![Poisson process](https://upload.wikimedia.org/wikipedia/commons/6/6e/Poisson_process.svg)
In
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
, statistics and related fields, a Poisson point process is a type of
random
In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wikt:order, order and does not follow an intelligible pattern or combination. Ind ...
mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical p ...
that consists of
points randomly located on a
mathematical space
In mathematics, a space is a set (sometimes called a universe) with some added structure.
While modern mathematics uses many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, ...
with the essential feature that the points occur independently of one another.
The Poisson point process is often called simply the Poisson process, but it is also called a Poisson random measure, Poisson random point field or Poisson point field. This
point process
In statistics and probability theory, a point process or point field is a collection of mathematical points randomly located on a mathematical space such as the real line or Euclidean space. Kallenberg, O. (1986). ''Random Measures'', 4th editio ...
has convenient mathematical properties,
which has led to its being frequently defined in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
and used as a
mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
for seemingly random processes in numerous disciplines such as
astronomy
Astronomy () is a natural science that studies astronomical object, celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and chronology of the Universe, evolution. Objects of interest ...
,
[G. J. Babu and E. D. Feigelson. Spatial point processes in astronomy. ''Journal of statistical planning and inference'', 50(3):311–326, 1996.] biology
Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...
,
[H. G. Othmer, S. R. Dunbar, and W. Alt. Models of dispersal in biological systems. ''Journal of mathematical biology'', 26(3):263–298, 1988.] ecology,
[H. Thompson. Spatial point processes, with applications to ecology. ''Biometrika'', 42(1/2):102–115, 1955.] geology,
[C. B. Connor and B. E. Hill. Three nonhomogeneous poisson models for the probability of basaltic volcanism: application to the yucca mountain region, nevada. ''Journal of Geophysical Research: Solid Earth (1978–2012)'', 100(B6):10107–10125, 1995.] seismology
Seismology (; from Ancient Greek σεισμός (''seismós'') meaning "earthquake" and -λογία (''-logía'') meaning "study of") is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other ...
,
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
,
[J. D. Scargle. Studies in astronomical time series analysis. v. bayesian blocks, a new method to analyze structure in photon counting data. ''The Astrophysical Journal'', 504(1):405, 1998.] economics,
[P. Aghion and P. Howitt. A Model of Growth through Creative Destruction. ''Econometrica'', 60(2). 323–351, 1992.] image processing
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
,
[M. Bertero, P. Boccacci, G. Desidera, and G. Vicidomini. Image deblurring with poisson data: from cells to galaxies. ''Inverse Problems'', 25(12):123006, 2009.] and telecommunications.
[F. Baccelli and B. Błaszczyszyn. ''Stochastic Geometry and Wireless Networks, Volume II- Applications'', volume 4, No 1–2 of ''Foundations and Trends in Networking''. NoW Publishers, 2009.][M. Haenggi, J. Andrews, F. Baccelli, O. Dousse, and M. Franceschetti. Stochastic geometry and random graphs for the analysis and design of wireless networks. ''IEEE JSAC'', 27(7):1029–1046, September 2009.]
The process is named after French mathematician
Siméon Denis Poisson
Baron Siméon Denis Poisson FRS FRSE (; 21 June 1781 – 25 April 1840) was a French mathematician and physicist who worked on statistics, complex analysis, partial differential equations, the calculus of variations, analytical mechanics, electri ...
despite Poisson's never having studied the process. Its name derives from the fact that if a collection of random points in some space forms a Poisson process, then the number of points in a region of finite size is a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
with a
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 or space if these events occur with a known co ...
. The process was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and insurance mathematics.
The Poisson point process is often defined on the
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
, where it can be considered as a
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
. In this setting, it is used, for example, in
queueing theory
Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the ...
to model random events, such as the arrival of customers at a store, phone calls at an exchange or occurrence of earthquakes, distributed in time. In the
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
, the point process, also known as a spatial Poisson process,
can represent the locations of scattered objects such as transmitters in a
wireless network
A wireless network is a computer network that uses wireless data connections between network nodes.
Wireless networking is a method by which homes, telecommunications networks and business installations avoid the costly process of introducing c ...
,
[J. G. Andrews, R. K. Ganti, M. Haenggi, N. Jindal, and S. Weber. A primer on spatial modeling and analysis in wireless networks. ''Communications Magazine, IEEE'', 48(11):156–163, 2010.][F. Baccelli and B. Błaszczyszyn. ''Stochastic Geometry and Wireless Networks, Volume I – Theory'', volume 3, No 3–4 of ''Foundations and Trends in Networking''. NoW Publishers, 2009.] particles
In the physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass.
