Smoothed Relative Density of Pairs of Covariate Values
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
- 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
- Additional arguments passed to
density.pppto smooth the scatterplots.
- Character string determining the smoothing method. See Details.
This is a bivariate version of
object is a point pattern, this command
produces a smoothed version of the scatterplot of
the values of the covariates
observed at the points of the point pattern.
cov1,cov2 must have continuous values.
object is a fitted point process model, suppose
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
method determines how the density estimates will be
combined to obtain an estimate of $\rho(z_1, z_2)$:
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$(Z_1,Z_2)$.
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$(Z_1,Z_2)$.
- A pixel image (object of class
"im"). Also belongs to the special class
"rho2hat"which has a plot method.
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.
data(bei) attach(bei.extra) plot(rho2hat(bei, elev, grad)) fit <- ppm(bei, ~polynom(elev, 3), covariates=bei.extra) plot(rho2hat(fit, elev, grad)) plot(rho2hat(fit, elev, grad, method="reweight"))