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In probability, statistics and related fields, a Poisson point process or Poisson process (also called Poisson random measure, Poisson random point field or Poisson point field) is a type of random mathematical object known as a point process or point field that consists of randomly located points located on some underlying mathematical space.〔D. Stoyan, W. S. Kendall, and J. Mecke. ''Stochastic geometry and its applications'', volume 2. Wiley, 1995.〕 The process has convenient mathematical properties,〔J. F. C. Kingman. ''Poisson processes'', volume 3. Oxford university press, 1992.〕 which has led to it being frequently defined in Euclidean space and used as a mathematical model for seemingly random processes in numerous disciplines such as astronomy,〔G. J. Babu and E. D. Feigelson. Spatial point processes in astronomy. ''Journal of statistical planning and inference'', 50(3):311-326, 1996.〕 biology,〔H. G. Othmer, S. R. Dunbar, and W. Alt. Models of dispersal in biological systems. ''Journal of mathematical biology'', 26(3):263-298, 1988.〕 ecology,〔H. Thompson. Spatial point processes, with applications to ecology. ''Biometrika'', 42(1/2):102-115, 1955.〕 geology,〔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.〕 physics,〔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.〕 image processing,〔M. Bertero, P. Boccacci, G. Desidera, and G. Vicidomini. Image deblurring with poisson data: from cells to galaxies. ''Inverse Problems'', 25(12):123006, 2009.〕 and telecommunications.〔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.〕〔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.〕 The Poisson point process is often defined on the real line playing an important role in the field of queueing theory 〔L. Kleinrock. ''Theory, volume 1, Queueing systems''. Wiley-interscience, 1975.〕 where it is used to model certain random events happening in time such as the arrival of customers at a store or phone calls at an exchange. In the plane, the point process, also known as a spatial Poisson process,〔A. Baddeley, I. Bárány, and R. Schneider. Spatial point processes and their applications. ''Stochastic Geometry: Lectures given at the CIME Summer School held in Martina Franca, Italy, September 13–18, 2004'', pages 1-75, 2007.〕 may represent scattered objects such as transmitters in a wireless network,〔〔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.〕〔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.〕〔M. Haenggi. ''Stochastic geometry for wireless networks''. Cambridge University Press, 2012.〕 particles colliding into a detector, or trees in a forest.〔 In this setting, the process is often used in mathematical models and in the related fields of spatial point processes,〔〔D. J. Daley and D. Vere-Jones. ''An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods'', Springer, New York, second edition, 2003.〕 stochastic geometry,〔 spatial statistics 〔〔J. Møller and R. P. Waagepetersen. ''Statistical inference and simulation for spatial point processes''. CRC Press, 2003.〕 and continuum percolation theory.〔R. Meester and R. Roy. Continuum percolation, volume 119 of cambridge tracts in mathematics, 1996.〕 In more abstract spaces, the Poisson point process serves as an object of mathematical study in its own right.〔 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.〔 Despite its wide use as a stochastic model of phenomena representable as points, the inherent nature of the process implies that it does not adequately describe phenomena in which there is sufficiently strong interaction between the points. This has sometimes led to the overuse of the point process in mathematical models,〔〔〔 and has inspired other point processes, some of which are constructed via the Poisson point process, that seek to capture this interaction.〔 The process is named after French mathematician Siméon Denis Poisson owing to the fact that if a collection of random points in some space form a Poisson process, then the number points in a region of finite size is directly related to the Poisson distribution, but Poisson, however, did not study the process, which independently arose in several different settings.〔〔D. Stirzaker. Advice to hedgehogs, or, constants can vary. ''The Mathematical Gazette'', 84(500):197-210, 2000.sing 〕〔P. Guttorp and T. L. Thorarinsdottir. What happened to discrete chaos, the quenouille process, and the sharp markov property? some history of stochastic point processes. ''International Statistical Review'', 80(2):253-268, 2012.〕 The process is defined with a single non-negative mathematical object, which, depending on the context, may be a constant, an integrable function or, in more general settings, a Radon measure.〔〔 If this object is a constant, then the resulting process is called a homogeneous 〔 or stationary 〔 Poisson point process. Otherwise, the parameter depends on its location in the underlying space, which leads to the inhomogeneous or nonhomogeneous Poisson point process'.〔 The word ''point'' is often omitted, but there are other ''Poisson processes'' of objects, which, instead of points, consist of more complicated mathematical objects such as lines and polygons, and such processes can be based on the Poisson point process.〔 ==History== 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Poisson point process」の詳細全文を読む スポンサード リンク
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