rhohat.Rd
\name{rhohat}
\alias{rhohat}
\alias{rhohat.ppp}
\alias{rhohat.quad}
\alias{rhohat.ppm}
\title{
Smoothing Estimate of Covariate Transformation
}
\description{
Computes a smoothing estimate of the intensity of a point process,
as a function of a (continuous) spatial covariate.
}
\usage{
rhohat(object, covariate, ...)
\method{rhohat}{ppp}(object, covariate, ...,
method=c("ratio", "reweight", "transform"),
smoother=c("kernel", "local"),
dimyx=NULL, eps=NULL,
n = 512, bw = "nrd0", adjust=1, from = NULL, to = NULL,
bwref=bw,
covname, confidence=0.95)
\method{rhohat}{quad}(object, covariate, ...,
method=c("ratio", "reweight", "transform"),
smoother=c("kernel", "local"),
dimyx=NULL, eps=NULL,
n = 512, bw = "nrd0", adjust=1, from = NULL, to = NULL,
bwref=bw,
covname, confidence=0.95)
\method{rhohat}{ppm}(object, covariate, ...,
method=c("ratio", "reweight", "transform"),
smoother=c("kernel", "local"),
dimyx=NULL, eps=NULL,
n = 512, bw = "nrd0", adjust=1, from = NULL, to = NULL,
bwref=bw,
covname, confidence=0.95)
}
\arguments{
\item{object}{
A point pattern (object of class \code{"ppp"}),
a quadrature scheme (object of class \code{"quad"})
or a fitted point process model (object of class \code{"ppm"}).
}
\item{covariate}{
Either a \code{function(x,y)} or a pixel image (object of
class \code{"im"}) providing the values of the covariate at any
location.
Alternatively one of the strings \code{"x"} or \code{"y"}
signifying the Cartesian coordinates.
}
\item{method}{
Character string determining the smoothing method. See Details.
}
\item{smoother}{
Character string determining the smoothing algorithm. See Details.
}
\item{dimyx,eps}{
Arguments passed to \code{\link{as.mask}} to control the pixel
resolution at which the covariate will be evaluated.
}
\item{bw}{
Smoothing bandwidth or bandwidth rule
(passed to \code{\link{density.default}}).
}
\item{adjust}{
Smoothing bandwidth adjustment factor
(passed to \code{\link{density.default}}).
}
\item{n, from, to}{
Arguments passed to \code{\link{density.default}} to
control the number and range of values at which the function
will be estimated.
}
\item{bwref}{
Optional. An alternative value of \code{bw} to use when smoothing
the reference density (the density of the covariate values
observed at all locations in the window).
}
\item{\dots}{
Additional arguments passed to \code{\link{density.default}}
or \code{\link[locfit]{locfit}}.
}
\item{covname}{
Optional. Character string to use as the name of the covariate.
}
\item{confidence}{
Confidence level for confidence intervals.
A number between 0 and 1.
}
}
\details{
If \code{object} is a point pattern, this command assumes that
\code{object} is a realisation of a Poisson point process with
intensity function \eqn{\lambda(u)}{lambda(u)} of the form
\deqn{\lambda(u) = \rho(Z(u))}{lambda(u) = rho(Z(u))} where
\eqn{Z} is the spatial covariate function given by \code{covariate},
and \eqn{\rho(z)}{rho(z)} is a function to be estimated.
This command computes estimators of \eqn{\rho(z)}{rho(z)}
proposed by Baddeley and Turner (2005)
and Baddeley et al (2012).
The covariate \eqn{Z} must have continuous values.
If \code{object} is a fitted point process model, suppose \code{X} is
the original data point pattern to which the model was fitted. Then
this command assumes \code{X} is a realisation of a Poisson point
process with intensity function of the form
\deqn{
\lambda(u) = \rho(Z(u)) \kappa(u)
}{
lambda(u) = rho(Z(u)) * kappa(u)
}
where \eqn{\kappa(u)}{kappa(u)} is the intensity of the fitted model
\code{object}. A smoothing estimator of \eqn{\rho(z)}{rho(z)} is computed.
