Post on 22-Feb-2016
description
Parameter identifiability, Parameter identifiability, constraints, and equifinality constraints, and equifinality
in data assimilation with in data assimilation with ecosystem modelsecosystem models
Dr. Yiqi LuoDr. Yiqi LuoBotany and microbiology Botany and microbiology
departmentdepartmentUniversity of Oklahoma, USAUniversity of Oklahoma, USA
Land surface models and FluxNET dataEdinburgh, 4-6 June 2008
(Luo et al. Ecol Appl. (Luo et al. Ecol Appl. In pressIn press))
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6
x 10-4
0
50
100
150
200
250
300
350
400
Histogram of generated samples for c2
Range of c2
Sam
plin
g fre
quen
cy
0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024 0.0260
100
200
300
400
500
600
Histogram of generated samples for c3
Range of c3
Sam
plin
g fre
quen
cy
Observed Data
Prior knowledge Posterior distribution
3 3.5 4 4.5 5 5.5 6 6.5
x 10-3
0
100
200
300
400
500
600
700
800
Histogram of generated samples for c5
Range of c5
Sam
plin
g fre
quen
cy
Parameter identifiability
Inverse model
Constrained
Edge-hitting
Equifinality
Wang et al. (2001) ------ a maximum of Wang et al. (2001) ------ a maximum of 3 or 43 or 4 p parameters can be determined.arameters can be determined. Braswell et al. (2005) ------ Braswell et al. (2005) ------ 13 out of 2313 out of 23 parame parameters were well-constrained.ters were well-constrained. Xu et al. (2006) ------ Xu et al. (2006) ------ 4 or 3 out of 74 or 3 out of 7 parameters parameters can be constrained, respectively under ambiecan be constrained, respectively under ambient and elevated COnt and elevated CO22..
Identiable parameters
Three methods to exaThree methods to examine parameter identimine parameter identifiabilityfiability1.1. Search methodSearch method2.2. Model Model
structurestructure3.3. Data variabilityData variability
Harvard Forest EMS-Tower
Eddy flux data Eddy flux data
COCO22 flux flux HH22O fluxO flux Wind speedWind speed TemperatureTemperature PARPAR Relative humidityRelative humidityHourly or half-hourlyHourly or half-hourly
Eddy flux technology
Leaf-level Photosynthesis
Sub-model
Canopy-level Photosynthesis
Sub-model
System-level C balanceSub-model
ModelModel
Table 1 Parameters informationTable 1 Parameters information
Develop prior distributionDevelop prior distribution
Apply Metropolis-Hasting algorithmApply Metropolis-Hasting algorithm
a) generate candidate a) generate candidate pp from sample space from sample spaceb) input to model and calculate cost functionb) input to model and calculate cost functionc) select according to decision criterionc) select according to decision criteriond) repeatd) repeat
Construct posterior distributionConstruct posterior distribution
Bayesian inversionBayesian inversion
Conditional Bayesian inversionConditional Bayesian inversion
Bayesian inversion
Bayesian inversion
Bayesian inversion
Bayesian inversion
Fig. 2 Decrease of cost function with each step of conditional inversion
ConclusionsConclusions Conditional inversion can Conditional inversion can
substantially increase the number of substantially increase the number of constrained parameters.constrained parameters.
Cost function and information loss Cost function and information loss decrease with each step of conditional decrease with each step of conditional inversion.inversion.
