Locally stationary Hawkes processes

Abstract

This paper addresses the generalisation of stationary Hawkes processes in order to allow for a time-evolving second-order analysis. Motivated by the concept of locally stationary autoregressive processes, we apply however inherently different techniques to describe the time-varying dynamics of self-exciting point processes. In particular we derive a stationary approximation of the Laplace transform of a locally stationary Hawkes process. This allows us to define a local intensity function and a local Bartlett spectrum which can be used to compute approximations of first and second order moments of the process. We complete the paper by some insightful simulation studies.

Temporally Smoothed Wavelet Coherence for Multivariate Point-Processes and Neuron-Firing

“…Further work is needed here to assess the behaviour of the estimator when correlation structure is present. For instance under self-exciting Hawkes, or Cox processes [Hawkes, 1971, Cox, 1955, or even locally stationary processes [Park et al, 2014, Gibberd and Nelson, 2016, Roueff et al, 2016.…”

Fast multi-resolution segmentation for nonstationary Hawkes process using cumulants

“…Plenty of state of the arts have performed inference for a timechanging baseline intensity with a stationary triggering kernel [13,14,27]. For both baseline intensity and triggering kernel being nonstationary, the authors of [24] and [23] provided a general nonparametric estimation theory for the firstand second-order cumulants of a locally stationary Hawkes process. However, this method is inefficient in computation complexity because every point on the two-dimensional covariance function Cov(τ, t) has to be estimated and it is not applicable to real applications.…”

“…However, this method is inefficient in computation complexity because every point on the two-dimensional covariance function Cov(τ, t) has to be estimated and it is not applicable to real applications. In this sense, our MRS algorithm can be considered as a "coarser" version of the work in [24]: it combines adjacent small sectors with similar statistical properties into a larger segment and only outputs more heterogeneous segments. Although it is "coarser," the computation complexity is drastically reduced to make it practical.…”

Testing the causality of Hawkes processes with time reversal

“…A similar procedure is developed in [17]. In the case of Figure 3: Difference of the log-likelihood scaled by T between forward and backward time arrows for a HP with an exponential kernel for λ0 = {0.001, 0.0025, 0.0050, 0.0075, 0.0100} and α = {0.010, 0.025, 0.050, 0.075, 0.100}, while β is adjusted to match the desired n. The data points are grouped according to their endogeneity and averaged over 100 runs for each parameter permutation.…”