distmat: a distance matrix. Usually the result of a call to nearest.dist.
Covariance: function defining the covariance. See example.
beta: parameters of the trend (fixed effects).
theta: parameters of the covariance structure.
Rstruct: the Cholesky structure of the covariance matrix.
beta0,theta0: inital values.
thetalower,thetaupper: lower and upper bounds of the parameter theta.
optim.control: arguments passed to optim.
hessian: Logical. Should a numerically differentiated Hessian matrix be returned?
cov.args: additional arguments passed to Covariance.
...: additional arguments passed to chol.
Returns
The negative-2-loglikelihood or the output from the function optim.
Details
We provide functions to calculate the negative-2-log-likelihood and maximum likelihood estimates for the model
y ~ N_n( X beta, Sigma(h;theta) )
in the case of a sparse or ordinary covariance matrices.
In the case of the *.spam versions, the covariance function has to return a spam object. In the other case, the methods are correctly overloaded and work either way, slightly slower than the *.spam counterparts though.
When working on the sphere, the distance matrix has to be transformed by
h -> R / 2 sin(h/2)
where R is the radius of the sphere.
The covariance function requires that the first argument is the distance matrix and the second the parameters. One can image cases in which the covariance function does not take the entire distance matrix but only some partial information thereof. (An example is the use of a kronecker type covariance structure.) In case of a sparse covariance construction where the argument Rstruct is not given, the first parameter element needs to be the range parameter. (This results from the fact, that a sparse structure is constructed that is independent of the parameter values to exploit the fast Choleski decomposition.)
In the zero-mean case, the neg2loglikelihood is calculated by setting the parameters X or beta to zero.
See Also
covmat, rmvnorm.spam
Examples
# True parameter values:truebeta <- c(1,2,.2)# beta = (intercept, linear in x, linear in y)truetheta <- c(.5,2,.02)# theta = (range, sill, nugget)# We now define a grid, distance matrix, and a sample:x <- seq(0,1,l=5)locs <- expand.grid( x, x)X <- as.matrix( cbind(1,locs))# design matrixdistmat <- nearest.dist( locs, upper=NULL)# distance matrixSigma <- cov.sph( distmat, truetheta)# true covariance matrixset.seed(15)y <- c(rmvnorm.spam(1,X %*% truebeta,Sigma))# construct sample# Here is the negative 2 log likelihood:neg2loglikelihood.spam( y, X, distmat, cov.sph, truebeta, truetheta)# We pass now to the mle:res <- mle.spam(y, X, distmat, cov.sph, truebeta, truetheta,thetalower=c(0,0,0),thetaupper=c(1,Inf,Inf))# Similar parameter estimates here, of course:mle.nomean.spam(y-X%*%res$par[1:3], distmat, cov.sph, truetheta, thetalower=c(0,0,0), thetaupper=c(1,Inf,Inf))