|
In probability and statistics, a 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 (raw) moments of random variables, hence arise often in the study of point processes and related fields.〔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. 〕 An example of a moment measure is the ''first moment measure'' or ''intensity measure'' of a point process, which gives the expected or average number of points of the point process being located in some region of space.〔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.〕 In other words, if the number of points of a point process located in some region of space is a random variable, then the first moment measure corresponds to the first 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.〕 Moment measures feature prominently in the study of point processes〔〔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. 〕〔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. 〕 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. 〕 whose applications are found in numerous 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 , 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「moment measure」の詳細全文を読む スポンサード リンク
|