Process-based modelling of Process-based modelling of vegetations and uncertainty vegetations and uncertainty quantification quantification Marcel van Oijen (CEH- Edinburgh) Statistics for Environmental Evaluation Glasgow, 2010-09-01
Process-based modelling of vegetations and uncertainty quantification Marcel van Oijen (CEH-Edinburgh) Statistics for Environmental Evaluation Glasgow,
Mar 28, 2015
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Process-based modelling of Process-based modelling of vegetations and uncertainty vegetations and uncertainty
quantificationquantification
Process-based modelling of Process-based modelling of vegetations and uncertainty vegetations and uncertainty
quantificationquantification
Marcel van Oijen (CEH-Edinburgh)
Statistics for Environmental EvaluationGlasgow, 2010-09-01
ContentsContentsContentsContents
1. Process-based modelling
2. The Bayesian approach
3. Bayesian Calibration (BC) of process-based models
4. Bayesian Model Comparison (BMC)
5. Examples of BC & BMC in other sciences
6. The future of BC & BMC?
7. References, Summary, Discussion
1. Process-based modelling
1. Process-based modelling
1.1 Ecosystem PBMs simulate 1.1 Ecosystem PBMs simulate biogeochemistrybiogeochemistry
1.1 Ecosystem PBMs simulate 1.1 Ecosystem PBMs simulate biogeochemistrybiogeochemistry
Atmosphere
Tree
Soil
Subsoil
H2OH2O
H2O
H2OC
C
C
N
N
N
NPhotosynthesis Source
Light CO2 TemperatureLight CO2 Temperature
SinkMIN(S,S)
Shoot growth
Reserves
Root growth
Reserves at maximum
Tillering
(+)
feedback
on sinksSink
limitation
Source-sink balance
Cascade of carbohydrates
Source-sink balance
Cascade of carbohydrates
ProtectedSOM
metabolic cellulose lignin
C, N variable C/N = 150 C/N = 100
Surface litter
metabolic cellulose lignin
C, N variable C/N = 150 C/N = 100
Soil litter
Nitrate N
UnprotectedSOM
C, N variable
C, N variable
Soil biomass
C/N = 8
StabilisedSOM
C, N variable
Soluble C
Ammonium N
CO2CO2
CO2
CO2
CO2 CO2
CO2
CO2
CO2
Root litter
Root exudate
Fertilizer, root exudate,Atmospheric deposition,Nitrogen fixation
Plant uptake
Fertilizer, atmos. deposition
Shoot litter
LeachingVolatilisation
Nitrif ication
Growth
Death
LeachingDenitrification
Mineralization
Immobilization
N
N
ProtectedSOM
metabolic cellulose lignin
C, N variable C/N = 150 C/N = 100
Surface litter
metabolic cellulose lignin
C, N variable C/N = 150 C/N = 100
Surface litter
metabolic cellulose lignin
C, N variable C/N = 150 C/N = 100
Soil litter
metabolic cellulose lignin
C, N variable C/N = 150 C/N = 100
Soil litter
Nitrate N
UnprotectedSOM
C, N variable
UnprotectedSOM
C, N variable
C, N variable
Soil biomass
C/N = 8
Soil biomass
C/N = 8
StabilisedSOM
C, N variable
StabilisedSOM
C, N variable
Soluble C
Ammonium N
CO2CO2
CO2
CO2
CO2 CO2
CO2
CO2
CO2
Root litter
Root exudate
Fertilizer, root exudate,Atmospheric deposition,Nitrogen fixation
Plant uptake
Fertilizer, atmos. deposition
Shoot litter
LeachingVolatilisation
Nitrif ication
Growth
Death
LeachingDenitrification
Mineralization
Immobilization
N
N
1.2 I/O of PBMs1.2 I/O of PBMs1.2 I/O of PBMs1.2 I/O of PBMs
Atmosphere
Tree
Soil
Subsoil
H2OH2O
H2O
H2OC
C
C
N
N
N
N
Atmosphere
Tree
Soil
Subsoil
H2OH2O
H2O
H2OC
C
C
N
N
N
N
Wind speed
Humidity
Rain
Temperature
Radiation
CO2
N-deposition
Wind speed
Humidity
Rain
Temperature
Radiation
CO2
N-deposition
Parameters & initial constants vegetation
Parameters & initial constants soil
Atm
os
ph
eri
c
dri
ve
rs
Input Model Output
Management & land use
