Boolean model (probability theory)

Realization of Boolean model with random-radii discs.

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 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. More precisely, the parameters are λ and a probability distribution on compact sets; for each point ξ of the Poisson point process we pick a set Cξ from the distribution, and then define ℬ as the union ∪ξ(ξ+Cξ) of translated sets.

To illustrate tractability with one simple formula, the mean density of ℬ equals 1−exp⁡(−λA) where Γ denotes the area of Cξ and A=E⁡(Γ). 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]

## References

1. ↑ Stoyan, D.; Kendall, W.S.; Mecke, J. (1987). Stochastic geometry and its applications. Wiley.
2. ↑ Schneider, R.; Weil, W. (2008). Stochastic and Integral Geometry. Springer.
3. ↑ Meester, R.; Roy, R. (2008). Continuum Percolation. Cambridge University Press.
4. ↑ Aldous, D. (1988). Probability Approximations via the Poisson Clumping Heuristic. Springer.

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