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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Laplace functional ： ウィキペディア英語版 Laplace functional In probability theory, a Laplace functional refers to one of two possible mathematical functions of functions or, more precisely, functionals that serve as mathematical tools for studying either point processes or concentration of measure properties of metric spaces. One type of Laplace functional,〔D. Stoyan, W. S. Kendall, and J. Mecke. ''Stochastic geometry and its applications'', volume 2. Wiley, 1995.〕〔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.〕 also known as a characteristic functional is defined in relation to a point process, which can be interpreted as random counting measures, and has applications in characterizing and deriving results on point processes.〔Barrett J. F. The use of characteristic functionals and cumulant generating functionals to discuss the effect of noise in linear systems, J. Sound & Vibration 1964 vol.1, no.3, pp. 229-238〕 Its is definition is analogous to a characteristic function for a random variable. The other Laplace functional is for probability spaces equipped with metrics, and is used to study the concentration of measure properties of the space. ==Definition for point processes== For a general point process $\backslash textstyle\; N$ defined on $\backslash textstyle\; \backslash textbf^d$, the Laplace functional is defined as:〔F. Baccelli and B. Baszczyszyn. ''Stochastic Geometry and Wireless Networks, Volume I - Theory'', volume 3, No 3-4 of ''Foundations and Trends in Networking''. NoW Publishers, 2009.〕 :$L\_(f)=E(f(x)(dx)\}),$ where $\backslash textstyle\; f$ is any measurable non-negative function on $\backslash textstyle\; \backslash textbf^d$ and :$\backslash int\_\; f(x)(dx)=\backslash sum\backslash limits\_\; f(x\_i).$ where the notation $N(dx)$ interprets the point process as a random counting measure; see Point process notation. 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「Laplace functional」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.