Hawkes background estimation improved by projected Wasserstein-Fisher-Rao flows
Projected Wasserstein-Fisher-Rao Flows for Hawkes Background-Intensity Estimation
Machine Learning
Summary
Estimating the background activity in Hawkes processes, which model events triggering other events, is challenging when the causing mechanism is known but the baseline rate is unknown. The authors develop a new mathematical approach using Wasserstein-Fisher-Rao geometry to estimate this background intensity more accurately. They translate complicated mathematical dynamics into a finite-dimensional framework, make the solution uniquely identifiable, and prove that their estimator is consistent under certain conditions. The method is then specified explicitly for Gaussian mixture models, making it computationally practical.
What this means in practice
- •For quantitative analysts: Estimate baseline event rates in financial markets when event-triggering relationships are known.
- •For network operations teams: Model network traffic background intensity separately from triggering events for better anomaly detection.
Authors
Isaias Banales, Arturo Jaramillo
Abstract
We consider the problem of estimating the background intensity of a Hawkes process when the triggering mechanism is known, treating the background as an unknown finite positive measure with a regular density. We propose a likelihood-based variational approach based on the Wasserstein-Fisher-Rao geometry. In this setting, the relevant first variations are naturally distribution-valued and therefore fall outside the classical gradient-flow formulation. We develop a distributional framework for the resulting dynamics and give them a finite-dimensional meaning through projections onto statistical manifolds. We identify the true background as the unique zero of a limiting generalized gradient and, under suitable stability, compactness, integrability and approximation conditions, establish a conditional consistency theorem for measurable estimators satisfying the empirical projected equations on events whose probabilities tend to one. Finally, we specialize the construction to Gaussian-mixture manifolds and obtain an explicit finite-dimensional scheme.