Poisson point process

In probability, statistics and related fields, a Poisson point process or Poisson process (also called a Poisson random measure, Poisson random point field or Poisson point field) is a type of random mathematical object known as a point process or point field that consists of randomly located points located on some underlying mathematical space.^[1] The process has convenient mathematical properties,^[2] which has led to it being frequently defined in Euclidean space and used as a mathematical model for seemingly random processes in numerous disciplines such as astronomy,^[3] biology,^[4] ecology,^[5] geology,^[6] physics,^[7] image processing,^[8] and telecommunications.^[9]^[10]

The Poisson point process is often defined on the real line playing an important role in the field of queueing theory ^[11] where it is used to model certain random events happening in time such as the arrival of customers at a store or phone calls at an exchange. In the plane, the point process, also known as a spatial Poisson process,^[12] may represent scattered objects such as transmitters in a wireless network,^[9]^[13]^[14]^[15] particles colliding into a detector, or trees in a forest.^[1] In this setting, the process is often used in mathematical models and in the related fields of spatial point processes,^[12]^[16] stochastic geometry,^[1] spatial statistics ^[12]^[17] and continuum percolation theory.^[18] In more abstract spaces, the Poisson point process serves as an object of mathematical study in its own right.^[2]

In all settings, the Poisson point process has the property that each point is stochastically independent to all the other points in the process, which is why it is sometimes called a purely or completely random process.^[16] 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 in which there is sufficiently strong interaction between the points. This has sometimes led to the overuse of the point process in mathematical models,^[1]^[2]^[13] and has inspired other point processes, some of which are constructed via the Poisson point process, that seek to capture this interaction.^[1]

The process is named after French mathematician Siméon Denis Poisson owing to the fact that if a collection of random points in some space form a Poisson process, then the number points in a region of finite size is directly related to the Poisson distribution, but Poisson never studied the process, which independently arose in several different settings.^[16]^[19]^[20] The process is defined with a single non-negative mathematical object, which, depending on the context, may be a constant, an integrable function or, in more general settings, a Radon measure.^[1]^[16] If this object is a constant, then the resulting process is called a homogeneous ^[2] or stationary ^[1] Poisson point process. Otherwise, the parameter depends on its location in the underlying space, which leads to the inhomogeneous or nonhomogeneous Poisson point process'.^[16] 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.^[2]

History

Poisson distribution

Despite its name, the Poisson point process was not discovered nor studied by the French mathematician Siméon Denis Poisson, and the name is cited as an example of Stigler's law.^[19]^[20] The name stems from its inherent relation to the Poisson distribution, derived by Poisson as a limiting case of the binomial distribution,^[21] which describes the probability of the sum of $\textstyle n$ Bernoulli trials with probability $\textstyle p$ , often likened to the number of heads (or tails) after $\textstyle n$ biased flips of a coin with the probability of a head (or tail) occurring being $\textstyle p$ . For some positive constant $\textstyle \Lambda>0$ , as $\textstyle n$ increases towards infinity and $\textstyle p$ decreases towards zero such that the product $\textstyle np=\Lambda$ is fixed, the Poisson distribution more closely approximates that of the binomial.^[22] Poisson derived the Poisson distribution, published in 1841, by examining the binomial distribution in the limit of $\textstyle p$ (to zero) and $\textstyle n$ (to infinity). It only appears once in all of Poisson's work,^[23] and the result was not well-known during his time,^[16] even though over the following years a number of people would use the distribution without citing Poisson including Philipp Ludwig von Seidel and Ernst Abbe.^[16]^[19] The distribution would be studied years after Poisson at the end of the 19th century in a different setting by Ladislaus Bortkiewicz who did cite Poisson and used the distribution with real data to study the number of deaths from horse kicks in the Prussian army.^[21]^[24]

Discovery

There are a number of claims for early uses or discoveries of the Poisson point process.^[19]^[20] It has been proposed that the earliest use of the Poisson point process was by John Michell in 1767, a decade before Poisson was born. Michell was interested in the probability a star being within a certain region of another star under the assumption that the stars were "scattered by mere chance", and studied an example consisting of the six brightest stars in the Pleiades, without deriving the Poisson distribution. This work inspired Simon Newcomb to study the problem and to calculate the Poisson distribution as an approximation for the binomial distribution.^[20]

