Monte Carlo and Possibilistic Methods for Uncertainty Analysis...Example 1: Uncertainty in Event Tree Analysis Safety system 1 Safety system 2 Piero Baraldi Example 2: Reliability

Monte Carlo and Possibilistic Methods for

Uncertainty AnalysisDr. P. Baraldi

Piero Baraldi

2‘The Uncertainty Factor’

OUTPUT

…Model parameters:

α1, α2,..

Uncertainty representation

Uncertainty propagation

u

y1

y2

yn

Model

Piero Baraldi

3

p1

p2

Seq1

Seq2

Seq3

Seq4

Example 1: Uncertainty in Event Tree

analysis

Safety system 1

(fire sprinkler system)

p1

p2

P(Seq4|fire)=p1p2MODEL

Safety system 2

(closing door)

Piero Baraldi

4

p1

p2

•How to characterize the uncertainty on the probability of occurrenceof the events (p1, p2)?•How to propagate the uncertainties from the single events to the sequences (p(seq4|fire))?

Seq1

Seq2

Seq3

Seq4

Example 1: Uncertainty in Event Tree

Analysis

Safety system 1 Safety system 2

Piero Baraldi

5Example 2: Reliability Assessment of a

Degrading Structure

R(Tm)

0.5

0

1

STRUCTURE

Monitored

parameter

h(t)

h(t) = actual crack depth

h(t)> hmax structure fails

( )dh

e C hdt

MODEL

h = actual crack depth

𝜔 = random variable that describes the stochasticity

of the degradation process

C, , = constants related to the material properties

t =time

MODEL:

Piero Baraldi


Degrading Structure

R(Tm)

0.5

0

1

STRUCTURE

Monitored

parameter

h(t)

h(t) = actual crack depth

h(t)> hmax structure fails

2~ (0, )N

hmax: “between 9 and 11,

with a preference for 10”


𝜔 = random variable that describes the stochasticity

of the degradation process

C, , = constants related to the material properties

t =time

( )dh

e C hdt

MODEL

Piero Baraldi

7In This Lecture


Probability distributions

Possibility distributions

Uncertainty propagation

Level 1:

• Purely probabilistic method

• Purely possibilistic

• Hybrid probabilistic and possibilistic method

Level 2

• Purely probabilistic method

Applications

Event tree uncertainty analysis

Reliability assessment of a degrading component

Piero Baraldi

8

PART I: Uncertainty Representation

Piero Baraldi

9

t

t

X

X

Uncertainty classification

Aleatory uncertainty: randomness due to inherent variability in the

system

Epistemic uncertainty: imprecision due to lack of knowledge and

information on the system

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10

• Aleatory uncertainty

probability distributions

• Epistemic uncertainty


Possibility distributions


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11

Epistemic Uncertainty Representation

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12

Sufficient informative data:

e.g. N experiments (y1,y2,…, yN)


1

y y

f(y) F (y)


(Sufficient Informative Data)

y1 y2

2

1

1 2,

y

y

P y y y f y dy

y2

2 2( )F y P y y

Piero Baraldi

13

Scarce information and of qualitative nature on the parameter value.

Expert: “the value of Y is between 9 and 11, with a preference for 10”

Information difficult to catch with a probability distribution

y

f (y)


(Scarce Information)

9 10 11

1= possibility distribution =

Degree of possibility that the uncertain variable Y be equal to

y

y

Possibility representation of the uncertainty

9 10 11

?

y y

Piero Baraldi

14Possibility Distribution: definitions

9 10 11 y

For any interval I=[y1,y2]

1

I

Possibility Measure:

Necessity Measure

0.5

y1 y2

yIIy

max

yIINIy

max11

« to what extent I is

consistent with the knowledge π? »

«to what extent I is certainly

implied by the knowledge π?»

y

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15Example 1

π(y)

9 10 11 y


1

I


Necessity Measure

0.5

y1 y2

? I

?IN

Piero Baraldi

16Example 2

I=[9.8, 10.5]

π(y)

9 10 10.5 11 y

1


Necessity Measure

? I

?IN

Piero Baraldi

17Interpretation of the Possibility Distribution

Lower limiting probability value ≤ ≤ Upper limiting probability value IYP

π(y)

9 10 11 y


1

I

Possibility Measure, Π(I)Necessity Measure, N(I)

0.5

y1 y2

yIYPyIyIy

maxmax1

Piero Baraldi

18Example 2: Interpretation

I=[9.8, 10.5]

