Point process
Encyclopedia
In statistics
and probability theory
, a point process is a type of random process for which any one realisation consists of a set of isolated points either in time or geographical space, or in even more general spaces. For example, the occurrence of lightning strikes might be considered as a point process in both time and geographical space if each is recorded according to its location in time and space.
Point processes are well studied objects in probability theory
and the subject of powerful tools in statistics
for modeling and analyzing spatial data, which is of interest in such diverse disciplines as forestry, plant ecology, epidemiology, geography, seismology, materials science, astronomy, telecommunications, economics and others.
Point processes on the real line form an important special case that is particularly amenable to study, because the different points are ordered in a natural way, and the whole point process can be described completely by the (random) intervals between the points. These point processes are frequently used as models for random events in time, such as the arrival of customers in a queue (queueing theory
), of impulses in a neuron (computational neuroscience
), particles in a Geiger counter
, location of radio stations in a telecommunication network or of searches on the world-wide web.
whose values are "point patterns" on a set S. While in the exact mathematical definition a point pattern is specified as a locally finite counting measure
, it is sufficient for more applied purposes to think of a point pattern as a countable
subset of S that has no limit point
s.
second countable
Hausdorff space
equipped with its Borel σ-algebra B(S). Write
for the set of locally finite
counting measures on S and
for the smallest σ-algebra
on
that renders all the point counts
for relatively compact sets B in B measurable.
A point process on S is a measurable map
from a probability space
to the measurable space 
.
By this definition, a point process is a special case of a random measure.
The most common example for the state space S is the Euclidean space Rn or a subset thereof, where a particularly interesting special case is given by the real half-line [0,∞). However, point processes are not limited to these examples and may among other things also be used if the points are themselves compact subsets of Rn, in which case ξ is usually referred to as a particle process.
It has been noted that the term point process is not a very good one if S is not a subset of the real line, as it might suggest that ξ is a stochastic process
. However, the term is well established and uncontested even in the general case.
where
denotes the Dirac measure, N is a integer-valued random variable and 
are random elements of S. If 
's are almost surely
distinct (or equivalently, almost surely
for all 
), then the point process is known as simple.
of a point process N is a
map from the set of all positive valued functions f on the state space of N, to
defined as follows:
They play a similar role as the characteristic functions
for random variable
. One important theorem says that: two point processes have the same law iff their Laplace functionals are equal.
th power of a point process, 
is defined on the product space 
as follows :
By monotone class theorem, this uniquely defines the product measure on
The expectation 
is called
the
th moment measure. The first moment measure is the mean measure.
Let
. The joint intensities of a point process 
w.r.t. the Lebesgue measure
are functions
such that for any disjoint bounded Borel subsets 
Joint intensities do not always exist for point processes. Given that moments
of a random variable
determine the random variable in many cases, a similar result is to be expected for joint intensities. Indeed, this has been shown in many cases.
is said to be stationary if 
has the same distribution as 
for all 
For a stationary point process, the mean measure 
for some constant 
and where 
stands for the Lebesgue measure. This 
is called the intensity of the point process. A stationary point process on 
has almost surely either 0 or an infinite number of points in total. For more on stationary point processes and random measure, refer to Chapter 12 of Daley & Vere-Jones. It is to be noted that stationarity has been defined and studied for point processes in more general spaces than 
.
. A Poisson process on the line can be characterised by two properties : the number of points (or events) in disjoint intervals are independent and have a Poisson distribution
. A Poisson point process can also be defined using these two properties. Namely, we say that a point process
is a Poisson point process if the following two conditions hold
1)
are independent for disjoint subsets
2) For any bounded subset
, 
has a Poisson distribution
with parameter
where
denotes the Lebesgue measure
.
The two conditions can be combined together and written as follows : For any disjoint bounded subsets
and non-negative integers 
we have that
The constant
is called the intensity of the Poisson point process. Note that the Poisson point process is characterised by the single parameter 
It is a simple, stationary point process.
To be more specific one calls the above point process, an homogeneous Poisson point process. An inhomogeneous Poisson point process is defined as above but by replacing
with 
where 
is a non-negative function on 
. These generalise the Poisson point process in that we use random measures in place of
. More formally, let 
be a random measure. A Cox point process driven by the random measure 
is the point process 
with the following two properties :
It is easy to see that Poisson point process (homogeneous and inhomogeneous) follow as special cases of Cox point processes. The mean measure of a Cox point process is
and thus in the special case of a Poisson point process, it is 
For a Cox point process,
is called the intensity measure. Further, if 
has a (random) density (Radon-Nikodyn derivative
)
i.e.,
then
is called the intensity field of the Cox point process. Stationarity of the intensity measures or intensity fields imply the stationarity of the corresponding Cox point processes.
