mKrig.MLE {fields}

R Documentation

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)

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

Arguments

x	Matrix of unique spatial locations (or in print or surface the returned mKrig object.)
y	Vector or matrix of observations at spatial locations, missing values are not allowed! Or in mKrig.coef a new vector of observations. If y is a matrix the columns are assumed to be independent observations vectors generated from the same covariance and measurement error model.
...	Additional arguments that would also be included in a call to mKrig to specify the covariance model and fixed model covariables.
lambda	If lambda.profile=FALSE the values of lambda to evaluate the likelihood if TRUE the starting values for the optimization. If lambda is NA then the optimum value from previous search is used as the starting value. If lambda is NA and it is the first value the starting value defaults to 1.0.
par.grid	A list or data frame with components being parameters for different covariance models. A typical component is theta comprising a vector of scale parameters to try. If par.grid is NULL then the covariance model is fixed at values that are given in ....
lambda.profile	If TRUE maximize likelihood over lambda.
relative.tolerance	Tolerance used to declare convergence when maximizing likelihood over lambda.
theta	Range parameter for compact Wendland covariance. (see fastTps)
verbose	If TRUE print out interesting intermediate results.
weights	Same weighting argument as used in mKrig.
Z	Same argument as mKrig to include spatial covariates.

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.

Note that fastTps.MLE is a convenient variant of this more general version to use directly with fastTps.

Value

A list with the components:

summary	A matrix giving the results for evaluating the likelihood for each covariance model.
par.grid	The par.grid argument used.
cov.args.MLE	The 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.
call	The calling arguments to this function.

Author(s)

Douglas W. Nychka

See Also

Krig.MLE, mKrig

Examples

# 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
## Not run: 
  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)

## End(Not run)

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