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
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
|
\name{jss14}
\alias{jss14}
%\alias{information}
%\alias{pars}
%\alias{pars.model}
%\alias{whole}
%\alias{whole.model}
\title{Covariance models for multivariate and vector-valued fields}
\description{
Here, the code of the paper on \sQuote{Analysis, simulation and prediction of
multivariate random fields with package RandomFields} is given.
}
\me
\references{
Schlather, M., Malinowski, A., Menck, P.J., Oesting, M. and
Strokorb, K. (2015)
Analysis, simulation and prediction of multivariate
random fields with package \pkg{RandomFields}. \emph{
Journal of Statistical Software}, \bold{63} (8), 1-25,
url = \sQuote{http://www.jstatsoft.org/v63/i08/}
}
\examples{\dontshow{StartExample()}
\dontshow{if (RFoptions()$internal$examples_reduced) RFoptions(modus_operandi="sloppy")}
\dontrun{
###########################################
## SECTION 4: UNCONDITIONAL SIMULATION ##
###########################################
RFoptions(seed = 0, height = 4)
## seed=0: *ANY* simulation will have the random seed 0; set
## RFoptions(seed=NA) to make them all random again
## height : height of X11
## always_close_device=FALSE: the pictures are kept on the screen
## Fig. 1: linear model of coregionalization
M1 <- c(0.9, 0.6)
M2 <- c(sqrt(0.19), 0.8)
model <- RMmatrix(M = M1, RMwhittle(nu = 0.3)) +
RMmatrix(M = M2, RMwhittle(nu = 2))
x <- y <- seq(-10, 10, 0.2)
simu <- RFsimulate(model, x, y)
plot(simu)
## Fig. 2: Wackernagel's delay model
model <- RMdelay(RMstable(alpha = 1.9, scale = 2), s = c(4, 4))
simu <- RFsimulate(model, x, y)
plot(simu, zlim = 'joint')
%dev.copy2pdf(file="delay0.pdf")
## Fig. 3: extended Wackernagel's delay model
model <- RMdelay(RMstable(alpha = 1.9, scale = 2), s = c(0, 4)) +
RMdelay(RMstable(alpha = 1.9, scale = 2), s = c(4, 0))
simu <- RFsimulate(model, x, y)
plot(simu, zlim = 'joint')
plot(model, dim = 2, xlim = c(-5, 5), main = "Covariance function",
cex = 1.5, col = "brown")
%dev.copy2pdf(file="delay_cov.pdf")
## Fig. 4: latent dimension model
## MARGIN.slices has the effect of choosing the third dimension
## as latent dimension
## n.slices has the effect of choosing a bivariate model
model <- RMgencauchy(alpha = 1.5, beta = 3)
simu <- RFsimulate(model, x, y, z = c(0,1))
plot(simu, MARGIN.slices = 3, n.slices = 2)
## Fig. 5: Gneiting's bivariate Whittle-Matern model
model <- RMbiwm(nudiag = c(1, 2), nured = 1, rhored = 1, cdiag = c(1, 5),
s = c(1, 1, 2))
simu <- RFsimulate(model, x, y)
plot(simu)
## Fig. 6: anisotropic linear model of coregionalization
M1 <- c(0.9, 0.6)
M2 <- c(sqrt(0.19), 0.8)
A1 <- RMangle(angle = pi/4, diag = c(0.1, 0.5))
A2 <- RMangle(angle = 0, diag = c(0.1, 0.5))
model <- RMmatrix(M = M1, RMgengneiting(kappa = 0, mu = 2, Aniso = A1)) +
RMmatrix(M = M2, RMgengneiting(kappa = 3, mu = 2, Aniso = A2))
simu <- RFsimulate(model, x, y)
plot(simu)
## Fig. 7: random vector field whose paths are curl free
## A 4-variate field is simulated, where the first component
## refers to the potential field, the second and third component
## to the curl free vector field and the forth component to the
## field of sinks and sources
model <- RMcurlfree(RMmatern(nu = 5), scale = 4)
simu <- RFsimulate(model, x, y)
