Nothing

```
# fields is a package for analysis of spatial data written for
# the R software environment .
# Copyright (C) 2018
# University Corporation for Atmospheric Research (UCAR)
# Contact: Douglas Nychka, nychka@ucar.edu,
# National Center for Atmospheric Research, PO Box 3000, Boulder, CO 80307-3000
#
# This program is free software; you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation; either version 2 of the License, or
# (at your option) any later version.
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with the R software environment if not, write to the Free Software
# Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
# or see http://www.r-project.org/Licenses/GPL-2
mKrig.trace <- function(object, iseed, NtrA) {
# do not reset the random seed if NA.
if( !is.na(iseed)){
set.seed(iseed)
}
# if more MonteCarlo samples > number of data points just
# find A exactly using np calls to predict.
np<- object$np
if (NtrA >= object$np) {
Ey <- diag(1, np)
NtrA <- np
hold <- diag(predict.mKrig(object, ynew = Ey, collapseFixedEffect=FALSE))
trA.info<- NA
trA.est <- sum(hold)
}
else {
# if fewer tests then use random trace method
# find fitted.values for iid N(0,1) 'data' to calculate the
# the Monte Carlo estimate of tr A(lambda)
# basically repeat the steps above but take some
# short cuts because we only need fitted.values
# create random normal 'data'
Ey <- matrix(rnorm(np * NtrA), nrow = np,
ncol = NtrA)
trA.info <- colSums(Ey * (predict.mKrig(object, ynew = Ey,
collapseFixedEffect=FALSE)))
trA.est <- mean(trA.info)
}
if (NtrA < np) {
MSE<-(sum(object$residuals^2)/np)
GCV <- MSE/(1 - trA.est /np)^2
GCV.info <- MSE/( 1 - trA.info/np)^2
}
else{
GCV<- NA
GCV.info <- NA
}
return(
list(trA.info = trA.info, eff.df = trA.est,
GCV= GCV, GCV.info=GCV.info)
)
}
mKrig.coef <- function(object, y, collapseFixedEffect=TRUE) {
# given new data y and the matrix decompositions in the
# mKrig object find coficients beta and c.
# beta are the coefficients for the fixed part
# in this case hard coded for a low order polynomial
# c are coefficients for the basis functions derived from the
# covariance function.
#
# see mKrig itself for more comments on the linear algebra
#
# Note that all these expressions make sense if y is a matrix
# of several data sets and one is solving for the coefficients
# of all of these at once. In this case beta and c.coef are matrices
#
# generalized least squares for d
if( any(is.na(y))){
stop("mKrig can not omit missing values in observation vecotor")
}
if( object$nt>0){
beta <- as.matrix(qr.coef(object$qr.VT, forwardsolve(object$Mc,
transpose = TRUE, y, upper.tri = TRUE)))
betaMeans<- rowMeans( beta)
if( collapseFixedEffect){
beta<- matrix( betaMeans, ncol=ncol(beta), nrow= nrow( beta))
}
# residuals from subtracting off fixed part
# of model as m-1 order polynomial
resid <- y - object$Tmatrix %*% beta
}
else{
beta<- NULL
resid <- y
}
# and now find c.
c.coef <- forwardsolve(object$Mc, transpose = TRUE, resid,
upper.tri = TRUE)
c.coef <- as.matrix(backsolve(object$Mc, c.coef))
out <- list( beta = (beta), c.coef = c.coef )
return(out)
}
```

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