The estimation procedure is determined by the character strings
\code{method} and \code{smoother}.
The estimation procedure involves computing several density estimates
and combining them.
The algorithm used to compute density estimates is
determined by \code{smoother}:
\itemize{
\item If \code{smoother="kernel"},
each the smoothing procedure is based on
fixed-bandwidth kernel density estimation,
performed by \code{\link{density.default}}.
\item If \code{smoother="local"}, the smoothing procedure
is based on local likelihood density estimation, performed by
\code{\link[locfit]{locfit}}.
}
The \code{method} determines how the density estimates will be
combined to obtain an estimate of \eqn{\rho(z)}{rho(z)}:
\itemize{
\item
If \code{method="ratio"}, then \eqn{\rho(z)}{rho(z)} is
estimated by the ratio of two density estimates.
The numerator is a (rescaled) density estimate obtained by
smoothing the values \eqn{Z(y_i)}{Z(y[i])} of the covariate
\eqn{Z} observed at the data points \eqn{y_i}{y[i]}. The denominator
is a density estimate of the reference distribution of \eqn{Z}.
\item
If \code{method="reweight"}, then \eqn{\rho(z)}{rho(z)} is
estimated by applying density estimation to the
values \eqn{Z(y_i)}{Z(y[i])} of the covariate
\eqn{Z} observed at the data points \eqn{y_i}{y[i]},
with weights inversely proportional to the reference density of
\eqn{Z}.
\item
If \code{method="transform"},
the smoothing method is variable-bandwidth kernel
smoothing, implemented by applying the Probability Integral Transform
to the covariate values, yielding values in the range 0 to 1,
then applying edge-corrected density estimation on the interval
\eqn{[0,1]}, and back-transforming.
}
}
\value{
A function value table (object of class \code{"fv"})
containing the estimated values of \eqn{\rho}{rho} for a sequence
of values of \eqn{Z}.
Also belongs to the class \code{"rhohat"}
which has special methods for \code{print}, \code{plot}
and \code{predict}.
}
\section{Categorical and discrete covariates}{
This technique assumes that the covariate has continuous values.
It is not applicable to covariates with categorical (factor) values
or discrete values such as small integers.
For a categorical covariate, use
\code{\link{quadratcount}(X, tess=covariate)}
}
\references{
Baddeley, A., Chang, Y.-M., Song, Y. and Turner, R. (2012)
Nonparametric estimation of the dependence of a point
process on spatial covariates.
\emph{Statistics and Its Interface} \bold{5} (2), 221--236.
Baddeley, A. and Turner, R. (2005)
Modelling spatial point patterns in R.
In: A. Baddeley, P. Gregori, J. Mateu, R. Stoica, and D. Stoyan,
editors, \emph{Case Studies in Spatial Point Pattern Modelling},
Lecture Notes in Statistics number 185. Pages 23--74.
Springer-Verlag, New York, 2006.
ISBN: 0-387-28311-0.
}
\author{
Adrian Baddeley
\email{Adrian.Baddeley@csiro.au}
\url{http://www.maths.uwa.edu.au/~adrian/}
and Rolf Turner
\email{r.turner@auckland.ac.nz}
}
\seealso{
\code{\link{rho2hat}},
\code{\link{methods.rhohat}}
}
\examples{
X <- rpoispp(function(x,y){exp(3+3*x)})
rho <- rhohat(X, "x")
rho <- rhohat(X, function(x,y){x})
plot(rho)
curve(exp(3+3*x), lty=3, col=2, add=TRUE)
rhoB <- rhohat(X, "x", method="reweight")
rhoC <- rhohat(X, "x", method="transform")
fit <- ppm(X, ~x)
rr <- rhohat(fit, "y")
}
\keyword{spatial}
\keyword{models}
