\name{Kest.fft}
\alias{Kest.fft}
\title{K-function using FFT}
\description{
Estimates the reduced second moment function \eqn{K(r)}
from a point pattern in a window of arbitrary shape,
using the Fast Fourier Transform.
}
\usage{
Kest.fft(X, sigma, r=NULL, breaks=NULL)
}
\arguments{
\item{X}{The observed point pattern,
from which an estimate of \eqn{K(r)} will be computed.
An object of class \code{"ppp"}, or data
in any format acceptable to \code{\link{as.ppp}()}.
}
\item{sigma}{
standard deviation of the isotropic Gaussian
smoothing kernel.
}
\item{r}{
vector of values for the argument \eqn{r} at which \eqn{K(r)}
should be evaluated. There is a sensible default.
}
\item{breaks}{
An alternative to the argument \code{r}.
Not normally invoked by the user.
See Details.
}
}
\value{
An object of class \code{"fv"} (see \code{\link{fv.object}}).
Essentially a data frame containing columns
\item{r}{the vector of values of the argument \eqn{r}
at which the function \eqn{K} has been estimated
}
\item{border}{the estimates of \eqn{K(r)} for these values of \eqn{r}
}
\item{theo}{the theoretical value \eqn{K(r) = \pi r^2}{K(r) = pi * r^2}
for a stationary Poisson process
}
}
\details{
This is an alternative to the function \code{\link{Kest}}
for estimating the \eqn{K} function. It may be useful for
very large patterns of points.
Whereas \code{\link{Kest}} computes the distance between
each pair of points analytically, this function discretises the
point pattern onto a rectangular pixel raster and applies
Fast Fourier Transform techniques to estimate \eqn{K(t)}.
The hard work is done by the function \code{\link{Kmeasure}}.
The result is an approximation whose accuracy depends on the
resolution of the pixel raster. The resolution is controlled
by setting the parameter \code{npixel} in
\code{\link{spatstat.options}}.
}
\references{
Cressie, N.A.C. \emph{Statistics for spatial data}.
John Wiley and Sons, 1991.
Diggle, P.J. \emph{Statistical analysis of spatial point patterns}.
Academic Press, 1983.
Ohser, J. (1983)
On estimators for the reduced second moment measure of
point processes. \emph{Mathematische Operationsforschung und
Statistik, series Statistics}, \bold{14}, 63 -- 71.
Ripley, B.D. \emph{Statistical inference for spatial processes}.
Cambridge University Press, 1988.
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{Kest}},
\code{\link{Kmeasure}},
\code{\link{spatstat.options}}
}
\examples{
pp <- runifpoint(10000)
\dontrun{
spatstat.options(npixel=512)
}
\testonly{
spatstat.options(npixel=125)
}
Kpp <- Kest.fft(pp, 0.01)
plot(Kpp)
}
\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}