# variofit

##### Variogram Based Parameter Estimation

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

- Keywords
- spatial

##### Usage

```
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, …)
```

##### 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`

. 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.

##### 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: $$LOSS(\theta) = \sum_k n_k [(\hat{\gamma}_k) - \gamma_k(\theta)]^2$$`"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.

##### Value

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.

##### 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 (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.

##### See Also

`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`

.

##### Examples

```
# 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 {
# }
```

*Documentation reproduced from package geoR, version 1.8-1, License: GPL (>= 2)*