Revision d606122dc24b56ecf537d55eda38f4bf5ac4de1f authored by Adrian Baddeley on 25 October 2010, 10:40:51 UTC, committed by cran-robot on 25 October 2010, 10:40:51 UTC
1 parent 66bc933
pcfinhom.Rd
\name{pcfinhom}
\alias{pcfinhom}
\title{
Inhomogeneous Pair Correlation Function
}
\description{
Estimates the inhomogeneous pair correlation function of
a point pattern using kernel methods.
}
\usage{
pcfinhom(X, lambda = NULL, ..., r = NULL, kernel = "epanechnikov", bw = NULL, stoyan = 0.15, correction = c("translate", "Ripley"), sigma = NULL, varcov = NULL)
}
\arguments{
\item{X}{
A point pattern (object of class \code{"ppp"}).
}
\item{lambda}{
Optional.
Values of the estimated intensity function.
Either a vector giving the intensity values
at the points of the pattern \code{X},
a pixel image (object of class \code{"im"}) giving the
intensity values at all locations, or a \code{function(x,y)} which
can be evaluated to give the intensity value at any location.
}
\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{\dots}{
Other arguments passed to the kernel density estimation
function \code{\link{density.default}}.
}
\item{stoyan}{
Bandwidth coefficient; see Details.
}
\item{correction}{
Choice of edge correction.
}
\item{sigma,varcov}{
Optional arguments passed to \code{\link{density.ppp}}
to control the smoothing bandwidth, when \code{lambda} is
estimated by kernel smoothing.
}
}
\details{
The inhomogeneous pair correlation function \eqn{g_{\rm inhom}(r)}{ginhom(r)}
is a summary of the dependence between points in a spatial point
process that does not have a uniform density of points.
The best intuitive interpretation is the following: the probability
\eqn{p(r)} of finding two points at locations \eqn{x} and \eqn{y}
separated by a distance \eqn{r} is equal to
\deqn{
p(r) = \lambda(x) lambda(y) g(r) \,{\rm d}x \, {\rm d}y
}{
p(r) = lambda(x) * lambda(y) * g(r) dx dy
}
where \eqn{\lambda}{lambda} is the intensity function
of the point process.
For a Poisson point process with intensity function
\eqn{\lambda}{lambda}, this probability is
\eqn{p(r) = \lambda(x) \lambda(y)}{p(r) = lambda(x) * lambda(y)}
so \eqn{g_{\rm inhom}(r) = 1}{ginhom(r) = 1}.
The inhomogeneous pair correlation function
is related to the inhomogeneous \eqn{K} function through
\deqn{
g_{\rm inhom}(r) = \frac{K'_{\rm inhom}(r)}{2\pi r}
}{
ginhom(r) = Kinhom'(r)/ ( 2 * pi * r)
}
where \eqn{K'_{\rm inhom}(r)}{Kinhom'(r)}
is the derivative of \eqn{K_{\rm inhom}(r)}{Kinhom(r)}, the
inhomogeneous \eqn{K} function. See \code{\link{Kinhom}} for information
about \eqn{K_{\rm inhom}(r)}{Kinhom(r)}.
The command \code{pcfinhom} estimates the inhomogeneous
pair correlation using a modified version of
the algorithm in \code{\link{pcf.ppp}}.
}
\value{
A function value table (object of class \code{"fv"}).
Essentially a data frame containing the variables
\item{r}{
the vector of values of the argument \eqn{r}
at which the inhomogeneous pair correlation function
\eqn{g_{\rm inhom}(r)}{ginhom(r)} has been estimated
}
\item{theo}{vector of values equal to 1,
the theoretical value of \eqn{g_{\rm inhom}(r)}{ginhom(r)}
for the Poisson process
}
\item{trans}{vector of values of \eqn{g_{\rm inhom}(r)}{ginhom(r)}
estimated by translation correction
}
\item{iso}{vector of values of \eqn{g_{\rm inhom}(r)}{ginhom(r)}
estimated by Ripley isotropic correction
}
as required.
}
\seealso{
\code{\link{pcf}},
\code{\link{pcf.ppp}},
\code{\link{Kinhom}}
}
\examples{
data(residualspaper)
X <- residualspaper$Fig4b
plot(pcfinhom(X, stoyan=0.2, sigma=0.1))
}
\author{Adrian Baddeley
\email{adrian@maths.uwa.edu.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
and Rolf Turner
\email{r.turner@auckland.ac.nz}
}
\keyword{spatial}
\keyword{nonparametric}
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