AN INTRODUCTION TO SPATIAL-TEMPORAL POINT PROCESSES
ALEXANDRE SILVA, MARISA PEREIRA, AND RICARDO SIMÕES
Instituto Superior Técnico
Universidade de Lisboa
A BSTRACT. This paper gives an introduction to the concepts of spatial and spatial-temporal point processes.
A brief review on one-dimensional (time) point process is given in order to present a natural generalization of
this concept to an abstract space S, establishing a comparison between temporal, spatial and spatial-temporal
models and their properties. Here, it is introduced the concept of intensity, second moment density, the natural analogues of the mean and auto-covariance functions, and the pair correlation. Some visualizations and
applications, such as the Poisson Cluster Process are also presented.
C ONTENTS
1. Introduction
2. Spatial Point Processes
2.1. Properties of Spatial Point Processes
3. Spatial-Temporal Point Processes
3.1. Properties of Spatial-Temporal Point Process
4. Applications
References
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1. I NTRODUCTION
Stochastic Processes are mathematical random models that evolve in time. Two of the most studied so
far, Bernoulli and Poisson Processes, are particular examples of a wide spectrum of models called OneDimensional (Time) Point Processes. These ones model a sequence of random times of when a particular
event occurs, with different assumptions depending on the problem at hand. However, they reveal themselves
insufficient when there is a need to consider a spatial distribution for these random points, giving birth to a
wide variety of models, called Spatial Point Processes. These ones model a random pattern of points in a
n-dimensional space, where n ≥ 2. The randomness associated with this kind of processes comes not only
from the number of points, but also their location. However, not all models can be seen as purely spatial or
temporal point processes. It is of most interest to record both the locations and the times of the occurrence
of certain events. These models are known as Spatial-Temporal Point Processes.
2. S PATIAL P OINT P ROCESSES
Point Processes are a collection of mathematical points randomly located on a mathematical space. When
one studies one-dimensional (time) point processes, there are several ways to approach these problems: we
may study the arrival times, the inter-arrival times, the cumulative counting process or, alternatively,
the interval counts. However, at the spatial level, there is no natural ordering of the points, so that there
is no natural analogue of the inter-arrival times nor of the counting process. Instead, the most useful way
to handle a spatial point process, assumed simple, without atoms and finite1, is to generalize the interval
counts to region counts. Hereinafter, let N(B) be the number of points falling in some bounded region
B ⊂ Rn . We start by introducing a simple generalization of the Poisson Process:
1Roughly speaking, we are focused on an arbitrary set where two events can’t overlap in space and we observe them with
probability 0 at any predefined location. See for more information [4].
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ALEXANDRE SILVA, MARISA PEREIRA, AND RICARDO SIMÕES
Definition 2.1. The Homogeneous Spatial Poisson Process, with intensity λ > 0 , is a point process in Rn
such that:
(1) for every bounded closed set B, N(B) follows a Poisson distribution with mean λ µn (B)2;
(2) if B1 , . . . , Bm are disjoint sets, then N (B1 ) , . . . , N (Bm ) are mutually independent.
One naturally asks if this definition can not be further generalized to a more general space S. For most
models this often suffices, but Rn is not always the most natural space to consider. For example, since the
Earth is approximately spherical, we may perceive it as S2 . Also, given that N represents some kind of
measure, it is more than normal that we can generalize these processes to any type of measure defined over
the space S. However, given a subset B ⊂ S, it is clear that can take different values. Because of this, we need
to introduce the concept of random measure. A more precise definition is introduced in [6], but informally
speaking, a random measure is a measure valued random element. It defines a measure in the sense that it
gives meaning of size (quantifiability) to a given object. The random element part comes for the fact that
given a subset of the space, it is possible to obtain a range of different values for the same subset.
Definition 2.2. Let S a locally compact metric space3, and Λ a random measure on S. A Spatial Point
Process X is a collection of points randomly located in S, together with a random measure Λ. The Homogeneous Spatial Poisson Process on S with intensity measure Λ is a point process on S such that:
(1) for every compact4 set B ⊂ S, the count N(B) has a Poisson distribution with mean Λ(B).
