Gradient-based estimation of linear Hawkes processes with general kernels

TL;DR

This paper introduces an efficient gradient estimation method for linear multivariate Hawkes processes with general kernels, enabling scalable parameter estimation on large datasets by reducing computational complexity.

Contribution

The authors develop an adaptive stratified sampling estimator for the gradient of the least squares error, improving the efficiency of parameter estimation for Hawkes processes with general kernels.

Findings

The proposed method significantly reduces computational complexity.

It outperforms existing methods on large datasets.

It enables practical application of gradient-based estimation for complex Hawkes models.

Abstract

Linear multivariate Hawkes processes (MHP) are a fundamental class of point processes with self-excitation. When estimating parameters for these processes, a difficulty is that the two main error functionals, the log-likelihood and the least squares error (LSE), as well as the evaluation of their gradients, have a quadratic complexity in the number of observed events. In practice, this prohibits the use of exact gradient-based algorithms for parameter estimation. We construct an adaptive stratified sampling estimator of the gradient of the LSE. This results in a fast parametric estimation method for MHP with general kernels, applicable to large datasets, which compares favourably with existing methods.