At the beginning of the 20th century the Poisson point process would arise independently during the same period in three different situations.^[19]^[22] In 1909 the Danish mathematician and engineer A.K. Erlang derived the Poisson distribution when developing a mathematical model for the number of incoming phone calls in a finite time interval. Erlang, not at the time aware of Poisson's earlier work, assumed that the number phone calls arriving in each interval of time were independent to each other, and then found the limiting case, which is effectively recasting the Poisson distribution as a limit of the binomial distribution.^[19] In 1910 physicists Ernest Rutherford and Hans Geiger, after conducting an experiment in counting the number of alpha particles, published their results in which English mathematician Harry Bateman derived the Poisson probabilities as a solution to a family of differential equations, although Bateman acknowledged that the solutions had been previously solved by others.^[16] This experimental work by Rutherford and Geiger partly inspired physicist Norman Campbell who in 1909 and 1910 published two key papers on thermionic noise, also known as shot noise, in vacuum tubes,^[16]^[19] where it is believed he independently discovered and used the Poisson process.^[22] In Campbell's work, he also outlined a form of Campbell's theorem,^[19] a key result in the theory of point processes,^[2]^[12]^[16] but Campbell credited the proof to the mathematician G. H. Hardy.^[19] The three above discoveries and applications of the Poisson point process has motivated some to say that 1909 should be considered the discovery year of the Poisson point process.^[19]^[22]

Early applications

The years after 1909 led to a number of studies and applications of the Poisson point process, however, its early history is complex, which has been explained by the various applications of the process in numerous fields by biologists, ecologists, engineers and others working in the physical sciences. The early results were published in different languages and in different settings, with no standard terminology and notation used.^[19] For example, in 1922 Swedish chemist and Nobel Laureate Theodor Svedberg proposed a model in which a spatial Poisson point process is the underlying process in order to study how plants are distributed in plant communities.^[25] A number of mathematicians started studying the process in the early 1930s, and important contributions were made by Andrey Kolmogorov, William Feller and Aleksandr Khinchin,^[19] among others.^[26] As an application, Kolmogorov used a spatial Poisson point process to model the formation of crystals in metals.^[1] In the field of teletraffic engineering, where a lot of the early researchers were Danes, such as Erlang, and Swedes, mathematicians and statisticians studied and used Poisson and other point processes.^[27]

History of terms

The Swede Conny Palm in his 1943 dissertation studied the Poisson and other point processes in the one-dimensional setting by examining them in terms of the statistical or stochastic dependence between the points in time.^[16]^[27] In his work exists the first known recorded use of the term point process as Punktprozesse in German.^[16]^[20]

It is believed ^[19] that William Feller, who also made the term random variable popular over a competing term chance variable through a coin flip with Joseph Doob, was the first in print to refer to it as the Poisson process in a 1940 paper. Although the Swedish statistician Ove Lundberg used the term Poisson process in his 1940 PhD dissertation,^[20] in which Feller was acknowledged as an influence,^[28] it has been claimed that Feller coined the term before 1940.^[22] It has been remarked that both Feller and Lundberg used the term as though it were well-known, implying it was already in spoken use.^[20] Feller worked from 1936 to 1939 alongside Swedish mathematician and statistician Harald Cramér at Stockholm University, where Lundberg was a PhD student under Cramér who did not use the term Poisson process in a book by him, finished in 1936, but did in subsequent editions, which his has led to the speculation that the term Poisson process was coined sometime between 1936 and 1939 at the Stockholm University.^[20]

Overview of definitions

The Poisson point process is one of the most studied stochastic processes in both the field of point processes and in more applied disciplines concerning random phenomena ^[16] due to its convenient properties as a mathematical model as well as being mathematically interesting.^[2] Depending on the setting, the process has several equivalent definitions ^[29] as well definitions of varying generality owing to its many applications and characterizations.^[16] It may be defined, studied and used in one dimension (on the real line) where it can be interpreted as a counting process or part of a queueing model;^[29]^[30] in higher dimensions such as the plane where it plays a role in stochastic geometry and spatial statistics;^[1]^[31] or on more abstract mathematical spaces.^[32] 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.^[1]^[16] Despite its different forms and varying generality, the Poisson point process has two key properties.

First key property: Poisson distributed number of points

The Poisson point process is related to the Poisson distribution, which implies that the probability a Poisson random variable $\textstyle N$ is equal to $\textstyle n$ is given by:

P\{N=n\}=\frac{\Lambda^n}{n!}e^{-\Lambda}

where $\textstyle n!$ denotes $\textstyle n$ factorial and $\textstyle \Lambda$ is the single Poisson parameter that is used to define the Poisson distribution. If a Poisson point process is defined on some underlying mathematical space, called a state space ^[2]^[33] or carrier space,^[34]^[35] then the number of points in a bounded region of the space will be a Poisson random variable with some parameter whose form will depend on the setting.^[2]

