Imprecise Reliability Assessment and Decision-Making when the Type of the Probability Distribution of the Random Variables is Unknown

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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

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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)

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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

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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
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Outline

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

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Polynomial Chaos Expansion (PCE) Approximation

• Random variable X weighted sum of basis
  functions

Hermite polynomials
Standard normal variable
Fourier coefficients
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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

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Finding PDF of random variable
Standard normal PDF
Slope of x(?)
nr4, mean value 98,000, standard deviation
6,000, skewness -1.31 and kurtosis 5.35
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Finding bounds of failure probability

• Dual optimization problem formulation
• Find the Fourier coefficients b
  
  To Maximize (Minimize) PF(b)
• Such that .

Shape measures, quantiles
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Efficient Probabilistic Re-analysis

• First, calculate the failure probability, PF(?),
  for one sampling PDF. Then calculate the failure
  probability, PF(??), for many sets of values of
  the parameters ?? by re weighting the same
  sample

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

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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

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Family of admissible stress PDFs
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Strength PDF
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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
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Results
Cannot decide which rod is better
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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
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Results
Decision Design 2 is better than 1
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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.