The function fit.variogram.model fits a variogram model to a sample
variogram by (weighted) non-linear least squares. There are print,
summary and
lines methods for summarizing and displaying fitted variogram
models.
fit.variogram.model(sv,
variogram.model = c("RMexp", "RMaskey", "RMbessel", "RMcauchy",
"RMcircular", "RMcubic", "RMdagum", "RMdampedcos", "RMdewijsian",
"RMfbm", "RMgauss", "RMgencauchy", "RMgenfbm", "RMgengneiting",
"RMgneiting", "RMlgd", "RMmatern", "RMpenta", "RMqexp",
"RMspheric", "RMstable", "RMwave", "RMwhittle"),
param, fit.param = default.fit.param()[names(param)],
aniso = default.aniso(), fit.aniso = default.fit.aniso(),
variogram.object = NULL,
max.lag = max(sv[["lag.dist"]]), min.npairs = 30,
weighting.method = c("cressie", "equal", "npairs"), hessian = TRUE,
verbose = 0, ...)
# S3 method for fitted.variogram
print(x, digits = max(3, getOption("digits") - 3), ...)# S3 method for fitted.variogram
summary(object, correlation = FALSE, signif = 0.95, ...)
# S3 method for fitted.variogram
lines(x, what = c("variogram", "covariance", "correlation"),
from = 1.e-6, to, n = 501, xy.angle = 90, xz.angle = 90,
col = 1:length(xy.angle), pch = 1:length(xz.angle), lty = "solid", ...)
an object of class sample.variogram, see
sample.variogram.
a named numeric vector with initial values of the variogram
parameters. The following parameter names are allowed (see
Details of georob and georobIntro for
information about the parametrization of variogram models):
variance: variance (sill \(\sigma^2\)) of the
auto-correlated component of the Gaussian random field
\(B(\mbox{\boldmath$s$\unboldmath})\).
snugget: variance
(spatial nugget \(\sigma^2_{\mathrm{n}}\))
of the seemingly spatially uncorrelated component of
\(B(\mbox{\boldmath$s$\unboldmath})\)
(micro-scale spatial variation; default value snugget = 0).
nugget: variance (nugget \(\tau^2\)) of the
independent errors
\(\varepsilon(\mbox{\boldmath$s$\unboldmath})\).
scale: range parameter (\(\alpha\)) of the variogram.
names of additional variogram parameters such as the
smoothness parameter \(\nu\) of the Whittle-Mat\'ern model (see
RMmodel and param.names).
a named logical vector (or a function such as
default.fit.param that creates this vector) with the same
names as used for param, defining which parameters are adjusted
(TRUE) and which are kept fixed at their initial values
(FALSE) when fitting the model.
a named numeric vector with initial values (or a function such as
default.aniso that creates this vector) for fitting
geometrically anisotropic variogram models. The names of aniso
are matched against the following names (see Details and
georobIntro for information about the parametrization of
variogram models):
f1: ratio \(f_1\) of lengths of second and first
semi-principal axes of an ellipsoidal surface with constant
semi-variance in \(\mathrm{I}\!\mathrm{R}^3\) (default f1 = 1).
f2: ratio \(f_2\) of lengths of third and first
semi-principal axes of the semi-variance ellipsoid (default f2 = 1).
omega: azimuth in degrees of the first semi-principal axis
of the semi-variance ellipsoid (default omega = 90).
phi: 90 degrees minus altitude of the first semi-principal axis
of the semi-variance ellipsoid (default phi = 90).
zeta: angle in degrees between the second semi-principal
axis and the direction of the line defined by the intersection
between the \(x\)-\(y\)-plane and the plane orthogonal to the
first semi-principal axis of the semi-variance ellipsoid through the
origin (default zeta = 0).
a named logical vector (or a function such as
default.fit.aniso that creates this vector) with the same
names as used for aniso, defining which parameters are adjusted
(TRUE) and which are kept fixed at their initial values
(FALSE) when fitting the model.
an optional list that defines a possibly nested variogram model. Each component is itself a list with the following components:
variogram.model: a character keyword defining the variogram model, see respective argument above.
param: a named numeric vector with initial values of the variogram parameters, see respective argument above.
fit.param: a named logical vector defining which parameters are adjusted, see respective argument above.
aniso: a named numeric vector with initial values for fitting geometrically anisotropic variogram models, see respective argument above.
fit.param: a named logical vector defining which anisotropy parameters are adjusted, see respective argument above.
