geoR (version 1.0-0)

wlsfit: Variogram Fitting by Weighted Least Squares

Description

Fits a variogram model to an empirical variogram. Variogram parameters are estimated by weighted least squares.

Usage

wlsfit(vario, ini.cov.pars,
       cov.model = c("exponential", "matern", "gaussian",
                     "spherical", "circular", "cubic", "wave",
                     "powered.exponential", "cauchy", "gneiting",
                     "gneiting.matern", "pure.nugget"),
       fix.nugget = FALSE, nugget = 0, kappa = NULL,
       simul.number = NULL, max.dist = "all",
       minimisation.function = c("optim", "nlm"),  lower = 0,
       weight = c("npairs", "cressie"), messages.screen = TRUE)

Arguments

vario

an object of the class "variogram", typically an output of the function variog. The object is a list with information about the empirical variogram.

ini.cov.pars

initial values for the covariance parameters: $\sigma^2$ (partial sill) and $\phi$ (range parameter). See DETAILS below.

cov.model

a string with the name of the correlation function. For further details see documentation for cov.spatial. Defaults to "exponential".

fix.nugget

logical, indicating whether the parameter $\tau^2$ (nugget variance) should be regarded as fixed (fix.nugget = TRUE) or should be estimated (fix.nugget = FALSE). Defaults to FALSE.

nugget

value for the nugget parameter. Regarded as a fixed values if fix.nugget = TRUE or as a initial value for the minimization algorithm if fix.nugget = FALSE. Defaults to zero.

kappa

fixed value of the smoothness parameter, only required by the following correlation functions: "matern", "powered.exponential", "gneiting" and "gneiting.matern".

simul.number

number of simulation. To be used when the object passed to the argument vario has empirical variograms for more than one data-set (or simulation). Indicates to which one the model will be fitted.

max.dist

maximum distance considered when fitting the variogram. Defaults to vario$max.dist.

minimisation.function

minimization function used to estimate the parameters. Options are "optim" and "nlm". Defaults to "optim".

lower

lower limits for the parameter values. Defaults to zero.

weight

type weights used in the loss function. See DETAILS below.

messages.screen

logical. Indicates whether or not status messages are printed on the screen (or other output device) while the function is running.

...

further parameters to be passed to the minimization function. Typically arguments of the type control() which controls the behavior of the minimization algorithm. See documentation for the selected minimization function for furth

Value

• An object of the class "variomodel" which is list with the following components:
• nugget
  value of the nugget parameter. An estimated value if fix.nugget = FALSE or a fixed value if fix.nugget = TRUE.
• cov.pars
  a 2 elements vector with estimated values of the covariance parameters $\sigma^2$ and $\phi$, respectively.
• cov.model
  a string with the name of the correlation function.
• kappa
  fixed value of the smoothness parameter.
• value
  minimized value of the loss function.
• max.dist
  maximum distance considered in the variogram fitting.
• minimisation.function
  minimization function used.
• message
  status messages returned by the function.
• method
  a string "WLS" indicating the estimation method.
• call
  the function call.

Details

Initial values The algorithms for minimization functions require initial values of the parameters. A unique initial value is used if a vector is provided in the argument ini.cov.pars. The elements are initial values for $\sigma^2$ and $\phi$, respectively. This vector is concatenated with the value of the argument nugget if fix.nugget = FALSE. Specification of multiple initial values is also possible. If this is the case, the function searches for the one which minimizes the loss function and uses this as the initial value for the minimization algorithm. Multiple initial values are specified by providing a matrix in the argument ini.cov.pars and/or, if fix.nugget = FALSE, providing a vector with length greater than one for the argument nugget. If ini.cov.pars is a matrix, the first column has values of $\sigma^2$ and the second has values of $\phi$. Weights Two different types of weights can be used within the loss function: [object Object],[object Object],See also Barry, Crowder and Diggle (1997) for a discussion on the methods to estimate variogram parameters.

References

Barry, J.T., Crowder, M.J. and Diggle, P.J. (1997) Parametric estimation of the variogram. Tech. Report, Dept Maths & Stats, Lancaster University.

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

Further information about geoR can be found at: http://www.maths.lancs.ac.uk/~ribeiro/geoR.html.

See Also

cov.spatial for a detailed description of the available correlation (variogram) functions, olsfit for ordinary least squares variogram fit, likfit for maximum and restricted maximum likelihood estimation, lines.variomodel for graphical output of the fitted model. For details on the minimization functions see documentation for optim and nlm.

Examples

if(is.R()) data(s100)
vario100 <- variog(s100, max.dist=1)
ini.vals <- expand.grid(seq(0,1,l=5), seq(0,1,l=5))
wls <- wlsfit(vario100, ini=ini.vals, fix.nug=TRUE)
summary(wls)
plot(vario100)
lines(wls)