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\name{densityHeat.lpp}
\alias{densityHeat.lpp}
\title{
Kernel Density on a Network using Heat Equation
}
\description{
Given a point pattern on a linear network,
compute a kernel estimate of intensity,
by solving the heat equation.
}
\usage{
\method{densityHeat}{lpp}(x, sigma=NULL, \dots,
at=c("pixels", "points"), leaveoneout=TRUE,
weights = NULL,
dx = NULL, dt = NULL, iterMax = 1e+06,
finespacing = TRUE, verbose=FALSE)
}
\arguments{
\item{x}{
Point pattern on a linear network (object of class \code{"lpp"})
to be smoothed.
}
\item{sigma}{
Smoothing bandwidth (standard deviation of the kernel).
A numeric value in the same units as the spatial coordinates of \code{x}.
Alternatively \code{sigma} may be a function which selects a
bandwidth when applied to \code{X},
for example, \code{\link[spatstat.explore]{bw.scott.iso}} or \code{\link{bw.lppl}}.
There is a sensible default.
}
\item{\dots}{
Arguments passed to \code{\link[spatstat.geom]{as.mask}} determining the
resolution of the result. (Any other arguments are ignored.)
}
\item{at}{
String specifying whether to compute the intensity values
at a fine grid of pixel locations on the network
(\code{at="pixels"}, the default) or
only at the data points of \code{x} (\code{at="points"}).
}
\item{leaveoneout}{
Logical value indicating whether to compute a leave-one-out
estimator. Applicable only when \code{at="points"}.
}
\item{weights}{
Optional. Numeric vector of weights associated with the
points of \code{x}. Weights may be positive, negative or zero.
}
\item{dx}{
Optional. Spacing of the sampling points along the network.
A single number giving a distance value in the same units as \code{x}.
}
\item{dt}{
Optional. Time step in the heat equation solver.
A single number.
}
\item{iterMax}{
Maximum number of iterations.
}
\item{finespacing}{
Logical value specifying whether the discrete approximation is
required to be accurate along every segment of the network,
no matter how short the segment is. See the section on Discretisation.
}
\item{verbose}{
Logical value specifying whether to print progress reports.
}
}
\details{
The function \code{\link[spatstat.explore]{densityHeat}} is generic.
This is the method for the class \code{"lpp"} of points on a linear
network.
Kernel smoothing is applied to the points of \code{x}
using a kernel based on path distances in the network.
If \code{at="pixels"} (the default),
the result is a pixel image on the linear network (class
\code{"linim"}) which can be plotted.
If \code{at="points"} the result is a numeric vector giving the
density estimates at the data points of \code{x}.
The smoothing operation is equivalent to the
\dQuote{equal-split continuous} rule described in
Section 9.2.3 of Okabe and Sugihara (2012).
However, the actual computation is performed rapidly, by solving the classical
time-dependent heat equation on the network,
as described in McSwiggan et al (2016).
Computational time is short, but increases quadratically with
\code{sigma}.
If \code{at="points"} and \code{leaveoneout=TRUE},
a leave-one-out estimate is computed at each data point
(that is, the estimate at each data point \code{x[i]} is based
on all of the points except \code{x[i]})
using the truncated series approximation
of McSwiggan et al (2019).
The argument \code{sigma} specifies the smoothing bandwidth.
If \code{sigma} is missing or \code{NULL},
the default is one-eighth of the length of the shortest side
of the bounding box of \code{x}.
If \code{sigma} is a function in the \R language, it is assumed
to be a bandwidth selection rule, and it will be applied to \code{x}
to compute the bandwidth value.
}
\section{Infinite bandwidth}{
If \code{sigma=Inf}, the resulting density estimate is
constant over all locations,
and is equal to the average density of points per unit length.
(If the network is not connected, then this rule
is applied separately to each connected component of the network).
}
\section{Discretisation and Error Messages}{
The arguments \code{dx}, \code{dt} and \code{iterMax}
determine the discretisation of the network, according to
a set of rules. The argument \code{finespacing} determines
which rule will be applied.
The arguments \code{dx}, \code{dt}, \code{iterMax}
are connected by several constraints; specifying one of these
arguments will affect the default values of the other two arguments.
The argument \code{finespacing} specifies whether
a very fine spacing of sample points is required, in order to attain
hihg accuracy.
\itemize{
\item
If \code{finespacing=TRUE} (the default), then the
sample point spacing \code{dx} must not exceed
one-third of the length of the shortest
segment of the network. This ensures that the discrete approximation
is accurate along every segment, no matter how short the segment
is. However, this may not be feasible
if it implies a very large number of sample points, or a large number
of iterations: in such cases, the code may terminate with an error
about illegal values of \code{dx}, \code{dt} or \code{iterMax}.
\item
If \code{finespacing=FALSE}, then
the sample point spacing \code{dx} will be about one-half the width
of a pixel in the default pixellation of the window of \code{x}.
This is usually a much coarser resolution than the one
selected by \code{finespacing=TRUE}. If it is too coarse,
the pixel resolution can be refined using
the arguments \code{dimyx}, \code{eps} or \code{xy}
passed to \code{\link[spatstat.geom]{as.mask}}. For example,
\code{dimyx=512} would specify a 512 x 512 pixel grid.
The default pixel resolution can be changed for the remainder of
the \R session by \code{\link[spatstat.geom]{spatstat.options}('npixel')}.
}
}
\value{
If \code{at="pixels"} (the default),
a pixel image on the linear network (object of class \code{"linim"}).
If \code{at="points"}, a numeric vector with one entry for each point
of \code{x}.
}
\references{
McSwiggan, G., Baddeley, A. and Nair, G. (2016)
Kernel density estimation on a linear network.
\emph{Scandinavian Journal of Statistics} \bold{44}, 324--345.
McSwiggan, G., Baddeley, A. and Nair, G. (2019)
Estimation of relative risk for events on a linear network.
\emph{Statistics and Computing} \bold{30}, 469--484.
Okabe, A. and Sugihara, K. (2012)
\emph{Spatial analysis along networks}.
Wiley.
}
\author{
\adrian and Greg McSwiggan.
}
\seealso{
\code{\link{density.lpp}}
}
\examples{
X <- runiflpp(3, simplenet)
D <- densityHeat(X, 0.2)
plot(D, style="w", main="", adjust=2)
densityHeat.lpp(X, 0.2, at="points")
Dw <- densityHeat(X, 0.2, weights=c(1,2,-1))
}
\keyword{spatial}
\keyword{methods}
\keyword{smooth}
\concept{Linear network}
|
