R: Log-Likelihood for a Gaussian Random Field

loglik.GRF {geoR} | R Documentation

## Log-Likelihood for a Gaussian Random Field

### Description

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

### 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 x 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) = -(n/2) * log(2 * pi) - 0.5 * log|V| - 0.5 * (y - F b)' V^{-1} (y - F b),

where n is the size of the data vector y, b is the mean (vector) parameter with dimention p, V 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(θ) = -((n-p)/2) * log (2 * pi) + 0.5 * log |F'F| - 0.5 * log |V| - 0.5 * log |F'VF| - 0.5 * (y - Fb)' V^(-1) (y - Fb).

### Value

The numerical value of the log-likelihood.

### Author(s)

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

### References

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

### See Also

`likfit` for likelihood-based parameter estimation.

### Examples

    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)

[Package geoR version 1.7-5.2 Index]