Cox-Hawkes: doubly stochastic spatiotemporal Poisson processes
Authors: Xenia Miscouridou
Samir Bhatt
George Mohler
Seth Flaxman
Swapnil Mishra
Venue: Accepted by TMLR ()

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 effects). 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 observations. 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.

Published in Transactions on Machine Learning Research (05/2023)
Cox-Hawkes: doubly stochastic spatiotemporal Poisson processes
Xenia Miscouridou'
amiscouriddou@@mppiii....c
I-X and Deportment of Mathematics, Imperial College London
Samir Bhatt
b.baatt@@mperialle
Department of Public Health, University of Copenhagen, School Public Health, Imperial College London
George Mohler
gmohler@kupui. edu
Department of Computer Science, Boston College
Seth Flaxman
seth.Bazman&css.rr..o
Depariment of Computer Science, University of Ouford
Swapnil Mishrat
swapnil mishra Gruus. www.s.eu.............mmmm edu.sg
Saw Swee Hock School of Public Health and Institute of Data Science, National University of Singapore and National University Health System
Corresponding author
Equol Contribution
Reviewed OpenReview: https:// oppps:/e/penrvvveeu. net/ forum? id= ddCC900444
Abstract
Hawkes processes are point process models that have been used to capture self-excitatory behaviour social interactions, neural activity, earthquakes and viral epidemics. They can model the occurrence of the times and locations of events. We develop new class of spa- tiotemporal Hawkes processes that can capture both triggering and clustering behaviour and provide efficient method for performing inference. We use log- Gaussian Cox process (LGCP) as prior for the background rate of the Hawkes process which gives arbitrary flexibil- ity to capture a wide range of underlying background effects (for infectious diseases these are called endemic effects). The Hawkes process and LGCP are computationally expensive duc to the former having a likelihood with quadratic complexity in the number of observations and the latter involving inversion of the precision matrix which cubic in observations. We propose a novel approach 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 datasct of reported crimes in the US.
Keywords: Gaussian process, self-excitation, clustering, Bayesian inference

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Published in Transactions on Machine Learning Research (05/2023)
1 Introduction
Hawkes processes are class of point processes that model self or mutual excitation between 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 external addition of a event, while endogenous events are self-excited from previous events by a triggering kernel. An example of the difference bet ween these two mechanisms is in disease transmission an exogenous event could be zoonosis event such as the transmission of Influenza from birds, while endogenous events are subsequent human to human transmission. Due to their ficxibility and mathematical tractability, Hawkes processes have been extensively used in the literature in series of applications. They have modelled among others, neural activity (Linderman et al.) 2014), earthquakes (Ogata, 1988). violence (Loeffler & Flaxman 2018 Holbrook et al. 2021) and social interactions (Miscouridou et al. 2018
The majority of 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 motivated by several factors, first, natural phenomena that self-excite tend to do so both spatial and temporally e.g. infectious discases, crime or diffusion over network. Second, natural processes tend to cluster closely in space and time (Tobler 1970). Third, in parametric formulations residual variation persists and this is often structured in both space and time (Diggle & Ribciro 2007).
A wide body of research exists in modelling spatial phenomena ranging from Kriging (Matheron 1962) lo
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 mcan function and a covariance function that allows control over the degree of clustering as well 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. [1988) 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-excitation. When examining the processes' endogenous and exogenous drivers, the lack of a self-exciting mechanism can potentially lead to spurious scientific conclusions even 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 mechanism - 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 straightforward how to also allow for exogenous spaco-time events simultancously. In the vast majority of previous applications, exogenous events occur at constant rate in both space and time. Non constant approaches exist but have usually been case specific. For example the use of periodic function had been effective in modelling seasonal malaria (Un- win et al. 2021). Some studies do provide nonparametric approaches for the background rate: Lewis & Mohler (2011) provide an estimation procedure for the background and kernel of the Hawkes process when no parametric form assumed for either of the two. Donnet et. al. (2020) and [Sulmm et al. (202T) use non- parametric estimation on the Hawkes kernel whereas Miscouridou et al. (2018) use a nonparametric prior on the background based on completely random measures to construct Hawkes processes that build directed
graphs.
Other recent approaches use neural networks to estimate the rate but the majority of these approaches are in purely temporal setlings such as Omi et al. (2019). There are lew that consider marked point processes such as Du et al. (2016). However, a marked temporal point process which can use marks as locations would not capture the spatial correlations we are interested in. More recently Zhou et al. (2022) and Chen et al. (2021) provided a way to model spatiotemporal settings with neural networks. However, both of them still lack the ability to capture more flexible background where background intensity is not just a constant
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but by itself produces clustering in time and space (we like to emphasise here that this type of clustering is different from the one emerging from the self-exciting nature of the process).
Beyond neural network approaches, there exist non deep models that deal with non-linear intensities. An example is Zhou et al. (2020) who proposed a sigmoid Gaussian Hawkes process with baseline intensity and triggering kernel drawn from a Gaussian process prior, passed through sigmoid link function to al. (2019); Sulem et al. (2021);
guarantee non-negativity. Similarly, Apostolopoulou
et al. 2021 propose point process models with non-linear component that allows both excitatory and inhibitory relationships in continuous time.
1971 with log- Gaussian Cox processes (Moller et al., 1998 Diggle el al. 2013). This synthesis allows us, for the first time, to have an exogenous background intensity process with self-excitation that is stochastic and able to vary in both space and time. We provide a suite of now methods for simulation and computationally tractable inference. Our methods 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 utilise a recently developed Gaussian process approximation (Mishra et al, 2022 Semenova et al.j 2022) that obliviates the need for repeated matrix inversions. Our work represents step towards more general, scalable, point process framework that encodes more flexible and plausible mechanisms to represent natural and physical phenomena.
Here we propose a novel space-time approach that combines Hawkes processes (Hawkes
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 utilisation of LCGPs for the exogenous background rate is extremely flexible and follows from the current state-of-tho-art in spatial statistics (Diggle et al. 2013). (ii) In contrast to previous work such as Loeffler & Flaxman (2018), our framework admits a generative model that can produce stochastic realisations at arbitrary set of locations. We provide 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 c.g. for natural or social phenomena such as crime, discascs, environment, or human behaviour.
2 Related methods
As mentioned before, modelling space through Hawkes processes was first 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- Nog-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 L.GPP background is not discussed in the review.
LGCP to capture the background underlying effects (these are called endemic effects in infoctious discase modelling but here we will use this term broadly for other applications too) and can model the exact spatial and time locations.
Our approach is the first to
Loeffler & Flaxman (2018) aim to understand whether gun violence in Chicago is contagious or merely clusters in space and time. To this end, they use a spatiotemporal Hawkes model and space-time test to distinguish between the two. The model uses a kernel density estimator for the background (endemic) effects and kernel for the epidemic events that is separable in space and time. Their model has different
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construction it does not admit generative procedure since the background rate is estimated using kernel density estimators.
Similarly to Loeffler & Flaxman 2018), Holbrook et al. (2221) build a scalable inference algorithm for para- metric spatiotemporal self-exciting processes. Their proposed model is the one of Loeffler & Flaxman (2018) which based on a Gaussian kernel smoother for the background. The main contribution to overcome the bottleneck of the quadratic computational complexity of such a point process. The authors develop high- nigh-performance computing statistical framework to do Bayesian analysis with Metropolis-Hie Hastingss us- ing contemporary hardware. They apply it gunfire dataset which covers larger dataset and more fine-grained than the one in Loeffler & Flaxman (2018).
