loglik.GRF

From geoR v1.8-1 by Paulo Ribeiro Jr

Log-Likelihood for a Gaussian Random Field

This function computes the value of the log-likelihood for a Gaussian random field.

Keywords: spatial

Usage

loglik.GRF(geodata, coords = geodata$coords, data = geodata$data,
           obj.model = NULL, cov.model = "exp", cov.pars, nugget = 0,
           kappa = 0.5, lambda = 1, psiR = 1, psiA = 0,
           trend = "cte", method.lik = "ML", compute.dists = TRUE,
           realisations = NULL)

Arguments

geodata: a list containing elements coords and data as described next. Typically an object of the class "geodata" - a geoR data-set. If not provided the arguments coords and data must be provided instead.
coords: an $n \times 2$ matrix, each row containing Euclidean coordinates of the n data locations. By default it takes the element coords of the argument geodata.
data: a vector with data values. By default it takes the element data of the argument geodata.
obj.model: a object of the class variomodel with a fitted model. Tipically an output of likfit or variofit.
cov.model: a string specifying the model for the correlation function. For further details see documentation for cov.spatial.
cov.pars: a vector with 2 elements with values of the covariance parameters $\sigma^2$ (partial sill) and $\phi$ (range parameter).
nugget: value of the nugget parameter. Defaults to $0$.
kappa: value of the smoothness parameter. Defaults to $0.5$.
lambda: value of the Box-Cox tranformation parameter. Defaults to $1$.
psiR: value of the anisotropy ratio parameter. Defaults to $1$, corresponding to isotropy.
psiA: value (in radians) of the anisotropy rotation parameter. Defaults to zero.
trend: specifies the mean part of the model. The options are: "cte" (constant mean), "1st" (a first order polynomial on the coordinates), "2nd" (a second order polynomial on the coordinates), or a formula of the type ~X where X is a matrix with the covariates (external trend). Defaults to "cte".
method.lik: options are "ML" for likelihood and "REML" for restricted likelihood. Defaults to "ML".
compute.dists: for internal use with other function. Don't change the default unless you know what you are doing.
realisations: optional. A vector indicating replication number for each data. For more details see as.geodata.

Details

The expression log-likelihood is: $$l(\theta) = -\frac{n}{2} \log (2\pi) - \frac{1}{2} \log |\Sigma| - \frac{1}{2} (y - F\beta)' \Sigma^{-1} (y - F\beta),$$ where $n$ is the size of the data vector $y$, $\beta$ is the mean (vector) parameter with dimention $p$, $\Sigma$ is the covariance matrix and $F$ is the matrix with the values of the covariates (a vector of $1$'s if the mean is constant.

The expression restricted log-likelihood is: $$rl(\theta) = -\frac{n-p}{2} \log (2\pi) + \frac{1}{2} \log |F' F| - \frac{1}{2} \log |\Sigma| - \frac{1}{2} \log |F' \Sigma F| - \frac{1}{2} (y - F\beta)' \Sigma^{-1} (y - F\beta).$$

Value

The numerical value of the log-likelihood.

References

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

Examples

# NOT RUN {
loglik.GRF(s100, cov.pars=c(0.8, .25), nugget=0.2)
loglik.GRF(s100, cov.pars=c(0.8, .25), nugget=0.2, met="RML")

## Computing the likelihood of a model fitted by ML
s100.ml <- likfit(s100, ini=c(1, .5))
s100.ml
s100.ml$loglik
loglik.GRF(s100, obj=s100.ml)

## Computing the likelihood of a variogram fitted model
s100.v <- variog(s100, max.dist=1)
s100.vf <- variofit(s100.v, ini=c(1, .5))
s100.vf
loglik.GRF(s100, obj=s100.vf)
# }

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

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