https://github.com/cran/spatstat
Revision 4fe059206e698a4b7135d792f3d533b173ecfe77 authored by Adrian Baddeley on 16 May 2012, 12:44:15 UTC, committed by cran-robot on 16 May 2012, 12:44:15 UTC
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Tip revision: 4fe059206e698a4b7135d792f3d533b173ecfe77 authored by Adrian Baddeley on 16 May 2012, 12:44:15 UTC
version 1.27-0
version 1.27-0
Tip revision: 4fe0592
F3est.Rd
\name{F3est}
\Rdversion{1.1}
\alias{F3est}
\title{
Empty Space Function of a Three-Dimensional Point Pattern
}
\description{
Estimates the empty space function \eqn{F_3(r)}{F3(r)} from
a three-dimensional point pattern.
}
\usage{
F3est(X, ..., rmax = NULL, nrval = 128, vside = NULL, correction = c("rs", "km", "cs"))
}
\arguments{
\item{X}{
Three-dimensional point pattern (object of class \code{"pp3"}).
}
\item{\dots}{
Ignored.
}
\item{rmax}{
Optional. Maximum value of argument \eqn{r} for which
\eqn{F_3(r)}{F3(r)} will be estimated.
}
\item{nrval}{
Optional. Number of values of \eqn{r} for which
\eqn{F_3(r)}{F3(r)} will be estimated. A large value of \code{nrval}
is required to avoid discretisation effects.
}
\item{vside}{
Optional.
Side length of the voxels in the discrete approximation.
}
\item{correction}{
Optional. Character vector specifying the edge correction(s)
to be applied. See Details.
}
}
\details{
For a stationary point process \eqn{\Phi}{Phi} in three-dimensional
space, the empty space function is
\deqn{
F_3(r) = P(d(0,\Phi) \le r)
}{
F3(r) = P(d(0,Phi) <= r)
}
where \eqn{d(0,\Phi)}{d(0,Phi)} denotes the distance from a fixed
origin \eqn{0} to the nearest point of \eqn{\Phi}{Phi}.
The three-dimensional point pattern \code{X} is assumed to be a
partial realisation of a stationary point process \eqn{\Phi}{Phi}.
The empty space function of \eqn{\Phi}{Phi} can then be estimated using
techniques described in the References.
The box containing the point
pattern is discretised into cubic voxels of side length \code{vside}.
The distance function \eqn{d(u,\Phi)}{d(u,Phi)} is computed for
every voxel centre point
\eqn{u} using a three-dimensional version of the distance transform
algorithm (Borgefors, 1986). The empirical cumulative distribution
function of these values, with appropriate edge corrections, is the
estimate of \eqn{F_3(r)}{F3(r)}.
The available edge corrections are:
\describe{
\item{\code{"rs"}:}{
the reduced sample (aka minus sampling, border correction)
estimator (Baddeley et al, 1993)
}
\item{\code{"km"}:}{
the three-dimensional version of the
Kaplan-Meier estimator (Baddeley and Gill, 1997)
}
\item{\code{"cs"}:}{
the three-dimensional generalisation of
the Chiu-Stoyan or Hanisch estimator (Chiu and Stoyan, 1998).
}
}
}
\value{
A function value table (object of class \code{"fv"}) that can be
plotted, printed or coerced to a data frame containing the function values.
}
\references{
Baddeley, A.J, Moyeed, R.A., Howard, C.V. and Boyde, A.
Analysis of a three-dimensional point pattern with replication.
\emph{Applied Statistics} \bold{42} (1993) 641--668.
Baddeley, A.J. and Gill, R.D. (1997)
Kaplan-Meier estimators of interpoint distance
distributions for spatial point processes.
\emph{Annals of Statistics} \bold{25}, 263--292.
Borgefors, G. (1986)
Distance transformations in digital images.
\emph{Computer Vision, Graphics and Image Processing}
\bold{34}, 344--371.
Chiu, S.N. and Stoyan, D. (1998)
Estimators of distance distributions for spatial patterns.
\emph{Statistica Neerlandica} \bold{52}, 239--246.
}
\author{
Adrian Baddeley
\email{Adrian.Baddeley@csiro.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
and Rana Moyeed.
}
\section{Warnings}{
A large value of \code{nrval} is required in order to avoid
discretisation effects (due to the use of histograms in the
calculation).
}
\seealso{
\code{\link{G3est}},
\code{\link{K3est}}.
}
\examples{
X <- rpoispp3(42)
Z <- F3est(X)
if(interactive()) plot(Z)
}
\keyword{spatial}
\keyword{nonparametric}
Computing file changes ...