π(y)

9 10 10.5 11 y

1

18.01

maxmax1

IYP

yIYPyIyIy

Piero Baraldi

19Probability and Possibility: Comparison

y

f (y)

y1 y2yy1 y2

2

1

y

y

f y dy 1 ,max maxy y

y I y I

Normalization 1f y dy

max 1( )y

y

Probability Possibility

21, yyYP

Piero Baraldi

20Possibility Distribution Limiting Cumulative

Distributions

F(y)(y)

Lower cumulative

distribution

9 10 11

1

y

Upper cumulative

distribution

y

,I u

1 ( )max maxy F u y

y I y I

5.0)5.9(0 F

yuYPyIyIy

maxmax1

u

Piero Baraldi

21Possibility Distribution Limiting Cumulative

Distributions

F(y)(y)

Lower cumulative

distribution

9 10 11

1

y

Upper cumulative

distribution

y

,I u

1 ( )max maxy F u y

y I y I

yuYPyIyIy

maxmax1

u

Family of probability

distributions

Piero Baraldi

22Definition of an α-cut

π(y)

1

y

α = 0.30

A0.30

yyIA :

11

1max1

maxmax1

AYP

AYPy

yAYPy

Ay

AyAy

Interpretation:

“Aα is certain to degree 1-α”

Piero Baraldi

23Example

π(y)

1

y

α = 0.30

A0.30

yyAI :

0.30

0.30 0.30

1 max maxy P y A y

y A y A

)(7.0 3.0AyP Y

Y

Piero Baraldi

24Possibility distribution: Core

π(y)

1

yA1

10 ( ) 1P y A

π possibility distribution

A1 core

Y

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25Possibility Distribution: Support

π(y)

1

yA0

1)( 0 AYP

π possibility distribution

A0 support

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26Constructing a possibility distribution 26

AVAILABLE INFORMATION• physical limits [a, b]

• most likely value c

(E.g., expert: “the true value of Y is between a and b, with a preference for c”)

π(y

)

-10 0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ks

(K

s)

a bc

= family of all the probability distributions

with support [a, b] and mode c

Piero Baraldi

Constructing a possibility distribution: Chebyshev

inequality27

• mean value μx

• standard deviation σx

1for k ,1

12k

kXkP xxxx

-10 0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

π(x

) = family of all the probability

distributions with mean μx and

standard deviation σx

AVAILABLE INFORMATION

1for k211

11],[

22 kAXPkkA

k

xxxx

k

Chebyshev

inequality

Piero Baraldi

28

LIMIT CASE: No available information on a model parameter y except that

its value is located somewhere between a value ymin and ymax

Laplace principle of insufficient reason:

“all that is equally plausible is equally probable”

f(y)

ymin ymax

No information available No relation between and

minmax

1

yy

2

minmax yyym

ym

maxmin ,, yyyPyyyP mm

myyYP ,min max, yyYP m

Comparison of Probabilistic/Possibilistic

Approach in Case of Scarce Information (1)

This is a paradox!!!

Piero Baraldi

29

LIMIT CASE: No available information on a model parameter y except that

its value is located somewhere between a value ymin and ymax

(y)

ymin ymax

For any Interval

the only information available is that:

Degree of possibility

that the parameter y

be equal to Y

1

maxmin , yyB

10 BP

Comparison Probabilistic/Possibilistic Approach

in Case of Scarce Information (2)

1)(0

max)(max1

BYP

yBYPyByBy

Piero Baraldi

Uncertainty representation: conclusions 30

Randomness(aleatory)

Lack of knowledge(epistemic)

Probability

distributions

Probability

distributions

Possibility

distributions

Types:

Representation:

“The value of Ks is between 5 and

60, with a preference for 30”Gaussian pdf

Ks(Ks)

Ks

5 30 60

1

(Sufficient statistical

information, i.e., data)

(Scarce information and/or of

qualitative nature, e.g., expert

opinion)

UNCERTAINTY

Example:

Gumbel pdf

-10 0 10 20 30 40 50 60 700

0.01

0.02

0.03

0.04

0.05

0.06f Q(q)

0 1000 2000 3000 4000 5000 60000

1

2

3

4

5

6

7

8x 10

-4

q

fKs(Ks)

Ks

Piero Baraldi

31

PART II: Uncertainty propagation

u =g(y1,…,yn)

Piero Baraldi

32Uncertainty propagation: objective

y1

y2

MODEL

g(y1,…,yn)

f(u)

f(y1)

u

yn

f(y2)

f(yn)

