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In probability and statistics, a nearest neighbour function, nearest neighbour distance distribution,〔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.〕 nearest-neighbour distribution function or nearest neighbour distribution is a mathematical function that is defined in relation to random mathematical objects known as point processes, which are types of stochastic processes often used as mathematical models of physical phenomena representable as randomly positioned points in time, space or both.〔D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf. ''Stochastic geometry and its applications'', volume 2. Wiley Chichester, 1995.〕〔D. J. Daley and D. Vere-Jones. ''An introduction to the theory of point processes. Vol. I''. Probability and its Applications (New York). Springer, New York, second edition, 2003.〕 More specifically, nearest neighbour functions are defined with respect to some point in the point process as being the probability distribution of the distance from this point to its nearest neighbouring point in the same point process, hence they are used to describe the probability of another point existing within some distance of a point. A nearest neighbour function can be contrasted with a spherical contact distribution function, which is not defined in reference to some initial point but rather as the probability distribution of the radius of a sphere when it first encounters or makes contact with a point of a point process. Nearest neighbour function are used in the study of point processes〔〔〔D. J. Daley and D. Vere-Jones. ''An introduction to the theory of point processes. Vol. . Probability and its Applications (New York). Springer, New York, second edition, 2008.〕 as well as the related fields of stochastic geometry〔 and spatial statistics,〔〔J. Moller and R. P. Waagepetersen. ''Statistical inference and simulation for spatial point processes''. CRC Press, 2003.〕 which are applied in various scientific and engineering disciplines such as biology, geology, physics, and telecommunications.〔〔〔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.〕〔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.〕 ==Point process notation== (詳細はmathematical space. Since these processes are often used to represent collections of points randomly scattered in space, time or both, the underlying space is usually ''d''-dimensional Euclidean space denoted here by , but they can be defined on more abstract mathematical spaces.〔 Point processes have a number of interpretations, which is reflected by the various types of point process notation.〔〔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. 〕 For example, if a point belongs to or is a member of a point process, denoted by , then this can be written as:〔 : and represents the point process being interpreted as a random set. Alternatively, the number of points of located in some Borel set is often written as:〔〔 : which reflects a random measure interpretation for point processes. These two notations are often used in parallel or interchangeably.〔〔 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「Nearest neighbour distribution」の詳細全文を読む スポンサード リンク
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