Poisson point process

In probability theory, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one another.^[1] The Poisson point process is also called a Poisson random measure, Poisson random point field and Poisson point field. When the process is defined on the real number line, it is often called simply the Poisson process.

Mean

$a_{0,t}=\int _{0}^{t}\lambda (\alpha )d\alpha$

$a_{0,t}+(a_{0,t})^{2}-(a_{0,t})^{2}=a_{0,t}$
since $R_{x}(t_{1},t_{2})=a_{0,min(t_{1},t_{2})}+a_{0,t_{1}}a_{0,t_{2}}$

where for

E\{X^{2}\}=R_{x}(t,t)=a_{0,t}+(a_{0,t})^{2}

This point process has convenient mathematical properties,^[2] which has led to its being frequently defined in Euclidean space and used as a mathematical model for seemingly random processes in numerous disciplines including astronomy,^[3] biology,^[4] ecology,^[5]geology,^[6] seismology,^[7] physics,^[8] economics,^[9] image processing,^[10]^[11] and telecommunications.^[12]^[13]

The process's name derives from the fact that if a collection of random points form a Poisson process, then the number of points in a region of finite size is a random variable with 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.^[14]^[15]

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^[16] 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,^[17] can represent the locations of scattered objects such as transmitters in a wireless network,^[12]^[18]^[19]^[20] particles colliding into a detector or trees in a forest.^[21] The process is often used in mathematical models and in the related fields of spatial point processes,^[22] stochastic geometry,^[1] spatial statistics^[22]^[23] and continuum percolation theory.^[24]

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.^[2] 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.^[25] Modeling a system as a Poisson process is insufficient when the point-to-point interactions are too strong (that is, the points are not stochastically independent). Such a system may be better modeled with a different point process.^[26]

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.^[27] 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.^[28] 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.^[29] The word point is often omitted,^[2] 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.^[30] Both the homogeneous and nonhomogeneous Poisson point processes are particular cases of the generalized renewal process.

${\textstyle N(0)=0;}$

has ; and

independent increments

the number of events (or points) in any interval of length ${\textstyle t}$ is a Poisson random variable with parameter (or mean) ${\textstyle \lambda t}$ .

$\textstyle N(0)=0;$

has ;

independent increments

$\textstyle \Pr\{N(t+h)-N(t)=1\}=\lambda (t)h+o(h);$ and

$\textstyle \Pr\{N(t+h)-N(t)\geq 2\}=o(h),$

the number of points in a bounded $\textstyle B$ is a Poisson random variable with mean $\textstyle \Lambda (B)$ . In other words, denote the total number of points located in $\textstyle B$ by $\textstyle {N}(B)$ , then the probability of random variable $\textstyle {N}(B)$ being equal to $\textstyle n$ is given by:

Borel set

In measure theory, the Poisson point process can be further generalized to what is sometimes known as the general Poisson point process^[21]^[89] or general Poisson process^[76] by using a Radon measure $\textstyle \Lambda$ , which is a locally finite measure. In general, this Radon measure $\textstyle \Lambda$ can be atomic, which means multiple points of the Poisson point process can exist in the same location of the underlying space. In this situation, the number of points at $\textstyle x$ is a Poisson random variable with mean $\textstyle \Lambda ({x})$ .^[89] But sometimes the converse is assumed, so the Radon measure $\textstyle \Lambda$ is diffuse or non-atomic.^[21]

A point process $\textstyle {N}$ is a general Poisson point process with intensity $\textstyle \Lambda$ if it has the two following properties:^[21]

The Radon measure $\textstyle \Lambda$ maintains its previous interpretation of being the expected number of points of $\textstyle {N}$ located in the bounded region $\textstyle B$ , namely

Furthermore, if $\textstyle \Lambda$ is absolutely continuous such that it has a density (which is the Radon–Nikodym density or derivative) with respect to the Lebesgue measure, then for all Borel sets $\textstyle B$ it can be written as:

where the density $\textstyle \lambda (x)$ is known, among other terms, as the intensity function.

History[edit]

Poisson distribution[edit]

Despite its name, the Poisson point process was neither discovered nor studied by its namesake. It is cited as an example of Stigler's law of eponymy.^[14]^[15] The name arises from the process's inherent relation to the Poisson distribution, derived by Poisson as a limiting case of the binomial distribution.^[90] It 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 coin flips 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.^[91]

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,^[92] and the result was not well known during his time. Over the following years others used the distribution without citing Poisson, including Philipp Ludwig von Seidel and Ernst Abbe.^[93] ^[14] At the end of the 19th century, Ladislaus Bortkiewicz studied the distribution, citing Poisson, using real data on the number of deaths from horse kicks in the Prussian army.^[90]^[94]

Discovery[edit]

There are a number of claims for early uses or discoveries of the Poisson point process.^[14]^[15] For example, John Michell in 1767, a decade before Poisson was born, was interested in the probability a star being within a certain region of another star under the erroneous 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 in 1860.^[15]

At the beginning of the 20th century the Poisson process (in one dimension) would arise independently in different situations.^[14]^[15] In Sweden 1903, Filip Lundberg published a thesis containing work, now considered fundamental and pioneering, where he proposed to model insurance claims with a homogeneous Poisson process.^[95]^[96]

