Cox-Hawkes doubly stochastic spatiotemporal Poisson processes Xenia Miscouridou1 Samir Bhatt12 George Mohler3

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Cox-Hawkes: doubly stochastic

spatiotemporal Poisson processes

Xenia Miscouridou1∗

, Samir Bhatt1,2, George Mohler3,

Seth Flaxman4†, Swapnil Mishra2†

1Imperial College London 2University of Copenhagen 3Boston College

4University of Oxford

Abstract

Hawkes processes are point process models that have been used to capture self-

excitatory behaviour in social interactions, neural activity, earthquakes and viral

epidemics. They can model the occurrence of the times and locations of events. Here

we develop a new class of spatiotemporal Hawkes processes that can capture both

triggering and clustering behaviour and we provide an eﬃcient method for performing

inference. We use a log-Gaussian Cox process (LGCP) as prior for the background

rate of the Hawkes process which gives arbitrary ﬂexibility to capture a wide range

of underlying background eﬀects (for infectious diseases these are called endemic ef-

fects). The Hawkes process and LGCP are computationally expensive due to the

former having a likelihood with quadratic complexity in the number of observations

and the latter involving inversion of the precision matrix which is cubic in observa-

tions. Here we propose a novel approach to perform MCMC sampling for our Hawkes

process with LGCP background, using pre-trained Gaussian Process generators which

provide direct and cheap access to samples during inference. We show the eﬃcacy

and ﬂexibility of our approach in experiments on simulated data and use our methods

to uncover the trends in a dataset of reported crimes in the US.

Keywords: Gaussian process, self-excitation, Bayesian inference, space-time

∗Correspondence to x.miscouridou@imperial.ac.uk

†Equal contribution

arXiv:2210.11844v1 [stat.ML] 21 Oct 2022

1 Introduction

Hawkes processes are a class of point processes that can model self or mutual excitation be-

tween events, in which the occurrence of one event triggers additional events, for example:

a violent event in one geographical area on a given day encourages another violent event in

an area nearby the next day. A unique feature of Hawkes processes is their ability to model

exogenous and endogenous ”causes” of events. An exogenous cause happens by the exter-

nal addition of a event, while endogenous events are self-excited from previous events by a

triggering kernel. An example of the diﬀerence between these two mechanisms is in disease

transmission - an exogenous event could be a zoonosis event such as the transmission of

Inﬂuenza from birds, while endogenous events are subsequent human to human transmis-

sion. Due to their ﬂexibility and mathematical tractability, Hawkes processes have been

extensively used in the literature in a series of applications. They have modelled among

others, neural activity (Linderman et al. 2014), earthquakes (Ogata 1988), violence (Lo-

eﬄer & Flaxman 2018, Holbrook et al. 2021) and social interactions (Miscouridou et al.

2018).

The majority of applied research on Hawkes processes focuses on the purely temporal

settings where events occur and are subsequently triggered only in time. However, many

practical problems require the inclusion of a spatial dimension. This inclusion is motivated

by several factors, ﬁrst, natural phenomena that self-excite tend to do so both spatial and

temporally e.g. infectious diseases, crime or diﬀusion over a network. Second, natural pro-

cesses tend to cluster closely in space and time (Tobler 1970). Third, in parametric formula-

tions residual variation persists and this is often structured in both space and time (Diggle

& Ribeiro 2007). A wide body of research exists in modelling spatial phenomena ranging

from Kriging (Matheron 1962) to model based estimates (Diggle & Ribeiro 2007) using

Gaussian processes. In the more general Gaussian process, which provides a prior function

class, spatial phenomena are modelled through a mean function and a covariance function

that allows control over the degree of clustering as well as the smoothness of the underlying

functions. Speciﬁcally for applications for spatial point patterns, an elegant formulation

using log-Gaussian Cox processes (LGCP), Møller et al. (1998) is commonly used (Diggle

et al. 2013). LGCPs can capture complex spatial structure but at a fundamental level are

unequipped with a mechanism to model self-excitement. When examining the processes’

endogenous and exogenous drivers, the lack of a self-exciting mechanism can potentially

lead to spurious scientiﬁc conclusions even if prediction accuracy is high. For example,

appealing again to the Inﬂuenza example, only modelling the distribution of cases using

an LGCP will ignore the complex interplay of zoonosis events and secondary transmission

events, both of which require diﬀerent policy actions.

