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In probability and statistics, a factorial moment measure is a mathematical quantity, function or, more precisely, measure that is defined in relation to 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. Moment measures generalize the idea of factorial moments, which are useful for studying non-negative integer-valued random variables.〔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. 〕 The first factorial moment measure of a point process coincides with its first moment measure or ''intensity measure'',〔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. 〕 which gives the expected or average number of points of the point process located in some region of space. In general, if the number of points in some region is considered as a random variable, then the moment factorial measure of this region is the factorial moment of this random variable.〔D. Stoyan, W. S. Kendall, J. Mecke, and L. Ruschendorf. ''Stochastic geometry and its applications'', volume 2. Wiley Chichester, 1995.〕 Factorial moment measures completely characterize a wide class of point processes, which means they can be used to uniquely identify a point process. If a factorial moment measure is absolutely continuous, then with respect to the Lebesgue measure it is said to have a ''density'' (which is a generalized form of a derivative), and this density is known by a number of names including factorial moment density, product density, coincidence density,〔 joint intensity , correlation function or multivariate frequency spectrum〔K. Handa. The two-parameter point process. ''Bernoulli'', 15(4):1082–1116, 2009. 〕 The first and second factorial moment densities of a point process are used in the definition of the ''pair correlation function'', which gives a way to statistically quantify the strength of interaction or correlation between points of a point process.〔A. Baddeley, I. Brny, 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. 〕 Factorial moment measures serve as useful tools 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 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 Rd, 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 ''N'', then this can be written as:〔 : and represents the point process being interpreted as a random set. Alternatively, the number of points of ''N'' located in some Borel set ''B'' 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「factorial moment measure」の詳細全文を読む スポンサード リンク
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