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 $$\lambda$$ 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 $${\mathcal B}$$. 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 $${\mathcal B}$$ as the union $$\cup_\xi (\xi + C_\xi)$$ of translated sets.

To illustrate tractability with one simple formula, the mean density of $${\mathcal B}$$ equals $$1 - \exp(- \lambda A)$$ where $$\Gamma$$ denotes the area of $$C_\xi$$ and $$A=\operatorname{E} (\Gamma).$$ 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.