The combination of a Hawkes process with an LGCP is found in Linderman & Adams (2015) where the authors propose a purely temporal multivariate Hawkes process with LGCP in the background with the goal infer a latent network structure given observed sequences of events. This approach based on Linderman & Adams 2014 but in discrete time and with an improved inference scheme based in mini batches. However both of these two have different scope to work and work only with temporal data.
Finally, Mohler 2013) develops a purely temporal Hawkes process model with LGCP background for count (aggregated) events. Mohler (2013) builds a Metropolis adjusted Langevin algorithm for estimation and uses the algorithms to disentangle the sources of clustering in crime and security data. We are instead interested in modelling and predicting exact event times and locations.
Few neural network models [uu et al., 2016 Mei & Eisner 2017 Omi et al. 2019 Zhang et al. 2022 have been suggested but they only temporal in nature. A few of them provide way to model spatial domain, by treating space locations as discrete marks. However, using marks doesn't 't allow to model spatial correlations, as observed in many settings. Okawa et al. (2019) extend neural networks to spatial settings but lack the ability to predict the next event in space and time. Work most similar to ours is Chen ctt al. (2021) and Zhou et al.] (2022), where they apply noural network inspired point process models for spatiotemporal data. However, they still lack the ability account for anything more than a constant background intensity.
3
Model
3.1 Point process intensity
A Hawkes process is inhomogeneous Poisson point process defined in terms of counting measure and intensity function rate. For generic spatiotemporal inhomogeneous Poisson point process the domain X' x [o. T), for X C Id we denote the counting mcasure of the process by N and the conditional intensity by A. The definition of a generic inhomogeneons point process intensity in space and time is as below. For s E X C Rdd (generally d here represents Euclidean or Cartesian coordinates) and E [0, 7)
[[N[(t, At) x B(s, As)] HH)]
s(1,s)
lim 40,
A (ss As)
where H4 denotes the history of all events of the process up to time t, ((A) is the counting mcasure of events over the set C X T)) and [B(s, As) is the Lebesgue measure of the ball B(s, As) with radius As > O. The intensity À has to be non-negative ie. À 2 0. Note that the spatial locations can be univariate, referring for example to regions or countries, or bivariate such as geographical coordinates of longitude and latitude or even multivariate depending on the context.
3.2 Hawkes process intensity
Hawkes processes were originally proposed by Hawkes [971 as temporal point processes. The intensity is conditional on the history of the process such that the current rate of events depends on previous events. We focus on self-exciting Hawkes processes, in which historic events encourage the appearance of future events. We develop spatiotemporal self-exciting processes which can predict the rate of events happening at specific
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locations and times. The conditional intensity defined as in equation addmis the form
{((,sHH,) - s((,s)+ + Σ (1 - 44 s;),
(2)
where (f1, En) denotes the ordered sequence of the times of the observed events and (5), S2, s.) their corresponding spatial locations. Events arise either from the background rate put,s) (exogenous non excitation effects) or from the triggering function/kernel g (endogenous or self-excitation effects). j is non-negative to ensure that the initial intensity is non-negative and we take 9 non-negative as we consider excitation effects and do not deal with inhibition. For the scope of our work, we are interested in excitation, however for other applications such as neural connectivity patterns where inhibition needed, one can read for example Cai et al. (0022).
9 can be parametric it can be estimated using full nonparametric assumptions, as done for example in Don- net et al. 22200: (Sulem et al. (2021). Similarly, it can take form of separable (additive or multiplicative) non-separable kernels in space and time. There exists lot of work covering all these cases for purely temporal processes but not in spatiotemporal settings. To give some background, in purely temporal cases there exist guarantees on the estimation and under certain conditions there are consistency results as well identifiability results. However once we add the spatial component, the results do not necessarily extend. Therefore, we consider here the simple case a separable form of a product of an exponential kernel time and a Gaussian kernel in space. For multivariate linear purely temporal Hawkes processes with constant background, it is known result. that one can recover the parameters, i.c. the process is identifiable and one can also prove consistency results: Donnet et al. (2020) prove this and give posterior concentration rates. Some results also exist for non-linear Hawkes processes with constant backgrounds: [rémaad & Massoulić (1966) provide results on the uniqueness of the stationary solution but they do not study estimation of the paramcters. Similarly Sulem et al. [2021] study general non-linear and nonparametric Hawkes processes and provide conditions on the Bayesian methods to estimate the parameters with a (presumably optimal) con- centration rate. Other approaches with time-varying backgrounds exist (e.g. Unwin et al. (2021) Zhou et al.) (2000)) but there are no theoretical results that apply directly in that case (lincar or non-lincar). We think it is an interesting direction to study the theoretical properties of spatiotemporal hawkes processes and we would encourage research in that direction. We would like to note though that in any Hawkes process with a temporally or spatiotemporally varying background, stationarity is not relevant anymore as the background is changing in time and therefore the expectation of the intensity cannot be constant.
The process with intensity defined in equation 2 can be treated as a Poisson cluster process, with mean number of offsprings given by fx fo g(t,, ds). To ensure that the cluster sizes are almost surely finite, we require that b E 10, each generation of offsprings follows a geometric progression, with expected for b > 1 the process explodes and we would have an infinite number of events in finite time. To see how explosion emerges, we refer the reader to of (Grimmett & Stirzaker 2001 Chapter 5) which give the calculations on the expected number of descendants of one atom. More on the implications of the values of can be found in Asmussen & Glynn 2200)).
total cluster size of I-B. For b 0 we have Cox process where
The triggering function 9, centered at the triggering event, is the intensity function for the oflspring process. Properly normalised, it induces a probability distribution for the location and times of the offspring events. The cluster process representation of the Hawkes process (Hawkes & Oakes 1974 will prove crucial to the
efficient simulation of self-exciting processes which we give in section 4.1
We give below the triggering kernel that admits a separable form of a product of an exponential kernel in time and a Gaussian kernel in space. Both of these choices are relevant for the applications we consider in the paper as we know from previous work Loeffler & Flaxman (2018): [Holboook et al.) (2021) that the decay and variation in crime data can can be well explained by the decay prescribed by an exponential and Gaussian kernel respectively. For t > 0 and X c Idd the self-exciting part of the rate is given by
g(t,s) = aB exp (-8t)
exp (-s" s--ss
(3)
2m 2
where a 0,8 3 0 and E a semi-positive definite matrix.
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For the temporal part use the widely used exponential kernel, originally proposed by Hawkes 1971 giving exponential decay which is suitable for the applications we are interested in. Note that an exponential kernel is not always a reasonable choice, for instance in infectious diseases one would prefer the Rayleigh kernel (e.g. see [nwin et al. (2021)). )). For the spatial part, we use a Gaussian kernel which is suitable for modelling spatial localions especially y for social violence settings as first proposed and used by other authors in literature. Specifically, (Loeffter & Flaxman 2018 Holbrook et al.) 2021) analyse the public policy and crime implications of larger gunshot dataset which includes the data used in this paper and choose these forms for the kernels.
