fields (version 6.8)

REML.test: Maximum Likelihood estimates for some Matern covariance parameters.

Description

For a fixed smoothness (shape) parameter these functions provide different ways of estimating and testing restricted and profile likehiloods for the Martern covariance parameters. MLE.Matern is a simple function that finds the restricted maximum likelihood (REML) estimates of the sill, nugget and range parameters (rho, sigma2 and theta) of the Matern covariance functions. The remaining functions are primarily for testing.

Usage

MLE.Matern(x, y, smoothness, theta.grid = NULL, ngrid = 20,
                 verbose = FALSE, niter = 25, tol = 1e-05,
                 Distance = "rdist", m = 2, Dmax = NULL, ...)
MLE.Matern.fast(x, y, smoothness, theta.grid = NULL, ngrid=20, verbose=FALSE,
                                         m=2, ...)
MLE.objective.fn( ltheta,info, value=TRUE)
MaternGLSProfile.test(x, y, smoothness = 1.5, init = log(c(0.05,1)))
MaternGLS.test(x, y, smoothness = 1.5, init = log(c(1, 0.2, 0.1)))
MaternQR.test (x, y, smoothness = 1.5, init = log(c(1, 0.2, 0.1))) 
MaternQRProfile.test (x, y, smoothness = 1.5, init = log(c(1))) 
REML.test(x, y, rho, sigma2, theta, nu = 1.5)

Value

For MLE.Matern (and MLE.Matern.fast)smoothnessValue of the smoothness functionparsMLE for rho, theta and sigmaREMLValue of minus the log restricted Profile likelihood at the maxmimumtrAEffective degrees of freedom in the predicted surface based on the MLE parameters.REML.gridMatrix with values of theta and the log likelihood from the initial grid search.

Rdversion

1.1

Details

MLE.Matern is a simple function to find the maximum likelihood estimates of using the restricted and profiled likeilihood that is intrinsic to the ccomputations in Krig. The idea is that the likelihood is concentrated to the parameters lambda and theta. (where lambda = sigma2/rho). For fixed theta then this is maximized over lambda using Krig and thus concetrates the likelihood on theta. The final maximization over theta is implemented as a golden section search and assumes a convex function. All that is needed is for three theta grid points where the middle point has a larger likelihood than the endpoints. In practice the theta grid defualts to a 20 points equally spaced between the .03 and .97 quantiles of the distribution of the pairwise distances. The likelihood is evaluated at these points and a possible triple is identified. If no exists from the grid search the function returns with NAs for the parameter estimates. Note that due to the setup of the golden section search the computation actually minimizes minus the log likelihood. MLE.Matern.fast is a similar function but replaces the optimaiztion step computed by Krig to a tighter set of code in the function MLE.objective.fn. See also mKrig.MLE for an alternative and streamlined function using mKrig rather than Krig.

Examples

# Just look at one day from the ozone2 
data(ozone2)

out<- MLE.Matern( ozone2$lon.lat,ozone2$y[16,],1.5, ngrid=8)
plot( out$REML.grid)
points( out$pars[2], out$REML, cex=2)
xline( out$pars[2], col="blue", lwd=2)
# to get a finer grid on initial search:
out<- MLE.Matern( ozone2$lon.lat,ozone2$y[16,],1.5,
                      theta.grid=c(.3,2), ngrid=40) 

# simulated data  200 points uniformly distributed
set.seed( 123)
x<- matrix( runif( 2*200), ncol=2)
n<- nrow(x)
rho= 2.0
sigma= .05
theta=.5

Cov.mat<-  rho* Matern( rdist(x,x), smoothness=1.0, range=theta)
A<- chol( Cov.mat)
gtrue<- t(A) %*% rnorm(n)
gtrue<- c( gtrue)
err<-  rnorm(n)*sigma
y<- gtrue + err
out0<- MLE.Matern( x,y,smoothness=1.0) # the bullet proof version
# the MLEs and -log likelihood at maximum
print( out0$pars)
print( out0$REML)

out<- MLE.Matern.fast( x,y, smoothness=1.0) # for the impatient
# the MLEs:
print( out$pars) 
print( out$REML)


# MLE for fixed theta (actually the MLE from out0) 
# that uses MLE.objective.fn directly
info<- list( x=x,y=y,smoothness=1.0, ngrid=80)
# the MLEs:
out2<- MLE.objective.fn(log(out0$pars[2]), info, value=FALSE)
print( out2$pars)

# Now back to Midwest ozone pollution ...
# Find the MLEs for ozone data and evaluate the Kriging surface.
  data(ozone2)
  out<- MLE.Matern.fast( ozone2$lon.lat,ozone2$y[16,],1.5)
#use these parameters to fit surface ....
  lambda.MLE<- out$pars[3]/out$pars[1]
  out2<- Krig( ozone2$lon.lat,ozone2$y[16,] , Covariance="Matern",
              theta=out$pars[2], smoothness=1.5, lambda= lambda.MLE)
  surface( out2) # uses default lambda -- which is the right one.

# here is another way to do this where the new lambda is given in 
# the predict step
  out2<- Krig( ozone2$lon.lat,ozone2$y[16,] , Covariance="Matern",
               theta=out$pars[2], smoothness=1.5)
# The default lambda is that found by GCV
# predict on a grid but use the MLE value for lambda:
  out.p<- predict.surface(out2, lambda= lambda.MLE)
  surface(out.p) # same surface!

# One could also use mKrig with a fixed lambda to compute the surface. 

# looping  through all the days of the ozone data set.
  data( ozone2)
  x<- ozone2$lon.lat
  y<- ozone2$y
  out.pars<- matrix( NA, ncol=3, nrow=89)

  for ( k in 1:89){
    hold<- MLE.Matern.fast( x,c(y[k,]), 1.5)$pars
    cat( "day", k," :", hold, fill=TRUE)
    out.pars[k,]<- hold }