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]], statistics and related fields, a '''Poisson point process''' is a type of [[random]] [[mathematical object]] that consists of [[Point (geometry)|points]] randomly located on a [[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 Poisson point process is also called a '''Poisson random measure''', '''Poisson random point field''' or '''Poisson point field'''. When the process is defined on the [[real line]], it is often called simply the '''Poisson process.'''

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>

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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===