Measurement errors Measurement errors and parameter identiand parameter identifiabilityfiability
Leaves X1 Woody X2 Fine Roots X3
Metabolic Litter X4 Structural Litter X5
Microbes X6
Slow SOM X7
Passive SOM X8
GPP
TECO – biogeochemical model
)(
100000010000
1000000101000010000001000000001000000001
8786
7675
68676564
5351
4341
cdiagC
ffff
ffffffff
A
0)0(
)()()(
XtX
tBPtACXtXdtd
TbbbB )00000( 321
No. of parameter
8
12
8
3
c1
010
2030
40
0.0E+00 1.0E-03 2.0E-03 3.0E-03 4.0E-03 5.0E-03
2.0-SD
1.0-SD
0.5-SD
c2
0
5
10
15
0 0.00005 0.0001 0.00015 0.0002 0.00025
2.0-SD
1.0-SD
0.5-SD
c3
05
10152025
0 0.002 0.004 0.006 0.008
2.0-SD
1.0-SD
0.5-SD
c4
0
1
2
3
4
0 0.01 0.02 0.03 0.04
c5
0
2
4
6
0 0.001 0.002 0.003
c6
0
20
40
60
80
0 0.1 0.2 0.3 0.4 0.5
c7
0
2
4
6
8
10
12
0 0.0005 0.001 0.0015
c8
0
0.5
1
1.5
2
2.5
0 2E-06 4E-06 6E-06 8E-06 0.00001
Exit rates
A1
0
1
2
3
0 0.5 1 1.5
2.0-SD
1.0-SD
0.5-SD
A4
012345
0 0.2 0.4 0.6 0.8 1
2.0-SD
1.0-SD
0.5-SD
A5
0
1
2
3
4
0 0.2 0.4 0.6
2.0-SD
1.0-SD
0.5-SD
A6
0
1
2
3
0 0.2 0.4 0.6
A7
0
1
2
3
0 0.2 0.4 0.6 0.8
A8
0
1
2
3
4
0 0.1 0.2 0.3 0.4
A9
0
1
2
3
0 0.2 0.4 0.6 0.8
A10
0
1
2
3
0 0.1 0.2 0.3 0.4
A11
0
1
2
3
0 0.5 1 1.5
Transfer coefficients
Xo1
02
46
8
0 100 200 300 400 500
2.0-SD
1.0-SD
0.5-SD
Xo2
0
510
15
20
0 2000 4000 6000
2.0-SD
1.0-SD
0.5-SD
Xo3
0
12
3
4
0 100 200 300 400
2.0-SD
1.0-SD
0.5-SD
Xo4
00.5
11.5
22.5
0 20 40 60 80
Xo5
0
1
2
3
4
0 100 200 300 400 500
Xo6
0
1
2
3
0 50 100 150
Xo7
0
5
10
15
20
0 1000 2000 3000 4000
Xo8
0
5
10
15
20
0 200 400 600 800
Initial values
X1
500 1000 1500 2000
Freq
uenc
y 10
2
05
10152025
X2
5000 6000 7000 8000 900001234
X3
500 1000 1500 2000
Freq
uenc
y 10
2
05
10152025 X4
500 1000 1500 20000
10203040
X5
500 1000 1500 2000
Freq
uenc
y 10
2
02468
10 X6
50 100 150 2000
20
40
X7
Carbon content (g C m-2)1000 2000 3000 4000
Freq
uenc
y 10
2
02468
10X8
Carbon content (g C m-2)
450 500 550 600 65002468
Pool sizes without data
X1
300 400 500 600 700 800
Freq
uenc
y 10
2
05
10152025
X2
5000 6000 7000 8000 900005
10152025
X3
100 200 300 400 500
Freq
uenc
y 10
2
05
10152025
X4
100 200 300 400 50005
10152025
X5
0 500 1000 1500 2000
Freq
uenc
y 10
2
05
10152025 X6
0 50 100 150 20005
10152025
X7
Carbon content (g C m-2)1500 2000 2500 3000 3500
Freq
uenc
y 10
2
05
10152025
halved SDambient SDdoubled SD
X8
Carbon content (g C m-2)
700 800 900 1000 1100 1200 130005
10152025
Pool sizes with data and different SD
ConclusionConclusion
