pcf.fv.Rd
\name{pcf.fv}
\alias{pcf.fv}
\title{Pair Correlation Function obtained from K Function}
\description{
Estimates the pair correlation function of
a point pattern, given an estimate of the K function.
}
\usage{
\method{pcf}{fv}(X, \dots, method="c")
}
\arguments{
\item{X}{
An estimate of the \eqn{K} function
or one of its variants.
An object of class \code{"fv"}.
}
\item{\dots}{
Arguments controlling the smoothing spline
function \code{smooth.spline}.
}
\item{method}{
Letter \code{"a"}, \code{"b"}, \code{"c"} or \code{"d"} indicating the
method for deriving the pair correlation function from the
\code{K} function.
}
}
\value{
A function value table
(object of class \code{"fv"}, see \code{\link{fv.object}})
representing a pair correlation function.
Essentially a data frame containing (at least) 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{pcf}{vector of values of \eqn{g(r)}
}
}
\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)}
from an estimate of \eqn{K(r)} or its variants,
using smoothing splines to approximate the derivative.
It is a method for the generic function \code{\link{pcf}}
for the class \code{"fv"}.
The argument \code{X} should be an estimated \eqn{K} function,
given as a function value table (object of class \code{"fv"},
see \code{\link{fv.object}}).
This object should be the value returned by
\code{\link{Kest}}, \code{\link{Kcross}}, \code{\link{Kmulti}}
or \code{\link{Kinhom}}.
The smoothing spline operations are performed by
\code{\link{smooth.spline}} and \code{\link{predict.smooth.spline}}
from the \code{modreg} library.
Four numerical methods are available:
\itemize{
\item
\bold{"a"} apply smoothing to \eqn{K(r)},
estimate its derivative, and plug in to the formula above;
\item
\bold{"b"} apply smoothing to
\eqn{Y(r) = \frac{K(r)}{2 \pi r}}{Y(r) = K(r)/(2 * pi * r)}
constraining \eqn{Y(0) = 0},
estimate the derivative of \eqn{Y}, and solve;
\item
\bold{"c"} apply smoothing to
\eqn{Z(r) = \frac{K(r)}{\pi r^2}}{Y(r) = K(r)/(pi * r^2)}
constraining \eqn{Z(0)=1},
estimate its derivative, and solve.
\item
\bold{"d"} apply smoothing to
\eqn{V(r) = \sqrt{K(r)}}{V(r) = sqrt(K(r))},
estimate its derivative, and solve.
}
Method \code{"c"} seems to be the best at
suppressing variability for small values of \eqn{r}.
However it effectively constrains \eqn{g(0) = 1}.
If the point pattern seems to have inhibition at small distances,
you may wish to experiment with method \code{"b"} which effectively
constrains \eqn{g(0)=0}. Method \code{"a"} seems
comparatively unreliable.
Useful arguments to control the splines
include the smoothing tradeoff parameter \code{spar}
and the degrees of freedom \code{df}. See \code{\link{smooth.spline}}
for details.
}
\references{
Stoyan, D, Kendall, W.S. and Mecke, J. (1995)
\emph{Stochastic geometry and its applications}.
2nd edition. Springer Verlag.
Stoyan, D. and Stoyan, H. (1994)
Fractals, random shapes and point fields:
methods of geometrical statistics.
John Wiley and Sons.
}
\seealso{
\code{\link{pcf}},
\code{\link{pcf.ppp}},
\code{\link{Kest}},
\code{\link{Kinhom}},
\code{\link{Kcross}},
\code{\link{Kdot}},
\code{\link{Kmulti}},
\code{\link{alltypes}},
\code{\link{smooth.spline}},
\code{\link{predict.smooth.spline}}
}
\examples{
# univariate point pattern
data(simdat)
\testonly{
simdat <- simdat[seq(1,simdat$n, by=4)]
}
K <- Kest(simdat)
p <- pcf.fv(K, spar=0.5, method="b")
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}