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\name{linearKinhom}
\alias{linearKinhom}
\title{
Inhomogeneous Linear K Function
}
\description{
Computes an estimate of the inhomogeneous linear \eqn{K} function
for a point pattern on a linear network.
}
\usage{
linearKinhom(X, lambda=NULL, r=NULL, ..., correction="Ang",
normalise=TRUE, normpower=1,
update=TRUE, leaveoneout=TRUE, sigma=NULL, 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 \code{"linim"}) or
a fitted point process model (object of class \code{"ppm"}
or \code{"lppm"}) or \code{NULL}.
}
\item{r}{
Optional. Numeric vector of values of the function argument \eqn{r}.
There is a sensible default. Users are advised not to specify
\code{r} in normal usage.
}
\item{\dots}{
Ignored.
}
\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 Details.
}
\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{ratio}{
Logical.
If \code{TRUE}, the numerator and denominator of
the estimate will also be saved,
for use in analysing replicated point patterns.
}
}
\details{
This command computes the inhomogeneous version of the
linear \eqn{K} 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).
Each estimate is initially computed as
\deqn{
\widehat K_{\rm inhom}(r) = \frac{1}{\mbox{length}(L)}
\sum_i \sum_j \frac{1\{d_{ij} \le r\}
e(x_i,x_j)}{\lambda(x_i)\lambda(x_j)}
}{
K^inhom(r)= (1/length(L)) sum[i] sum[j] 1(d[i,j] <= r) *
e(x[i],x[j])/(lambda(x[i]) * lambda(x[j]))
}
where \code{L} is the linear network,
\eqn{d_{ij}}{d[i,j]} is the distance between points
\eqn{x_i}{x[i]} and \eqn{x_j}{x[j]}, and
\eqn{e(x_i,x_j)}{e(x[i],x[j])} is a weight.
If \code{correction="none"} then this weight is equal to 1,
while if \code{correction="Ang"} the weight is
\eqn{e(x_i,x_j,r) = 1/m(x_i, d_{ij})}{e(x[i],x[j],r) = 1/m(x[i],d[i,j])}
where \eqn{m(u,t)} is the number of locations on the network that lie
exactly \eqn{t} units distant from location \eqn{u} by the shortest
path.
If \code{normalise=TRUE} (the default), then the estimates
described above
are multiplied by \eqn{c^{\mbox{normpower}}}{c^normpower} where
\eqn{
c = \mbox{length}(L)/\sum (1/\lambda(x_i)).
}{
c = length(L)/sum[i] (1/lambda(x[i])).
}
This rescaling reduces the variability and bias of the estimate
in small samples and in cases of very strong inhomogeneity.
The default value of \code{normpower} is 1 (for consistency with
previous versions of \pkg{spatstat})
but the most sensible value is 2, which would correspond to rescaling
the \code{lambda} values so that
\eqn{
\sum (1/\lambda(x_i)) = \mbox{area}(W).
}{
sum[i] (1/lambda(x[i])) = area(W).
}
}
\section{Warning}{
Older versions of \code{\link{linearKinhom}} interpreted
\code{lambda=NULL} to mean that the homogeneous function
\code{\link{linearK}} 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"}).
}
\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.
}
\seealso{
\code{\link{lpp}}
}
\examples{
X <- rpoislpp(5, simplenet)
fit <- lppm(X ~x)
K <- linearKinhom(X, lambda=fit)
plot(K)
Ke <- linearKinhom(X, sigma=bw.lppl)
plot(Ke)
}
\keyword{spatial}
\keyword{nonparametric}
\concept{Linear network}
|
