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

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

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.