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A Framework to Estimate Uncertain Random Variables

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. Outline.

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A Framework to Estimate Uncertain Random Variables

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  1. 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

  2. Outline • Background & Motivation • Central tenet: Distribution Systems • Application to Synthetic Data

  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

  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)

  5. Vague Information: Example • Compute Pr[z>1.17] where • Variables x & y are independent uncertain parameters • Consider 3 cases: Case 1 Case 2 Case 3 X X X 0.1 1 0.1 1 0.1 1 Y y1 y1 0.0 1 0.6 0.5 0.7 y2 y2 0.6 0.4 0.8 y3 y3 0.4 0.1 0.7 y4 y4 0.5 0.3 0.7 Oberkampf, et al, “Challenge Problems: Uncertainty in System Response Given Uncertain Parameters,” Technical report, Sandia National Laboratories, August 2002.

  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

  7. Domain of Application

  8. Outline • Background & Motivation • Central tenet: Distribution Systems • Application to Synthetic Data

  9. Parametric Distributions: The Normal • Normal distribution • Mean • Std Deviation • Upon standardization no parameters left in equation

  10. Standard Beta distribution: Various shapes Two parameters Still fixed shape for given mean and standard deviation Beta PDF: More Flexibility • Need generalization: 4 parameter Beta family

  11. Parametric Distribution Systems • Have additional parameters • Location • Scale • Shape • Parameters remaining in PDFequation after standardization • This idea is not new: • Systems for extremes or tails • Systems for bulk of data

  12. Tail Distribution System • Three EVD Types • Gumbel • Frechet • Weibull • Generalized EVD

  13. Bulk Distribution Systems • Various systems have been developed and being considered: • Exponential Power • Pearson System • Others: • Johnson Transformation • Gram-Charlier Expansions • …

  14. Exponential Power Family • Generalization of Normal distribution: exponent

  15. impossible II bounded b2 semi- bounded unbounded b1 Pearson Distribution System • Seven Types • 4 parameters • Contains popular PDFs: Beta, Normal, Gamma, Student-t • Classification based on • Skewness b1 • Kurtosis b2

  16. Outline • Background & Motivation • Central tenet: Distribution Systems • Application to Synthetic Data

  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

  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

  19. Pearson Type Classification

  20. Exponential Power Family

  21. Results

  22. Comparison • Both families retrieve Normal if sample is large enough

  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

  24. Thank You! Luc Huyse & Ben Thacker Southwest Research Institute San Antonio, TX

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