georob (version 0.2-3)

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

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(), max.lag = max(sv[["lag.dist"]]), min.npairs = 30,  weighting.method = c("cressie", "equal", "npairs"), hessian = TRUE,  verbose = 0, ...) 
"print"(x, digits = max(3, getOption("digits") - 3), ...)
"summary"(object, correlation = FALSE, signif = 0.95, ...) 
"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 $\sigma^2$) of the auto-correlated component of the Gaussian random field $B(s)$.
• snugget: variance (spatial nugget $\sigma^2_n$) of the seemingly spatially uncorrelated component of $B(s)$ (micro-scale spatial variation; default value snugget = 0).
• nugget: variance (nugget $\tau^2$) of the independent errors $\epsilon(s)$.
• 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).

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 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 the first semi-principal axis of the semivariance ellipsoid (default omega = 90).
• phi: 90 degrees minus altitude of the 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 (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.

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.

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.

fit.param

logical vector indicating which variogram parameters were fitted.

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.
• 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 $(C_1, C_2, C_3)$ (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).

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.

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.dist.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 = default.fit.param(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.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")## 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 = 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 = "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)