Maximum Likelihood estimates for some Matern covariance parameters.

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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

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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)

Arguments

Dmax

Maximum distance for grid used to evaluate the fitted covariance function.

Distance

Distance function used in finding covariance.

x

A matrix of spatial locations with rows indexing location and columns the dimension (e.g. longitude/latitude)

y

Spatial observations

smoothness

Value of the Matern shape parameter.

theta.grid

Grid of theta parameter values to use for grid search in maximizing the Likelilood. The defualt is do an initial grid search on ngrid points with the range at the 3 an d 97 quantiles of the pairwise distances.If only two points are passed then this is used as the range for a sequence of ngrid points.

ngrid

Number of points in grid search.

init

Initial values of the parameters for optimization. For the first three functions these are in the order rho, theta sigma2 and in a log scale. For MaternQRProfile.test initial value is just log(theta).

verbose

If TRUE prints more information.

rho

Marginal variance of Matern process (the "sill")

sigma2

Variance of measurement error (the "nugget")

theta

Scale parameter (the "range")

nu

Smoothness parameter

ltheta

log of range parameter

info

A list with components x,y, smoothness, ngrid that pass the information to the optimizer. See details below.

value

If TRUE only reports minus log Profile likelihood with profile on the range parameter. If FALSE returns a list of information.

m

Polynomial of degree (m-1) will be included in model as a fixed part.

niter

Maximum number of interations in golden section search.

tol

Tolerance for convergence in golden section search.

...

Additional arguments that are passed to the Krig function in evaluating the profile likelihood.

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.

Value

For MLE.Matern (and MLE.Matern.fast)

smoothness

Value of the smoothness function

pars

MLE for rho, theta and sigma

REML

Value of minus the log restricted Profile likelihood at the maxmimum

trA

Effective degrees of freedom in the predicted surface based on the MLE parameters.

REML.grid

Matrix with values of theta and the log likelihood from the initial grid search.

Note

See the script REMLest.test.R and Likelihood.test.R in the tests directory to see how these functions are used to check the likelihood expressions.

Author(s)

Doug Nychka

Examples

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# 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)
## Not run: 
# 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)

## End(Not run)

## Not run: 
# 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<- predictSurface(out2, lambda= lambda.MLE)
  surface(out.p) # same surface!

## End(Not run)

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

## Not run: 
# 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 }

## End(Not run)

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