Boolean model (probability theory)
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probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, 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 In mathematics, stochastic geometry is the study of random spatial patterns. At the heart of the subject lies the study of random point patterns. This leads to the theory of spatial point processes, hence notions of Palm conditioning, which exten ...
. Take a
Poisson point process In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
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 In mathematics, stochastic geometry is the study of random spatial patterns. At the heart of the subject lies the study of random point patterns. This leads to the theory of spatial point processes, hence notions of Palm conditioning, which exten ...
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

Spatial processes {{probability-stub