Piero Baraldi

33Level 1 uncertainty propagation

• Level 1:

• y1,…,yn are uncertain parameters described by the probability

distributions fθi (yi)

• the parameters θi of the probability distributions fθi (yi) are

known

u =g (y1,…,yn)

Piero Baraldi

34Level 1 uncertainty propagation: Example

Model Input

y1=TA = failure time

of A

•y2=TB = failure time

of B

BA

Model output:

u =Tsys= failure time of

the system

= g(TA ,TB)

MODEL

Piero Baraldi


• Level 1:


distributions fθi(yi )

• the parameters θi of the probability distributions fθi (yi) are

known

u =g (y1,…,yn)

Model Input:

•y1=tA = failure time of A

•y2=tB = failure time of B

BA

Model output:

•u=tsys= failure time of the system

= g(tA ,tB)

input uncertainty (aleatory)

BB

B

AA

A

t

BBB

t

AAA

etfT

etfT

)(

)(~

~

Parameter of f

0008.0

001.0

B

A

LEVEL 1 uncertainty

Piero Baraldi


• Level 2:


distributions fθi (y1 )

• the parameters θi of the probability distributions fθi (y1 ) are

uncertain (epistemic uncertainty)

u =g (y1,…,yn)

Model Input:

•y1=tA = failure time of A

•y2=tB = failure time of B

BA

Model output:

•u=tsys= failure time of the system

= g(tA ,tB)

input uncertainty (aleatory)

BB

B

AA

A

t

BBB

t

AAA

etfT

etfT

)(

)(~

~

Parameter of f

LEVEL 1

uncertainty

),(

),(

BBB

AAA

N

N

~

~

LEVEL 2

uncertainty

Piero Baraldi

37

Uncertainty propagation – Level 1

u =g(y1,…,yk, yk+1,…,yn)

1. Purely probabilistic method• y1,…,yn = probability distributions

2. Purely possibilistic method• y1,…,yn = possibility distributions

3. Hybrid Monte Carlo and Possibilistic method• y1,…,yk = probability distribution

• yk+1,…,yn = possibility distribution

Piero Baraldi

38


u =g(y1,…,yn)

1. Purely probabilistic method• y1,…,yn are uncertain parameters

described by the probability distributions

f(y1),…, f(yn)

Piero Baraldi

39Purely probabilistic uncertainty propagation

)(

)(

),(

12

11

21

1

1

yfY

yfY

yygu

Y

Y

???)(uFU

212121

),(:,212121

)()(),(

)()(),()(

21

212121

dydyyfyfyyI

dydyyfyfuYYgPuUPuF

YYg

uyygyyYYU

otherwise

uyyfifyyI g

0

),(1),(

21

21with

MC analog dart game:

• sample from

• corresponding award is:

Consider N trials :

),( 21

ii yy )(),( 11 11yfyf YY

),( 21

ii

g yyI

),(),...,,(),,( 21

2

2

2

1

1

2

1

1

NN yyyyyy

N

yyI

uFNi

ii

g

U

,...,1

21 ),(

)(

Piero Baraldi

40

i=0

i=i+1

Sample a realization of the uncertain

parameters from their probability distributions

i

n

ii yyy ,...,, 21

nYYY yfyfyfn

,...,, 21 21

Compute:

),...,,( 21

i

n

iii yyygu

i=N?NO

Estimate the cumulative distribution F(u)

from: Nuu ,...,1

Purely probabilistic uncertainty propagation:

the procedure

YES

F(u)

u

Piero Baraldi

41


u =g(y1,…,yn)

1. Purely possibilistic method• y1,…,yn are uncertain parameters

described by the possibility distributions

π(y1),…, π(yn)

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Purely Possibilistic method: Uncertainty

Propagation through U=g(Y)

Extension principle of fuzzy set theory

• Simple case:

• 1 input quantity, ,whose uncertainty is described by the

possibility distribution π(y)

• 1 output quantity, , whose uncertainty is described

by the possibility distribution πU(u):

Basic idea: extend the function g

• From a function from to

• to a function from and to the class of all the possibility distributions

defined on

42

uygyYU yu

,

sup

Y

UYgU ,)(

Piero Baraldi

• For a given u, we proceed in three steps:

• Select the set of y values such that g(y)=u

• Compute the corresponding set of πY(y)

• Choose for πU(u) the maximum among πY(y)

43Extension principle of fuzzy set theory

g(y)

g(y)

yy1 y2 y3

πY(y)

πY(y3) πY(y2)

πY(y2)

πY(y

3)

πU(u

) u

uygyYU yu

,

sup

Piero Baraldi

Extension Principle: Example

For a chemical reactor, we consider a physical model 𝑔 used for

evaluating the consequence of a catastrophic failure event. We

assume that the model specifies that the consequences 𝐶 (in arbitrary

units) of a catastrophic failure are a quadratic function of the quantity

of toxic gas release 𝑅 (in arbitrary units), i.e.

