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{{Short description|Type of random mathematical object}}
{{Use dmy dates|date=July 2021}}
{{Infobox probability distribution
[[File:Poisson process.svg|thumb|alt=Poisson point process|A visual depiction of a Poisson point process starting from 0, in which increments occur continuously and independently at rate ''λ''.]]▼
|name=Poisson Process
|pdf_image=[[File:Poisson Process.png|325px]]
|mean=<math>a_{0, t} = \int_{0}^{t} \lambda(\alpha) d\alpha</math>
|variance=<math>a_{0, t} + (a_{0, t})^2 - (a_{0, t})^2 = a_{0, t}</math> <br>
since <math>R_x(t_1,t_2) = a_{0, min(t_1,t_2)} + a_{0, t_1} a_{0, t_2}</math>
where for <math>E\{X^2\} = R_x(t,t) = a_{0, t} + (a_{0, t})^2</math>
|type=multivariate
}}
▲[[File:Poisson process.svg|thumb|alt=Poisson point process|A visual depiction of a Poisson point process starting
This [[point process]] has convenient mathematical properties,<ref name="Kingman1992">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|date=17 December 1992|publisher=Clarendon Press|isbn=978-0-19-159124-2}}</ref> which has led to its being frequently defined in [[Euclidean space]] and used as a [[mathematical model]] for seemingly random processes in numerous disciplines such as [[astronomy]],<ref name="babu1996spatial">G. J. Babu and E. D. Feigelson. Spatial point processes in astronomy. ''Journal of statistical planning and inference'', 50(3):311–326, 1996.</ref> [[biology]],<ref name="othmer1988models">H. G. Othmer, S. R. Dunbar, and W. Alt. Models of dispersal in biological systems. ''Journal of mathematical biology'', 26(3):263–298, 1988.</ref> ecology,<ref name="thompson1955spatial">H. Thompson. Spatial point processes, with applications to ecology. ''Biometrika'', 42(1/2):102–115, 1955.</ref> geology,<ref name="connor1995three">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.</ref> [[seismology]],<ref>{{Cite journal|last1=Gardner|first1=J. K.|last2=Knopoff|first2=L.|date=1974|title=Is the sequence of earthquakes in Southern California, with aftershocks removed, Poissonian?|url=https://pubs.geoscienceworld.org/ssa/bssa/article-abstract/64/5/1363/117341/is-the-sequence-of-earthquakes-in-southern|journal=Bulletin of the Seismological Society of America|volume=64|issue=5 |pages=1363–1367|doi=10.1785/BSSA0640051363 |bibcode=1974BuSSA..64.1363G |s2cid=131035597 }}</ref> [[physics]],<ref name="scargle1998studies">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.</ref> economics,<ref name="AghionHowitt1992">P. Aghion and P. Howitt. A Model of Growth through Creative Destruction. ''Econometrica'', 60(2). 323–351, 1992.</ref> [[image processing]],<ref name="bertero2009image">M. Bertero, P. Boccacci, G. Desidera, and G. Vicidomini. Image deblurring with poisson data: from cells to galaxies. ''Inverse Problems'', 25(12):123006, 2009.</ref><ref>{{cite web | url=https://caseymuratori.com/blog_0010 | title=The Color of Noise }}</ref> and telecommunications.<ref name="baccelli2009stochastic2">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.</ref><ref name="Haenggi2009">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.</ref>▼
▲This
▲The process is named after French mathematician [[Siméon Denis Poisson]] 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]] with a [[Poisson distribution]]. The process was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and insurance mathematics.<ref name="Stirzaker2000">{{cite journal|last1=Stirzaker|first1=David|title=Advice to Hedgehogs, or, Constants Can Vary|journal=The Mathematical Gazette|volume=84|issue=500|year=2000|pages=197–210|issn=0025-5572|doi=10.2307/3621649|jstor=3621649|s2cid=125163415}}</ref><ref name="GuttorpThorarinsdottir2012">{{cite journal|last1=Guttorp|first1=Peter|last2=Thorarinsdottir|first2=Thordis L.|title=What Happened to Discrete Chaos, the Quenouille Process, and the Sharp Markov Property? Some History of Stochastic Point Processes|journal=International Statistical Review|volume=80|issue=2|year=2012|pages=253–268|issn=0306-7734|doi=10.1111/j.1751-5823.2012.00181.x|s2cid=80836 }}</ref>
The Poisson point process is often defined on the
The point process depends on a single mathematical object, which, depending on the context, may be a [[Constant (mathematics)|constant]], a [[locally integrable function]] or, in more general settings, a [[Radon measure]].<ref name="ChiuStoyan2013page41and51">{{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|date=27 June 2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3|pages=41 and 51 }}</ref> 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'''.<ref name="ChiuStoyan2013page41">{{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|date=27 June 2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3|pages=41–42}}</ref> 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.{{sfnp|Daley|Vere-Jones|2003|page=22}} The word ''point'' is often omitted,<ref name="Kingman1992">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|date=17 December 1992|publisher=Clarendon Press|isbn=978-0-19-159124-2}}</ref> but there are other ''Poisson processes'' of objects, which, instead of points, consist of more complicated mathematical objects such as [[line (geometry)|line]]s and [[polygon]]s, and such processes can be based on the Poisson point process.<ref name="Kingman1992page73to76">{{cite book|author=J. F. C. Kingman|title=Poisson Processes|url=https://books.google.com/books?id=VEiM-OtwDHkC|date=17 December 1992|publisher=Clarendon Press|isbn=978-0-19-159124-2|pages=73–76}}</ref> Both the homogeneous and nonhomogeneous Poisson point processes are particular cases of the [[generalized renewal process]].
