Boolean model (probability theory)
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The 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.
To illustrate tractability with one simple formula, the mean density of equals [Unparseable or potentially dangerous latex formula. Error 2 ] where denotes the area of . The classical theory of stochastic geometry develops many further formulas – see .
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. and Weil, W. (2008). Stochastic and Integral Geometry. Springer.
- ^ Meester, R. and Roy, R. (2008). Continuum Percolation. Cambridge University Press.
- ^ Aldous, D. (1988). Probability Approximations via the Poisson Clumping Heuristic. Springer.