Boolean model (probability theory)

Realization of Boolean model with random-radii discs.

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. ^[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. and 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.