cencovariance {lacunaritycovariance}

Centred covariance estimation

Description

This function estimates the centred covariance of a stationary RACS. Available estimators are the plug-in moment centred covariance estimator, two 'balanced' estimators suggested by Picka (2000), and a third 'balanced' estimator inspired by one of Picka's pair-correlation estimators.

Usage

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

cencovariance.cvchat(
  cvchat,
  cpp1 = NULL,
  phat = NULL,
  setcov_boundarythresh = 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.

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.

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 usual 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 centred covariance of a stationary RACS is

\kappa(v) = C(v) - p^2.

The estimators available are (see (Section 3.4, Hingee, 2019) for more information):

• plugin the plug-in moment centred covariance estimator

• mattfeldt an estimator inspired by an 'intrinsically' balanced pair-correlation estimator from Picka (1997) that was later studied in an isotropic situation by Mattfeldt and Stoyan (Mattfeldt and Stoyan, 2000)

• pickaint Picka's 'intrinsically' balanced centred covariance estimator (Picka, 2000).

• pickaH Picka's 'additively' balanced centred covariance estimator (Picka, 2000).

Currently computes centred covariance using racscovariance.

Value

If drop = TRUE and only one estimator is requested then a im object containing the centred covariance estimate is returned. Otherwise a named imlist of im objects containing the centred covariance estimates for each requested estimator.

Functions

• cencovariance(): Centred covariance estimates from a binary map.

• cencovariance.cvchat(): Generates centred covariances estimates from a plug-in moment estimate of covariance, Picka's reduced window estimate of coverage probability, and the plug-in moment estimate of coverage probability. If these estimates already exist, then cencovariance.cvchat saves significant computation time over cencovariance.

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 <- heather$coarse
obswin <- Frame(xi)
cencovariance(xi, obswin, estimators = "all")