In Denmark A.K. Erlang derived the Poisson distribution in 1909 when developing a mathematical model for the number of incoming phone calls in a finite time interval. Erlang unaware of Poisson's earlier work and assumed that the number phone calls arriving in each interval of time were independent of each other. He then found the limiting case, which is effectively recasting the Poisson distribution as a limit of the binomial distribution.^[14]

In 1910 Ernest Rutherford and Hans Geiger published experimental results on counting alpha particles. Their experimental work had mathematical contributions from Harry Bateman, who derived Poisson probabilities as a solution to a family of differential equations, though the solution had been derived earlier, resulting in the independent discovery of the Poisson process.^[14] After this time, there were many studies and applications of the Poisson process, but its early history is complicated, which has been explained by the various applications of the process in numerous fields by biologists, ecologists, engineers and various physical scientists.^[14]

Early applications[edit]

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.^[14] 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 to study how plants are distributed in plant communities.^[97] 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,^[14] among others.^[98] In the field of teletraffic engineering, mathematicians and statisticians studied and used Poisson and other point processes.^[99]

History of terms[edit]

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.^[100]^[99] In his work exists the first known recorded use of the term point processes as Punktprozesse in German.^[100]^[15]

It is believed^[14] that William Feller was the first in print to refer to it as the Poisson process in a 1940 paper. Although the Swede Ove Lundberg used the term Poisson process in his 1940 PhD dissertation,^[15] in which Feller was acknowledged as an influence,^[101] it has been claimed that Feller coined the term before 1940.^[91] 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 by then.^[15] Feller worked from 1936 to 1939 alongside 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.^[15]

Terminology[edit]

The terminology of point process theory in general has been criticized for being too varied.^[15] In addition to the word point often being omitted,^[65]^[2] the homogeneous Poisson (point) process is also called a stationary Poisson (point) process,^[49] as well as uniform Poisson (point) process.^[44] The inhomogeneous Poisson point process, as well as being called nonhomogeneous,^[49] is also referred to as the non-stationary Poisson process.^[74]^[102]

The term point process has been criticized, as the term process can suggest over time and space, so random point field,^[103] resulting in the terms Poisson random point field or Poisson point field being also used.^[104] A point process is considered, and sometimes called, a random counting measure,^[105] hence the Poisson point process is also referred to as a Poisson random measure,^[106] a term used in the study of Lévy processes,^[106]^[107] but some choose to use the two terms for Poisson points processes defined on two different underlying spaces.^[108]

The underlying mathematical space of the Poisson point process is called a carrier space,^[109]^[110] or state space, though the latter term has a different meaning in the context of stochastic processes. In the context of point processes, the term "state space" can mean the space on which the point process is defined such as the real line,^[111]^[112] which corresponds to the index set^[113] or parameter set^[114] in stochastic process terminology.

The measure $\textstyle \Lambda$ is called the intensity measure,^[115] mean measure,^[38] or parameter measure,^[69] as there are no standard terms.^[38] If $\textstyle \Lambda$ has a derivative or density, denoted by $\textstyle \lambda (x)$ , is called the intensity function of the Poisson point process.^[21] For the homogeneous Poisson point process, the derivative of the intensity measure is simply a constant $\textstyle \lambda >0$ , which can be referred to as the rate, usually when the underlying space is the real line, or the intensity.^[44] It is also called the mean rate or the mean density^[116] or rate .^[34] For $\textstyle \lambda =1$ , the corresponding process is sometimes referred to as the standard Poisson (point) process.^[45]^[59]^[117]

The extent of the Poisson point process is sometimes called the exposure.^[118]^[119]

Convergence to a Poisson point process[edit]

In general, when an operation is applied to a general point process the resulting process is usually not a Poisson point process. For example, if a point process, other than a Poisson, has its points randomly and independently displaced, then the process would not necessarily be a Poisson point process. However, under certain mathematical conditions for both the original point process and the random displacement, it has been shown via limit theorems that if the points of a point process are repeatedly displaced in a random and independent manner, then the finite-distribution of the point process will converge (weakly) to that of a Poisson point process.^[147]

Similar convergence results have been developed for thinning and superposition operations^[147] that show that such repeated operations on point processes can, under certain conditions, result in the process converging to a Poisson point processes, provided a suitable rescaling of the intensity measure (otherwise values of the intensity measure of the resulting point processes would approach zero or infinity). Such convergence work is directly related to the results known as the Palm–Khinchin^[f] equations, which has its origins in the work of Conny Palm and Aleksandr Khinchin,^[148] and help explains why the Poisson process can often be used as a mathematical model of various random phenomena.^[147]

Boolean model (probability theory)

Continuum percolation theory

Compound Poisson process

Cox process

Point process

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Stochastic geometry models of wireless networks

Markovian arrival processes

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Poisson point process

Mean

Mean

Variance

independent increments

independent increments

Borel set

History[edit]

Poisson distribution[edit]

Discovery[edit]

Early applications[edit]

History of terms[edit]

Terminology[edit]

Convergence to a Poisson point process[edit]

Boolean model (probability theory)

Continuum percolation theory

Compound Poisson process

Cox process

Point process

Stochastic geometry

Stochastic geometry models of wireless networks

Markovian arrival processes

Stochastic Geometry: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 13–18, 2004

Cox, D. R.

ISBN

ISBN

Kingman, John Frank

ISBN

ISBN

ISBN

ISBN

ISBN

A First Course in Stochastic Models