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, ...,
transform=FALSE,
dimyx=NULL, eps=NULL,
n = 512, bw = "nrd0", adjust=1, from = NULL, to = NULL,
bwref=bw,
covname)
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
transform
Logical value determining the smoothing method. 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.
covname
Optional. Character string to use as the name of the covariate.
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 an estimator of $\rho(z)$ proposed by Baddeley and Turner (2005).

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 modified version of the Baddeley-Turner (2005) smoothing estimator is computed.

If transform=FALSE, the smoothing method is fixed bandwidth kernel smoothing, using density.default. If transform=TRUE, 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 fixed-bandwidth smoothing 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. and Chang, Y.-M. and Song, Y. and Turner, R. Diagnostics for transformation of covariates in spatial Poisson point process models. Submitted for publication. 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.

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")