Given a point pattern and two spatial covariates \(Z_1\) and \(Z_2\), construct a smooth estimate of the relative risk of the pair \((Z_1,Z_2)\).
rho2hat(object, cov1, cov2, ..., method=c("ratio", "reweight"))A point pattern (object of class "ppp"),
a quadrature scheme (object of class "quad")
or a fitted point process model (object of class "ppm").
The two covariates.
Each argument is either a function(x,y) or a pixel image (object of
class "im") providing the values of the covariate at any
location, or one of the strings "x" or "y"
signifying the Cartesian coordinates.
Additional arguments passed to density.ppp to smooth
the scatterplots.
Character string determining the smoothing method. See Details.
A pixel image (object of class "im"). Also
belongs to the special class "rho2hat" which has a plot method.
This is a bivariate version of rhohat.
If object is a point pattern, this command
produces a smoothed version of the scatterplot of
the values of the covariates cov1 and cov2
observed at the points of the point pattern.
The covariates cov1,cov2 must have continuous values.
If object is a fitted point process model, suppose X is
the original data point pattern to which the model was fitted. Then
this command assumes X is a realisation of a Poisson point
process with intensity function of the form
$$
\lambda(u) = \rho(Z_1(u), Z_2(u)) \kappa(u)
$$
where \(\kappa(u)\) is the intensity of the fitted model
object, and \(\rho(z_1,z_2)\) is a function
to be estimated. The algorithm computes a smooth estimate of the
function \(\rho\).
The method determines how the density estimates will be
combined to obtain an estimate of \(\rho(z_1, z_2)\):
If method="ratio", then \(\rho(z_1, z_2)\) is
estimated by the ratio of two density estimates.
The numerator is a (rescaled) density estimate obtained by
smoothing the points \((Z_1(y_i), Z_2(y_i))\)
obtained by evaluating the two covariate \(Z_1, Z_2\)
at the data points \(y_i\). The denominator
is a density estimate of the reference distribution of
\((Z_1,Z_2)\).
If method="reweight", then \(\rho(z_1, z_2)\) is
estimated by applying density estimation to the
points \((Z_1(y_i), Z_2(y_i))\)
obtained by evaluating the two covariate \(Z_1, Z_2\)
at the data points \(y_i\),
with weights inversely proportional to the reference density of
\((Z_1,Z_2)\).
Baddeley, A., Chang, Y.-M., Song, Y. and Turner, R. (2012) Nonparametric estimation of the dependence of a point process on spatial covariates. Statistics and Its Interface 5 (2), 221--236.
# NOT RUN {
data(bei)
attach(bei.extra)
plot(rho2hat(bei, elev, grad))
fit <- ppm(bei, ~elev, covariates=bei.extra)
# plot(rho2hat(fit, elev, grad))
plot(rho2hat(fit, elev, grad, method="reweight"))
# }
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