K3est.Rd
\name{K3est}
\Rdversion{1.1}
\alias{K3est}
\title{
K-function of a Three-Dimensional Point Pattern
}
\description{
Estimates the \eqn{K}-function from a three-dimensional point pattern.
}
\usage{
K3est(X, ..., rmax = NULL, nrval = 128, correction = c("translation", "isotropic"))
}
\arguments{
\item{X}{
Three-dimensional point pattern (object of class \code{"pp3"}).
}
\item{\dots}{
Ignored.
}
\item{rmax}{
Optional. Maximum value of argument \eqn{r} for which
\eqn{K_3(r)}{K3(r)} will be estimated.
}
\item{nrval}{
Optional. Number of values of \eqn{r} for which
\eqn{K_3(r)}{K3(r)} will be estimated. A large value of \code{nrval}
is required to avoid discretisation effects.
}
\item{correction}{
Optional. Character vector specifying the edge correction(s)
to be applied. See Details.
}
}
\details{
For a stationary point process \eqn{\Phi}{Phi} in three-dimensional
space, the three-dimensional \eqn{K} function
is
\deqn{
K_3(r) = \frac 1 \lambda E(N(\Phi, x, r) \mid x \in \Phi)
}{
K3(r) = (1/lambda) E(N(Phi,x,r) | x in Phi)
}
where \eqn{\lambda}{lambda} is the intensity of the process
(the expected number of points per unit volume) and
\eqn{N(\Phi,x,r)}{N(Phi,x,r)} is the number of points of
\eqn{\Phi}{Phi}, other than \eqn{x} itself, which fall within a
distance \eqn{r} of \eqn{x}. This is the three-dimensional
generalisation of Ripley's \eqn{K} function for two-dimensional
point processes (Ripley, 1977).
The three-dimensional point pattern \code{X} is assumed to be a
partial realisation of a stationary point process \eqn{\Phi}{Phi}.
The distance between each pair of distinct points is computed.
The empirical cumulative distribution
function of these values, with appropriate edge corrections, is
renormalised to give the estimate of \eqn{K_3(r)}{K3(r)}.
The available edge corrections are:
\describe{
\item{\code{"translation"}:}{
the Ohser translation correction estimator
(Ohser, 1983; Baddeley et al, 1993)
}
\item{\code{"isotropic"}:}{
the three-dimensional counterpart of
Ripley's isotropic edge correction (Ripley, 1977; Baddeley et al, 1993).
}
}
}
\value{
A function value table (object of class \code{"fv"}) that can be
plotted, printed or coerced to a data frame containing the function values.
}
\references{
Baddeley, A.J, Moyeed, R.A., Howard, C.V. and Boyde, A. (1993)
Analysis of a three-dimensional point pattern with replication.
\emph{Applied Statistics} \bold{42}, 641--668.
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. (1977)
Modelling spatial patterns (with discussion).
\emph{Journal of the Royal Statistical Society, Series B},
\bold{39}, 172 -- 212.
}
\author{
Adrian Baddeley
\email{Adrian.Baddeley@csiro.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
and Rana Moyeed.
}
\seealso{
\code{\link{F3est}},
\code{\link{G3est}},
\code{\link{pcf3est}}
}
\examples{
X <- rpoispp3(42)
Z <- K3est(X)
if(interactive()) plot(Z)
}
\keyword{spatial}
\keyword{nonparametric}