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", "RMbessel", "RMcauchy", 
      "RMcircular", "RMcubic", "RMdagum", "RMdampedcos", "RMdewijsian", "RMfbm",
      "RMgauss", "RMgenfbm", "RMgencauchy", "RMgengneiting", "RMgneiting", "RMlgd",
      "RMmatern", "RMpenta", "RMaskey", "RMqexp", "RMspheric", "RMstable",
      "RMwave", "RMwhittle"
    ), 
    param, 
    fit.param = c( variance = TRUE, snugget = FALSE, nugget = TRUE, scale = TRUE, 
      alpha = FALSE, beta = FALSE, delta = FALSE, 
      gamma = FALSE, kappa = FALSE, lambda = FALSE, mu = FALSE, nu = FALSE
    )[names(param)], 
    aniso = c(f1 = 1, f2 = 1, omega = 90, phi = 90, zeta = 0), 
    fit.aniso = c(f1 = FALSE, f2 = FALSE, omega = FALSE, 
        phi = FALSE, zeta = FALSE),
    max.lag = max(sv[["lag.dist"]]), min.npairs = 30, 
    weighting.method = c("cressie", "equal", "npairs"), hessian = TRUE, 
    verbose = 0, ...)
    
## 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 Variogram).
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 as passed by the argument param to Variogram, see param.names.
fit.param	a named logical 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 for fitting geometrically anisotropic variogram models. The following parameter names are allowed (see Details of georob and georobIntro for information about the parametrization of variogram models): f1: ratio f_1 of lengths of second and first second semi-principal axes of an ellipsoidal surface with constant semivariance in R^3 (default f1 = 1). f2: ratio f_2 of lengths of third and first semi-principal axes of the semivariance ellipsoid (default f2 = 1). omega: azimuth in degrees of first semi-principal axis of the semivariance ellipsoid (default omega = 90). phi: 90 degrees minus altitude of first semi-principal axis of the semivariance 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 semivariance ellipsoid through the origin (default zeta = 0).
fit.aniso	a named logical 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.
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).
hessian	logical controlling whether the hessian is computed by optim.
verbose	positive integer controlling logging of diagnostic messages to the console during model fitting.
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 semivariances 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 semivariances should be plotted.
xz.angle	numeric (vector) with angles in x-z-plane (in degrees, clockwise positive from zenith to south) for which semivariances 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 optim or 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.

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.model	the name of the fitted parametric variogram model.
param	a named vector with the (estimated) variogram parameters of the fitted model.
aniso	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. 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.
call	the matched call.
residuals	a numeric vector with the residuals, that is the sample semivariance minus the fitted values.
fitted	a numeric vector with the modelled semivariances.
weights	a numeric vector with the weights used for fitting.
hessian	a symmetric matrix giving an estimate of the Hessian at the solution (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; sample.variogram for computing sample variograms.

Examples

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))

r.irf0.iso <- fit.variogram.model(r.sv.iso, variogram.model = "RMfbm",
    param = c(variance = 100, nugget = 1000, scale = 1., alpha = 1.),
    fit.param = c( variance = TRUE, nugget = TRUE, scale = FALSE, alpha = TRUE),
    method = "Nelder-Mead", hessian = FALSE, control = list(maxit = 5000))  
summary(r.irf0.iso, correlation = TRUE)

## Not run: 
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(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")
## End(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 = c(variance = TRUE, nugget = TRUE, scale = FALSE, alpha = TRUE),
    aniso = c(f1 = 0.4, f2 = 1., omega = 135, phi = 90., zeta = 0.),
    fit.aniso = c(f1 = TRUE, f2 = FALSE, omega = TRUE, phi = FALSE, zeta = FALSE),
    method = "Nelder-Mead", hessian = TRUE, control = list(maxit = 5000))
summary(r.irf0.aniso, correlation = TRUE)

## Not run: 
lines(r.irf0.aniso, xy.angle = seq( 0, 135, by = 45))
## End(Not run)

georob documentation built on May 2, 2019, 6:53 p.m.