bayesian methods for calibrating and comparing process-based vegetation models marcel van oijen...

Bayesian methods for calibrating and Bayesian methods for calibrating and comparing process-based vegetation comparing process-based vegetation

models models

Bayesian methods for calibrating and Bayesian methods for calibrating and comparing process-based vegetation comparing process-based vegetation

models models

Marcel van Oijen (CEH-Edinburgh)

ContentsContentsContentsContents

1. Process-based modelling of forests and uncertainties

2. Bayes’ Theorem (BT)

3. Bayesian Calibration (BC) of process-based models

4. Bayesian Model Comparison (BMC)

5. BC & BMC in NitroEurope

6. Examples of BC & BMC in other sciences

7. BC & BMC as tools to develop theory

8. References, Summary, Discussion

1. Introduction:Process-based modelling of forests and

uncertainties

1. Introduction:Process-based modelling of forests and

uncertainties

1.1 Forest growth in Europe1.1 Forest growth in Europe1.1 Forest growth in Europe1.1 Forest growth in Europe

22 sites

Empirical methods + process-based modelling

Modelling groups in UK, Sweden and Finland (2), coordinated by CEH-Edinburgh

Forests across Europe have started to grow faster in the 20th century:

Causes? Future trend?

Previous observations RECOGNITION

-

+/0

+

0

+

+

+

++ + + +/-+ + +/0 +

0

0

ps-/0/+

0

+

-

+/0

+

0

+

+

+

++ + + +/-+ + +/0 +

0

0

ps-/0/+

--

+/0+/0

++

00

++

++

++

++++ ++ ++ +/-+/-++ ++ +/0+/0 ++

00

00

psps-/0/+-/0/+

00

++

Project RECOGNITION (FAIRCT98-4124):

15 partner countries across Europe

1.2 Forest growth in Europe1.2 Forest growth in Europe1.2 Forest growth in Europe1.2 Forest growth in Europe

R2 = 0.4364

0

5

10

15

20

25

30

35

45 50 55 60 65 70

Latitude (°)

NP

P (

t D

M h

a-1

y-1

)

NPP before growth rate increase (1920)

% C

hange in NP

P -5

0

5

10

15

20

25

Latitude

N-dep.

CO2

Climate

CausalFactors:

Change in NPP(% increase)

% C

hange in NP

P -5

0

5

10

15

20

25

% C

hange in NP

P -5

0

5

10

15

20

25

Latitude

N-dep.

CO2

Climate

CausalFactors:

N-dep.

CO2

Climate

N-dep.

CO2

Climate

CausalFactors:

Change in NPP(% increase)

CONCLUSION 20th century

Growth accelerated byN-deposition.

0

40

1920 1960 2000Year

(deg

. C),

(kg

N h

a-1 y

-1)

0

700

(ppm

CO

2)

N-deposition CO2

Temperature

Environmental change 2000-2080:Effects on NPP

HOGPFZ

HELKAR

PUSRAJ

PFFSOL

BRILO

PTRI

GA2GA1

ALT AALSKO

BLA JAD

PUNKAN

KEMKOL

% C

han

ge in

NP

P -10

-5

0

5

10

15

20

25

CO2

ClimateN-depositionCUMULATIVE EFFECTS

Latitude

EFMEnvironmental change 2000-2080:

Effects on NPP

HOGPFZ

HELKAR

PUSRAJ

PFFSOL

BRILO

PTRI

GA2GA1

ALT AALSKO

BLA JAD

PUNKAN

KEMKOL

% C

han

ge in

NP

P -10

-5

0

5

10

15

20

25

Latitude

CONCLUSION 21st century:

Growth likely to be accelerated by climate change and increasing [CO2].

1.3 Reality check !1.3 Reality check !1.3 Reality check !1.3 Reality check !

How reliable is the European forest study:• Sufficient data for model parameterization?• Sufficient data for model input?• Would another model have given different

results?

In every study using systems analysis and simulation:Model parameters, inputs and structure are uncertain

How to deal with uncertainties optimally?

1.4 Forest models and uncertainty1.4 Forest models and uncertainty1.4 Forest models and uncertainty1.4 Forest models and uncertainty

Soil

Trees

H2OC

Atmosphere

H2O

H2OC

Nutr.

Subsoil (or run-off)

H2OC

Nutr.

Nutr.

Nutr.

Model

Jmax

-100 0 100 200 300 400 500

Fre

quen

cy

0.00

0.04

0.08

0.12

0.16

Vmax

-50 0 50 100 150 200 250 300

0.00

0.05

0.10

0.15

0.20

0.25

umax,root

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.00

0.05

0.10

0.15

0.20

0.25

0.30

froot

-0.5 0.0 0.5 1.0 1.5

0.00

0.05

0.10

0.15

0.20

0.25

Initial Csoluble

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

0.00

0.05

0.10

0.15

0.20

Initial Cstarch

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

0.00

0.05

0.10

0.15

0.20

Initial Wtotal

Value

-0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

0.0

0.1

0.2

0.3

0.4

0.5

Initial Nsoluble

Value

-0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

0.00

0.05

0.10

0.15

0.20

Photosynthesis

Fre

qu

ency

Parameter value

Parameter value

Allocation

C-pools

N-pools

Jmax

-100 0 100 200 300 400 500

Fre

quen

cy

0.00

0.04

0.08

0.12

0.16

Vmax

-50 0 50 100 150 200 250 300

0.00

0.05

0.10

0.15

0.20

0.25

umax,root

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.00

0.05

0.10

0.15

0.20

0.25

0.30

froot

-0.5 0.0 0.5 1.0 1.5

0.00

0.05

0.10

0.15

0.20

0.25

Initial Csoluble

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

0.00

0.05

0.10

0.15

0.20

Initial Cstarch

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

0.00

0.05

0.10

0.15

0.20

Initial Wtotal

Value

-0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

0.0

0.1

0.2

0.3

0.4

0.5

Initial Nsoluble

Value

-0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

0.00

0.05

0.10

0.15

0.20

Photosynthesis

Fre

qu

ency

Parameter value

Parameter value

Allocation

C-pools

N-pools

[Levy et al, 2004]

1.4 Forest models and uncertainty1.4 Forest models and uncertainty1.4 Forest models and uncertainty1.4 Forest models and uncertainty

Jmax

-100 0 100 200 300 400 500

Fre

quen

cy

0.00

0.04

0.08

0.12

0.16

Vmax

-50 0 50 100 150 200 250 300

0.00

0.05

0.10

0.15

0.20

0.25

umax,root

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.00

0.05

0.10

0.15

0.20

0.25

0.30

froot

-0.5 0.0 0.5 1.0 1.5

0.00

0.05

0.10

0.15

0.20

0.25

Initial Csoluble

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

0.00

0.05

0.10

0.15

0.20

Initial Cstarch

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

0.00

0.05

0.10

0.15

0.20

Initial Wtotal

Value

-0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

0.0

0.1

0.2

0.3

0.4

0.5

Initial Nsoluble

Value

-0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

0.00

0.05

0.10

0.15

0.20

Photosynthesis

Fre

qu

ency

Parameter value

Parameter value

Allocation

C-pools

N-pools

Jmax

-100 0 100 200 300 400 500

Fre

quen

cy

0.00

0.04

0.08

0.12

0.16

Vmax

-50 0 50 100 150 200 250 300

0.00

0.05

0.10

0.15

0.20

0.25

umax,root

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30

0.00

0.05

0.10

0.15

0.20

0.25

0.30

froot

-0.5 0.0 0.5 1.0 1.5

0.00

0.05

0.10

0.15

0.20

0.25

Initial Csoluble

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

0.00

0.05

0.10

0.15

0.20

Initial Cstarch

-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

0.00

0.05

0.10

0.15

0.20

Initial Wtotal

Value

-0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

0.0

0.1

0.2

0.3

0.4

0.5

Initial Nsoluble

Value

-0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

0.00

0.05

0.10

0.15

0.20

Photosynthesis

Fre

qu

ency

Parameter value

Parameter value

Allocation

C-pools

N-pools

bgc

century

hybrid

bgc

0.0

0.1

0.2

0.3

0.4

century

Freq

uenc

y

0.0

0.1

0.2

0.3

0.4

hybrid

-40 -20 0 20 40 60 80

0.0

0.1

0.2

0.3

0.4

Ctotal / Ndepositedkg C (kg N)-1NdepUE (kg C kg-1 N)

[Levy et al, 2004]

1.5 Model-data fusion1.5 Model-data fusion1.5 Model-data fusion1.5 Model-data fusion

Uncertainties are everywhere: Models (environmental inputs, parameters, structure), Data

Uncertainties can be expressed as probability distributions (pdf’s)

We need methods that:• Quantify all uncertainties• Show how to reduce them• Efficiently transfer information: data

models model application

Calculating with uncertainties (pdf’s) = Probability Theory

2. Bayes’ Theorem2. Bayes’ Theorem2. Bayes’ Theorem2. Bayes’ Theorem

2.1 Dealing with uncertainty: Medical diagnostics2.1 Dealing with uncertainty: Medical diagnostics2.1 Dealing with uncertainty: Medical diagnostics2.1 Dealing with uncertainty: Medical diagnostics

A flu epidemic occurs: one percent of people is ill

Diagnostic test, 99% reliable

Test result is positive (bad news!)What is P(diseased|test positive)?

(a) 0.50(b) 0.98(c) 0.99

P(dis) = 0.01

P(pos|hlth) = 0.01

P(pos|dis) = 0.99

P(dis|pos) = P(pos|dis) P(dis) / P(pos)

Bayes’ Theorem

2.2 Bayesian updating of probabilities2.2 Bayesian updating of probabilities2.2 Bayesian updating of probabilities2.2 Bayesian updating of probabilities

Model parameterization: P(params) → P(params|data)Model selection: P(models) → P(model|data)

SPAM-killer: P(SPAM) → P(SPAM|E-mail header)Weather forecasting: …Climate change prediction: …Oil field discovery: …GHG-emission estimation: …Jurisprudence:… …

Bayes’ Theorem: Prior probability → Posterior prob.

Medical diagnostics: P(disease) → P(disease|test result)

2.2 Bayesian updating of probabilities2.2 Bayesian updating of probabilities2.2 Bayesian updating of probabilities2.2 Bayesian updating of probabilities

Model parameterization: P(params) → P(params|data)Model selection: P(models) → P(model|data)

Bayes’ Theorem: Prior probability → Posterior prob.

Application of Bayes’ Theorem to process-based models (not analytically solvable):

Markov Chain Monte-Carlo (Metropolis algorithm)

2.3 What and why?2.3 What and why?2.3 What and why?2.3 What and why?

• We want to use data and models to explain and predict ecosystem behaviour

• Data as well as model inputs, parameters and outputs are uncertain

• No prediction is complete without quantifying the uncertainty. No explanation is complete without analysing the uncertainty

• Uncertainties can be expressed as probability density functions (pdf’s)

• Probability theory tells us how to work with pdf’s: Bayes Theorem (BT) tells us how a pdf changes when new information arrives

• BT: Prior pdf Posterior pdf

• BT: Posterior = Prior x Likelihood / Evidence

• BT: P(θ|D) = P(θ) P(D|θ) / P(D)

• BT: P(θ|D) P(θ) P(D|θ)

3. Bayesian Calibration (BC)3. Bayesian Calibration (BC)of process-based modelsof process-based models

3. Bayesian Calibration (BC)3. Bayesian Calibration (BC)of process-based modelsof process-based models

3.1 Process-based forest models3.1 Process-based forest models3.1 Process-based forest models3.1 Process-based forest models

Soil

Trees

H2OC

Atmosphere

H2O

H2OC

Nutr.


H2OC

Nutr.

Nutr.

Nutr.

Soil C

NPP

HeightEnvironmental scenarios

Initial values

Parameters

Model

3.2 Process-based forest model BASFOR3.2 Process-based forest model BASFOR3.2 Process-based forest model BASFOR3.2 Process-based forest model BASFOR

Soil

Trees

H2OC

Atmosphere

H2O

H2OC

Nutr.


