bayesian ebdo savannah mourelatos feb2008 v2

8/13/2019 Bayesian EBDO Savannah Mourelatos Feb2008 V2

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REC 2008; Zissimos P. Mourelatos

Design under Uncertainty usingEvidence Theory and a Bayesian

Approach

Jun ZhouZissimos P. Mourelatos

Mechanical Engineering DepartmentOakland University

Rochester, MI 48309, [email protected]


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REC 2008; Zissimos P. Mourelatos 2

Overview

Introduction

Design under uncertainty

Uncertainty theories

Evidence Based Design Optimization ( EBDO )

Fundamentals of Bayesian Approach

Bayesian Approach to Design Optimization ( BADO )

A Combined Bayesian and EBDO Approach Example

General conclusions


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Design Under Uncertainty

Analysis /SimulationInput Output

Uncertainty(Quantified)

Uncertainty

(Calculated)

Propagation

Design

1. Quantification

2. Propagation

3. Design


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Aleatory Uncertainty (Irreducible, Stochastic)

Probabilistic distributions Bayesian updating

Epistemic Uncertainty (Reducible, Subjective,

Ignorance, Lack of Information) Fuzzy Sets; Possibility methods (non-conflicting information)

Evidence theory (conflicting information)

Uncertainty Types

25% 30%45%

Probability distribution Expert Opinion Finite samples Interval

Decreasing level of information


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Evidence-Based Design

Optimization ( EBDO )


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Set Notation

Universe

(X)

Power Set(All sets) Element

A B CAB

A Pl A P A Bel )(Evidence Theory



Evidence-Based Design Optimization (EBDO)Basic Probability Assignment (BPA): m(A)

X AIf m(A)>0 for then A is a focal element

0.2 0.1 0.4 0.3Y

5 6.2 8 9.5 11

Expert A

50.3 0.3 0.4

Y7 8.7 11

Expert B

Y0.x1

7 8.7 119.586.2

0.x20.x3 0.x60.x5

0.x4

5

Combining Rule

(Dempster Shafer)



Evidence-Based Design Optimization (EBDO)

S

a

b

BPA

S

a

b

S

a

b

BPA

BPA structure for a two-input problem




If we define,

C baccba yba f g g G cc ,,,,0,: 0then

G Pl G P G Bel )(where

0)( g P G P

and

G B

BmG Bel 0G B

BmG Pl




ming

ma xg0g

0g 0g

ming

ma xg0g

0g 0g

ming

maxg0g

0g

0g

XXXX

g g g g max,min, maxmin

Position of a focal element w.r.t. limit state

Contributes toBelief

0)0(0 g Pl g P g Bel

Contributes toPlausibility


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Evidence-Based Design Optimization (EBDO)Design Principle

Therefore,

f p g P 0 f p g Pl 0is satisfied if

R g P 0 R g Bel 0is satisfied if

OR

If non-negative null form is used for feasibility,0 g

0)0(0 g Pl g P g Bel

0)0(0 g Pl g P g Bel failure


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N N f N

p,xd,xd,

min

i f i

p g Pl 0,, PXd , ni ,...,1

U L ddd N U

N N L xxx


Formulation


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0 g

0 g

0 g

Calculation of f p g Pl 0


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Feasible Region

x2

x1

g1(x1,x2)=0

g2(x1,x2)=0ObjectiveReduces

initial design

point

frame ofdiscernment

EBDO optimum

deterministicoptimum

Evidence-Based Design Optimization (EBDO)Implementation


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25% 30%45%

Probability distribution Expert Opinion Finite samples Interval

Decreasing level of information

Available Information


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Fundamentals of Bayesian

Approach


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1120/1230/5,2

212

2121 x x x x x x g

Example

0, 21 x x g : Success 0, 21 x x g : Failure

3.0,0.3~1 N X

2 X : Bayesian uncertain variable


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Example

0, 221 X X X g P Want to calculate :

Number of sample points


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Probabilistic Constraint :

f p R g P 10,,,, PXZYd

Determ. Random Bayesian

f p R g P

10,,, PXZYd

0,,, PXZYd g

R =0.78

Confidence Percentile = 85%Probability = 0.78


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Probabilistic Constraint :