As mentioned above we consider here separable form of g. Note that non-separable kernel approaches exist in literature, such as Jun & Cook (202)), in which temporal patterns differ according to location. However one cannot naively use those without properly assessing identifiability concerns. The current construction can be extended to cover those, however this would be beyond the scope of the novelty of the current method and the scope of the applications we consider here.
The other part of the intensity is s),,s), which is the background rate of the process. It is nonnegative function with initial nonzero value that captures the underlying patterns in space and time that encourage the clustering of events in those time and space locations. It often takes the form of constant for simplicity, or a parametric form such as periodic as assumed in Unwin et al. (2021) or can even have a nonparametric prior constructed on random measures as in Miscouridou et al. (2018). As further explained in more detail below, we assume a log-Ga oog-Gausiaan process prior on p(t, s) which to our knowledge has not been used before in the literature of spatiotemporal Hawkes processes.
3.3 Latent log Gaussian process for background rate
We use a latent Gaussian process (GP) to determine the background rate of events in time E R and space Rd. This means that the background rate takes the form
(tt,, s) = exp (ʃ (t)
(4)
where ((,,s) is function realisation from Gaussian process prior in space and time. Formally, Gaussian process is collection of random variables, such that any finite collection of them is Gaussian distributed. GPs are a class of Bayesian nonparametric models that define a prior over functions which in our case are functions over time and space. Similarly lo a probability distribution that describes random variables which are scalars or vectors (for multivariate distributions), a Gaussian process is distribution over functions and belongs in the family of stochastic processes.
GPs are a powerful tool in machine learning, for learning complex functions with applications in regression and classification problems. We refer the reader to (Rasmussen & Williams 2005 Chapter 2) for details on Gaussian processes and their properties.
A Gaussian process on RD, for any D > 0 is completely specified by its mean function m(-) and covariance function k(..). We will denote a draw from a Gaussian process as
f() GP (m--),k(:.))).
The Gaussian process is centered around its mean function, with the correlation structure (how similar two points are) of the residuals specified via the covariance kernel. Propertics of the underlying function space such snoothness, differentiability and periodicily can be contedlled by the choice of kernel. One of the most popular choices of covariance kernel, and the one we choose to introduce the model with, is the Gaussian kernel (also commonly called the squared exponential kernel), defined for u' E RDD by the covariance function
(-1) 1 I -rpe) 212
Cou (f ( (u, w² exp
where |u| denotes the Euclidean norm, i.c. it is equal to |u| Ex
C uf if u is a vector (D > c.g. the (D= = e.g. timestamps). w² > 0 defines the
spatial locations) and to the absolute value of u if u is a scalar

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kernel's variance scale and >> 0 is length scale parameter that specifies how nearsighted the correlation between pairs of events is. The hyperparameters can be varied, thus also known as free parameters. The kernel and mean of the GP together fully specify the prior distribution over functions.
We will consider an additive separable kernel with a bivariate spatial dimension s = (z. y) and univariate temporal dimension t. Note that one could naively consider a. joint fst with no assumptions of additivity (or other form of structure) at all. However this would not be advisable as it would be impossible in this recover the underlying latent functions. When there is not enough structure
to guarantee that in the form of the background it is much more difficult study the identifiability of the latent functions fa and fs. In non-identifiable cases, the prior dominates the estimation and the estimated ft, fs will be heavily influenced by the prior and not by the data. We consider here the additive structure as a minimum type of structure to assume on the background latent process which is still very generic, able to capture arbitrary background trends.
In order to ensure a nonnegalive background we exponentiate the additive kernel. From this kernel specifi- cation the background intensity p(t,s) s) follows a log- Gog-Gansian Cox process (Moller et al. 1998 Diggle et al. 2013) over space and time
p((,s) exp (Ja (S) (s) (t) fc GP (ma, ks)
(6)
fa GP (my,
where my and my are the GP mean functions and ki, Ks are the kernels defined by the hyperparameters wf, wh, 4,
3.4 Full Model Likelihood
To model the spatial coordinates s (z, y) and time stamps t, we use a Hawkes kernel (ts = ge (s) from equation 3 and the log-Gaussian Cox process μ(t, exp (fa(s + fa(t)) from equation 6 Without loss of generality we will assume here that the Gaussian processes have zero mean. The joint model consider
is a Hawkes process with composite rate A(t, y) which is the sum of the intensities of an LGCP process and a Hawkes process
A(1, = (Xs (I, + {s
Σ gelt - tg)ga (I Iz, 9i)
+
<
exp (fs (I,, v) + ft()))
1
x; (a- (_(r
(y 91)² 20 y
cBcxp (-B(t tz)) 2T
exp
(7)
20 xr.
<
One could see this as an intercept coming from constant contributions by both the temporal and spatial background processes as these are only identifiable through their sum and not separately. Given a set of observed ordered times (f1, .x,
[O,T) and the corresponding locations (si, 82,,,,,..., E X, let D denote the full data D (t,, s and L(D) the likelihood. Following equation (7.1.2) in DDaley & Vore-Jones 2008 Chapter 7) the likelihood is given by
II A(t,, si) exp (-/x A(t, s(t,s)dtds
L(D)
A(t, Dy, 91) exp (-/x
1' A(t, x, (llxx,))dda ddddddy
(8)
We give below details on how simulate from the process with the rate defined in equation y and how perform Bayesian inference using the likelihood from equation

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4 Methods
4.1 Simulation
By construction our model admits a generative process facilitating simulation. This is an important and nuanced advantage over previous spatiotemporal models (Loeffler & Flaxman 2018 Holbrook et al. 2021 which were not fully generative due to a deterministic parameterisation of the exogenous component. Note that the model of [Mohler (2013) does admit a generative model but only for purely temporal model for aggregated (count) data. In general, Hawkes processes be simulated in two ways: through intensity based approach
cluster based approach. We give below Algorithm | to simulate from model via the latter approach, i.e. through simulating the background first and then the generations of subsequent offsprings. Note that for the hyperparameters 44,6, w/ w one can either fix them to known value or (hyper)priors on them.
Algorithm Cluster based generative algorithm for Hawkes process simulation
Require: Fix T 0,2 Draw 4,4 1g p² (:)
Draw wf w5 (.)
Draw ~ p(.)
Draw fi ~ GP(O, kt), fs GP(O, ka) Set p(t,s) s) = exp (fe(t) + fa(s) + aa)
Draw No ~ Pois (oo Jx p (t,s) dtds
Draw (4 :5
(0) (0)
No
from a Poisson Process with rate p(t, where si
I
(U)
(0)
i=1
Ui
No while G. + do
Set Go (0) s
(0)
-
i=1
for i = 1 to N/ do
Draw C; ~ Pois (Io fx Stalt, s)dtds). the number of offsprings of event i
for j = 1 to Cg do
Draw (((41) vid Exp((3 + (,00
((+1) sid Normal Draw 8j
x
(1)
0
(1)
Set O₂ = (((tt)) s (e+1))
end for end for + =
G = f Oy.... Oa....,.l|
end while return Gr
We clustering approach (Hawkes & Oakes 1974 process where each event has mean number of offspring b (see section 3.2 and relies the following idea: for each immigrant i, the times and locations of the first-generation offspring arrivals given the knowledge of the total nunber of them are each i.i.d. distributed. We provide the simulation in Algorithm I In Algorithm D we introduce ds to denote the total mean the background rate as the GPs have zero mean. p+(+) refers to probability distribution on the positive real line and p(-) distribution on the real line. As test check for making sure that our Hawkes process simulations are correct we employ an approximate Kolmogorov-Smimoo type of test adapting Algorithm 7.4.V from Daley & Vere-Jones 20088.