Simulation of time series of plant and soil variables
1.3 I/O of empirical models1.3 I/O of empirical models1.3 I/O of empirical models1.3 I/O of empirical models
Two parameters:P1 = slopeP2 = intercept
Input Model Output
Y = P1 + P2 * t
1.4 Environmental evaluation: increasing 1.4 Environmental evaluation: increasing use of PBMsuse of PBMs
1.4 Environmental evaluation: increasing 1.4 Environmental evaluation: increasing use of PBMsuse of PBMs
C-sequestration (model output for
1920-2000)
Uncertainty of C-sequestration
1.5 Forest models and uncertainty1.5 Forest models and uncertainty1.5 Forest models and uncertainty1.5 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]
1.6 Forest models and uncertainty1.6 Forest models and uncertainty1.6 Forest models and uncertainty1.6 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]
1.7 Many models!1.7 Many models!1.7 Many models!1.7 Many models!
Status: 680 models (21.05.10)
Search models (by free-text-search)Result of query :List of words : soil, carbon 96 models found logical operator: and type of search: word
ANIMO: Agricultural NItrogen MOdel BETHY: Biosphere Energy-Transfer Hydrology scheme BIOMASS: Forest canopy carbon and water balance model BIOME-BGC: Biome model - BioGeochemical Cycles BIOME3: Biome model BLUEGRAMA: BLUE GRAMA CANDY: Carbon and Nitrogen Dynamics in soils CARBON: Wageningen Carbon Cycle Model CARBON_IN_SOILS: TURNOVER OF CARBON IN SOIL CARDYN: CARbon DYNamics CASA: Carnegie-Ames-Stanford Approach (CASA) Biosphere model CENTURY: grassland and agroecosystem dynamics model CERES_CANOLA : CERES-Canola 3.0 CHEMRANK: Interactive Model for Ranking the Potential of Organic Chemicals to Contaminte Groundwater COUPMODEL: Coupled heat and mass transfer model for soil-plant-atmosphere system
(…)
http://ecobas.org/www-server/index.html
1.8 Reality check !1.8 Reality check !1.8 Reality check !1.8 Reality check !
How reliable are these model studies:• Sufficient data for model parameterization?• Sufficient data for model input?• Would other models 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?
2. The Bayesian approach2. The Bayesian approach2. The Bayesian approach2. The Bayesian approach
Probability TheoryProbability TheoryProbability TheoryProbability Theory
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
“
”
The Bayesian approach: reasoning using The Bayesian approach: reasoning using probability theoryprobability theory
The Bayesian approach: reasoning using The Bayesian approach: reasoning using probability theoryprobability theory
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
2.1 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)
= P(pos|dis) P(dis)P(pos|dis) P(dis) + P(pos|hlth) P(hlth)
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)
= P(pos|dis) P(dis)P(pos|dis) P(dis) + P(pos|hlth) P(hlth)
= 0.99 0.01 0.99 0.01 + 0.01 0.99
= 0.50
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)
2.3 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
Bayesian updating of probabilities for process-Bayesian updating of probabilities for process-based modelsbased models
Bayesian updating of probabilities for process-Bayesian updating of probabilities for process-based modelsbased models
Model parameterization: P(params) → P(params|data)Model selection: P(models) → P(model|data)
Bayes’ Theorem: Prior probability → Posterior prob.
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.
Subsoil (or run-off)
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.
