Poisson point process: Difference between revisions

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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 [[real line]], where it can be considered as a [[stochastic process]]. In this setting, 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, such as the arrival of customers at a store, phone calls at an exchange or occurrence of earthquakes, distributed in time. 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"/> In this setting, 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 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"/> 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.{{sfnp|Daley|Vere-Jones|2003|page=27}} DespiteModeling its widea usesystem as a stochasticPoisson modelProcess ofis phenomenainsufficient representable as points,when the inherentpoint-to-point natureinteractions ofare the process implies that it does not adequately describe phenomena where there is sufficientlytoo strong interaction between(i.e. the points. Thisare hasnot inspiredstochastically theindependent). proposalSuch ofa othersystem pointmay processes,be some of which arebetter constructedmodeled with thea Poissondifferent point process, that seek to capture such interaction.<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]].