R g P 0,,, PXZYd

0.78

0.85

Confidence Percentile

0,,, PXZYd g

Confidence Percentile = 85%Probability = 0.78

R = 0.78


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Confidence Percentile Using ExtremeDistribution of Smallest Value

0, 221 X X X g P

R=0.465

Co nfidence Percentile usingBeta Distribution = 99.87%

Confidence Percentile usingExtreme Distribution = 57.3%

Beta

Extreme Value


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Bayesian Approach to Design

Optimization ( BADO )


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BADO Formulation

N N

Y N

min p,xd, ZYxd,

,,f ,

R ) g ( P ii

0,,, PXZYd ni ,...,1

U L ddd U L YYY

U N

L xxx

s.t.

Target Confidence Percentile

ConfidencePercentile


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Thin-Walled Pressure Vessel Example

Rt

g

t R g

t R L g

Y t Rt SF t Rt R P

g

tY SF t R P

g

50.1)(

120.1)(

6022

0.1)(

)2()22(

0.1)(

2)5.0(

0.1)(

5

4

3

2

22

2

1

X

X

X

X

X

4810

240.6

0.225.0

L

R

t

L N N R

R R f t L N

23

,, 34

max

5,...,10, j R g P ii XZYs.t.

yielding

t L,Y Y P ,Z

RX


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A Combined Bayesian and

EBDO Approach


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Illustration with the Thin-WalledPressure Vessel Example

Figure 6. Histogram of sample points.

01

49

2

# o f s a m p

l e s

45

3

Variables

# o f s a m p

l e s

6R N 5.4R N 3R N NR 3R N 4.5R N 6R N

Figure 6. Histogram of sample points.

01

49

2

# o f s a m p

l e s

45

3

Variables

# o f s a m p

l e s

6R N 5.4R N 3R N NR 3R N 4.5R N 6R N

100 Sample Points

P=3/100=0.03

??


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Define random variable X where,

X is # of sample points in 3,5.4 N N R R segment.

X ~ Beta (3+1, 100-3+1) = Beta (4, 98)

Then:

Using extreme distribution of smallest value with 8.0 : 0054.08.018.035.4 1001 X N N F R R R P


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2.2e-5

6R N 5.4R N 3R N NR 3NR 4.5R N 6R N

0.0054 0.3155 0.3523 0.0025 0.00068

0.3236

2.2e-5

6R N 5.4R N 3R N NR 3NR 4.5R N 6R N

0.0054 0.3155 0.3523 0.0025 0.00068

0.3236

R# of Sample

Points

BPA(Extreme

Value)[ N R - 6.0 , N R - 4.5] 0 2.2e-5

[ N R - 4.5 , N R - 3.0] 3 0.0054

[ N R - 3.0 , N R ] 45 0.3155

[ N R , N R + 3.0] 49 0.3523[ N R + 3.0 , N R + 4.5] 2 0.0025

[ N R + 4.5 , N R + 6.0] 1 0.00068

[ N R - 6.0 , N R + 6.0] --- 0.3236



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General Conclusions

There are formal design optimization tools depending onamount and form of available information; e.g. RBDO,EBDO, PBDO, BADO, Bayesian + Evidence.

There is a trade-off between conservative designs (loss ofoptimality) and available information.

Deterministic RBDO EBDO BADOLess Information

More Conservative Design

PBDO

Bayesian + Evidence


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Q & A


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Evidence-Based Design Optimization (EBDO)Implementation

Feasible Region

x2

x1

g1(x1,x2)=0

g2(x1,x2)=0

ObjectiveReduces

hyper-ellipseinitial design

point

frame of

discernment

B

MPP for g 1=0

deterministicoptimum


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REC 2008 Zi i P M l

Bayesian Approach Design Optimization (BADO)1. Uncertainty is in the form of sample points.

2. The probability distribution is Beta distribution.

Beta(5,5)

Beta(22,12)

Prior and posterior distributions Confidence percentileExtreme value and Beta distribution

N N

f Y N p,xd, ZYxd, ,,min,

)PXd( t i g P 0,, ni ,...,1s.t. U L ddd

U LYYY

U N

L xxx ,

,

3. BADO formulation

bayesian ebdo savannah mourelatos feb2008 v2

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