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
Kest.Rd
\name{Kest}
\alias{Kest}
\title{K-function}
\description{
Estimates the reduced second moment function \eqn{K(r)}
from a point pattern in a window of arbitrary shape.
}
\synopsis{
Kest(X, r=NULL, breaks=NULL, slow,
correction=c("border", "isotropic", "Ripley", "translate"), \dots)
}
\usage{
Kest(X)
Kest(X, r)
Kest(X, r, correction=c("border", "isotropic", "Ripley", "translate"))
Kest(X, breaks=breaks)
}
\arguments{
\item{X}{The observed point pattern,
from which an estimate of \eqn{K(r)} will be computed.
An object of class \code{"ppp"}, or data
in any format acceptable to \code{\link{as.ppp}()}.
}
\item{r}{
vector of values for the argument \eqn{r} at which \eqn{K(r)}
should be evaluated. There is a sensible default.
}
\item{breaks}{
An alternative to the argument \code{r}.
Not normally invoked by the user.
See Details.
}
\item{correction}{
A character vector containing any selection of the
options \code{"border"}, \code{"bord.modif"},
\code{"isotropic"}, \code{"Ripley"} or \code{"translate"}.
It specifies the edge correction(s) to be applied.
}
}
\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 columns
\item{r}{the vector of values of the argument \eqn{r}
at which the function \eqn{K} has been estimated
}
\item{theo}{the theoretical value \eqn{K(r) = \pi r^2}{K(r) = pi * r^2}
for a stationary Poisson process
}
together with columns named
\code{"border"}, \code{"bord.modif"},
\code{"iso"} and/or \code{"trans"},
according to the selected edge corrections. These columns contain
estimates of the function \eqn{K(r)} obtained by the edge corrections
named.
}
\details{
The \eqn{K} function (variously called ``Ripley's K-function''
and the ``reduced second moment function'')
of a stationary point process \eqn{X} is defined so that
\eqn{\lambda K(r)}{lambda K(r)} equals the expected number of
additional random points within a distance \eqn{r} of a
typical random point of \eqn{X}. Here \eqn{\lambda}{lambda}
is the intensity of the process,
i.e. the expected number of points of \eqn{X} per unit area.
The \eqn{K} function is determined by the
second order moment properties of \eqn{X}.
An estimate of \eqn{K} 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{K} is a useful statistic
summarising aspects of inter-point ``dependence'' and ``clustering''.
For inferential purposes, the estimate of \eqn{K} is usually compared to the
true value of \eqn{K} for a completely random (Poisson) point process,
which is \eqn{K(r) = \pi r^2}{K(r) = pi * r^2}.
Deviations between the empirical and theoretical \eqn{K} curves
may suggest spatial clustering or spatial regularity.
This routine \code{Kest} estimates the \eqn{K} function
of a stationary point process, given observation of the process
inside a known, bounded window.
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{K} 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 corrections implemented here are
\describe{
\item{border}{the border method or
``reduced sample'' estimator (see Ripley, 1988). This is
the least efficient (statistically) and the fastest to compute.
It can be computed for a window of arbitrary shape.
}
\item{isotropic/Ripley}{Ripley's isotropic correction
(see Ripley, 1988; Ohser, 1983).
This is currently implemented only for rectangular windows.
}
\item{translate}{Translation correction (Ohser, 1983).
Implemented for all window geometries, but slow for
complex windows.
}
}
Note that the estimator assumes the process is stationary (spatially
homogeneous). For inhomogeneous point patterns, see
\code{\link{Kinhom}}.
The estimator \code{Kest} ignores marks.
Its counterparts for multitype point patterns
are \code{\link{Kcross}}, \code{\link{Kdot}},
and for general marked point patterns
see \code{\link{Kmulti}}.
Some writers, particularly Stoyan (1994, 1995) advocate the use of
the ``pair correlation function''
\deqn{
g(r) = \frac{K'(r)}{2\pi r}
}{
g(r) = K'(r)/ ( 2 * pi * r)
}
where \eqn{K'(r)} is the derivative of \eqn{K(r)}.
See \code{\link{pcf}} on how to estimate this function.
}
\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.
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.
Ohser, J. (1983)
On estimators for the reduced second moment measure of
point processes. \emph{Mathematische Operationsforschung und
Statistik, series Statistics}, \bold{14}, 63 -- 71.
Ripley, B.D. \emph{Statistical inference for spatial processes}.
Cambridge University Press, 1988.
Stoyan, D, Kendall, W.S. and Mecke, J. (1995)
\emph{Stochastic geometry and its applications}.
2nd edition. Springer Verlag.
Stoyan, D. and Stoyan, H. (1994)
Fractals, random shapes and point fields:
methods of geometrical statistics.
John Wiley and Sons.
}
\section{Warnings}{
The estimator of \eqn{K(r)} is approximately unbiased for each fixed \eqn{r}.
Bias increases with \eqn{r} and depends on the window geometry.
For a rectangular window it is prudent to restrict the \eqn{r} values to
a maximum of \eqn{1/4} of the smaller side length of the rectangle.
Bias may become appreciable for point patterns consisting of
fewer than 15 points.
While \eqn{K(r)} is always a non-decreasing function, the estimator
of \eqn{K} is not guaranteed to be non-decreasing. This is rarely
a problem in practice.
}
\seealso{
\code{\link{Fest}},
\code{\link{Gest}},
\code{\link{Jest}},
\code{\link{pcf}},
\code{\link{reduced.sample}},
\code{\link{Kcross}},
\code{\link{Kdot}},
\code{\link{Kinhom}},
\code{\link{Kmulti}}
}
\examples{
pp <- runifpoint(50)
K <- Kest(pp)
data(cells)
K <- Kest(cells, correction="isotropic")
plot(K)
plot(K, main="K function for cells")
# plot the L function
plot(K, sqrt(iso/pi) ~ r)
plot(K, sqrt(./pi) ~ r, ylab="L(r)", main="L function for cells")
}
\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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