There have been many specific classes of Cox point processes that have been studied in detail such as:
By Jensen's inequality, one can verify that Cox point processes satisfy the following inequality: for all bounded Borel subsets
,
where
stands for a Poisson point process with intensity measure 
Thus points are distributed with greater variability in a Cox point process compared to a Poisson point process. This is sometimes called clustering or attractive property of the Cox point process.
, random matrix theory, and combinatorics
, is that of determinantal point process
es.
Point processes on R+ are typically described by giving the sequence of their (random) inter-event times (T1, T2,...), from which the actual sequence (X1, X2,...) of event times can be obtained as
If the inter-event times are independent and identically distributed, the point process obtained is called a renewal process
.
where Ht denotes the history of event times preceding time t.
in the 
-dimensional Euclidean space 
is defined as:
where
is the ball centered at 
of a radius 
,
and
denotes the information of the point process 
outside
.
The need to use point processes to model these kinds of data lies in their inherent spatial structure. Accordingly, a first question of interest is often whether the given data exhibit complete spatial randomness
(i.e. are a realization of a spatial Poisson process
) as opposed to exhibiting either spatial aggregation or spatial inhibition.
In contrast, many datasets considered in classical multivariate statistics
consist of indepently generated datapoints that may be governed by one or several covariates (typically non-spatial).
Apart from the applications in spatial statistics, point processes are one of the fundamental objects in stochastic geometry
. Research has also focussed extensively on various models built on point processes such as Voronoi Tessellations, Random geometric graphs, Boolean model etc.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
and probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
, a point process is a type of random process for which any one realisation consists of a set of isolated points either in time or geographical space, or in even more general spaces. For example, the occurrence of lightning strikes might be considered as a point process in both time and geographical space if each is recorded according to its location in time and space.
Point processes are well studied objects in probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
and the subject of powerful tools in statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
for modeling and analyzing spatial data, which is of interest in such diverse disciplines as forestry, plant ecology, epidemiology, geography, seismology, materials science, astronomy, telecommunications, economics and others.
Point processes on the real line form an important special case that is particularly amenable to study, because the different points are ordered in a natural way, and the whole point process can be described completely by the (random) intervals between the points. These point processes are frequently used as models for random events in time, such as the arrival of customers in a queue (queueing theory
Queueing theory
Queueing theory is the mathematical study of waiting lines, or queues. The theory enables mathematical analysis of several related processes, including arriving at the queue, waiting in the queue , and being served at the front of the queue...
), of impulses in a neuron (computational neuroscience
Computational neuroscience
Computational neuroscience is the study of brain function in terms of the information processing properties of the structures that make up the nervous system...
), particles in a Geiger counter
Geiger counter
A Geiger counter, also called a Geiger–Müller counter, is a type of particle detector that measures ionizing radiation. They detect the emission of nuclear radiation: alpha particles, beta particles or gamma rays. A Geiger counter detects radiation by ionization produced in a low-pressure gas in a...
, location of radio stations in a telecommunication network or of searches on the world-wide web.
General point process theory
In mathematics, a point process is a random elementRandom element
In probability theory, random element is a generalization of the concept of random variable to more complicated spaces than the simple real line...
whose values are "point patterns" on a set S. While in the exact mathematical definition a point pattern is specified as a locally finite counting measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
, it is sufficient for more applied purposes to think of a point pattern as a countable
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
subset of S that has no limit point
Limit point
In mathematics, a limit point of a set S in a topological space X is a point x in X that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. Note that x does not have to be an element of S...
s.
Definition
Let S be locally compactLocally compact space
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.-Formal definition:...
second countable
Second-countable space
In topology, a second-countable space, also called a completely separable space, is a topological space satisfying the second axiom of countability. A space is said to be second-countable if its topology has a countable base...
Hausdorff space
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
equipped with its Borel σ-algebra B(S). Write
for the set of locally finiteLocally finite measure
In mathematics, a locally finite measure is a measure for which every point of the measure space has a neighbourhood of finite measure.-Definition:...
counting measures on S and
for the smallest σ-algebraSigma-algebra
In mathematics, a σ-algebra is a technical concept for a collection of sets satisfying certain properties. The main use of σ-algebras is in the definition of measures; specifically, the collection of sets over which a measure is defined is a σ-algebra...
on
that renders all the point countsfor relatively compact sets B in B measurable.
