rhohat

0th

Percentile

Smoothing Estimate of Covariate Transformation

Computes a smoothing estimate of the intensity of a point process, as a function of a spatial covariate.

Keywords
models, spatial
Usage
rhohat(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
object
A point pattern (object of class "ppp"), a quadrature scheme (object of class "quad") or a fitted point process model (object of class "ppm").
covariate
Either a function(x,y) or a pixel image (object of class "im") providing the values of the covariate at any location. Alternatively one of the strings "x" or "y" signifying the Cartesian
method
Character string determining the smoothing method. See Details.
smoother
Character string determining the smoothing algorithm. See Details.
dimyx,eps
Arguments passed to as.mask to control the pixel resolution at which the covariate will be evaluated.
bw
Smoothing bandwidth or bandwidth rule (passed to density.default).
Smoothing bandwidth adjustment factor (passed to density.default).
n, from, to
Arguments passed to density.default to control the number and range of values at which the function will be estimated.
bwref
Optional. An alternative value of bw to use when smoothing the reference density (the density of the covariate values observed at all locations in the window).
...
Additional arguments passed to density.default or locfit.
covname
Optional. Character string to use as the name of the covariate.
confidence
Confidence level for confidence intervals. A number between 0 and 1.
Details

If object is a point pattern, this command assumes that object is a realisation of a Poisson point process with intensity function $\lambda(u)$ of the form $$\lambda(u) = \rho(Z(u))$$ where $Z$ is the spatial covariate function given by covariate, and $\rho(z)$ is a function to be estimated. This command computes estimators of $\rho(z)$ proposed by Baddeley and Turner (2005) and Baddeley et al (2012).

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 $$\lambda(u) = \rho(Z(u)) \kappa(u)$$ where $\kappa(u)$ is the intensity of the fitted model object. A smoothing estimator of $\rho(z)$ is computed.

The estimation procedure is determined by the character strings method and smoother. The estimation procedure involves computing several density estimates and combining them. The algorithm used to compute density estimates is determined by smoother:

• Ifsmoother="kernel", each the smoothing procedure is based on fixed-bandwidth kernel density estimation, performed bydensity.default.
• Ifsmoother="local", the smoothing procedure is based on local likelihood density estimation, performed bylocfit.
The method determines how the density estimates will be combined to obtain an estimate of $\rho(z)$:
• Ifmethod="ratio", then$\rho(z)$is estimated by the ratio of two density estimates. The numerator is a (rescaled) density estimate obtained by smoothing the values$Z(y_i)$of the covariate$Z$observed at the data points$y_i$. The denominator is a density estimate of the reference distribution of$Z$.
• Ifmethod="reweight", then$\rho(z)$is estimated by applying density estimation to the values$Z(y_i)$of the covariate$Z$observed at the data points$y_i$, with weights inversely proportional to the reference density of$Z$.
• Ifmethod="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$[0,1]$, and back-transforming.

Value

• A function value table (object of class "fv") containing the estimated values of $\rho$ for a sequence of values of $Z$. Also belongs to the class "rhohat" which has special methods for print, plot and predict.

References

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, in press. 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, 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.

rho2hat, methods.rhohat

• rhohat
Examples
X <-  rpoispp(function(x,y){exp(3+3*x)})
rho <- rhohat(X, "x")
rho <- rhohat(X, function(x,y){x})
plot(rho)
rr <- rhohat(fit, "y")