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0.00
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1.00
−2
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1
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0 5 10 15 20 0 5 10 15 20h x
C y
drawDraw 1
Draw 2
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0.50
0.75
1.00
−4
−2
0
2
−3
−2
−1
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1
−2
0
2
0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00
0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00
h x
x x
C y
y y
CovExpPow Exp(p=1.5)Sq Exp
Exponential
Powered Exponential (p=1.5) Square Exponential
Covariance − l=12, sigma2=1
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1.00
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0
1
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−3
−2
−1
0
1
−1
0
1
2
0 2 4 6 0 2 4 6
0 2 4 6 0 2 4 6
h x
x x
C y
y y
v=1/2
v=3/2
v=5/2
Matern − v=1/2
Matern − v=3/2 Matern − v=5/2
Covariance − l=2, sigma2=1
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-
N
not> 7 l 777 U or P
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0.50
0.75
1.00
−2
−1
0
1
−1
0
1
2
−1
0
1
0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00
0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00
h x
x x
y
y y
alpha=1
alpha=3
alpha=10
Rational Quadratic − alpha=1
Rational Quadratic − alpha=10 Rational Quadratic − alpha=100
Covariance − l=12, sigma2=1
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0.00
0.25
0.50
0.75
1.00
−3
−2
−1
0
1
−2
0
2
−2
−1
0
1
2
0.0 0.3 0.6 0.9 0.0 0.3 0.6 0.9
0.0 0.3 0.6 0.9 0.0 0.3 0.6 0.9
h x
x x
y
y y
l=1
l=3
l=10
Spherical − l=1
Spherical − l=3 Spherical − l=10
Covariance − sigma2=1
0
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0.00
0.25
0.50
0.75
1.00
−1
0
1
−2
−1
0
1
−2
−1
0
1
2
0 1 2 3 4 0 2 4 6
0 2 4 6 0 2 4 6
h x
x x
y
y y
forcats::as_factor(Cov)p=1
p=2
p=3
Periodic − p=1
Periodic − p=2 Periodic − p=3
Covariance − l=2, sigma2=1
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−1.0
−0.5
0.0
0.5
1.0
0.00 0.25 0.50 0.75 1.00x
y
-
−10
−5
0
5
−2 −1 0 1 2x
y
Cov_a * Cov_b
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−4
−2
0
2
0 1 2 3x
y
Cov_a * Cov_b
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−5
−4
−3
−2
−1
0
0 1 2 3x
y
drawDraw 1
Draw 2
Cov_a + Cov_b
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−2
−1
0
1
2
0 2 4 6x
y
Cov_a * Cov_b
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−2
−1
0
1
0.0 2.5 5.0 7.5 10.0x
y
Cov_a + Cov_b
-
Cov_A (short) Cov_B (long) Cov_A + Cov_B
0.0 2.5 5.0 7.5 10.0 0.0 2.5 5.0 7.5 10.0 0.0 2.5 5.0 7.5 10.0
−3
−2
−1
0
1
2
x
y
-
.
-
330
360
390
1960 1980 2000x
y
SourceNOAA
Scripps (co2 in R)
-
alpha
l[1] l[2] l[3] l[4] l[5]
sigma2[1] sigma2[2] sigma2[3] sigma2[4] sigma2[5]
0 2505007501000
0 2505007501000 0 2505007501000 0 2505007501000 0 2505007501000
0.02
0.03
0.04
0.0
0.3
0.6
0.9
0.0
0.2
0.4
0.6
0.8
2
4
6
0.00.51.01.52.0
0.6
0.8
1.0
1.2
010203040
0.005
0.010
0.015
0.020
0
2000
4000
6000
0.02
0.04
0.06
0
2
4
6
8
.iteration
estim
ate
-
Sigma_3 Sigma_4
Sigma_1 Sigma_2
1960 1970 1980 1990 1960 1970 1980 1990
−4
0
4
8
−1.5
−1.0
−0.5
0.0
0.5
1.0
−20
−10
0
10
20
30
−2
−1
0
1
2
x
post_m
ean
covSigma_1
Sigma_2
Sigma_3
Sigma_4
x
-
330
360
390
1960 1980 2000x
post_m
ean
O
-
360
370
380
390
400
410
2000 2005 2010 2015x
post_m
ean
-
330
360
390
1960 1980 2000Time
co2
level80
95
Forecasts from ARIMA(1,1,1)(1,1,2)[12]
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-
Sigma_3 Sigma_4
Sigma_1 Sigma_2
2000 2005 2010 2015 2000 2005 2010 2015
−5
0
5
−1.0
−0.5
0.0
0.5
1.0
30
40
50
60
−1
0
1
x
post_m
ean
covSigma_1
Sigma_2
Sigma_3
Sigma_4