Second key property: complete independence

The other key property is that for a collection of disjoint and bounded subregions of the underlying space, the number of points in each bounded subregion will be completely independent to all the others. This property is known under several names such as complete randomness, complete independence,^[16] or independent scattering ^[1]^[17]^[33] 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,^[36] which motivates the Poisson process being sometimes called a purely or completely random process.^[16]

Different definitions

The Poisson point process is often defined on the real line in the homogeneous setting, and then extended to a more general settings with more mathematical rigour.^[16]^[36] For all the instances of the Poisson point process, the two key properties^{[lower-alpha 1]} of the Poisson distribution and complete independence play an important role.^[1]

Homogeneous Poisson point process

If a Poisson point process has a constant parameter, say, $\textstyle \lambda$ , then it 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.^[2]^[17] In fact, the parameter $\textstyle \lambda$ 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, hence it is sometimes called the mean density;^[16] see Terminology; the extent is sometimes called the exposure.^[37]^[38]

Defined on the real line

Consider two real numbers $\textstyle a$ and $\textstyle b$ , where $\textstyle a\le b$ , and which may represent points in time. Denote by $\textstyle N(a,b]$ the random number of points of a homogeneous Poisson point process existing with values greater than $\textstyle a$ but less than or equal to $\textstyle b$ , or in other words, the number of points of the process in the interval $\textstyle (a,b]$ . If the points form or belong to a homogeneous Poisson process with parameter $\textstyle \lambda>0$ , then the probability of $\textstyle n$ points existing in the above interval $\textstyle (a,b]$ is given by:

P\{N(a,b]=n\}=\frac{[\lambda(b-a)]^n}{n!}e^{-\lambda (b-a)},

In other words, $\textstyle N(a,b]$ is a Poisson random variable with mean $\textstyle \lambda(b-a)$ . Furthermore, the number of points in any two subjoint intervals, say, $\textstyle (a_1,b_1]$ and $\textstyle (a_2,b_2]$ are independent to each other, and this extends to any finite number of subjoint intervals.^[16] In the queueing theory context, one can consider a point existing (in an interval) as an event, but this is different to the word event in the probability theory sense.^{[lower-alpha 2]} It follows that $\textstyle \lambda$ is the expected number of arrivals that occur per unit of time, and it is sometimes called the rate parameter.^[30]

For a more formal definition of a stochastic process, such as a point process, one can use the Kolmogorov theorem, which essentially says a stochastic process is characterized (or uniquely defined) by its finite-dimensional distribution, which in this context gives the joint probability of some number of points existing in each disjoint finite interval. More specifically, let $\textstyle N(a_i,b]$ denote the number of points of (a point process) happening in the half-open interval $\textstyle (a_i,b_i]$ , where the real numbers $\textstyle a_i<b_i\leq a_{i+1}$ . Then for some positive integer $\textstyle k$ , the homogeneous Poisson point process on the real line with parameter $\textstyle \lambda>0$ is defined with the finite-dimensional distribution:^[16]

P\{N(a_i,b_i]=n_i, i=1, \dots, k\}=\prod_{i=1}^k\frac{[\lambda(b_i-a_i)]^{n_i}}{n_i!}e^{-\lambda(b_i-a_i)},

Key properties

The above definition has two important features pertaining to the Poisson point processes in general:

the number of points in each finite interval has a Poisson distribution;
the number of points in disjoint intervals are independent random variables.

Furthermore, it has a third feature related to just the homogeneous Poisson process:

the distribution of each interval $\textstyle(a+t,b+t]$ only depends on the interval's length $\textstyle b-a$ .

In other words, for any finite $\textstyle t>0$ , the random variable $\textstyle N(a+t,b+t]$ is independent of $\textstyle t$ ,^[16] and, hence, the process is Stationary process, which is why it is sometimes called the stationary Poisson process.

Law of large numbers

The quantity $\textstyle \lambda(b_i-a_i)$ can be interpreted as the expected or average number of points occurring in the interval $\textstyle (a_i,b_i]$ , namely:

E\{N(a_i,b_i] \}=\lambda(b_i-a_i),

where $\textstyle E$ denotes the expectation operator. In other words, the parameter $\textstyle \lambda$ of the Poisson process coincides with the density of points. Furthermore, the homogeneous Poisson point process adheres to its own form of the (strong) law of large numbers.^[2] More specifically, with probability one:

\lim_{t\rightarrow \infty} \frac{N (t) }{t} =\lambda,

where $\textstyle \lim$ denotes the limit of a function.

