variofit: Variogram Based Parameter Estimation
In rundel/geoR: Analysis of Geostatistical Data

Description Usage Arguments Details Value Author(s) References See Also Examples

Estimate covariance parameters by fitting a parametric model to a empirical variogram. Variograms models can be fitted by using weighted or ordinary least squares.

variofit(vario, ini.cov.pars, cov.model,
         fix.nugget = FALSE, nugget = 0,
         fix.kappa = TRUE, kappa = 0.5,
         simul.number = NULL, max.dist = vario$max.dist,
         weights, minimisation.function,
         limits = pars.limits(), messages, ...)

`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`. For the linear model use `cov.model = "linear"`. Read values from `variomodel` object passed `ini.cov.pars`, otherwise default is the exponential model.
`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.
`fix.kappa`	logical, indicating whether the parameter kappa should be regarded as fixed or be estimated. Defaults to `TRUE`.
`kappa`	value of the smoothness parameter. Regarded as a fixed values if `fix.kappa = TRUE` or as a initial value for the minimization algorithm if `fix.kappa = FALSE`. Only required if one of the following correlation functions is used: `"matern"`, `"powered.exponential"`, `"cauchy"` and `"gneiting.matern"`. Defaults to 0.5.
`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`.
`weights`	type weights used in the loss function. See `DETAILS` below.
`limits`	values defining lower and upper limits for the model parameters used in the numerical minimisation. Only valid if `minimisation.function = "optim"`. The auxiliary function `pars.limits` is called to set the limits.
`minimisation.function`	minimization function used to estimate the parameters. Options are `"optim"`, `"nlm"`. If `weights = "equal"` the option `"nls"` is also valid and det as default. Otherwise defaults to `"optim"`.
`messages`	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 further details.

Numerical minimization

The parameter values are found by numerical optimization using one of the functions: optim, nlm and nls. In given circunstances the algorithm may not converge to correct parameter values when called with default options and the user may need to pass extra options for the optimizers. For instance the function optim takes a control argument. The user should try different initial values and if the parameters have different orders of magnitude may need to use options to scale the parameters. Some possible workarounds in case of problems include:

rescale you data values (dividing by a constant, say)
rescale your coordinates (subtracting values and/or dividing by constants)
Use the mechanism to pass control() options for the optimiser internally

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 and kappa if fix.kappa = TRUE.

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, vectors in the arguments nugget and kappa (if included in the estimation). If ini.cov.pars is a matrix, the first column has values of sigma^2 and the second has values of phi.

Alternatively the argument ini.cov.pars can take an object of the class eyefit or variomodel. This allows the usage of an output of the functions eyefit, variofit or likfit be used as initial value.

If minimisation.function = "nls" only the values of phi and kappa (if this is included in the estimation) are used. Values for the remaning are not need by the algorithm.

If cov.model = "linear" only the value of sigma^2 is used. Values for the remaning are not need by this algorithm.

If cov.model = "pure.nugget" no initial values are needed since no minimisation function is used.

Weights

The different options for the argument weights are used to define the loss function to be minimised. The available options are as follows.

"npairs"

indicates that the weights are given by the number of pairs in each bin. This is the default option unless variog$output.type == "cloud". The loss function is:

LOSS(theta) = sum_k n_k (hat(gamma) - gamma(theta))^2

"cressie"

weights as suggested by Cressie (1985).

LOSS(theta) = ∑_k n_k [(hat(gamma_k) - gamma_k(theta))/{gamma_k(theta)}]^2

"equal"

equal values for the weights. For this case the estimation corresponds to the ordinary least squares variogram fitting. This is the default option if variog$output.type == "cloud".

LOSS(θ) = ∑_k (\hat(γ) - γ(θ))^2

Where theta is the vector with the variogram parameters and for each kth-bin n_k is the number of pairs, hat(gamma_k) is the value of the empirical variogram and gamma_k(theta) is the value of the theoretical variogram.

See also Cressie (1993) and Barry, Crowder and Diggle (1997) for further discussions on methods to estimate the variogram parameters.

An object of the class "variomodel" and "variofit" 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 two 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.
`weights`	a string indicating the type of weights used for the variogram fitting.
`method`	a string indicating the type of variogram fitting method (OLS or WLS).
`fix.kappa`	logical indicating whether the parameter kappa was fixed.
`fix.nugget`	logical indicating whether the nugget parameter was fixed.
`lambda`	transformation parameters inherith from the object provided in the argument `vario`.
`message`	status messages returned by the function.
`call`	the function call.

Paulo Justiniano Ribeiro Jr. paulojus@leg.ufpr.br,
Peter J. Diggle p.diggle@lancaster.ac.uk.

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 (1985) Mathematical Geology. 17, 563-586.

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

Further information on the package geoR can be found at:
http://www.leg.ufpr.br/geoR.

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

vario100 <- variog(s100, max.dist=1)
ini.vals <- expand.grid(seq(0,1,l=5), seq(0,1,l=5))
ols <- variofit(vario100, ini=ini.vals, fix.nug=TRUE, wei="equal")
summary(ols)
wls <- variofit(vario100, ini=ini.vals, fix.nug=TRUE)
summary(wls)
plot(vario100)
lines(wls)
lines(ols, lty=2)