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, ..., r = NULL,
kernel = "epanechnikov", bw = NULL, stoyan = 0.15,
correction = c("isotropic", "Ripley", "translate"),
divisor = c("r", "d"))
}
\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}{
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; see Details.
}
\item{correction}{
Choice of edge correction.
}
\item{divisor}{
Choice of divisor in the estimation formula:
either \code{"r"} (the default) or \code{"d"}. See Details.
}
}
\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.
\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 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). That is,
\eqn{h = c/\sqrt{\lambda}}{h = c/sqrt(lambda)},
where \eqn{\lambda}{lambda} is the (estimated) intensity of the
unmarked point process,
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}.
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}}, \code{\link{pcfmulti}}.
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
and \rolf
}
\keyword{spatial}
\keyword{nonparametric}