Boolean model (probability theory)

File:Boolean model.svg

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 $\lambda$ 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.

{{cite book

| author = Stoyan, D.

| author2 = Kendall, W.S.

| author3 = Mecke, J.

| name-list-style = amp

| title = Stochastic geometry and its applications

| year = 1987

| publisher = Wiley}}

{{cite book

|author1=Schneider, R. |author2=Weil, W.

|name-list-style=amp | title = Stochastic and Integral Geometry

| year = 2008

| publisher = Springer}}

As related topics, the case of constant-sized discs is the basic model of continuum percolation{{cite book

|author1=Meester, R. |author2=Roy, R.

|name-list-style=amp | title = Continuum Percolation

| year = 2008

| publisher = Cambridge University Press}}

and the low-density Boolean models serve as a first-order approximations in the

study of extremes in many models.{{cite book

| author = Aldous, D.

| title = Probability Approximations via the Poisson Clumping Heuristic

| year = 1988

| publisher = Springer}}

References

Category:Spatial processes