H2OC

Nutr.

Nutr.

Nutr.

BASFOR

40+ parameters 12+ output variables

3.3 BASFOR: outputs3.3 BASFOR: outputs3.3 BASFOR: outputs3.3 BASFOR: outputs

0 0.5 1 1.5 2 2.5 3

x 104

0

200

400

600

Vo

lTo

t

0 0.5 1 1.5 2 2.5 3

x 104

0

100

200

300

Vo

l

Model "basforc9"

0 0.5 1 1.5 2 2.5 3

x 104

0

5

10

15

Ctr

ee

To

t

0 0.5 1 1.5 2 2.5 3

x 104

0

2

4

6

8

Ctr

ee

0 0.5 1 1.5 2 2.5 3

x 104

0

2

4

6

Cs

tem

0 0.5 1 1.5 2 2.5 3

x 104

0

0.5

1

1.5

2

Cb

ran

ch

0 0.5 1 1.5 2 2.5 3

x 104

0

0.05

0.1

0.15

0.2

Cle

af

0 0.5 1 1.5 2 2.5 3

x 104

0

0.5

1

1.5

Cro

ot

0 0.5 1 1.5 2 2.5 3

x 104

0

5

10

15

20

h

0 0.5 1 1.5 2 2.5 3

x 104

0

0.5

1

1.5

2

LA

I

Time0 0.5 1 1.5 2 2.5 3

x 104

8

10

12

14

Cs

oil

Time0 0.5 1 1.5 2 2.5 3

x 104

0.35

0.4

0.45

Ns

oil

Time

Volume(standing)

Carbon in trees(standing + thinned)

Carbon in soil

3.4 BASFOR: parameter uncertainty3.4 BASFOR: parameter uncertainty3.4 BASFOR: parameter uncertainty3.4 BASFOR: parameter uncertainty

0 5

x 10-3

0

2000

4000

CB0T0 0.005 0.01

0

2000

4000

CL0T0 0.005 0.01

0

2000

4000

CR0T

Prior parameter marginal probability distributions (beta)

0 5

x 10-3

0

2000

4000

CS0T0.4 0.6 0.80

1000

2000

BETA300 350 4000

1000

2000

CO20

0.25 0.3 0.350

1000

2000

FB0.25 0.3 0.350

1000

2000

FLMAX0.25 0.3 0.350

1000

2000

FS0.4 0.6 0.80

1000

2000

GAMMA5 10 15

0

1000

2000

KCA0.35 0.4 0.45

0

1000

2000

KCAEXP

0 2 4

x 10-4

0

1000

2000

KDBT0 0.5 1

x 10-3

0

1000

2000

KDRT0 10 20

0

2000

4000

KH0.2 0.3 0.40

1000

2000

KHEXP0 1 2

x 10-3

0

1000

2000

KNMINT0 1 2

x 10-3

0

1000

2000

KNUPTT

0.02 0.03 0.040

1000

2000

KTA10 20 30

0

1000

2000

KTB0 0.5 1

0

1000

2000

KEXTT4 6 8

0

2000

4000

LAIMAXT1 2 3

x 10-3

0

1000

2000

LUET0.01 0.02 0.030

1000

2000

NCLMINT

0.02 0.04 0.060

1000

2000

NCLMAXT0.02 0.03 0.040

1000

2000

NCRT0 1 2

x 10-3

0

1000

2000

NCWT0 20 40

0

2000

4000

SLAT4 6 8

0

1000

2000

TRANCOT150 200 2500

1000

2000

WOODDENS

0 0.5 10

1000

2000

CLITT06 8 10

0

1000

2000

CSOMF01 2 3

0

1000

2000

CSOMS00 0.01 0.02

0

1000

2000

NLITT00.2 0.3 0.40

1000

2000

NSOMF00 0.1 0.2

0

1000

2000

NSOMS0

0 1 2

x 10-3

0

1000

2000

NMIN00.4 0.6 0.80

1000

2000

FLITTSOMF0 0.05 0.1

0

2000

4000

FSOMFSOMS0 2 4

x 10-3

0

1000

2000

KDLITT0 1 2

x 10-4

0

1000

2000

KDSOMF0 1 2

x 10-5

0

1000

2000

KDSOMS

3.5 BASFOR: prior output uncertainty3.5 BASFOR: prior output uncertainty3.5 BASFOR: prior output uncertainty3.5 BASFOR: prior output uncertainty

0 1 2 3

x 104

0

500

1000V

olT

ot

(m3

ha

-1)

0 1 2 3

x 104

0

500

1000

Vo

l (m

3 h

a-1

)

0 1 2 3

x 104

0

10

20

30

Ctr

ee

To

t (k

g m

-2)

0 1 2 3

x 104

0

10

20

30

Ctr

ee

(k

g m

-2)

0 1 2 3

x 104

0

5

10

Cs

tem

(k

g m

-2)

0 1 2 3

x 104

0

1

2

Cb

ran

ch

(k

g m

-2)

0 1 2 3

x 104

0

0.5

1

Cle

af

(kg

m-2

)

0 1 2 3

x 104

0

2

4

Cro

ot

(kg

m-2

)

0 1 2 3

x 104

0

10

20

30

h (

m)

0 1 2 3

x 104

0

2

4

LA

I (m

2 m

-2)

Time0 1 2 3

x 104

0

5

10

15

Cs

oil

(kg

m-2

)

Time0 1 2 3

x 104

0

0.2

0.4

0.6

Ns

oil

(kg

m-2

)

Time

Volume(standing)


Carbon in soil

3.6 Data Dodd Wood (R. Matthews, Forest 3.6 Data Dodd Wood (R. Matthews, Forest Research)Research)

3.6 Data Dodd Wood (R. Matthews, Forest 3.6 Data Dodd Wood (R. Matthews, Forest Research)Research)

0 1 2 3

x 104

0

500

1000V

olT

ot

(m3

ha-

1)

0 1 2 3

x 104

0

500

1000

Vo

l (m

3 h

a-1

)

0 1 2 3

x 104

0

10

20

30

Ctr

eeT

ot

(kg

m-2

)

0 1 2 3

x 104

0

10

20

30

Ctr

ee

(kg

m-2

)

0 1 2 3

x 104

0

5

10

Cs

tem

(kg

m-2

)0 1 2 3

x 104

0

1

2

Cb

ran

ch (

kg m

-2)

0 1 2 3

x 104

0

0.5

1

Cle

af (

kg m

-2)

0 1 2 3

x 104

0

2

4

Cro

ot (

kg

m-2

)

0 1 2 3

x 104

0

10

20

30

h (m

)

0 1 2 3

x 104

0

2

4

LAI

(m2

m-2

)

Time0 1 2 3

x 104

0

5

10

15

Cs

oil

(kg

m-2

)

Time0 1 2 3

x 104

0

0.2

0.4

0.6

Ns

oil

(kg

m-2

)

Time

Volume(standing)


Carbon in soil

Dodd WoodDodd Wood

3.7 Using data in Bayesian calibration of BASFOR3.7 Using data in Bayesian calibration of BASFOR3.7 Using data in Bayesian calibration of BASFOR3.7 Using data in Bayesian calibration of BASFOR

0 1 2 3

x 104

0

500

1000

Vo

lTo

t (m

3 h

a-1

)

0 1 2 3

x 104

0

500

1000

Vo

l (m

3 h

a-1

)

0 1 2 3

x 104

0

10

20

30

Ctr

ee

To

t (k

g m

-2)

0 1 2 3

x 104

0

10

20

30

Ctr

ee

(k

g m

-2)

0 1 2 3

x 104

0

5

10

Cs

tem

(k

g m

-2)

0 1 2 3

x 104

0

1

2

Cb

ran

ch

(k

g m

-2)

0 1 2 3

x 104

0

0.5

1

Cle

af

(kg

m-2

)

0 1 2 3

x 104

0

2

4

Cro

ot

(kg

m-2

)

0 1 2 3

x 104

0

10

20

30

h (

m)

0 1 2 3

x 104

0

2

4

LA

I (m

2 m

-2)

Time0 1 2 3

x 104

0

5

10

15

Cs

oil

(kg

m-2

)

Time0 1 2 3

x 104

0

0.2

0.4

0.6

Ns

oil

(kg

m-2

)

Time

0 0.5 1

x 10-3

0

1000

2000

CB0T0 1 2

x 10-3

0

1000

2000

CL0T0 2 4

x 10-3

0

1000

2000

CR0T

Parameter marginal probability distributions

0 1 2

x 10-3

0

500

1000

CS0T0.4 0.6 0.80

500

1000

BETA300 350 4000

500

1000

CO20

0.25 0.3 0.350

500

1000

FB0.25 0.3 0.350

1000

2000

FLMAX0.25 0.3 0.350

1000

2000

FS0.4 0.6 0.80

500

1000

GAMMA5 10 15

0

500

1000

KCA0.35 0.4 0.45

0

500

1000

KCAEXP

0 2 4

x 10-4

0

500

1000

KDBT0 5

x 10-4

0

1000

2000

KDRT0 5 10

0

500

1000

KH0.2 0.3 0.40

500

1000

KHEXP0 1 2

x 10-3

0

500

1000

KNMINT0 1 2

x 10-3

0

500

1000

KNUPTT

0.02 0.03 0.040

500

1000

KTA10 20 30

0

500

1000

KTB0 0.5 1

0

1000

2000

KEXTT4 6 8

0

1000

2000

LAIMAXT1 2 3

x 10-3

0

500

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Prior pdf

Posterior pdf

DataBayesiancalibration

Dodd WoodDodd Wood

3.8 Bayesian calibration: posterior 3.8 Bayesian calibration: posterior uncertaintyuncertainty

3.8 Bayesian calibration: posterior 3.8 Bayesian calibration: posterior uncertaintyuncertainty

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Carbon in soil

3.9 How does BC work again?3.9 How does BC work again?3.9 How does BC work again?3.9 How does BC work again?

P(|D) = P() P(D| ) / P(D) P() P(D|f())

“Posterior distribution of parameters”

“Prior distribution of parameters”

“Likelihood” of data, given mismatch with

model output

f = the model, e.g. BASFOR

Bayesian calibration in action!Bayesian calibration in action!Bayesian calibration in action!Bayesian calibration in action!

OutputParameter prob. distr.