Magnitudes of measurement errMagnitudes of measurement errors do not affect parameter idenors do not affect parameter identifiability but influence relative tifiability but influence relative constraints of parametersconstraints of parameters
Base modelBase modelGPP
Leaves X1 Stems X2 Roots X3
Metabolic L. X4 Struct. L. X5
Microbes X6
Slow SOM X7
Passive SOM X8
Simplified modelsSimplified models
Plant C
Litter C
GPP CO2
Soil C
Plant C
Litter C
GPP CO2
O Soil C
Miner. C
3P model 4P model
Simplified modelsSimplified models6P model 7P model
GPP
Leaves X1 Stems X2 Roots X3
Litter X4
Slow C X5
Miner. Soil C X6
GPP
Leaves X1 Stems X2 Roots X3
Metabolic L. X4
Struct. L. X5
Microbes X6
Soil C X7
3P model-parameter 3P model-parameter constraintsconstraints
c1
0
100
200
300
400
500
600
2.08
E-04
2.27
E-04
2.46
E-04
2.65
E-04
2.83
E-04
3.02
E-0
4
3.21
E-04
3.40
E-04
3.58
E-0
4
3.77
E-04
3.96
E-04
4.15
E-0
4
4.33
E-04
4.52
E-04
4.71
E-0
4
c2
0100200300400500600700
1.56
E-03
1.89
E-03
2.22
E-0
3
2.55
E-03
2.88
E-03
3.21
E-03
3.54
E-03
3.87
E-03
4.20
E-03
4.53
E-03
4.86
E-03
5.19
E-03
5.53
E-03
5.86
E-03
6.19
E-03
c3
0
100
200
300
400
500
600
4.80
E-0
5
6.79
E-05
8.78
E-05
1.08
E-04
1.28
E-04
1.48
E-04
1.67
E-0
4
1.87
E-04
2.07
E-04
2.27
E-04
2.47
E-04
2.67
E-0
4
2.87
E-04
3.07
E-04
3.27
E-0
4
Plant C Litter C
Soil C
4P model-parameter 4P model-parameter constraintsconstraints
Plant C Litter C
Slow Soil C
c1
0100200300400500600700
2.81
E-0
4
3.08
E-04
3.34
E-0
4
3.61
E-04
3.88
E-0
4
4.15
E-04
4.42
E-04
4.69
E-04
4.96
E-04
5.23
E-04
c2
0200400600800
10001200
2.46
E-03
3.20
E-03
3.94
E-0
3
4.68
E-03
5.42
E-03
6.16
E-03
6.90
E-03
7.64
E-03
8.38
E-03
9.12
E-03
c3
0200400600800
1000
3.16
E-04
3.51
E-04
3.86
E-0
4
4.22
E-04
4.57
E-04
4.93
E-04
5.28
E-04
5.63
E-04
5.99
E-04
6.34
E-04
c4
0
500
1000
1500
2000
4.00
E-06
1.61
E-05
2.82
E-0
5
4.03
E-05
5.23
E-05
6.44
E-05
7.65
E-05
8.85
E-05
1.01
E-04
1.13
E-04
Passive Soil C
6P model-parameter 6P model-parameter constraintsconstraints
Foliage
Litter C Slow Soil C Passive Soil C
c1
050
100
150200
1.26
E-03
1.33
E-03
1.39
E-03
1.46
E-03
1.53
E-03
1.59
E-03
1.66
E-03
1.73
E-03
1.79
E-03
1.86
E-03
c2
050
100150200250
1.16
E-05
2.87
E-0
5
4.58
E-05
6.30
E-05
8.01
E-05
9.72
E-05
1.14
E-04
1.31
E-04
1.49
E-0
4
1.66
E-04
c3
050
100150200250
4.37
E-03
4.54
E-03
4.70
E-03
4.87
E-03
5.04
E-03
5.21
E-03
5.37
E-03
5.54
E-03
5.71
E-03
5.88
E-03
Woody Fine roots
c4
050
100150200250300
2.52
E-03
3.00
E-03
3.49
E-03
3.98
E-0
3
4.47
E-03
4.96
E-03
5.45
