Estimate the mark variogram of a marked point pattern.
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
"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
- Arguments passed to the density estimation routine
sm.density) selected by
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.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
"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.
Mark correlation function
markcorr for numeric marks.
# Longleaf Pine data # marks represent tree diameter data(longleaf) # Subset of this large pattern swcorner <- owin(c(0,100),c(0,100)) sub <- longleaf[ , swcorner] # mark correlation function mv <- markvario(sub) plot(mv)