paircorr {lacunaritycovariance}

Balanced estimation of pair-correlation.

Description

Estimates the pair-correlation function of a stationary RACS. The plug-in moment pair-correlation estimator and three 'balanced' estimators suggested by Picka (2000) are available.

Usage

paircorr(
  xi,
  obswin = NULL,
  setcov_boundarythresh = NULL,
  estimators = "all",
  drop = FALSE
)

paircorr.cvchat(
  cvchat,
  cpp1 = NULL,
  phat = NULL,
  estimators = "all",
  drop = FALSE
)

Arguments

xi

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. See lacunaritycovariance-package for details.

obswin

If xi is an owin object then obswin is an owin object that specifies the observation window.

setcov_boundarythresh

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 covariance is given a value of NA.

estimators

A list of strings specifying estimators to use. See details. estimators = "all" will select all available estimators.

drop

If TRUE and one estimator selected then the returned value will be a single im object and not a list of im object. estimators = "all" will select all inbuilt estimators. See details.

cvchat

The plug-in moment estimate of covariance as an im object. Typically created with plugincvc.

cpp1

Picka's reduced window estimate of coverage probability as an im object - used in improved (balanced) covariance estimators. Can be generated using cppicka.

phat

The plug-in moment estimate of coverage probability, which is the observed foreground area in xi divided by the total area of the observation window. See coverageprob for more information.

Details

The pair-correlation of a stationary RACS is

g(v) = C(v) / p^2.

The estimators available are (see (Hingee, 2019) for more information):

• plugin the plug-in moment pair-correlation estimator which is Chat(v) / (phat^2), where Chat and phat are the plug-in moment estimate of covariance and the usual estimate of coverage probability, respectively.

• mattfeldt an 'intrinsically' balanced pair-correlation estimator suggested by Picka (1997). A similar isotropic pair-correlation estimator was later studied by Mattfeldt and Stoyan (2000).

• pickaint Picka's 'intrinsically' balanced pair-correlation estimator (Picka, 2000).

• pickaH Picka's 'additively' balanced pair-correlation estimator (Picka, 2000).

Value

If drop = TRUE and a single estimator is requested then an im object containing the pair-correlation estimate is returned. Otherwise a named imlist of im objects containing the pair-correlation estimates for each requested estimator.

Functions

• paircorr(): Estimates pair-correlation from a binary map.

• paircorr.cvchat(): Generates pair-correlation estimates from the plug-in moment estimates of covariance, Picka's reduced window estimate of coverage probability, and the coverage fraction (which is an unbiased estimate of the coverage probability). If these estimates already exist then paircorr.cvchat can save significant computation time.

Author(s)

Kassel Liam Hingee

References

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.

Examples

xi <- as.im(heather$coarse, na.replace = 0, eps = 4 * heather$coarse$xstep)

# Estimate pair correlation from a binary map
pclns_directest <- paircorr(xi, estimators = "all")

phat <- coverageprob(xi)
cvchat <- plugincvc(xi)
cpp1 <- cppicka(xi)

# Compute pair correlation estimates from estimates covariance,
# coverage probability and Picka's reduced-window coverage probability.
pclns_fromcvc <- paircorr.cvchat(cvchat, cpp1, phat, estimators = "all")