(2) if B1 , . . . , Bm are disjoint compact sets, then N (B1 ) , . . . , N (Bm ) are mutually independent.
Example 2.3. (Homogeneous Spatial Poisson process on the sphere). Take S to be the surface of the unit
ball in R3 and Λ = λ µ2 , where λ > 0 and µ2 is the usual area measure on S equal to 4π. This yields the
homogeneous Poisson point process on the unit sphere, with intensity λ . The total number of points N(S)
is a Poisson random variable with mean Λ(S) = λ µ2 (S) = 4πλ .
2.1. Properties of Spatial Point Processes. Now we describe, for point processes, the equivalent ideas
of expected value and variance of a random variable. We will focus in the case S = Rn , even though this
is generalized to LCSH spaces. The intensity or first moment is the analogue of the expected value of a
random variable, and can be defined as follows:
Definition 2.4. Let X be a point process on S. Writing ν(B) = E [N(B)], for some B ⊂ S, it defines a measure
ν on S, called the intensity measure of X, provided that ν(B) < ∞ for all compact B. If it satisfies
Z
ν(B) =
λ (s)ds
B
for some function λ , we call λ the intensity function of X. Informally, we can think of the intensity function
as the mean number of events per unit of measure5. That is
E[N(B(s, r))]
.
r→0
r
The second moment measure is related to the variance or covariance of random variables.
λ (s) = lim
Definition 2.5. Let us define the intensity measure ν2 as ν2 (A × B) = E [N(A)N(B)]. ν2 is called the second
moment measure of X. A point process X on S is said to have second moment density λ2 if
Z
ν2 (B) =
B
λ2 (s, u)ds du
for any compact B ⊂ S × S. Informally, this density gives the joint probability that there will be points of X
at a given location s and u. So, we can think of λ2 as
E[N(B(s, r1 ))N(B(u, r2 ))]
λ2 (s, u) = lim
r1 ,r2 →0
r1 r2
2µ is the usual Lebesgue measure in Rn .
n
3In short, each small portion of the space looks like a compact space, and has a metric attached to it. However, point processes
theory is already treated in locally compact second-countable Hausdorff (LCSH) spaces (metrizable).
4natural extension of bounded and closed set in Rn .
5B(s, r) = {a ∈ S : ∥s − a∥ ≤ r}.
AN INTRODUCTION TO SPATIAL-TEMPORAL POINT PROCESSES
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Since cov [N (B1 ) , N (B2 )] = E [N (B1 ) N (B2 )] − E [N (B1 )] E [N (B2 )], the second moment measure contains all information about the variances and covariances of the variables N(B). We also introduce the following property derived from the second-moment, which have many applications in the statistical analysis
and inference of point processes.
Definition 2.6. Suppose X is a point process on S which has an intensity function λ (s) and a second moment
density λ2 (s, u). Then we define the pair correlation function of X by
ρ2 (s, u) =
λ2 (s, u)
.
λ (s)λ (u)
Note that the pair correlation function for a homogeneous Poisson process is 1. So we can interpret this
in the following way: values far away from 1 indicate that is less likely that a pair of events will occur at the
specified locations than in a homogeneous Poisson process with the same intensity.
At last, we introduce some statistical properties that these processes may have. Like the processes we
are used to study, the concept of stationary arises, being a way to encode the idea that no point in the point
process is special.
Definition 2.7. A spatial point process X in S is called:
• stationary if, for any fixed vector s ∈ S, the distribution of the shifted point process X + s is the same
to the distribution of X.
• first and second-order stationary if λ (s) ≡ λ and λ2 (s, s′ ) = λ2 (s − s′ , 0).
• (spatially) isotropic if, for any rotation r around the origin, the rotated point process rX = {rx : x ∈
X} has the same distribution as X.
3. S PATIAL -T EMPORAL P OINT P ROCESSES
We are now able to present the following definition:
Definition 3.1. A Spatial-Temporal Point Process X is a collection of points randomly located on a locally
compact metric space S × T , together with a random measure Λ, such that Λ : S × T → Z+
0 . Given A ⊂ S
and B ⊂ T , Λ(A × B) usually represents the number of points falling in region A during time B.