for simulation which makes use of the Poisson cluster
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To simulate from model proposed above, i.. Cox-Hawkes process we need to draw from GP. Since GPs are infinitely dimensional objects, in order to simulate them we have to resort to finite approximations. The most common approach is to implement them through finitelyddmeeniional multivariate Gaussian dis- tributions. This is the approach we take as well for simulating our GPs. In order to sample points from the LGCP background of the process, we draw an (approximate) realisation from the GP prior and then use rejection sampling to sample the exact temporal and spatial locations. The algorithm can be found Appendix A.1
4.2 Inference
Given a set of n observed data D = 14,8& over a period [o, T) and a spatial area denoted by X, we are interested in a Bayesian approach to infer the paramcters and hyperparameters of the model. Denote by 0 and & set of the parameters of the background rate u((,s) and the triggering rale g(t,8) respectively. This gives (oo, f((t), (s)) and Φ (a, B, 02. og).
The posterior is then given by
( OID) (0)) x L(D)
(9)
=((((())((f((()))))) (())(())(( L(D).
where L(D) is the likelihood defined in equation and with some abuse of notation, use to denote both prior and posterior for all the parameters. For ag use Normal prior. The prior for T()), xB),, x x and (o))) is Truncated Normal (restricted on the positive real line) to ensure the positivity of these parameters. For the experiments we demonstrate below we have also tried Gamma and Exponential priors for and B, however in that case the MCMC chains showed worse mixing in comparison to the case of the Truncated Normal prior. Note that the prior on the functions ft and fa can be further defined by the priors on the hyperparameters ~ InverseGamma, w LogNormal for the temporal process and l InverseGamma, w LogNormal for the spatial.
Our objective is to approximate the total posterior distribution D) using MCMC sampling. A classical Hawkes process has quadratic complexity for computing the likelihood. Only in special cases such that of a purely temporal exponential kernel the complexity is reduced from quadratic to linear as it admits a rocursive construction. Soe Dassios & Zhao (2013) for an explanation. Note however we cannot apply this our case as it does not hold when we add (on top of the temporal exponential kernel) the Gaussian spatial kernel. Inference is in general cumbersome and people tend to either resort approximations or high performance computing techniques such as Holbrook al. (2021).
A naive formulation of combining log- Gaussian cox processes the background intensity function in the spatiotemporal Hawkes process will increase the computational complexity for the inference. This happens because in addition to the quadratic complexity arising from the triggering kernel the exogeneous formulation naively introduces a cubic complexity for LGCP (Diggle et al. 2013).
We propose to circumvent. the computational issues through a reduced rank approximation of a Gaussian process (Semenova et al. 2022) through variational autoencoders (VAE). This approach relies on pre- pre-training VAE on samples from Gaussian process to create reduced rank generative model. Once this VAE is trained, the decoder can be used to generate new samples for Bayesian inference. More specifically, in this framework one should first train a VAE to approximate a class of GP priors (the class of GP priors Iearned varies from conlext to context. depending on our prior belief about the problem space) and then utilises the trained decoder to produce approximate samples from the GP. This step reduces the inference time and complexity as drawing from a standard normal distribution 2 ~ AN( I) with uncorrelated 2, is much more efficient than drawing from a highly correlated multivariate normal A (0,E) with dense Σ. For more details section 2.5 in Semenova et al. (0022). Here will denote this approximation to the Gaussian
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15
observed
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+++ -- -+--0 +* HP+
Figure 1: Plot for the temporal Gaussian process (t) simulated data. The red line is the simulated draw of the Gaussian process, the green line is the mean posterior and the yellow shaded area is the 90% credible interval. The red marks on the x-axis are the exact simulated times from the background process.
Process prior by n. Hence, we obtain overall the Bayesian hierarchical model
( 0)D) (())())
2 t (f((t) (AAAAAAAAAAAAOTTTTT
(10)
The code for simulation and inference for this class of models of Cox processss implemeete and numpyro (Phan et al. 2019) can be found at httpp:///ggtuuu..ooooa/a////ttts Hawkes
in python innke
5 Experiments
We demonstrate the applicability of our methods both simulated and real data. For simulations our goal is twofold: (i) to show that
accurately estimate the parameters of both the background and self- exciting components thereby showing that we recover the true underlying mechanisms and (ii) to show that our method performs well under model misspecification, thereby showing our model is sufficiently general to be used in real data situations where the true underlying data generating mechanism is unknown. On real settings we apply our methods to gunfire data used in Loefller Flaxman (2018) detected by an acoustic gunshot locator system to uncover the underlying patterns of crime contagion space and time. We show how model can be used effects in important settings. Note that throughout the following refer to the model used for simulating data the true model.
actionable tool by practitioners to understand and measure contagion
5.1 Experiment 1: Simulated Data
We simulate data from Hawkes process with rate as given in equation 7 on the domain (0, 7] [0, 50).
[0, 1] [0, 1]. For the background rate which governs the exogenous events we simulate a realisation of the latent (separable) spatiotemporal Gaussian process with covariance kernels defined in equation using 4 10, wf 1, 0.25, u 1. The simulated ft(t) from the temporal Gaussian process can be seen in Figure [ red and the temporal events drawn from this background are also shown in red on the x-axis. The simulated two-dimensional spatial Gaussian process can be seen at the left plot of Figure 2 Note that we also use an explicit constant intercept of ag = 0.8 giving an overall background rate of exp(ap + ft + fs) and inference we use a Normal prior on it.
For the diffusion effect we use a triggering spatiotemporal kernel of the form in equation with values 0.5, 3 0.7 for the exponential kernel. For the Gaussian spatial kernel we will assume common
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Simulated fs
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Figure 2: Simulated draw and the posterior predictive distribution for the 2-dimensional spatial Gaussian process. The simulated (2, y) is shown on the left on a regular grid and the mean predictive distribution is shown on the right with the simulated locations in red.
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Figure 3: MCMC trace for ao, a, 3, on simulated data where the red line shows the simulated value for the experiments. The samples shown are collected from chains.
parameter o² for both of and og which we will assume to 0.5. This gives a set of around n 210 spatiotemporal points (1s, Ne, y of which the ratio of background to offspring events is roughly 1 1.
For inference run 3 chains with 1,500 samples each of which 500 were discarded burn in, using thinning size of 1. In Figure 3 we report the trace plots for the parameters a, B, which define the triggering kernel that governs excitation. We also report do which we used as the total mean of the latent Gaussian process p(t, v, exp (ft(t) + o + fa(æ, v)). We use Normal prior on ag. In all cases the simulated values shown in red are within the trace coverage. In our experiments we combine the samples (after removing the warmup iterations) from all the chains. The plots overall show good convergence with good mixing between the chains and no multimodal behaviour. Regarding the Gaussian process fitting, we show the posterior predictive plots in Figures and 2 For the one-dimensional temporal Gaussian process we plot the simulated draw ft( in Figure [ in red. The blue line is the mean posterior of fe(t) and the yellow shaded area is the 90% credible interval obtained from the posterior predictive distribution. The red dots on the x-axis are the exact simulated time events drawn from the process. The 90% credible interval covers well the simulated function, and the mean posterior predictive is very close to the simulated one, showing good model fit.
