
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
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
and a probability distribution on compact sets; for each point
of the Poisson point process we pick a set
from the distribution, and then define
as the union
of translated sets.
To illustrate tractability with one simple formula, the mean density of
equals
where
denotes the area of
and
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
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