3.1. Properties of Spatial-Temporal Point Process. Analogously to the Spatial Point Processes, we resume next the definitions and statistical properties inherent to this process.
Definition 3.2. Let X be a spatial-temporal point process in S × T . Then:
(1) Provided that the limit exists, we define the intensity function as6
E[N(B((s,t), r, τ)]
.
λ (s,t) = lim
r,τ→0
rτ
(2) The second moment measure can be expressed as:
E[N(B((s1 ,t1 ), r1 , τ1 ))N(B((s2 ,t2 ), r2 , τ2 ))]
λ2 ((s1 ,t1 ), (s2 ,t2 )) =
lim
r1 ,τ1 ,r2 ,τ2 ,→0
r1 τ2 r2 τ2
(3) it is called stationary if, for any fixed vector (s,t) ∈ S ×T , the distribution of the shifted point process
X + (s,t) is the same to the distribution of X.
(4) first and second-order stationary:
• in space, if: λ (s,t) ≡ λ (t) and λ2 ((s,t), (s′ ,t)) = λ2 (s − s′ ,t).
• in time, if: λ (s,t) ≡ λ (s) and λ2 ((s,t), (s,t ′ )) = λ2 (s,t − t ′ ).
• in both space and time, if: λ (s,t) = λ and λ2 ((s,t), (s′ ,t ′ )) = λ2 (s − s′ ,t − t ′ ).
(5) it is (spatially) isotropic, if for all t ∈ T , the result in Definition 2.7 is verified.
(6) The pair correlation function can be expressed as:
λ2 ((s1 ,t1 ), (s2 ,t2 ))
ρ2 ((s1 ,t1 ), (s2 ,t2 )) =
λ (s1 ,t1 )λ (s2 ,t2 )
.
6B ((s,t), r, τ) = B(s, r) × [t − τ,t + τ]
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ALEXANDRE SILVA, MARISA PEREIRA, AND RICARDO SIMÕES
All the interpretations are the same as before, with the single difference that things now happen also in a
time window. We also introduce the concept of first order separability.
Definition 3.3. A spatial-temporal point process X is first-order separable if its intensity λ (s,t) can be
factorized as:
λ (s,t) = µ(s)η(t), ∀(s,t) ∈ S × T.
Other separability orders can be defined, as in [5], so we left it as for the readers curiosity. At last, and
for the sake of the examples, we present the Poisson cluster process. This model is a Poisson process in
which each point of a point process is replaced by a random finite set of points, called the cluster associated
and the original point is considered as the cluster center. The process is referred as a Neyman-Scott process,
when the points in the cluster are independently and identically distributed of the cluster center.
Definition 3.4. A spatial-temporal Poisson cluster process can be defined as:
(1) The Parent process is a Poisson process with intensity λ p (s,t).
(2) Each parent has a random number Nc of offspring, from a Poisson distribution with mean λc and
independently distributed for each parent.
(3) Related to a probability density function f(·) on Rn × R+
0 , we have the positions and times of the
offspring independently and identically distributed relative to their parents.
(4) The final process is composed only by offspring.
4. A PPLICATIONS
Having introduced the fundamentals of spatial and spatial-temporal Point Processes, we will now present
some simulations using the functions introduced in the paper in study, from the stpp package from R. They
allow the user to generate very simple, dynamic animations that grant us the capacity to observe particular
moments in time to extract valuable insights, or to even estimate parameters of space-time processes. These
examples are based on the paper [3].
Example 4.1. Visualization of Foot-and-Mouth Outbreak in Cumbria (United Kingdom, 2001)
The first example is related to the visualization of the Foot-and-Mouth Outbreak that happened in the UK,
in 2001. The dataset fmd ([2]) contains items characterized by the location on the map of Cumbria where
the case was reported, and a temporal reference indicating when that case was reported.
We start by generating a static visualization of the spatial-temporal data using the plot.stpp function (see
Figure 1). The image on the left is a spatial representation of this outbreak, where time is treated as a mark
(for more information, see Marked Poisson Processes in [1]) and the darker the point, the more recently the
event occurred. If this option is disabled, then an additional plot with the cumulative number of events over
time is produced, as shown in image at the right. The S shape is common in epidemiological events.
