Jmulti.Rd
\name{Jmulti}
\alias{Jmulti}
\title{
Marked J Function
}
\description{
For a marked point pattern,
estimate the multitype \eqn{J} function
summarising dependence between the
points in subset \eqn{I}
and those in subset \eqn{J}.
}
\usage{
Jmulti(X, I, J, eps=NULL, r=NULL, breaks=NULL, \dots, disjoint=NULL,
correction=NULL)
}
\arguments{
\item{X}{The observed point pattern,
from which an estimate of the multitype distance distribution function
\eqn{J_{IJ}(r)}{J[IJ](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. See Details.
}
\item{J}{Subset of points in \code{X} to which distances are measured.
See Details.
}
\item{eps}{A positive number.
The pixel resolution of the discrete approximation to Euclidean
distance (see \code{\link{Jest}}). There is a sensible default.
}
\item{r}{numeric vector. The values of the argument \eqn{r}
at which the distribution function
\eqn{J_{IJ}(r)}{J[IJ](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.
}
\item{\dots}{Ignored.}
\item{disjoint}{Optional flag indicating whether
the subsets \code{I} and \code{J} are disjoint.
If missing, this value will be computed by inspecting the
vectors \code{I} and \code{J}.
}
\item{correction}{
Optional. Character string specifying the edge correction(s)
to be used. Options are \code{"none"}, \code{"rs"}, \code{"km"},
\code{"Hanisch"} and \code{"best"}.
}
}
\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{J_{IJ}(r)}{J[IJ](r)} has been estimated
}
\item{rs}{the ``reduced sample'' or ``border correction''
estimator of \eqn{J_{IJ}(r)}{J[IJ](r)}
}
\item{km}{the spatial Kaplan-Meier estimator of \eqn{J_{IJ}(r)}{J[IJ](r)}
}
\item{han}{the Hanisch-style estimator of \eqn{J_{IJ}(r)}{J[IJ](r)}
}
\item{un}{the uncorrected estimate of \eqn{J_{IJ}(r)}{J[IJ](r)},
formed by taking the ratio of uncorrected empirical estimators
of \eqn{1 - G_{IJ}(r)}{1 - G[IJ](r)}
and \eqn{1 - F_{J}(r)}{1 - F[J](r)}, see
\code{\link{Gdot}} and \code{\link{Fest}}.
}
\item{theo}{the theoretical value of \eqn{J_{IJ}(r)}{J[IJ](r)}
for a marked Poisson process with the same estimated intensity,
namely 1.
}
}
\details{
The function \code{Jmulti}
generalises \code{\link{Jest}} (for unmarked point
patterns) and \code{\link{Jdot}} and \code{\link{Jcross}} (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. Define
\deqn{J_{IJ}(r) = \frac{1 - G_{IJ}(r)}{1 - F_J(r)}}{
J[IJ](r) = (1 - G[IJ](r))/(1 - F[J](r))}
where \eqn{F_J(r)}{F[J](r)} is the cumulative distribution function of
the distance from a fixed location to the nearest point
of \eqn{X_J}{X[J]}, and \eqn{G_{IJ}(r)}{GJ(r)}
is the distribution function 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 any type of subset indices, for example,
logical vectors of length equal to \code{npoints(X)},
or integer vectors with entries in the range 1 to
\code{npoints(X)}, or negative integer vectors.
Alternatively, \code{I} and \code{J} may be \bold{functions}
that will be applied to the point pattern \code{X} to obtain
index vectors. If \code{I} is a function, then evaluating
\code{I(X)} should yield a valid subset index. This option
is useful when generating simulation envelopes using
\code{\link{envelope}}.
It is assumed 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$window})
may have arbitrary shape.
Biases due to edge effects are
treated in the same manner as in \code{\link{Jest}}.
The argument \code{r} is the vector of values for the
distance \eqn{r} at which \eqn{J_{IJ}(r)}{J[IJ](r)} should be evaluated.
It is also used to determine the breakpoints
(in the sense of \code{\link{hist}})
for the computation of histograms of distances. The reduced-sample and
Kaplan-Meier estimators are computed from histogram counts.
In the case of the Kaplan-Meier estimator this introduces a discretisation
error which is controlled by the fineness of the breakpoints.
First-time users would be strongly advised not to specify \code{r}.
However, if it is specified, \code{r} must satisfy \code{r[1] = 0},
and \code{max(r)} must be larger than the radius of the largest disc
contained in the window. Furthermore, the successive entries of \code{r}
must be finely spaced.
}
\references{
Van Lieshout, M.N.M. and Baddeley, A.J. (1999)
Indices of dependence between types in multivariate point patterns.
\emph{Scandinavian Journal of Statistics} \bold{26}, 511--532.
}
\seealso{
\code{\link{Jcross}},
\code{\link{Jdot}},
\code{\link{Jest}}
}
\examples{
data(longleaf)
# Longleaf Pine data: marks represent diameter
\testonly{
longleaf <- longleaf[seq(1,longleaf$n, by=50), ]
}
Jm <- Jmulti(longleaf, marks(longleaf) <= 15, marks(longleaf) >= 25)
plot(Jm)
}
\author{Adrian Baddeley
\email{Adrian.Baddeley@csiro.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
and Rolf Turner
\email{r.turner@auckland.ac.nz}
}
\keyword{spatial}
\keyword{nonparametric}