Subsoil (or run-off)
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 trees(standing + thinned)
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 trees(standing + thinned)
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
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x 104
0
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1000
Vo
lTo
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l (m
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0
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20
30
Ctr
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10
20
30
Ctr
ee
(k
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0 1 2 3
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Cb
ran
ch
(k
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0 1 2 3
x 104
0
0.5
1
Cle
af
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4
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ot
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0 1 2 3
x 104
0
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h (
m)
0 1 2 3
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LA
I (m
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oil
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oil
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0
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CB0T0 1 2
x 10-3
0
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CL0T0 2 4
x 10-3
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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
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GAMMA5 10 15
0
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1000
KCA0.35 0.4 0.45
0
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1000
KCAEXP
0 2 4
x 10-4
0
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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
NCWT6 8 10
0
1000
2000
SLAT4 6 8
0
500
1000
TRANCOT150 200 2500
1000
2000
WOODDENS
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
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1000
Vo
lTo
t (m
3 h
a-1
)
0 1 2 3
x 104
0
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1000
Vo
l (m
3 h
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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 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
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
0 1 2 3
x 104
0
500
1000V
olT
ot
(m3
ha-
1)
0 1 2 3
x 104
0
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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 trees(standing + thinned)
Carbon in soil
3.9 Calculating the posterior using MCMC3.9 Calculating the posterior using MCMC3.9 Calculating the posterior using MCMC3.9 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
2
4x 10
-3
CL0
0 5000 10000
2
4
6x 10
-3
CR0
0 5000 10000
2468
x 10-3Parameter trace plots
CW0
0 5000 10000
0.450.5
0.55 BETA
0 5000 10000
330340350360370 CO20
0 5000 100000.260.280.3
0.320.34 FLMAX
0 5000 10000
0.55
0.6 FW
0 5000 10000
0.450.5
0.55 GAMMA
0 5000 10000468
101214
KCA
0 5000 100000.350.4
0.45 KCAEXP
0 5000 100000.8
11.21.41.61.8
x 10-3
KDL
0 5000 10000
2
4
x 10-4
KDR
0 5000 10000
68
101214
x 10-5
KDW
0 5000 10000
4
6 KH
0 5000 10000
0.220.240.260.280.3
0.32KHEXP
0 5000 1000034567
x 10-3
KLAIMAX
0 5000 100000.60.8
11.21.41.61.8
x 10-3
KNMIN
0 5000 100000.60.8
11.21.41.61.8
x 10-3
KNUPT
0 5000 10000
0.0250.03
0.035 KTA
0 5000 1000015
20
25 KTB
0 5000 10000
0.4
0.5
0.6 KTREE
0 5000 100001.5
2
2.5
x 10-3
LUE0
0 5000 10000
0.015
0.02
0.025NLCONMIN
0 5000 10000
0.0350.04
0.045 NLCONMAX
0 5000 10000
0.0250.03
0.035 NRCON
0 5000 100000.60.8
11.21.41.61.8
x 10-3
NWCON
0 5000 1000068
101214 SLA
0 5000 100000.2
0.4CLITT0
0 5000 10000
6
8CSOMF0
0 5000 10000
1.52
2.5 CSOMS0
0 5000 100000.0060.0080.01
0.0120.0140.0160.018 NLITT0
0 5000 10000
0.250.3
0.35 NSOMF0
0 5000 100000.060.080.1
0.120.140.160.18 NSOMS0
0 5000 10000
0.51
1.5
x 10-3
Iteration
NMIN0
0 5000 10000
0.50.60.7
Iteration
FLITTSOMF