Bayes’ Theorem:P( |D) P() P(D|(f())

Data

3.10 Calculating the posterior using MCMC3.10 Calculating the posterior using MCMC3.10 Calculating the posterior using MCMC3.10 Calculating the posterior using MCMC

Sample of 104 -105 parameter vectors from the posterior distribution P(|D) for the parameters

P(|D) P() P(D|f())

1. Start anywhere in parameter-space: p1..39(i=0)

2. Randomly choose p(i+1) = p(i) + δ

3. IF: [ P(p(i+1)) P(D|f(p(i+1))) ] / [ P(p(i)) P(D|f(p(i))) ] > Random[0,1]THEN: accept p(i+1)ELSE: reject p(i+1)i=i+1

4. IF i < 104 GOTO 2 Metropolis et al (1953)

0 5000 10000

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330340350360370 CO20

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MCMC trace plots

BC3D.AVI

3.11 MCMC in action3.11 MCMC in action3.11 MCMC in action3.11 MCMC in action

C:\users\mvano\BAYESIAN MODELLING & Uncertainty\MCMC in MATLAB - EXAMPLES (BASFOR,EXPOLINEAR)\demo_EXPOLINEAR\expol3dm.mdl

3.12 Using data in Bayesian calibration of BASFOR3.12 Using data in Bayesian calibration of BASFOR3.12 Using data in Bayesian calibration of BASFOR3.12 Using data in Bayesian calibration of BASFOR

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Prior pdf


Posterior pdf

Dodd WoodDodd Wood

3.13 Parameter correlations3.13 Parameter correlations3.13 Parameter correlations3.13 Parameter correlations

CL

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CL0 1.00 0.60 -0.67 -0.58 0.25 -0.16 0.51 0.46 0.26 0.12 0.64 0.59 0.38 -0.42 -0.07 0.71 -0.28 0.17 -0.64 -0.32 -0.58 0.23 0.55 0.52 0.12 0.50 -0.58 0.10 0.50 -0.66 -0.57 0.55 0.62

CR0 0.60 1.00 -0.49 -0.54 0.17 0.40 0.01 0.24 0.51 0.56 0.49 0.96 -0.19 -0.09 0.06 0.55 0.07 0.83 -0.60 -0.81 -0.21 -0.17 0.61 0.67 0.20 0.65 -0.54 -0.05 0.33 -0.29 0.05 0.46 0.61

CW0 -0.67 -0.49 1.00 0.91 0.24 0.45 -0.70 -0.82 -0.23 0.03 -0.74 -0.57 -0.74 0.77 -0.31 -0.98 0.76 -0.10 0.85 0.14 0.78 -0.61 -0.84 -0.91 0.51 -0.81 0.77 -0.30 -0.38 0.84 0.33 -0.88 -0.90

BETA -0.58 -0.54 0.91 1.00 0.30 0.42 -0.78 -0.79 -0.46 -0.08 -0.79 -0.61 -0.66 0.81 0.04 -0.95 0.60 -0.32 0.94 0.17 0.61 -0.59 -0.98 -0.95 0.29 -0.94 0.84 0.01 -0.46 0.83 -0.01 -0.94 -0.96

CO20 0.25 0.17 0.24 0.30 1.00 0.05 -0.26 -0.41 -0.33 -0.28 0.11 0.09 -0.35 0.67 -0.02 -0.21 0.62 0.00 0.37 0.06 -0.22 -0.76 -0.33 -0.37 0.15 -0.19 0.57 -0.33 -0.34 -0.02 -0.28 -0.54 -0.36

FLMAX -0.16 0.40 0.45 0.42 0.05 1.00 -0.69 -0.62 0.43 0.82 -0.56 0.25 -0.87 0.54 -0.05 -0.40 0.59 0.64 0.19 -0.81 0.74 -0.49 -0.31 -0.18 0.61 -0.33 0.06 -0.14 0.21 0.75 0.36 -0.35 -0.21

FW 0.51 0.01 -0.70 -0.78 -0.26 -0.69 1.00 0.61 0.32 -0.18 0.56 0.05 0.86 -0.83 -0.28 0.77 -0.60 -0.16 -0.75 0.26 -0.55 0.76 0.68 0.58 -0.25 0.58 -0.63 -0.17 0.54 -0.77 -0.13 0.72 0.72

GAMMA 0.46 0.24 -0.82 -0.79 -0.41 -0.62 0.61 1.00 -0.05 -0.28 0.82 0.45 0.78 -0.82 0.19 0.75 -0.81 -0.06 -0.64 0.14 -0.72 0.63 0.80 0.73 -0.46 0.78 -0.65 0.49 0.06 -0.85 -0.31 0.87 0.67

KCA 0.26 0.51 -0.23 -0.46 -0.33 0.43 0.32 -0.05 1.00 0.84 -0.01 0.38 -0.10 -0.34 -0.49 0.39 0.07 0.72 -0.68 -0.69 0.35 0.30 0.49 0.51 0.47 0.37 -0.69 -0.49 0.86 0.05 0.54 0.45 0.62

KCAEXP 0.12 0.56 0.03 -0.08 -0.28 0.82 -0.18 -0.28 0.84 1.00 -0.30 0.41 -0.48 0.00 -0.24 0.07 0.24 0.76 -0.36 -0.91 0.59 0.01 0.16 0.27 0.59 0.06 -0.48 -0.22 0.68 0.42 0.44 0.16 0.32

KDL 0.64 0.49 -0.74 -0.79 0.11 -0.56 0.56 0.82 -0.01 -0.30 1.00 0.64 0.56 -0.53 -0.03 0.73 -0.39 0.17 -0.61 0.07 -0.81 0.21 0.81 0.67 -0.25 0.88 -0.48 0.10 -0.02 -0.93 -0.25 0.70 0.63

KDR 0.59 0.96 -0.57 -0.61 0.09 0.25 0.05 0.45 0.38 0.41 0.64 1.00 -0.06 -0.20 0.12 0.59 -0.07 0.75 -0.61 -0.69 -0.34 -0.10 0.70 0.72 0.09 0.75 -0.57 0.10 0.19 -0.42 -0.01 0.57 0.63

KDW 0.38 -0.19 -0.74 -0.66 -0.35 -0.87 0.86 0.78 -0.10 -0.48 0.56 -0.06 1.00 -0.84 0.12 0.70 -0.86 -0.49 -0.54 0.49 -0.73 0.81 0.54 0.50 -0.60 0.47 -0.48 0.29 0.21 -0.81 -0.41 0.67 0.56

KH -0.42 -0.09 0.77 0.81 0.67 0.54 -0.83 -0.82 -0.34 0.00 -0.53 -0.20 -0.84 1.00 0.07 -0.78 0.85 0.08 0.80 -0.07 0.44 -0.93 -0.77 -0.73 0.30 -0.64 0.84 -0.25 -0.52 0.68 0.12 -0.92 -0.79

KHEXP -0.07 0.06 -0.31 0.04 -0.02 -0.05 -0.28 0.19 -0.49 -0.24 -0.03 0.12 0.12 0.07 1.00 0.14 -0.43 -0.26 0.14 0.00 -0.40 -0.01 -0.12 0.15 -0.76 -0.05 0.12 0.72 -0.37 -0.05 -0.47 -0.02 0.00

KLAIMAX 0.71 0.55 -0.98 -0.95 -0.21 -0.40 0.77 0.75 0.39 0.07 0.73 0.59 0.70 -0.78 0.14 1.00 -0.67 0.21 -0.93 -0.21 -0.70 0.60 0.88 0.93 -0.38 0.83 -0.82 0.11 0.51 -0.83 -0.22 0.89 0.96

KNMIN -0.28 0.07 0.76 0.60 0.62 0.59 -0.60 -0.81 0.07 0.24 -0.39 -0.07 -0.86 0.85 -0.43 -0.67 1.00 0.38 0.53 -0.22 0.58 -0.86 -0.52 -0.59 0.66 -0.42 0.60 -0.63 -0.22 0.61 0.42 -0.73 -0.58

KNUPT 0.17 0.83 -0.10 -0.32 0.00 0.64 -0.16 -0.06 0.72 0.76 0.17 0.75 -0.49 0.08 -0.26 0.21 0.38 1.00 -0.43 -0.83 0.28 -0.27 0.45 0.46 0.47 0.48 -0.41 -0.38 0.33 0.10 0.58 0.26 0.41

KTA -0.64 -0.60 0.85 0.94 0.37 0.19 -0.75 -0.64 -0.68 -0.36 -0.61 -0.61 -0.54 0.80 0.14 -0.93 0.53 -0.43 1.00 0.39 0.40 -0.64 -0.92 -0.93 0.08 -0.83 0.94 0.07 -0.71 0.66 -0.05 -0.92 -0.99

KTB -0.32 -0.81 0.14 0.17 0.06 -0.81 0.26 0.14 -0.69 -0.91 0.07 -0.69 0.49 -0.07 0.00 -0.21 -0.22 -0.83 0.39 1.00 -0.33 0.16 -0.25 -0.39 -0.46 -0.21 0.47 0.05 -0.52 -0.25 -0.22 -0.21 -0.38

KTREE -0.58 -0.21 0.78 0.61 -0.22 0.74 -0.55 -0.72 0.35 0.59 -0.81 -0.34 -0.73 0.44 -0.40 -0.70 0.58 0.28 0.40 -0.33 1.00 -0.26 -0.52 -0.51 0.66 -0.58 0.24 -0.32 0.15 0.91 0.60 -0.50 -0.48

LUE0 0.23 -0.17 -0.61 -0.59 -0.76 -0.49 0.76 0.63 0.30 0.01 0.21 -0.10 0.81 -0.93 -0.01 0.60 -0.86 -0.27 -0.64 0.16 -0.26 1.00 0.52 0.53 -0.33 0.35 -0.72 0.28 0.56 -0.45 -0.13 0.73 0.62

NLCONMIN 0.55 0.61 -0.84 -0.98 -0.33 -0.31 0.68 0.80 0.49 0.16 0.81 0.70 0.54 -0.77 -0.12 0.88 -0.52 0.45 -0.92 -0.25 -0.52 0.52 1.00 0.94 -0.16 0.97 -0.85 0.00 0.41 -0.77 0.10 0.95 0.92

NLCONMAX 0.52 0.67 -0.91 -0.95 -0.37 -0.18 0.58 0.73 0.51 0.27 0.67 0.72 0.50 -0.73 0.15 0.93 -0.59 0.46 -0.93 -0.39 -0.51 0.53 0.94 1.00 -0.32 0.91 -0.87 0.11 0.46 -0.67 0.05 0.92 0.96

NRCON 0.12 0.20 0.51 0.29 0.15 0.61 -0.25 -0.46 0.47 0.59 -0.25 0.09 -0.60 0.30 -0.76 -0.38 0.66 0.47 0.08 -0.46 0.66 -0.33 -0.16 -0.32 1.00 -0.22 -0.01 -0.46 0.34 0.44 0.31 -0.23 -0.21

NWCON 0.50 0.65 -0.81 -0.94 -0.19 -0.33 0.58 0.78 0.37 0.06 0.88 0.75 0.47 -0.64 -0.05 0.83 -0.42 0.48 -0.83 -0.21 -0.58 0.35 0.97 0.91 -0.22 1.00 -0.72 -0.03 0.23 -0.79 0.12 0.86 0.85

SLA -0.58 -0.54 0.77 0.84 0.57 0.06 -0.63 -0.65 -0.69 -0.48 -0.48 -0.57 -0.48 0.84 0.12 -0.82 0.60 -0.41 0.94 0.47 0.24 -0.72 -0.85 -0.87 -0.01 -0.72 1.00 -0.13 -0.75 0.51 -0.03 -0.93 -0.92

CLITT0 0.10 -0.05 -0.30 0.01 -0.33 -0.14 -0.17 0.49 -0.49 -0.22 0.10 0.10 0.29 -0.25 0.72 0.11 -0.63 -0.38 0.07 0.05 -0.32 0.28 0.00 0.11 -0.46 -0.03 -0.13 1.00 -0.25 -0.15 -0.64 0.22 0.00

CSOMF0 0.50 0.33 -0.38 -0.46 -0.34 0.21 0.54 0.06 0.86 0.68 -0.02 0.19 0.21 -0.52 -0.37 0.51 -0.22 0.33 -0.71 -0.52 0.15 0.56 0.41 0.46 0.34 0.23 -0.75 -0.25 1.00 -0.10 0.09 0.50 0.65

CSOMS0 -0.66 -0.29 0.84 0.83 -0.02 0.75 -0.77 -0.85 0.05 0.42 -0.93 -0.42 -0.81 0.68 -0.05 -0.83 0.61 0.10 0.66 -0.25 0.91 -0.45 -0.77 -0.67 0.44 -0.79 0.51 -0.15 -0.10 1.00 0.39 -0.74 -0.68

NLITT0 -0.57 0.05 0.33 -0.01 -0.28 0.36 -0.13 -0.31 0.54 0.44 -0.25 -0.01 -0.41 0.12 -0.47 -0.22 0.42 0.58 -0.05 -0.22 0.60 -0.13 0.10 0.05 0.31 0.12 -0.03 -0.64 0.09 0.39 1.00 -0.05 0.01

NSOMF0 0.55 0.46 -0.88 -0.94 -0.54 -0.35 0.72 0.87 0.45 0.16 0.70 0.57 0.67 -0.92 -0.02 0.89 -0.73 0.26 -0.92 -0.21 -0.50 0.73 0.95 0.92 -0.23 0.86 -0.93 0.22 0.50 -0.74 -0.05 1.00 0.92