A point process on S is a measurable map
from a probability space
Probability space
In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind...
to the measurable space .By this definition, a point process is a special case of a random measure.
The most common example for the state space S is the Euclidean space Rn or a subset thereof, where a particularly interesting special case is given by the real half-line [0,∞). However, point processes are not limited to these examples and may among other things also be used if the points are themselves compact subsets of Rn, in which case ξ is usually referred to as a particle process.
It has been noted that the term point process is not a very good one if S is not a subset of the real line, as it might suggest that ξ is a stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
. However, the term is well established and uncontested even in the general case.
Representation
Every point process ξ can be represented aswhere
denotes the Dirac measure, N is a integer-valued random variable and are random elements of S. If 's are almost surelyAlmost surely
In probability theory, one says that an event happens almost surely if it happens with probability one. The concept is analogous to the concept of "almost everywhere" in measure theory...
distinct (or equivalently, almost surely
for all ), then the point process is known as simple.Expectation measure
The expectation measure Eξ (also known as mean measure) of a point process ξ is a measure on S that assigns to every Borel subset B of S the expected number of points of ξ in B. That is,Laplace functional
The Laplace functional of a point process N is amap from the set of all positive valued functions f on the state space of N, to
defined as follows:They play a similar role as the characteristic functions
Characteristic function (probability theory)
In probability theory and statistics, the characteristic function of any random variable completely defines its probability distribution. Thus it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative...
for random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
. One important theorem says that: two point processes have the same law iff their Laplace functionals are equal.
Moment measures
Theth power of a point process, is defined on the product space as follows :By monotone class theorem, this uniquely defines the product measure on
The expectation is calledthe
th moment measure. The first moment measure is the mean measure.Let
. The joint intensities of a point process w.r.t. the Lebesgue measureLebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
are functions
such that for any disjoint bounded Borel subsets Joint intensities do not always exist for point processes. Given that moments
Moment (mathematics)
In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...
of a random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
determine the random variable in many cases, a similar result is to be expected for joint intensities. Indeed, this has been shown in many cases.
Stationarity
A point process is said to be stationary if has the same distribution as for all For a stationary point process, the mean measure for some constant and where stands for the Lebesgue measure. This is called the intensity of the point process. A stationary point process on has almost surely either 0 or an infinite number of points in total. For more on stationary point processes and random measure, refer to Chapter 12 of Daley & Vere-Jones. It is to be noted that stationarity has been defined and studied for point processes in more general spaces than .Examples of point processes
We shall see some examples of point processes inPoisson point process
The simplest and ubiquitous example of a point process is the Poisson point process, which is a spatial generalisation of the Poisson processPoisson process
A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...
. A Poisson process on the line can be characterised by two properties : the number of points (or events) in disjoint intervals are independent and have a Poisson distribution
Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...
. A Poisson point process can also be defined using these two properties. Namely, we say that a point process
is a Poisson point process if the following two conditions hold1)
are independent for disjoint subsets2) For any bounded subset
, has a Poisson distributionPoisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...
with parameter
where denotes the Lebesgue measureLebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
.
The two conditions can be combined together and written as follows : For any disjoint bounded subsets
and non-negative integers we have thatThe constant
is called the intensity of the Poisson point process. Note that the Poisson point process is characterised by the single parameter It is a simple, stationary point process.To be more specific one calls the above point process, an homogeneous Poisson point process. An inhomogeneous Poisson point process is defined as above but by replacing
with where is a non-negative function on Cox point process
This class of point processes are named after Sir David CoxDavid Cox (statistician)
Sir David Roxbee Cox FRS is a prominent British statistician.-Early years:Cox studied mathematics at St. John's College, Cambridge and obtained his PhD from the University of Leeds in 1949, advised by Henry Daniels and Bernard Welch.-Career:He was employed from 1944 to 1946 at the Royal Aircraft...
. These generalise the Poisson point process in that we use random measures in place of
. More formally, let be a random measure. A Cox point process driven by the random measure is the point process with the following two properties :- Given
, 
is Poisson distributed with parameter 
for any bounded subset 
- For any finite collection of disjoint subsets
and conditioned on 
we have that 
are independent.
