mKrig.MLE: maximizes likelihood for the process marginal variance (rho) and nugget standard deviation (sigma) parameters (e.g. lambda) over a list of covariance models or a grid of covariance parameter values.

Description

This function is designed to explore the likelihood surface for different covariance parameters with the option of maximizing over sigma and rho.

Usage

mKrig.MLE( x, y, weights = rep(1, nrow(x)), Z = NULL, ...,
                 par.grid = NULL, lambda = NULL, lambda.profile = TRUE,
                 verbose = FALSE, relative.tolerance = 1e-04)

Arguments

Value

A list with the componentssummaryA matrix giving the results for evaluating the likelihood for each covariance model.par.gridThe par.grid argument used.cov.args.MLEThe list of covariance arguments (except for lambda) that have the largest likelihood over the list covariance models. To fit the surface at the largest likelihood among those tried

do.call( "mKrig", c(obj$mKrig.args, obj$cov.args.MLE,list(lambda=obj$lambda.opt)) )

where obj is the list returned by this function.

callThe calling arguments to this function.

Details

The observational model follows the same as that described in the Krig function and thus the two primary covariance parameters for a stationary model are the nugget standard deviation (sigma) and the marginal variance of the process (rho). It is useful to reparametrize as rho and lambda= sigma^2/rho. The likelihood can be maximized analytically over rho and the parameters in the fixed part of the model the estimate of rho can be substituted back into the likelihood to give a expression that is just a function of lambda and the remaining covariance parameters. It is this expression that is then maximized numerically over lambda when lambda.profile = TRUE.

Examples

Run this code

# some synthetic data
  N<- 100
  set.seed(123)
  x<- matrix(runif(2*N), N,2)
  theta<- .2
  Sigma<-  Matern( rdist(x,x)/theta , smoothness=1.0)
  Sigma.5<- chol( Sigma)
  sigma<- .1
  M<-5 #  Five (5) independent spatial data sets
  F.true<- t( Sigma.5)%*% matrix( rnorm(N*M), N,M)
  Y<-  F.true +  sigma* matrix( rnorm(N*M), N,M)
# find MLE for lambda with range and smoothness fixed in Matern for first data set
  obj<- mKrig.MLE( x,Y[,1], Covariance="Matern", theta=.2, smoothness=1.0)
  obj$summary # take a look
  fit<- mKrig( x,Y[,1], Covariance="Matern", theta=.2, smoothness=1.0, lambda= obj$lambda.best) 
#
# search over the range parameter and use all 5 replications for combined likelihood
par.grid<- list( theta= seq(.1,.25,,6))
# default starting value for lambda is .02 subsequent ones use previous optimum.
  obj<- mKrig.MLE( x,Y, Covariance="Matern",lambda=c(.02,rep(NA,4)), smoothness=1.0, par.grid=par.grid)