View source: R/racscovariance.R
racscovariance | R Documentation |
Estimates the covariance of a stationary RACS. The plug-in moment covariance estimator and newer balanced estimators based on (Picka, 1997; Picka, 2000) are available.
racscovariance( xi, obswin = NULL, setcov_boundarythresh = NULL, estimators = "all", drop = FALSE ) racscovariance.cvchat( cvchat, cpp1 = NULL, phat = NULL, estimators = "all", drop = FALSE )
xi |
A binary map. Either an |
obswin |
The observation window as an |
setcov_boundarythresh |
To avoid instabilities caused by dividing by very small quantities, if the set covariance of the observation window
is smaller than |
estimators |
A list of strings specifying covariance estimators to use.
See details.
Passing |
drop |
If TRUE and one estimator is selected then the returned value will be a single |
cvchat |
The plug-in moment estimate of covariance as an |
cpp1 |
Picka's reduced window estimate of coverage probability as an |
phat |
The classical estimate of coverage probability,
which is the observed area in |
The covariance of a RACS is also known as the two-point coverage probability, and is closely related to the semivariogram. The covariance of a stationary RACS Ξ given a vector v is the probability that two points separated by a vector v are covered by Ξ.
Given a vector v, the plug-in moment covariance estimate from a binary map is the volume of the set of points, x, such that both
x and x+v are observed to be in the foreground
relative to the volume of points, x, for which both x and x+v
are in the observation window (Hingee, 2019).
Picka (1997, 2000) suggested a number of improvements to centred
covariance estimation (see cencovariance
) that 'balanced' the
data used to estimate covariance with the data used to estimate coverage
probability. These lead to covariance estimators that give
estimates for the covariance of Ξ that are a constant offset from
covariance estimates for the complement of Ξ (note the constant offset
depends on the coverage probability), which
appears to avoid some surprising behaviour that the plug-in moment covariance estimator
suffers (Hingee, 2019).
These estimators are called pickaint
and pickaH
in this package.
Another improved estimator, inspired by an 'intrinsic modification' briefly mentioned by Picka (1997)
for pair-correlation estimators, is also available.
We have called this estimator mattfeldt
as a similar isotropic estimator for pair-correlation
was studied by Mattfeldt and Stoyan (2000).
The estimators available are (see (Hingee, 2019) for more information):
plugin
the plug-in moment covariance estimator
mattfeldt
an estimator inspired by an
'intrinsically' balanced pair-correlation estimator from Picka that was later studied in an
isotropic situation by Mattfeldt and Stoyan (2000)
pickaint
an estimator inspired by an
'intrinsically' balanced centred covariance estimator from Picka (2000).
pickaH
an estimator inspired by the
'additively' balanced centred covariance estimator from Picka (2000).
If drop = TRUE
and only one estimator is requested then
an im
object containing the covariance estimate.
Otherwise a named imlist
of covariance estimates corresponding to each requested estimator.
racscovariance
: Estimates covariance from a binary map.
racscovariance.cvchat
: Computes covariance estimates from
a plug-in moment estimate of covariance, Picka's reduced window estimate of coverage probability,
and the usual estimate of coverage probability.
If these estimates already exist then racscovariance.cvchat
can save significant computation time.
Kassel Liam Hingee
Hingee, K.L. (2019) Spatial Statistics of Random Closed Sets for Earth Observations. PhD: Perth, Western Australia: University of Western Australia. Submitted.
Mattfeldt, T. and Stoyan, D. (2000) Improved estimation of the pair correlation function of random sets. Journal of Microscopy, 200, 158-173.
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$coarse obswin <- Frame(xi) # Estimate from a binary map balancedcvchats_direct <- racscovariance(xi, obswin = obswin, estimators = "all") phat <- coverageprob(xi, obswin = obswin) cvchat <- plugincvc(xi, obswin) cpp1 <- cppicka(xi, obswin = Frame(heather$coarse)) harmonised <- harmonise.im(cvchat = cvchat, cpp1 = cpp1) cvchat <- harmonised$cvchat cpp1 <- harmonised$cpp1 # Compute balanced estimate of covariance from other estimates balancedcvchats_fromplugincvc <- racscovariance.cvchat(cvchat, cpp1, phat, estimators = "pickaH", drop = TRUE)
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