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#
# fields is a package for analysis of spatial data written for
# the R software environment.
# Copyright (C) 2022 Colorado School of Mines
# 1500 Illinois St., Golden, CO 80401
# Contact: Douglas Nychka, douglasnychka@gmail.edu,
#
# 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
##END HEADER
##END HEADER
# this is a test script to verify the likelihood computations are
# correct with the eigen decomposition format used in Krig
# see Krig.flplike for the concise computation.
#
suppressMessages(library(fields))
options( echo=FALSE)
test.for.zero.flag<- 1
# utility function foor testing
REML.test <- function(x, y, rho, sigma2, theta, nu = 1.5) {
Tmatrix <- fields.mkpoly(x, 2)
qr.T <- qr(Tmatrix)
N <- length(y)
Q2 <- qr.yq2(qr.T, diag(1, N))
ys <- t(Q2) %*% y
N2 <- length(ys)
A <- (rho * Matern(rdist(x, x), range = theta, smoothness = nu) +
sigma2 * diag(1, N))
A <- t(Q2) %*% A %*% Q2
Ac <- chol(A)
w <- backsolve(Ac, ys, transpose = TRUE)
REML.like <- (N2/2) * log(2 * pi) + (1/2) * 2 * sum(log(diag(Ac))) +
(1/2) * t(w) %*% w
REML.like <- -1 * REML.like
ccoef <- rho * Q2 %*% solve(A) %*% ys
return(list(REML.like = REML.like, A = A, ccoef = ccoef,
quad.form = t(w) %*% w, rhohat = (t(w) %*% w/N2) * rho,
det = 2 * sum(log(diag(Ac))), N2 = N2))
}
data( ozone2)
x<- ozone2$lon.lat
y<- ozone2$y[16,]
is.good <- !is.na( y)
x<- x[is.good,]
y<- y[is.good]
aRange<- 2.0
# check log likelihood calculation
nu<- 1.5
lambda<- .2
out<- mKrig( x,y, aRange=aRange,Covariance="Matern", smoothness=nu, lambda=lambda)
# peg sigma and tau as MLEs from mKrig
sigma <- out$summary["sigma2"]
tau2<- sigma*lambda
N<- length( y)
dd<- rdist( x,x)
M<- sigma* Matern( dd, range= aRange, smoothness=nu) + tau2* diag( 1, N)
X<- fields.mkpoly( x, 2)
Mi<- solve( M)
betahat<- solve(t(X)%*%Mi%*%X)%*% t(X)%*% Mi%*% y
res<- y - X%*%betahat
ccoef<- ( Mi%*% ( res))*sigma
# sanity check that estimates are the same
test.for.zero( ccoef, out$c.coef, tag="check ccoef")
# find full log likelihood
chol(M)-> cM
lLike<- -(N/2)*log(2*pi) - (1/2)* (2*sum( log( diag(cM)))) - (1/2)* t(res)%*% Mi %*% res
test.for.zero( lLike, out$summary["lnProfileLike.FULL"], tag="llike profile from mKrig")
# formula for full likelihood using peices from mKrig
# lLike.test<- -(N/2)*log(2*pi) - (1/2)* out$lnDetCov - (1/2)*(N)*log( sigma) - (1/2)*out$quad.form/sigma
# test.for.zero( lLike, lLike.test, tag="llike full verses sigmahat")
# REML check
nu<- 1.5
aRange<- .6
obj<- Krig( x,y, aRange=aRange,Covariance="Matern", smoothness=nu )
# sanity check that c coefficients agree with Krig
sigma<- 500
lambda<- .2
tau2<- lambda*sigma
hold<- REML.test( x,y,sigma, tau2, aRange, nu=1.5)
ccoef2<- Krig.coef( obj, lambda)$c
test.for.zero( hold$ccoef, ccoef2, tag="ccoefs")
# check RSS with Krig decomposition.
RSS1<- sum( (lambda*ccoef2)**2)
lD <- obj$matrices$D * lambda
RSS2 <- sum(((obj$matrices$u * lD)/(1 + lD))^2)
test.for.zero( RSS2, RSS1, tag=" RSS using matrices")
# check quadratic form with Krig
D.temp<- obj$matrices$D[ obj$matrices$D>0]
A3test<- (1/lambda)* obj$matrices$V %*% diag((D.temp*lambda)/ (1 +D.temp*lambda) )%*% t( obj$matrices$V)
test.for.zero(solve(A3test), hold$A/sigma, tol=5e-8)
Quad3<- sum( D.temp*(obj$matrices$u[obj$matrices$D>0])^2/(1+lambda*D.temp))
test.for.zero( hold$quad.form, Quad3/sigma, tag="quad form")
# test determinants
N2<- length( D.temp)
det4<- -sum( log(D.temp/(1 + D.temp*lambda)) )
det1<- sum( log(eigen( hold$A/sigma)$values))
test.for.zero( det1, det4, tag="det" )
# test REML Likelihood
lLikeREML.test<--1*( (N2/2)*log(2*pi) - (1/2)*(sum( log(D.temp/(1 + D.temp*lambda)) ) - N2*log(sigma)) +
(1/2)*sum( lD*(obj$matrices$u)^2/(1+lD)) /(lambda*sigma) )
test.for.zero( hold$REML.like, lLikeREML.test, tag="REML using matrices")
cat("all done with likelihood tests", fill=TRUE)
options( echo=TRUE)
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