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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

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Uncertainty & Sensitivity analysis Techniques Design of experiments method Sampling-based method Fourier Amplitude Sensitivity Test (FAST) Sobol’s method ANOVA method Moment independent approaches

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FAST Search function: Fourier transformation: Variance decomposition: Sum spectrum

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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;

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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; 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. for each parameter. ↑

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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 = x 1 + x 2, correlation=0.7, characteristic frequency is 5 and 23 respectively. Independent sample orderCorrelated sample order of X 1 Response Variable Y

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Reorder for correlation: Model Y = x 1 + x 2, correlation=0.7,common characteristic frequency is 5 for both x 1 and x 2 Independent sample order Correlated sample order of X 1 Response Variable Y

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correlation=0.0 correlation=0.2 correlation=0.5correlation=0.9

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FAST sample with common characteristic frequency Reordered sample Model outputs based on reordered sample Sample for FAST analysis of x 1 Sample for FAST analysis of x 2 Sample for the common variable s 1) 2) 3) 4) 5) 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 ModelCharacteristic Frequency Sample SizeMaximum harmonic order Test case one2392114 Test case two23461 4 for x 1 -x 4 2 for others Test case three23461 4 for x 1 -x 3 2 for others Test case four23461 4 for x 2 -x 5 2 for others

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Test case one: Y=2x 1 +3x 2, 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) Circles are based on correlation ratio method by Saltelli(2001) based McKay’s one-way ANOVA. Nonparametric method suitable for nonlinear and monotonic models. 50,000 model runs (=100 replications x 500 samples) Rank correlation

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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 ParameterLabelLower boundUpper bound x1x1industrial output per capita desired315385 x2x2industrial capital output ratio before 19952.73.3 x3 fraction of industrial output allocated to consumption before 1995 0.3870.473 X4 fraction of industrial output allocated to consumption after 1995 0.3870.473 X5 average life of industrial capital before 1995 12.615.4 X6 average life of industrial capital after 1995 16.219.8 x7x7initial industrial capital1.89(10+11)2.31(10+11) Rank correlation of.6 between x3 and x4 and.4 between x5 and x6

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x2x2 x3x3 x4x4 x5x5 Year Sensitivity (Correlation ratio method (CRM). 50,000 model runs (=100 replications x 500 samples)

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Assume Independence Assume correlation

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Application: Uncertainty in forest landscape response to global warming PnET-II Temperature Precipitation ANPP SEP LANDIS-II Landscape composition CO2 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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(c) (d) (b) (a) 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)Intergovernmental Panel on Climate Change VariableNMeanStd DevMedianMinimumMaximum Temperature277.435762.360657.457143.9733712.58936 PAR27571.2629832.19750566.47531522.93955633.86171 Precipitation2792.536736.9325592.7303979.19724105.41716 CO227689.34074111.10015699.40000546.80000923.25000 PAR is photosynthetic active radiation

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Rank correlation structure Spearman Correlation Coefficients, N = 27 Prob > |r| under H0: Rho=0 TemperaturePARPrecipitationCO2 Temperature1.000000.69780 <.0001 0.23871 0.2305 0.40934 0.0340 PAR0.69780 <.0001 1.000000.26862 0.1755 0.06963 0.7300 Precipitation0.23871 0.2305 0.26862 0.1755 1.000000.05495 0.7854 CO20.40934 0.0340 0.06963 0.7300 0.05495 0.7854 1.00000 PAR is photosynthetic active radiation

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Uncertainty Spruce-firPine Aspen-birch Maple-ash Simulation year Uncertainty

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Sensitivity Spruce-firPine Aspen-birch Maple-ash Simulation year Sensitivity

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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!

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