization - numerical assessment
  14. On the statistical error
  15. Approximation of the bottleneck precomputation term
  16. ...and 13 more sections

Key Result

proposition 1

Let $\mathcal{H}_T$ and $\widetilde{\mathcal{H}}_T$ be respectively a set of events (drawn from a spatio-temporal Hawkes process) and its discretized version on the grid $\mathcal{G}$ with stepsize $\Delta=(\Delta_{\mathcal{X}}, \Delta_{\mathcal{Y}}, \Delta_T)$. Suppose ass:1 to be satisfied. Thus, and where $C(\lambda_i)$ is a constant depending only on the regularity of $\lambda_i$.

Figures (6)

  • Figure 1: Median and $25$%-$75$% quantiles of the $\ell_2$-norm between true and estimated parameters (left), and computational time with respect to $\Delta$ (right), for various $T$ and $S$.
  • Figure 2: Median and $25$%-$75$% quantiles of the $\ell_2$-norm between true and estimated parameters (left), and computational time with respect to $T$ (right), for various $S$.
  • Figure B.1: Median and $25$%-$75$% quantiles of the $\ell_2$-norm between true and estimated parameters for a truncated Inverse Power Law spatial kernel and a Kumaraswamy temporal kernel, with respect to $\Delta$, for various $T$ and $S$.
  • Figure B.2: Median and $25$%-$75$% quantiles of the $\ell_2$-norm between true and estimated parameters for a truncated Exponential (left), and a truncated Gaussian (right) temporal kernels, with respect to $\Delta$, for various $T$ and $S$.
  • Figure B.3: Square error on parameters for the Kumaraswamy temporal kernel, as a function of $T$, $S$ and $\Delta$.
  • ...and 1 more figures

Theorems & Definitions (2)

  • proposition 1
  • proposition 2