1. 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

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

3. 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. 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. 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

6. 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. 7 Domain of Application

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

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

10. 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

11. 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

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

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

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

15. 15 Pearson Distribution System  Seven Types  4 parameters  Contains popular PDFs: Beta, Normal, Gamma, Student-t  Classification based on  Skewness  1  Kurtosis  2 II 11 22 unbounded bounded impossible semi- bounded

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

17. 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. 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. 19 Pearson Type Classification

20. 20 Exponential Power Family

21. 21 Results

22. 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)

23. 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. 24 Thank You! Luc Huyse & Ben Thacker Southwest Research Institute San Antonio, TX