0 5000 100000.020.040.060.08
Iteration
FSOMFSOMS
0 5000 10000
11.5
22.5
x 10-3
Iteration
KDLITT
0 5000 10000
5
10x 10
-5
Iteration
KDSOMF
0 5000 10000
5
10x 10
-6
Iteration
KDSOMS
MCMC trace plots
3.10 BC using MCMC: an example in EXCEL3.10 BC using MCMC: an example in EXCEL3.10 BC using MCMC: an example in EXCEL3.10 BC using MCMC: an example in EXCEL
Click here for BC_MCMC1.xls
install.packages("mvtnorm")require(mvtnorm)chainLength = 10000 data <- matrix(c(10,6.09,1.83, 20,8.81,2.64, 30,10.66,3.27), nrow=3, ncol=3, byrow=T) param <- matrix(c(0,5,10, 0,0.5,1) , nrow=2, ncol=3, byrow=T)pMinima <- c(param[1,1], param[2,1])pMaxima <- c(param[1,3], param[2,3])logli <- matrix(, nrow=3, ncol=1)vcovProposal = diag( (0.05*(pMaxima-pMinima)) ^2 )pValues <- c(param[1,2], param[2,2])pChain <- matrix(0, nrow=chainLength, ncol = length(pValues)+1)logPrior0 <- sum(log(dunif(pValues, min=pMinima, max=pMaxima)))model <- function (times,intercept,slope) {y <- intercept+slope*times return(y)}for (i in 1:3) {logli[i] <- -0.5*((model(data[i,1],pValues[1],pValues[2])- data[i,2])/data[i,3])^2 - log(data[i,3])}logL0 <- sum(logli)pChain[1,] <- c(pValues, logL0) # Keep first valuesfor (c in (2 : chainLength)){ candidatepValues <- rmvnorm(n=1, mean=pValues, sigma=vcovProposal)if (all(candidatepValues>pMinima) && all(candidatepValues<pMaxima)) {Prior1 <- prod(dunif(candidatepValues, pMinima, pMaxima))}else {Prior1 <- 0}if (Prior1 > 0) { for (i in 1:3){logli[i] <- -0.5*((model(data[i,1],candidatepValues[1],candidatepValues[2])- data[i,2])/data[i,3])^2 - log(data[i,3])} logL1 <- sum(logli)logalpha <- (log(Prior1)+logL1) - (logPrior0+logL0)if ( log(runif(1, min = 0, max =1)) < logalpha ) { pValues <- candidatepValues logPrior0 <- log(Prior1) logL0 <- logL1}} pChain[c,1:2] <- pValues pChain[c,3] <- logL0 }nAccepted = length(unique(pChain[,1])) acceptance = (paste(nAccepted, "out of ", chainLength, "candidates accepted ( = ", round(100*nAccepted/chainLength), "%)"))print(acceptance)mp <- apply(pChain, 2, mean)print(mp)pCovMatrix <- cov(pChain)print(pCovMatrix)
MC
MC
in R
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
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
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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)
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x 104
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10
15
Cs
oil
(kg
m-2
)
Time0 1 2 3
x 104
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0.2
0.4
0.6
Ns
oil
(kg
m-2
)
Time
0 0.5 1
x 10-3
0
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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
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
NCWT6 8 10
0
1000
2000
SLAT4 6 8
0
500
1000
TRANCOT150 200 2500
1000
2000
WOODDENS
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
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1000
KDSOMF0 1 2
x 10-5
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1000
KDSOMS
0 1 2 3
x 104
0
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1000
Vo
lTo
t (m
3 h
a-1
)
0 1 2 3
x 104
0
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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)
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x 104
0
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4
LA
I (m
2 m
-2)
Time0 1 2 3
x 104
0
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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 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
Prior pdf
DataBayesiancalibration
Posterior pdf
Dodd WoodDodd Wood
3.13 Parameter correlations3.13 Parameter correlations3.13 Parameter correlations3.13 Parameter correlations
CL
0
CR
0
CW
0
BE
TA
CO
20
FL
MA
X
FW
GA
MM
A
KC
A
KC
AE
XP
KD
L
KD
R
KD
W
KH
KH
EX
P
KL
AIM
AX
KN
MIN
KN
UP
T
KTA
KT
B
KT
RE
E
LU
E0
NL
CO
NM
IN
NL
CO
NM
AX
NR
CO
N
NW
CO
N
SL
A
CL
ITT
0
CS
OM
F0
CS
OM
S0
NL
ITT
0
NS
OM
F0
NS
OM
S0
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
