\name{Kmeasure}
\alias{Kmeasure}
\title{Reduced Second Moment Measure}
\description{
Estimates the reduced second moment measure \eqn{\kappa}{Kappa}
from a point pattern in a window of arbitrary shape.
}
\usage{
Kmeasure(X, sigma, edge=TRUE, ..., varcov=NULL)
}
\arguments{
\item{X}{The observed point pattern,
from which an estimate of \eqn{\kappa}{Kappa} will be computed.
An object of class \code{"ppp"}, or data
in any format acceptable to \code{\link{as.ppp}()}.
}
\item{sigma}{
Standard deviation \eqn{\sigma}{sigma} of the Gaussian
smoothing kernel. Incompatible with \code{varcov}.
}
\item{edge}{
logical value indicating whether an edge correction
should be applied.
}
\item{\dots}{Ignored.}
\item{varcov}{
Variance-covariance matrix of the Gaussian smoothing kernel.
Incompatible with \code{sigma}.
}
}
\value{
A real-valued pixel image (an object of class \code{"im"},
see \code{\link{im.object}}) whose pixel values are estimates
of the value of the reduced second moment measure for each pixel
(i.e. estimates of the integral of the second moment density
over each pixel).
}
\details{
The reduced second moment measure \eqn{\kappa}{Kappa}
of a stationary point process \eqn{X} is defined so that,
for a `typical' point \eqn{x} of the process,
the expected number of other points \eqn{y} of the process
such that the vector \eqn{y - x} lies in a region \eqn{A},
equals \eqn{\lambda \kappa(A)}{lambda Kappa(A)}.
Here \eqn{\lambda}{lambda}
is the intensity of the process,
i.e. the expected number of points of \eqn{X} per unit area.
The more familiar
K-function \eqn{K(t)} is just the value of the reduced second moment measure
for each disc centred at the origin; that is,
\eqn{K(t) = \kappa(b(0,t))}{K(t) = Kappa(b(0,t))}.
An estimate of \eqn{\kappa}{Kappa} derived from a spatial point
pattern dataset can be useful in exploratory data analysis.
Its advantage over the K-function is that it is also sensitive
to anisotropy and directional effects.
This function computes an estimate of \eqn{\kappa}{Kappa}
from a point pattern dataset \code{X},
which is assumed to be a realisation of a stationary point process,
observed inside a known, bounded window. Marks are ignored.
The algorithm approximates the point pattern and its window by binary pixel
images, introduces a Gaussian smoothing kernel
and uses the Fast Fourier Transform \code{\link{fft}}
to form a density estimate of \eqn{\kappa}{Kappa}. The calculation
corresponds to the edge correction known as the ``translation
correction''.
The Gaussian smoothing kernel may be specified by either of the
arguments \code{sigma} or \code{varcov}. If \code{sigma} is a single
number, this specifies an isotropic Gaussian kernel
with standard deviation \code{sigma} on each coordinate axis.
If \code{sigma} is a vector of two numbers, this specifies a Gaussian
kernel with standard deviation \code{sigma[1]} on the \eqn{x} axis,
standard deviation \code{sigma[2]} on the \eqn{y} axis, and zero
correlation between the \eqn{x} and \eqn{y} axes. If \code{varcov} is
given, this specifies the variance-covariance matrix of the
Gaussian kernel. There do not seem to be any well-established rules
for selecting the smoothing kernel in this context.
The density estimate of \eqn{\kappa}{Kappa}
is returned in the form of a real-valued pixel image.
Pixel values are estimates of the
integral of the second moment density over the pixel.
(The uniform Poisson process would have values identically equal to
\eqn{a} where \eqn{a} is the area of a pixel.)
Sums of pixel values over a desired region \eqn{A} are estimates of the
value of \eqn{\kappa(A)}{Kappa(A)}. The image \code{x} and \code{y}
coordinates are on the same scale as vector displacements in the
original point pattern window. The point \code{x=0, y=0} corresponds
to the `typical point'.
A peak in the image near \code{(0,0)} suggests clustering;
a dip in the image near \code{(0,0)} suggests inhibition;
peaks or dips at other positions suggest possible periodicity.
}
\references{
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.
}
\seealso{
\code{\link{Kest}},
\code{\link{spatstat.options}},
\code{\link{im.object}}
}
\examples{
data(cells)
image(Kmeasure(cells, 0.05))
# shows pronounced dip around origin consistent with strong inhibition
data(redwood)
image(Kmeasure(redwood, 0.03), col=grey(seq(1,0,length=32)))
# shows peaks at several places, reflecting clustering and ?periodicity
}
\author{Adrian Baddeley
\email{adrian@maths.uwa.edu.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
and Rolf Turner
\email{r.turner@auckland.ac.nz}
}
\keyword{spatial}
\keyword{nonparametric}