R: Maximum Likelihood estimates for some Matern covariance...
REML.test
R Documentation
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)
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
# 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)