https://github.com/cran/spatstat
Tip revision: 5380692d89c0728c413d9b9fd1103c2b65bfc205 authored by Adrian Baddeley on 28 July 2005, 23:03:00 UTC
version 1.7-11
version 1.7-11
Tip revision: 5380692
Kcross.Rd
\name{Kcross}
\alias{Kcross}
\title{
Multitype K Function (Cross-type)
}
\description{
For a multitype point pattern,
estimate the multitype \eqn{K} function
which counts the expected number of points of type \eqn{j}
within a given distance of a point of type \eqn{i}.
}
\synopsis{
Kcross(X, i=1, j=2, r=NULL, breaks=NULL, correction, \dots)
}
\usage{
Kcross(X, i=1, j=2)
Kcross(X, i=1, j=2, correction=c("border", "isotropic", "Ripley", "translate"))
Kcross(X, i=1, j=2, r, correction)
Kcross(X, i=1, j=2, breaks)
}
\arguments{
\item{X}{The observed point pattern,
from which an estimate of the cross type \eqn{K} function
\eqn{K_{ij}(r)}{Kij(r)} will be computed.
It must be a multitype point pattern (a marked point pattern
whose marks are a factor). See under Details.
}
\item{i}{Number or character string identifying the type (mark value)
of the points in \code{X} from which distances are measured.
}
\item{j}{Number or character string identifying the type (mark value)
of the points in \code{X} to which distances are measured.
}
\item{r}{numeric vector. The values of the argument \eqn{r}
at which the distribution function
\eqn{K_{ij}(r)}{Kij(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.
}
\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}}).
Essentially a data frame containing numeric columns
\item{r}{the values of the argument \eqn{r}
at which the function \eqn{K_{ij}(r)}{Kij(r)} has been estimated
}
\item{theo}{the theoretical value of \eqn{K_{ij}(r)}{Kij(r)}
for a marked Poisson process, namely \eqn{\pi r^2}{pi * r^2}
}
together with a column or 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_{ij}(r)}{Kij(r)}
obtained by the edge corrections named.
}
\details{
This function \code{Kcross} and its companions
\code{\link{Kdot}} and \code{\link{Kmulti}}
are generalisations of the function \code{\link{Kest}}
to multitype point patterns.
A multitype point pattern is a spatial pattern of
points classified into a finite number of possible
``colours'' or ``types''. In the \code{spatstat} package,
a multitype pattern is represented as a single
point pattern object in which the points carry marks,
and the mark value attached to each point
determines the type of that point.
The argument \code{X} must be a point pattern (object of class
\code{"ppp"}) or any data that are acceptable to \code{\link{as.ppp}}.
It must be a marked point pattern, and the mark vector
\code{X$marks} must be a factor.
The arguments \code{i} and \code{j} will be interpreted as
levels of the factor \code{X$marks}. (Warning: this means that
an integer value \code{i=3} will be interpreted as the 3rd smallest level,
not the number 3).
The ``cross-type'' (type \eqn{i} to type \eqn{j})
\eqn{K} function
of a stationary multitype point process \eqn{X} is defined so that
\eqn{\lambda_j K_{ij}(r)}{lambda[j] Kij(r)} equals the expected number of
additional random points of type \eqn{j}
within a distance \eqn{r} of a
typical point of type \eqn{i} in the process \eqn{X}.
Here \eqn{\lambda_j}{lambda[j]}
is the intensity of the type \eqn{j} points,
i.e. the expected number of points of type \eqn{j} per unit area.
The function \eqn{K_{ij}}{Kij} is determined by the
second order moment properties of \eqn{X}.
An estimate of \eqn{K_{ij}(r)}{Kij(r)}
is a useful summary statistic in exploratory data analysis
of a multitype point pattern.
If the process of type \eqn{i} points
were independent of the process of type \eqn{j} points,
then \eqn{K_{ij}(r)}{Kij(r)} would equal \eqn{\pi r^2}{pi * r^2}.
Deviations between the empirical \eqn{K_{ij}}{Kij} curve
and the theoretical curve \eqn{\pi r^2}{pi * r^2}
may suggest dependence between the points of types \eqn{i} and \eqn{j}.
This algorithm estimates the distribution function \eqn{K_{ij}(r)}{Kij(r)}
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$window})
may have arbitrary shape.
Biases due to edge effects are
treated in the same manner as in \code{\link{Kest}},
using the border correction.
The argument \code{r} is the vector of values for the
distance \eqn{r} at which \eqn{K_{ij}(r)}{Kij(r)} should be evaluated.
The values of \eqn{r} must be increasing nonnegative numbers
and the maximum \eqn{r} value must exceed the radius of the
largest disc contained in the window.
The pair correlation function can also be applied to the
result of \code{Kcross}; see \code{\link{pcf}}.
}
\references{
Cressie, N.A.C. \emph{Statistics for spatial data}.
John Wiley and Sons, 1991.
Diggle, P.J. \emph{Statistical analysis of spatial point patterns}.
Academic Press, 1983.
Harkness, R.D and Isham, V. (1983)
A bivariate spatial point pattern of ants' nests.
\emph{Applied Statistics} \bold{32}, 293--303
Lotwick, H. W. and Silverman, B. W. (1982).
Methods for analysing spatial processes of several types of points.
\emph{J. Royal Statist. Soc. Ser. B} \bold{44}, 406--413.
Ripley, B.D. \emph{Statistical inference for spatial processes}.
Cambridge University Press, 1988.
Stoyan, D, Kendall, W.S. and Mecke, J.
\emph{Stochastic geometry and its applications}.
2nd edition. Springer Verlag, 1995.
}
\section{Warnings}{
The arguments \code{i} and \code{j} are interpreted as
levels of the factor \code{X$marks}. Beware of the usual
trap with factors: numerical values are not
interpreted in the same way as character values. See the first example.
The reduced sample estimator of \eqn{K_{ij}}{Kij} 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]}.
}
\seealso{
\code{\link{Kdot}},
\code{\link{Kest}},
\code{\link{Kmulti}},
\code{\link{pcf}}
}
\examples{
data(betacells)
# cat retina data
K01 <- Kcross(betacells, "off", "on")
plot(K01)
K10 <- Kcross(betacells, "on", "off")
# synthetic example
pp <- runifpoispp(50)
pp <- pp \%mark\% factor(sample(0:1, pp$n, replace=TRUE))
K <- Kcross(pp, "0", "1") # note: "0" not 0
}
\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}