This function provides estimates of coverage probability from subsets of the observation window, which are a key component of balanced estimators of covariance, centred covariance, pair-correlation and gliding box lacunarity.
cppicka(xi, obswin = NULL, setcov_boundarythresh = NULL)An im object. Pixel values correspond to estimates of the coverage probability
from the subregion of the observation window, \(W\), that is the intersection of \(W\) and \(W\) shifted by vector \(v\), where \(v\) is the pixel location.
An observation of a RACS of interest as a full binary map (as an im object) or as the foreground set (as an owin object).
In the latter case the observation window, obswin, must be supplied.
If xi is an owin object then obswin is an
owin object that specifies the observation window.
To avoid instabilities caused by dividing by very small quantities, if the set covariance of the observation window
is smaller than setcov_boundarythresh, then the returned pixel value is NA.
Kassel Liam Hingee
The plug-in moment covariance estimator (plugincvc) uses less of the observation window than the usual coverage probability estimators.
Picka (1997, 2000) created new 'balanced' estimators of centred covariance and pair-correlation
that accounted for this difference.
A key component of Picka's estimators is an estimate of the coverage probability from the subregion of the binary map that is
the intersection between \(W\) and \(W\) shifted by vector \(v\), where \(W\) is the observation window (p.~687, Picka, 2000).
If we treat \(X\) and \(W\) as indicator functions representing the foreground and observation window respectively,
this coverage probability estimator used by Picka is
$$ \frac{\int X(u) W(u) W(u - v) du} {\int W(u) W(u - v) du}. $$
cppicka produces these estimates for an array of vectors \(v\) using fast Fourier transforms.
Picka, J.D. (1997) Variance-Reducing Modifications for Estimators of Dependence in Random Sets. Ph.D.: Illinois, USA: The University of Chicago.
Picka, J.D. (2000) Variance reducing modifications for estimators of standardized moments of random sets. Advances in Applied Probability, 32, 682-700.
xi <- heather$coarse
obswindow <- Frame(heather$coarse)
cp <- coverageprob(xi, obswindow)
cpp1 <- cppicka(xi, obswindow)
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