https://github.com/cran/spatstat
Tip revision: f6c1811e0b8755c851c04178019e0c07db04e697 authored by Adrian Baddeley on 22 December 2016, 09:53:41 UTC
version 1.48-0
version 1.48-0
Tip revision: f6c1811
Kdot.inhom.Rd
\name{Kdot.inhom}
\alias{Kdot.inhom}
\title{
Inhomogeneous Multitype K Dot Function
}
\description{
For a multitype point pattern,
estimate the inhomogeneous version of the dot \eqn{K} function,
which counts the expected number of points of any type
within a given distance of a point of type \eqn{i},
adjusted for spatially varying intensity.
}
\usage{
Kdot.inhom(X, i, lambdaI=NULL, lambdadot=NULL, \dots, r=NULL, breaks=NULL,
correction = c("border", "isotropic", "Ripley", "translate"),
sigma=NULL, varcov=NULL, lambdaIdot=NULL,
lambdaX=NULL, update=TRUE, leaveoneout=TRUE)
}
\arguments{
\item{X}{The observed point pattern,
from which an estimate of the inhomogeneous cross type \eqn{K} function
\eqn{K_{i\bullet}(r)}{Ki.(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}{The type (mark value)
of the points in \code{X} from which distances are measured.
A character string (or something that will be converted to a
character string).
Defaults to the first level of \code{marks(X)}.
}
\item{lambdaI}{
Optional.
Values of the estimated intensity of the sub-process of
points of type \code{i}.
Either a pixel image (object of class \code{"im"}),
a numeric vector containing the intensity values
at each of the type \code{i} points in \code{X},
a fitted point process model
(object of class \code{"ppm"} or \code{"kppm"} or \code{"dppm"}),
or a \code{function(x,y)} which
can be evaluated to give the intensity value at any location.
}
\item{lambdadot}{
Optional.
Values of the estimated intensity of the entire point process,
Either a pixel image (object of class \code{"im"}),
a numeric vector containing the intensity values at each of the
points in \code{X}, a fitted point process model
(object of class \code{"ppm"} or \code{"kppm"} or \code{"dppm"}),
or a \code{function(x,y)} which
can be evaluated to give the intensity value at any location.
}
\item{\dots}{
Ignored.
}
\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}{
This argument is for internal use only.
}
\item{correction}{
A character vector containing any selection of the
options \code{"border"}, \code{"bord.modif"},
\code{"isotropic"}, \code{"Ripley"}, \code{"translate"},
\code{"translation"},
\code{"none"} or \code{"best"}.
It specifies the edge correction(s) to be applied.
Alternatively \code{correction="all"} selects all options.
}
\item{sigma}{
Standard deviation of isotropic Gaussian smoothing kernel,
used in computing leave-one-out kernel estimates of
\code{lambdaI}, \code{lambdadot} if they are omitted.
}
\item{varcov}{
Variance-covariance matrix of anisotropic Gaussian kernel,
used in computing leave-one-out kernel estimates of
\code{lambdaI}, \code{lambdadot} if they are omitted.
Incompatible with \code{sigma}.
}
\item{lambdaIdot}{
Optional. A matrix containing estimates of the
product of the intensities \code{lambdaI} and \code{lambdadot}
for each pair of points, the first point of type \code{i} and
the second of any type.
}
\item{lambdaX}{
Optional. Values of the intensity for all points of \code{X}.
Either a pixel image (object of class \code{"im"}),
a numeric vector containing the intensity values
at each of the points in \code{X},
a fitted point process model
(object of class \code{"ppm"} or \code{"kppm"} or \code{"dppm"}),
or a \code{function(x,y)} which
can be evaluated to give the intensity value at any location.
If present, this argument overrides both \code{lambdaI} and
\code{lambdadot}.
}
\item{update}{
Logical value indicating what to do when
\code{lambdaI}, \code{lambdadot} or \code{lambdaX}
is a fitted point process model
(class \code{"ppm"}, \code{"kppm"} or \code{"dppm"}).
If \code{update=TRUE} (the default),
the model will first be refitted to the data \code{X}
(using \code{\link{update.ppm}} or \code{\link{update.kppm}})
before the fitted intensity is computed.
If \code{update=FALSE}, the fitted intensity of the
model will be computed without re-fitting it to \code{X}.
}
\item{leaveoneout}{
Logical value (passed to \code{\link{density.ppp}} or
\code{\link{fitted.ppm}}) specifying whether to use a
leave-one-out rule when calculating the intensity.
}
}
\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_{i\bullet}(r)}{Ki.(r)} has been estimated
}
\item{theo}{the theoretical value of \eqn{K_{i\bullet}(r)}{Ki.(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_{i\bullet}(r)}{Ki.(r)}
obtained by the edge corrections named.
}
\details{
This is a generalisation of the function \code{\link{Kdot}}
to include an adjustment for spatially inhomogeneous intensity,
in a manner similar to the function \code{\link{Kinhom}}.
Briefly, given a multitype point process, consider the points without
their types, and suppose this unmarked point process
has intensity function
\eqn{\lambda(u)}{lambda(u)} at spatial locations \eqn{u}.
