0.00

0.25

0.50

0.75

1.00

−2

−1

0

1

2

0 5 10 15 20 0 5 10 15 20h x

C y

drawDraw 1

Draw 2

0.00

0.25

0.50

0.75

1.00

−4

−2

0

2

−3

−2

−1

0

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

0.00

0.25

0.50

0.75

1.00

−2

−1

0

1

2

−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

-

N

not> 7 l 777 U or P

0.00

0.25

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

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

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

−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

−4

−2

0

2

0 1 2 3x

y

Cov_a * Cov_b

−5

−4

−3

−2

−1

0

0 1 2 3x

y

drawDraw 1

Draw 2

Cov_a + Cov_b

−2

−1

0

1

2

0 2 4 6x

y

Cov_a * Cov_b

−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]

-

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