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Depending on the setting, the process has several equivalent definitions<ref name="Tijms2003page1">{{cite book|author=H. C. Tijms|title=A First Course in Stochastic Models|url=https://books.google.com/books?id=RK9yFrNxom8C|date=18 April 2003|publisher=John Wiley & Sons|isbn=978-0-471-49880-3|pages=1–2}}</ref> as well as definitions of varying generality owing to its many applications and characterizations.{{sfnp|Daley|Vere-Jones|2003|pages=26–37}} 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;<ref name="Tijms2003page1and9">{{cite book|author=H. C. Tijms|title=A First Course in Stochastic Models|url=https://books.google.com/books?id=RK9yFrNxom8C|date=18 April 2003|publisher=John Wiley & Sons|isbn=978-0-471-49880-3|pages=1 and 9}}</ref><ref name="Ross1996page59">{{cite book|author=Sheldon M. Ross|title=Stochastic processes|url=https://books.google.com/books?id=ImUPAQAAMAAJ|year=1996|publisher=Wiley|isbn=978-0-471-12062-9|pages=59–60}}</ref> in higher dimensions such as the plane where it plays a role in [[stochastic geometry]]<ref name="ChiuStoyan2013"/> and [[spatial statistics]];<ref name="baddeley1999crash">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.</ref> or on more general mathematical spaces.<ref name="DaleyVere-Jones2007page1">{{cite book|author1=D.J. Daley|author2=David Vere-Jones|title=An Introduction to the Theory of Point Processes: Volume II: General Theory and Structure|url=https://books.google.com/books?id=nPENXKw5kwcC|date=12 November 2007|publisher=Springer Science & Business Media|isbn=978-0-387-21337-8|pages=1–2}}</ref> 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.<ref name="ChiuStoyan2013page110to111">{{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|date=27 June 2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3|pages=110–111 }}</ref>
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.<ref name="ChiuStoyan2013page41and51"/><ref name="Kingman1992page11"/> The two properties are not logically independent; indeed, the Poisson distribution of point counts implies the independence property
===Poisson distribution of point counts===
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:<math> \Pr \{N(t)=n\}=\frac{(\lambda t)^n}{n!} e^{-\lambda t}. </math>
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 <math display=inline> 1/\lambda</math>.<ref name="Tijms2003"/> The time differences between the events or arrivals are known as '''interarrival''' <ref name="Ross1996page64">{{cite book|author=Sheldon M. Ross|title=Stochastic processes|url=https://books.google.com/books?id=ImUPAQAAMAAJ|year=1996|publisher=Wiley|isbn=978-0-471-12062-9|page=64}}</ref> or '''
===Interpreted as a point process on the real line===
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For the inhomogeneous case, a couple of different methods can be used depending on the nature of the intensity function <math>\textstyle \lambda(x)</math>.<ref name="ChiuStoyan2013page53to55"/> If the intensity function is sufficiently simple, then independent and random non-uniform (Cartesian or other) coordinates of the points can be generated. For example, simulating a Poisson point process on a circular window can be done for an isotropic intensity function (in polar coordinates <math>\textstyle r</math> and <math>\textstyle \theta</math>), implying it is rotationally variant or independent of <math>\textstyle \theta</math> but dependent on <math>\textstyle r</math>, by a change of variable in <math>\textstyle r</math> if the intensity function is sufficiently simple.<ref name="ChiuStoyan2013page53to55"/>
For more complicated intensity functions, one can use an [[Rejection sampling|acceptance-rejection method]], which consists of using (or 'accepting') only certain random points and not using (or 'rejecting') the other points, based on the ratio:.<ref name="Streit2010page14">{{cite book|author=Roy L. Streit|title=Poisson Point Processes: Imaging, Tracking, and Sensing|url=https://books.google.com/books?id=KAWmFYUJ5zsC|date=15 September 2010|publisher=Springer Science & Business Media|isbn=978-1-4419-6923-1|pages=14–16}}</ref>
:<math> \frac{\lambda(x_i)}{\Lambda(W)}=\frac{\lambda(x_i)}{\int_W\lambda(x)\,\mathrm dx. } </math>
where <math>\textstyle x_i</math> is the point under consideration for acceptance or rejection.
That is, a ___location is uniformly randomly selected for consideration, then to determine whether to place a sample at that ___location a uniformly randomly drawn number in <math> [0,1] </math> is compared to the probability density function <math> \frac{\lambda(x)}{\Lambda(W)} </math> , accepting if it is smaller than the probability density function, and repeating until the previously chosen number of samples have been drawn.
