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# 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.
# 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
data( ozone2)
x<- ozone2$lon.lat
y<- ozone2$y[16,]
is.good <- !is.na( y)
x<- x[is.good,]
y<- y[is.good]
theta<- 2.0
# check log likelihood calculation
nu<- 1.5
lambda<- .2
out<- mKrig( x,y, theta=theta,Covariance="Matern", smoothness=nu, lambda=lambda)
# peg rho and sigma as MLEs from mKrig
rho <- out$rho.MLE
sigma2<- rho*lambda
N<- length( y)
dd<- rdist( x,x)
M<- rho* Matern( dd, range= theta, smoothness=nu) + sigma2* diag( 1, N)
X<- fields.mkpoly( x, 2)
Mi<- solve( M)
betahat<- solve(t(X)%*%Mi%*%X)%*% t(X)%*% Mi%*% y
res<- y - X%*%betahat
ccoef<- ( Mi%*% ( res))*rho
# sanity check that estimates are the same
test.for.zero( ccoef, out$c, 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
# formula for full likelihood using peices from mKrig
lLike.test<- -(N/2)*log(2*pi) - (1/2)* out$lnDetCov - (1/2)*(N)*log( rho) - (1/2)*out$quad.form/rho
test.for.zero( lLike, lLike.test, tag="llike full verses rhohat")
test.for.zero( lLike, out$lnProfileLike, tag="llike profile from mKrig")
# REML check
nu<- 1.5
theta<- .6
obj<- Krig( x,y, theta=theta,Covariance="Matern", smoothness=nu )
# sanity check that c coefficients agree with Krig
rho<- 500
lambda<- .2
sigma2<- lambda*rho
hold<- REML.test( x,y,rho, sigma2, theta, nu=1.5)
ccoef2<- Krig.coef( obj, lambda)$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/rho, tol=5e-8)
Quad3<- sum( D.temp*(obj$matrices$u[obj$matrices$D>0])^2/(1+lambda*D.temp))
test.for.zero( hold$quad.form, Quad3/rho, tag="quad form")
# test determinants
N2<- length( D.temp)
det4<- -sum( log(D.temp/(1 + D.temp*lambda)) )
det1<- sum( log(eigen( hold$A/rho)$values))
test.for.zero( det1, det4, tag="det" )
# test REML Likelihood
lLikeREML.test<--1*( (N2/2)*log(2*pi) - (1/2)*(sum( log(D.temp/(1 + D.temp*lambda)) ) - N2*log(rho)) +
(1/2)*sum( lD*(obj$matrices$u)^2/(1+lD)) /(lambda*rho) )
test.for.zero( hold$REML.like, lLikeREML.test, tag="REML using matrices")
# profile likelihood
# lnProfileLike <- (-np/2 - log(2*pi)*(np/2)
# - (np/2)*log(rho.MLE) - (1/2) * lnDetCov)
# test using full REML likelihood.
nu<- 1.5
rho<- 7000
lambda<- .02
sigma2<- lambda*rho
theta<- 2.0
obj<- Krig( x,y, theta=theta,Covariance="Matern", smoothness=nu )
hold<- REML.test(x,y,rho, sigma2, theta, nu=1.5)
np<- hold$N2
rho.MLE<- c(hold$rhohat)
lnDetCov<-sum( log(eigen( hold$A/rho)$values))
l0<- REML.test(x,y,rho.MLE, rho.MLE*lambda, theta, nu=1.5)$REML.like
l1<- (-np/2 - log(2*pi)*(np/2)- (np/2)*log(rho.MLE) - (1/2) * lnDetCov)
l2<- (-1)*Krig.flplike( lambda, obj)
test.for.zero( l0,l2, tag="REML profile flplike")
test.for.zero( l1,l2, tag="REML profile flplike")
cat("all done with likelihood tests", fill=TRUE)
options( echo=TRUE)
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