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, …)an object of the class "variogram", typically an output of the function
variog. The object is a list with information about the
empirical variogram.
initial values for the covariance parameters:
\(\sigma^2\) (partial sill) and \(\phi\) (range
parameter). See DETAILS below.
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.
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.
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.
logical, indicating whether the parameter
\(\kappa\) should be regarded as fixed or
be estimated. Defaults to TRUE.
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 \(0.5\).
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.
maximum distance considered when fitting the
variogram. Defaults to vario$max.dist.
type weights used in the loss function. See
DETAILS below.
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.
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".
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.
An object of the class "variomodel" and "variofit" which is list with the following components:
value of the nugget parameter. An estimated value if
fix.nugget = FALSE or a fixed value if fix.nugget = TRUE.
a two elements vector with estimated values of the covariance parameters \(\sigma^2\) and \(\phi\), respectively.
a string with the name of the correlation function.
fixed value of the smoothness parameter.
minimized value of the loss function.
maximum distance considered in the variogram fitting.
minimization function used.
a string indicating the type of weights used for the variogram fitting.
a string indicating the type of variogram fitting method (OLS or WLS).
logical indicating whether the parameter \(\kappa\) was fixed.
logical indicating whether the nugget parameter was fixed.
transformation parameters inherith from the object
provided in the argument vario.
status messages returned by the function.
the function call.
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:
$$LOSS(\theta) = \sum_k n_k [(\hat{\gamma}_k) -
\gamma_k(\theta)]^2$$
"cressie"weights as suggested by Cressie (1985). $$LOSS(\theta) = \sum_k n_k [\frac{\hat{\gamma}_k - \gamma_k(\theta)}{\gamma_k(\theta)}]^2$$
"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".
$$LOSS(\theta) = \sum_k [(\hat{\gamma}_k) -
\gamma_k(\theta)]^2$$
Where \thetatheta is the vector with the variogram parameters and for each k^{th}kth-bin n_kn_k is the number of pairs, (\hat{\gamma}_k)hat(gamma_k) is the value of the empirical variogram and \gamma_k(\theta)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.
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.
# NOT RUN {
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)
# }
# NOT RUN {
# }
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