Revision 2cce766bb3e656de2c829da29ab8c51681004900 authored by Adrian Baddeley on 11 November 2010, 00:00:00 UTC, committed by Gabor Csardi on 11 November 2010, 00:00:00 UTC
1 parent 6bd38f3
pcf.Rd
\name{pcf}
\alias{pcf}
\title{Pair Correlation Function}
\description{
Estimate the pair correlation function.
}
\usage{
pcf(X, \dots)
}
\arguments{
\item{X}{
Either the observed data point pattern,
or an estimate of its \eqn{K} function,
or an array of multitype \eqn{K} functions
(see Details).
}
\item{\dots}{
Other arguments passed to the appropriate method.
}
}
\value{
Either a function value table
(object of class \code{"fv"}, see \code{\link{fv.object}})
representing a pair correlation function,
or a function array (object of class \code{"fasp"},
see \code{\link{fasp.object}})
representing an array of pair correlation functions.
}
\details{
The pair correlation function of a stationary point process is
\deqn{
g(r) = \frac{K'(r)}{2\pi r}
}{
g(r) = K'(r)/ ( 2 * pi * r)
}
where \eqn{K'(r)} is the derivative of \eqn{K(r)}, the
reduced second moment function (aka ``Ripley's \eqn{K} function'')
of the point process. See \code{\link{Kest}} for information
about \eqn{K(r)}. For a stationary Poisson process, the
pair correlation function is identically equal to 1. Values
\eqn{g(r) < 1} suggest inhibition between points;
values greater than 1 suggest clustering.
We also apply the same definition to
other variants of the classical \eqn{K} function,
such as the multitype \eqn{K} functions
(see \code{\link{Kcross}}, \code{\link{Kdot}}) and the
inhomogeneous \eqn{K} function (see \code{\link{Kinhom}}).
For all these variants, the benchmark value of
\eqn{K(r) = \pi r^2}{K(r) = pi * r^2} corresponds to
\eqn{g(r) = 1}.
This routine computes an estimate of \eqn{g(r)}
either directly from a point pattern,
or indirectly from an estimate of \eqn{K(r)} or one of its variants.
This function is generic, with methods for
the classes \code{"ppp"}, \code{"fv"} and \code{"fasp"}.
If \code{X} is a point pattern (object of class \code{"ppp"})
then the pair correlation function is estimated using
a traditional kernel smoothing method (Stoyan and Stoyan, 1994).
See \code{\link{pcf.ppp}} for details.
If \code{X} is a function value table (object of class \code{"fv"}),
then it is assumed to contain estimates of the \eqn{K} function
or one of its variants (typically obtained from \code{Kest} or
\code{Kinhom}).
This routine computes an estimate of \eqn{g(r)}
using smoothing splines to approximate the derivative.
See \code{\link{pcf.fv}} for details.
If \code{X} is a function value array (object of class \code{"fasp"}),
then it is assumed to contain estimates of several \eqn{K} functions
(typically obtained from \code{\link{Kmulti}} or
\code{\link{alltypes}}). This routine computes
an estimate of \eqn{g(r)} for each cell in the array,
using smoothing splines to approximate the derivatives.
See \code{\link{pcf.fasp}} for details.
}
\references{
Stoyan, D. and Stoyan, H. (1994)
Fractals, random shapes and point fields:
methods of geometrical statistics.
John Wiley and Sons.
}
\seealso{
\code{\link{pcf.ppp}},
\code{\link{pcf.fv}},
\code{\link{pcf.fasp}},
\code{\link{Kest}},
\code{\link{Kinhom}},
\code{\link{Kcross}},
\code{\link{Kdot}},
\code{\link{Kmulti}},
\code{\link{alltypes}}
}
\examples{
# ppp object
data(simdat)
\testonly{
simdat <- simdat[seq(1,simdat$n, by=4)]
}
p <- pcf(simdat)
plot(p)
# fv object
K <- Kest(simdat)
p2 <- pcf(K, spar=0.8, method="b")
plot(p2)
# multitype pattern; fasp object
data(betacells)
\testonly{
betacells <- betacells[seq(1,betacells$n, by=10)]
}
betaK <- alltypes(betacells, "K")
betap <- pcf(betaK, spar=1, method="b")
plot(betap)
}
\author{Adrian Baddeley
\email{Adrian.Baddeley@csiro.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
and Rolf Turner
\email{r.turner@auckland.ac.nz}
}
\keyword{spatial}
\keyword{nonparametric}
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