For the two- dhodeninnii spatial Gaussian process we plot the simulated draw function fa 8) at the left plot of Figure 2 The the mean posterior predictive distribution is shown in the centre and the mean predictive
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distribution with the true simulated locations embedded on it is shown the right. The color scale on the right shows the relative values ranging from dark blue (smallest) to yellow (highest). The simulated and the mean of the posterior predictive distribution, are relatively similar when compared visually which shows a good fit for the model.
To quantify convergence we require good A diagnostics. For all our results the R diagnostics returned by the sampler were in [1, 1.002] for all the estimated parameters. This, combined with visual inspection of the MCMC trace show good evidence of convergence and good mixing behaviour.
To further validate our method we perform a goodness of fit test. From the estimated model parameters we obtain above, we compute the estimated intensity and then apply the time-rescaling theorem (Meyer 1969 Papangelou, 1972) which states that if the process is correct, the rescaled times are independent exponentially distributed random variables with unit rate. We can thus compare the empirical and theoretical distributions to assess how well the chosen model agrees with the empirical data. We make this comparison using a test. We provide in Appendix A.2 in Figure g the QQ plot which shows good fit the best fit line has slope of 1.1 and intercept of 0.2 with the coefficient of determination that measures the correlation between the two axes to be 0.98. Secondly, we perform hypothesis testing using the Kolmogorov-Smirno statistic to test the mull hypothesis that two iistiibuiions were drawn from the same distribution. Our p-value is > 0.05 ing that at a confidence level 95% we cannot reject the null hypothesis. Given the above, we conclude a good fit of our process to the data.
quantile-qaantile (QQ) plot and Kolmogorov-
5.2 Experiment 2: Model misspecification
Our second experiment on simulated data compares and contrasts our method (LGCP-Hawkes) to Hawkes process with constant background and pure log Gaussian Cox Process. The intensity for our (GCP-Hawkes model is equation 7 for Hawkes it is equation with constant stt,s) = and for LGCP it is equation 0
We simulate data from these three inhomogeneous point process models and then fit each model on every dataset on a train set and perform prediction on a test set. Note that we also fit under a homogencous Poisson model it's the baseline giving the simplest spatiotemporal model that exists. We show that our model LGCk is reasonable approach even when there is model mismatch (i.e. when the data are drawn from pure Hawkes pure LGCP). It is therefore a good approach to use in real data scenarios when the underling data generating mechanism is unknown.
Tt is in general challenging to evaluate the quality of model fit from different point process models.
We use two ways to evaluate the model fit and generalisation ability of our model. We first adopt procodure to predict the temporal and spatial locations of future events under the inference model and then compute the combined error between those events and events generated under the true generating model that was used for simulation. This procedure mirrors the properties that practitioners would desire from their model in real world settings.
The metrics we use to test the generalisation ability of the model are (a) the combined root mean square error (RMSE) between the exact simulated and predicted events and (b) the normalised negative log-likelihood of the test events. The formula we use is the following RMSE RMSE, + RMSE, where RMSE, = - E (1/ 0,)² for t,, yi here denoting the true events and 7, T., Ti denoting the predicted ones. The RMSE with its standard error is demonstrated in Figure E (top). Similarly we evaluate the normalised negative log- ligeiiill (NNL) on the test data in each case given the setup explained below and report the mean and standard error in Figure g (bottom). The normalisation is given through the division of the original negative log-likelihood by the number of datapoints involved (here the number of test. points).
V Teeast Eja (t, 13--ty 2222 and RMSE. = V neant Ejer }2 + Thenst
The experimental setup is as follow. We simulate 100 datasets (cach of which give on average 300 events) over fixed time window and a fixed spatial domain and then do train-test split. We repeat this using as generating model each of the models.
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Simulate LGCP
Simulate LGCP Hawkes
Simulate Hawkes
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LGCP LGCP*Hawkkes Hawkes
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Polsson
Inference model
Figure Plots to support the model misspecification experiment. Top: Average RMSE and its standard error (combined for space and time) reported for the model misspecification experiment. Bottom: Average negative normalised log-likelln likelihooo and its standard error on test data. The left plot corresponds simulated dataset from an LGCP model, the middle to LGCP-Hawker and the right from Hawkes model. In all three cases we perform inference under all LGCP, LGCP-Hawker Hawkes as well as Poisson (baseline).
For every dataset, we then perform inference under our MCMC scheme under every model. Given the estimated parameters, we predict 200 times the next 10 future events which we compare to those of the test set. We compute the error between the true and estimated events across the 200 predictions and the 100 simulations. We report the mean and standard error of the RMSE and normalised negative log-likelihood (NNL) graphically in Figure 4 This plot shows how good each model is in predicting the near future.
As shown in Figure and as expected, the RMSE error is always lowest when the true model is used for inference, however in all cases the next best model is LGCP-Hawkes although the differences are not always statistically significant. This provides evidence for our model's ability to flexibly capture a wide range of underlying patterns. Looking the RMSE error all cases the worst model the Poisson baseline, as constant intensity in space and time cannot capture the inhomogencitics in the data. These results highlight that when the true data generating process is unknown, which is the default scenario in real world settings, our model is likely to be a robust choice. The NNL results suggest the same conclusion, NNL is always lowest when the true model is used for inference, and in all cases the next best model is The difference with the RMSE results is that under an LGCP-Hawkes model and a Hawkes model the log-likelihood of an LGCP deviates quite a lot from the L.CP-Hawkes and Hawkes model.
in all cases
see how the I.GCP-Hakkes model under study correctly recovers the other processes, namely Poisson, LGCP and Hawkes. Recall that intensity consists of two parts, the background and the excitation. We will report in each case two ratios, rp: the ratio of the background to
Furthermore, what of interest
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the total intensity and Tg: the ratio of the excitation to the total intensity to explain how model recovers correctly the mechanism which drives the appearance of the events. We will also consider the coefficient for the excitation part of the intensity, which we denote by = x B and the parameter of the background, and will report their respective mean estimates.
Applying our T.GC--Hakees model (that assumes an intensity as equation 7 on simulated data obtained from the LGCP with true value ag 0.5, and using the posterior mean of the parameters, we estimate In 0.91, 0.09. The mean estimates for the excitation coefficient and dg 0.02 and 0.58 re- spectively. These suggest that the model has correctly inferred that there is negligible contribution to the intensity coming from the self-excitation part, meaning that events arise because of spatiotemporal back- ground variation and not contagion.
Applying our LaWkesawkes model on simulated data obtained from the Poisson process with ag = 0.8, we estimate rg 0.85, TE 0.15. This suggests that most of the contribution comes from the background and small part from the excitation. The mean estimate for e and ag 0.09 and 0.52 respectively. These suggest that the model has correctly inferred that there is very little contribution to the intensity coming from the self-excitation part, meaning that events arise because of spatiotemporal background variation and not contagion.
Applying our I.GPPHawkess model on simulated data from the Hawkes process with dg = 0.5 and c a x 3 0.4, we estimate rB = 0.43, rg = 0.57 which suggests that the events of this process are due to both the background and the self-exciting kernel. The mean estimate for the excitation coefficient c and og are 0.42 and 0.52 respectively, which shows that the model has properly recovered the contribution from both background effects and self-cxciting behaviour.
