markvario
Mark Variogram
Estimate the mark variogram of a marked point pattern.
- Keywords
- spatial, nonparametric
Usage
markvario(X, correction = c("isotropic", "Ripley", "translate"),
r = NULL, method = "density", ..., normalise=FALSE)
Arguments
- X
- The observed point pattern.
An object of class
"ppp"
or something acceptable toas.ppp
. It must have marks which are numeric. - correction
- A character vector containing any selection of the
options
"isotropic"
,"Ripley"
or"translate"
. It specifies the edge correction(s) to be applied. - r
- numeric vector. The values of the argument $r$ at which the mark variogram $\gamma(r)$ should be evaluated. There is a sensible default.
- method
- 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
orsm.density
) selected bymethod
. - normalise
- If
TRUE
, normalise the variogram by dividing it by the estimated mark variance.
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).
Value
- An object of class
"fv"
(seefv.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.
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
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)