Multivariate Representations of Univariate Marked Hawkes Processes

TL;DR

This paper establishes a connection between univariate marked Hawkes processes and multivariate Hawkes processes, enabling more flexible inference and approximation of complex phenomena.

Contribution

It introduces multivariate unmarked Hawkes representations to parameterize univariate marked Hawkes processes, with theoretical guarantees and practical validation.

Findings

Multivariate representations can asymptotically approximate a wide class of univariate marked Hawkes processes.

The proposed framework is stationary under certain conditions.

Simulation results demonstrate the effectiveness and provide error bounds for the approach.

Abstract

Univariate marked Hawkes processes are used to model a range of real-world phenomena including earthquake aftershock sequences, contagious disease spread, content diffusion on social media platforms, and order book dynamics. This paper illustrates a fundamental connection between univariate marked Hawkes processes and multivariate Hawkes processes. Exploiting this connection renders a framework that can be built upon for expressive and flexible inference on diverse data. Specifically, multivariate unmarked Hawkes representations are introduced as a tool to parameterize univariate marked Hawkes processes. We show that such multivariate representations can asymptotically approximate a large class of univariate marked Hawkes processes, are stationary given the approximated process is stationary, and that resultant conditional intensity parameters are identifiable. A simulation study demonstrates the efficacy of this approach, and provides heuristic bounds for error induced by the relatively larger parameter space of multivariate Hawkes processes.