Boolean model (probability theory)

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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[edit]

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