rbdd  R Documentation 
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.
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)
lambda 
Intensity of the germ process (which is a Poisson point process) 
discr 
Radius of the discs 
window 
The window to simulate in (an 
seed 
Optional input (default in NULL). Is an integer passed to

coverp 
Coverage probability of the Boolean model 
r 
is the radius to calculate covariance 
xrange 
range of x values for 
yrange 
range of y values for 
eps 
list of length 2 of the steps between samples points in x and y respectively for 
See Functions section.
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.
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).
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.
# 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 < 2.2064E3
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)
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