fit.variogram.model: Fitting Model Functions to Sample Variograms

In georob: Robust Geostatistical Analysis of Spatial Data

Description

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.

Usage

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"),
    control = control.fit.variogram.model(),
    verbose = 0)

control.fit.variogram.model(maximizer = c("nlminb", "optim"),
    param.tf = param.transf(), fwd.tf = fwd.transf(),
    deriv.fwd.tf = dfwd.transf(), bwd.tf = bwd.transf(),
    hessian = TRUE, optim = control.optim(), nlminb = control.nlminb())

## S3 method for class 'fitted.variogram'
print(x, digits = max(3, getOption("digits") - 3), ...)

## S3 method for class 'fitted.variogram'
summary(object, correlation = FALSE, signif = 0.95, ...)

## S3 method for class '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", ...)

Arguments

sv	an object of class sample.variogram, see sample.variogram.
variogram.model	a character keyword defining the variogram model to be fitted. Currently, most basic variogram models provided by the package RandomFields can be fitted (see Details of georob and RMmodel).
param	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 σ^2) of the auto-correlated component of the Gaussian random field B(s). snugget: variance (spatial nugget σ^2_n) of the seemingly spatially uncorrelated component of B(s) (micro-scale spatial variation; default value snugget = 0). nugget: variance (nugget τ^2) of the independent errors ε(s). scale: range parameter (α) of the variogram. names of additional variogram parameters such as the smoothness parameter ν of the Whittle-Mat\'ern model (see RMmodel and param.names).
fit.param	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.
aniso	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 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).
fit.aniso	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.
variogram.object	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.
max.lag	a positive numeric defining the maximum lag distance to be used for fitting or plotting variogram models (default all lag classes).
min.npairs	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).
weighting.method	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).
verbose	positive integer controlling logging of diagnostic messages to the console during model fitting.
control	a list with the components maximizer, param.tf, fwd.tf, bwd.tf, hessian, optim and nlminb or a function such as control.fit.variogram.model that generates such a list. See codecontrol.georob for information on codemaximizer, param.tf, fwd.tf, bwd.tf, hessian, optim and nlminb.
maximizer	character keyword defining the optimizer for nonlinear least squares. Possible values are nlminb (default) or optim.
hessian	logical scalar controlling whether the Hessian should be computed at the nonlinear least squares estimates.
param.tf	a function such as param.transf, which returns a named vector of character strings that define the transformations to be applied to the variogram parameters for model fitting, see control.georob.
fwd.tf	a function such as fwd.transf, which returns a named list of invertible functions to be used to transform variogram parameters, see control.georob.
deriv.fwd.tf	a function such as dfwd.transf, which returns a named list of functions corresponding to the first derivatives of fwd.tf, see Details.
bwd.tf	a function such as bwd.transf, which returns the named list of inverse functions corresponding to fwd.tf, see see control.georob.
nlminb	a list of arguments passed to nlminb or a function such as control.nlminb that generates such a list (see nlminb for allowed arguments).
optim	a list of arguments passed to optim or a function such as control.optim that generates such a list (see optim for allowed arguments).
object, x	an object of class fitted.variogram.
digits	positive integer indicating the number of decimal digits to print.
correlation	logical controlling whether the correlation matrix of the fitted variogram parameters is computed (default FALSE).
signif	confidence level for computing confidence intervals for variogram parameters (default 0.95).
what	the quantity that should be displayed (default "variogram").
from	numeric, minimal lag distance used in plotting variogram models.
to	numeric, maximum lag distance used in plotting variogram models (default: largest lag distance of current plot).
n	positive integer specifying the number of equally spaced lag distances for which semi-variances are evaluated in plotting variogram models (default 501).
xy.angle	numeric (vector) with azimuth angles (in degrees, clockwise positive from north) in x-y-plane for which semi-variances should be plotted.
xz.angle	numeric (vector) with angles in x-z-plane (in degrees, clockwise positive from zenith to south) for which semi-variances should be plotted.
col	color of curves to distinguish curves relating to different azimuth angles in x-y-plane.
pch	type of plotting symbols added to lines to distinguish curves relating to different angles in x-z-plane.
lty	line type for plotting variogram models.
...	additional arguments passed to methods.

