Poisson point process: Difference between revisions

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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 distribution of the number of points regionsin ofany thegiven samefinite sizeregion hasfollows 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 }}</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>