Mark Variogram

Estimate the mark variogram of a marked point pattern.

spatial, nonparametric
markvario(X, correction = c("isotropic", "Ripley", "translate"),
r = NULL, method = "density", ..., normalise=FALSE)
The observed point pattern. An object of class "ppp" or something acceptable to as.ppp. It must have marks which are numeric.
A character vector containing any selection of the options "isotropic", "Ripley" or "translate". It specifies the edge correction(s) to be applied.
numeric vector. The values of the argument $r$ at which the mark variogram $\gamma(r)$ should be evaluated. There is a sensible default.
A character vector indicating the user's choice of density estimation technique to be used. Options are "density", "loess", "sm" and "smrep".
Arguments passed to the density estimation routine (density, loess or sm.density) selected by method.
If TRUE, normalise the variogram by dividing it by the estimated mark variance.

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).


  • An object of class "fv" (see fv.object). Essentially a data frame containing numeric columns
  • rthe values of the argument $r$ at which the mark variogram $\gamma(r)$ has been estimated
  • theothe 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.


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.

  • markvario
# 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)
Documentation reproduced from package spatstat, version 1.41-1, License: GPL (>= 2)

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