Estimation of the Hawkes Process With Renewal Immigration Using the EM Algorithm

TL;DR

This paper develops two EM algorithms to estimate a generalized Hawkes process with renewal immigration, allowing for dependence between clusters and overdispersion, and demonstrates their effectiveness and importance of correct model specification.

Contribution

It introduces a novel Hawkes process with renewal immigration and two EM algorithms for its statistical estimation, extending existing methods to more flexible models.

Findings

Both EM algorithms are consistent in simulations.

Misspecification of the immigration process causes significant bias.

The second algorithm has linear time complexity.

Abstract

We introduce the Hawkes process with renewal immigration and make its statistical estimation possible with two Expectation Maximization (EM) algorithms. The standard Hawkes process introduces immigrant points via a Poisson process, and each immigrant has a subsequent cluster of associated offspring of multiple generations. We generalize the immigration to come from a Renewal process; introducing dependence between neighbouring clusters, and allowing for over/under dispersion in cluster locations. This complicates evaluation of the likelihood since one needs to know which subset of the observed points are immigrants. Two EM algorithms enable estimation here: The first is an extension of an existing algorithm that treats the entire branching structure - which points are immigrants, and which point is the parent of each offspring - as missing data. The second considers only if a point is an immigrant or not as missing data and can be implemented with linear time complexity. Both algorithms are found to be consistent in simulation studies. Further, we show that misspecifying the immigration process introduces signficant bias into model estimation-- especially the branching ratio, which quantifies the strength of self excitation. Thus, this extended model provides a valuable alternative model in practice.