Hest.Rd
\name{Hest}
\alias{Hest}
\title{Spherical Contact Distribution Function}
\description{
Estimates the spherical contact distribution function of a
random set.
}
\usage{
Hest(X, r=NULL, breaks=NULL, ...,
correction=c("km", "rs", "han"),
conditional=TRUE)
}
\arguments{
\item{X}{The observed random set.
An object of class \code{"ppp"}, \code{"psp"} or \code{"owin"}.
}
\item{r}{
Optional. Vector of values for the argument \eqn{r} at which \eqn{H(r)}
should be evaluated. Users are advised \emph{not} to specify this
argument; there is a sensible default.
}
\item{breaks}{
Optional. An alternative to the argument \code{r}.
Not normally invoked by the user.
}
\item{\dots}{Arguments passed to \code{\link{as.mask}}
to control the discretisation.
}
\item{correction}{
Optional.
The edge correction(s) to be used to estimate \eqn{H(r)}.
A vector of character strings selected from
\code{"none"}, \code{"rs"}, \code{"km"}, \code{"han"}
and \code{"best"}.
}
\item{conditional}{
Logical value indicating whether to compute the
conditional or unconditional distribution. See Details.
}
}
\details{
The spherical contact distribution function
of a stationary random set \eqn{X}
is the cumulative distribution function \eqn{H} of the distance
from a fixed point in space to the nearest point of \eqn{X},
given that the point lies outside \eqn{X}.
That is, \eqn{H(r)} equals
the probability that \code{X} lies closer than \eqn{r} units away
from the fixed point \eqn{x}, given that \code{X} does not cover \eqn{x}.
Let \eqn{D = d(x,X)} be the shortest distance from an arbitrary
point \eqn{x} to the set \code{X}. Then the spherical contact
distribution function is
\deqn{H(r) = P(D \le r \mid D > 0)}{H(r) = P(D <= r | D > 0)}
For a point process, the spherical contact distribution function
is the same as the empty space function \eqn{F} discussed
in \code{\link{Fest}}.
The argument \code{X} may be a point pattern
(object of class \code{"ppp"}), a line segment pattern
(object of class \code{"psp"}) or a window (object of class
\code{"owin"}). It is assumed to be a realisation of a stationary
random set.
The algorithm first calls \code{\link{distmap}} to compute the
distance transform of \code{X}, then computes the Kaplan-Meier
and reduced-sample estimates of the cumulative distribution
following Hansen et al (1999).
If \code{conditional=TRUE} (the default) the algorithm
returns an estimate of the spherical contact function
\eqn{H(r)} as defined above.
If \code{conditional=FALSE}, it instead returns an estimate of the
cumulative distribution function
\eqn{H^\ast(r) = P(D \le r)}{H*(r) = P(D <= r)}
which includes a jump at \eqn{r=0} if \code{X} has nonzero area.
}
\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 up to six columns:
\item{r}{the values of the argument \eqn{r}
at which the function \eqn{H(r)} has been estimated
}
\item{rs}{the ``reduced sample'' or ``border correction''
estimator of \eqn{H(r)}
}
\item{km}{the spatial Kaplan-Meier estimator of \eqn{H(r)}
}
\item{hazard}{the hazard rate \eqn{\lambda(r)}{lambda(r)}
of \eqn{H(r)} by the spatial Kaplan-Meier method
}
\item{han}{the spatial Hanisch-Chiu-Stoyan estimator of \eqn{H(r)}
}
\item{raw}{the uncorrected estimate of \eqn{H(r)},
i.e. the empirical distribution of the distance from
a fixed point in the window to the nearest point of \code{X}
}
}
\references{
Baddeley, A.J. Spatial sampling and censoring.
In O.E. Barndorff-Nielsen, W.S. Kendall and
M.N.M. van Lieshout (eds)
\emph{Stochastic Geometry: Likelihood and Computation}.
Chapman and Hall, 1998.
Chapter 2, pages 37-78.
Baddeley, A.J. and Gill, R.D.
The empty space hazard of a spatial pattern.
Research Report 1994/3, Department of Mathematics,
University of Western Australia, May 1994.
Hansen, M.B., Baddeley, A.J. and Gill, R.D.
First contact distributions for spatial patterns:
regularity and estimation.
\emph{Advances in Applied Probability} \bold{31} (1999) 15-33.
Ripley, B.D. \emph{Statistical inference for spatial processes}.
Cambridge University Press, 1988.
Stoyan, D, Kendall, W.S. and Mecke, J.
\emph{Stochastic geometry and its applications}.
2nd edition. Springer Verlag, 1995.
}
\seealso{\code{\link{Fest}}}
\examples{
X <- runifpoint(42)
H <- Hest(X)
Y <- rpoisline(10)
H <- Hest(Y)
data(heather)
H <- Hest(heather$coarse)
H <- Hest(heather$coarse, conditional=FALSE)
}
\author{Adrian Baddeley
\email{Adrian.Baddeley@csiro.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
and Rolf Turner
\email{r.turner@auckland.ac.nz}
}
\keyword{spatial}
\keyword{nonparametric}