plot(simu, select.variables = list(1, 2 : 3, 4))
plot(model, dim = 2, xlim = c(-3, 3), main = "", cex = 2.3, col="brown")
%dev.copy2pdf(file="curlfree_cov.pdf")
## Fig. 8: Kolmogorov's model of turbulence
## See the physical literature for its derivation and details
x <- y <- seq(-2, 2, len = 20)
model <- RMkolmogorov()
simu <- RFsimulate(model, x, y, z = 1)
plot(simu, select.variables = list(1 : 2), col = c("red"))
plot(model, dim = 3, xlim = c(-3, 3), MARGIN = 1 : 2, cex = 2.3,
fixed.MARGIN = 1.0, main = "", col = "brown")
%dev.copy2pdf(file="kolmogorov_cov.pdf")
###########################################
## SECTION 5: DATA ANALYSIS ##
###########################################
## get the data
data("weather")
PT <- weather[ , 1 : 2] ## full data set takes more than
## half an hour to be analysed
## transformation of earth coordinates to Euclidean coordinates
Dist.mat <- as.vector(RFearth2dist(weather[ , 3 : 4]))
All <- TRUE
\dontshow{if(RFoptions()$internal$examples_reduced){warning("reduced data set")
All <- 1:10
PT <- weather[All , 1 : 2]
Dist.mat <- as.vector(RFearth2dist(weather[All , 3 : 4]))
}}
#################################
## model definition ##
#################################
## bivariate pure nugget effect:
nug <- RMmatrix(M = matrix(nc = 2, c(NA, 0, 0, NA)), RMnugget())
## parsimonious bivariate Matern model
pars.model <- nug + RMbiwm(nudiag = c(NA, NA), scale = NA, cdiag = c(NA, NA),
rhored = NA)
## whole bivariate Matern model
whole.model <- nug + RMbiwm(nudiag = c(NA, NA), nured = NA, s = rep(NA, 3),
cdiag = c(NA, NA), rhored = NA)
#################################
## model fitting and testing ##
#################################
## 'parsimonious model'
## fitting takes a while ( > 10 min)
pars <- RFfit(pars.model, distances = Dist.mat, dim = 3, data = PT)
print(pars)
print(pars, full=TRUE)
RFratiotest(pars)
#RFcrossvalidate(pars, x = weather[All , 3 : 4], data = PT, full = TRUE)
## 'whole model'
## fitting takes a while ( > 10 min)
whole <- RFfit(whole.model, distances = Dist.mat, dim = 3, data = PT)
print(whole, full=TRUE)
RFratiotest(whole)
#RFcrossvalidate(whole, x = weather[All , 3 : 4], data = PT, full = TRUE)
## compare parsimonious and whole
RFratiotest(nullmodel = pars, alternative = whole)
#################################
## kriging ##
#################################
## First, the coordinates are projected on a plane
a <- colMeans(weather[All , 3 : 4]) * pi / 180
P <- cbind(c(-sin(a[1]), cos(a[1]), 0),
c(-cos(a[1]) * sin(a[2]), -sin(a[1]) * sin(a[2]), cos(a[2])),
c(cos(a[1]) * cos(a[2]), sin(a[1]) * cos(a[2]), sin(a[2])))
x <- RFearth2cartesian(weather[All , 3 : 4])
plane <- (t(x) \%*\%P)[ , 1 : 2]
## here, kriging is performed on a rectangular that covers
## the projected points above. The rectangular is approximated
## by a grid of length 200 in each direction.
n <- 200
r <- apply(plane, 2, range)
dta <- cbind(plane, weather[All , 1 : 2])
z <- RFinterpolate(pars, data = dta, dim = 2,
x = seq(r[1, 1], r[2, 1], length = n),
y = seq(r[1, 2], r[2, 2], length = n),
varunits = c("Pa", "K"), spConform = TRUE)
plot(z)
}
\dontshow{RFoptions(modus_operandi="normal")}
\dontshow{FinalizeExample()}}
\seealso{
\link{weather}, \link{SS12}, \link{S10}.
}
\keyword{spatial}
|
