distcdf.Rd
\name{distcdf}
\alias{distcdf}
\title{Distribution Function of Interpoint Distance }
\description{
Computes the cumulative distribution function of the distance
between two independent random points in a given window
or windows.
}
\usage{
distcdf(W, V=W, \dots, dW=1, dV=dW, nr=1024, regularise=TRUE)
}
\arguments{
\item{W}{
A window (object of class \code{"owin"}) containing the
first random point.
}
\item{V}{
Optional. Another window containing the second random point.
Defaults to \code{W}.
}
\item{\dots}{
Arguments passed to \code{\link{as.mask}} to determine the
pixel resolution for the calculation.
}
\item{dV, dW}{
Optional. Probability densities (not necessarily normalised)
for the first and second random points respectively.
Data in any format acceptable
to \code{\link{as.im}}, for example, a \code{function(x,y)}
or a pixel image or a numeric value. The default
corresponds to a uniform distribution over the window.
}
\item{nr}{
Integer. The number of values of interpoint distance \eqn{r}
for which the CDF will be computed.
Should be a large value!
}
\item{regularise}{
Logical value indicating whether to smooth the results
for very small distances, to avoid discretisation artefacts.
}
}
\value{
An object of class \code{"fv"}, see \code{\link{fv.object}},
which can be plotted directly using \code{\link{plot.fv}}.
}
\details{
This command computes the Cumulative Distribution Function
\eqn{
CDF(r) = Prob(T \le r)
}{
CDF(r) = Prob(T \le r)
}
of the Euclidean distance \eqn{T = \|X_1 - X_2\|}{T = |X1-X2|}
between two independent random points \eqn{X_1}{X1} and \eqn{X_2}{X2}.
In the simplest case, the command \code{distcdf(W)}, the random points are
assumed to be uniformly distributed in the same
window \code{W}.
Alternatively the two random points may be
uniformly distributed in two different windows \code{W} and \code{V}.
In the most general case the first point \eqn{X_1}{X1} is random
in the window \code{W} with a probability density proportional to
\code{dW}, and the second point \eqn{X_2}{X2} is random in
a different window \code{V} with probability density proportional
to \code{dV}. The values of \code{dW} and \code{dV} must be
finite and nonnegative.
The calculation is performed by numerical integration of the set covariance
function \code{\link{setcov}} for uniformly distributed points, and
by computing the covariance function \code{\link{imcov}} in the
general case. The accuracy of the result depends on
the pixel resolution used to represent the windows: this is controlled
by the arguments \code{\dots} which are passed to \code{\link{as.mask}}.
For example use \code{eps=0.1} to specify pixels of size 0.1 units.
The arguments \code{W} or \code{V} may also be point patterns
(objects of class \code{"ppp"}).
The result is the cumulative distribution function
of the distance from a randomly selected point in the point pattern,
to a randomly selected point in the other point pattern or window.
If \code{regularise=TRUE} (the default), values of the cumulative
distribution function for very short distances are smoothed to avoid
discretisation artefacts. Smoothing is applied to all distances
shorter than the width of 7 pixels.
}
\seealso{
\code{\link{setcov}},
\code{\link{as.mask}}.
}
\examples{
# The unit disc
B <- disc()
plot(distcdf(B))
}
\author{\adrian
and \rolf
}
\keyword{spatial}
\keyword{math}