3.14 Continued calibration when new data become 3.14 Continued calibration when new data become availableavailable
DataBayesiancalibration
DataBayesiancalibration
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
3.14 Continued calibration when new data become 3.14 Continued calibration when new data become availableavailable
3.14 Continued calibration when new data become 3.14 Continued calibration when new data become availableavailable
DataBayesiancalibration
DataBayesiancalibration
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
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
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(NitroEurope team)
• Data Assimilation forest EC data (David Cameron, Mat Williams)
• Risk of frost damage in grassland (Stig Morten Thorsen, Anne-Grete Roer, MvO)
• Uncertainty in agricultural soil models (Lehuger, Reinds, MvO)
• Uncertainty in UK C-sequestration(MvO, Jonty Rougier, Ron Smith, Tommy Brown, Amanda Thomson)
Parameterization and uncertainty
quantification of 3-PG model of forest
growth & C-stock(Genevieve Patenaude,
Ronnie Milne, M. v.Oijen)Uncertainty in
earth system resilience
(Clare Britton & David Cameron)
[CO2]
Time
3.16 BASFOR: forest C-sequestration 1920-3.16 BASFOR: forest C-sequestration 1920-20002000
3.16 BASFOR: forest C-sequestration 1920-3.16 BASFOR: forest C-sequestration 1920-20002000
0.797-1.39
1.39-1.97
1.97-2.56
2.56-3.15
3.15-3.74
3.74-4.33
4.33-4.92
4.92-5.51
5.51-6.09
6.09-6.68
N(soil) (kg/m2)
- Uncertainty due to model parameters only, NOT uncertainty in inputs / upscaling
Soil N-content C-sequestration Uncertainty of C-sequestration
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
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
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
Posterior uncertainty (using height data)
Height data Skogaby
3.20 Prior & posterior uncertainty: use of height 3.20 Prior & posterior uncertainty: use of height datadata
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
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
Posterior uncertainty (using height data)
Height data (hypothet.)
3.20 Prior & posterior uncertainty: use of height 3.20 Prior & posterior uncertainty: use of height datadata
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
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
1.5N
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
Height BiomassPrior pred. uncertainty
Posterior uncertainty (using height data)
Posterior uncertainty (using precision height data)
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 Multiple models -> structural 4.1 Multiple models -> structural uncertaintyuncertainty
4.1 Multiple models -> structural 4.1 Multiple models -> structural uncertaintyuncertainty
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]
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.
Subsoil (or run-off)
H2OC
Nutr.
Nutr.
Nutr.
Model 1
Soil
Trees
H2OC
Atmosphere
H2O
H2OC
Nutr.
Subsoil (or run-off)
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
Parameter marginal probability distributions (truncated normal)
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
Parameter marginal probability distributions (truncated normal)
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
Parameter marginal probability distributions (truncated normal)
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
Parameter marginal probability distributions (truncated normal)
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: what do we need , what do 4.6 Summary of BMC: what do we need , what do we do?we do?
4.6 Summary of BMC: what do we need , what do 4.6 Summary of BMC: what do we need , what do we do?we do?
What do we need to carry out a BMC?
1. Multiple models: M1, … , Mn
2. For each model, a list of its parameters: θ1, … , θn
3. Data: D
What do we do with the models, parameters and data?
1. We express our uncertainty about the correctness of models, parameter values and data by means of probability distributions.
