pcfdot.Rd
\name{pcfdot}
\alias{pcfdot}
\title{Multitype pair correlation function (i-to-any)}
\description{
Calculates an estimate of the multitype pair correlation function
(from points of type \code{i} to points of any type)
for a multitype point pattern.
}
\usage{
pcfdot(X, i, ...)
}
\arguments{
\item{X}{The observed point pattern,
from which an estimate of the dot-type pair correlation function
\eqn{g_{i\bullet}(r)}{gdot[i](r)} will be computed.
It must be a multitype point pattern (a marked point pattern
whose marks are a factor).
}
\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{\dots}{
Arguments passed to \code{\link{pcf.ppp}}.
}
}
\details{
This is a generalisation of the pair correlation function \code{\link{pcf}}
to multitype point patterns.
For two locations \eqn{x} and \eqn{y} separated by a nonzero
distance \eqn{r},
the probability \eqn{p(r)} of finding a point of type \eqn{i} at location
\eqn{x} and a point of any type at location \eqn{y} is
\deqn{
p(r) = \lambda_i \lambda g_{i\bullet}(r) \,{\rm d}x \, {\rm d}y
}{
p(r) = lambda[i] * lambda * gdot[i](r) dx dy
}
where \eqn{\lambda}{lambda} is the intensity of all points,
and \eqn{\lambda_i}{lambda[i]} is the intensity of the points
of type \eqn{i}.
For a completely random Poisson marked point process,
\eqn{p(r) = \lambda_i \lambda}{p(r) = lambda[i] * lambda}
so \eqn{g_{i\bullet}(r) = 1}{gdot[i](r) = 1}.
For a stationary multitype point process, the
type-\code{i}-to-any-type pair correlation
function between marks \eqn{i} and \eqn{j} is formally defined as
\deqn{
g_{i\bullet}(r) = \frac{K_{i\bullet}^\prime(r)}{2\pi r}
}{
g(r) = Kdot[i]'(r)/ ( 2 * pi * r)
}
where \eqn{K_{i\bullet}^\prime}{Kdot[i]'(r)} is the derivative of
the type-\code{i}-to-any-type \eqn{K} function
\eqn{K_{i\bullet}(r)}{Kdot[i](r)}.
of the point process. See \code{\link{Kdot}} for information
about \eqn{K_{i\bullet}(r)}{Kdot[i](r)}.
The command \code{pcfdot} computes a kernel estimate of
the multitype pair correlation function from points of type \eqn{i}
to points of any type.
It uses \code{\link{pcf.ppp}} to compute kernel estimates
of the pair correlation functions for several unmarked point patterns,
and uses the bilinear properties of second moments to obtain the
multitype pair correlation.
See \code{\link{pcf.ppp}} for a list of arguments that control
the kernel estimation.
The companion function \code{\link{pcfcross}} computes the
corresponding analogue of \code{\link{Kcross}}.
}
\value{
An object of class \code{"fv"}, see \code{\link{fv.object}},
which can be plotted directly using \code{\link{plot.fv}}.
Essentially a data frame containing columns
\item{r}{the vector of values of the argument \eqn{r}
at which the function \eqn{g_{i\bullet}}{gdot[i]} has been estimated
}
\item{theo}{the theoretical value \eqn{g_{i\bullet}(r) = 1}{gdot[i](r) = r}
for independent marks.
}
together with columns named
\code{"border"}, \code{"bord.modif"},
\code{"iso"} and/or \code{"trans"},
according to the selected edge corrections. These columns contain
estimates of the function \eqn{g_{i,j}}{g[i,j]}
obtained by the edge corrections named.
}
\seealso{
Mark connection function \code{\link{markconnect}}.
Multitype pair correlation \code{\link{pcfcross}}.
Pair correlation \code{\link{pcf}},\code{\link{pcf.ppp}}.
\code{\link{Kdot}}
}
\examples{
data(amacrine)
p <- pcfdot(amacrine, "on")
p <- pcfdot(amacrine, "on", stoyan=0.1)
plot(p)
}
\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}