Memoryless property

The distance between two consecutive points of a point process on the real line will be an exponential random variable with parameter $\textstyle \lambda$ (or equivalently, mean $\textstyle 1/\lambda$ ). This implies that the points have the memoryless property: the existence of one point existing in a finite interval does not affect the probability (distribution) of other points existing. This property is directly related to the complete independence of the Poisson process, however, it has no natural equivalence when the Poisson process is defined in higher dimensions.^[2]

Orderliness and simplicity

A stochastic process with stationary increments is sometimes said to be orderly,^[39] ordinary ^[32] or regular ^[30] if

P\{N(t,t+\delta]>1\}= o(\delta),

where little-o notation is 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 state space) is zero. For point processes in general on the real line, the (probability distribution) property of orderliness implies that the process is simple ^[39] or has the (sample path) property of simplicity,^[32] which is the case for the homogeneous Poisson point process.

Relationship to other processes

On the real line, the Poisson point process is a type of continuous-time Markov process known as a birth-death process (with just births and zero deaths) and is called a pure ^[30] or simple birth process.^[40] More complicated processes with the Markov property, such as Markov arrival processes, have been defined where the Poisson process is a special case.^[29]^[41]

Counting process interpretation

The homogeneous Poisson point process, when considered on the positive half-line, is sometimes defined as a counting process, which can be denoted as $\textstyle \{N(t), t\geq 0\}$ .^[29]^[30] A counting process represents the total number of occurrences or events that have happened up to and including time $\textstyle t$ . A counting process is a Poisson counting process with rate $\textstyle \lambda>0$ if it has the following three properties:

$\textstyle N(0)=0$ ;
has independent increments; and
the number of events (or points) in any interval of length $\textstyle t$ is a Poisson random variable with parameter (or mean) $\textstyle \lambda t$ .

The last property implies

E[N(t)]=\lambda t.

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 $\textstyle 1/\lambda$ .^[29] The time differences between the events or arrivals are known as interrarrival ^[30]^[36] or interoccurence times.^[29] These two definitions of the Poisson counting process agree with the previous definition of the 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

N(-\infty,t]-t,

is a martingale.^[42]

Restricted to the half-line

If the homogeneous Poisson point process is considered just on the half-line $\textstyle [0,\infty)$ , which is often the case when $\textstyle t$ represents time, as it does for the previous counting process,^[29]^[30] then the resulting process is not truly invariant under translation.^[2] In that case the process is no longer stationary, according to some definitions of stationarity.^[1]^[2]^[39]

Applications

There have been many applications of the homogeneous Poisson point process on the real line in an attempt to model seemingly random and independent events occurring. It has a fundamental role in queueing theory, which is the probability field of developing suitable stochastic models to represent the random arrival and departure of certain phenomena.^[11]^[29]^[30]^[36] For example, customers arriving and being served or phone calls arriving at a phone exchange can be both studied with techniques from queueing theory. In the original paper proposing the online payment system known as Bitcoin featured a mathematical model based on a homogeneous Poisson point process.^[43]

Generalizations

The Poisson counting process or, more generally, the homogeneous Poisson point process on the real line is considered one of the simplest stochastic processes for counting random numbers of points.^[39]^[44] The process can be generalized in a number of ways. One possible generalization is to extend the distribution of interarrival times from the exponential distribution to other distributions, which introduces the stochastic process known as a renewal process. Another generalization is to define it on higher dimensional spaces such as the plane.^[16]

Spatial Poisson point process

A spatial Poisson process is a Poisson point process defined in the plane $\textstyle \textbf{R}^2$ .^[42]^[45] For its definition, consider a bounded, open or closed (or more precisely, Borel measurable) region $\textstyle B$ of the plane. Denote by $\textstyle N(B)$ the (random) number of points of $\textstyle N$ existing in this region $\textstyle B\subset \textbf{R}^2$ . If the points belong to a homogeneous Poisson process with parameter $\textstyle \lambda>0$ , then the probability of $\textstyle n$ points existing in $\textstyle B$ is given by:

P\{N(B)=n\}=\frac{(\lambda|B|)^n}{n!}e^{-\lambda|B|}

where $\textstyle |B|$ denotes the area of $\textstyle B$ .

More formally, for some some finite integer $\textstyle k\geq 1$ , consider a collection of disjoint, bounded Borel (measurable) sets $\textstyle B_1,\dots,B_k$ . Let $\textstyle N(B_i)$ denote the number of points of existing in $\textstyle B_i$ . Then the homogeneous Poisson point process with parameter $\textstyle \lambda>0$ has the finite-dimensional distribution ^[16]

P\{N(B_i)=n_i, i=1, \dots, k\}=\prod_{i=1}^k\frac{(\lambda|B_i|)^{n_i}}{n_i!}e^{-\lambda|B_i|}.

