NEURAL SPATIO-TEMPORAL POINT PROCESSES

Abstract

We propose a new class of parameterizations for spatio-temporal point processes which leverage Neural ODEs as a computational method and enable flexible, highfidelity models of discrete events that are localized in continuous time and space. Central to our approach is a combination of continuous-time neural networks with two novel neural architectures, i.e., Jump and Attentive Continuous-time Normalizing Flows. This approach allows us to learn complex distributions for both the spatial and temporal domain and to condition non-trivially on the observed event history. We validate our models on data sets from a wide variety of contexts such as seismology, epidemiology, urban mobility, and neuroscience.

1. INTRODUCTION

Modeling discrete events that are localized in continuous time and space is an important task in many scientific fields and applications. Spatio-temporal point processes (STPPs) are a versatile and principled framework for modeling such event data and have, consequently, found many applications in a diverse range of fields. This includes, for instance, modeling earthquakes and aftershocks (Ogata, 1988; 1998) , the occurrence and propagation of wildfires (Hering et al., 2009) , epidemics and infectious diseases (Meyer et al., 2012; Schoenberg et al., 2019) , urban mobility (Du et al., 2016) , the spread of invasive species (Balderama et al., 2012) , and brain activity (Tagliazucchi et al., 2012) . It is of great interest in all of these areas to learn high-fidelity models which can jointly capture spatial and temporal dependencies and their propagation effects. However, existing parameterizations of STPPs are strongly restricted in this regard due to computational considerations: In its general form, STPPs require solving multivariate integrals for computing likelihood values and thus have primarily been studied within the context of different approximations and model restrictions. This includes, for instance, restricting the model class to parameterizations with known closed-form solutions (e.g., exponential Hawkes processes (Ozaki, 1979) ), to restrict dependencies between the spatial and temporal domain (e.g., independent and unpredictable marks (Daley & Vere-Jones, 2003)), or to discretize continuous time and space (Ogata, 1998) . These restrictions and approximations-which can lead to mis-specified models and loss of information-motivated the development of neural temporal point processes such as Neural Hawkes Processes (Mei & Eisner, 2017) and Neural Jump SDEs (Jia & Benson, 2019) . While these methods are more flexible, they can still require approximations such as Monte-Carlo sampling of the likelihood (Mei & Eisner, 2017; Nickel & Le, 2020) and, most importantly, only model restricted spatial distributions (Jia & Benson, 2019) . x 0 5 t * (t) To overcome these issues, we propose a new class of parameterizations for spatio-temporal point processes which leverage Neural ODEs as a computational method and allows us to define flexible, * Work done while at Facebook AI Research. 1



Figure 1: Color is used to denote p(x|t), which can be evaluated for Neural STPPs. After observing an event in one mode, the model is instantaneously updated as it strongly expects an event in the next mode. After a period of no observations, the model smoothly reverts back to the marginal distribution.

