geoR (version 1.5-6)

likfit: Likelihood Based Parameter Estimation for Gaussian Random Fields

Description

Maximum likelihood (ML) or restricted maximum likelihood (REML) parameter estimation for (transformed) Gaussian random fields.

Usage

likfit(geodata, coords = geodata$coords, data = geodata$data,
       trend = "cte", ini.cov.pars, fix.nugget = FALSE, nugget = 0,
       fix.kappa = TRUE, kappa = 0.5, fix.lambda = TRUE, lambda = 1,
       fix.psiA = TRUE, psiA = 0, fix.psiR = TRUE, psiR = 1, 
       cov.model = "matern", realisations,
       method.lik = "ML", components = FALSE,
       nospatial = TRUE, limits = pars.limits(),
       print.pars = FALSE, messages, ...)

Arguments

geodata

a list containing elements coords and data as described next. Typically an object of the class "geodata". If not provided the arguments coords and data must be provided

coords

an $n \times 2$ matrix where each row has the 2-D coordinates of the $n$ data locations. By default it takes the component coords of the argument geodata, if provided.

data

a vector with n data values. By default it takes the component data of the argument geodata, if provided.

trend

specifies the mean part of the model. See documentation of trend.spatial for further details. Defaults to "cte".

ini.cov.pars

initial values for the covariance parameters: $\sigma^2$ (partial sill) and $\phi$ (range parameter). Typically a vector with two components. However a matrix can be used to provide several initial values. See DETAILS below.

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 of the nugget parameter. Regarded as a fixed value if fix.nugget = TRUE otherwise as the initial value for the minimisation algorithm. Defaults to zero.

fix.kappa

logical, indicating whether the extra parameter $\kappa$ should be regarded as fixed (fix.kappa = TRUE) or should be estimated (fix.kappa = FALSE). Defaults to TRUE.

kappa

value of the extra parameter $\kappa$. Regarded as a fixed value if fix.kappa = TRUE otherwise as the initial value for the minimisation algorithm. Defaults to $0.5$. This parameter is valid only if the covariance function is one

fix.lambda

logical, indicating whether the Box-Cox transformation parameter $\lambda$ should be regarded as fixed (fix.lambda = TRUE) or should be be estimated (fix.lambda = FALSE). Defaults to TRUE.

lambda

value of the Box-Cox transformation parameter $\lambda$. Regarded as a fixed value if fix.lambda = TRUE otherwise as the initial value for the minimisation algorithm. Defaults to $1$. Two particular cases are $\lambda = 1$

fix.psiA

logical, indicating whether the anisotropy angle parameter $\psi_R$ should be regarded as fixed (fix.psiA = TRUE) or should be estimated (fix.psiA = FALSE). Defaults to TRUE.

psiA

value (in radians) for the anisotropy angle parameter $\psi_A$. Regarded as a fixed value if fix.psiA = TRUE otherwise as the initial value for the minimisation algorithm. Defaults to $0$. See

fix.psiR

logical, indicating whether the anisotropy ratio parameter $\psi_R$ should be regarded as fixed (fix.psiR = TRUE) or should be estimated (fix.psiR = FALSE). Defaults to TRUE.

psiR

value, always greater than 1, for the anisotropy ratio parameter $\psi_R$. Regarded as a fixed value if fix.psiR = TRUE otherwise as the initial value for the minimisation algorithm. Defaults to $1$. See

cov.model

a string specifying the model for the correlation function. For further details see documentation for cov.spatial. Defaults are equivalent to the exponential model.

realisations

optional. A vector indicating the number of replication for each datum. For further information see DETAILS below and documentation for as.geodata.

method.lik

options are "ML" for maximum likelihood and "REML" for restricted maximum likelihood. Defaults to "ML".

components

an $n \times 3$ data-frame with fitted values for the three model components: trend, spatial and residuals. See the section DETAILS below for the model specification.

nospatial

logical. If TRUE parameter estimates for the model without spatial component are included in the output.

limits

values defining lower and upper limits for the model parameters used in the numerical minimisation. The auxiliary function pars.limits is called to set the limits. See also Limits

print.pars

logical. If TRUE the parameters and the value of the negative log-likelihood (up to a constant) are printed each time the function to be minimised is called.

messages

logical. Indicates whether status messages should be printed on the screen (or output device) while the function is running.

...

additional parameters to be passed to the minimisation function. Typically arguments of the type control() which controls the behavior of the minimisation algorithm. For further details see documentation for the minimisation fu

