Boolean model (probability theory)

For statistics 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 ${\displaystyle \lambda }${\displaystyle \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 ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}}. More precisely, the parameters are ${\displaystyle \lambda }${\displaystyle \lambda } and a probability distribution on compact sets; for each point ${\displaystyle \xi }${\displaystyle \xi } of the Poisson point process we pick a set ${\displaystyle C_{\xi }}${\displaystyle C_{\xi }} from the distribution, and then define ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}} as the union ${\displaystyle \cup _{\xi }(\xi +C_{\xi })}${\displaystyle \cup _{\xi }(\xi +C_{\xi })} of translated sets.

To illustrate tractability with one simple formula, the mean density of ${\displaystyle {\mathcal {B}}}${\displaystyle {\mathcal {B}}} equals ${\displaystyle 1-\exp(-\lambda A)}${\displaystyle 1-\exp(-\lambda A)} where ${\displaystyle \Gamma }${\displaystyle \Gamma } denotes the area of ${\displaystyle C_{\xi }}${\displaystyle C_{\xi }} and ${\displaystyle A=\operatorname {E} (\Gamma ).}${\displaystyle A=\operatorname {E} (\Gamma ).} The classical theory of stochastic geometry develops many further formulae. ^{[1] [2]}

As related topics, the case of constant-sized discs is the basic model of continuum percolation ^[3] and the low-density Boolean models serve as a first-order approximations in the study of extremes in many models.^[4]

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