2. We apply the rules of probability theory to transfer the information from the data to the probability distributions for models and parameters
3. The result tells us which model is the most plausible, and what its parameter values are likely to be
5. Examples of BC & BMC in other 5. Examples of BC & BMC in other sciencessciences
5. Examples of BC & BMC in other 5. 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
5.1 Bayes in other disguises5.1 Bayes in other disguises5.1 Bayes in other disguises5.1 Bayes in other disguises
- Inverse modelling (e.g. to estimate emission rates from concentrations)
- Geostatistics, e.g. Bayesian kriging
- Data Assimilation (KF, EnKF etc.)
5.2 Bayes in other disguises (cont.)5.2 Bayes in other disguises (cont.)5.2 Bayes in other disguises (cont.)5.2 Bayes in other disguises (cont.)
5.3 Regional application of plot-scale models5.3 Regional application of plot-scale models5.3 Regional application of plot-scale models5.3 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
6. The future of BC & 6. The future of BC & BMC?BMC?
6. The future of BC & 6. The future of BC & BMC?BMC?
6.1 Trends6.1 Trends6.1 Trends6.1 Trends
• More use of Bayesian approaches in all areas of environmental science
• Improvements in computational techniques for BC & BMC of slow process-based models
• Increasing use of hierarchical models (to represent complex prior pdf’s, or to represent spatial relationships)
• Replacement of informal methods (or methods that only approximate the full probability approach) by BMC
Bayes in climate scienceBayes in climate scienceBayes in climate scienceBayes in climate science
Improvements in Markov Chain Monte Carlo Improvements in Markov Chain Monte Carlo algorithmsalgorithms
Improvements in Markov Chain Monte Carlo Improvements in Markov Chain Monte Carlo algorithmsalgorithms
Hierarchical Bayesian modelling in ecologyHierarchical Bayesian modelling in ecologyHierarchical Bayesian modelling in ecologyHierarchical Bayesian modelling in ecology
See also:
Ogle, K. and J.J. Barber (2008) "Bayesian data-model integration in plant physiological and ecosystem ecology." Progress in Botany 69:281-311
Using BC to make model spin-up Using BC to make model spin-up unnecessaryunnecessary
Using BC to make model spin-up Using BC to make model spin-up unnecessaryunnecessary
(subm.)
Bayes & spaceBayes & spaceBayes & spaceBayes & space
Van Oijen, Thomson & Ewert (2009)
7. Summary, References, 7. Summary, References, DiscussionDiscussion
7. Summary, References, 7. Summary, References, DiscussionDiscussion
7.1 Summary of BC&BMC: What is the Bayesian 7.1 Summary of BC&BMC: What is the Bayesian approach?approach?
7.1 Summary of BC&BMC: What is the Bayesian 7.1 Summary of BC&BMC: What is the Bayesian approach?approach?
1. Express all uncertainties probabilistically Assign probability distributions to (1) data, (2) the collection of models, (3) the parameter-set of each individual model
2. Use the rules of probability theory to transfer the information from the data to the probability distributions for models and parameters
Main tool from probability theory to do this: Bayes’ Theorem
P(α|D) P(α) P(D|α)
Posterior is proportional to prior times likelihood
α = parameter set parameterisation (“Bayesian Calibration”, BC)
α = model set model evaluation (“Bayesian Model Comparison”, BMC)
7.2 Bayesian methods: References7.2 Bayesian methods: References7.2 Bayesian methods: References7.2 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
Bayesian Calibration (BC) and Bayesian Model Comparison (BMC) of process-based models: Theory, implementation and guidelines
Freely downloadable from http://nora.nerc.ac.uk/6087
/
7.4 Discussion statements / Conclusions7.4 Discussion statements / Conclusions7.4 Discussion statements / Conclusions7.4 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
7.4 Discussion statements / Conclusions7.4 Discussion statements / Conclusions7.4 Discussion statements / Conclusions7.4 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
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
Appendix A: How to do BCAppendix A: How to do BCAppendix A: How to do BCAppendix A: 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?
Data have information content, which is additiveData have information content, which is additiveData have information content, which is additiveData have information content, which is additive
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
= +
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