Nonlinear Hawkes Process with Gaussian Process Self Effects

TL;DR

This paper introduces a flexible nonlinear Hawkes process model with Gaussian Process self-effects, enabling effective inference with limited data and capturing both excitatory and inhibitory influences.

Contribution

It presents a novel Gaussian Process-based extension of Hawkes processes that simplifies inference and enhances flexibility over previous models.

Findings

Model effectively captures complex self-effects.

Performs well with limited data.

Demonstrated on diverse real-world datasets.

Abstract

Traditionally, Hawkes processes are used to model time--continuous point processes with history dependence. Here we propose an extended model where the self--effects are of both excitatory and inhibitory type and follow a Gaussian Process. Whereas previous work either relies on a less flexible parameterization of the model, or requires a large amount of data, our formulation allows for both a flexible model and learning when data are scarce. We continue the line of work of Bayesian inference for Hawkes processes, and our approach dispenses with the necessity of estimating a branching structure for the posterior, as we perform inference on an aggregated sum of Gaussian Processes. Efficient approximate Bayesian inference is achieved via data augmentation, and we describe a mean--field variational inference approach to learn the model parameters. To demonstrate the flexibility of the model we apply our methodology on data from three different domains and compare it to previously reported results.