1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
|
#
# 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
# tests of predictSE
# against direct linear algebra
suppressMessages(library(fields))
options( echo=FALSE)
test.for.zero.flag<- TRUE
x0<- cbind( 0,4)
Krig( ChicagoO3$x, ChicagoO3$y, cov.function = "Exp.cov", aRange=50,
lambda=.06, GCV=FALSE)-> out
# direct calculation
Krig.Amatrix( out, x=x0)-> A
test.for.zero( A%*%ChicagoO3$y, predict( out, x0),tag="Amatrix vs. predict")
Sigma0<- out$sigmahat*Exp.cov( ChicagoO3$x, ChicagoO3$x, aRange=50)
S0<- out$sigmahat*c(Exp.cov( x0, x0, aRange=50))
S1<- out$sigmahat*Exp.cov( out$x, x0, aRange=50)
#yhat= Ay
#var( f0 - yhat)= var( f0) - 2 cov( f0,yhat)+ cov( yhat)
look<- S0 - t(S1)%*% t(A) - A%*%S1 +
A%*% ( Sigma0 + diag(out$tauHat.MLE**2/out$weightsM))%*% t(A)
#
#compare to
# diagonal elements
test2<- predictSE( out, x= x0)
test.for.zero( sqrt(diag( look)), test2,tag="Marginal predictSE")
# now test shortcut formula that leverages the prediction step for Kriging
#
Sigma<- Exp.cov( ChicagoO3$x, ChicagoO3$x, aRange=50) +
diag(out$lambda/out$weightsM)
#Sigma<- ( Sigma0 + diag(out$tauHat.MLE**2/out$weightsM))
Tmatrix <- do.call(out$null.function.name, c(out$null.args,
list(x = out$xM, Z = out$ZM)))
Omega<- solve( t(Tmatrix)%*% solve( Sigma)%*% Tmatrix)
Id<- diag( 1, nrow( Tmatrix))
qr.R( out$matrices$qr.VT) -> Rmat
Omega.test<- solve(t(Rmat)%*% (Rmat))
# Omega is the GLS covariance matrix for estimated parameters in fixed part of
# spatial model (the d coefficients). These are usually the "spatial drift" -- a
# low order polynomial
test.for.zero( Omega, Omega.test, tag="comparing Omega")
# M1 and M2 are matrices that go from obs to the estimated coefficients (d,c)
M1<- Omega%*% t(Tmatrix)%*% solve( Sigma)
M2<- solve( Sigma)%*% ( Id - Tmatrix%*% M1)
x0<- cbind( 0,4)
k0<- Exp.cov( out$x, x0, aRange=50)
#k0<- S1
t0<- c( 1, c(x0))
hold<- t( t0)%*%M1 + t(k0)%*% M2
test.for.zero( hold, A)
test.for.zero( M2%*%Sigma%*%t( M2), M2)
# benchmark using standard predictSE function
SE0<- predictSE( out, x=x0)
# shortcut formula explicitly
MSE<- S0 + out$sigmahat*t(t0)%*% Omega %*%t0 -
out$sigmahat*(t(k0)%*% M2 %*% k0 + t(t0)%*% M1%*% k0) -
out$sigmahat*t(t0)%*% M1%*% k0
# collecting terms to make this look like two predict steps.
MSE2<- S0 + out$sigmahat*t(t0)%*% Omega %*%t0 -
out$sigmahat* predict( out, yM= k0, x=x0) -
out$sigmahat* predict( out, yM= k0, x=x0, just.fixed=TRUE)
hold<- Krig.coef(out, y=k0)
tempc<- t(k0)%*% hold$c
tempd<- t(t0)%*%hold$d
MSE4<- S0 + out$sigmahat*t(t0)%*% Omega %*%t0 -
out$sigmahat * (tempc +2*tempd)
test.for.zero(SE0, sqrt( MSE4), tag="test of formula with explicit d and c")
# test of new function
Krig( ChicagoO3$x, ChicagoO3$y, cov.function = "Exp.cov", aRange=50,lambda=.06)-> out0
SE0<- predictSE.Krig( out0, x=x0)
mKrig( ChicagoO3$x, ChicagoO3$y, cov.function = "Exp.cov", aRange=50, lambda=.06)-> out2
SE3<- predictSE.mKrig( out2, xnew=x0)
test.for.zero(SE0, sqrt( MSE), tag="Krig function and direct formula")
test.for.zero(sqrt(MSE), sqrt( MSE2),
tag="new predict formula and direct formula")
test.for.zero( SE3, SE0, tag="New se _function_ and old Krig _function_")
#
# test of vectors of locations.
# receate object
out0<- Krig( ChicagoO3$x, ChicagoO3$y, cov.function = "Exp.cov", aRange=50, lambda=.06)
out<- mKrig( ChicagoO3$x, ChicagoO3$y, cov.function = "Exp.cov", aRange=50, lambda=.06)
x0<-rep( c( -20, -10,10,20),4)
x0 <- cbind( x0 , sort( x0))
x0<- rbind( c(0,4), x0)
k0<- Exp.cov( ChicagoO3$x,x0, aRange=50)
t0<- t(fields.mkpoly(x0, m=out$m))
hold<- Krig.coef(out0, y=k0)
MSE5<- (rep( S0,nrow(x0)) +
out0$sigmahat * colSums( t0 *(out0$matrices$Omega%*%t0))
-out0$sigmahat* colSums((k0)*hold$c) -
2*out0$sigmahat*colSums(t0*hold$d))
hold<- mKrig.coef(out, y=k0, collapse=FALSE)
sigmahat<- out$summary["sigma2"]
MSE6<- (rep( S0, nrow(x0)) +
sigmahat * colSums( t0 *(out$Omega%*%t0))
-sigmahat* colSums((k0)*hold$c.coef) -
2*sigmahat*colSums(t0*hold$beta))
test.for.zero( predictSE( out0, x0), sqrt(MSE5),
tag="Benchmark of formula")
test.for.zero( predictSE( out0, x0), sqrt(MSE6),
tag="Benchmark of formula mKrig coefs")
test.for.zero( predictSE( out, x0), predictSE.mKrig(out, x0),
tag="test function with several locations Krig mKrig functions" )
|
