Boolean model (probability theory)
||This article provides insufficient context for those unfamiliar with the subject. Learn how and when to remove this template message) (May 2012) (|
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 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 . More precisely, the parameters are and a probability distribution on compact sets; for each point of the Poisson point process we pick a set from the distribution, and then define as the union of translated sets.
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.
- Stoyan, D., Kendall, W.S. and Mecke, J. (1987). Stochastic geometry and its applications. Wiley.
- Schneider, R. & Weil, W. (2008). Stochastic and Integral Geometry. Springer.
- Meester, R. & Roy, R. (2008). Continuum Percolation. Cambridge University Press.
- Aldous, D. (1988). Probability Approximations via the Poisson Clumping Heuristic. Springer.