NSOMS0 0.62 0.61 -0.90 -0.96 -0.36 -0.21 0.72 0.67 0.62 0.32 0.63 0.63 0.56 -0.79 0.00 0.96 -0.58 0.41 -0.99 -0.38 -0.48 0.62 0.92 0.96 -0.21 0.85 -0.92 0.00 0.65 -0.68 0.01 0.92 1.00

NMIN0 -0.16 -0.31 -0.47 -0.41 -0.64 -0.43 0.56 0.33 0.16 -0.09 -0.06 -0.30 0.66 -0.63 0.29 0.45 -0.72 -0.33 -0.40 0.25 -0.21 0.79 0.27 0.41 -0.66 0.16 -0.39 0.14 0.33 -0.23 0.06 0.42 0.45

FLITTSOMF 0.48 0.60 -0.01 0.08 0.61 0.31 -0.43 0.03 -0.22 0.05 0.36 0.63 -0.39 0.40 0.15 -0.02 0.34 0.33 0.12 -0.39 -0.22 -0.62 0.01 -0.02 0.29 0.13 0.12 0.23 -0.28 -0.11 -0.40 -0.10 -0.10

FSOMFSOMS -0.66 -0.28 0.86 0.83 0.08 0.55 -0.89 -0.56 -0.33 0.08 -0.58 -0.27 -0.78 0.72 -0.04 -0.91 0.61 0.04 0.81 -0.03 0.69 -0.63 -0.69 -0.72 0.41 -0.62 0.65 0.07 -0.55 0.78 0.27 -0.70 -0.83

KDLITT 0.42 0.28 -0.93 -0.89 -0.55 -0.51 0.73 0.87 0.25 -0.04 0.62 0.39 0.81 -0.91 0.26 0.90 -0.88 0.02 -0.83 -0.01 -0.63 0.80 0.84 0.88 -0.56 0.77 -0.80 0.34 0.37 -0.75 -0.16 0.92 0.87

KDSOMF 0.15 -0.43 -0.39 -0.31 -0.08 -0.70 0.75 0.19 -0.03 -0.42 0.09 -0.46 0.75 -0.46 0.03 0.41 -0.49 -0.59 -0.27 0.55 -0.45 0.60 0.12 0.14 -0.51 0.04 -0.13 -0.14 0.29 -0.43 -0.25 0.20 0.29

KDSOMS -0.55 -0.18 0.83 0.81 0.13 0.80 -0.75 -0.92 0.12 0.47 -0.89 -0.35 -0.86 0.75 -0.12 -0.79 0.72 0.18 0.62 -0.32 0.89 -0.54 -0.76 -0.66 0.52 -0.77 0.51 -0.28 -0.03 0.98 0.39 -0.77 -0.65

39 parameters3

9 p

ara

me

ters

3.14 Continued calibration when new data become 3.14 Continued calibration when new data become availableavailable




0 5

x 10-3

0

2000

4000

CB0T0 0.005 0.01

0

2000

4000

CL0T0 0.005 0.01

0

2000

4000

CR0T

Prior param e te r m arginal probability distributions (be ta)

0 5

x 10-3

0

2000

4000

CS0T0.4 0.6 0.80

1000

2000

B ETA300 350 4000

1000

2000

CO20

0.25 0.3 0.350

1000

2000

FB0.25 0.3 0.350

1000

2000

FLM AX0.25 0.3 0.350

1000

2000

FS0.4 0.6 0.80

1000

2000

GAM M A5 10 15

0

1000

2000

KCA0.35 0.4 0.45

0

1000

2000

K CA EXP

0 2 4

x 10-4

0

1000

2000

K DB T0 0.5 1

x 10-3

0

1000

2000

K DRT0 10 20

0

2000

4000

K H0.2 0.3 0.40

1000

2000

K HE XP0 1 2

x 10-3

0

1000

2000

K NM INT0 1 2

x 10-3

0

1000

2000

K NUPTT

0.02 0.03 0.040

1000

2000

K TA10 20 30

0

1000

2000

K TB0 0.5 1

0

1000

2000

K EXTT4 6 8

0

2000

4000

LA IM AXT1 2 3

x 10-3

0

1000

2000

LUET0.01 0.02 0.030

1000

2000

NCLM INT

0.02 0.04 0.060

1000

2000

NCLM AXT0.02 0.03 0.040

1000

2000

NCRT0 1 2

x 10-3

0

1000

2000

NCW T0 20 40

0

2000

4000

S LA T4 6 8

0

1000

2000

TRA NCOT150 200 2500

1000

2000

W OODDENS

0 0.5 10

1000

2000

CLITT06 8 10

0

1000

2000

CSOM F01 2 3

0

1000

2000

CS OMS 00 0.01 0.02

0

1000

2000

NLITT00.2 0.3 0.40

1000

2000

NSOM F00 0.1 0.2

0

1000

2000

NSOM S 0

0 1 2

x 10-3

0

1000

2000

NMIN00.4 0.6 0.80

1000

2000

FLITTS OM F0 0.05 0.1

0

2000

4000

FS OMFS OM S0 2 4

x 10-3

0

1000

2000

K DLITT0 1 2

x 10-4

0

1000

2000

KDS OM F0 1 2

x 10-5

0

1000

2000

KDS OM S

0 1 2 3

x 104

0

500

1000

Vo

lT

ot (m

3 h

a-1

)

0 1 2 3

x 104

0

500

1000

Vo

l (m

3 h

a-1

)

0 1 2 3

x 104

0

10

20

30

Ctre

eT

ot (k

g m

-2

)

0 1 2 3

x 104

0

10

20

30

Ctre

e (k

g m

-2

)

0 1 2 3

x 104

0

5

10

Cs

te

m (k

g m

-2

)

0 1 2 3

x 104

0

1

2

Cb

ra

nc

h (k

g m

-2

)

0 1 2 3

x 104

0

0.5

1

Cle

af (k

g m

-2

)

0 1 2 3

x 104

0

2

4

Cro

ot (k

g m

-2

)

0 1 2 3

x 104

0

10

20

30

h (m

)

0 1 2 3

x 104

0

2

4

LA

I (m

2 m

-2

)

Time0 1 2 3

x 104

0

5

10

15

Cs

oil

(k

g m

-2

)

Time0 1 2 3

x 104

0

0.2

0.4

0.6

Ns

oil (k

g m

-2

)

Time

0 0.5 1

x 10-3

0

1000

2000

CB 0T0 1 2

x 10-3

0

1000

2000

CL0T0 2 4

x 10-3

0

1000

2000

CR0T

Param e ter m arginal probability distributions

0 1 2

x 10-3

0

500

1000

CS 0T0.4 0.6 0.80

500

1000

BE TA300 350 4000

500

1000

CO20

0.25 0.3 0.350

500

1000

FB0.25 0.3 0.350

1000

2000

FLMAX0.25 0.3 0.350

1000

2000

FS0.4 0.6 0.80

500

1000

GAM MA5 10 15

0

500

1000

KCA0.35 0.4 0.45

0

500

1000

K CAE XP

0 2 4

x 10-4

0

500

1000

KDBT0 5

x 10-4

0

1000

2000

KDRT0 5 10

0

500

1000

KH0.2 0.3 0.40

500

1000

KHEXP0 1 2

x 10-3

0

500

1000

KNM INT0 1 2

x 10-3

0

500

1000

K NUPTT

0.02 0.03 0.040

500

1000

K TA10 20 30

0

500

1000

K TB0 0.5 1

0

1000

2000

K EXTT4 6 8

0

1000

2000

LAIMA XT1 2 3

x 10-3

0

500

1000

LUET0.01 0.02 0.030

1000

2000

NCLM INT

0.02 0.04 0.060

500

1000

NCLMA XT0.02 0.03 0.040

1000

2000

NCRT0.5 1 1.5

x 10-3

0

500

1000

NCW T6 8 10

0

1000

2000

S LAT4 6 8

0

500

1000

TRANCOT150 200 2500

1000

2000

W OODDENS

0 0.5 10

500

1000

CLITT06 8 10

0

500

1000

CSOM F01 2 3

0

500

1000

CSOMS00 0.01 0.02

0

500

1000

NLITT00.2 0.3 0.40

1000

2000

NS OM F00 0.1 0.2

0

500

1000

NS OM S0

0 1 2

x 10-3

0

1000

2000

NMIN00.4 0.6 0.80

500

1000

FLITTSOMF0 0.05 0.1

0

500

1000

FS OM FSOMS0 2 4

x 10-3

0

1000

2000

KDLITT0 1 2

x 10-4

0

500

1000

KDSOM F0 1 2

x 10-5

0

500

1000

KDSOM S

0 1 2 3

x 104

0

500

1000

Vo

lTo

t (m

3 h

a-1

)

0 1 2 3

x 104

0

500

1000

Vo

l (m

3 h

a-1

)

0 1 2 3

x 104

0

10

20

30

Ctre

eT

ot (k

g m

-2

)

0 1 2 3

x 104

0

10

20

30

Ctre

e (k

g m

-2

)

0 1 2 3

x 104

0

5

10

Cs

te

m (k

g m

-2

)

0 1 2 3

x 104

0

1

2

Cb

ra

nc

h (k

g m

-2

)

0 1 2 3

x 104

0

0.5

1

Cle

af (k

g m

-2

)

0 1 2 3

x 104

0

2

4

Cro

ot (k

g m

-2

)

0 1 2 3

x 104

0

10

20

30

h (m

)

0 1 2 3

x 104

0

2

4

LA

I (m

2 m

-2

)

Time0 1 2 3

x 104

0

5

10

15

Cs

oil (k

g m

-2

)

Time0 1 2 3

x 104

0

0.2

0.4

0.6

Ns

oil

(k

g m

-2

)

Time

Prior pdf

Posterior pdf

Bayesiancalibration

Prior pdf

Dodd WoodDodd Wood

Newdata

RheolaRheola





0 5

x 10-3

0

2000

4000

CB0T0 0.005 0.01

0

2000

4000

CL0T0 0.005 0.01

0

2000

4000

CR0T

Prior param eter m arginal probability distributions (beta)

0 5

x 10-3

0

2000

4000

CS0T0.4 0.6 0.80

1000

2000

BETA300 350 4000

1000

2000

CO20

0.25 0.3 0.350

1000

2000

FB0.25 0.3 0.350

1000

2000

FLMAX0.25 0.3 0.350

1000

2000

FS0.4 0.6 0.80

1000

2000

GAMMA5 10 15

0

1000

2000

KCA0.35 0.4 0.45

0

1000

2000

KCAEXP

0 2 4

x 10-4

0

1000

2000

KDBT0 0.5 1

x 10-3

0

1000

2000

KDRT0 10 20

0

2000

4000

KH0.2 0.3 0.40

1000

2000

KHEXP0 1 2

x 10-3

0

1000

2000

KNMINT0 1 2

x 10-3

0

1000

2000

KNUPTT

0.02 0.03 0.040

1000

2000

KTA10 20 30

0

1000

2000

KTB0 0.5 1

0

1000

2000

KEXTT4 6 8

0

2000

4000

LAIMAXT1 2 3

x 10-3

0

1000

2000

LUET0.01 0.02 0.030

1000

2000

NCLMINT

0.02 0.04 0.060

1000

2000

NCLMAXT0.02 0.03 0.040

1000

2000

NCRT0 1 2

x 10-3

0

1000

2000

NCW T0 20 40

0

2000

4000

SLAT4 6 8

0

1000

2000

TRANCOT150 200 2500

1000

2000

W OODDENS

0 0.5 10

1000

2000

CLITT06 8 10

0

1000

2000

CSOMF01 2 3

0

1000

2000

CSOMS00 0.01 0.02

0

1000

2000

NLITT00.2 0.3 0.40

1000

2000

NSOMF00 0.1 0.2

0

1000

2000

NSOMS0

0 1 2

x 10-3

0

1000

2000

NMIN00.4 0.6 0.80

1000

2000

FLITTSOMF0 0.05 0.1

0

2000

4000

FSOMFSOMS0 2 4

x 10-3

0

1000

2000

KDLITT0 1 2

x 10-4

0

1000

2000

KDSOMF0 1 2

x 10-5

0

1000

2000

KDSOMS

0 1 2 3

x 104

0

500

1000

Vo

lT

ot (m

3 h

a-1

)