==General Poisson point process==
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===Poisson distribution===
Despite its name, the Poisson point process was neither discovered nor studied by
Poisson derived the Poisson distribution, published in 1841, by examining the binomial distribution in the [[Limit (mathematics)|limit]] of <math>\textstyle p</math> (to zero) and <math>\textstyle n</math> (to infinity). It only appears once in all of Poisson's work,<ref name="stigler1982poisson">{{cite journal |first=S. M. |last=Stigler |title=Poisson on the Poisson Distribution |journal=Statistics & Probability Letters |volume=1 |issue=1 |pages=33–35 |year=1982 |doi=10.1016/0167-7152(82)90010-4 }}</ref> and the result was not well known during his time. Over the following years
<ref name="Stirzaker2000" /> At the end of the 19th century, [[Ladislaus Bortkiewicz]]
===Discovery===
There are a number of claims for early uses or discoveries of the Poisson point process.<ref name="Stirzaker2000"/><ref name="GuttorpThorarinsdottir2012"/> 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 [[star]]s 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.<ref name="GuttorpThorarinsdottir2012"/>
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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.<ref name="EmbrechtsFrey2001page367">{{cite book|last1=Embrechts|first1=Paul|title=Stochastic Processes: Theory and Methods|last2=Frey|first2=Rüdiger|last3=Furrer|first3=Hansjörg|chapter=Stochastic processes in insurance and finance|volume=19|year=2001|page=367|issn=0169-7161|doi=10.1016/S0169-7161(01)19014-0|series=Handbook of Statistics|isbn=9780444500144}}</ref><ref name="Cramér1969">{{cite journal|last1=Cramér|first1=Harald|title=Historical review of Filip Lundberg's works on risk theory|journal=Scandinavian Actuarial Journal|volume=1969|issue=sup3|year=1969|pages=6–12|issn=0346-1238|doi=10.1080/03461238.1969.10404602}}</ref>
In [[Denmark]]
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.<ref name="Stirzaker2000"/> 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.<ref name="Stirzaker2000"/>
===Early applications===
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| publisher = [[R (programming language)#CRAN|Comprehensive R Archive Network]]
| doi = 10.18637/jss.v078.i10
| doi-access = free| arxiv = 1612.01907
| s2cid = 14379617
| url = https://cran.r-project.org/web/packages/KFAS/vignettes/KFAS.pdf
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===The Mecke equation===
The Mecke equation characterizes the Poisson point process. Let <math>\mathbb{N}_\sigma</math> be the space of all <math>\sigma</math>-finite measures on some general space <math>\mathcal{Q}</math>. A point process <math>\eta</math> with intensity <math>\lambda</math> on <math>\mathcal{Q}</math> is a Poisson point process if and only if for all measurable functions <math>f:\mathcal{Q}\times\mathbb{N}_\sigma\to \mathbb{R}_+</math> the following holds
:<math>
For further details see.<ref name="Proper Point Process">{{cite book|author1=Günter Last|author2=Mathew Penrose|title=Lectures on the Poisson Process|url=http://www.math.kit.edu/stoch/~last/seite/lectures_on_the_poisson_process/media/lastpenrose2017.pdf|date=8 August 2017}}</ref>
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==Avoidance function==
The '''avoidance function'''
:<math> v(B)=\Pr \{N(B)=0\}. </math>
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===Stein's method===
[[Stein's method]] is a mathematical technique originally developed for approximating random variables such as [[Gaussian distribution|Gaussian]] and Poisson variables, which has also been applied to point processes. Stein's method can be used to derive upper bounds on [[probability metric]]s, which give way to quantify how different two random mathematical objects vary stochastically.<ref name="chen2013approximating"/><ref name="barbour1992stein">A. D. Barbour and T. C. Brown. Stein's method and point process approximation. ''Stochastic Processes and their Applications'', 43(1):9–31, 1992.</ref> Upperbounds on probability metrics such as [[total variation]] and [[Wasserstein distance]] have been derived.<ref name="chen2013approximating"/>
Researchers have applied Stein's method to Poisson point processes in a number of ways,<ref name="chen2013approximating"/> such as using [[Palm calculus]].<ref name="chen2004stein"/> Techniques based on Stein's method have been developed to factor into the upper bounds the effects of certain [[point process operation]]s such as thinning and superposition.<ref name="schuhmacher2005super">D. Schuhmacher. Distance estimates for dependent superpositions of point processes. ''Stochastic processes and their applications'', 115(11):1819–1837, 2005.</ref><ref name="schuhmacher2005thinnings">D. Schuhmacher. Distance estimates for poisson process approximations of dependent thinnings. ''Electronic Journal of Probability'', 10:165–201, 2005.</ref> Stein's method has also been used to derive upper bounds on metrics of Poisson and other processes such as the [[Cox point process]], which is a Poisson process with a random intensity measure.<ref name="chen2013approximating"/>
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