The epistemic uncertainty related to 𝑅 is described by the triangular

possibility distribution:

44

2)( RRgC

𝜋𝑅 𝑟

= 0 if 𝑟 < 0 or 𝑟 > 2𝑟 if 0 ≤ 𝑟 ≤ 12 − 𝑟 if 1 < 𝑟 ≤ 2

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

r

R

(r)

Piero Baraldi

For a fixed value 𝑐 > 0 of the consequences:

• find the quantity of toxic gas release 𝑟 such that 𝑐 = 𝑟2

. In this

particular case, the solutions are 𝑟1 = − 𝑐 and 𝑟2 = 𝑐.

• evaluate the possibility distribution of the quantity of toxic gas release,

𝜋𝑅, at the identified 𝑟 values, i.e. 𝜋𝑅 𝑟1 and 𝜋𝑅 𝑟2 ,

• assign to the possibility distribution of the consequence, 𝜋𝐶, the

maximum of 𝜋𝑅 𝑟1 and 𝜋𝑅 𝑟2 . Since, in this case, 𝜋𝑅 𝑟1 = 𝜋𝑅 − 𝑐is equal to 0, the maximum value is obtained as 𝜋𝑅 𝑟1 = 𝜋𝑅 𝑐which, according to the definition of 𝜋𝑅 𝑟 is given by:

𝜋𝐶 𝑐 = 𝜋𝑅 𝑟2 = 𝜋𝑅 𝑐 =𝑐 𝑖𝑓 0 < 𝑐 ≤ 1

2 − 𝑐 𝑖𝑓 1 < 𝑐 ≤ 40 𝑖𝑓 𝑐 > 4.

45Extension Principle: Example

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46Extension principle of fuzzy set theory

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

c

C(c

)

Piero Baraldi

Extension principle of fuzzy set theory 47

Multi-input case: U=g(Y1,Y2 …,Yn)

• input quantities: Y1,Y2 …,Yn, whose uncertainty is described by the

possibility distribution π(y1), π(y2),…, π(yn)

• 1 output quantity, U=g(Y1,Y2 …,Yn) whose uncertainty is described

by the possibility distribution πU(u):

Basic idea: extend the function g

• The use of the minimum operator is justified by:

nYYY

uyyygyyy

U yyyun

nn

,...,,min 11

,...,,,,...,,

11

2121

sup

nYYYnYYY yyyyyynn

,...,,min,....,, 1121,...,, 1111

Piero Baraldi

Extension Principle: alternative formulation

• Output possibility distribution is represented in the form of a nested

set of intervals (alpha-cut)

• Indicating as 𝑌1𝛼 , … , 𝑌𝑛𝛼 the α-cuts of the input quantities 𝑌1, … , 𝑌𝑁, the

extension principle, for a given value of 𝛼 in [0,1], becomes:

48

uuuuA U:,

𝑢𝛼 = inf 𝑔 𝑦1, … , 𝑦𝑛 , 𝑦1 ∈ 𝑌1𝛼 … . , 𝑦𝑛 ∈ 𝑌𝑛𝛼 ,

𝑢𝛼 = sup 𝑔 𝑦1, … , 𝑦𝑛 , 𝑦1 ∈ 𝑌1𝛼 … . , 𝑦𝑛 ∈ 𝑌𝑛𝛼 .

Piero Baraldi

49


u =g(y1,…,yk, yk+1,…,yn)

2. Hybrid Monte Carlo and Possibilistic method• y1,…,yk = probability distributions:

p(y1), …,p(yk)

• yk+1,…,yn = possibility distributions:

π(yk+1), …, π(yn)

Piero Baraldi

50

Repeat

N

simulations

Monte Carlo sampling of the variables described by probability distributions

𝑦1𝑗~ f1(y1), ..., 𝑦𝑘

𝑗~ fk(yk)

extension principle to process uncertainty described by possibility

distributions

Possibility distribution 𝜋𝑗(𝑢) of u =g(y1,…,yk, yk+1,…,yn)

0

1

0

1

Simulation 1 Simulation N

...uu

𝜋1 (𝑢)

Hybrid Monte Carlo and Possibilistic method

𝜋𝑁(𝑢)

Piero Baraldi

51

...(yk+1)α

yk+1

α=0

α=α+Δα

0

1

u

(u)

Outer loop: probabilistic parameters

Inner loop: possibilisticparameters α=1?

