Poisson point process: Difference between revisions

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{{Short description|Type of random mathematical object}}

{{Use dmy dates|date=July 2021}}
{{Infobox probability distribution
|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]]

In [[probability theory]], [[statistics]] and related fields, a '''Poisson point process''' (also known as: '''Poisson random measure''', '''Poisson random point field''' and '''Poisson point field''') is a type of [[mathematical object]] that consists of [[Point (geometry)|points]] randomly located on a [[Space (mathematics)|mathematical space]] with the essential feature that the points occur independently of one another.<ref name="ChiuStoyan2013">{{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}}</ref> 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]].<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>

This point process is used as a [[mathematical model]] for seemingly random processes in numerous disciplines including [[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 |url-access=subscription}}</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>

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]]<ref name="Kleinrock1976">{{cite book|author=Leonard Kleinrock|title=Queueing Systems: Theory|url=https://archive.org/details/queueingsystems01klei|url-access=registration|year=1976|publisher=Wiley|isbn=978-0-471-49110-1}}</ref> 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 (geometry)|plane]], the point process, also known as a '''spatial Poisson process''',<ref name="BaddeleyBárány2006page10">{{cite book|author1=A. Baddeley|author2=I. Bárány|author3=R. Schneider|title=Stochastic Geometry: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 13–18, 2004|url=https://books.google.com/books?id=X-m5BQAAQBAJ|date=26 October 2006|publisher=Springer|isbn=978-3-540-38175-4|page=10}}</ref> can represent the locations of scattered objects such as transmitters in a [[wireless network]],<ref name="baccelli2009stochastic2"/><ref name="andrews2010primer">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.</ref><ref name="baccelli2009stochastic1">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.</ref><ref name="Haenggi2013">{{cite book|author=Martin Haenggi|title=Stochastic Geometry for Wireless Networks|url=https://books.google.com/books?id=CLtDhblwWEgC|year=2013|publisher=Cambridge University Press|isbn=978-1-107-01469-5}}</ref> [[particles]] colliding into a detector or trees in a forest.<ref name="ChiuStoyan2013page51"/> The process is often used in mathematical models and in the related fields of spatial point processes,<ref name="BaddeleyBárány2006">{{cite book|author1=A. Baddeley|author2=I. Bárány|author3=R. Schneider|title=Stochastic Geometry: Lectures given at the C.I.M.E. Summer School held in Martina Franca, Italy, September 13–18, 2004|url=https://books.google.com/books?id=X-m5BQAAQBAJ|date=26 October 2006|publisher=Springer|isbn=978-3-540-38175-4}}</ref> [[stochastic geometry]],<ref name="ChiuStoyan2013"/> [[spatial statistics]]<ref name="BaddeleyBárány2006"/><ref name="MollerWaagepetersen2003">{{cite book|author1=Jesper Moller|author2=Rasmus Plenge Waagepetersen|title=Statistical Inference and Simulation for Spatial Point Processes|url=https://books.google.com/books?id=dBNOHvElXZ4C|date=25 September 2003|publisher=CRC Press|isbn=978-0-203-49693-0}}</ref> and [[continuum percolation theory]].<ref name="meester1996continuum">R. Meester and R. Roy. Continuum percolation, volume 119 of cambridge tracts in mathematics, 1996.</ref>

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]].

The Poisson point process can be defined on more [[Abstraction (mathematics)|abstract]] spaces. Beyond applications, the Poisson point process is an object of mathematical study in its own right.<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> 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.{{sfnp|Daley|Vere-Jones|2003|page=27}} 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.<ref name="ChiuStoyan2013page35">{{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=35–36}}</ref>

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" /> 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]].

{{toclimit|limit=3}}

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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, {{efn|See Section 2.3.2 of Chiu, Stoyan, Kendall, Mecke<ref name="ChiuStoyan2013"/> or Section 1.3 of Kingman.<ref name="Kingman1992"/>}} while in the converse direction the assumptions that: (i) the point process is simple, (ii) has no fixed atoms, and (iii) is a.s. boundedly finite are required. {{sfnp|Daley|Vere-Jones|2003|pages=34–39}}

===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 '''interoccurenceinteroccurrence''' times.<ref name="Tijms2003"/>

===Interpreted as a point process on the real line===

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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>\PrE \left[\int f(x,\eta)\eta(\mathrm{d}x)\right]=\int \PrE \left[ f(x,\eta+\delta_x) \right] \lambda(\mathrm{d}x)</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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===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"/>