Revision 6ebd6ff5fb7707ef87961f50ee794d8eb7143d0a authored by Adrian Baddeley on 10 August 2005, 00:00:00 UTC, committed by Gabor Csardi on 10 August 2005, 00:00:00 UTC
1 parent 5493ce5
Gest.Rd
\name{Gest}
\alias{Gest}
\alias{nearest.neighbour}
\title{
Nearest Neighbour Distance Function G
}
\description{
Estimates the nearest neighbour distance distribution
function \eqn{G(r)} from a point pattern in a
window of arbitrary shape.
}
\synopsis{
Gest(X, r=NULL, breaks=NULL, \dots)
}
\usage{
Gest(X)
Gest(X, r)
Gest(X, breaks)
nearest.neighbour(X)
}
\arguments{
\item{X}{The observed point pattern,
from which an estimate of \eqn{G(r)} will be computed.
An object of class \code{ppp}, or data
in any format acceptable to \code{\link{as.ppp}()}.
}
\item{r}{numeric vector. The values of the argument \eqn{r}
at which \eqn{G(r)} should be evaluated. There is a sensible default.
First-time users are strongly advised not to specify this argument.
See below for important conditions on \eqn{r}.
}
\item{breaks}{An alternative to the argument \code{r}.
Not normally invoked by the user. See the \bold{Details} section.
}
}
\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 six columns:
\item{r}{the values of the argument \eqn{r}
at which the function \eqn{G(r)} has been estimated
}
\item{rs}{the ``reduced sample'' or ``border correction''
estimator of \eqn{G(r)}
}
\item{km}{the spatial Kaplan-Meier estimator of \eqn{G(r)}
}
\item{hazard}{the hazard rate \eqn{\lambda(r)}{lambda(r)}
of \eqn{G(r)} by the spatial Kaplan-Meier method
}
\item{raw}{the uncorrected estimate of \eqn{G(r)},
i.e. the empirical distribution of the distances from
each point in the pattern \code{X} to the nearest other point of
the pattern
}
\item{theo}{the theoretical value of \eqn{G(r)}
for a stationary Poisson process of the same estimated intensity.
}
}
\details{
The nearest neighbour distance distribution function
(also called the ``\emph{event-to-event}'' or
``\emph{inter-event}'' distribution)
of a point process \eqn{X}
is the cumulative distribution function \eqn{G} of the distance
from a typical random point of \eqn{X} to
the nearest other point of \eqn{X}.
An estimate of \eqn{G} derived from a spatial point pattern dataset
can be used in exploratory data analysis and formal inference
about the pattern (Cressie, 1991; Diggle, 1983; Ripley, 1988).
In exploratory analyses, the estimate of \eqn{G} is a useful statistic
summarising one aspect of the ``clustering'' of points.
For inferential purposes, the estimate of \eqn{G} is usually compared to the
true value of \eqn{G} for a completely random (Poisson) point process,
which is
\deqn{G(r) = 1 - e^{ - \lambda \pi r^2} }{%
G(r) = 1 - exp( - lambda * pi * r^2)}
where \eqn{\lambda}{lambda} is the intensity
(expected number of points per unit area).
Deviations between the empirical and theoretical \eqn{G} curves
may suggest spatial clustering or spatial regularity.
This algorithm estimates the nearest neighbour distance distribution
function \eqn{G}
from the point pattern \code{X}. It assumes that \code{X} can be treated
as a realisation of a stationary (spatially homogeneous)
random spatial point process in the plane, observed through
a bounded window.
The window (which is specified in \code{X} as \code{X$window})
may have arbitrary shape.
The argument \code{X} is interpreted as a point pattern object
(of class \code{"ppp"}, see \code{\link{ppp.object}}) and can
be supplied in any of the formats recognised
by \code{\link{as.ppp}()}.
The estimation of \eqn{G} is hampered by edge effects arising from
the unobservability of points of the random pattern outside the window.
An edge correction is needed to reduce bias (Baddeley, 1998; Ripley, 1988).
The two edge corrections implemented here are the border method or
``\emph{reduced sample}'' estimator, and the spatial Kaplan-Meier estimator
(Baddeley and Gill, 1997).
The argument \code{r} is the vector of values for the
distance \eqn{r} at which \eqn{G(r)} should be evaluated.
It is also used to determine the breakpoints
(in the sense of \code{\link{hist}})
for the computation of histograms of distances. The reduced-sample and
Kaplan-Meier estimators are computed from histogram counts.
In the case of the Kaplan-Meier estimator this introduces a discretisation
error which is controlled by the fineness of the breakpoints.
First-time users would be strongly advised not to specify \code{r}.
However, if it is specified, \code{r} must satisfy \code{r[1] = 0},
and \code{max(r)} must be larger than the radius of the largest disc
contained in the window. Furthermore, the successive entries of \code{r}
must be finely spaced.
The algorithm also returns an estimate of the hazard rate function,
\eqn{\lambda(r)}{lambda(r)}, of \eqn{G(r)}. The hazard rate is
defined as the derivative
\deqn{\lambda(r) = - \frac{d}{dr} \log (1 - G(r))}{%
lambda(r) = - (d/dr) log(1 - G(r))}
This estimate should be used with caution as \eqn{G} is not necessarily
differentiable.
The naive empirical distribution of distances from each point of
the pattern \code{X} to the nearest other point of the pattern,
is a biased estimate of \eqn{G}.
However this is also returned by the algorithm, as it is sometimes
useful in other contexts. Care should be taken not to use the uncorrected
empirical \eqn{G} as if it were an unbiased estimator of \eqn{G}.
}
\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.
Kaplan-Meier estimators of interpoint distance
distributions for spatial point processes.
\emph{Annals of Statistics} \bold{25} (1997) 263-292.
Cressie, N.A.C. \emph{Statistics for spatial data}.
John Wiley and Sons, 1991.
Diggle, P.J. \emph{Statistical analysis of spatial point patterns}.
Academic Press, 1983.
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.
}
\section{Warnings}{
The function \eqn{G} does not necessarily have a density.
Any valid c.d.f. may appear as the nearest neighbour distance
distribution function of a stationary point process.
The reduced sample estimator of \eqn{G} is pointwise approximately
unbiased, but need not be a valid distribution function; it may
not be a nondecreasing function of \eqn{r}. Its range is always
within \eqn{[0,1]}.
The spatial Kaplan-Meier estimator of \eqn{G} is always nondecreasing
but its maximum value may be less than \eqn{1}.
}
\seealso{
\code{\link{Fest}},
\code{\link{Jest}},
\code{\link{Kest}},
\code{\link{km.rs}},
\code{\link{reduced.sample}},
\code{\link{kaplan.meier}}
}
\examples{
data(cells)
G <- Gest(cells)
plot(G)
# P-P style plot
plot(G, cbind(km,theo) ~ theo)
# the empirical G is below the Poisson G,
# indicating an inhibited pattern
\dontrun{
plot(G, . ~ r)
plot(G, . ~ theo)
plot(G, asin(sqrt(.)) ~ asin(sqrt(theo)))
}
}
\author{Adrian Baddeley
\email{adrian@maths.uwa.edu.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
and Rolf Turner
\email{rolf@math.unb.ca}
\url{http://www.math.unb.ca/~rolf}
}
\keyword{spatial}
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