rbdd {lacunaritycovariance}

Simulation of Boolean Model of Deterministic Discs

Description

Functions for simulating a Boolean model with grains that are discs of fixed constant radius (the abbreviation 'bdd' is short for Boolean model with Deterministic Discs). A Boolean model is a two stage model, first the locations (called germs) of grains are randomly distributed according to a Poisson point process, then a random grain is placed on each germ independently. Introductions to Boolean models are available in many stochastic geometry books (Chiu et al., 2013). Also described here are functions for calculating the coverage probability, germ intensity, and covariance from model parameters for a Boolean model with deterministic discs.

Usage

rbdd(lambda, discr, window, seed = NULL)

bddcoverageprob(lambda, discr)

bddlambda(coverp, discr)

bdddiscr(coverp, lambda)

bddcovar.iso(r, lambda, discr)

bddcovar(xrange, yrange, eps, lambda, discr)

Arguments

lambda

Intensity of the germ process (which is a Poisson point process)

discr

Radius of the discs

window

The window to simulate in (an owin object)

seed

Optional input (default in NULL). Is an integer passed to set.seed. Used to reproduce patterns exactly.

coverp

Coverage probability of the Boolean model

r

is the radius to calculate covariance

xrange

range of x values for bddcovar

yrange

range of y values for bddcovar

eps

list of length 2 of the steps between samples points in x and y respectively for bddcovar. If eps is of length 1 then the steps between sample points in the x and y directions will both be equal to eps.

Value

See Functions section.

Functions

• rbdd(): Returns an owin that is a set generated by simulating a Boolean model with specified intensity and disc radius. The window information is not contained in this object. If the simulated set is empty then an empty owin object is returned. The point process of germs is generated using spatstat's rpoispp.

• bddcoverageprob(): Returns the true coverage probability given the intensity and disc radius.

• bddlambda(): Returns the germ intensity given coverage probability and disc radius.

• bdddiscr(): Returns the disc radius given coverage probability and germ intensity.

• bddcovar.iso(): Returns the true covariance of points separated by a distance r given the intensity, lambda and disc radius discr of the model.

• bddcovar(): Returns an image of the covariance as calculated from disc radius and intensity.

WARNING

The returned object of rbdd is an owin specifying the realisation of the Boolean model within the simulation window. The simulation window is not included, thus the object returned by rbdd can have much smaller extent than the simulation window (e.g. when the simulated set is empty).

References

Chiu, S.N., Stoyan, D., Kendall, W.S. and Mecke, J. (2013) Stochastic Geometry and Its Applications, 3rd ed. Chichester, United Kingdom: John Wiley & Sons.

Examples

# Simulate Boolean model with discs of radius 10.
# The coverage probability is very close to 0.5.
discr <- 10
w <- owin(xrange = c(0, 100), c(0, 100))
lambda <- 2.2064E-3 
xi <- rbdd(lambda, discr, w)

# Compute properties of Boolean model from parameters
cp <- bddcoverageprob(lambda, discr)
cvc <- bddcovar(c(-10, 10), c(-10, 10), c(0.2, 0.2), lambda, discr)