0 1 2 3

x 104

0

500

1000

Vo

l (m

3 h

a-1

)

0 1 2 3

x 104

0

10

20

30

Ctre

eT

ot (k

g m

-2

)

0 1 2 3

x 104

0

10

20

30

Ctre

e (k

g m

-2

)

0 1 2 3

x 104

0

5

10

Cs

te

m (k

g m

-2

)

0 1 2 3

x 104

0

1

2

Cb

ra

nc

h (k

g m

-2

)

0 1 2 3

x 104

0

0.5

1

Cle

af (k

g m

-2

)

0 1 2 3

x 104

0

2

4

Cro

ot (k

g m

-2

)

0 1 2 3

x 104

0

10

20

30

h (m

)

0 1 2 3

x 104

0

2

4

LA

I (m

2 m

-2

)

Time0 1 2 3

x 104

0

5

10

15

Cs

oil (k

g m

-2

)

Time0 1 2 3

x 104

0

0.2

0.4

0.6

Ns

oil (k

g m

-2

)

Time

0 0.5 1

x 10-3

0

1000

2000

CB0T0 1 2

x 10-3

0

1000

2000

CL0T0 2 4

x 10-3

0

1000

2000

CR0T


0 1 2

x 10-3

0

500

1000

CS0T0.4 0.6 0.80

500

1000

BETA300 350 4000

500

1000

CO20

0.25 0.3 0.350

500

1000

FB0.25 0.3 0.350

1000

2000

FLMAX0.25 0.3 0.350

1000

2000

FS0.4 0.6 0.80

500

1000

GAMMA5 10 15

0

500

1000

KCA0.35 0.4 0.45

0

500

1000

KCAEXP

0 2 4

x 10-4

0

500

1000

KDBT0 5

x 10-4

0

1000

2000

KDRT0 5 10

0

500

1000

KH0.2 0.3 0.40

500

1000

KHEXP0 1 2

x 10-3

0

500

1000

KNMINT0 1 2

x 10-3

0

500

1000

KNUPTT

0.02 0.03 0.040

500

1000

KTA10 20 30

0

500

1000

KTB0 0.5 1

0

1000

2000

KEXTT4 6 8

0

1000

2000

LAIMAXT1 2 3

x 10-3

0

500

1000

LUET0.01 0.02 0.030

1000

2000

NCLMINT

0.02 0.04 0.060

500

1000

NCLMAXT0.02 0.03 0.040

1000

2000

NCRT0.5 1 1.5

x 10-3

0

500

1000

NCW T6 8 10

0

1000

2000

SLAT4 6 8

0

500

1000

TRANCOT150 200 2500

1000

2000

W OODDENS

0 0.5 10

500

1000

CLITT06 8 10

0

500

1000

CSOMF01 2 3

0

500

1000

CSOMS00 0.01 0.02

0

500

1000

NLITT00.2 0.3 0.40

1000

2000

NSOMF00 0.1 0.2

0

500

1000

NSOMS0

0 1 2

x 10-3

0

1000

2000

NMIN00.4 0.6 0.80

500

1000

FLITTSOMF0 0.05 0.1

0

500

1000

FSOMFSOMS0 2 4

x 10-3

0

1000

2000

KDLITT0 1 2

x 10-4

0

500

1000

KDSOMF0 1 2

x 10-5

0

500

1000

KDSOMS

0 1 2 3

x 104

0

500

1000

Vo

lT

ot (m

3 h

a-1

)

0 1 2 3

x 104

0

500

1000

Vo

l (m

3 h

a-1

)

0 1 2 3

x 104

0

10

20

30

Ctre

eT

ot (k

g m

-2

)

0 1 2 3

x 104

0

10

20

30

Ctre

e (k

g m

-2

)

0 1 2 3

x 104

0

5

10

Cs

te

m (k

g m

-2

)

0 1 2 3

x 104

0

1

2

Cb

ra

nc

h (k

g m

-2

)

0 1 2 3

x 104

0

0.5

1

Cle

af (k

g m

-2

)

0 1 2 3

x 104

0

2

4

Cro

ot (k

g m

-2

)

0 1 2 3

x 104

0

10

20

30

h (m

)

0 1 2 3

x 104

0

2

4

LA

I (m

2 m

-2

)

Time0 1 2 3

x 104

0

5

10

15

Cs

oil (k

g m

-2

)

Time0 1 2 3

x 104

0

0.2

0.4

0.6

Ns

oil (k

g m

-2

)

Time

Newdata

Bayesiancalibration

Prior pdf

Posterior pdf

Prior pdf

Dodd WoodDodd Wood

0 0.5 1

x 10-3

0

500

CB0T0 1 2

x 10-3

0

500

CL0T0 5

x 10-3

0500

1000

CR0T

Parameter marginal probability distributions (truncated normal)

0 1 2

x 10-3

0

500

CS0T0.4 0.6 0.80

500

BETA300 350 4000

5001000

CO20

0.25 0.3 0.350

500

FB0.25 0.3 0.350

5001000

FLMAX0.25 0.3 0.350

500

FS0.4 0.6 0.80

500

GAMMA5 10 15

0

500

KCA0.350.4 0.45

0500

1000

KCAEXP

0 2 4

x 10-4

0

500

KDBT0 2 4

x 10-4

0

500

KDRT0 5 10

0

500

KH0.2 0.3 0.40

200400

KHEXP0 1 2

x 10-3

0

500

KNMINT0 1 2

x 10-3

0500

1000

KNUPTT

0.02 0.03 0.040

500

KTA10 20 30

0

500

KTB0 0.5 1

0

500

KEXTT4 6 8

0

500

LAIMAXT1 2 3

x 10-3

0500

1000

LUET0.01 0.02 0.030

5001000

NCLMINT

0.02 0.04 0.060

500

NCLMAXT0.02 0.025 0.030

200400

NCRT0.5 1 1.5

x 10-3

0

500

NCWT6 8 10

0

500

SLAT4 6 8

0

500

TRANCOT150 200 2500

500

WOODDENS

0 0.5 10

500

CLITT06 8 10

0200400

CSOMF01 2 3

0200400

CSOMS00 0.01 0.02

0

500

NLITT00.2 0.3 0.40

500

NSOMF00 0.1 0.2

0

500

NSOMS0

0 1 2

x 10-3

0500

1000

NMIN00.4 0.6 0.80

500

FLITTSOMF0 0.05 0.1

0500

1000

FSOMFSOMS0 2 4

x 10-3

0500

1000

KDLITT0 1 2

x 10-4

0

500

KDSOMF0 1 2

x 10-5

0500

1000

KDSOMS

RheolaRheola

0 0.5 1 1.5 2 2.5

x 104

0200

400

600800

VolT

ot

0 0.5 1 1.5 2 2.5

x 104

0

200

400

Vol

Model "basforc9": Expectation +- s.d. and MAP-output

0 0.5 1 1.5 2 2.5

x 104

0

10

20

30

Ctr

eeT

ot

0 0.5 1 1.5 2 2.5

x 104

0

5

10

15

Ctr

ee

0 0.5 1 1.5 2 2.5

x 104

0

2

4

6

8

Cste

m

0 0.5 1 1.5 2 2.5

x 104

0

0.5

1

1.5

Cbra

nch

0 0.5 1 1.5 2 2.5

x 104

0

0.2

0.4

0.6

Cle

af

0 0.5 1 1.5 2 2.5

x 104

0

2

4

Cro

ot

0 0.5 1 1.5 2 2.5

x 104

5

10

15

20

h

0 0.5 1 1.5 2 2.5

x 104

0

1

2

LA

I

Time0 0.5 1 1.5 2 2.5

x 104

8

10

12

14

Csoil

Time0 0.5 1 1.5 2 2.5

x 104

0.3

0.4

0.5

Nsoil

Time

3.15 Bayesian projects at CEH-Edinburgh3.15 Bayesian projects at CEH-Edinburgh3.15 Bayesian projects at CEH-Edinburgh3.15 Bayesian projects at CEH-Edinburgh

• Selection of forest models

• Data Assimilation forest EC data (David Cameron, Mat Williams, M.v.Oijen)

• Risk of frost damage in grassland

• Uncertainty in UK C-sequestration(Marcel van Oijen, Jonathan Rougier,Ron Smith, Tommy Brown, Amanda Thomson)

UKCIP

Change in annual mean Temperature

Change in potential C-seq.

Uncertainty in change of potential C-seq.

UKCIP


Change in potential C-seq.Change in potential C-seq.



Uncertainty in earth system resilience

(Clare Britton & David Cameron)

Parameterization and uncertainty quantification of 3-PG model of

forest growth & C-stock(Genevieve Patenaude, Ronnie Milne, M. v.Oijen)[CO2]

Time

3.16 BASFOR: forest C-sequestration 2005-20763.16 BASFOR: forest C-sequestration 2005-20763.16 BASFOR: forest C-sequestration 2005-20763.16 BASFOR: forest C-sequestration 2005-2076

UKCIP


Change in potential C-seq.


- Uncertainty due to model parameters only, NOT uncertainty in inputs / upscaling

3.17 Integrating RS-data (Patenaude et al.) 3.17 Integrating RS-data (Patenaude et al.) 3.17 Integrating RS-data (Patenaude et al.) 3.17 Integrating RS-data (Patenaude et al.)

Model 3-PG

BC

RS-data:Hyper-

spectral, LiDAR,

SAR

3.18 What kind of measurements 3.18 What kind of measurements would have reduced uncertainty would have reduced uncertainty

the most ?the most ?

3.18 What kind of measurements 3.18 What kind of measurements would have reduced uncertainty would have reduced uncertainty

the most ?the most ?

3.19 Prior predictive uncertainty & height-data3.19 Prior predictive uncertainty & height-data3.19 Prior predictive uncertainty & height-data3.19 Prior predictive uncertainty & height-data

0 5000 10000 150000

5

10

15

20

h

0 5000 10000 150000

5

10

Cw

BASFOR: Predictive uncertainty

0 5000 10000 150000

0.5

1

1.5

Cl

0 5000 10000 150000

1

2

3

Cr

0 5000 10000 15000-0.5

0

0.5

1

1.5

NP

Py

0 5000 10000 150000

5

10

LAI

0 5000 10000 150000

0.05

0.1

0.15

Ntr

ee

0 5000 10000 150000

0.02

0.04

0.06

NC

l

0 5000 10000 150000

5

10

Cso

il

0 5000 10000 150000

0.2

0.4

0.6

Nso

il

Time0 5000 10000 15000

0

0.05

0.1

0.15

0.2

Nm

in

Time0 5000 10000 15000

0

50

100

150

Min

y

Time

Height BiomassPrior pred. uncertainty

Height data Skogaby

3.20 Prior & posterior uncertainty: use of height 3.20 Prior & posterior uncertainty: use of height datadata


0 5000 10000 150000

5

10

15

20

h

0 5000 10000 150000

5

10

Cw


0 5000 10000 150000

0.5

1

1.5

Cl

0 5000 10000 150000

1

2

3

Cr

0 5000 10000 15000-0.5

0

0.5

1

1.5

NP

Py

0 5000 10000 150000

5

10

LAI

0 5000 10000 150000

0.05

0.1

0.15

Ntr

ee

0 5000 10000 150000

0.02

0.04

0.06

NC

l

0 5000 10000 150000

5

10

Cso

il

0 5000 10000 150000

0.2

0.4

0.6

Nso

il

Time0 5000 10000 15000

0

0.05

0.1

0.15

0.2

Nm

in

Time0 5000 10000 15000

0

50

100

150

Min

y

Time


Posterior uncertainty (using height data)

Height data Skogaby



0 5000 10000 150000

5

10

15

20

h

0 5000 10000 150000

5

10

Cw


0 5000 10000 150000

0.5

1

1.5

Cl

0 5000 10000 150000

1

2

3

Cr

0 5000 10000 15000-0.5

0

0.5

1

1.5

NP

Py

0 5000 10000 150000

5

10

LAI

0 5000 10000 150000

0.05

0.1

0.15

Ntr

ee

0 5000 10000 150000

0.02

0.04

0.06

NC

l

0 5000 10000 150000

5

10

Cso

il

0 5000 10000 150000

0.2

0.4

0.6

Nso

il

Time0 5000 10000 15000

0

0.05

0.1

0.15

0.2

Nm

in

Time0 5000 10000 15000

0

50

100

150

Min

y

Time



Height data (hypothet.)