Value

• An object of the classes "likGRF" and "variomodel". The function summary.likGRF is used to print a summary of the fitted model. The object is a list with the following components:
• cov.model
  a string with the name of the correlation function.
• nugget
  value of the nugget parameter $\tau^2$. This is an estimate if fix.nugget = FALSE otherwise, a fixed value.
• cov.pars
  a vector with the estimates of the parameters $\sigma^2$ and $\phi$, respectively.
• kappa
  value of the smoothness parameter. Valid only if the correlation function is one of: "matern", "powered.exponential", "cauchy" or "gneiting.matern".
• beta
  estimate of mean parameter $\beta$. This can be a scalar or vector depending on the trend (covariates) specified in the model.
• beta.var
  estimated variance (or covariance matrix) for the mean parameter $\beta$.
• lambda
  values of the Box-Cox transformation parameter. A fixed value if fix.lambda = TRUE otherwise the estimate value.
• aniso.pars
  fixed values or estimates of the anisotropy parameters, according to the function call.
• method.lik
  estimation method used, "ML" (maximum likelihood) or "REML" (restricted maximum likelihood).
• loglik
  the value of the maximized likelihood.
• npars
  number of estimated parameters.
• AIC
  value of the Akaike Information Criteria, $AIC=-2 ln(L) + 2 p$ where $L$ is the maximised likelihood and p is the number of parameters in the model.
• BIC
  value of the Bayesian Information Criteria, $BIC=-2ln(L) + p log(n)$, where $n$ is the number of data, $L,p$ as for AIC above.
• parameters.summary
  a data-frame with all model parameters, their status (estimated or fixed) and values.
• info.minimisation
  results returned by the minimisation function.
• max.dist
  maximum distance between 2 data points. This information relevant for other functions which use outputs from likfit.
• trend
  the trend (covariates) matrix $X$.
• log.jacobian
  numerical value of the logarithm of the Jacobian of the transformation.
• nospatial
  estimates for the model without the spatial component.
• call
  the function call.

concept

variogram parameter estimation

Details

This function estimate the parameters of the Gaussian random field model, specified as: $$Y(x) = \mu(x) + S(x) + e$$ where

• $x$defines a spatial location. Typically Euclidean coordinates on a plane.
• $Y$is the variable been observed.
• $\mu(x) = X\beta$is the mean component of the model (trend).
• $S(x)$is a stationary Gaussian process with variance$\sigma^2$(partial sill) and a correlation function parametrized in its simplest form by$\phi$(the range parameter). Possible extra parameters for the correlation function are the smoothness parameter$\kappa$and the anisotropy parameters$\phi_R$and$\phi_A$(anisotropy ratio and angle, respectively).
• $e$is the error term with variance parameter$\tau^2$(nugget variance).

The additional parameter $\lambda$ allows for the Box-Cox transformation of the response variable. If used (i.e. if $\lambda \neq 1$) $Y(x)$ above is replaced by $g(Y(x))$ such that $$g(Y(x)) = \frac{Y^\lambda(x) - 1}{\lambda}.$$

Two particular cases are $\lambda = 1$ which indicates no transformation and $\lambda = 0$ indicating the log-transformation. Numerical minimization

In general parameter estimation is performed numerically using the Rfunction optim to minimise the negative log-likelihood computed by the function negloglik.GRF. If the nugget, anisotropy ($\psi_A, \psi_R$), smoothness ($\kappa$) and transformation ($\lambda$) parameters are held fixed then the numerical minimisation can be reduced to one-dimension and the function optimize is used instead of optim. In this case initial values are irrelevant.

Limits Lower and upper limits for parameter values can be individually specified using the function link{pars.limits}. For example, including the following in the function call: limits = pars.limits(phi=c(0, 10), lambda=c(-2.5, 2.5)), will change the limits for the parameters $\phi$ and $\lambda$. Default values are used if the argument limits is not provided.

There are internal reparametrisation depending on the options for parameters to be estimated. For instance for the common situation when fix.nugget=FALSE the minimisation is performed in a reduced parameter space using $\tau^2_{rel} = \frac{\tau^2}{\sigma^2}$. In this case values of $\sigma^2$ and $\beta$ are then given by analytical expressions which are function of the two parameters remaining parameters and limits for these two parameters will be ignored.

Since parameter values are found by numerical optimization using the function optim, 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 optimizer. 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 passcontrol()options for the optimiser internally

Transformation If the fix.lambda = FALSE and nospatial = FALSE the Box-Cox parameter for the model without the spatial component is obtained numerically, with log-likelihood computed by the function boxcox.ns.

Multiple initial values can be specified providing a $n \time 2$ matrix for the argument ini.cov.pars and/or providing a vector for the values of the remaining model parameters. In this case the log-likelihood is computed for all combinations of the model parameters. The parameter set which maximises the value of the log-likelihood is then used to start the minimisation algorithm.

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.

The argument realisations allows replicated data to be used. For instance, data collected at different times at least partially the same locations can be pooled together in the parameter estimation if independence is assumed between time points. The argument realisations takes a vector indicating the replication number (e.g. the times). The log-likelihoods are computed for each replication and added together. Notice that this assumes independence among the replications.

References

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

See Also

summary.likGRF for summary of the results, plot.variogram, lines.variogram and lines.variomodel for graphical output, proflik for computing profile likelihoods, variofit and for other estimation methods, and optim for the numerical minimisation function.

Examples

data(s100)
ml <- likfit(s100, ini=c(0.5, 0.5), fix.nug = TRUE)
ml
summary(ml)
reml <- likfit(s100, ini=c(0.5, 0.5), fix.nug = TRUE, met = "REML")
summary(reml)
plot(variog(s100))
lines(ml)
lines(reml, lty = 2)