They vary greatly in size or quantity, from s ...
colliding into a detector, or trees in a forest.
In this setting, the process is often used in mathematical models and in the related fields of spatial point processes,
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. This leads to the theory of spatial point processes, hence notions of Palm conditioning, which exten ...
,
spatial statistics
Spatial analysis or spatial statistics includes any of the formal techniques which studies entities using their topological, geometric, or geographic properties. Spatial analysis includes a variety of techniques, many still in their early dev ...
and
continuum percolation theory In mathematics and probability theory, continuum percolation theory is a branch of mathematics that extends discrete percolation theory to continuous space (often Euclidean space ). More specifically, the underlying points of discrete percolation f ...
.
[R. Meester and R. Roy. Continuum percolation, volume 119 of cambridge tracts in mathematics, 1996.] The Poisson point process can be defined on more
abstract spaces. Beyond applications, the Poisson point process is an object of mathematical study in its own right.
In all settings, the Poisson point process has the property that each point is
stochastically independent
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence o ...
to all the other points in the process, which is why it is sometimes called a ''purely'' or ''completely'' random process. Despite its wide use as a stochastic model of phenomena representable as points, the inherent nature of the process implies that it does not adequately describe phenomena where there is sufficiently strong interaction between the points. This has inspired the proposal of other point processes, some of which are constructed with the Poisson point process, that seek to capture such interaction.
The point process depends on a single mathematical object, which, depending on the context, may be a
constant, a
locally integrable function In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions lies ...
or, in more general settings, a
Radon measure
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel se ...
.
In the first case, the constant, known as the rate or intensity, is the average
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
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
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
s, 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
In mathematics, stochastic geometry is the study of random spatial patterns. At the heart of the subject lies the study of random point patterns. This leads to the theory of spatial point processes, hence notions of Palm conditioning, which exten ...
and
spatial statistics
Spatial analysis or spatial statistics includes any of the formal techniques which studies entities using their topological, geometric, or geographic properties. Spatial analysis includes a variety of techniques, many still in their early dev ...
;
[A. Baddeley. A crash course in stochastic geometry. ''Stochastic Geometry: Likelihood and Computation Eds OE Barndorff-Nielsen, WS Kendall, HNN van Lieshout (London: Chapman and Hall)'', pages 1–35, 1999.] 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, independence implies the Poisson distribution of point counts, but not the converse.
Poisson distribution of point counts
A Poisson point process is characterized via the
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 or space if these events occur with a known co ...
. The Poisson distribution is the probability distribution of a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
(called a ''Poisson random variable'') such that the probability that
equals
is given by:
:
where
denotes
factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) \t ...
and the parameter
determines the shape of the distribution. (In fact,
equals the expected value of
.)
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,
[W. Feller. Introduction to probability theory and its applications, vol. ii pod. 1974.] 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 (that is, it assigns length, area, or volume to sets) 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 (or average) 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
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Interna ...
, area,
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
, or time, depending on the underlying mathematical space, and it is also called the ''mean density'' or ''mean rate''; see
Terminology
Terminology is a group of specialized words and respective meanings in a particular field, and also the study of such terms and their use; the latter meaning is also known as terminology science. A ''term'' is a word, compound word, or multi-wor ...
.
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 counting process is a stochastic process with values that are non-negative, integer, and non-decreasing:
# ''N''(''t'') ≥ 0.
# ''N''(''t'') is an integer.
# If ''s'' ≤ ''t'' then ''N''(''s'') ≤ ''N''(''t'').
If ''s'' < ''t'', then ''N''(' ...
, 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:
*
* has
independent increments In probability theory, independent increments are a property of stochastic processes and random measures. Most of the time, a process or random measure has independent increments by definition, which underlines their importance. Some of the stochas ...
; and
* the number of events (or points) in any interval of length
is a Poisson random variable with parameter (or mean)
.
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 interoccurence times.
Interpreted as a point process on the real line
Interpreted as a
point process
In statistics and probability theory, a point process or point field is a collection of mathematical points randomly located on a mathematical space such as the real line or Euclidean space. Kallenberg, O. (1986). ''Random Measures'', 4th editio ...
, a Poisson point process can be defined on the
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
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
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...
is given by:
:
For some positive integer
, the homogeneous Poisson point process has the finite-dimensional distribution given by:
:
where the real numbers