Transcript Presentation - University of Toledo

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
1
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)
2
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
3
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
4
Outline
• Polynomial Chaos Expansion (PCE)
Approximation of a Probability Density
Function
• Finding bounds of failure probability
• Example
• Conclusion
5
Polynomial Chaos Expansion
(PCE) Approximation
• Random variable X weighted sum of basis
functions:
Hermite polynomials

X   b i Γ i (ξ)
i 1
Standard normal variable
Fourier coefficients
6
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
7
Finding PDF of random variable
.
nr f ( )
f X ( x)    i
i 1 dx
d   1( x)
i
Slope of x()
6 10
4
4 10
4
2 10
4
PDF
Standard normal PDF
0
4
8.5 10
9 10
4
9.5 10
1 10
Stress
4
5
1.05 10
5
1.1 10
5
nr=4, mean value 98,000, standard
deviation 6,000, skewness -1.31 and
kurtosis 5.35
8
Finding bounds of failure probability
• Dual optimization problem formulation
• Find the Fourier coefficients b
To Maximize (Minimize) PF(b)
• Such that: θ  [θ, θ].
Shape measures, quantiles
9
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:
1
If PF (θ)   I i (x i ), x i drawn from f Xs (x i , θ)
n i
then
f X (x i , θ)
1
PF (θ)   I i (x i )
n i
f Xs (x i , θ)
10
Weighting a sample to calculate failure probability
for many values of distribution parameters
.
.
Weighting a sample, 1000 replications
Sample values that caused failure
Natural Frequency of Absorber
1.3
Absorber natural frequency
1.2
1.1
1
0.9
0.8
1.2
1
0.8
0.7
0.6
0.6
0.7
0.8
0.9
1
1.1
1.2
Primary system natural frequency
Orginal Sample
Weighted
Weighted
1.3
0.6
0.8
0.85
0.9 0.95
1
1.05 1.1
Natural Frequency of Primary System
1.15
Sampling PDF
Weighted
11
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
12
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
13
Family of admissible stress PDFs
5
810
Skew=0, Kurtosis=3
Skew=-0.83, Kurtosis=4
Skew=0.83, Kurtosis=4
Expert judgment :
skewness  [-1,1]
PDF
mean  98,000 psi,
st. dev.  6,000 psi,
5
610
5
410
5
210
kurtosis  [3,4]
0
4
610
810
4
5
110
1.210
5
5
1.410
Stress (psi)
14
Strength PDF
5
510
Rod 1
Rod 2
5
410
PDF
5
310
5
210
5
110
0
4
510
5
110
5
1.510
5
210
Strength (psi)
15
Decision rule: compare failure
probabilities
Rod 2
Rod 1
PFmax
PFmin
PFmin PFmax
Select Rod 1
Rod 1
PFmin
PFmax
Rod 2
PFmax
PFmin
Indecision
16
Results
Minimum and maximum failure probabilities of the two rods and the corresponding
utilities
95% confidence intervals are provided under each estimate
Rod
Minimum pf
1
7.21×10-3
-3
6.94×10
2
Maximum pf
-3
7.48×10
2.36×10-3
-3
2.30×10
0.014
Minimum
U(pf)
0.557
Maximum
U(pf)
0.438
0.0127 0.0153
0.564 0.551
0.456 0.421
0.728
0.509
0.732 0.724
0.545 0.479
9.47×10-3
-3
2.43×10
-3
7.73×10
0.011×10
-3
Cannot decide which rod is better
17
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
18
Results
Maximum pf1-pf2
Minimum
U(pf1)-U(pf2)
Minimum pf1-pf2
0.014 - 9.5×10-3
-0.171
7.2 ×10-3 -2.36×10-3
Maximum
U(pf1)-U(pf2)
-0.072
Decision: Design 2 is better than 1
19
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.
20