0 5000 10000 150000

5

10

15

20

h

0 5000 10000 150000

5

10

Cw


0 5000 10000 150000

0.5

1

1.5

Cl

0 5000 10000 150000

1

2

3

Cr

0 5000 10000 15000-0.5

0

0.5

1

1.5N

PP

y

0 5000 10000 150000

5

10

LAI

0 5000 10000 150000

0.05

0.1

0.15

Ntr

ee

0 5000 10000 150000

0.02

0.04

0.06

NC

l

0 5000 10000 150000

5

10

Cso

il

0 5000 10000 150000

0.2

0.4

0.6

Nso

il

Time0 5000 10000 15000

0

0.05

0.1

0.15

0.2

Nm

in

Time0 5000 10000 15000

0

50

100

150

Min

y

Time



Posterior uncertainty (using precision height data)

3.21 Summary for BC procedure3.21 Summary for BC procedure3.21 Summary for BC procedure3.21 Summary for BC procedure

Data D ± σModel fPrior P()

Calibrated parameters, with covariances

Uncertainty of model output

Sensitivity analysis of model parameters

“Error function” e.g. N(0, σ)

MCMC

Samples of (104 – 105)

Samples of f()(104 – 105)

P(D|f())Posterior P(|D) PCC

3.22 Summary for BC vs tuning3.22 Summary for BC vs tuning3.22 Summary for BC vs tuning3.22 Summary for BC vs tuning

Model tuning1. Define parameter ranges

(permitted values)2. Select parameter values

that give model output closest (r2, RMSE, …) to data

3. Do the model study with the tuned parameters (i.e. no model output uncertainty)

Bayesian calibration1. Define parameter pdf’s2. Define data pdf’s

(probable measurement errors)

3. Use Bayes’ Theorem to calculate posterior parameter pdf

4. Do all future model runs with samples from the parameter pdf (i.e. quantify uncertainty of model results)

BC can use data to reduce parameter

uncertainty for any process-based model

4. Bayesian Model Comparison (BMC)4. Bayesian Model Comparison (BMC)4. Bayesian Model Comparison (BMC)4. Bayesian Model Comparison (BMC)

4.1 RECOGNITION revisited: model uncertainty4.1 RECOGNITION revisited: model uncertainty4.1 RECOGNITION revisited: model uncertainty4.1 RECOGNITION revisited: model uncertainty

0

5

10

15

20

25

Latitude

EFM

4.1 RECOGNITION revisited: model uncertainty4.1 RECOGNITION revisited: model uncertainty4.1 RECOGNITION revisited: model uncertainty4.1 RECOGNITION revisited: model uncertainty

HOGPFZ

HELKAR

PUSRAJ

PFFSOL

BRILOP

TRIGA2

GA1ALT

AALSKO

BLAJA

DPUN

KANKEM

KOL-5

0

5

10

15

20

25

Latitude

HOGPFZ

HELKAR

PUSRAJ

PFFSOL

BRILOP

TRIGA2

GA1ALT

AALSKO

BLAJA

DPUN

KANKEM

KOL

0

10

20

30

40

Latitude

-10

-5

0

5

10

15

20

HOGPFZ

HELKAR

PUSRAJ

PFFSOL

BRILOP

TRIGA2

GA1ALT

AALSKO

BLAJA

DPUN

KANKEM

KOL-10

0

10

20

Latitude

EFM

EFIMODFinnFor

Q

4.2 Bayesian comparison of two models4.2 Bayesian comparison of two models4.2 Bayesian comparison of two models4.2 Bayesian comparison of two models

Bayes Theorem for model probab.:P(M|D) = P(M) P(D|M) / P(D)

The “Integrated likelihood” P(D|Mi) can be approximated from the MCMC sample of

outputs for model Mi (*)

Soil

Trees

H2OC

Atmosphere

H2O

H2OC

Nutr.


H2OC

Nutr.

Nutr.

Nutr.

Model 1

Soil

Trees

H2OC

Atmosphere

H2O

H2OC

Nutr.


H2OC

Nutr.

Nutr.

Nutr.

Model 2

P(M2|D) / P(M1|D) = P(D|M2) / P(D|M1)

The “Bayes Factor” P(D|M2) / P(D|M1) quantifies how the data D change the

odds of M2 over M1

P(M1) = P(M2) = ½

(*)

MCMCMCMC

MCMC

MCMCi

MCMCi

M

DP

n

DPPP

DPDPP

P

w

DPwdDPPMDP

)|(1

)|()()(

)|()|()(

)()|(

)|()()|()(

harmonic mean of likelihoods in MCMC-sample (Kass & Raftery, 1995)

4.3 BMC: Tuomi et al. 20074.3 BMC: Tuomi et al. 20074.3 BMC: Tuomi et al. 20074.3 BMC: Tuomi et al. 2007

4.4 Bayes Factor for two big forest models4.4 Bayes Factor for two big forest models4.4 Bayes Factor for two big forest models4.4 Bayes Factor for two big forest models

MCMC 5000 steps

MCMC 5000 steps

0 2 4

x 10-3

0200400

CL02 4 6

x 10-3

0100200

CR00 0.005 0.01

0200400

CW0


0.4 0.6 0.80

100200

BETA300 350 4000

100200

CO200.25 0.3 0.350

100200

FLMAX

0.5 0.6 0.70

100200

FW0.4 0.6 0.80

100200

GAMMA0 2 4

0100200

KCA0 0.5 1

0100200

KCAEXP0 0.5 1

x 10-3

0100200

KDL0 0.5 1

x 10-3

0100200

KDR

2 4 6

x 10-5

0100200

KDW3 4 5

0100200

KH0.2 0.3 0.40

100200

KHEXP4 6 8

x 10-3

0100200

KLAIMAX0 1 2

x 10-3

0100200

KNMIN0 1 2

x 10-3

0100200

KNUPT

0.02 0.03 0.040

100200

KTA10 20 30

0100200

KTB0.4 0.6 0.80

100200

KTREE2 2.5 3

x 10-3

0100200

LUE00.01 0.015 0.020

100200

NLCONMIN0.04 0.05 0.060

100200

NLCONMAX

0.02 0.03 0.040

100200

NRCON0 1 2

x 10-3

0100200

NWCON0 20 40

0100200

SLA0 0.5 1

0100200

CLITT06 8 10

0100200

CSOMF01 2 3

0100200

CSOMS0

0 0.01 0.020

100200

NLITT00.2 0.3 0.40

100200

NSOMF00 0.1 0.2

0100200

NSOMS00 1 2

x 10-3

0200400

NMIN00.4 0.6 0.80

100200

FLITTSOMF0 0.05 0.1

0200400

FSOMFSOMS

0 2 4

x 10-3

0200400

KDLITT0 0.5 1

x 10-4

0100200

KDSOMF0 1 2

x 10-5

0100200

KDSOMS

0 1 2

x 10-3

0100200

CB0T0 5

x 10-3

0100200

CL0T0 2 4

x 10-3

0100200

CR0T


0 1 2

x 10-3

0100200

CS0T0.4 0.6 0.80

100200

BETA300 350 4000

100200

CO20

0.25 0.3 0.350

100200

FB0.25 0.3 0.350

100200

FLMAX0.25 0.3 0.350

100200

FS0.4 0.6 0.80

100200

GAMMA0 2 4

0100200

KCA0.4 0.6 0.80

100200

KCAEXP

0.5 1 1.5

x 10-4

0100200

KDBT0 5

x 10-4

0100200

KDRT2 4 6

0100200

KH0.2 0.3 0.40

50100

KHEXP0 1 2

x 10-3

0100200

KNMINT0 1 2

x 10-3

0100200

KNUPTT

0.02 0.03 0.040

100200

KTA10 20 30

0100200

KTB0.4 0.6 0.80

100200

KEXTT4 6 8

0100200

LAIMAXT2 2.5 3

x 10-3

0100200

LUET0.01 0.015 0.020

100200

NCLMINT

0.04 0.05 0.060

50100

NCLMAXT0.02 0.03 0.040

100200

NCRT0 1 2

x 10-3

050

100

NCWT10 20 30

050

100

SLAT4 6 8

0100200

TRANCOT0 0.5 1

0100200

CLITT0

6 8 100

100200

CSOMF01 2 3

050

100

CSOMS00 0.01 0.02

0100200

NLITT00.2 0.3 0.40

100200

NSOMF00 0.1 0.2

0100200

NSOMS00 1 2

x 10-3

0100200

NMIN0

0.4 0.6 0.80

50100

FLITTSOMF0 0.05 0.1

0100200

FSOMFSOMS0 2 4

x 10-3

0100200

KDLITT0 1 2

x 10-4

0100200

KDSOMF0 0.5 1

x 10-5

0100200

KDSOMS

Calculation of P(D|BASFOR)

Skogaby

Rajec

Skogaby

Rajec

Calculation of P(D|BASFOR+)

Data Rajec: Emil Klimo

4.5 Bayes Factor for two big forest models4.5 Bayes Factor for two big forest models4.5 Bayes Factor for two big forest models4.5 Bayes Factor for two big forest models

MCMC 5000 steps

MCMC 5000 steps

0 2 4

x 10-3

0200400

CL02 4 6

x 10-3

0100200

CR00 0.005 0.01

0200400

CW0


0.4 0.6 0.80

100200

BETA300 350 4000

100200

CO200.25 0.3 0.350

100200

FLMAX

0.5 0.6 0.70

100200

FW0.4 0.6 0.80

100200

GAMMA0 2 4

0100200

KCA0 0.5 1

0100200

KCAEXP0 0.5 1

x 10-3

0100200

KDL0 0.5 1

x 10-3

0100200

KDR

2 4 6

x 10-5

0100200

KDW3 4 5

0100200

KH0.2 0.3 0.40

100200

KHEXP4 6 8

x 10-3

0100200

KLAIMAX0 1 2

x 10-3

0100200

KNMIN0 1 2

x 10-3

0100200

KNUPT

0.02 0.03 0.040

100200

KTA10 20 30

0100200

KTB0.4 0.6 0.80

100200

KTREE2 2.5 3

x 10-3

0100200

LUE00.01 0.015 0.020

100200

NLCONMIN0.04 0.05 0.060

100200

NLCONMAX

0.02 0.03 0.040

100200

NRCON0 1 2

x 10-3

0100200

NWCON0 20 40

0100200

SLA0 0.5 1

0100200

CLITT06 8 10

0100200

CSOMF01 2 3

0100200

CSOMS0

0 0.01 0.020

100200

NLITT00.2 0.3 0.40

100200

NSOMF00 0.1 0.2

0100200

NSOMS00 1 2

x 10-3

0200400

NMIN00.4 0.6 0.80

100200

FLITTSOMF0 0.05 0.1

0200400

FSOMFSOMS

0 2 4

x 10-3

0200400

KDLITT0 0.5 1

x 10-4

0100200

KDSOMF0 1 2

x 10-5

0100200

KDSOMS

0 1 2

x 10-3

0100200

CB0T0 5

x 10-3

0100200

CL0T0 2 4

x 10-3

0100200

CR0T


0 1 2

x 10-3

0100200

CS0T0.4 0.6 0.80

100200

BETA300 350 4000

100200

CO20

0.25 0.3 0.350

100200

FB0.25 0.3 0.350

100200

FLMAX0.25 0.3 0.350

100200

FS0.4 0.6 0.80

100200

GAMMA0 2 4

0100200

KCA0.4 0.6 0.80

100200

KCAEXP

0.5 1 1.5

x 10-4

0100200

KDBT0 5

x 10-4

0100200

KDRT2 4 6

0100200

KH0.2 0.3 0.40

50100

KHEXP0 1 2

x 10-3

0100200

KNMINT0 1 2

x 10-3

0100200

KNUPTT

0.02 0.03 0.040

100200

KTA10 20 30

0100200

KTB0.4 0.6 0.80

100200

KEXTT4 6 8

0100200

LAIMAXT2 2.5 3

x 10-3

0100200

LUET0.01 0.015 0.020

100200

NCLMINT

0.04 0.05 0.060

50100

NCLMAXT0.02 0.03 0.040

100200

NCRT0 1 2

x 10-3

050

100

NCWT10 20 30

050

100

SLAT4 6 8

0100200

TRANCOT0 0.5 1

0100200

CLITT0

6 8 100

100200

CSOMF01 2 3

050

100

CSOMS00 0.01 0.02

0100200

NLITT00.2 0.3 0.40

100200

NSOMF00 0.1 0.2

0100200

NSOMS00 1 2

x 10-3

0100200

NMIN0

0.4 0.6 0.80

50100

FLITTSOMF0 0.05 0.1

0100200

FSOMFSOMS0 2 4

x 10-3

0100200

KDLITT0 1 2

x 10-4

0100200

KDSOMF0 0.5 1

x 10-5

0100200

KDSOMS

Calculation of P(D|BASFOR)