It is easy to see that Poisson point process (homogeneous and inhomogeneous) follow as special cases of Cox point processes. The mean measure of a Cox point process is
and thus in the special case of a Poisson point process, it is For a Cox point process,
is called the intensity measure. Further, if has a (random) density (Radon-Nikodyn derivativeRadon–Nikodym theorem
In mathematics, the Radon–Nikodym theorem is a result in measure theory that states that, given a measurable space , if a σ-finite measure ν on is absolutely continuous with respect to a σ-finite measure μ on , then there is a measurable function f on X and taking values in [0,∞), such that\nu =...
)
i.e.,then
is called the intensity field of the Cox point process. Stationarity of the intensity measures or intensity fields imply the stationarity of the corresponding Cox point processes.There have been many specific classes of Cox point processes that have been studied in detail such as:
- Log Gaussian Cox point processes:
for a Gaussian random field 
- Shot noise Cox point processes:,
for a Poisson point process 
and kernel 
- Generalised shot noise Cox point processes:
for a point process 
and kernel 
- Lévy based Cox point processes:
for a Lévy basis 
and kernel 
, and - Permanental Cox point processes:
for k independent Gaussian random fields 
's - Sigmoidal Gaussian Cox point processes:
for a Gaussian random field 
and random 
By Jensen's inequality, one can verify that Cox point processes satisfy the following inequality: for all bounded Borel subsets
,where
stands for a Poisson point process with intensity measure Thus points are distributed with greater variability in a Cox point process compared to a Poisson point process. This is sometimes called clustering or attractive property of the Cox point process.Determinantal point processes
An important class of point processes, with applications to physicsPhysics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, random matrix theory, and combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
, is that of determinantal point process
Determinantal point process
In mathematics, a determinantal point process is a stochastic point process, the probability distribution of which is characterized as a determinant of some function. Such processes arise as important tools in random matrix theory, combinatorics, and physics....
es.
Point processes on the real half-line
Historically the first point processes that were studied had the real half line R+ = [0,∞) as their state space, which in this context is usually interpreted as time. These studies were motivated by the wish to model telecommunication systems, in which the points represented events in time, such as calls to a telephone exchange.Point processes on R+ are typically described by giving the sequence of their (random) inter-event times (T1, T2,...), from which the actual sequence (X1, X2,...) of event times can be obtained as
If the inter-event times are independent and identically distributed, the point process obtained is called a renewal process
Renewal theory
Renewal theory is the branch of probability theory that generalizes Poisson processes for arbitrary holding times. Applications include calculating the expected time for a monkey who is randomly tapping at a keyboard to type the word Macbeth and comparing the long-term benefits of different...
.
Conditional intensity function
The conditional intensity function of a point process on the real half-line is a function λ(t|Ht) defined aswhere Ht denotes the history of event times preceding time t.
Papangelou intensity function
The Papangelou intensity function of a point process in the -dimensional Euclidean space is defined as:
where
is the ball centered at of a radius ,and
denotes the information of the point process outside
.Point processes in spatial statistics
The analysis of point pattern data in a compact subset S of Rn is a major object of study within spatial statistics. Such data appear in a broad range of disciplines, amongst which are- forestry and plant ecology (positions of trees or plants in general)
- epidemiology (home locations of infected patients)
- zoology (burrows or nests of animals)
- geography (positions of human settlements, towns or cities)
- seismology (epicenters of earthquakes)
- materials science (positions of defects in industrial materials)
- astronomy (locations of stars or galaxies)
- computational neuroscience (spikes of neurons).
The need to use point processes to model these kinds of data lies in their inherent spatial structure. Accordingly, a first question of interest is often whether the given data exhibit complete spatial randomness
Complete spatial randomness
Complete spatial randomness describes a point process whereby point events occur within a given study area in a completely random fashion. Such a process is often modeled using only one parameter, i.e. the density of points, \rho within the defined area...
(i.e. are a realization of a spatial Poisson process
Poisson process
A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...
) as opposed to exhibiting either spatial aggregation or spatial inhibition.
In contrast, many datasets considered in classical multivariate statistics
Multivariate statistics
Multivariate statistics is a form of statistics encompassing the simultaneous observation and analysis of more than one statistical variable. The application of multivariate statistics is multivariate analysis...
consist of indepently generated datapoints that may be governed by one or several covariates (typically non-spatial).
Apart from the applications in spatial statistics, point processes are one of the fundamental objects in stochastic geometry
Stochastic geometry
In mathematics, stochastic geometry is the study of random spatial patterns. At the heart of the subject lies the study of random point patterns...
. Research has also focussed extensively on various models built on point processes such as Voronoi Tessellations, Random geometric graphs, Boolean model etc.