Note that the arguments variogram.model, param,
fit.param, aniso and fit.aniso are ignored when
variogram.object is passed to fit.variogram.model.
a positive numeric defining the maximum lag distance to be used for fitting or plotting variogram models (default all lag classes).
a positive integer defining the minimum number of data
pairs required so that a lag class is used for fitting a variogram
model (default 30).
a character keyword defining the weights for non-linear least squares. Possible values are:
"equal": no weighting ,
"npairs": weighting by number of data pairs in a lag class,
"cressie": “Cressie's weights” (default, see Cressie, 1993,
sec. 2.6.2).
logical controlling whether the hessian is computed by
optim.
positive integer controlling logging of diagnostic messages to the console during model fitting.
an object of class fitted.variogram.
positive integer indicating the number of decimal digits to print.
logical controlling whether the correlation matrix of
the fitted variogram parameters is computed (default FALSE).
confidence level for computing confidence intervals for
variogram parameters (default 0.95).
the quantity that should be displayed (default "variogram").
numeric, minimal lag distance used in plotting variogram models.
numeric, maximum lag distance used in plotting variogram models (default: largest lag distance of current plot).
positive integer specifying the number of equally spaced lag
distances for which semi-variances are evaluated in plotting variogram
models (default 501).
numeric (vector) with azimuth angles (in degrees, clockwise positive from north) in \(x\)-\(y\)-plane for which semi-variances should be plotted.
numeric (vector) with angles in \(x\)-\(z\)-plane (in degrees, clockwise positive from zenith to south) for which semi-variances should be plotted.
color of curves to distinguish curves relating to different azimuth angles in \(x\)-\(y\)-plane.
type of plotting symbols added to lines to distinguish curves relating to different angles in \(x\)-\(z\)-plane.
line type for plotting variogram models.
additional arguments passed to optim or
to methods.
The function fit.variogram.model generates an object of class
fitted.variogram which is a list with the following components:
the value of the object function (weighted residual sum of squares) evaluated at the solution.
the name of the fitted parametric variogram model.
a named vector with the (estimated) variogram parameters of the fitted model.
logical vector indicating which variogram parameters were fitted.
a list with the following components:
isotropic: logical indicating whether an isotropic
variogram was fitted.
aniso: a named numeric vector with the (estimated)
anisotropy parameters of the fitted model.
fit.aniso: logical vector indicating which anisotropy
parameters were fitted.
sincos: a list with sin and cos of the
angles \(\omega\), \(\phi\) and \(\zeta\) that define the
orientation of the anisotropy ellipsoid.
rotmat: the matrix
\((\mbox{\boldmath$C$\unboldmath}_1,
\mbox{\boldmath$C$\unboldmath}_2,
\mbox{\boldmath$C$\unboldmath}_3)\) (see
georobIntro).
sclmat: a vector with the elements 1, \(1/f_1\),
\(1/f_2\) (see georobIntro).
a character vector listing the transformations of the variogram parameters used for model fitting.
a list of functions for variogram parameter transformations.
a list of functions for inverse variogram parameter transformations.
logical indicating whether numerical maximization by
optim converged.
a diagnostic integer issued by
optim (component convergence) about convergence.
the matched call.
a numeric vector with the residuals, that is the sample semi-variance minus the fitted values.
a numeric vector with the modelled semi-variances.
a numeric vector with the weights used for fitting.
a symmetric matrix giving an estimate of the Hessian at
the solution (missing if hessian is false).
The parametrization of geometrically anisotropic variograms is
described in detail in georobIntro, and the section
Details of georob describes how the parameter
estimates are constrained to permissible ranges. The same
mechanisms are used in fit.variogram.model.
Cressie, N. A. C. (1993) Statistics for Spatial Data. New York: John Wiley & Sons.
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 {
r.irf0.iso <- fit.variogram.model(r.sv.iso, variogram.model = "RMfbm",
param = c(variance = 100, nugget = 1000, scale = 1., alpha = 1.),
fit.param = default.fit.param(scale = FALSE, alpha = TRUE),
method = "Nelder-Mead", hessian = FALSE, control = list(maxit = 5000))
summary(r.irf0.iso, correlation = TRUE)
plot(r.sv.iso, type = "l")
lines(r.irf0.iso, line.col = "red")
# }
# 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 {
r.irf0.aniso <- fit.variogram.model(r.sv.aniso, variogram.model = "RMfbm",
param = c(variance = 100, nugget = 1000, scale = 1., alpha = 1.5),
fit.param = default.fit.param(scale = FALSE, alpha = TRUE),
aniso = default.aniso(f1 = 0.4, omega = 135.),
fit.aniso = default.fit.aniso(f1 = TRUE, omega = TRUE),
method = "BFGS", hessian = TRUE, control = list(maxit = 5000))
summary(r.irf0.aniso, correlation = TRUE)
plot(r.sv.aniso, type = "l")
lines(r.irf0.aniso, xy.angle = seq(0, 135, by = 45))
# }
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