sample.variogram: Computing (Robust) Sample Variograms of Spatial Data
In georob: Robust Geostatistical Analysis of Spatial Data

Description Usage Arguments Details Value Author(s) References See Also Examples

The function sample.variogram computes the sample empirical) variogram of a spatial variable by the method-of-moment and three robust estimators. Both omnidirectional and direction-dependent variograms can be computed, the latter for observation locations in s three-dimensionsal domain. There are summary and plot methods for summarizing and displaying sample variograms.

sample.variogram(response, locations, lag.class.def, 
    xy.angle.def = c(0, 180), xz.angle.def = c(0, 180), max.lag = Inf, 
    estimator = c("qn", "mad", "matheron", "ch"), mean.angle = TRUE)
   
## S3 method for class 'sample.variogram'
summary(object, ...)

## S3 method for class 'sample.variogram'
plot(x, type = "p", add = FALSE, xlim = c(0, max(x[["lag.dist"]])),
    ylim = c(0, 1.1 * max(x[["gamma"]])), col, pch, cex = 0.8, 
    xlab = "lag distance", ylab = "semivariance", 
    annotate.npairs = FALSE, npairs.pos = 3, npairs.cex = 0.7, 
    legend = nlevels(x[["xy.angle"]]) > 1 || nlevels(x[["xz.angle"]]) > 1, 
    legend.pos = "topleft", ...)

`response`	a numeric vector with the values of the response for which the sample variogram should be computed.
`locations`	a numeric matrix with the coordinates of the locations where the response was observed. May have an arbitrary number of columns for an omnidirectional variogram, but at most 3 columns if a directional variogram is computed.
`lag.class.def`	a numeric scalar defining a constant bin width for grouping the lag distances or a numeric vector with the upper bounds of a set of contiguous bins.
`xy.angle.def`	an numeric vector defining angular classes in the horizontal plane for computing directional variograms. `xy.angle.def` must contain an ascending sequence of azimuth angles in degrees from north (positive clockwise to south), see Details. Omnidirectional variograms are computed with the default `c(0,180)`.
`xz.angle.def`	an numeric vector defining angular classes in the x-z-plane for computing directional variograms. `xz.angle.def` must contain an ascending sequence of angles in degrees from zenith (positive clockwise to nadir), see Details. Omnidirectional variograms are computed with the default `c(0,180)`.
`max.lag`	positive numeric defining the largest lag distance for which semivariances should be computed (default no restriction).
`estimator`	character keyword defining the estimator for computing the sample variogram. Possible values are: `"qn"`: Genton's robust `Qn`-estimator (default, Genton, 1998), `"mad"`: Dowd's robust MAD-estimator (Dowd, 1984), `"matheron"`: non-robust method-of-moments estimator, `"ch"`: robust Cressie-Hawkins estimator (Cressie and Hawkins, 1980).
`mean.angle`	logical controlling whether the mean lag vector (per combination of lag distance and angular class) is computed from the mean angles of all the lag vectors falling into a given class (`TRUE`, default) or from the mid-angles of the respective angular classes (`FALSE`).
`object, x`	an object of class `sample.variogram`

`type, xlim, ylim, xlab, ylab`	see respective arguments of `plot.default`.
`add`	logical controlling whether a new plot should be generated (`FALSE`, default) or whether the information should be added to the current plot (`TRUE`).
`col`	the color of plotting symbols for distinguishing semivariances for angular classes in the x-y-plane.
`pch`	the type of plotting symbols for distinguishing semivariances for angular classes in the x-z-plane.
`cex`	character expansion factor for plotting symbols.
`annotate.npairs`	logical controlling whether the plotting symbols should be annotated by the number of data pairs per lag class.
`npairs.pos`	integer defining the position where text annotation about number of pairs should be plotted, see `text`.
`npairs.cex`	numeric defining the character expansion for text annotation about number of pairs.
`legend`	logical controlling whether a `legend` should be plotted.
`legend.pos`	a character keyword defining where to place the legend, see `legend` for possible values.
`...`	additional arguments passed to `plot.sample.variogram` and `lines.georob`.

The angular classes in the x-y- and x-z-plane are defined by vectors of ascending angles on the half circle. The ith angular class is defined by the vector elements, say l and u, with indices i and i+1. A lag vector belongs to the ith angular class if its azimuth (or angle from zenith), say \varphi, satisfies l < φ <= u. If the first and the last angles of xy.angle.def or xz.angle.def are equal to 0 and 180 degrees, respectively, then the first and the last angular class are “joined”, i.e., if there are K angles, there will be only K-2 angular classes and the first class is defined by the interval ( xy.angle.def[K-1]-180, xy.angle.def[2] ] and the last class by ( xy.angle.def[K-2], xy.angle.def[K-1] ].

An object of class sample.variogram, which is a data frame with the following components:

`lag.dist`	the mean lag distance of the lag class,
`xy.angle`	the angular class in the x-y-plane,
`xz.angle`	the angular class in the x-z-plane,
`gamma`	the estimated semivariance of the lag class,
`npairs`	the number of data pairs in the lag class,
`lag.x`	the x-component of the mean lag vector of the lag class,
`lag.x`	the y-component of the mean lag vector of the lag class,
`lag.z`	the z-component of the mean lag vector of the lag class.

Andreas Papritz andreas.papritz@env.ethz.ch.

Cressie, N. and Hawkins, D. M. (1980) Robust Estimation of the Variogram: I. Mathematical Geology, 12, 115–125.

Dowd, P. A. (1984) The variogram and kriging: Robust and resistant estimators. In Geostatistics for Natural Resources Characterization, Verly, G., David, M., Journel, A. and Marechal, A. (Eds.) Dordrecht: D. Reidel Publishing Company, Part I, 1, 91–106.

Genton, M. (1998) Highly Robust Variogram Estimation. Mathematical Geology, 30, 213–220.

georobIntro for a description of the model and a brief summary of the algorithms; georob for (robust) fitting of spatial linear models; fit.variogram.model for fitting variogram models to sample variograms.

data(wolfcamp, package = "geoR")

## fitting an isotropic IRF(0) model
r.sv.iso <- sample.variogram(wolfcamp[["data"]], locations = wolfcamp[[1]], 
    lag.class.def = seq(0, 200, by = 15))

## Not run: 
plot( r.sv.iso, type = "l")
## End(Not run)

## fitting an anisotropic IRF(0) model
r.sv.aniso <- sample.variogram(wolfcamp[["data"]],
    locations = wolfcamp[[1]], lag.class.def = seq(0, 200, by = 15),
    xy.angle.def = c(0., 22.5, 67.5, 112.5, 157.5, 180.))
## Not run: 
plot(r.sv.aniso, type = "l", add = TRUE, col = 2:5)
## End(Not run)