Smoothed Relative Density of Pairs of Covariate Values
Given a point pattern and two spatial covariates Z1 and Z2, construct a smooth estimate of the relative risk of the pair (Z1, Z2).
rho2hat(object, cov1, cov2, ..., method=c("ratio", "reweight"))object |
A point pattern (object of class |
cov1,cov2 |
The two covariates.
Each argument is either a |
... |
Additional arguments passed to |
method |
Character string determining the smoothing method. See Details. |
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(Z1(u), Z2(u)) * kappa(u)
where kappa(u) is the intensity of the fitted model
object, and rho(z1, z2) 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(z1, z2):
If method="ratio", then rho(z1,z2) is
estimated by the ratio of two density estimates.
The numerator is a (rescaled) density estimate obtained by
smoothing the points (Z1(y[i]), Z2(y[i]))
obtained by evaluating the two covariate Z1, Z2
at the data points y[i]. The denominator
is a density estimate of the reference distribution of
(Z1, Z2).
If method="reweight", then rho(z1,z2) is
estimated by applying density estimation to the
points (Z1(y[i]), Z2(y[i]))
obtained by evaluating the two covariate Z1, Z2
at the data points y[i],
with weights inversely proportional to the reference density of
(Z1, Z2).
A pixel image (object of class "im"). Also
belongs to the special class "rho2hat" which has a plot method.
Adrian Baddeley Adrian.Baddeley@curtin.edu.au
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, ~elev, covariates=bei.extra) # plot(rho2hat(fit, elev, grad)) plot(rho2hat(fit, elev, grad, method="reweight"))
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