5.3 Experiment 3: Gunshot Dataset
100
Weekly gunshots
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(b)
Figure 5: Spatial (a) and temporal (b) distribution of the gunfire data in Washington DC over the year 2013. The spatial locations the exact geographical coordinates and the temporal locations are shown weekly.
We use gunshot data in 2013 recorded by an acoustic gunshot locator system (AGLS) in Washington DC and follow Loeffler Flaxman (2018) for data preprocessing. There were 1,171 gunshots recorded in total. Spatial locations were rounded produce approximately 100m spatial resolution and see temporal resolution. Visualisations of the temporal and spatial distributions of the data. shown in Figure 5( and (b).
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Figure 6: MCMC trace for the parameters (a) ao, ao,a, 3 and (b) o2, og when collecting the MCMC samples from all chains and discarding warmup.
We perform inference with the HMC (Neal 2011 | routines of numpyro which uses the NUTS (Hoffman & Gelman 2014 algorithm. We used chains cach with 4, 000 samples from which 2000 are discarded warmup. We join logether the samples from the two chains and report the combined MCMC trace for each of the parameters. Note that did some prior sensitivity analysis to assess the robustness of our results. We used different parameters the priors for the parameters and we observed that the posterior distributions of the parameters were similar, giving posterior mean estimates very close to each other. We also tried different priors such as Gamma and Exponential but the convergence of the chains was better when using Truncated Normal distributions. Note that we have rescaled accordingly the temporal and spatial locations of the events, in order simplify their use inference and be appropriate for the domain of our pre-trained GP generators. Rescaling, is standard practice in point process modelling. In Figure G(a) we report the MCMC trace plots for the parameters and B and ag and in Figure ((b) we report 02 oy which define the lengthscales of the spatial Gaussian kernel (x and distance) that governs excitation in space.
Regarding the Gaussian process fitting, we show the posterior predictive plots in Figures 7 and g For the one-dimensional temporal Gaussian process we plot the estimated function ft(t) in Figure 7 The green line
is the mean posterior of [+(1) and the yellow shaded area the 90% credible interval obtained from the posterior predictive distribution. The red marks indicate the observed true events.
For the spatial Gaussian process plot the estimated function fa y) at the left plot of Figure The mean predictive distribution with the true locations embedded on it is shown the right. The plots overall
extimated 50%
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Figure 7: Posterior predictive for fe(t) with the posterior mean in green and the 90% credible interval in the yellow shaded area, with true time stamps of the events on the x-axis.
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Estimated
Estimated with true locations
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Figure 8: Mean of the posterior predictive for fa(2, y) (left) and similarly with the true locations (right).
show good convergence with good mixing between the chains and multimodal behaviour. This is also quantified by the convergence diagnostics å which were equal to for all the parameters estimated.
We report an estimate and 90% credible intervals of âg = 0.53 --446,...477 á = 0.73 (0.68, 0.78), 0.18 (6, 0.21), 1/3 5.35(4.64, 6.11), 02 9.26e-5(P2) (7.900 1.07e, a3 5.65e (4.78e 6.67e-5)
which can be interpreted as below, following the way of interpretation of Loeffler & Flaxman (018).. The average number of shootings triggered by one shooting is around 0.73. Then, rounding to the nearest minute or meter correspondingly, the temporal lengthscale for the exponential triggering kernel estimated to around minutes, the spatial triggering lengthscale æ distance, denoted by or around 10m and for the y distance 8m. This means that for every 100 shootings that occur, these create most another 73. Using the right upper bound of the uncertainty intervals, the period in which diffusion takes place is within less than 6 minutes and the area is within 10 meters in distance and 8 meters in the distance. Regarding the background effects the posterior mean of the spatiotemporal Gaussian process is estimated to be 0.53. The results have some differences from the ones reported by Loeiller & Flaxman (2018) and Holbrook et al. 2021h but the model assumed here has different form and we have applied it on a different subset of the gunshot dataset.
Table 1: Fist row: Average RMSE on test data with its standard error in brackets computed when predicting future unseon temporal and spatial events under the five models. Sccond row: Average NNL with its standard error in brackets computed when predicting future unseen temporal and spatial events under the five models. Third row:
Average NNL with standard error bracketss computed on training data under the five models.
Hawkes- Hawkes-LGCP
Hawkes
LGCP
Poisson
DeepSTPP
RMSE (test)
7.33 (0.11)
8.14 (0.13)
7.90 (0.09)
14.2 (0.29)
7.86 (0.25)
NNL (test)
-2.534(0.09)
-1.789(0.08)
-0.309 (0.039)
-0.26(0.024)
-2.24(0.11)
NNL (train)
3.42(0.0007) 3.15(0.0007)
2.47(00.026)
1.99(0.0003)
-2.76(0.009)
We compare our model to the LGCP model, Hawkes model. bascline Poisson and the state-of-the-art neural network based spatiotemporal model of (Zhou el al. 2022). In our results, we name the model of (Zhou ett al.) p022) as DeP. We report Table [ the average RMSE on test data, normalised negative log- likelihood (NNL) on both training and test data. As scen from Table [ LGCP-Hawker gives the best predictive performance both in terms of RMSE and NNL. The normalisation of the likeliiol is done dividing by the number of training and test data points respectively. The training in inference times
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45 minutes for Hawkes-LGCP, 8 minutes for Hawkes, 25 seconds for LGCP, 11 seconds for Poisson and 12 minutes for DeepSTPP.
6 Conclusion
We presented a novel model combining Hawkes processes with Gaussian processes, and used it to identify patlerns in gun violence in Washington DC. Mechodologically, ours is the first. model of its kind to have such flexibility in capturing underlying patterns in the rate of occurrence of events, combining a powerful nonparametric statistical model with interpretable mechanistic self-exciting point process model. This combination means that it can be used across a range of real world spatiotemporal problems in which the underlying data mechanism is unknown. Applications could include social networks, biology, economics and epidemiology. Its general and practical form make it an actionable tool for practitioners that can be used to design interventions and for policy making.
There many directions for future research. One could study the properties of this model in theoretical level to define the implications of different forms of the background rate and whether they are identifiable. Additionally, this model can be further exlended to include additional covariates. These take the role of marks in a Hawkes process construction and can bring more information in infectious disease applications in which one wants to characterise the discase transmission and quantify the sources that govern infection in space and times. Deviating from univariate setting, one can consider interacting Hawkes processes model events in different states or regions where the intensity of events in one regions depends on the intensity in another region. This model could be useful for crime data, and also in neuroscience, where multiple neural trains interact across different parts of the brain. Compatationally, this may prove to be difficult extension. In scenarios where the background trends are polentially coming from different sources, incorporating transformations of Gaussian process could make the framework even more flexible and able to capture multimodal distributions. Finally, one can extend this in a generic flexible framework for Hawkes processes with non-linear intensities that can potentially capture inhibition.
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A Appendix
A.1 Simulation
In Algorithm 2 we show how to simulate from the LGCP background of the process with intensity p(t,s) exp( + 40 + (+S)). For simulation purposes we wrile explicilly 4n = onz + dos but only estimate them via their sum (i.e. ao) as aoe and 40s are only identifiable through their sum. pt denotes a probability distribution on the positive real line whercas p denotes a probability on the real line. We take the prior on & and &g to be Inverse Gamma, and the one on w/ and w to be Log Normal but other options are possible. For the prior age and dos we Normal distribution.
