https://github.com/cran/spatstat
Tip revision: c75a36fce807d4a1105403c1f8628a196a3e54da authored by Adrian Baddeley on 24 October 2008, 14:28:47 UTC
version 1.14-5
version 1.14-5
Tip revision: c75a36f
Iest.Rd
\name{Iest}
\alias{Iest}
\title{Estimate the I-function}
\description{
Estimates the summary function \eqn{I(r)} for a multitype point pattern.
}
\usage{
Iest(X, eps=NULL, r=NULL, breaks=NULL)
}
\arguments{
\item{X}{The observed point pattern,
from which an estimate of \eqn{I(r)} will be computed.
An object of class \code{"ppp"}, or data
in any format acceptable to \code{\link{as.ppp}()}.
}
\item{eps}{
the resolution of the discrete approximation to Euclidean distance
(see below). There is a sensible default.
}
\item{r}{Optional. Numeric vector of values for the argument \eqn{r}
at which \eqn{I(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 \code{r}.
}
\item{breaks}{
An alternative to the argument \code{r}. Not normally invoked by the user.
See Details section.
}
}
\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
\item{r}{the vector of values of the argument \eqn{r}
at which the function \eqn{I} has been estimated}
\item{rs}{the ``reduced sample'' or ``border correction''
estimator of \eqn{I(r)} computed from
the border-corrected estimates of \eqn{J} functions}
\item{km}{the spatial Kaplan-Meier estimator of \eqn{I(r)} computed from
the Kaplan-Meier estimates of \eqn{J} functions}
\item{un}{the uncorrected estimate of \eqn{I(r)}
computed from the uncorrected estimates of \eqn{J}
}
\item{theo}{the theoretical value of \eqn{I(r)}
for a stationary Poisson process: identically equal to \eqn{0}
}
}
\note{
Sizeable amounts of memory may be needed during the calculation.
}
\details{
The \eqn{I} function
summarises the dependence between types in a multitype point process
(Van Lieshout and Baddeley, 1999)
It is based on the concept of the \eqn{J} function for an
unmarked point process (Van Lieshout and Baddeley, 1996).
See \code{\link{Jest}} for information about the \eqn{J} function.
The \eqn{I} function is defined as
\deqn{ %
I(r) = \sum_{i=1}^m p_i J_{ii}(r) %
- J_{\bullet\bullet}(r)}{ %
I(r) = (sum p[i] Jii(r)) - J(r)
}
where \eqn{J_{\bullet\bullet}}{J} is the \eqn{J} function for
the entire point process ignoring the marks, while
\eqn{J_{ii}}{Jii} is the \eqn{J} function for the
process consisting of points of type \eqn{i} only,
and \eqn{p_i}{p[i]} is the proportion of points which are of type \eqn{i}.
The \eqn{I} function is designed to measure dependence between
points of different types, even if the points are
not Poisson. Let \eqn{X} be a stationary multitype point process,
and write \eqn{X_i}{X[i]} for the process of points of type \eqn{i}.
If the processes \eqn{X_i}{X[i]} are independent of each other,
then the \eqn{I}-function is identically equal to \eqn{0}.
Deviations \eqn{I(r) < 1} or \eqn{I(r) > 1}
typically indicate negative and positive association, respectively,
between types.
See Van Lieshout and Baddeley (1999)
for further information.
An estimate of \eqn{I} derived from a multitype spatial point pattern dataset
can be used in exploratory data analysis and formal inference
about the pattern. The estimate of \eqn{I(r)} is compared against the
constant function \eqn{0}.
Deviations \eqn{I(r) < 1} or \eqn{I(r) > 1}
may suggest negative and positive association, respectively.
This algorithm estimates the \eqn{I}-function
from the multitype point pattern \code{X}.
It assumes that \code{X} can be treated
as a realisation of a stationary (spatially homogeneous)
random spatial marked point process in the plane, observed through
a 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}()}. It must be a multitype point pattern
(it must have a \code{marks} vector which is a \code{factor}).
The function \code{\link{Jest}} is called to
compute estimates of the \eqn{J} functions in the formula above.
In fact three different estimates are computed
using different edge corrections. See \code{\link{Jest}} for
information.
}
\references{
Van Lieshout, M.N.M. and Baddeley, A.J. (1996)
A nonparametric measure of spatial interaction in point patterns.
\emph{Statistica Neerlandica} \bold{50}, 344--361.
Van Lieshout, M.N.M. and Baddeley, A.J. (1999)
Indices of dependence between types in multivariate point patterns.
\emph{Scandinavian Journal of Statistics} \bold{26}, 511--532.
}
\seealso{
\code{\link{Jest}}
}
\examples{
data(amacrine)
Ic <- Iest(amacrine)
plot(Ic, main="Amacrine Cells data")
# values are below I= 0, suggesting negative association
# between 'on' and 'off' cells.
}
\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}