Calculation of P(D|BASFOR+)

Data Rajec: Emil Klimo

P(D|M1) = 7.2e-016

P(D|M2) = 5.8e-15

Bayes Factor = 7.8, so BASFOR+ supported by

the data

0 1 2 3 4

x 104

0

20

40

h

0 1 2 3 4

x 104

0

10

20

Cw

Model "BASFORC6e": Expectation +- s.d. and MAP-output

0 1 2 3 4

x 104

0

0.5

1

1.5

Cl

0 1 2 3 4

x 104

0

1

2

3C

r

0 1 2 3 4

x 104

0

0.5

1

1.5

NP

Py

0 1 2 3 4

x 104

0

10

20

30

LAI

0 1 2 3 4

x 104

0

0.05

0.1

Ntre

e

0 1 2 3 4

x 104

0

0.02

0.04

0.06

NC

l

0 1 2 3 4

x 104

0

10

20

30

Cso

il

0 1 2 3 4

x 104

0.2

0.4

0.6

0.8

Nso

il

Time0 1 2 3 4

x 104

-0.01

0

0.01

0.02

Nm

in

Time0 1 2 3 4

x 104

0

50

100

150

Min

y

Time

4.6 Summary of BMC procedure4.6 Summary of BMC procedure4.6 Summary of BMC procedure4.6 Summary of BMC procedure

Data DPrior P(1)

Updated parameters

MCMC

Samples of 1

(104 – 105)

Posterior P(1|D)

Model 1

MCMC

Prior P(2)

Model 2

Samples of 2

(104 – 105)

Posterior P(2|D)Updated parameters

P(D|M1) P(D|M2)

Bayes factorUpdated model odds

5. B5. BC & BMC in NitroEurope5. B5. BC & BMC in NitroEurope

C 3 NEU Plot Scale

ModellingC2 NEU Ecosystem

Manipulation

C1 NEU

Flux Network

C 4NEU Landscape

Analysis

C 5 NEU European

Integration

C 6 NEU Verification

Other EU & nationalactivities, including

CarboEurope IP

C 3 NEU Plot Scale

ModellingC2 NEU Ecosystem

Manipulation

C1 NEU

Flux Network

C2 NEU Ecosystem

Manipulation

C1 NEU

Flux Network

C 4NEU Landscape

Analysis

C 5 NEU European

Integration

C 4NEU Landscape

Analysis

C 5 NEU European

Integration

C 6 NEU Verification

Other EU & nationalactivities, including

CarboEurope IP

A8.2NEU Advisory Group

Stakeholders

WP 8NEU Management

Coordinator

WP 1 NEU

Flux NetworkSutton, UK

A1.3 NEU Advanced Network:Fluxes, pools & budgets

Campbell, UK

A1.1 Advanced N flux

measurement methods Nemitz, UK

A1.2 Long-term N flux

methods & applicationPilegaard, DK

A.1.4 Plant & soil pools,

processes & interactionsCotrufo, I

A.1.5 Inferential N fluxes and C interactions

Sutton, UK

WP 2 NEU Ecosystem

ManipulationBeier, DK

A2.1 Forest change

(inc. afforestation)Gundersen, DK

A2.2 Shrubland change

(& natural wetlands)Beier, DK

A2.4 Arable change

(inc. drainage effects)Rees, UK

A2.3 Grassland change

(inc grazing interactions)Soussana, FR

A2.5 Manipulation Synthesis

Beier, DK

A10.4ESF N Assessment &

UNEP - INIErisman, ECN, NL

A10.3 COST Atmos-Biosphere& multiple N strategies

Domburg, NL

A10.5Input to EC,

FCCC & CLRTAPSeufert, JRC, I.

A10.2IGBP - iLEAPS

Nemitz, UK / Vesala FI

Figure 3. NitroEurope IP: Science and management structure

A8.4General AssemblyAll NEU Partners

A8.5Review & Assessment

SSC +AG

A8.3Financial

ManagementCoordinator + partners

A10.1 Innovation Highlights

& NEU PortalCEH, Sutton, UK

A8.1Science Management

Scientific Steering Committee

WP 5 NEU European

Integrationde Vries, NL

A5.1 GIS-based assembly

of input dataSeufert, JRC, I

A5.2 Deriving past, present

& future scenariosObersteiner, IIASA, A

A5.3 Developmt. of integratedmulti-component model

de Vries, NL

A5.4 Application of European-scale ecosystem models

P. Smith, UK

A5.5 Application of the multi-

component modelKros, NL

WP 10 NEU Dissemination

IP Secretariat

WP 7NEU Standards and Data Management

IP Secretariat

A9.1Summer Schools

Zechmeister, A/ Rees, UK

A7.3NEU Data Centres & TF Data ManagementBADC, de Rudder, UK

A.7.1 TF Common

measurement protocolsBeier, DK

A7.2TF Common

modelling protocolsde Vries, NL

WP 9NEU Training

IP Secretariat

A9.2Executive Training

Erisman/Domburg NL

RTD & Innovation Activities

Work Package Activity

Training Activities

Management Activities

WP 3 NEU Plot Scale

ModellingButterbach-Bahl, D

A3.1 Assessment of models & uncertainty analysis

van Oijen, UK

A3.2 Development of

core modelsButterbach-Bahl, D

A3.3 Interpretn. & simulation

of flux measurementsCalanca, CH

A3.4 Effect of past & present management decisions & adaptation strategies

P. Smith, UK

WP 4NEU Landscape

AnalysisCellier, FR

A4.1 Landscape Inventories

Cellier, FR

A4.3Landscape validation

measurementsTheobald, UK

A4.2Development & applic’n

of landscape modelCellier, FR

A4.4Whole-farm and

landscape decisionsOlesen, DK

WP 6 NEU Verification

Erisman, NL

A6.1 Verification & uncertnty:bottom-up NEU models

van Oijen, UK

A6.4 Improvement of IPCC methods & inventories

van Amstel, NL

A6.3 Verification of official UNFCCC inventories

Erisman, NL

A6.2 Independent inverse-

modelling of European N2O & CH4 emissions

Bergamaschi, JRC, I

A8.2NEU Advisory Group

Stakeholders

WP 8NEU Management

Coordinator

WP 1 NEU



Campbell, UK








Sutton, UK

WP 1 NEU



Campbell, UK








Sutton, UK

WP 2 NEU Ecosystem


A2.1 Forest change




A2.4 Arable change





Beier, DK

WP 2 NEU Ecosystem


A2.1 Forest change




A2.4 Arable change





Beier, DK

A10.4ESF N Assessment &

UNEP - INIErisman, ECN, NL

A10.3 COST Atmos-Biosphere& multiple N strategies

Domburg, NL

A10.5Input to EC,

FCCC & CLRTAPSeufert, JRC, I.

A10.2IGBP - iLEAPS

Nemitz, UK / Vesala FI

Figure 3. NitroEurope IP: Science and management structure

A8.4General AssemblyAll NEU Partners

A8.5Review & Assessment

SSC +AG

A8.3Financial

ManagementCoordinator + partners

A10.1 Innovation Highlights

& NEU PortalCEH, Sutton, UK

A8.1Science Management

Scientific Steering Committee

WP 5 NEU European







de Vries, NL


P. Smith, UK



WP 5 NEU European







de Vries, NL


P. Smith, UK



WP 10 NEU Dissemination

IP Secretariat

WP 7NEU Standards and Data Management

IP Secretariat

A9.1Summer Schools

Zechmeister, A/ Rees, UK

A7.3NEU Data Centres & TF Data ManagementBADC, de Rudder, UK

A.7.1 TF Common

measurement protocolsBeier, DK

A7.2TF Common

modelling protocolsde Vries, NL

WP 9NEU Training

IP Secretariat

A9.2Executive Training

Erisman/Domburg NL

RTD & Innovation Activities


Training Activities



Training Activities


WP 3 NEU Plot Scale



van Oijen, UK

A3.2 Development of





P. Smith, UK

WP 4NEU Landscape

AnalysisCellier, FR


Cellier, FR





A4.4Whole-farm and


WP 3 NEU Plot Scale



van Oijen, UK

A3.2 Development of





P. Smith, UK

WP 3 NEU Plot Scale



van Oijen, UK

A3.2 Development of





P. Smith, UK

WP 4NEU Landscape

AnalysisCellier, FR


Cellier, FR





A4.4Whole-farm and


WP 4NEU Landscape

AnalysisCellier, FR


Cellier, FR





A4.4Whole-farm and



Erisman, NL


van Oijen, UK


van Amstel, NL


Erisman, NL



Bergamaschi, JRC, I


Erisman, NL


van Oijen, UK


van Amstel, NL


Erisman, NL



Bergamaschi, JRC, I

What is the effect of reactive nitrogen supply on the direction and magnitude of net greenhouse gas budgets for Europe?

This CEH co-ordinated IP builds on CEH’s involvement in other previous and current European GHG projects such as GREENGRASS, CarboMont and CarboEurope IP

5.1 NitroEurope & uncertainty5.1 NitroEurope & uncertainty5.1 NitroEurope & uncertainty5.1 NitroEurope & uncertainty

5.2 NitroEurope & Uncertainty5.2 NitroEurope & Uncertainty5.2 NitroEurope & Uncertainty5.2 NitroEurope & Uncertainty

Modellers NEU (2006)

NitroEurope (NEU):

• non-CO2 GHG Europe

• experiments at plot-scale, observations at regional scale

• models at plot- and regional scale

• protocols for good-modelling practice and for uncertainty quantification and analysis (collab. with CEU in JUTF)

5.3 Uncertainty assessment NEU models5.3 Uncertainty assessment NEU models5.3 Uncertainty assessment NEU models5.3 Uncertainty assessment NEU models

Plot scale forest model added in 2007:

DAYCENT BC Yes

BC

BC

5.4 NEU – Forest model comparison 2007-85.4 NEU – Forest model comparison 2007-85.4 NEU – Forest model comparison 2007-85.4 NEU – Forest model comparison 2007-8

• 4 models (DNDC, BASFOR, COUP, DayCENT)• Models frozen 30-11-2007• Calibration of models using data Höglwald

(D) {Mainly N2O & NO-emission rates}

• Comparison of models using data AU & DK

Bayesian Calibration(BC)

Bayesian Model Comparison

(BMC)

Bayesian Calibration (BC) and Bayesian Model Comparison (BMC) of process-based models in NitroEurope: Theory, implementation and guidelines

Bayesian Calibration (BC) and Bayesian Model Comparison (BMC) of process-based models in NitroEurope: Theory, implementation and guidelines

• Theory of BC and BMC

• Methods for doing BC: MCMC and Accept-Reject•3.1 Standard Metropolis algorithm•3.2 Metropolis with a modified proposal generating mechanism (“Reflection method”)•3.3 Accept-Reject algorithm

