Multivariate Hawkes Processes for Large-scale Inference

Abstract

In this paper, we present a framework for fitting multivariate Hawkes processes for large-scale problems both in the number of events in the observed history n and the number of event types d (i.e. dimensions). The proposed Low-Rank Hawkes Process (LRHP) framework introduces a low-rank approximation of the kernel matrix that allows to perform the nonparametric learning of the d 2 triggering kernels using at most O(ndr 2 ) operations, where r is the rank of the approximation (r d, n). This comes as a major improvement to the existing state-of-the-art inference algorithms that are in O(nd 2 ). Furthermore, the low-rank approximation allows LRHP to learn representative patterns of interaction between event types, which may be valuable for the analysis of such complex processes in real world datasets. The efficiency and scalability of our approach is illustrated with numerical experiments on simulated as well as real datasets.

A neural network based model for multi-dimensional nonlinear Hawkes processes

“…• Maximum likelihood estimation (MLE) It is the most commonly used approach for estimating the kernels of the Hawkes process, as employed by many methods such as Bonnet et al [2021], , and Lemonnier et al [2016]. However, MLE methods have high computational costs as their time complexity increases quadratically with the number of arrivals.…”

“…Lemonnier and Vayatis [2014] develop the Markovian Estimation of Mutually Interacting Process (MEMIP) method, which utilizes weighted exponential functions to determine kernels for the nonlinear Hawkes process. To extend the MEMIP for large dimensional datasets, Lemonnier et al [2016] introduce dimensionality reduction features. However, the above approaches require the kernel function to be smooth, which is a drawback.…”