HOME

TheInfoList



OR:

For statistics in
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 o ...
, 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 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 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