rhohat
Smoothing Estimate of Covariate Transformation
Computes a smoothing estimate of the intensity of a point process, as a function of a spatial covariate.
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
). - adjust
- 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 forprint
,plot
andpredict
.
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.
See Also
Examples
X <- rpoispp(function(x,y){exp(3+3*x)})
rho <- rhohat(X, "x")
rho <- rhohat(X, function(x,y){x})
plot(rho)
curve(exp(3+3*x), lty=3, col=2, add=TRUE)
fit <- ppm(X, ~x)
rr <- rhohat(fit, "y")