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Efstratios Nikolaidis The University of Toledo Zissimos P. Mourelatos Oakland University

Imprecise Reliability Assessment and Decision-Making when the Type of the Probability Distribution of the Random Variables is Unknown. Efstratios Nikolaidis The University of Toledo Zissimos P. Mourelatos Oakland University. Introduction. Decision under uncertainty with limited data

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Efstratios Nikolaidis The University of Toledo Zissimos P. Mourelatos Oakland University

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  1. Imprecise Reliability Assessment and Decision-Making when the Type of the Probability Distribution of the Random Variables is Unknown Efstratios Nikolaidis The University of Toledo Zissimos P. Mourelatos Oakland University

  2. Introduction Decision under uncertainty with limited data • Complete probabilistic model of inputs: joint PDF • Uncertainty in PDF: • Distribution parameters • Type • Estimate of reliability of a design and selection of best design depends on assumed PDF • Approaches for modeling uncertainty • Guidelines to select type: maximum entropy, insufficient reason principle • Non parametric PDF: Gaussian process, Polynomial Chaos Expansion (PCE)

  3. Problem definition • Given statistical summaries (shape measures, credible intervals) find reliability bounds • Scope: Independent random variables • Representation of dependence • Perfect and opposite dependence • Copulas • Nataf transformation

  4. Approach Judgment and data Statistical summaries (credible intervals and shape measures) Family of PDFs consistent with data Optimizer Approximation of reliability Reliability (failure probability) bounds Selection of best design

  5. Outline • Polynomial Chaos Expansion (PCE) Approximation of a Probability Density Function • Finding bounds of failure probability • Example • Conclusion

  6. Polynomial Chaos Expansion (PCE) Approximation • Random variable X weighted sum of basis functions: Hermite polynomials Standard normal variable Fourier coefficients

  7. Polynomial Chaos Expansion (PCE) Approximation Pros: • Flexible: can represent a rich class of variables • Sufficiently general to represent variables with arbitrary PDFs • Values of Fourier coefficients can be found efficiently using information about statistical summaries (moments, credible intervals, percentiles) • Easy to generate sample values of approximated random variable Limitations: • No closed form expression of PDF • Difficult to represent heavy tailed PDFs (large probabilities of values that are many ’s away from the mean • PDF can exhibit irregularities for some combination of values of statistical summaries Alternative representations of unknown PDF: basis vectors can be Askey, Laguerre, Jacobi, or Legendre polynomials

  8. Finding PDF of random variable Standard normal PDF Slope of x() nr=4, mean value 98,000, standard deviation 6,000, skewness -1.31 and kurtosis 5.35

  9. Finding bounds of failure probability • Dual optimization problem formulation • Find the Fourier coefficients bTo Maximize (Minimize) PF(b) • Such that: . Shape measures, quantiles

  10. Efficient Probabilistic Re-analysis First, calculate the failure probability, PF(θ), for one sampling PDF. Then calculate the failure probability, PF(θ), for many sets of values of the parameters θby re weighting the same sample:

  11. Weighting a sample to calculate failure probability for many values of distribution parameters

  12. Properties of estimated failure probability • Can quantify accuracy of failure probability estimate; standard deviation and confidence intervals • Analytical expressions for sensitivity derivatives of failure probability • Estimate of failure probability varies smoothly with distribution parameters

  13. Example • Select one of two rods: • Strength, known PDF, Weibull • Unknown PDF of stress, know mean value, standard deviation and ranges for skewness and kurtosis • Criterion: failure probability

  14. Family of admissible stress PDFs

  15. Strength PDF

  16. Decision rule: compare failure probabilities Rod 2 Rod 1 PFmax PFmin PFmax PFmin Select Rod 1 Rod 1 Rod 2 PFmin PFmax PFmax PFmin Indecision

  17. Results Cannot decide which rod is better

  18. Alternative decision rule: Calculate and compare probability difference 0 PF1-PF2 Design 2 better than 1 because designer is always better off exchanging design 2 for 1. Stochastic (state-by-state) dominance. 0 Still cannot decide which design is better

  19. Results Decision: Design 2 is better than 1

  20. Conclusion • Challenge: make decisions when type of PDF of random variables is unknown • Proposed approach • Model family of PDFs consistent with available evidence by PCE • Presented and demonstrated procedure for making design decisions • Comparing alternatives in terms of failure probabilities may lead to indecision. Can break tie by considering difference in failure probabilities.

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