Details

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.

The method summary computes confidence intervals of the estimated variogram and anisotropy parameters from the Hessian matrix of the residual sums of squares, based on the asymptotic normal distribution of least squares estimate estimates. Note that the Hessian matrix with respect to the transformed variogram and anisotropy parameters is used for this. Hence the inverse Hessian matrix is the covariance matrix of the transformed parameters, confidence intervals are first computed for the transformed parameters and the limits of these intervals are transformed back to the orginal scale of the parameters. Optionally, summary reports the correlation matrix of the transformed parameters, also computed from the Hessian matrix.

Value

The function fit.variogram.model generates an object of class fitted.variogram which is a list with the following components:

sse	the value of the object function (weighted residual sum of squares) evaluated at the solution.
variogram.object	the estimated parameters of a possibly nested variograms model. This is a list that contains for each variogram model structure the following components: variogram.model: the name of the fitted parametric variogram model. param: a named numeric vector with the (estimated) variogram parameters. fit.param: a named logical vector with the flags defining what variogram parameters were estimated. isotropic: logical indicating whether an isotropic variogram was fitted. aniso: a named numeric vector with the (estimated) anisotropy parameters. fit.aniso: a named logical vector with the flags defining what anisotropy parameters were estimated. sincos: a list with sin and cos of the angles ω, φ and ζ that define the orientation of the anisotropy ellipsoid. rotmat: the matrix (C_1, C_2, C_3) (see georobIntro). sclmat: a vector with the elements 1, 1/f_1, 1/f_2 (see georobIntro).
param.tf	a character vector listing the transformations of the variogram parameters used for model fitting.
fwd.tf	a list of functions for variogram parameter transformations.
bwd.tf	a list of functions for inverse variogram parameter transformations.
converged	logical indicating whether numerical maximization by optim converged.
convergence.code	a diagnostic integer issued by optim (component convergence) about convergence.
iter	a named integer vector of length two with the number of function and gradient evaluations by optim.
call	the matched call.
residuals	a numeric vector with the residuals, that is the sample semi-variance minus the fitted values.
fitted	a numeric vector with the modelled semi-variances.
weights	a numeric vector with the weights used for fitting.
hessian	a symmetric matrix with the Hessian at the solution (missing if hessian is false).
hessian.tfpa	a symmetric matrix with the Hessian at the solution with respect to the transformed variogram and anisotropy parameters (missing if hessian is false). This Hessian is used by summary.fitted.variogram to compute confidence intervals for the estimated parameters.
hessian.ntfpa	a symmetric matrix with the Hessian at the solution with respect to the non-transformed variogram and anisotropy parameters (missing if hessian is false).

Author(s)

Andreas Papritz [e-mail].

References

Cressie, N. A. C. (1993) Statistics for Spatial Data. New York: John Wiley & Sons.

See Also

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.

Examples

data(wolfcamp)

## fitting an isotropic IRF(0) model
r.sv.iso <- sample.variogram(pressure~1, data = wolfcamp,
    locations = ~x + y, 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))
summary(r.irf0.iso, correlation = TRUE)

plot(r.sv.iso, type = "l")
lines(r.irf0.iso, line.col = "red")
## End(Not run)

## fitting an anisotropic IRF(0) model
r.sv.aniso <- sample.variogram(pressure~1, data = wolfcamp,
    locations = ~x + y, 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),
    control = control.fit.variogram.model(
      maximizer = "optim",
      optim = control.optim(
        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))
## End(Not run)

georob documentation built on June 4, 2021, 5:06 p.m.