A Metric Space for Point Process Excitations
This work addresses the challenge of high-dimensional estimation in point processes for researchers in fields like epidemiology and finance, though it appears incremental as it generalizes existing spatiotemporal frameworks.
The authors tackled the problem of estimating interactions in multivariate Hawkes processes by proposing a Hidden Hawkes Geometry (HHG) model that embeds event types in a hidden metric space, and demonstrated its effectiveness in uncovering interactions across applications like COVID-19 dynamics and options markets.
A multivariate Hawkes process enables self- and cross-excitations through a triggering matrix that behaves like an asymmetrical covariance structure, characterizing pairwise interactions between the event types. Full-rank estimation of all interactions is often infeasible in empirical settings. Models that specialize on a spatiotemporal application alleviate this obstacle by exploiting spatial locality, allowing the dyadic relationships between events to depend only on separation in time and relative distances in real Euclidean space. Here we generalize this framework to any multivariate Hawkes process, and harness it as a vessel for embedding arbitrary event types in a hidden metric space. Specifically, we propose a Hidden Hawkes Geometry (HHG) model to uncover the hidden geometry between event excitations in a multivariate point process. The low dimensionality of the embedding regularizes the structure of the inferred interactions. We develop a number of estimators and validate the model by conducting several experiments. In particular, we investigate regional infectivity dynamics of COVID-19 in an early South Korean record and recent Los Angeles confirmed cases. By additionally performing synthetic experiments on short records as well as explorations into options markets and the Ebola epidemic, we demonstrate that learning the embedding alongside a point process uncovers salient interactions in a broad range of applications.