Fast multi-resolution segmentation for nonstationary Hawkes process using cumulants

Abstract

The stationarity is assumed in the vanilla Hawkes process, which reduces the model complexity but introduces a strong assumption. In this paper, we propose a fast multi-resolution segmentation algorithm to capture the time-varying characteristics of the nonstationary Hawkes process. The proposed algorithm is based on the first-and second-order cumulants. Except for the computation efficiency, the algorithm can provide a hierarchical view of the segmentation at different resolutions. We extensively investigate the impact of hyperparameters on the performance of this algorithm. To ease the choice of hyperparameter, we propose a refined Gaussian process-based segmentation algorithm, which is proved to be a robust method. The proposed algorithm is applied to a real vehicle collision dataset, and the outcome shows some interesting hierarchical dynamic time-varying characteristics.

Hawkes Processes Modeling, Inference and Control: An Overview

“…The works with Multi-Armed Bandits [14], randomized kernels [32], graph neural networks for temporal knowledge graphs [24] and composition of HP-like Point Processes with Warping functions defined over the time event sequences [85] can be considered in this category; • Improving the speed of inference or sampling, so as to reduce the time spent in model estimation an aspect which may be critical for some real-world applications. The works of [31] in Bayesian mitigation of spatial coarsening, [99] in multi-resolution segmentation for nonstationary Hawkes process using cumulants, [45] on thinning of event sequences for accelerating inference steps, [50] on the use of Lambert-W functions of improving sequence sampling, [11] on perfect sampling are examples of such, and [61] on recursive computation of HP moments;…”

Nonlinear Hawkes Processes in Time-Varying System

“…Our proposed model permits the latent interaction structure of HPs to change with time, e.g., the presynaptic neuron excites the postsynaptic one in one state but inhibits it in another state, which previous models are incapable of inferring. [18] × × Zhou et al [40] × × Bacry & Muzy [2] × × Zhang et al [34] × × Zhou [35] × × Zhou et al [37] × × Gerhard et al [12] × × Apostolopoulou et al [1] × × Zhou et al [39] × Wang et al [32] × × Wu et al [33] × × Zhou et al [38] × Morariu-Patrichi et al [25] × × our work Contribution Our contributions are: (1) we propose a novel flexible, nonlinear and nonhomogeneous HPs variant that has flexible influence patterns, is able to handle inhibitive effects, and has state process driven (time-varying) parameters simultaneously; and (2) we develop two efficient Bayesian inference algorithms, a Gibbs sampler and a mean-field variational inference method, that leverage latent variable augmentation techniques [7; 19; 29] to obtain closed-form iterative updates. It is worth noting that, although some work also used the state process to describe time-varying parameters, it cannot simultaneously address the three limitations as our model does.…”

Sequential change-point detection for mutually exciting point processes over networks

“…In Rambaldi et al (2018), a model selection scheme was proposed to identify the presence of exogenous events that increase the intensity of the Hawkes process for a given time period. A cumulant based multi-resolution segmentation algorithm was proposed in Zhou et al (2020) to find the optimal partition of the nonstationary Hawkes process into several nonoverlapping segments. On the contrary, we focus on sequential detection to detect the change as quickly as possible.…”