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Boolean model (probability theory) : ウィキペディア英語版
Boolean model (probability theory)

In probability theory, the Boolean-Poisson model or simply Boolean model for a random subset of the plane (or higher dimensions, analogously) is one of the simplest and most tractable models in stochastic geometry. Take a Poisson point process of rate \lambda in the plane and make each point be the center of a random set; the resulting union of overlapping sets is a realization of the Boolean model . More precisely, the parameters are \lambda and a probability distribution on compact sets; for each point \xi of the Poisson point process we pick a set C_\xi from the distribution, and then define as the union
\cup_\xi (\xi + C_\xi) of translated sets.
To illustrate tractability with one simple formula, the mean density of equals 1 - \exp(- \lambda A) where \Gamma denotes the area of C_\xi and A=\operatorname (\Gamma). The classical theory of stochastic geometry develops many further formulae.

〕〔

As related topics, the case of constant-sized discs is the basic model of continuum percolation

and the low-density Boolean models serve as a first-order approximations in the
study of extremes in many models.〔

==References==


抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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