A general first-order global sensitivity analysis method

1
A general first-order global sensitivity analysis
method

• Chonggang Xu, George Z. Gertner
• Department of Natural Resources and Environmental
  Sciences,
• University of Illinois at Urbana-Champaign
• USA

2
Uncertainty Sensitivity analysis Techniques

• Design of experiments method
• Sampling-based method
• Fourier Amplitude Sensitivity Test (FAST)
• Sobols method
• ANOVA method
• Moment independent approaches

3
FAST
Search function
Fourier transformation
Sum spectrum
Variance decomposition
4
FAST advantages limitations

• Computationally efficient global sensitivity
  analysis method
• Suitable for nonlinear and non-monotonic models

• Aliasing effects for small sample
  sizes(frequency interference)
• Suitability for only models with independent
  parameters

5
Real applications

• Dependence among parameters
• Complex model with many parameters, which needs
  much computation times and large sample sizes in
  traditional FAST

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FAST improvement

• Reorder
• ? independent parameters limitation
• Random Balance Design (Tarantola et al. 2006)
  ? commom frequency for all parameters
• and permuting
• ? Overcome aliasing effect limitation

?
for each parameter.
Tarantola, S, Gatelli, D, Mara, TA. Random
balance designs for the estimation of first order
global sensitivity indices. Reliability
Engineering and System Safety, 2006 91(6)
717-727.
7
Details of synthesized FAST

• Xu, C. and G.Z. Gertner 2007. A general
  first-order global sensitivity analysis method.
  Reliability Engineering and System Safety (In
  press).

