markcrosscorr.Rd
\name{markcrosscorr}
\alias{markcrosscorr}
\title{
Mark Cross-Correlation Function
}
\description{
Given a spatial point pattern with several columns of marks,
this function computes the mark correlation function between
each pair of columns of marks.
}
\usage{
markcrosscorr(X, r = NULL,
correction = c("isotropic", "Ripley", "translate"),
method = "density", \dots, normalise = TRUE, Xname = NULL)
}
\arguments{
\item{X}{The observed point pattern.
An object of class \code{"ppp"} or something acceptable to
\code{\link{as.ppp}}.
}
\item{r}{Optional. Numeric vector. The values of the argument \eqn{r}
at which the mark correlation function
\eqn{k_f(r)}{k[f](r)} should be evaluated.
There is a sensible default.
}
\item{correction}{
A character vector containing any selection of the
options \code{"isotropic"}, \code{"Ripley"}, \code{"translate"},
\code{"translation"}, \code{"none"} or \code{"best"}.
It specifies the edge correction(s) to be applied.
Alternatively \code{correction="all"} selects all options.
}
\item{method}{
A character vector indicating the user's choice of
density estimation technique to be used. Options are
\code{"density"},
\code{"loess"},
\code{"sm"} and \code{"smrep"}.
}
\item{\dots}{
Arguments passed to the density estimation routine
(\code{\link{density}}, \code{\link{loess}} or \code{sm.density})
selected by \code{method}.
}
\item{normalise}{
If \code{normalise=FALSE},
compute only the numerator of the expression for the
mark correlation.
}
\item{Xname}{
Optional character string name for the dataset \code{X}.
}
}
\details{
First, all columns of marks are converted to numerical values.
A factor with \eqn{m} possible levels is converted to
\eqn{m} columns of dummy (indicator) values.
Next, each pair of columns is considered, and the mark
cross-correlation is defined as
\deqn{
k_{mm}(r) = \frac{E_{0u}[M_i(0) M_j(u)]}{E[M_i,M_j]}
}{
k[mm](r) = E[0u](M(i,0) * M(j,u))/E(Mi * Mj)
}
where \eqn{E_{0u}}{E[0u]} denotes the conditional expectation
given that there are points of the process at the locations
\eqn{0} and \eqn{u} separated by a distance \eqn{r}.
On the numerator,
\eqn{M_i(0)}{M(i,0)} and \eqn{M_j(u)}{M(j,u)}
are the marks attached to locations \eqn{0} and \eqn{u} respectively
in the \eqn{i}th and \eqn{j}th columns of marks respectively.
On the denominator, \eqn{M_i}{Mi} and \eqn{M_j}{Mj} are
independent random values drawn from the
\eqn{i}th and \eqn{j}th columns of marks, respectively,
and \eqn{E} is the usual expectation.
Note that \eqn{k_{mm}(r)}{k[mm](r)} is not a ``correlation''
in the usual statistical sense. It can take any
nonnegative real value. The value 1 suggests ``lack of correlation'':
if the marks attached to the points of \code{X} are independent
and identically distributed, then
\eqn{k_{mm}(r) \equiv 1}{k[mm](r) = 1}.
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.
The cross-correlations are estimated in the same manner as
for \code{\link{markcorr}}.
}
\value{
A function array (object of class \code{"fasp"}) containing
the mark cross-correlation functions for each possible pair
of columns of marks.
}
\author{
\adrian
\rolf
and \ege
}
\seealso{
\code{\link{markcorr}}
}
\examples{
# The dataset 'betacells' has two columns of marks:
# 'type' (factor)
# 'area' (numeric)
if(interactive()) plot(betacells)
plot(markcrosscorr(betacells))
}
\keyword{spatial}
\keyword{nonparametric}