\name{Gmulti}
\alias{Gmulti}
\title{
Marked Nearest Neighbour Distance Function
}
\description{
For a marked point pattern,
estimate the distribution of the distance
from a typical point in subset \code{I}
to the nearest point of subset \eqn{J}.
}
\synopsis{
Gmulti(X, I, J, r=NULL, breaks=NULL, \dots)
}
\usage{
Gmulti(X, I, J)
Gmulti(X, I, J, r)
Gmulti(X, I, J, breaks)
}
\arguments{
\item{X}{The observed point pattern,
from which an estimate of the multitype distance distribution function
\eqn{G_{IJ}(r)}{GIJ(r)} will be computed.
It must be a marked point pattern.
See under Details.
}
\item{I}{Subset of points of \code{X} from which distances are
measured.
}
\item{J}{Subset of points in \code{X} to which distances are measured.
}
\item{r}{numeric vector. The values of the argument \eqn{r}
at which the distribution function
\eqn{G_{IJ}(r)}{GIJ(r)} should be evaluated.
There is a sensible default.
First-time users are strongly advised not to specify this argument.
See below for important conditions on \eqn{r}.
}
\item{breaks}{An alternative to the argument \code{r}.
Not normally invoked by the user. See the \bold{Details} section.
}
}
\value{
An object of class \code{"fv"} (see \code{\link{fv.object}}).
Essentially a data frame containing six numeric columns
\item{r}{the values of the argument \eqn{r}
at which the function \eqn{G_{IJ}(r)}{GIJ(r)} has been estimated
}
\item{rs}{the ``reduced sample'' or ``border correction''
estimator of \eqn{G_{IJ}(r)}{GIJ(r)}
}
\item{km}{the spatial Kaplan-Meier estimator of \eqn{G_{IJ}(r)}{GIJ(r)}
}
\item{hazard}{the hazard rate \eqn{\lambda(r)}{lambda(r)}
of \eqn{G_{IJ}(r)}{GIJ(r)} by the spatial Kaplan-Meier method
}
\item{raw}{the uncorrected estimate of \eqn{G_{IJ}(r)}{GIJ(r)},
i.e. the empirical distribution of the distances from
each point of type \eqn{i} to the nearest point of type \eqn{j}
}
\item{theo}{the theoretical value of \eqn{G_{IJ}(r)}{GIJ(r)}
for a marked Poisson process with the same estimated intensity
}
}
\details{
The function \code{Gmulti}
generalises \code{\link{Gest}} (for unmarked point
patterns) and \code{\link{Gdot}} and \code{\link{Gcross}} (for
multitype point patterns) to arbitrary marked point patterns.
Suppose \eqn{X_I}{X[I]}, \eqn{X_J}{X[J]} are subsets, possibly
overlapping, of a marked point process. This function computes an
estimate of the cumulative
distribution function \eqn{G_{IJ}(r)}{GIJ(r)} of the distance
from a typical point of \eqn{X_I}{X[I]} to the nearest distinct point of
\eqn{X_J}{X[J]}.
The argument \code{X} must be a point pattern (object of class
\code{"ppp"}) or any data that are acceptable to \code{\link{as.ppp}}.
The arguments \code{I} and \code{J} specify two subsets of the
point pattern. They may be logical vectors of length equal to
\code{X$n}, or integer vectors with entries in the range 1 to
\code{X$n}, etc.
This algorithm estimates the distribution function \eqn{G_{IJ}(r)}{GIJ(r)}
from the point pattern \code{X}. It assumes that \code{X} can be treated
as a realisation of a stationary (spatially homogeneous)
random spatial point process in the plane, observed through
a bounded window.
The window (which is specified in \code{X} as \code{X$window})
may have arbitrary shape.
Biases due to edge effects are
treated in the same manner as in \code{\link{Gest}}.
The argument \code{r} is the vector of values for the
distance \eqn{r} at which \eqn{G_{IJ}(r)}{GIJ(r)} should be evaluated.
It is also used to determine the breakpoints
(in the sense of \code{\link{hist}})
for the computation of histograms of distances. The reduced-sample and
Kaplan-Meier estimators are computed from histogram counts.
In the case of the Kaplan-Meier estimator this introduces a discretisation
error which is controlled by the fineness of the breakpoints.
First-time users would be strongly advised not to specify \code{r}.
However, if it is specified, \code{r} must satisfy \code{r[1] = 0},
and \code{max(r)} must be larger than the radius of the largest disc
contained in the window. Furthermore, the successive entries of \code{r}
must be finely spaced.
The algorithm also returns an estimate of the hazard rate function,
\eqn{\lambda(r)}{lambda(r)}, of \eqn{G_{IJ}(r)}{GIJ(r)}.
This estimate should be used with caution as \eqn{G_{IJ}(r)}{GIJ(r)}
is not necessarily differentiable.
The naive empirical distribution of distances from each point of
the pattern \code{X} to the nearest other point of the pattern,
is a biased estimate of \eqn{G_{IJ}}{GIJ}.
However this is also returned by the algorithm, as it is sometimes
useful in other contexts. Care should be taken not to use the uncorrected
empirical \eqn{G_{IJ}}{GIJ} as if it were an unbiased estimator of
\eqn{G_{IJ}}{GIJ}.
}
\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.
Diggle, P. J. (1986).
Displaced amacrine cells in the retina of a
rabbit : analysis of a bivariate spatial point pattern.
\emph{J. Neurosci. Meth.} \bold{18}, 115--125.
Harkness, R.D and Isham, V. (1983)
A bivariate spatial point pattern of ants' nests.
\emph{Applied Statistics} \bold{32}, 293--303
Lotwick, H. W. and Silverman, B. W. (1982).
Methods for analysing spatial processes of several types of points.
\emph{J. Royal Statist. Soc. Ser. B} \bold{44}, 406--413.
Ripley, B.D. \emph{Statistical inference for spatial processes}.
Cambridge University Press, 1988.
Stoyan, D, Kendall, W.S. and Mecke, J.
\emph{Stochastic geometry and its applications}.
2nd edition. Springer Verlag, 1995.
Van Lieshout, M.N.M. and Baddeley, A.J. (1999)
Indices of dependence between types in multivariate point patterns.
\emph{Scandinavian Journal of Statistics} \bold{26}, 511--532.
}
\section{Warnings}{
The function \eqn{G_{IJ}}{GIJ} does not necessarily have a density.
The reduced sample estimator of \eqn{G_{IJ}}{GIJ} is pointwise approximately
unbiased, but need not be a valid distribution function; it may
not be a nondecreasing function of \eqn{r}. Its range is always
within \eqn{[0,1]}.
The spatial Kaplan-Meier estimator of \eqn{G_{IJ}}{GIJ}
is always nondecreasing
but its maximum value may be less than \eqn{1}.
}
\seealso{
\code{\link{Gcross}},
\code{\link{Gdot}},
\code{\link{Gest}}
}
\examples{
data(longleaf)
# Longleaf Pine data: marks represent diameter
\testonly{
longleaf <- longleaf[seq(1,longleaf$n, by=50), ]
}
Gm <- Gmulti(longleaf, longleaf$marks <= 15, longleaf$marks >= 25)
plot(Gm)
}
\author{Adrian Baddeley
\email{adrian@maths.uwa.edu.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
and Rolf Turner
\email{rolf@math.unb.ca}
\url{http://www.math.unb.ca/~rolf}
}
\keyword{spatial}