Poisson point process


In probability theory, statistics and related fields, a Poisson point process is a type of mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one another. The process's name derives from the fact that the number of points in any given finite region follows a Poisson distribution. The process and the distribution are named after French mathematician Siméon Denis Poisson. The process itself was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and actuarial science.
This point process is used as a mathematical model for seemingly random processes in numerous disciplines including astronomy, biology, ecology, geology, seismology, physics, economics, image processing, and telecommunications.
The Poisson point process is often defined on the real number line, where it can be considered a stochastic process. It is used, for example, in queueing theory to model random events distributed in time, such as the arrival of customers at a store, phone calls at an exchange or occurrence of earthquakes. In the plane, the point process, also known as a spatial Poisson process, can represent the locations of scattered objects such as transmitters in a wireless network, particles colliding into a detector or trees in a forest. The process is often used in mathematical models and in the related fields of spatial point processes, stochastic geometry, spatial statistics and continuum percolation theory.
The point process depends on a single mathematical object, which, depending on the context, may be a constant, a locally integrable function or, in more general settings, a Radon measure. In the first case, the constant, known as the rate or intensity, is the average density of the points in the Poisson process located in some region of space. The resulting point process is called a homogeneous or stationary Poisson point process. In the second case, the point process is called an inhomogeneous or nonhomogeneous 'Poisson point process', and the average density of points depend on the location of the underlying space of the Poisson point process. The word point is often omitted, but there are other Poisson processes of objects, which, instead of points, consist of more complicated mathematical objects such as lines and polygons, and such processes can be based on the Poisson point process. Both the homogeneous and nonhomogeneous Poisson point processes are particular cases of the generalized renewal process.

Overview of definitions

Depending on the setting, the process has several equivalent definitions as well as definitions of varying generality owing to its many applications and characterizations. The Poisson point process can be defined, studied and used in one dimension, for example, on the real line, where it can be interpreted as a counting process or part of a queueing model; in higher dimensions such as the plane where it plays a role in stochastic geometry and spatial statistics; or on more general mathematical spaces. Consequently, the notation, terminology and level of mathematical rigour used to define and study the Poisson point process and points processes in general vary according to the context.
Despite all this, the Poisson point process has two key properties—the Poisson property and the independence property— that play an essential role in all settings where the Poisson point process is used. The two properties are not logically independent; indeed, the Poisson distribution of point counts implies the independence property, while in the converse direction the assumptions that: the point process is simple, has no fixed atoms, and is a.s. boundedly finite are required.

Poisson distribution of point counts

A Poisson point process is characterized via the Poisson distribution. The Poisson distribution is the probability distribution of a random variable such that the probability that equals is given by:
where denotes factorial and the parameter determines the shape of the distribution.
By definition, a Poisson point process has the property that the number of points in a bounded region of the process's underlying space is a Poisson-distributed random variable.

Complete independence

Consider a collection of disjoint and bounded subregions of the underlying space. By definition, the number of points of a Poisson point process in each bounded subregion will be completely independent of all the others.
This property is known under several names such as complete randomness, complete independence, or independent scattering and is common to all Poisson point processes. In other words, there is a lack of interaction between different regions and the points in general, which motivates the Poisson process being sometimes called a purely or completely random process.

Homogeneous Poisson point process

If a Poisson point process has a parameter of the form, where is Lebesgue measure and is a constant, then the point process is called a homogeneous or stationary Poisson point process. The parameter, called rate or intensity, is related to the expected number of Poisson points existing in some bounded region, where rate is usually used when the underlying space has one dimension. The parameter can be interpreted as the average number of points per some unit of extent such as length, area, volume, or time, depending on the underlying mathematical space, and it is also called the mean density or mean rate; see [|Terminology].

Interpreted as a counting process

The homogeneous Poisson point process, when considered on the positive half-line, can be defined as a counting process, a type of stochastic process, which can be denoted as. A counting process represents the total number of occurrences or events that have happened up to and including time. A counting process is a homogeneous Poisson counting process with rate if it has the following three properties:
The last property implies:
In other words, the probability of the random variable being equal to is given by:
The Poisson counting process can also be defined by stating that the time differences between events of the counting process are exponential variables with mean. The time differences between the events or arrivals are known as interarrival or interoccurrence times.

Interpreted as a point process on the real line

Interpreted as a point process, a Poisson point process can be defined on the real line by considering the number of points of the process in the interval. For the homogeneous Poisson point process on the real line with parameter, the probability of this random number of points, written here as, being equal to some counting number is given by:
For some positive integer, the homogeneous Poisson point process has the finite-dimensional distribution given by:
where the real numbers.
In other words, is a Poisson random variable with mean, where. Furthermore, the number of points in any two disjoint intervals, say, and are independent of each other, and this extends to any finite number of disjoint intervals. In the queueing theory context, one can consider a point existing as an event, but this is different to the word event in the probability theory sense. It follows that is the expected number of arrivals that occur per unit of time.

Key properties

The previous definition has two important features shared by Poisson point processes in general:
  • the number of arrivals in each finite interval has a Poisson distribution;
  • the number of arrivals in disjoint intervals are independent random variables.
Furthermore, it has a third feature related to just the homogeneous Poisson point process:
  • the Poisson distribution of the number of arrivals in each interval only depends on the interval's length.
In other words, for any finite, the random variable is independent of, so it is also called a stationary Poisson process.

Law of large numbers

The quantity can be interpreted as the expected or average number of points occurring in the interval, namely:
where denotes the expectation operator. In other words, the parameter of the Poisson process coincides with the density of points. Furthermore, the homogeneous Poisson point process adheres to its own form of the law of large numbers. More specifically, with probability one:
where denotes the limit of a function, and is expected number of arrivals occurred per unit of time.

Memoryless property

The distance between two consecutive points of a point process on the real line will be an exponential random variable with parameter . This implies that the points have the memoryless property: the existence of one point existing in a finite interval does not affect the probability of other points existing, but this property has no natural equivalence when the Poisson process is defined on a space with higher dimensions.

Orderliness and simplicity

A point process with stationary increments is sometimes said to be orderly or regular if:
where little-o notation is being used. A point process is called a simple point process when the probability of any of its two points coinciding in the same position, on the underlying space, is zero. For point processes in general on the real line, the property of orderliness implies that the process is simple, which is the case for the homogeneous Poisson point process.

Martingale characterization

On the real line, the homogeneous Poisson point process has a connection to the theory of martingales via the following characterization: a point process is the homogeneous Poisson point process if and only if
is a martingale.