Modeling of Spatio-Temporal Hawkes Processes With Randomized Kernels

Abstract

We investigate spatio-temporal event analysis using point processes. Inferring the dynamics of event sequences spatiotemporally has many practical applications including crime prediction, social media analysis, and traffic forecasting. In particular, we focus on spatio-temporal Hawkes processes that are commonly used due to their capability to capture excitations between event occurrences. We introduce a novel inference framework based on randomized transformations and gradient descent to learn the process. We replace the spatial kernel calculations by randomized Fourier feature-based transformations. The introduced randomization by this representation provides flexibility while modeling the spatial excitation between events. Moreover, the system described by the process is expressed within closed-form in terms of scalable matrix operations. During the optimization, we use maximum likelihood estimation approach and gradient descent while properly handling positivity and orthonormality constraints. The experiment results show the improvements achieved by the introduced method in terms of fitting capability in synthetic and real-life datasets with respect to the conventional inference methods in the spatio-temporal Hawkes process literature. We also analyze the triggering interactions between event types and how their dynamics change in space and time through the interpretation of learned parameters.

What is the Fourier Transform of a Spatial Point Process?

“…The corresponding theory for spatial point processes has been neglected apart from some notable and not very recent exceptions [5], [6], [56]. For point processes machine learning researchers have just started to discover the utility of Fourier-based methods [38], [44]. Current state of the art for the spectral analysis of point processes is that we really do not even know what to compute, and know even less how to address its digital implementation.…”

“…Yet unlike random fields and time series, spectral analysis of point processes is still in its infancy, see also [5], [6], [56], and critically, the digital processing of a point process remains fully outstanding. Recent interest in Fourier features in machine learning based approaches for point patterns such as [38], [44] show the potential of using Fourier based information as features for estimation and detection. The work in this manuscript both establishes what Fourier features to calculate for homogeneous processes from a sampling perspective, and their large but finite sampling area properties, just like [33] determined the large but finite properties of Fourier representations for random fields.…”

Hawkes Processes Modeling, Inference and Control: An Overview

“…• Enriching HP variants (parametric, nonparametric, neural), or blending them with other ML approaches, so as to make them suitable for specific situations. 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;…”

Estructuras estocásticas notables en procesos puntuales espacio-temporales y medidas de riesgo bivariantes

“…En palabras sencillas, estos modelos estiman una función de intensidad que predice la tasa de eventos en cualquier localización espacial (x, y) y tiempo t. En las últimas décadas, se han realizado avances en la estimación, inferencia, simulación y herramientas de diagnóstico para los modelos de procesos puntuales en general. Estos avances han tomado mayor importancia en aplicaciones como medio ambiente (Siino et al, 2018;Nasirzadeh et al, 2021;Lieshout et al, 2012;Pratiwi et al, 2017;Reinhart, 2018), presencia de plantas o animales (Møller y Díaz-Avalos, 2010;Myllymäki y Penttinen, 2009;Illian et al, 2012;Serra et al, 2014;Sørbye et al, 2019;Flagg y Hoegh, 2022), salud (Beneš et al, 2002;Rostami et al, 2017;Johnson et al, 2019;Chiang et al, 2022;Heinen, 2003), finanzas (Aït-Sahalia et al, 2015;Hawkes, 2018;Moreno Trujillo, 2019;Heinen, 2003;Fokianos y Tjøstheim, 2011), análisis de medios sociales (Ilhan y Kozat, 2020), accidentes de tráfico (Tang et al, 2022) y otras múltiples áreas.…”