Nonlinear Hawkes Processes in Time-Varying System

TL;DR

This paper introduces a flexible nonlinear, nonhomogeneous Hawkes process model with a state process, enabling better modeling of time-varying systems, and develops efficient Bayesian inference algorithms for it.

Contribution

It proposes a novel, fully flexible Hawkes process model that overcomes previous limitations by incorporating nonlinearity and nonhomogeneity with a state process.

Findings

Model outperforms existing methods in experiments

Efficient Bayesian inference algorithms developed

Handles time-varying systems effectively

Abstract

Hawkes processes are a class of point processes that have the ability to model the self- and mutual-exciting phenomena. Although the classic Hawkes processes cover a wide range of applications, their expressive ability is limited due to three key hypotheses: parametric, linear and homogeneous. Recent work has attempted to address these limitations separately. This work aims to overcome all three assumptions simultaneously by proposing the flexible state-switching Hawkes processes: a flexible, nonlinear and nonhomogeneous variant where a state process is incorporated to interact with the point processes. The proposed model empowers Hawkes processes to be applied to time-varying systems. For inference, we utilize the latent variable augmentation technique to design two efficient Bayesian inference algorithms: Gibbs sampler and mean-field variational inference, with analytical iterative updates to estimate the posterior. In experiments, our model achieves superior performance compared to the state-of-the-art competitors.