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
localpcf.Rd
\name{localpcf}
\alias{localpcf}
\title{Local pair correlation function}
\description{
Computes individual contributions to the
pair correlation function from each data point.
}
\usage{
localpcf(X, ..., delta=NULL, rmax=NULL, nr=512, stoyan=0.15)
}
\arguments{
\item{X}{A point pattern (object of class \code{"ppp"}).}
\item{\dots}{Ignored.}
\item{delta}{
Smoothing bandwidth. The halfwidth of the Epanechnikov kernel.
}
\item{rmax}{
Optional. Maximum value of distance \eqn{r} for which
pair correlation values \eqn{g(r)} should be computed.
}
\item{nr}{
Optional. Number of values of distance \eqn{r} for which
pair correlation \eqn{g(r)} should be computed.
}
\item{stoyan}{
Optional. The value of the constant \eqn{c} in Stoyan's rule
of thumb for selecting the smoothing bandwidth \code{delta}.
}
}
\details{
\code{localpcf} computes the contribution, from each individual
data point in a point pattern \code{X}, to the
empirical pair correlation function of \code{X}.
These contributions are sometimes known as LISA (local indicator
of spatial association) functions based on pair correlation.
Given a spatial point pattern \code{X}, the local pcf
\eqn{g_i(r)}{g[i](r)} associated with the \eqn{i}th point
in \code{X} is computed by
\deqn{
g_i(r) = \frac a {2 \pi n} \sum_j k(d_{i,j} - r)
}{
g[i](r) = (a/(2 * pi * n) * sum[j] k(d[i,j] - r)
}
where the sum is over all points \eqn{j \neq i}{j != i},
\eqn{a} is the area of the observation window, \eqn{n} is the number
of points in \code{X}, and \eqn{d_{ij}}{d[i,j]} is the distance
between points \code{i} and \code{j}. Here \code{k} is the
Epanechnikov kernel,
\deqn{
k(t) = \frac 3 { 4\delta} \max(0, 1 - \frac{t^2}{\delta^2}).
}{
k(t) = (3/(4*delta)) * max(0, 1 - t^2/delta^2).
}
Edge correction is performed using the border method
(for the sake of computational efficiency):
the estimate \eqn{g_i(r)}{g[i](r)} is set to \code{NA} if
\eqn{r > b_i}{r > b[i]}, where \eqn{b_i}{b[i]}
is the distance from point \eqn{i} to the boundary of the
observation window.
The smoothing bandwidth \eqn{\delta}{delta} may be specified.
If not, it is chosen by Stoyan's rule of thumb
\eqn{\delta = c/\hat\lambda}{delta = c/lambda}
where \eqn{\hat\lambda = n/a}{lambda = n/a} is the estimated intensity
and \eqn{c} is a constant, usually taken to be 0.15.
The value of \eqn{c} is controlled by the argument \code{stoyan}.
}
\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{K} has been estimated
}
\item{theo}{the theoretical value \eqn{K(r) = \pi r^2}{K(r) = pi * r^2}
or \eqn{L(r)=r} for a stationary Poisson process
}
together with columns containing the values of the
local pair correlation function for each point in the pattern.
Column \code{i} corresponds to the \code{i}th point.
The last two columns contain the \code{r} and \code{theo} values.
}
\seealso{
\code{\link{localK}},
\code{\link{pcf}}
}
\examples{
data(ponderosa)
X <- ponderosa
g <- localpcf(X)
plot(g, main="local pair correlation functions for ponderosa", legend=FALSE)
# plot only the local pair correlation function for point number 7
plot(g, est007 ~ r)
}
\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}
Computing file changes ...