(a) plot function with marks
(b) plot function without marks
Figure 1. Visualization of a Spatial-Temporal process with plot.stpp function
AN INTRODUCTION TO SPATIAL-TEMPORAL POINT PROCESSES
5
However, if we want to generate a more dynamical plot in order to focus our study on events that have
occurred until a moment T , the stan function can be used (see Figure 2). Here, two sliders are presented,
the first one controls time and the other one can be set to W units in time, and highlights the points whose
time coordinate t satisfy the interval condition T −W < t < T . The range of these sliders were normalized
for the sake of simplification.
(a) at t=0
(b) at t=0.5
(c) at t=1
Figure 2. Visualization of a Spatio-Temporal process with stan function
Example 4.2. Generation of a Poisson Cluster Process
For this example we will make use of the rpcp function, that allow us to generate realizations of a Poisson Cluster Process, as presented in Definition 3.4. Let us define the parent process intensity function
as λ p (x, y,t) = a · e−4y−2t and the offspring as normally dispersed in both space and time. The constant
a = 2000/((1 − e−4 ) · (1 − e−2 )) is defined as such so that the mean number of points is 122. Below, in
Figure 3, we can find the realization process in three sequential time-steps.
(a) at t=0
(b) at t=0.5
(c) at t=1
Figure 3. Visualization of a Poisson Cluster process
Example 4.3. Estimation of Pair Correlation Function
For this particular example, we decided to use another function of the stpp package, the PCFhat, that allows
the estimation of the pair correlation function of a process. The dataset of the first example will again be
used. In order to proceed, we first need estimate the density function of the underlying process. For that, and
in order to simplify the problem, we will assume that it is stationary and first-order separable. We estimate
each component using the stats package. After this, we estimate the pair correlation function using the
following formula:
n
1
1 ks u − si − s j kt v − ti − t j
ρ̂2 (u, v) =
∑ ∑ wi j vi j
|S × T | i=1
λ (si ,ti ) λ (s j ,t j )
j̸=i
where ks and kt are kernels functions and wi j and vi j are edge corrections. The results obtained are shown in
figure 4. (for more information on this package and the estimator, see [3]). The horizontal axis represents
the distance (in lags between two points) and the vertical axis exhibits the time (in this case, in days, and
also in time lags). This pair correlation function shows that the interactions among the points are intense for
small values of u and v and for larger values they seem to vanish. They tend to stabilize around a value of 1.
Therefore, for large values of u and v this can be seen as a Spatial-Temporal Poisson Process.
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ALEXANDRE SILVA, MARISA PEREIRA, AND RICARDO SIMÕES
Figure 4. Estimation of the PCF on fmd data
R EFERENCES
[1] Adrian Baddeley, Imre Bárány, and Rolf Schneider. “Spatial point processes and their applications”.
In: Stochastic Geometry: Lectures Given at the CIME Summer School Held in Martina Franca, Italy,
September 13–18, 2004 (2007), pp. 1–75.
[2] Peter Diggle, Barry Rowlingson, and Ting-li Su. “Point process methodology for on-line spatio-temporal
disease surveillance”. In: Environmetrics: The official journal of the International Environmetrics Society 16.5 (2005), pp. 423–434.
[3] Edith Gabriel, Barry S Rowlingson, and Peter J Diggle. “stpp: an R package for plotting, simulating
and analyzing Spatio-Temporal Point Patterns”. In: Journal of Statistical Software 53 (2013), pp. 1–29.
[4] Charles J. Geyer. URL: https://www.stat.umn.edu/geyer/8501/.
[5] Jonatan A. González et al. “Spatio-temporal point process statistics: A review”. In: Spatial Statistics
18 (2016), pp. 505–544. ISSN: 2211-6753. DOI: https://doi.org/10.1016/j.spasta.2016.10.
002. URL: https://www.sciencedirect.com/science/article/pii/S2211675316301130.
[6] Olav Kallenberg. Random measures. Springer, 1976.
Email address: alexandremcsilva@tecnico.ulisboa.pt
Email address: marisampereira@tecnico.ulisboa.pt
Email address: ricardo.fernandes.simoes@tecnico.ulisboa.pt