Algorithm 2 Simulation of the LGCP events from the background
Require: T> 0
Draw anz. Aus ~ p(.)
Draw 4 ~ ~xppp
Draw wf.w? ~ (+(())
Form ki, Ki using equation
Draw fi GP(O, k)), fs (g(
Set r(t) = (t) + aot). r(s) cxp(fz(s) + ans)
Approximate (tt, So' r (t) r(s) dtds
Draw No Pois (t)
Draw [t) No
[0. from r(t) via rejection sampling Draw {s) X from r(s) via rejection sampling
return Go (ti, s
A.2 Simulation experiment
Ordered Values
Theoretical quantiles
Figure Quantile-quantile plot to compare the quantiles of the empirical and theoretical intensity.
To further support Experiment in section 5. we consider goodness of fit tests. We provide a QQ plot and a Kolmogoroo-Smirnov test. The QQ plot shown in Figure g measures the agreement between the observed and the theoretical quantiles of the intensity function. The theoretical quantiles plotted x-axis are those from the exponential distribution whereas the empirical ones in y-axis are those from the fitted intensity under I.GCP- model. The best fit line cocfficient of determination is 0.98 suggesting almost perfect correlation between the two. This suggests good agreement between the point process model and the experimental data as the points lie on 45-degree line. Secondly, we measure the discrepancy between the two using a Kolmogorov-Sminn a value > 0.05 clcarly not rejecting the null hypothesis that the two samples are coming from the same distribution.
data. has slope of 1.1 and intercept -0.2. The
test which gives
20


Reviews:
[{"review_id": "uAdkqfe4St", "reviewer": "Reviewer_qAT6", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- LCGP prior for the background allows for a fully generative process that can be used to simulate from Hawkes processes with a variety of background rates compared to kernel density estimation approaches (e.g. Loeffler & Flaxman, 2018).\n- Model is very flexible and improves prediction accuracy, yet the parameters are still very interpretable, making it useful for exploratory analysis.\n\nWeaknesses:\n- Many (mostly minor) presentation issues scattered throughout the manuscript. It almost feels like it has never been proofread. I have tried to point out most of these below.\n- Broader impacts of the proposed work should be discussed. This is particularly important because the authors analyze a gunshot dataset to draw potential conclusions about gunshots triggering other gunshots. Their conclusions may have implications for public policy, so limitations of their conclusions need to be made very clear.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors propose the Hawkes-LGCP point process model for timestamped events taking place at different locations in space. It uses a Hawkes process with triggering mechanisms for both time and space to allow past events to encourage future events nearby in space and in the near future. It further incorporates a log-Gaussian Cox process (LGCP) prior for the Hawkes process background rate. The authors apply their model to analyze a data set of gunshots in Washington DC and find that one gunshot typically triggers 0.7 future shots and within a roughly 10x8 meter area in the next 6 minutes.\n\nThe main contributions I observe are as follows:\n- Novel formulation for a self-exciting spatiotemporal point process by incorporating an LCGP prior for the background rate in a spatiotemporal Hawkes process.\n- Efficient algorithms to sample from this Hawkes-LGCP model and to perform Bayesian inference.\n- Interesting analysis on gunshot data that attempts to identify how much influence a gunshot has on future gunshots nearby.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Major issues:\n- Section 3.2, 3rd last paragraph: \"$N(t) - N(\\mathbf{s}) = \\infty$ for $t - \\mathbf{s} < \\infty$.\": This is confusing because $\\mathbf{s}$ has been used to represent space and is a vector. In this equation, I believe $s$ should be a scalar representing time, so I suggest a different variable name. Furthermore, I am not sure if this paragraph fits given the previous paragraph. The previous paragraph is expressing almost the same content but more specific to the exponential kernel and Gaussian form taken in this paper.\n- Section 4.1, last paragraph: Provide more details (in an appendix) or at least a reference on how to sample from the LGCP by drawing an approximate realization then using rejection sampling.\n- RMSE computation for Figure 4: clarify whether the RMSE is just over the (x,y) spatial locations or whether it includes error in timestamp as well.\n- Either specify the rescaling that was performed on time and space in the gunshot data set so that the reader can better interpret the parameter estimates (e.g., how large of a range does $\\sigma_x^2 = 9.26e^{-5}$ really correspond to?). Alternatively reverse scale the parameter estimates back into human interpretable units, such as meters for distance and minutes for time.\n\nMinor issues:\n- All references that are published in conference proceedings seem to be missing the name of the proceedings, e.g. Linderman & Adams (2014), which was published in the ICML proceedings.\n- Equation references seem incorrectly formatted, e.g. \"Eq equation 7\" and \"Eq equation 8\".\n- Page 2, second last paragraph: \"prior on t he background\": t he -> the\n- Section 3.1, first paragraph: \"the appearance of future events. s\": Remove extra \"s\".\n- Section 3.1, first paragraph: \"$|B(\\mathbf s, \\mathbf s)|$ is the Lebesgue measure of the ball $B(\\mathbf s, \\mathbf s)$ with radius $\\mathbf s > 0$.\": I suggest using different letters for the center of the ball and the radius, e.g. $\\mathbf s$ and $\\Delta \\mathbf s$, as in equation (1).\n- Section 3.2, paragraph after equation (2): However, we The condition -> However, the condition\n- In Section 3.4, the spatial coordinates now switch to $s$, not $\\mathbf s$. Stay consistent in your notation.\n- Section 4.2, second paragraph after equation (9): linearas -> linear as\n- Section 4.2, second paragraph after equation (9): lgcp -> LGCP\n- Figure 2 is blurry and should be created in vector format or higher resolution raster.\n\n", "broader_impact_concerns": "No broader impact statement is provided, and I think one is necessary. See my discussion in the Weaknesses section.\n", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "TRrPuxPk0U", "reviewer": "Reviewer_3Wae", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n\nIn general, it is an easy-to-read paper. The visualizations in the paper also help the reader comprehend the functionality of the model. I also appreciate the source code the authors are providing. The provided experiments show promise for use of their model inI w real-world applications.\n\n**Weaknesses**\n\nI will expound more below. In general:\n\n(1) some points in the presentation should be corrected.\n\n(2) both the introduction and the experiments should be expanded to include relevant work.\n\n(3) Some inference details ( metropolis hasting ratios), simulation details, and identifiability results should be explicitly stated to make the article self-contained (without having to resort to the source code for mathematical details).\n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors propose a variant of a spatio-temporal Hawkes process which combines a Hawkes process for inducing mutually exciting effects and a Gaussian process for capturing clustering effects in the background rate . Based on that they provide some synthetic experiments for demonstrating their model. They also test their model on the gunshot data. Their model outperforms simpler baseline point process models in terms of RMSE in the event predictions.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "**(1) Some references should be included.**\n\nIt would strengthen the paper if the authors also compare with similar (mathematically or functionally ) similar models [1] [2]. Especially [2] is a very similar model. The authors extend [2] to include the space domain while adding a Hawkes process intensity. However, the use of a GP in the intensity of a point process detracts from the mathematical novelty of the article.\n\nMoreover, one of the main motivation of authors work is the lack of expressive enough background rates. However, the intensity of [2] offers such an expressive background rate. Similarly, in [3] the background rate is a constant term multiplied by a probability term capable of modeling complex (inhibitory) effects. Finally, when the authors refer to the cluster-based Hawkes process simulation they should cite [5]\n\n\n**(2) Some points in the presentation should be corrected.