The inclusion of space has a long history via the Hawkes process triggering mecha-

nism - ﬁstly modelled using the Epidemic Type Aftershock Sequence (ETAS) kernel Ogata

(1988) but many subsequent approaches now exist. However, to our knowledge, very few

approaches consider spatial and temporal events in both the exogenous and endogenous

Hawkes process mechanisms - that is where events can occur in space and time, and then

these events trigger new events also in space and time. Many mechanisms have been

proposed for space-time triggering kernels Reinhart (2018), but it is not clear nor straight-

forward how to also allow for exogenous space-time events simultaneously. In the vast

majority of previous applications, exogenous events occur at a constant rate in both space

and time or with highly speciﬁc forms that depend on the setting e.g. periodic functions

for seasonal malaria data (Unwin et al. 2021). Some studies do provide nonparametric

approaches for the background rate: Lewis & Mohler (2011) provide an estimation pro-

cedure for the background and kernel of the Hawkes process when no parametric form is

assumed for either of the two. Miscouridou et al. (2018) use a nonparametric prior based

on completely random measures to construct the discrete background rate for the Hawkes

processes that build directed networks. Other recent approaches use neural networks to es-

timate the rate (Omi et al. 2019). However, these nonparametric approaches do not provide

a compelling stochastic mechanism, such as LGCPs, that yield a generative process.

Here we propose a novel time space approach that combines Hawkes processes (Hawkes

1971) with log-Gaussian Cox processes (Møller et al. 1998, Diggle et al. 2013). This syn-

thesis allows us, for the ﬁrst time, to have a exogenous background intensity process with

self-excitation that is stochastic and able to vary in both space and time. We provide a

suite of new methods for simulation and computationally tractable inference. Our meth-

ods leverage modern computational techniques that are scalable and can eﬃciently learn

complex spatiotemporal data. We apply our approach on both simulated and real data.

Our novel addition of an LGCP prior in both space and time is accompanied with new

computational challenges: a Hawkes process is quadratic in complexity due to a double

summation in the likelihood, and LGCPs incur cubic complexity from matrix inversions.

To ensure our approach is scalable and still competitive with standard Hawkes processes we

utilize a recently developed Gaussian process approximation (Mishra et al. 2020, Semenova

et al. 2022) that obliviates the need for repeated matrix inversions. Our work represents

a step towards more general, scalable, point process framework that encodes more ﬂexible

and plausible mechanisms to represent natural and physical phenomena.

Our contributions

A summary of the contributions of our work is: (i) We provide a novel model formulation

for a highly ﬂexible self-exciting process that can capture endogenous and exogenous events

in both space and time. Our utilization of LCGPs for the exogenous background rate is

extremely ﬂexible and follows from the current state-of-the-art in spatial statistics (Diggle

et al. 2013), (ii) in contrast to previous work (e.g. Loeﬄer & Flaxman (2018)) our framework

admits a generative model that can produce stochastic realizations at an arbitrary set of

locations. We provide a novel algorithm to sample from this generative process, (iii) we

oﬀer an eﬃcient Bayesian inference approach that ensures our more ﬂexible model is still as

scalable as standard Hawkes processes and straightforward to implement computationally,

(iv) our framework is directly applicable to numerous spatiotemporal problems where there

are both endogenous and exogenous causes e.g. for natural or social phenomena such as

crime, diseases, environment, or human behaviour.

2 Related methods

As mentioned before, modelling space through Hawkes processes was ﬁst used with the

Epidemic Type Aftershock Sequence (ETAS) kernel (Ogata 1988) and other approaches

followed some of which exist in Reinhart (2018). For modelling spatial point patterns

without self-excitation, log-Gaussian Cox processes (LGCP) Møller et al. (1998) provide

an elegant approach as explained in Diggle et al. (2013).

Reinhart (2018) provide an overview on spatiotemporal Hawkes processes explaining

various options for the form of the intensity, the kernels and the corresponding simulating

algorithm. However, the case of an LGCP background is not discussed in the review or

elsewhere.

Our approach is the ﬁrst to use an LGCP to capture the background underlying eﬀects

(these are called endemic eﬀects in infectious disease modelling but here we will use this

term broadly for other applications too) and can model the exact spatial and time locations.

Loeﬄer & Flaxman (2018) aim to understand whether gun violence in Chicago is conta-

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Cox-Hawkes:doublystochasticspatiotemporalPoissonprocessesXeniaMiscouridou1*,SamirBhatt1;2,GeorgeMohler3,SethFlaxman4y,SwapnilMishra21ImperialCollegeLondon2UniversityofCopenhagen3BostonCollege4UniversityofOxfordAbstractHawkesprocessesarepointprocessmodelsthathavebeenusedtocaptureself-excitatorybehav...

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Cox-Hawkes doubly stochastic spatiotemporal Poisson processes Xenia Miscouridou1 Samir Bhatt12 George Mohler3

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