Log-Gaussian Cox Processes on General Metric Graphs

TL;DR

This paper introduces a scalable Bayesian framework for modeling spatial point processes on complex networks like road systems using log-Gaussian Cox processes and Gaussian Whittle-Matérn fields, with efficient inference and real-world application.

Contribution

It develops a novel, computationally efficient approach for Bayesian inference of point processes on general metric graphs, integrating fractional stochastic differential equations and exact likelihood approximations.

Findings

Successfully applied to large-scale road accident data with over 150,000 segments.

Identified high-risk road segments and hotspots using exceedance probabilities.

Demonstrated scalability and theoretical robustness of the method.

Abstract

The modeling of spatial point processes has advanced considerably, yet extending these models to non-Euclidean domains, such as road networks, remains a challenging problem. We propose a novel framework for log-Gaussian Cox processes on general compact metric graphs by leveraging the Gaussian Whittle-Mat\'ern fields, which are solutions to fractional-order stochastic differential equations on metric graphs. To achieve computationally efficient likelihood-based inference, we introduce a numerical approximation of the likelihood that eliminates the need to approximate the Gaussian process. This method, coupled with the exact evaluation of finite-dimensional distributions for Whittle-Mat\'ern fields with integer smoothness, ensures scalability and theoretical rigour, with derived convergence rates for posterior distributions. The framework is implemented in the open-source MetricGraph R package, which integrates seamlessly with R-INLA to support fully Bayesian inference. We demonstrate the applicability and scalability of this approach through an analysis of road accident data from Al-Ahsa, Saudi Arabia, consisting of over 150,000 road segments. By identifying high-risk road segments using exceedance probabilities and excursion sets, our framework provides localized insights into accident hotspots and offers a powerful tool for modeling spatial point processes directly on complex networks.