Suppose we place a mass of \eqn{1/\lambda(\zeta)}{1/lambda(z)}
at each point \eqn{\zeta}{z} of the process. Then the expected total
mass per unit area is 1. The
inhomogeneous ``dot-type'' \eqn{K} function
\eqn{K_{i\bullet}^{\mbox{inhom}}(r)}{K[i.]inhom(r)} equals the expected
total mass within a radius \eqn{r} of a point of the process
of type \eqn{i}, discounting this point itself.
If the process of type \eqn{i} points
were independent of the points of other types,
then \eqn{K_{i\bullet}^{\mbox{inhom}}(r)}{K[i.]inhom(r)}
would equal \eqn{\pi r^2}{pi * r^2}.
Deviations between the empirical \eqn{K_{i\bullet}}{Ki.} curve
and the theoretical curve \eqn{\pi r^2}{pi * r^2}
suggest dependence between the points of types \eqn{i} and \eqn{j} for
\eqn{j\neq i}{j != i}.
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$marks} must be a factor.
The argument \code{i} will be interpreted as a
level of the factor \code{X$marks}. (Warning: this means that
an integer value \code{i=3} will be interpreted as the number 3,
\bold{not} the 3rd smallest level).
If \code{i} is missing, it defaults to the first
level of the marks factor, \code{i = levels(X$marks)[1]}.
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}.
}
\item{a function}{
of the form \code{function(x,y)}
which can be evaluated to give values of the intensity at
any locations.
}
\item{a fitted point process model}{
(object of class \code{"ppm"}, \code{"kppm"} or \code{"dppm"})
whose fitted \emph{trend} can be used as the fitted intensity.
(If \code{update=TRUE} the model will first be refitted to the
data \code{X} before the trend is computed.)
}
\item{omitted:}{
if \code{lambdaI} is omitted then it will be estimated
using a leave-one-out kernel smoother.
}
}
If \code{lambdaI} is omitted, then it will be estimated using
a `leave-one-out' kernel smoother, as described in Baddeley,
\Moller
and Waagepetersen (2000). The estimate of \code{lambdaI} for a given
point is computed by removing the point from the
point pattern, applying kernel smoothing to the remaining points using
\code{\link{density.ppp}}, and evaluating the smoothed intensity
at the point in question. The smoothing kernel bandwidth is controlled
by the arguments \code{sigma} and \code{varcov}, which are passed to
\code{\link{density.ppp}} along with any extra arguments.
Similarly the argument \code{lambdadot} should contain
estimated values of the intensity of the entire point process.
It may be either a pixel image, a numeric vector of length equal
to the number of points in \code{X}, a function, or omitted.
Alternatively if the argument \code{lambdaX} is given, then it specifies
the intensity values for all points of \code{X}, and the
arguments \code{lambdaI}, \code{lambdadot} will be ignored.
(The two arguments \code{lambdaI}, \code{lambdadot} allow the user
to specify two different methods for calculating the intensities of
the two kinds of points, while \code{lambdaX} ensures that the same
method is used for both kinds of points.)
For advanced use only, the optional argument \code{lambdaIdot}
is a matrix containing estimated
values of the products of these two intensities for each pair of
points, the first point of type \code{i} and the second of any type.
The argument \code{r} is the vector of values for the
distance \eqn{r} at which \eqn{K_{i\bullet}(r)}{Ki.(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{
\ifelse{latex}{\out{M\o ller}}{Moller}, J. and Waagepetersen, R.
Statistical Inference and Simulation for Spatial Point Processes
Chapman and Hall/CRC
Boca Raton, 2003.
}
\section{Warnings}{
The argument \code{i} is interpreted as
a level of the factor \code{X$marks}. It is converted to a character
string if it is not already a character string.
The value \code{i=1} does \bold{not}
refer to the first level of the factor.
}
\seealso{
\code{\link{Kdot}},
\code{\link{Kinhom}},
\code{\link{Kcross.inhom}},
\code{\link{Kmulti.inhom}},
\code{\link{pcf}}
}
\examples{
# Lansing Woods data
woods <- lansing
woods <- woods[seq(1,npoints(woods), by=10)]
ma <- split(woods)$maple
lg <- unmark(woods)
# Estimate intensities by nonparametric smoothing
lambdaM <- density.ppp(ma, sigma=0.15, at="points")
lambdadot <- density.ppp(lg, sigma=0.15, at="points")
K <- Kdot.inhom(woods, "maple", lambdaI=lambdaM,
lambdadot=lambdadot)
# Equivalent
K <- Kdot.inhom(woods, "maple", sigma=0.15)
# Fit model
fit <- ppm(woods ~ marks * polynom(x,y,2))
K <- Kdot.inhom(woods, "maple", lambdaX=fit, update=FALSE)
# synthetic example: type A points have intensity 50,
# type B points have intensity 50 + 100 * x
lamB <- as.im(function(x,y){50 + 100 * x}, owin())
lamdot <- as.im(function(x,y) { 100 + 100 * x}, owin())
X <- superimpose(A=runifpoispp(50), B=rpoispp(lamB))
K <- Kdot.inhom(X, "B", lambdaI=lamB, lambdadot=lamdot)
}
\author{
\spatstatAuthors
}
\keyword{spatial}
\keyword{nonparametric}