pcf.ppp.Rd
\name{pcf.ppp}
\alias{pcf.ppp}
\title{Pair Correlation Function of Point Pattern}
\description{
Estimates the pair correlation function of
a point pattern using kernel methods.
}
\usage{
\method{pcf}{ppp}(X, \dots, r = NULL, kernel="epanechnikov", bw=NULL, stoyan=0.15,
correction=c("translate", "ripley"))
}
\arguments{
\item{X}{
A point pattern (object of class \code{"ppp"}).
}
\item{r}{
Vector of values for the argument \eqn{r} at which \eqn{g(r)}
should be evaluated. There is a sensible default.
}
\item{kernel}{
Choice of smoothing kernel,
passed to \code{density}.
}
\item{bw}{
Bandwidth for smoothing kernel, passed to \code{density}.
}
\item{\dots}{
Other arguments passed to the kernel density estimation
function \code{density}.
}
\item{stoyan}{
Bandwidth coefficient; see Details.
}
\item{correction}{
Choice of edge correction.
}
}
\value{
A function value table
(object of class \code{"fv"}).
Essentially a data frame containing the variables
\item{r}{the vector of values of the argument \eqn{r}
at which the pair correlation function \eqn{g(r)} has been estimated
}
\item{theo}{vector of values equal to 1,
the theoretical value of \eqn{g(r)} for the Poisson process
}
\item{trans}{vector of values of \eqn{g(r)}
estimated by translation correction
}
\item{ripley}{vector of values of \eqn{g(r)}
estimated by Ripley isotropic correction
}
as required.
}
\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.
This routine computes an estimate of \eqn{g(r)}
by the kernel smoothing method (Stoyan and Stoyan (1994),
pages 284--285). By default, their recommendations
are followed exactly.
If \code{correction="translate"} then the translation correction
is used. The estimate is given in equation (15.15), page 284 of
Stoyan and Stoyan (1994).
If \code{correction="ripley"} then Ripley's isotropic edge correction
is used; the estimate is given in equation (15.18), page 285
of Stoyan and Stoyan (1994).
If \code{correction=c("translate", "ripley")} then both estimates
will be computed.
The choice of smoothing kernel is controlled by the
argument \code{kernel} which is passed to \code{\link{density}}.
The default is the Epanechnikov kernel, recommended by
Stoyan and Stoyan (1994, page 285).
The bandwidth of the smoothing kernel can be controlled by the
argument \code{bw}. Its precise interpretation
is explained in the documentation for \code{\link{density}}.
For the Epanechnikov kernel, the argument \code{bw} is
equivalent to \eqn{h/\sqrt{5}}{h/sqrt(5)}.
Stoyan and Stoyan (1994, page 285) recommend using the Epanechnikov
kernel with support \eqn{[-h,h]} chosen by the rule of thumn
\eqn{h = c/\sqrt{\lambda}}{h = c/sqrt(lambda)},
where \eqn{\lambda}{lambda} is the (estimated) intensity of the
point process, and \eqn{c} is a constant in the range from 0.1 to 0.2.
See equation (15.16).
If \code{bw} is missing, then this rule of thumb will be applied.
The argument \code{stoyan} determines the value of \eqn{c}.
The argument \code{r} is the vector of values for the
distance \eqn{r} at which \eqn{g(r)} should be evaluated.
There is a sensible default.
If it is specified, \code{r} must be a vector of increasing numbers
starting from \code{r[1] = 0},
and \code{max(r)} must not exceed half the diameter of
the window.
}
\references{
Stoyan, D. and Stoyan, H. (1994)
Fractals, random shapes and point fields:
methods of geometrical statistics.
John Wiley and Sons.
}
\seealso{
\code{\link{Kest}},
\code{\link{pcf}},
\code{\link{density}}
}
\examples{
data(simdat)
\testonly{
simdat <- simdat[seq(1,simdat$n, by=4)]
}
p <- pcf(simdat)
plot(p, main="pair correlation function for simdat")
# indicates inhibition at distances r < 0.3
}
\author{Adrian Baddeley
\email{adrian@maths.uwa.edu.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
and Rolf Turner
\email{r.turner@auckland.ac.nz}
}
\keyword{spatial}
\keyword{nonparametric}