A Multivariate Hawkes Process with Gaps in Observations

TL;DR

This paper introduces a novel multivariate Hawkes process model that accounts for gaps in observational data, enabling the detection of hidden directional relationships among entities despite sparse and incomplete observations.

Contribution

It proposes a variational framework for learning sparse causal relationships in multivariate Hawkes processes with missing data, incorporating boundary conditions to handle observation gaps.

Findings

Robust recovery of hidden relationships is possible with 10-30% observed data.

Knowledge of observation gaps and boundary conditions improves pattern discovery.

Numerical simulations demonstrate effectiveness even with small observed intervals.

Abstract

Given a collection of entities (or nodes) in a network and our intermittent observations of activities from each entity, an important problem is to learn the hidden edges depicting directional relationships among these entities. Here, we study causal relationships (excitations) that are realized by a multivariate Hawkes process. The multivariate Hawkes process (MHP) and its variations (spatio-temporal point processes) have been used to study contagion in earthquakes, crimes, neural spiking activities, the stock and foreign exchange markets, etc. In this paper, we consider the multivariate Hawkes process with gaps in observations (MHPG). We propose a variational problem for detecting sparsely hidden relationships with a multivariate Hawkes process that takes into account the gaps from each entity. We bypass the problem of dealing with a large amount of missing events by introducing a small number of unknown boundary conditions. In the case where our observations are sparse (e.g. from 10% to 30%), we show through numerical simulations that robust recovery with MHPG is still possible even if the lengths of the observed intervals are small but they are chosen accordingly. The numerical results also show that the knowledge of gaps and imposing the right boundary conditions are very crucial in discovering the underlying patterns and hidden relationships.