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\name{linearpcfinhom}
\alias{linearpcfinhom}
\title{
Inhomogeneous Linear Pair Correlation Function
}
\description{
Computes an estimate of the inhomogeneous linear pair correlation function
for a point pattern on a linear network.
}
\usage{
linearpcfinhom(X, lambda=NULL, r=NULL, ..., correction="Ang",
normalise=TRUE, normpower=1,
update = TRUE, leaveoneout = TRUE,
sigma=NULL, adjust.sigma=1,
bw="nrd0", adjust.bw=1,
ratio = FALSE)
}
\arguments{
\item{X}{
Point pattern on linear network (object of class \code{"lpp"}).
}
\item{lambda}{
Intensity values for the point pattern. Either a numeric vector,
a \code{function}, a pixel image (object of class \code{"im"}) or
a fitted point process model (object of class \code{"ppm"}
or \code{"lppm"}).
}
\item{r}{
Optional. Numeric vector of values of the function argument \eqn{r}.
There is a sensible default.
}
\item{\dots}{
Arguments passed to \code{\link{density.default}}
to control the smoothing of the estimates of pair correlation.
}
\item{correction}{
Geometry correction.
Either \code{"none"} or \code{"Ang"}. See Details.
}
\item{normalise}{
Logical. If \code{TRUE} (the default), the denominator of the estimator is
data-dependent (equal to the sum of the reciprocal intensities at the data
points, raised to \code{normpower}), which reduces the sampling variability.
If \code{FALSE}, the denominator is the length of the network.
}
\item{normpower}{
Integer (usually either 1 or 2).
Normalisation power. See explanation in \code{\link{linearKinhom}}.
}
\item{update}{
Logical value indicating what to do when \code{lambda} is a fitted model
(class \code{"lppm"} or \code{"ppm"}).
If \code{update=TRUE} (the default),
the model will first be refitted to the data \code{X}
(using \code{\link{update.lppm}} or \code{\link[spatstat.model]{update.ppm}})
before the fitted intensity is computed.
If \code{update=FALSE}, the fitted intensity of the
model will be computed without re-fitting it to \code{X}.
}
\item{leaveoneout}{
Logical value specifying whether to use a
leave-one-out rule when calculating the intensity.
See Details.
}
\item{sigma}{
Smoothing bandwidth (passed to \code{\link{density.lpp}})
for kernel density estimation of the intensity when
\code{lambda=NULL}.
}
\item{adjust.sigma}{
Numeric value. \code{sigma} will be multiplied by this value.
}
\item{bw}{
Smoothing bandwidth (passed to \code{\link[stats]{density.default}})
for one-dimensional kernel smoothing of the pair correlation function.
Either a numeric value, or a character string recognised
by \code{\link[stats]{density.default}}.
}
\item{adjust.bw}{
Numeric value. \code{bw} will be multiplied by this value.
}
\item{ratio}{
Logical.
If \code{TRUE}, the numerator and denominator of
each estimate will also be saved,
for use in analysing replicated point patterns.
}
}
\details{
This command computes the inhomogeneous version of the
linear pair correlation function from point pattern data on a linear network.
The argument \code{lambda} should provide estimated values
of the intensity of the point process at each point of \code{X}.
If \code{lambda=NULL}, the intensity will be estimated by kernel
smoothing by calling \code{\link{density.lpp}} with the smoothing
bandwidth \code{sigma}, and with any other relevant arguments
that might be present in \code{\dots}. A leave-one-out kernel estimate
will be computed if \code{leaveoneout=TRUE}.
If \code{lambda} is given,
it may be a numeric vector (of length equal to
the number of points in \code{X}), or a \code{function(x,y)} that will be
evaluated at the points of \code{X} to yield numeric values,
or a pixel image (object of class \code{"im"}) or a fitted point
process model (object of class \code{"ppm"} or \code{"lppm"}).
If \code{lambda} is a fitted point process model,
the default behaviour is to update the model by re-fitting it to
the data, before computing the fitted intensity.
This can be disabled by setting \code{update=FALSE}.
The intensity at data points will be computed
by \code{\link{fitted.lppm}} or \code{\link[spatstat.model]{fitted.ppm}}.
A leave-one-out estimate will be computed if \code{leaveoneout=TRUE}
and \code{update=TRUE}.
If \code{correction="none"}, the calculations do not include
any correction for the geometry of the linear network.
If \code{correction="Ang"}, the pair counts are weighted using
Ang's correction (Ang, 2010).
The bandwidth for smoothing the pairwise distances
is determined by arguments \code{\dots}
passed to \code{\link{density.default}}, mainly the arguments
\code{bw} and \code{adjust}. The default is
to choose the bandwidth by Silverman's rule of thumb
\code{bw="nrd0"} explained in \code{\link{density.default}}.
}
\section{Warning}{
Older versions of \code{\link{linearpcfinhom}} interpreted
\code{lambda=NULL} to mean that the homogeneous function
\code{\link{linearpcf}} should be computed. This was changed to the
current behaviour in version \code{3.1-0} of \pkg{spatstat.linnet}.
}
\value{
Function value table (object of class \code{"fv"}).
If \code{ratio=TRUE} then the return value also has two
attributes called \code{"numerator"} and \code{"denominator"}
which are \code{"fv"} objects
containing the numerators and denominators of each
estimate of \eqn{g(r)}.
}
\author{
\wei and
\adrian.
}
\references{
Ang, Q.W. (2010) Statistical methodology for spatial point patterns
on a linear network. MSc thesis, University of Western Australia.
Ang, Q.W., Baddeley, A. and Nair, G. (2012)
Geometrically corrected second-order analysis of
events on a linear network, with applications to
ecology and criminology.
\emph{Scandinavian Journal of Statistics} \bold{39}, 591--617.
Okabe, A. and Yamada, I. (2001) The K-function method on a network and
its computational implementation. \emph{Geographical Analysis}
\bold{33}, 271-290.
}
\seealso{
\code{\link{linearpcf}},
\code{\link{linearKinhom}},
\code{\link{lpp}}
}
\examples{
X <- rpoislpp(5, simplenet)
fit <- lppm(X ~x)
g <- linearpcfinhom(X, lambda=fit, update=FALSE)
plot(g)
ge <- linearpcfinhom(X, sigma=bw.lppl)
}
\keyword{spatial}
\keyword{nonparametric}
\concept{Linear network}
|