• FAQ – Bayesian Calibration

• References

• Appendix 1: MCMC code in MATLAB: the Metropolis algorithm• Appendix 2: MCMC code in MATLAB: Metropolis-with-Reflection• Appendix 3: ACCEPT-REJECT code in MATLAB• Appendix 4: MCMC code in R: the Metropolis algorithm

5.6 BASFOR changes for NEU5.6 BASFOR changes for NEU5.6 BASFOR changes for NEU5.6 BASFOR changes for NEU

1. Soil temperature calculated

2. Mineralisation of litter and SOM = f(Tsoil): Gaussian curve (Tuomi et al. 2007):

• f = exp[ (T-10) (2Tm-T-10) / 2σ2 ]

3. Nemission split up into N2O and NO: Hole-In-the-Pipe (HIP) approach (Davidson & Verchot, 2000):

• fN2O = 1 / ( 1 + exp[-r(WFPS-WFPS50)] )

0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

Water-Filled Pore Space (WFPS) (-)

fN2O (-)

5.12 BC results: Prior & Posterior5.12 BC results: Prior & Posterior5.12 BC results: Prior & Posterior5.12 BC results: Prior & Posterior

0 0.02 0.04 0.06 0.08 0.10

10

20

30

FSOMFSOMS

200 400 600 800 1000 1200 1400 16000

1

2

3x 10

-3

TCLITT

0 1 2 3 4

x 10-3

0

5000

10000

15000

KNEMIT

35 40 45 50 55 60 650

0.02

0.04

0.06

0.08

0.1

TMAXF

4 6 8 10 12 140

0.1

0.2

0.3

0.4

RFN2O

0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

WFPS50N2O

0.5 1 1.5 2 2.5 3 3.5 4

x 105

0

0.2

0.4

0.6

0.8

1x 10

-5

TCSOMS

0.4 0.5 0.6 0.7 0.80

2

4

6

8

10

FLITTSOMF

5.13 BC results: simulation uncertainty & 5.13 BC results: simulation uncertainty & datadata

5.13 BC results: simulation uncertainty & 5.13 BC results: simulation uncertainty & datadata

0 1000 2000 3000 4000 5000 6000 7000 8000 900020

30

40

50

Hei

ght

0 1000 2000 3000 4000 5000 6000 7000 8000 90000.02

0.03

0.04

0.05

NC

LT

0 1000 2000 3000 4000 5000 6000 7000 8000 9000100

150

200

250

Cw

ood

0 1000 2000 3000 4000 5000 6000 7000 8000 900020

40

60

80

Cro

ot0 1000 2000 3000 4000 5000 6000 7000 8000 9000

5

10

15

20

Cle

af

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

5

10

LAI

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-50

0

50

N2O

d10

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-50

0

50

100

NO

d10

5.15 Data have information content, which is 5.15 Data have information content, which is additiveadditive

5.15 Data have information content, which is 5.15 Data have information content, which is additiveadditive

0

1

2

3

CB0TCL0

TCR0T

CS0T FB

FLM

AXFS

GGAM

MA

KCA

KCAEXPKH

KHEXP

KNMIN

T

KNUPTT

KEXTT

KRNINTCT

LUET

NCLMAXT

FNCLM

INT

NCRT

NCWT

SLAT

TCCLM

AXT

FTCCLM

INT

TCCBT

TCCRT

TOPTT

TTOLT

TRANCO

T

WO

ODDENS

CLITT

0

CSOM0

FCSOM

F0

CNLITT0

CNSOM

F0

CNSOM

S0

NMIN

0

FLIT

TSOM

F

FSO

MFS

OMS

TCLI

TT

TCSOM

F

TCSOM

S

KNEMIT

TMAXF

TSIG

MAF

RFN2O

WFPS50

N2O

Data1&2: One stepData1&2: Two steps

0 1000 2000 3000 4000 5000 6000 7000 8000 900020

30

40

50

Hei

ght

0 1000 2000 3000 4000 5000 6000 7000 8000 90000.02

0.03

0.04

0.05

NC

LT

0 1000 2000 3000 4000 5000 6000 7000 8000 9000100

150

200

250

Cw

ood

0 1000 2000 3000 4000 5000 6000 7000 8000 900020

40

60

80C

root

0 1000 2000 3000 4000 5000 6000 7000 8000 90005

10

15

20

Cle

af

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

5

10

LAI

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-50

0

50

N2O

d10

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-50

0

50

100

NO

d10

0 1000 2000 3000 4000 5000 6000 7000 8000 900020

30

40

50

Hei

ght

0 1000 2000 3000 4000 5000 6000 7000 8000 90000.02

0.03

0.04

0.05

NC

LT

0 1000 2000 3000 4000 5000 6000 7000 8000 9000100

150

200

250

Cw

ood

0 1000 2000 3000 4000 5000 6000 7000 8000 900020

40

60

80

Cro

ot

0 1000 2000 3000 4000 5000 6000 7000 8000 90005

10

15

20

Cle

af

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

5

10

LAI

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-50

0

50

N2O

d10

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-50

0

50

100

NO

d10

0 1000 2000 3000 4000 5000 6000 7000 8000 900020

30

40

50

Hei

ght

0 1000 2000 3000 4000 5000 6000 7000 8000 90000.02

0.03

0.04

0.05

NC

LT

0 1000 2000 3000 4000 5000 6000 7000 8000 9000100

150

200

250

Cw

ood

0 1000 2000 3000 4000 5000 6000 7000 8000 900020

40

60

80

Cro

ot

0 1000 2000 3000 4000 5000 6000 7000 8000 90005

10

15

20C

leaf

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

5

10

LAI

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-50

0

50

N2O

d10

0 1000 2000 3000 4000 5000 6000 7000 8000 9000-50

0

50

100

NO

d10

= +

5.16 BMC5.16 BMC5.16 BMC5.16 BMC

BASFOR BASFOR with T-sensitivity

Data 1983-1997

Data 1998-2003

BF =1131.0

log P(D) = -614.4

log P(D) = -427.6

log P(D) = -607.4

log P(D) = -428.7

BF =0.33

6. Examples of BC & BMC in other 6. Examples of BC & BMC in other sciencessciences

6. Examples of BC & BMC in other 6. Examples of BC & BMC in other sciencessciences

Linear regression using least

squares

• Model: straight line• Prior: uniform

• Likelihood: Gaussian (iid)

BC, e.g. for spatiotemporal stochastic modelling with spatial correlations

included in the prior

=

Note:

• Realising that LS-regression is a special case of BC opens up possibilities to improve on it, e.g. by having more information in the prior or likelihood (Sivia 2005)

• All Maximum Likelihood estimation methods can be seen as limited forms of BC where the prior is ignored (uniform) and only the maximum value of the likelihood is identified (ignoring uncertainty)

Hierarchical modelling =

BC,except that uncertainty is ignored

6.1 Bayes in other disguises6.1 Bayes in other disguises6.1 Bayes in other disguises6.1 Bayes in other disguises

- Inverse modelling (e.g. to estimate emission rates from concentrations)

- Geostatistics, e.g. Bayesian kriging

- Data Assimilation (KF, EnKF etc.)

6.2 Bayes in other disguises (cont.)6.2 Bayes in other disguises (cont.)6.2 Bayes in other disguises (cont.)6.2 Bayes in other disguises (cont.)

6.3 Regional application of plot-scale models6.3 Regional application of plot-scale models6.3 Regional application of plot-scale models6.3 Regional application of plot-scale models

Upscaling method Model structure

Modelling uncertainty

1.

Stratify into homogeneous subregions & Apply

Unchanged P(θ) unchangedUpscaling unc.

2.

Apply to selected points (plots) & Interpolate

Unchanged (but extend w. geostatistical model)

P(θ) unchanged (Bayesian kriging only), Interpolation uncertainty

3.

Reinterpret the model as a regional one & Apply

Unchanged New BC using regional I-O data

4.

Summarise model behav. & Apply exhaustively (deterministic metamodel)

E.g. multivariate regression model or simple mechanistic

New BC needed of metamodel using plot-data

5.

As 4. (stochastic emulator)

E.g. Gaussian process emulator

Code uncertainty (Kennedy & O’H.)

6.

Summarise model behaviour & Embed in regional model

Unrelated new model

New BC using regional I-O data

7. References, Summary, 7. References, Summary, DiscussionDiscussion

7. References, Summary, 7. References, Summary, DiscussionDiscussion

7.1 Bayesian methods: References7.1 Bayesian methods: References7.1 Bayesian methods: References7.1 Bayesian methods: References

Bayes, T. (1763)

Metropolis, N. (1953)

Kass & Raftery (1995)

Green, E.J. / MacFarlane, D.W. / Valentine, H.T. , Strawderman, W.E. (1996, 1998, 1999, 2000)

Jansen, M. (1997)

Jaynes, E.T. (2003)

Van Oijen et al. (2005)

Bayes’ Theorem

MCMC

BMC

Forest models

Crop models

Probability theory

Complex process-based models, MCMC

7.2 Discussion statements / Conclusions7.2 Discussion statements / Conclusions7.2 Discussion statements / Conclusions7.2 Discussion statements / Conclusions

Uncertainty (= incomplete information) is described by pdf’s

1. Plausible reasoning implies probability theory (PT) (Cox, Jaynes)

2. Main tool from PT for updating pdf’s: Bayes Theorem3. Parameter estimation = quantifying joint parameter pdf4. Model evaluation = quantifying pdf in model space requires at

least two models

7.2 Discussion statements / Conclusions7.2 Discussion statements / Conclusions7.2 Discussion statements / Conclusions7.2 Discussion statements / ConclusionsUncertainty (= incomplete information) is described by pdf’s

1. Plausible reasoning implies probability theory (PT) (Cox, Jaynes)2. Main tool from PT for updating pdf’s: Bayes Theorem3. Parameter estimation = quantifying joint parameter pdf BC4. Model evaluation = quantifying pdf in model space requires at

least two models BMC

Practicalities:1. When new data arrive: MCMC provides a universal method for

calculating posterior pdf’s2. Quantifying the prior:

• Not a key issue in env. sci.: (1) many data, (2) prior is posterior from previous calibration

• MaxEnt can be used (Jaynes)3. Defining the likelihood:

• Normal pdf for measurement error usually describes our prior state of knowledge adequately (Jaynes)

4. Bayes Factor shows how new data change the odds of models, and is a by-product from Bayesian calibration (Kass & Raftery)

Overall: Uncertainty quantification often shows that our models are not very reliable

App2.1 How to do BCApp2.1 How to do BCApp2.1 How to do BCApp2.1 How to do BC

The problem: You have: (1) a prior pdf P(θ) for your model’s parameters, (2) new data. You also know how to calculate the likelihood P(D|θ). How do you now go about using BT to calculate the posterior P(θ|D)?

Methods of using BT to calculate P(θ|D):

1. Analytical. Only works when the prior and likelihood are conjugate (family-related). For example if prior and likelihood are normal pdf’s, then the posterior is normal too.

2. Numerical. Uses sampling. Three main methods:

1. MCMC (e.g. Metropolis, Gibbs)

• Sample directly from the posterior. Best for high-dimensional problems

2. Accept-Reject

• Sample from the prior, then reject some using the likelihood. Best for low-dimensional problems

3. Model emulation followed by MCMC or A-R

Should we measure the “sensitive Should we measure the “sensitive parameters”?parameters”?

Should we measure the “sensitive Should we measure the “sensitive parameters”?parameters”?

Yes, because the sensitive parameters:• are obviously important for prediction ?

No, because model parameters:• are correlated with each other, which we do not measure• cannot really be measured at all

So, it may be better to measure output variables, because they:• are what we are interested in• are better defined, in models and measurements• help determine parameter correlations if used in Bayesian

calibration

Key question: what data are most informative?

bayesian methods for calibrating and comparing process-based vegetation models marcel van oijen...

Documents

europe slide

discussion slide

probability theory slide

pposdis pdis pposdis

model parameters

processbased models

model inputs

data models model application