pcfmulti.Rd
\name{pcfmulti}
\alias{pcfmulti}
\title{
Marked pair correlation function
}
\description{
For a marked point pattern,
estimate the multitype pair correlation function
using kernel methods.
}
\usage{
pcfmulti(X, I, J, ..., r = NULL,
kernel = "epanechnikov", bw = NULL, stoyan = 0.15,
correction = c("translate", "Ripley"),
divisor = c("r", "d"),
Iname = "points satisfying condition I",
Jname = "points satisfying condition J")
}
\arguments{
\item{X}{The observed point pattern,
from which an estimate of the cross-type pair correlation function
\eqn{g_{ij}(r)}{g[i,j](r)} will be computed.
It must be a multitype point pattern (a marked point pattern
whose marks are a factor).
}
\item{I}{Subset index specifying the points of \code{X}
from which distances are measured.
}
\item{J}{Subset index specifying the points in \code{X} to which
distances are measured.
}
\item{\dots}{
Ignored.
}
\item{r}{
Vector of values for the argument \eqn{r} at which \eqn{g(r)}
should be evaluated. There is a sensible default.
}
\item{kernel}{
Choice of smoothing kernel,
passed to \code{\link{density.default}}.
}
\item{bw}{
Bandwidth for smoothing kernel,
passed to \code{\link{density.default}}.
}
\item{stoyan}{
Coefficient for default bandwidth rule.
}
\item{correction}{
Choice of edge correction.
}
\item{divisor}{
Choice of divisor in the estimation formula:
either \code{"r"} (the default) or \code{"d"}.
}
\item{Iname,Jname}{
Optional. Character strings describing the members of
the subsets \code{I} and \code{J}.
}
}
\details{
This is a generalisation of \code{\link{pcfcross}}
to arbitrary collections of points.
The algorithm measures the distance from each data point
in subset \code{I} to each data point in subset \code{J},
excluding identical pairs of points. The distances are
kernel-smoothed and renormalised to form a pair correlation
function.
\itemize{
\item
If \code{divisor="r"} (the default), then the multitype
counterpart of the standard
kernel estimator (Stoyan and Stoyan, 1994, pages 284--285)
is used. By default, the recommendations of Stoyan and Stoyan (1994)
are followed exactly.
\item
If \code{divisor="d"} then a modified estimator is used:
the contribution from
an interpoint distance \eqn{d_{ij}}{d[ij]} to the
estimate of \eqn{g(r)} is divided by \eqn{d_{ij}}{d[ij]}
instead of dividing by \eqn{r}. This usually improves the
bias of the estimator when \eqn{r} is close to zero.
}
There is also a choice of spatial edge corrections
(which are needed to avoid bias due to edge effects
associated with the boundary of the spatial window):
\code{correction="translate"} is the Ohser-Stoyan translation
correction, and \code{correction="isotropic"} or \code{"Ripley"}
is Ripley's isotropic correction.
The arguments \code{I} and \code{J} specify two subsets of the
point pattern \code{X}. 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}}.
The choice of smoothing kernel is controlled by the
argument \code{kernel} which is passed to \code{\link{density}}.
The default is the Epanechnikov kernel.
The bandwidth of the smoothing kernel can be controlled by the
argument \code{bw}. Its precise interpretation
is explained in the documentation for \code{\link{density.default}}.
For the Epanechnikov kernel with support \eqn{[-h,h]},
the argument \code{bw} is equivalent to \eqn{h/\sqrt{5}}{h/sqrt(5)}.
If \code{bw} is not specified, the default bandwidth
is determined by Stoyan's rule of thumb (Stoyan and Stoyan, 1994, page
285) applied to the points of type \code{j}. That is,
\eqn{h = c/\sqrt{\lambda}}{h = c/sqrt(lambda)},
where \eqn{\lambda}{lambda} is the (estimated) intensity of the
point process of type \code{j},
and \eqn{c} is a constant in the range from 0.1 to 0.2.
The argument \code{stoyan} determines the value of \eqn{c}.
}
\value{
An object of class \code{"fv"}.
}
\seealso{
\code{\link{pcfcross}},
\code{\link{pcfdot}},
\code{\link{pcf.ppp}}.
}
\examples{
adult <- (marks(longleaf) >= 30)
juvenile <- !adult
p <- pcfmulti(longleaf, adult, juvenile)
}
\author{\adrian
and \rolf
}
\keyword{spatial}
\keyword{nonparametric}