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. Travis V. Anderson Presentation Outline 2 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

3. Travis V. Anderson 3 Section 1: Introduction & Motivation

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

5. 5 F-35 Joint-Strike Fighter

6. Travis V. Anderson 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. Travis V. Anderson 7 Section 2: Uncertainty Analysis Methods

8. Travis V. Anderson Uncertainty Analysis Methods 8 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

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

10. Travis V. Anderson 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. Travis V. Anderson 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. Travis V. Anderson 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. Travis V. Anderson 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. Travis V. Anderson 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. Travis V. Anderson 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. Travis V. Anderson 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. Travis V. Anderson 17 Section 3: Propagation of Variance

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

19. Travis V. Anderson First Order Taylor Series 19

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

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

22. Travis V. Anderson Assumptions and Limitations 22 1.The approximation is generally more accurate for linear models  This Section 2.Only variance is propagated and higher-order statistics are neglected  Section 4 3.All inputs are assumed be Gaussian  Section 4 4.System outputs and output derivatives can be obtained 5.Taking the Taylor series expansion about a single point causes the approximation to be of local validity only 6.The input means and standard deviations must be known 7.All inputs are assumed to be independent

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

24. Travis V. Anderson Second-Order Error Propagation 24 Just as before: 1.Subtract the expectation of a second-order Taylor series from a second-order Taylor series 2.Square both sides, and take the expectation   Odd moments are zero Assumption: Inputs are Gaussian

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

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

28. Travis V. Anderson Computational Cost 28

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

30. Travis V. Anderson Predicting Truncation Error 30 Can Truncation Error Be Predicted?

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

32. Travis V. Anderson Trigonometric Correction Factor 32

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

34. Travis V. Anderson Correction Factors where: Exponential (1 st Order): y = b x 34

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

36. Travis V. Anderson Kinematic Motion of Flapping Wing 36

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

38. Travis V. Anderson Computational Cost 38 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

39. Travis V. Anderson 39 Section 4: Propagation of Skewness & Kurtosis

40. Travis V. Anderson Non-Gaussian Error Propagation 40 Predicted Gaussian Output Actual System Output Predicted Non-Gaussian Output Actual System Output

41. Travis V. Anderson Skewness 41 Measure of a distribution’s asymmetry A symmetric distribution has zero skewness

42. Travis V. Anderson Propagation of Skewness 42 Based on a second-order Taylor series

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

44. Travis V. Anderson Propagation of Kurtosis 44 Based on a second-order Taylor series

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

46. Travis V. Anderson Input Distribution 46

47. Travis V. Anderson Gaussian Error Propagation 47 Probability Overlap:53% Predicted Gaussian Output Actual System Output

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

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

50. Travis V. Anderson 50 Section 5: Conclusion & Future Work

51. Travis V. Anderson Conclusion Fourth-order accuracy in variance propagation can be achieved with only first- or second-order computational cost Designers do not need to assume Gaussian output.  A fully-described output distribution can be obtained without significant additional cost 51

52. Travis V. Anderson Future Work Develop predictable correction factors for other types of nonlinear functions and models (differential equations, state-space models, etc.) Apply correction factors to open-form models Can correction factors be obtained for skewness and kurtosis propagation? 52

53. Travis V. Anderson Questions? 53

54. Travis V. Anderson 54

55. Travis V. Anderson Variance Example: Whirlybird 55

56. Travis V. Anderson Variance Example: Whirlybird 56 Compositional Model System Model (Pitch)

57. Travis V. Anderson Higher-Order Stats Example: Thrust 57 Thrust Output

58. Travis V. Anderson Higher-Order Stats Example: Thrust 58 Input Distribution Gaussian Output Non-Gaussian Output Actual Output Overlap: 65% Overlap: 79%

59. Travis V. Anderson Non-Gaussian Proof 59 Propagation of Skewness Even Gaussian Inputs Produce Skewed Outputs If 2 nd Derivatives Are Non-Zero (Nonlinear Systems)