Nearly assumptionless screening for the mutually-exciting multivariate Hawkes process

Abstract

We consider the task of learning the structure of the graph underlying a mutually-exciting multivariate Hawkes process in the high-dimensional setting. We propose a simple and computationally inexpensive edge screening approach. Under a subset of the assumptions required for penalized estimation approaches to recover the graph, this edge screening approach has the sure screening property: with high probability, the screened edge set is a superset of the true edge set. Furthermore, the screened edge set is relatively small. We illustrate the performance of this new edge screening approach in simulation studies.

Renewal in Hawkes processes with self-excitation and inhibition

“…Brémaud and Massoulié [5] have devised efficient techniques based on Poisson point process thinning (or embedding) for this framework. The recent works [7] and [28] provide interesting contributions from this perspective, which will be further discussed at the end of Section 1.…”

“…The assumption will be used to exploit the coupling we will construct between the process with reproduction function h and a dominating process with reproduction function . Similar assumptions involving or are often made in the literature; see [7, Assumption 1] and [28, p. 6], for example.…”

“…The intensity functions are of the form under a number of assumptions, in particular that the are -Lipschitz, that the Perron–Frobenius eigenvalue of the matrix is strictly less than 1, and that either the functions have a common uniform bound or the signed functions vanish outside a common bounded interval [7, Assumption 1, Assumption 4]. In order to derive concentration inequalities for the Hawkes processes, the authors of [7] apply the theory of weakly dependent sequences and therefore develop specific coupling techniques in order to control time dependencies. The very recent work [28] provides renewal time points for rather general one-dimensional Hawkes processes with self-inhibition, using technical splitting methods requiring specific couplings.…”

“…Both papers [7] and [28] involve sets of assumptions on the reproduction functions that differ from the ones here. Note that (1.14) is the natural stability assumption involving the growth bound at infinity for the dominating process, and that in the present paper we do not need regularity or monotonicity assumptions on .…”

Rejoinder: A Review of Self-Exciting Spatio-Temporal Point Processes and Their Applications

“…There are now several competing methods for estimating the causality graph G and testing hypotheses about its edges (e.g. Chen, Witten and Shojaie, 2017;Xu, Farajtabar and Zha, 2016;Achab et al, 2017), again focusing on purely temporal processes. It would be quite interesting to see these methods extended to spatio-temporal mutually exciting processes and used in applications, where G may answer substantive scientific questions and the use of leading indicator processes could improve predictions.…”