Kcross.inhom.Rd
\name{Kcross.inhom}
\alias{Kcross.inhom}
\title{
Inhomogeneous Cross K Function
}
\description{
For a multitype point pattern,
estimate the inhomogeneous version of the cross \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},
adjusted for spatially varying intensity.
}
\usage{
Kcross.inhom(X, i=1, j=2, lambdaI, lambdaJ, \dots, r=NULL, breaks=NULL,
correction = c("border", "isotropic", "Ripley", "translate") ,
lambdaIJ=NULL)
}
\arguments{
\item{X}{The observed point pattern,
from which an estimate of the inhomogeneous 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{lambdaI}{
Values of the the estimated intensity of the sub-process of
points of type \code{i}.
Either a pixel image (object of class \code{"im"}),
or a numeric vector containing the intensity values
at each of the type \code{i} points in \code{X}.
}
\item{lambdaJ}{
Values of the the estimated intensity of the sub-process of
points of type \code{j}.
Either a pixel image (object of class \code{"im"}),
or a numeric vector containing the intensity values
at each of the type \code{j} points in \code{X}.
}
\item{r}{
Optional. Numeric vector giving the values of the argument \eqn{r}
at which the cross K 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}{
Optional. 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.
}
\item{\dots}{
Ignored.
}
\item{lambdaIJ}{
Optional. A matrix containing estimates of the
product of the intensities \code{lambdaI} and \code{lambdaJ}
for each pair of points of types \code{i} and \code{j} respectively.
}
}
\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 is a generalisation of the function \code{\link{Kcross}}
to include an adjustment for spatially inhomogeneous intensity,
in a manner similar to the function \code{\link{Kinhom}}.
The inhomogeneous cross-type \eqn{K} function is described by
Moller and Waagepetersen (2003, pages 48-49 and 51-53).
Briefly, given a multitype point process, suppose the sub-process
of points of type \eqn{j} has intensity function
\eqn{\lambda_j(u)}{lambda[j](u)} at spatial locations \eqn{u}.
Suppose we place a mass of \eqn{1/\lambda_j(\zeta)}{1/lambda[j](z)}
at each point \eqn{\zeta}{z} of type \eqn{j}. Then the expected total
mass per unit area is 1. The
inhomogeneous ``cross-type'' \eqn{K} function
\eqn{K_{ij}^{\mbox{inhom}}(r)}{K[ij]inhom(r)} equals the expected
total mass within a radius \eqn{r} of a point of the process
of type \eqn{i}.
If the process of type \eqn{i} points
were independent of the process of type \eqn{j} points,
then \eqn{K_{ij}^{\mbox{inhom}}(r)}{K[ij]inhom(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}
suggest dependence between the points of types \eqn{i} and \eqn{j}.
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 argument \code{lambdaI} supplies the values
of the intensity of the sub-process of points of type \code{i}.
It may be either
\describe{
\item{a pixel image}{(object of class \code{"im"}) which
gives the values of the type \code{i} intensity
at all locations in the window containing \code{X};
}
\item{a numeric vector}{containing the values of the
type \code{i} intensity evaluated only
at the data points of type \code{i}. The length of this vector
must equal the number of type \code{i} points in \code{X}.
}
}
Similarly \code{lambdaJ} should contain
estimated values of the intensity of the sub-process of points of
type \code{j}. It may be either a pixel image or a numeric vector.
The optional argument \code{lambdaIJ} is for advanced use only.
It is a matrix containing estimated
values of the products of these two intensities for each pair of
data points of types \code{i} and \code{j} respectively.
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 argument \code{correction} chooses the edge correction
as explained e.g. in \code{\link{Kest}}.
The pair correlation function can also be applied to the
result of \code{Kcross.inhom}; see \code{\link{pcf}}.
}
\references{
Moller, J. and Waagepetersen, R.
Statistical Inference and Simulation for Spatial Point Processes
Chapman and Hall/CRC
Boca Raton, 2003.
}
\section{Warnings}{
The arguments \code{i} and \code{j} are interpreted as
levels of the factor \code{X$marks}. Beware of the usual
trap with factors: numerical values are not
interpreted in the same way as character values.
}
\seealso{
\code{\link{Kcross}},
\code{\link{Kinhom}},
\code{\link{pcf}}
}
\examples{
# Lansing Woods data
data(lansing)
lansing <- lansing[seq(1,lansing$n, by=10)]
ma <- split(lansing)$maple
wh <- split(lansing)$whiteoak
# method (1): estimate intensities by nonparametric smoothing
lambdaM <- density.ppp(ma, sigma=0.15)
lambdaW <- density.ppp(wh, sigma=0.15)
K <- Kcross.inhom(lansing, "whiteoak", "maple", lambdaW[wh], lambdaM[ma])
K <- Kcross.inhom(lansing, "whiteoak", "maple", lambdaW, lambdaM)
# method (2): fit parametric intensity model
fit <- ppm(lansing, ~marks * polynom(x,y,2))
# evaluate fitted intensities at data points
# (these are the intensities of the sub-processes of each type)
inten <- fitted(fit, dataonly=TRUE)
# split according to types of points
lambda <- split(inten, lansing$marks)
K <- Kcross.inhom(lansing, "whiteoak", "maple",
lambda$whiteoak, lambda$maple)
# synthetic example: type A points have intensity 50,
# type B points have intensity 100 * x
lamB <- as.im(function(x,y){50 + 100 * x}, owin())
X <- superimpose(A=runifpoispp(50), B=rpoispp(lamB))
XA <- split(X)$A
XB <- split(X)$B
K <- Kcross.inhom(X, "A", "B",
lambdaI=rep(50, XA$n), lambdaJ=lamB[XB])
K <- Kcross.inhom(X, "A", "B",
lambdaI=as.im(50, X$window), lambdaJ=lamB)
}
\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}