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Reorder for correlation Model Y x1 x2,
correlation0.7, characteristic frequency is 5
and 23 respectively.
Independent sample order
Correlated sample order of X1
Response Variable Y
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Reorder for correlation Model Y x1
x2,correlation0.7,common characteristic
frequency is 5 for both x1 and x2
Correlated sample order of X1
Independent sample order
Response Variable Y
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correlation0.0
correlation0.2
correlation0.5
correlation0.9
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FAST sample with common characteristic frequency
6)
Sample for FAST analysis of x1
Reordered sample
5)
3)
Model outputs based on reordered sample
4)
2)
5)
1)
Sample for the common variable s
Sample for FAST analysis of x2
6)
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Maximum Harmonic order selection
Scaled characteristic spectrum
1. Random balance design may introduce random
error. 2. Assume that the low characteristic
amplitudes at high harmonic order are more
susceptible to the random error than relatively
high characteristic amplitudes at a low
harmonic order .
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Simulated annealing refinement for correlated
samples
PAR is photosynthetic active radiation
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TEST CASES Synthesized FAST specification for
test cases
Model Characteristic Frequency Sample Size Maximum harmonic order
Test case one 23 921 14
Test case two 23 461 4 for x1-x4 2 for others
Test case three 23 461 4 for x1-x3 2 for others
Test case four 23 461 4 for x2-x5 2 for others
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Test case one
Y2x13x2 , where x1 and x2 are standard normally
distributed with a Pearson correlation
coefficient of 0.7
SFAST is synthesized FAST Circles are analytical
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Test case two (Lu and Mohanty, 2001)
Rank correlation
Circles are based on correlation ratio method by
Saltelli(2001) based McKays one-way ANOVA.
Nonparametric method suitable for nonlinear and
monotonic models. 50,000 model runs (100
replications x 500 samples)
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Test case three
(G-function of Sobol. Non-monotonic test model)
Rank correlation
Circles are based on correlation ratio method .
50,000 model runs (100 replications x 500
samples)
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Test case four World 3 Model(Meadows et al.,
1992)
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Parameter uncertainty specification
Parameter Label Lower bound Upper bound
x1 industrial output per capita desired 315 385
x2 industrial capital output ratio before 1995 2.7 3.3
x3 fraction of industrial output allocated to consumption before 1995 0.387 0.473
X4 fraction of industrial output allocated to consumption after 1995 0.387 0.473
X5 average life of industrial capital before 1995 12.6 15.4
X6 average life of industrial capital after 1995 16.2 19.8
x7 initial industrial capital 1.89(1011) 2.31(1011)
Rank correlation of .6 between x3 and x4 and .4
between x5 and x6
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Sensitivity
Year
(Correlation ratio method (CRM). 50,000 model
runs (100 replications x 500 samples)
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Assume correlation
Assume Independence
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Application Uncertainty in forest landscape
response to global warming
PnET-II is a forest ecosystem process model
(LINKAGES) is a forest GAP model LANDIS is a
spatially dynamic forest landscape model)
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Example of data after 1994 is based on prediction
by Canadian Climate Center (CCC) in the Phase-II
Vegetation-Ecosystem Modeling and Analysis
Project (VEMAP). Data before 1994 is historical
data. We assume the climate stabilizes after year
2099.
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Parameter uncertainty
Based on 27 predicted climate data structure from the Intergovernmental Panel on Climate Change (IPCC) Third and Fourth Assessment Report , and the Phase-II Vegetation-Ecosystem Modeling and Analysis Project (VEMAP) Based on 27 predicted climate data structure from the Intergovernmental Panel on Climate Change (IPCC) Third and Fourth Assessment Report , and the Phase-II Vegetation-Ecosystem Modeling and Analysis Project (VEMAP) Based on 27 predicted climate data structure from the Intergovernmental Panel on Climate Change (IPCC) Third and Fourth Assessment Report , and the Phase-II Vegetation-Ecosystem Modeling and Analysis Project (VEMAP) Based on 27 predicted climate data structure from the Intergovernmental Panel on Climate Change (IPCC) Third and Fourth Assessment Report , and the Phase-II Vegetation-Ecosystem Modeling and Analysis Project (VEMAP) Based on 27 predicted climate data structure from the Intergovernmental Panel on Climate Change (IPCC) Third and Fourth Assessment Report , and the Phase-II Vegetation-Ecosystem Modeling and Analysis Project (VEMAP) Based on 27 predicted climate data structure from the Intergovernmental Panel on Climate Change (IPCC) Third and Fourth Assessment Report , and the Phase-II Vegetation-Ecosystem Modeling and Analysis Project (VEMAP) Based on 27 predicted climate data structure from the Intergovernmental Panel on Climate Change (IPCC) Third and Fourth Assessment Report , and the Phase-II Vegetation-Ecosystem Modeling and Analysis Project (VEMAP)
Variable N Mean Std Dev Median Minimum Maximum
Temperature 27 7.43576 2.36065 7.45714 3.97337 12.58936
PAR 27 571.26298 32.19750 566.47531 522.93955 633.86171
Precipitation 27 92.53673 6.93255 92.73039 79.19724 105.41716
CO2 27 689.34074 111.10015 699.40000 546.80000 923.25000
PAR is photosynthetic active radiation
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Rank correlation structure
Spearman Correlation Coefficients, N 27 Prob gt r under H0 Rho0 Spearman Correlation Coefficients, N 27 Prob gt r under H0 Rho0 Spearman Correlation Coefficients, N 27 Prob gt r under H0 Rho0 Spearman Correlation Coefficients, N 27 Prob gt r under H0 Rho0 Spearman Correlation Coefficients, N 27 Prob gt r under H0 Rho0
  Temperature PAR Precipitation CO2
Temperature 1.00000 0.69780lt.0001 0.238710.2305 0.409340.0340
PAR 0.69780lt.0001 1.00000 0.268620.1755 0.069630.7300
Precipitation 0.238710.2305 0.268620.1755 1.00000 0.054950.7854
CO2 0.409340.0340 0.069630.7300 0.054950.7854 1.00000
PAR is photosynthetic active radiation
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Uncertainty
Aspen-birch
Maple-ash
Uncertainty
Spruce-fir
Pine
Simulation year
27
Sensitivity
Aspen-birch
Maple-ash
Sensitivity
Spruce-fir
Pine
Simulation year
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Conclusion

• Proposes a general first-order global sensitivity
  approach for linear/nonlinear models with as many
  correlated or uncorrelated parameters as the user
  specifies
• FAST is computationally efficient and would be a
  good choice for uncertainty and sensitivity
  analysis for models with correlated parameters

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Conclusion

• Proposes a general first-order global sensitivity
  approach for linear/nonlinear models with as many
  correlated or uncorrelated parameters as the user
  specifies
• FAST is computationally efficient and would be a
  good choice for uncertainty and sensitivity
  analysis for models with correlated parameters

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Thank You!