**\n\na. Section 3.1 should be renamed to regular point process definition. The definitions of this section refer to the broad class of point processes characterized by an intensity function and not to a Hawkes process. Moreover, it should be stated explicitly that $\\lambda \\ge 0$. There is also a typo: $B(s,s)$ should be $B(s, \\Delta s)$ in the explanatory text below equation 1.\n\nb. In Algorithm 1:\nthe condition $G_l!=0$ does not appear correct since $G_l$ in the last line of the body of the loop seems to increase (to include all the generated events so far. I understand that the authors want to convey: unless there are no more generated events in the time window [0,T] . However, they should accurately formulate this. Moreover, in the simulation steps ( $C_i$, $O_i$), they should provide the exact distributions they are sampling from. Finally the notation ${t,s} \\sim \\mu(t,s)$ is also not correct since $\\mu$ refers to the background rate and not a distribution. Probably the distribution is a Poisson process characterized by $\\mu(s,t)$?\n\n\n**(3) Some technical details are missing:**\n\nThe authors should provide the exact MCMC updates similar to [6]. Regarding the identifiability results, the authors mention:\n\nIn purely temporal cases we have guarantees on the estimationand under certain conditions we have consistency results as well identifiability results. However once we addthe spatial component, the results do not necessarily extend. Therefore, we take here a separable form of aproduct of an exponential kernel in time and a Gaussian kernel in space.\n\nI think the authors should either cite accordingly or provide explicit derivations for supporting this statement.\n\n**(4)In the synthetic experiments, the model should be further validated**\n\na. It would be helpful if the authors should also provide goodness-of-fit tests (Kolmogorov-Smirnov), see [7], [3]\n\nb. clarification: in the synthetic examples, is the VAE-based GP approximation used? In either case, showing similar results for both cases could help justify the use of a VAE-based GP approximation in the point process context.\n\nc. In figure 4, it would help if authors also provide the mean values of the inferred parameters. Ideally, a LGCP-Hawkes should correctly recover either a pure Hawkes (zero covariance in the GP?) or a LGCP (zero excitation coefficients). However, there is a discrepancy in the error in these cases.\n\nd. a visualization of the GP background rate could provide further insights.\n\n**(5)In the gunshot experiments, some details are missing**\n\na. Can the authors provide inference time?\n\nb. I also think the authors should provide the loglikelihood (not only specific parameters) to demonstrate convergence of their inference algorithm\n\nc. The authors use truncated normal priors. Gamma or exponential priors are also commonly used. Demonstrating sensitivity wrt to the prior type would also strengthen the paper.\n\n\n\n\n\n\n[1] Chen, Ricky TQ, Brandon Amos, and Maximilian Nickel. \"Neural Spatio-temporal point processes.\n\n[2] Malem-Shinitski N, Ojeda C, Opper M. Flexible Temporal Point Processes Modeling with Nonlinear Hawkes Processes with Gaussian Processes Excitations and Inhibitions.\n\n[3] Apostolopoulou, I., Linderman, S., Miller, K., & Dubrawski, A. (2019). Mutually regressive point processes. Advances in Neural Information Processing Systems, 32.\n\n[4] RUBIN, Izhak. Regular point processes and their detection. IEEE Transactions on Information Theory, 1972, 18.5: 547-557.\n\n[5]. G. Hawkes and D. Oakes, “A cluster process representation of a self-exciting process,”\nJournal of Applied Probability, vol. 11, no. 3, pp. 493–503, 1974.\n\n[6] RASMUSSEN, Jakob Gulddahl. Bayesian inference for Hawkes processes. Methodology and Computing in Applied Probability, 2013, 15.3: 623-642.\n\n[7] Emery N Brown, Riccardo Barbieri, Valerie Ventura, Robert E Kass, and Loren M Frank. The time- rescaling theorem and its application to neural spike train data analysis. Neural computation, 14(2):325–346, 2002.\n\n", "broader_impact_concerns": "no concerns", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "HenyIsLw8l", "reviewer": "Reviewer_tzF9", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n(1) The idea and the description of the proposed method are clear. The derivation of the proposed method is easy to follow.\n\nWeaknesses:\n(1) The novelty of the proposed method is limited. The proposed Hawkes-LGCP model is a simple combination of two existing models (i.e., the LGCP model for the exogenous part and the Hawkes model for the endogenous part). The MCMC-based inference is well known as well. The simulation method is a straightforward application of the existing branch processing-based simulation method. In summary, in the aspect of methodology, the contribution of this submission is incremental. \n\n(2) The experimental part is too weak. (a) The baselines include Poisson, classic Hawkes, and LGCP. It is natural that the combination of Hawkes and LGCP (i.e., the proposed method) owns better capacity and performance. However, whether the proposed method can outperform other spatiotemporal point process (STPP) models, e.g., nonlinear Hawkes, self-correcting process, and neural network-based PP models, is not verified. (b) The authors claimed that using MCMC-based inference can reduce the complexity of the learning process. However, the runtime of the proposed method is not tested. Compared to other learning methods, e.g., variational inference, the advantage of the proposed method is not convincing. (c) In the aspect of evaluation, only RMSE is applied. The other commonly-used metrics, e.g., the log-likelihood of testing data, are ignored.\n\n(3) In the aspect of methodology, the proposed model did not consider high-dimensional features of events (which is common in practice). Additionally, for multi-variate STPP cases, especially, those high-dimensional ones, the efficiency of the MCMC-based learning method is questionable. The limitations of the current model and its learning algorithm are obvious and hard to solve.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "In this submission, the authors proposed an improved spatiotemporal Hawkes process, in which the extrinsic triggering part is modeled by a log-Gaussian Cox process (LGCP). The authors developed an MCMC-based inference method to reduce the computational complexity of the learning process. A cluster generation-based simulation method is developed. This simulation method is a natural extension of the branching process-based simulation method for the classic Hawkes process. Experiments on synthetic and real-world spatiotemporal data show that compared to Hawkes, Poisson, and LGCP, the proposed Hawkes-LGCP model has better performance.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "No", "audience_explanation": "", "requested_changes": "(1) According to the weaknesses above, more state-of-the-art methods should be considered as baselines, and more analytic experiments and more evaluation criteria should be considered. \n\n(2) The advantages of the proposed model and its MCMC-based inference should be demonstrated.\n", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: I am happy to recommend acceptance of this paper, concurring with the opinion of the three reviewers. The paper presents a new approach for point process modeling based on spatiotemporal Hawkes processes with an underlying log-Gaussian cox process. While this does combine well-studied components, the combination proves clearly useful and the proposed inference algorithm advances the literature in this area. The changes made in response to the reviewers has addressed their remaining concerns regarding run-time and comparisons, and the re-write has improved the overall presentation.

The reviewers still believe it would be worthwhile comparing against Chen et al, since some of the datasets they use seem comparable in size with the gunshot data used in this paper. However, while I encourage the authors to include this, I do not think it should be a prerequisite for acceptance, since the paper does include comparison with the more recent Zhou et al paper (which itself shows improvement over the Chen et al paper).

Decision: Accept (poster)