Details of the Hybrid Monte Carlo and

Possibilistic method

y1

j

kk

j

yy

yy

...

11

j=0

j =j+1

Monte Carlo sampling of:

from f1(y1), ..., yk~ fk(yk)

j

k

j yy ,...,1

)(uj

j =N?

Assumption: Extension principle

𝑢𝛼 = 𝑖𝑛𝑓𝑦𝑘+1∈𝐴𝛼

𝑌𝑘+1 ,…,𝑦𝑛∈𝐴𝛼𝑌𝑛

𝑔 𝑦1, … , 𝑦𝑛

𝑢𝛼 = 𝑠𝑢𝑝𝑦𝑘+1∈𝐴𝛼

𝑌𝑘+1 ,….,𝑦𝑛∈𝐴𝛼𝑌𝑛

𝑔 𝑦1, … , 𝑦𝑛

Piero Baraldi

52

...

(yk+1)α

yk+1

α=0

α=α+Δα

0

1

u

(u)


Inner loop: possibilistic parameters

α=1?



y1


from f1(y1), ..., yk~ fk(yk) )(uj



𝑔 𝑦1, … , 𝑦𝑛

j =N?

𝑢𝛼

input

output

j=0

j =j+1

j

k

j yy ,...,1

Piero Baraldi

53

...

(yk+1)α

yk+1

α=0

α=α+Δα

0

1

u

(u)



α=1?



y1





𝑔 𝑦1, … , 𝑦𝑛

j =N?

input

output

𝑢𝛼

j=0

j =j+1

j

k

j yy ,...,1

Piero Baraldi

54

...

(y)

α

yk+1

α=0

α=α+Δα

0

1

u

(u)



α=1?



y1



𝑔 𝑦1, … , 𝑦𝑛



𝑔 𝑦1, … , 𝑦𝑛



j =N?

j=0

j =j+1

j

k

j yy ,...,1

Piero Baraldi

55

...

0

1

u

(u)

α=0

α=α+Δα

0z



α=1?



y1

)(uj

j=0

j =j+1


from f1(y1), ..., yk~ fk(yk)

j

k

j yy ,...,1

)(uj

j =N?



𝑔 𝑦1, … , 𝑦𝑛



𝑔 𝑦1, … , 𝑦𝑛

Piero Baraldi

56

...

0

1

z

(u)

α=0

α=α+Δα

0



α=1?



y1



𝑔 𝑦1, … , 𝑦𝑛



𝑔 𝑦1, … , 𝑦𝑛



j=0

j =j+1 j =N?

j

k

j yy ,...,1

Piero Baraldi

57

...

0

1

Realization 1

u

(u)

0

1

Realization m

z

(u)

Find associated limiting

Cumulative distributionsUpper and lower cumulative

distributions of u : for any ucompute the average of the obtained cumulativedistributions

u0

Details of the aggregation (1)

0.6

u*

0.9

u0

1

u0

1

Piero Baraldi

58

Level 1 uncertainty propagation - applications:

1. Event tree analysis

2. Reliability Assessment of a Degrading Component

Piero Baraldi

59

Application 1)

Event tree analysis

(only epistemic uncertainty is considered)

Piero Baraldi

60Case study

Hardware-

Failure-

Dominated

(HFD) Events

Human-

Error-

Dominated

(HED)

Events

Anticipated Transient Without Scram (ATWS) in a PWR nuclear reactor

Objective: compute the probability of occurrence of a severe sequence

Piero Baraldi

61Example of HED event

12ADS inhibit XI HED

Automatic depressurization system (ADS) is designed to decrease

the pressure of the reactor in order to start the low-pressure system.

The low-pressure system will inject water into the reactor vessel to

protect the fuel. When an ATWS event happens, the reactor power is

controlled by the level of water in core. Since high-level water will

cause high power, the operator should inhibit all ADS valves

manually. If the operator fails to do so, event XI will occur.

