# fields, Tools for spatial data
# Copyright 2004-2011, Institute for Mathematics Applied Geosciences
# University Corporation for Atmospheric Research
# Licensed under the GPL -- www.gpl.org/licenses/gpl.html
library( fields)
# tests of predict.se
# against direct linear algebra
options( echo=FALSE)
test.for.zero.flag<- 1
x0<- expand.grid( c(-8,-4,0,20,30), c(10,8,4,0))
Krig( ozone$x, ozone$y, cov.function = "Exp.cov", theta=50)-> out
# direct calculation
Krig.Amatrix( out, x=x0)-> A
test.for.zero( A%*%ozone$y, predict( out, x0),tag="Amatrix vs. predict")
Sigma<- out$rhohat*Exp.cov( ozone$x, ozone$x, theta=50)
S0<- out$rhohat*c(Exp.cov( x0, x0, theta=50))
S1<- out$rhohat*Exp.cov( out$x, x0, theta=50)
#yhat= Ay
#var( f0 - yhat)= var( f0) - 2 cov( f0,yhat)+ cov( yhat)
look<- S0 - t(S1)%*% t(A) - A%*%S1 +
A%*% ( Sigma + diag(out$shat.MLE**2/out$weightsM))%*% t(A)
#
#compare to
# diagonal elements
test2<- predict.se( out, x= x0)
test.for.zero( sqrt(diag( look)), test2,tag="Marginal predict.se")
out2<- Krig( ozone$x, ozone$y, cov.function = "Exp.cov", theta=50,
lambda=out$lambda)
test2<- predict.se( out2, x= x0)
test.for.zero( sqrt(diag( look)), test2,tag="Marginal predict.se fixed ")
test<- predict.se( out, x= x0, cov=TRUE)
test.for.zero( look, test,tag="Full covariance predict.se")
# simulation based.
set.seed( 333)
sim.Krig.standard( out, x0,M=4e3)-> test
var(test)-> look
predict.se( out, x=x0)-> test2
mean( diag( look)/ test2**2)-> look2
test.for.zero(look2, 1.0, tol=1.5e-2, tag="Marginal standard Cond. Sim.")
predict.se( out, x=x0, cov=TRUE)-> test2
# multiply simulated values by inverse square root of covariance
# to make them white
eigen( test2, symmetric=TRUE)-> hold
hold$vectors%*% diag( 1/sqrt( hold$values))%*% t( hold$vectors)-> hold
cor(test%*% hold)-> hold2
# off diagonal elements of correlations -- expected values are zero.
abs(hold2[ col(hold2)> row( hold2)])-> hold3
test.for.zero( mean(hold3), 0, relative=FALSE, tol=.02,
tag="Full covariance standard Cond. Sim.")
# test of sim.Krig.grid.R
#
# first create and check a gridded test case.
data( ozone2)
as.image(ozone2$y[16,], x= ozone2$lon.lat, ncol=24, nrow=20,
na.rm=TRUE)-> dtemp
#
# A useful disctrtized version of ozone2 data
x<- dtemp$xd
y<- dtemp$z[ dtemp$ind]
weights<- dtemp$weights[ dtemp$ind]
Krig( x, y, Covariance="Matern",
theta=1.0, smoothness=1.0, weights=weights) -> out
set.seed(234)
ind0<- cbind( sample( 1:20, 5), sample( 1:24, 5))
x0<- cbind( dtemp$x[ind0[,1]], dtemp$y[ind0[,2]])
# an inline check plot(out$x, cex=2); points( x0, col="red", pch="+",cex=2)
# direct calculation as backup ( also checks weighted case)
Krig.Amatrix( out, x=x0)-> A
test.for.zero( A%*%out$yM, predict( out, x0),tag="Amatrix vs. predict")
Sigma<- out$rhohat*stationary.cov(
out$xM, out$xM, theta=1.0,smoothness=1.0, Covariance="Matern")
S0<- out$rhohat*stationary.cov(
x0, x0, theta=1.0,smoothness=1.0, Covariance="Matern")
S1<- out$rhohat*stationary.cov(
out$xM, x0, theta=1.0,smoothness=1.0, Covariance="Matern")
#yhat= Ay
#var( f0 - yhat)= var( f0) - 2 cov( f0,yhat)+ cov( yhat)
look<- S0 - t(S1)%*% t(A) - A%*%S1 +
A%*% ( Sigma + diag(out$shat.MLE**2/out$weightsM) )%*% t(A)
test<- predict.se( out, x0, cov=TRUE)
test.for.zero( c( look), c( test), tag="Weighted case and exact for ozone2 full
cov", tol=1e-8)
########################################################################
######### redo test with smaller grid to speed things up
#cat("Conditional simulation test -- this takes some time", fill=TRUE)
# redo data set to smaller grid size
N1<-10
N2<-12
as.image(ozone2$y[16,], x= ozone2$lon.lat, ncol=N2, nrow=N1,
na.rm=TRUE)-> dtemp
#
# A useful discretized version of ozone2 data
xd<- dtemp$xd
y<- dtemp$z[ dtemp$ind]
weights<- dtemp$weights[ dtemp$ind]
Krig( xd, y, Covariance="Matern",
theta=1.0, smoothness=1.0, weights=weights) -> out
xr<- range( dtemp$x)
yr<- range( dtemp$y)
M1<-N1
M2<- N2
glist<- list( x=seq( xr[1], xr[2],,M1) , y=seq( yr[1], yr[2],,M2))
set.seed( 233)
# with extrap TRUE this finesses problems with
# how NAs are handled in var below
sim.Krig.grid( out, grid= glist, M=100, extrap=TRUE)-> look
predict.surface.se( out, grid=glist,extrap=TRUE)-> test
look2<- matrix( NA, M1,M2)
for( k in 1:M2){
for ( j in 1:M1){
look2[j,k] <- sqrt(var( look$z[j,k,], na.rm=TRUE)) }
}
test.for.zero( 1-mean(c(look2/test$z), na.rm=TRUE), 0, relative=FALSE,
tol=.001, tag="Conditional simulation marginal se for grid")
cat("all done testing predict.se ", fill=TRUE)
options( echo=TRUE)