E-03
5.94
E-03
6.42
E-03
6.91
E-03
c5
050
100150200250
3.61
E-04
3.95
E-0
4
4.30
E-04
4.65
E-04
5.00
E-04
5.35
E-04
5.70
E-04
6.05
E-04
6.40
E-0
4
6.74
E-04
c6
0100
200300
400
3.10
E-06
1.23
E-05
2.15
E-05
3.07
E-05
3.99
E-05
4.91
E-05
5.83
E-05
6.75
E-05
7.67
E-05
8.58
E-05
7PM model-parameter 7PM model-parameter constraintsconstraintsFoliage
Metabolic L. C Structure L. C Microbes C
Woody Fine roots
Soil C
c1
050
100150200250
1.25
E-0
3
1.32
E-0
3
1.40
E-0
3
1.47
E-0
3
1.54
E-0
3
1.62
E-0
3
1.69
E-0
3
1.76
E-0
3
1.84
E-0
3
1.91
E-0
3
c2
050
100150200250300
1.16
E-0
5
2.85
E-0
5
4.53
E-05
6.22
E-0
5
7.90
E-05
9.59
E-0
5
1.13
E-04
1.30
E-04
1.47
E-0
4
1.63
E-0
4
c3
050
100150200250300
4.25
E-0
3
4.42
E-0
3
4.59
E-03
4.77
E-0
3
4.94
E-03
5.12
E-0
3
5.29
E-03
5.46
E-03
5.64
E-0
3
5.81
E-0
3
c4
0
50
100
150
3.26
E-0
3
5.97
E-0
3
8.69
E-03
1.14
E-0
2
1.41
E-02
1.68
E-0
2
1.95
E-02
2.23
E-02
2.50
E-0
2
2.77
E-0
2
c5
050
100150200250300
1.30
E-0
4
3.49
E-0
4
5.67
E-04
7.85
E-0
4
1.00
E-03
1.22
E-0
3
1.44
E-03
1.66
E-03
1.88
E-0
3
2.10
E-0
3
c6
050
100150200250300
5.24
E-03
6.19
E-03
7.14
E-03
8.08
E-03
9.03
E-03
9.98
E-03
1.09
E-02
1.19
E-02
1.28
E-02
1.38
E-02
c7
050
100150200250
8.40
E-06
2.66
E-05
4.48
E-05
6.30
E-05
8.12
E-05
9.94
E-05
1.18
E-04
1.36
E-04
1.54
E-04
1.72
E-04
8P model-parameter 8P model-parameter constraintsconstraints
c1
0200400600800
9.92
E-04
1.26
E-03
1.54
E-03
1.81
E-03
2.08
E-0
3
2.35
E-03
2.62
E-03
2.90
E-03
c2
0100200300400500600
1.47
E-05
5.04
E-05
8.60
E-05
1.22
E-04
1.57
E-0
4
1.93
E-04
2.29
E-0
4
2.64
E-0
4
c3
0200400600800
3.82
E-0
3
4.29
E-03
4.75
E-0
3
5.22
E-03
5.68
E-03
6.15
E-03
6.62
E-03
7.08
E-03
c4
050
100150200250300
2.45
E-03
6.18
E-03
9.91
E-03
1.36
E-0
2
1.74
E-02
2.11
E-02
2.48
E-02
2.86
E-02
c5
0100200300400
1.47
E-04
5.05
E-04
8.62
E-04
1.22
E-03
1.58
E-0
3
1.93
E-03
2.29
E-0
3
2.65
E-0
3
c6
0500
10001500200025003000
9.36
E-03
3.52
E-02
6.10
E-02
8.67
E-02
1.13
E-01
1.38
E-01
c7
0100200300400500600
4.28
E-05
1.21
E-04
2.00
E-04
2.78
E-04
3.57
E-04
4.35
E-04
5.13
E-04
5.92
E-04
c8
0100200300400
5.00
E-07
1.70
E-06
2.90
E-06
4.10
E-06
5.30
E-06
6.50
E-06
7.60
E-06
8.80
E-06
ConclusionConclusion
Differences in model structure Differences in model structure are corresponding to different are corresponding to different sets of parameters. The number sets of parameters. The number of constrained parameters varies of constrained parameters varies with data availability with data availability