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 λ {\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 B {\displaystyle {\mathcal {B}}} . More precisely, the parameters are λ {\displaystyle \lambda } and a probability distribution on compact sets; for each point ξ {\displaystyle \xi } of the Poisson point process we pick a set C ξ {\displaystyle C_{\xi }} from the distribution, and then define B {\displaystyle {\mathcal {B}}} as the union
∪ ξ ( ξ + C ξ ) {\displaystyle \cup _{\xi }(\xi +C_{\xi })} of translated sets. To illustrate tractability with one simple formula, the mean density of B {\displaystyle {\mathcal {B}}} equals 1 − exp ( − λ A ) {\displaystyle 1-\exp(-\lambda A)} where Γ {\displaystyle \Gamma } denotes the area of C ξ {\displaystyle C_{\xi }} and A = E ( Γ ) . {\displaystyle A=\operatorname {E} (\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