spatstat.explore (version 3.7-0)

markvario: Mark Variogram

Description

Estimate the mark variogram of a marked point pattern.

Usage

markvario(X, correction = c("isotropic", "Ripley", "translate"),
r = NULL, method = "density", ..., rmax=NULL, normalise=FALSE)

Value

An object of class "fv" (see fv.object).

Essentially a data frame containing numeric columns

r

the values of the argument \(r\) at which the mark variogram \(\gamma(r)\) has been estimated

theo

the theoretical value of \(\gamma(r)\) when the marks attached to different points are independent; equal to the sample variance of the marks

together with a column or columns named

"iso" and/or "trans", according to the selected edge corrections. These columns contain estimates of the function \(\gamma(r)\)

obtained by the edge corrections named.

The returned object also has an attribute "smooth.args" giving the smoothing parameters that were used to calculate the estimate.

Details

The mark variogram \(\gamma(r)\) of a marked point process \(X\) is a measure of the dependence between the marks of two points of the process a distance \(r\) apart. It is informally defined as $$ \gamma(r) = E[\frac 1 2 (M_1 - M_2)^2] $$ where \(E[ ]\) denotes expectation and \(M_1,M_2\) are the marks attached to two points of the process a distance \(r\) apart.

The mark variogram of a marked point process is analogous, but not equivalent, to the variogram of a random field in geostatistics. See Waelder and Stoyan (1996).

References

Cressie, N.A.C. (1991) Statistics for spatial data. John Wiley and Sons, 1991.

Mase, S. (1996) The threshold method for estimating annual rainfall. Annals of the Institute of Statistical Mathematics 48 (1996) 201-213.

Waelder, O. and Stoyan, D. (1996) On variograms in point process statistics. Biometrical Journal 38 (1996) 895-905.

See Also

Mark correlation function markcorr for numeric marks.

Mark connection function markconnect and multitype K-functions Kcross, Kdot for factor-valued marks.

Examples

    # Longleaf Pine data
    # marks represent tree diameter
    # Subset of this large pattern
    swcorner <- owin(c(0,100),c(0,100))
    sub <- longleaf[ , swcorner]
    # mark correlation function
    mv <- markvario(sub)
    plot(mv)