Simulation and Calibration of a Fully Bayesian Marked Multidimensional Hawkes Process with Dissimilar Decays

TL;DR

This paper introduces an exact, efficient simulation method for multidimensional Hawkes processes with different decay rates, and a modular inference procedure to analyze their excitation dynamics, demonstrated on real dark network data.

Contribution

It presents a novel superposition-based simulation approach and an inference algorithm for multidimensional Hawkes processes with dissimilar decays, improving speed and accuracy.

Findings

Significant speed improvements over existing methods

Exact simulation without approximation

Effective inference of excitation strengths in real networks

Abstract

We propose a simulation method for multidimensional Hawkes processes based on superposition theory of point processes. This formulation allows us to design efficient simulations for Hawkes processes with differing exponentially decaying intensities. We demonstrate that inter-arrival times can be decomposed into simpler auxiliary variables that can be sampled directly, giving exact simulation with no approximation. We establish that the auxiliary variables provides information on the parent process for each event time. The algorithm correctness is shown by verifying the simulated intensities with their theoretical moments. A modular inference procedure consisting of Gibbs samplers through the auxiliary variable augmentation and adaptive rejection sampling is presented. Finally, we compare our proposed simulation method against existing methods, and find significant improvement in terms of algorithm speed. Our inference algorithm is used to discover the strengths of mutually excitations in real dark networks.