rho2hat.Rd
\name{rho2hat}
\alias{rho2hat}
\title{
Smoothed Relative Density of Pairs of Covariate Values
}
\description{
Given a point pattern and two spatial covariates \eqn{Z_1}{Z1} and
\eqn{Z_2}{Z2}, construct a smooth estimate of the relative risk of
the pair \eqn{(Z_1,Z_2)}{(Z1, Z2)}.
}
\usage{
rho2hat(object, cov1, cov2, ..., method=c("ratio", "reweight"))
}
\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{cov1,cov2}{
The two covariates.
Each argument is either a \code{function(x,y)} or a pixel image (object of
class \code{"im"}) providing the values of the covariate at any
location, or one of the strings \code{"x"} or \code{"y"}
signifying the Cartesian coordinates.
}
\item{\dots}{
Additional arguments passed to \code{\link{density.ppp}} to smooth
the scatterplots.
}
\item{method}{
Character string determining the smoothing method. See Details.
}
}
\details{
This is a bivariate version of \code{\link{rhohat}}.
If \code{object} is a point pattern, this command
produces a smoothed version of the scatterplot of
the values of the covariates \code{cov1} and \code{cov2}
observed at the points of the point pattern.
The covariates \code{cov1,cov2} 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_1(u), Z_2(u)) \kappa(u)
}{
lambda(u) = rho(Z1(u), Z2(u)) * kappa(u)
}
where \eqn{\kappa(u)}{kappa(u)} is the intensity of the fitted model
\code{object}, and \eqn{\rho(z_1,z_2)}{rho(z1, z2)} is a function
to be estimated. The algorithm computes a smooth estimate of the
function \eqn{\rho}{rho}.
The \code{method} determines how the density estimates will be
combined to obtain an estimate of \eqn{\rho(z_1, z_2)}{rho(z1, z2)}:
\itemize{
\item
If \code{method="ratio"}, then \eqn{\rho(z_1, z_2)}{rho(z1,z2)} is
estimated by the ratio of two density estimates.
The numerator is a (rescaled) density estimate obtained by
smoothing the points \eqn{(Z_1(y_i), Z_2(y_i))}{(Z1(y[i]), Z2(y[i]))}
obtained by evaluating the two covariate \eqn{Z_1, Z_2}{Z1, Z2}
at the data points \eqn{y_i}{y[i]}. The denominator
is a density estimate of the reference distribution of
\eqn{(Z_1,Z_2)}{(Z1, Z2)}.
\item
If \code{method="reweight"}, then \eqn{\rho(z_1, z_2)}{rho(z1,z2)} is
estimated by applying density estimation to the
points \eqn{(Z_1(y_i), Z_2(y_i))}{(Z1(y[i]), Z2(y[i]))}
obtained by evaluating the two covariate \eqn{Z_1, Z_2}{Z1, Z2}
at the data points \eqn{y_i}{y[i]},
with weights inversely proportional to the reference density of
\eqn{(Z_1,Z_2)}{(Z1, Z2)}.
}
}
\value{
A pixel image (object of class \code{"im"}). Also
belongs to the special class \code{"rho2hat"} which has a plot method.
}
\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.
}
\author{
\adrian
}
\seealso{
\code{\link{rhohat}},
\code{\link{methods.rho2hat}}
}
\examples{
data(bei)
attach(bei.extra)
plot(rho2hat(bei, elev, grad))
fit <- ppm(bei, ~elev, covariates=bei.extra)
\dontrun{
plot(rho2hat(fit, elev, grad))
}
plot(rho2hat(fit, elev, grad, method="reweight"))
}
\keyword{spatial}
\keyword{models}