In statistics and probability theory, a point process is a collection of mathematical points randomly located on some underlying mathematical space such as the real line, the Cartesian plane, or more abstract spaces. Point processes can be used as mathematical models of phenomena or objects are representable as points in some type of space.
Contents
- General point process theory
- Definition
- Representation
- Expectation measure
- Laplace functional
- Moment measure
- Stationarity
- Examples of point processes
- Poisson point process
- Cox point process
- Determinantal point processes
- Point processes on the real half line
- Conditional intensity function
- Papangelou intensity function
- Point processes in spatial statistics
- References
There are different mathematical interpretations of a point process, such a random counting measure or a random set. Some authors regard a point process and stochastic process as two different objects such that a point process is a random object that arises from or is associated with a stochastic process, though it has been remarked that the difference between point processes and stochastic processes is not clear. Others consider a point process as a stochastic process, where the process is indexed by sets of the underlying space on which it is defined, such as the real line or
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, computational neuroscience, economics and others.
Point processes on the real line form an important special case that is particularly amenable to study, because the 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.
General point process theory
In mathematics, a point process is a random element 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 points.
Definition
Let S be a locally compact second countable Hausdorff space equipped with its Borel σ-algebra B(S). Write
measurable for all relatively compact sets B in B(S).
A point process on S is a measurable map
from a probability 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.
Representation
Every point process ξ can be represented as
where
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
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.
Moment measure
The
By monotone class theorem, this uniquely defines the product measure on
Let
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.
Stationarity
A point process
Examples of point processes
We shall see some examples of point processes in
Poisson point process
The simplest and most ubiquitous example of a point process is the Poisson point process, which is a spatial generalisation of the Poisson process. A Poisson (counting) 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
1)
2) For any bounded subset
The two conditions can be combined together and written as follows : For any disjoint bounded subsets
The constant
Cox point process
A Cox process (named after Sir David Cox) is a generalisation of the Poisson point process, in that we use random measures in place of
- Given
Λ ( ⋅ ) ,ξ ( B ) is Poisson distributed with parameterΛ ( B ) for any bounded subsetB . - For any finite collection of disjoint subsets
B 1 , … , B n Λ ( B 1 ) , … , Λ ( B n ) , we have thatξ ( B 1 ) , … , ξ ( B n ) 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
For a Cox point process,
then
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
Determinantal point processes
An important class of point processes, with applications to physics, random matrix theory, and combinatorics, is that of determinantal point processes.
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.
Conditional intensity function
The conditional intensity function of a point process on the real half-line is a function λ(t | Ht) defined as
where Ht denotes the history of event times preceding time t.
The compensator of a point process, also known as the dual-predictable projection, is the integrated conditional intensity function defined by
Papangelou intensity function
The Papangelou intensity function of a point process
where
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
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 independently 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.