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 a
three-dimensional domain. There are summary and plot
methods for summarizing and displaying sample variograms.
sample.variogram(object, ...)# S3 method for default
sample.variogram(object, locations, lag.dist.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 formula
sample.variogram(object, data, subset, na.action,
locations, lag.dist.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 georob
sample.variogram(object, lag.dist.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 sample.variogram
summary(object, ...)
# S3 method for 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, lty, 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", ...)
a numeric vector with the values of the response for which
the sample variogram should be computed
(sample.variogram.default), a formula, specifying in its left part
the response variable (right part of formula is ignored,
sample.variogram.formula), an object of class georob
(sample.variogram.georob) or an object of class
sample.variogram (summary.sample.variogram).
a numeric matrix with the coordinates of the locations
where the response was observed (sample.variogram.default) or a
one-sided formula specifying the coordinates
(sample.variogram.formula). The matrix may have an arbitrary
number of columns for an omnidirectional variogram, but at most 3 columns
if a directional variogram is computed.
an optional data frame, list or environment (or another
object coercible by as.data.frame to a data frame)
containing the response variable and the coordinates where the data
was recorded. If not found in data, the variables are taken from
environment(formula), typically the environment from which
sample.variogram is called.
an optional vector specifying a subset of observations to be used for estimating the variogram.
a function which indicates what should happen when the
data contain NAs. The default is set by the na.action
argument of options, and is na.fail if that is
unset. The “factory-fresh” default is na.omit.
Another possible value is NULL, no action. Value
na.exclude can be useful.
a numeric scalar defining a constant bin width for
grouping the lag distances or a numeric vector with the bounds of a set
of contiguous bins (upper bounds of bins except for the first element of
lag.dist.def which is the lower bound of the first bin).
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).
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).
positive numeric defining the largest lag distance for which semi variances should be computed (default no restriction).
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).
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).
an object of class sample.variogram.
see respective arguments of
plot.default.
logical controlling whether a new plot should be
generated (FALSE, default) or whether the information should be
added to the current plot (TRUE).
the color of plotting symbols for distinguishing semi variances for angular classes in the \(x\)-\(y\)-plane.
the type of plotting symbols for distinguishing semi variances for angular classes in the \(x\)-\(z\)-plane.
the line type.
character expansion factor for plotting symbols.
logical controlling whether the plotting symbols should be annotated by the number of data pairs per lag class.
integer defining the position where text annotation
about number of pairs should be plotted, see
text.
numeric defining the character expansion for text annotation about number of pairs.
logical controlling whether a
legend should be plotted.
a character keyword defining where to place the
legend, see legend for possible values.
additional arguments passed to
plot.formula.
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 semi-variance 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. |
The angular classes in the \(x\)-\(y\)- and \(x\)-\(z\)-plane are
defined by vectors of ascending angles on the half circle. The \(i\)th
angular class is defined by the vector elements, say l and u,
with indices \(i\) and \(i+1\). A lag vector belongs to the
\(i\)th angular class if its azimuth (or angle from zenith), say
\(\varphi\), satisfies \( l < \varphi \leq u\).
If the first and the last element 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]].
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;
georobObject for a description of the class georob;
profilelogLik for computing profiles of Gaussian likelihoods;
plot.georob for display of RE(ML) variogram estimates;
control.georob for controlling the behaviour of georob;
georobModelBuilding for stepwise building models of class georob;
cv.georob for assessing the goodness of a fit by georob;
georobMethods for further methods for the class georob;
predict.georob for computing robust Kriging predictions;
lgnpp for unbiased back-transformation of Kriging prediction
of log-transformed data;
georobSimulation for simulating realizations of a Gaussian process
from model fitted by georob.
# NOT RUN {
data(wolfcamp, package = "geoR")
## fitting an isotropic IRF(0) model
r.sv.iso <- sample.variogram(wolfcamp[["data"]], locations = wolfcamp[[1]],
lag.dist.def = seq(0, 200, by = 15))
# }
# NOT RUN {
plot(r.sv.iso, type = "l")
# }
# NOT RUN {
## fitting an anisotropic IRF(0) model
r.sv.aniso <- sample.variogram(wolfcamp[["data"]],
locations = wolfcamp[[1]], lag.dist.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)
# }
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