Travis V. Anderson July 26, 2011 Graduate Committee: Christopher A. Mattson

1. Efficient, Accurate, and Non-Gaussian Statistical Error Propagation Through Nonlinear System Models Travis V. Anderson July 26, 2011 Graduate Committee: Christopher A. Mattson David T. Fullwood Kenneth W. Chase

2. Presentation Outline Section 1: Introduction & Motivation Section 2: Uncertainty Analysis Methods Section 3: Propagation of Variance Section 4: Propagation of Skewness & Kurtosis Section 5: Conclusion & Future Work 2

3. Section 1: Introduction & Motivation 3

4. Engineering Disasters Tacoma Narrows Bridge Space Shuttle Challenger Hindenburg Chernobyl

5. F-35 Joint-Strike Fighter 5

6. Research Motivation • Allow the system designer to quantify system model accuracy more quickly and accurately • Allow the system designer to verify design decisions at the time they are made • Prevent unnecessary design iterations and system failures by creating better system designs 6

7. Section 2: Uncertainty Analysis Methods 7

8. Uncertainty Analysis Methods • Error Propagation via Taylor Series Expansion • Brute Force Non-Deterministic Analysis (Monte Carlo, Latin Hypercube, etc.) • Deterministic Model Composition • Error Budgets • Univariate Dimension Reduction • Interval Analysis • Bayesian Inference • Response Surface Methodologies • Anti-Optimizations 8

9. Brute Force Non-Deterministic Analysis • Fully-described, non-Gaussian output distribution can be obtained • Simulation must be executed again each time any input changes • Computationally expensive 9

10. Deterministic Model Composition • A compositional system model is created • Each component’s error is included in an error-augmented system model • Component error values are varied as the model is executed repeatedly to determine max/min error bounds 10

11. Error Budgets • Error in one component is perturbed at a time • Each perturbation’s effect on model output is observed • Either errors must be independent or a separate model of error interactions is required 11

12. Univariate Dimension Reduction • Data is transformed from a high-dimensional space to a lower-dimensional space • In some situations, analysis in reduced space may be more accurate than in the original space 12

13. Interval Analysis • Measurement and rounding errors are bounded • Arithmetic can be performed using intervals instead of a single nominal value • Many software languages, libraries, compilers, data types, and extensions support interval arithmetic • XSC, Profil/BIAS, Boost, Gaol, Frink, MATLAB (Intlab) • IEEE Interval Standard (P1788) 13

14. Bayesian Inference • Combines common-sense knowledge with observational evidence • Meaningful relationships are declared, all others are ignored • Attempts to eliminate needless model complexity 14

15. Response Surface Methodologies • Typically uses experimental data and design of experiments techniques • An n-dimensional response surface shows the output relationship between n-input variables 15

16. Anti-Optimizations • Two-tiered optimization problem • Uncertainty is anti-optimized on a lower level to find the worst-case scenario • The overall design is then optimized on a higher-level to find the best design 16

17. Section 3: Propagation of Variance 17

18. Central Moments • 0th Central Moment is 1 • 1st Central Moment is 0 • 2nd Central Moment is variance • 3rd Central Moment is used to calculate skewness • 4th Central Moment is used to calculate kurtosis 18

19. First Order Taylor Series 19

20. First-Order Formula Derivation Square and take the Expectation of both sides: Covariance Term • Assumption: • Inputs are independent 20

21. First-Order Error Propagation • Formula for error propagation most-often cited in literature • Frequently used “blindly” without an appreciation of its underlying assumptions and limitations 21

22. Assumptions and Limitations • The approximation is generally more accurate for linear models  This Section • Only variance is propagated and higher-order statistics are neglected  Section 4 • All inputs are assumed be Gaussian Section 4 • System outputs and output derivatives can be obtained • Taking the Taylor series expansion about a single point causes the approximation to be of local validity only • The input means and standard deviations must be known • All inputs are assumed to be independent 22

23. First-Order Accuracy Function: y = 1000sin(x) Input Variance: 0.2 100% Error Unacceptable! 23

24. Second-Order Error Propagation Just as before: • Subtract the expectation of a second-order Taylor series from a second-order Taylor series • Square both sides, and take the expectation  • Assumption: • Inputs are Gaussian  Odd moments are zero 24

25. Second-Order Error Propagation • Second-order formula for error propagation most-often cited in literature • Like the first-order approximation, the second-order approximation is also frequently used “blindly” without an appreciation of its underlying assumptions and limitations

26. Second-Order Accuracy Function: y = 1000sin(x) Input Variance: 0.2 26

27. Higher-Order Accuracy Function: y = 1000sin(x) Input Variance: 0.2 27

28. Computational Cost 28

29. Predicting Truncation Error • How can we achieve higher-order accuracy with lower-order cost? 29

30. Predicting Truncation Error • Can Truncation Error Be Predicted? 30

31. Adding A Correction Factor Trigonometric (2nd Order): y = sin(x) or y = cos(x) 31

32. Trigonometric Correction Factor 32

33. Correction Factors Natural Log (1st Order): y = ln(x) Exponential (1st Order): y = exp(x) 33

34. Correction Factors Exponential (1st Order): y = bx where: 34

35. So What Does All This Mean? • We can achieve higher-order accuracy with lower-order computational cost Average Error Computational Cost 35

36. Kinematic Motion of Flapping Wing 36

37. Accuracy of Variance Propagation Order 2nd: 3rd: 4th: CF: RMS Rel. Err. 40.97% 11.18% 1.32% 1.96% 37

38. Computational Cost Execution time was reduced from ~70 minutes to ~4 minutes  A computational cost reduction by 1750% Fourth-order accuracy was obtained with only second-order computational cost 38

39. Section 4: Propagation of Skewness & Kurtosis 39

40. Non-Gaussian Error Propagation Predicted Gaussian Output Actual System Output Predicted Non-Gaussian Output Actual System Output 40

41. Skewness • Measure of a distribution’s asymmetry • A symmetric distribution has zero skewness 41

42. Propagation of Skewness • Based on a second-order Taylor series 42

43. Kurtosis & Excess Kurtosis • Measure of a distribution’s “peakedness” or thickness of its tails Kurtosis Excess Kurtosis 43

44. Propagation of Kurtosis • Based on a second-order Taylor series 44

45. Flat Rolling Metalworking Process Maximum change in material thickness achieved in a single pass Roller Radius Coefficient of Friction 45

46. Input Distribution 46

47. Gaussian Error Propagation • Probability Overlap: 53% Predicted Gaussian Output Actual System Output 47

48. Non-Gaussian Error Propagation • Probability Overlap: 93% Predicted Non-Gaussian Output Actual System Output 48

49. Benefits of Higher-Order Statistics That’s a 263% reduction in the number of passes! Gaussian Non-Gaussian Accuracy: Max ΔH: (99.5% success rate) 53% 3.0 cm 93% 7.9 cm 49

50. Section 5: Conclusion & Future Work 50