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Southwest Research Institute, San Antonio, Texas A Framework to Estimate Uncertain Random Variables Luc Huyse and Ben H. Thacker Reliability and Materials Integrity Luc.Huyse@swri.org, Ben.Thacker@swri.org 45th Structures, Structural Dynamics and Materials (SDM) Conference 19-22 April 2004 Palm Springs, CA
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2 Outline Background & Motivation Central tenet: Distribution Systems Application to Synthetic Data
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3 Increased Reliance on Simulation Modern Applications Higher performance requirements Revolutionary designs Shorter design cycles Focused testing Application Drivers: Efficient Certification (DOD) Weapons Stockpile Stewardship (DOE) High Level Radioactive Waste Disposal (NRC) Gas Turbine Engine Certification (FAA/NASA) Industry (Aerospace, Automotive, Manufacturing) The “Shifting Paradigm” Old Models used to provide insight New Models used to make predictions
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4 Uncertainty Quantification Establishing credibility in model simulations requires uncertainty quantification Inherent uncertainty Cannot be reduced Probabilistic approach (known PDF) Propagate uncertainty through model to quantify uncertain outputs Epistemic Uncertainty Can be reduced Various approaches proposed (probabilistic, non-probabilistic) Most common source is lack of data (expert opinion)
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5 Vague Information: Example Compute Pr[z>1.17] where Variables x & y are independent uncertain parameters Consider 3 cases: Case 1 Case 2Case 3 Oberkampf, et al, “Challenge Problems: Uncertainty in System Response Given Uncertain Parameters,” Technical report, Sandia National Laboratories, August 2002. XXX Y y1y1 y2y2 y3y3 y4y4 y1y1 y2y2 y3y3 y4y4 0.11 1 1 0.6 0.4 0.5 0.4 0.1 0.3 0.7 0.8 0.7 0.01
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6 Dealing with Inherent and Epistemic Uncertainty New methodology being developed to handle case when only limited data are available Handles mix of point and interval estimates Deals with conflicting data PDF shape is treated as a uncertain Compatible within existing probabilistic analysis machinery
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7 Domain of Application
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8 Outline Background & Motivation Central tenet: Distribution Systems Application to Synthetic Data
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9 Parametric Distributions: The Normal Normal distribution Mean Std Deviation Upon standardization no parameters left in equation
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10 Beta PDF: More Flexibility Standard Beta distribution: Various shapes Two parameters Still fixed shape for given mean and standard deviation Need generalization: 4 parameter Beta family
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11 Parametric Distribution Systems Have additional parameters Location Scale Shape Parameters remaining in PDF equation after standardization This idea is not new: Systems for extremes or tails Systems for bulk of data
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12 Tail Distribution System Three EVD Types Gumbel Frechet Weibull Generalized EVD
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13 Bulk Distribution Systems Various systems have been developed and being considered: Exponential Power Pearson System Others: Johnson Transformation Gram-Charlier Expansions …
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14 Exponential Power Family Generalization of Normal distribution: exponent
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15 Pearson Distribution System Seven Types 4 parameters Contains popular PDFs: Beta, Normal, Gamma, Student-t Classification based on Skewness 1 Kurtosis 2 II 11 22 unbounded bounded impossible semi- bounded
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16 Outline Background & Motivation Central tenet: Distribution Systems Application to Synthetic Data
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17 Application Objective: determine how efficient these PDF families are Draw synthetic samples from Normal distribution Mean = 10 Standard deviation = 3 Sample sizes: n = 50, 250, 1500 Selected the Normal since it belongs to all PDF families
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18 Pearson Distribution System Normal distribution is limiting case for all 7 types Used first 4 moments to estimate coefficients Used bootstrap re-sampling to compute standard errors on coefficients (confidence intervals) Compare Pearson coefficients based on samples with the exact results for the Normal PDF
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19 Pearson Type Classification
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20 Exponential Power Family
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21 Results
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22 Comparison Both families retrieve Normal if sample is large enough 1500 dataNormalPearsonExp. Power Mean Std dev c 1 c 2 Shape 10 3 0 9.98 (0.13) 2.93 (0.88) 0.11 (0.43) -0.01 (0.02) - 9.99 (0.08) 3.01 (0.05) - -0.02 (0.07)
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23 Summary Probabilistic approach can be used even if only limited or vague data are available. Decision should be based on whether or not the variable is truly random, not the availability of data Use probabilistic sensitivity analysis to guide subsequent data collection efforts Uncertainty in PDF shape can be represented by PDF distribution systems Data determines the shape Power-Exponential system is well suited for symmetric data Improved fitting needed to use Pearson system for small data sets Uncertainty on PDF translated into uncertainty on risk
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24 Thank You! Luc Huyse & Ben Thacker Southwest Research Institute San Antonio, TX
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