Jcross.Rd
\name{Jcross}
\alias{Jcross}
\title{
Multitype J Function (i-to-j)
}
\description{
For a multitype point pattern,
estimate the multitype \eqn{J} function
summarising the interpoint dependence between
points of type \eqn{i} and of type \eqn{j}.
}
\usage{
Jcross(X, i, j, eps=NULL, r=NULL, breaks=NULL, \dots, correction=NULL)
}
\arguments{
\item{X}{The observed point pattern,
from which an estimate of the multitype \eqn{J} function
\eqn{J_{ij}(r)}{Jij(r)} will be computed.
It must be a multitype point pattern (a marked point pattern
whose marks are a factor). See under Details.
}
\item{i}{The type (mark value)
of the points in \code{X} from which distances are measured.
A character string (or something that will be converted to a
character string).
Defaults to the first level of \code{marks(X)}.
}
\item{j}{The type (mark value)
of the points in \code{X} to which distances are measured.
A character string (or something that will be
converted to a character string).
Defaults to the second level of \code{marks(X)}.
}
\item{eps}{A positive number.
The resolution of the discrete approximation to Euclidean
distance (see below). There is a sensible default.
}
\item{r}{Optional. Numeric vector. The values of the argument \eqn{r}
at which the function
\eqn{J_{ij}(r)}{Jij(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}{
This argument is for internal use only.
}
\item{\dots}{Ignored.}
\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"}.
Alternatively \code{correction="all"} selects all options.
}
}
\value{
An object of class \code{"fv"} (see \code{\link{fv.object}}).
Essentially a data frame containing six numeric columns
\item{J}{the recommended
estimator of \eqn{J_{ij}(r)}{Jij(r)},
currently the Kaplan-Meier estimator.
}
\item{r}{the values of the argument \eqn{r}
at which the function \eqn{J_{ij}(r)}{Jij(r)} has been estimated
}
\item{km}{the Kaplan-Meier
estimator of \eqn{J_{ij}(r)}{Jij(r)}
}
\item{rs}{the ``reduced sample'' or ``border correction''
estimator of \eqn{J_{ij}(r)}{Jij(r)}
}
\item{han}{the Hanisch-style
estimator of \eqn{J_{ij}(r)}{Jij(r)}
}
\item{un}{the ``uncorrected''
estimator of \eqn{J_{ij}(r)}{Jij(r)}
formed by taking the ratio of uncorrected empirical estimators
of \eqn{1 - G_{ij}(r)}{1 - Gij(r)}
and \eqn{1 - F_{j}(r)}{1 - Fj(r)}, see
\code{\link{Gdot}} and \code{\link{Fest}}.
}
\item{theo}{the theoretical value of \eqn{J_{ij}(r)}{Jij(r)}
for a marked Poisson process, namely 1.
}
The result also has two attributes \code{"G"} and \code{"F"}
which are respectively the outputs of \code{\link{Gcross}}
and \code{\link{Fest}} for the point pattern.
}
\details{
This function \code{Jcross} and its companions
\code{\link{Jdot}} and \code{\link{Jmulti}}
are generalisations of the function \code{\link{Jest}}
to multitype point patterns.
A multitype point pattern is a spatial pattern of
points classified into a finite number of possible
``colours'' or ``types''. In the \pkg{spatstat} package,
a multitype pattern is represented as a single
point pattern object in which the points carry marks,
and the mark value attached to each point
determines the type of that point.
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}}.
It must be a marked point pattern, and the mark vector
\code{X$marks} must be a factor.
The argument \code{i} will be interpreted as a
level of the factor \code{X$marks}. (Warning: this means that
an integer value \code{i=3} will be interpreted as the number 3,
\bold{not} the 3rd smallest level).
The ``type \eqn{i} to type \eqn{j}'' multitype \eqn{J} function
of a stationary multitype point process \eqn{X}
was introduced by Van lieshout and Baddeley (1999). It is defined by
\deqn{J_{ij}(r) = \frac{1 - G_{ij}(r)}{1 -
F_{j}(r)}}{Jij(r) = (1 - Gij(r))/(1-Fj(r))}
where \eqn{G_{ij}(r)}{Gij(r)} is the distribution function of
the distance from a type \eqn{i} point to the nearest point of type \eqn{j},
and \eqn{F_{j}(r)}{Fj(r)} is the distribution
function of the distance from a fixed point in space to the nearest
point of type \eqn{j} in the pattern.
An estimate of \eqn{J_{ij}(r)}{Jij(r)}
is a useful summary statistic in exploratory data analysis
of a multitype point pattern.
If the subprocess of type \eqn{i} points is independent
of the subprocess of points of type \eqn{j},
then \eqn{J_{ij}(r) \equiv 1}{Jij(r) = 1}.
Hence deviations of the empirical estimate of
\eqn{J_{ij}}{Jij} from the value 1
may suggest dependence between types.
This algorithm estimates \eqn{J_{ij}(r)}{Jij(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{Window(X)})
may have arbitrary shape.
Biases due to edge effects are
treated in the same manner as in \code{\link{Jest}},
using the Kaplan-Meier and border corrections.
The main work is done by \code{\link{Gmulti}} and \code{\link{Fest}}.
The argument \code{r} is the vector of values for the
distance \eqn{r} at which \eqn{J_{ij}(r)}{Jij(r)} should be evaluated.
The values of \eqn{r} must be increasing nonnegative numbers
and the maximum \eqn{r} value must exceed the radius of the
largest disc contained in the window.
}
\references{
Van Lieshout, M.N.M. and Baddeley, A.J. (1996)
A nonparametric measure of spatial interaction in point patterns.
\emph{Statistica Neerlandica} \bold{50}, 344--361.
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 arguments \code{i} and \code{j} are always interpreted as
levels of the factor \code{X$marks}. They are converted to character
strings if they are not already character strings.
The value \code{i=1} does \bold{not}
refer to the first level of the factor.
}
\seealso{
\code{\link{Jdot}},
\code{\link{Jest}},
\code{\link{Jmulti}}
}
\examples{
# Lansing woods data: 6 types of trees
woods <- lansing
\testonly{
woods <- woods[seq(1,npoints(woods), by=30)]
}
Jhm <- Jcross(woods, "hickory", "maple")
# diagnostic plot for independence between hickories and maples
plot(Jhm)
# synthetic example with two types "a" and "b"
pp <- runifpoint(30) \%mark\% factor(sample(c("a","b"), 30, replace=TRUE))
J <- Jcross(pp)
}
\author{
\spatstatAuthors.
}
\keyword{spatial}
\keyword{nonparametric}
