https://github.com/cran/spatstat
Tip revision: 64ebd4d0b6ec4fcd6f418895d6d2b7988e9605d1 authored by Adrian Baddeley on 23 October 2014, 08:43:49 UTC
version 1.39-0
version 1.39-0
Tip revision: 64ebd4d
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.
}
\usage{
Gest(X, r=NULL, breaks=NULL, \dots,
correction=c("rs", "km", "han"),
domain=NULL)
}
\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}{Optional. 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}{
This argument is for internal use only.
}
\item{\dots}{Ignored.}
\item{correction}{
Optional.
The edge correction(s) to be used to estimate \eqn{G(r)}.
A vector of character strings selected from
\code{"none"}, \code{"rs"}, \code{"km"}, \code{"Hanisch"}
and \code{"best"}.
}
\item{domain}{
Optional. Calculations will be restricted to this subset
of the window. See Details.
}
}
\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 some or all of the following
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{han}{the Hanisch correction estimator of \eqn{G(r)}
}
\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 edge corrections implemented here are the border method or
``\emph{reduced sample}'' estimator, the spatial Kaplan-Meier estimator
(Baddeley and Gill, 1997) and the Hanisch estimator (Hanisch, 1984).
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
estimators are computed from histogram counts.
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.
If the argument \code{domain} is given, the estimate of \eqn{G(r)}
will be based only on the nearest neighbour distances
measured from points falling inside \code{domain} (although their
nearest neighbours may lie outside \code{domain}).
This is useful in bootstrap techniques. The argument \code{domain}
should be a window (object of class \code{"owin"}) or something acceptable to
\code{\link{as.owin}}. It must be a subset of the
window of the point pattern \code{X}.
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 it is sometimes useful.
It can be returned by the algorithm, by selecting \code{correction="none"}.
Care should be taken not to use the uncorrected
empirical \eqn{G} as if it were an unbiased estimator of \eqn{G}.
To simply compute the nearest neighbour distance for each point in the
pattern, use \code{\link{nndist}}. To determine which point is the
nearest neighbour of a given point, use \code{\link{nnwhich}}.
}
\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.
Hanisch, K.-H. (1984) Some remarks on estimators of the distribution
function of nearest-neighbour distance in stationary spatial point
patterns. \emph{Mathematische Operationsforschung und Statistik,
series Statistics} \bold{15}, 409--412.
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{nndist}},
\code{\link{nnwhich}},
\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.Baddeley@uwa.edu.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
and Rolf Turner
\email{r.turner@auckland.ac.nz}
}
\keyword{spatial}
\keyword{nonparametric}