Applications

According to one statistical study, the positions of cellular or mobile phone base stations in the Australian city Sydney, pictured above, resemble a Poisson point process, while in many other cities around the world they do not and other point processes are required.^[46]

The spatial Poisson point process features prominently in spatial statistics, stochastic geometry, and continuum percolation theory. This process is applied in various physical sciences such as a model developed for alpha particles being detected.^[1] In recent years, it has been frequently used to model seemingly disordered spatial configurations of certain wireless communication networks.^[13]^[14]^[15] For example, models for cellular or mobile phone networks have been developed where it is assumed the phone network transmitters, known as base stations, are positioned according to a homogeneous Poisson point process.

Defined in higher dimensions

The previous homogeneous Poisson point process immediately extends to higher dimensions by replacing the notion of area with (high dimensional) volume. For some bounded region $\textstyle B$ of Euclidean space $\textstyle \textbf{R}^d$ , if the points form a homogeneous Poisson process with parameter $\textstyle \lambda>0$ , then the probability of $\textstyle n$ points existing in $\textstyle B\subset \textbf{R}^d$ is given by:

P\{N(B)=n\}=\frac{(\lambda|B|)^n}{n!}e^{-\lambda|B|}

where $\textstyle |B|$ now denotes the $\textstyle n$ -dimensional volume of $\textstyle B$ . Furthermore, for a collection of disjoint, bounded Borel sets $\textstyle B_1,\dots,B_k \subset \textbf{R}^d$ , let $\textstyle N(B_i)$ denote the number of points of $\textstyle N$ existing in $\textstyle B_i$ . Then the corresponding homogeneous Poisson point process with parameter $\textstyle \lambda>0$ has the finite-dimensional distribution ^[16]

P\{N(B_i)=n_i, i=1, \dots, k\}=\prod_{i=1}^k\frac{(\lambda|B_i|)^{n_i}}{n_i!}e^{-\lambda|B_i|}.

Homogeneous Poisson point processes do not depend on the position of the underlying state space through its parameter $\textstyle \lambda$ , which implies it is both a stationary process (invariant to translation) and an isotropic (invariant to rotation) stochastic process.^[1] Similarly to the one-dimensional case, the homogeneous point process is restricted to some bounded subset of $\textstyle \textbf{R}^d$ , then depending on some definitions of stationarity, the process is no longer stationary.^[1]^[39]

Points are uniformly distributed

If the homogeneous point process is defined on the real line as a mathematical model for occurrences of some phenomenon, then it has the characteristic that the positions of these occurrences or events on the real line (often interpreted as time) will be uniformly distributed. More specifically, if an event occurs (according to this process) in an interval $\textstyle (a-b]$ where $\textstyle a \leq b$ , then its location will be a uniform random variable defined on that interval.^[16] Furthermore, the homogeneous point process is sometimes called the uniform Poisson point process (see Terminology). This uniformity property extends to higher dimensions in the Cartesian coordinate, but it does not hold in other coordinate systems (for example, polar or spherical).

Inhomogeneous Poisson point process

The inhomogeneous or nonhomogeneous Poisson point process (see Terminology) is a Poisson point process with a Poisson parameter set as some location-dependent function in the underlying space on which the Poisson process is defined. For Euclidean space $\textstyle R^d$ , this is achieved by introducing a locally integrable positive function $\textstyle \lambda(x)$ , where $\textstyle x$ is a $\textstyle n$ -dimensional point located in $\textstyle R^d$ , such that for any bounded region $\textstyle B$ the ( $\textstyle d$ -dimensional) volume integral of $\textstyle \lambda (x)$ over region $\textstyle B$ is finite. In other words, if this integral, denoted by $\textstyle \Lambda (B)$ , is:^[17]

\Lambda (B)=\int_B \lambda(x) dx < \infty,

where $\textstyle dx$ is a ( $\textstyle d$ -dimensional) volume element,^{[lower-alpha 3]} then for any collection of disjoint bounded Borel measurable sets $\textstyle B_1,\dots,B_k$ , an inhomogeneous Poisson process with (intensity) function $\textstyle \lambda(x)$ has the finite-dimensional distribution:^[16]

P\{N(B_i)=n_i, i=1, \dots, k\}=\prod_{i=1}^k\frac{(\Lambda(B_i))^{n_i}}{n_i!}e^{-\Lambda(B_i)}.

Furthermore, $\textstyle \Lambda (B)$ has the interpretation of being the expected number of points of the Poisson process located in the bounded region $\textstyle B$ , namely

\Lambda (B)= E[N(B)] .