Piero Baraldi

62Epistemic uncertainty representation (mixed

information)

Uncertainty representation:

Probability of Hardware-Failure-Dominated (HFD) event

p.d.f

Probability of Human-Error-Dominated event (HED)

possibility distributions

Piero Baraldi

63Hardware-Failure-Dominated events

Piero Baraldi

64Human-Error-Dominated events

Piero Baraldi

65Epistemic uncertainty representation (mixed

information)

Uncertainty representation:

Probability of Hardware-Failure-Dominated (HFD) event

p.d.f

Probability of Human-Error-Dominated event (HED)

possibility distributions

Uncertainty propagation: Hybrid Monte Carlo and Possibilistic

method

Event sequence probability limiting cumulative distributions

Piero Baraldi

66Results

10-13

10-12

10-11

10-10

10-9

10-8

10-7

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Severe Consequence Accident

pSev

hybrid approach: Pl([0,u))

hybrid approach: Bel([0,u))

MC approach: cdf

fuzzy approach: Pl([0,u))

fuzzy approach: Bel([0,u))

Severe consequence accident

psev

Probabilistic approach

Upper cumulative distributions

Lower cumulative distributions

Pure probabilistic approach

Piero Baraldi

67Results: 90% percentile

10-13

10-12

10-11

10-10

10-9

10-8

10-7

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Severe Consequence Accident

pSev

hybrid approach: Pl([0,u))

hybrid approach: Bel([0,u))

MC approach: cdf

fuzzy approach: Pl([0,u))

fuzzy approach: Bel([0,u))

Severe consequence accident

psev

Probabilistic approach

Upper cumulative distributions

Lower cumulative distributions

Pure probabilistic approach

90

sevp

Pure probabilistic

approach:

With probability 90% we

can say that the

probability of having a

severe accident is lower

than 7 10^-7

Piero Baraldi

68Results

Hybrid approach (3.7810-8, 1.9510-7)

Probabilistic approach 1.0910-7

95% percentile for the probability of occurrence of a severe

consequence accident

Piero Baraldi

69

Application 2:

Reliability Assessment of a Degrading Structure

Piero Baraldi


Degrading Structure

R(Tm)=P(XS(Tm)=0)

0.5

0

1

STRUCTURE

Monitored

parameter

h 2~ (0, )N

Hmax: “between 9 and 11,

with a preference for 10”


C, , = constants related to the material properties( )dh

e C hdt

ASSUMPTIONS:

• h monitored parameter

•h>Hmax structure fails

Piero Baraldi

71

Time

h

TTF=?

Hmax

1pt

1pth

11

10

9

Uncertainty in the case study

aleatory with

epistemic Hmax: between 9 and 11, with preference for 10

2~ (0, )N ( )dh

e C hdL

Piero Baraldi

72

Time

h

TTF=?

TTF=?

Hmax

1pt

1pth

11

10

9

Uncertainty in the case study

aleatory with


2~ (0, )N ( )dh

e C hdL

Piero Baraldi

73

Time

h

TTF=?

TTF=?

Hmax

1pt

1pth

aleatory with


2~ (0, )N ( )dh

e C hdL

11

10

9

Epistemic uncertainty representation in the

case study

Piero Baraldi

74

Application of the Hybrid Monte Carlo and possibilistic

method

Piero Baraldi

75

(Hmax)

Available information on Hmax :

Possibility distribution

Lower cumulative

distribution

F (Hmax)

9 10 11

1

Hmax Hmax

[9, 11]

most likely value is 10

Upper cumulative

distribution

Epistemic uncertainty representation in the

case study

Piero Baraldi

76Uncertainty propagation

Probabilistic distributionsPossibilistic distributions

•

•

ω

working

failed

otherwise1

)(if0)),(()(

max

max

HThHThfTX

miss

missmissS

)0)(()( missSmiss TXPTR

Piero Baraldi

77

( ) ( 1) ( )th t h t e C K

Tmiss = 500

MC simulations of the degradation

evolution: h(Tmiss)

20,t N

Piero Baraldi

78Possibility distributions of the component

state: MC simulation 1

h1(TMiss) = 13.7(Hmax)

9 10 11

inputs

otherwise1

)(if0)),(()(

max

max

HThHThfTX

miss

missmissS

output

0 10

0.2

0.4

0.6

0.8

1

Xs

(Xs)

0 10

0.2

0.4

0.6

0.8

1

Xs

(Xs)

0 10

0.2

0.4

0.6

0.8

1

Xs

(Xs)

0 10

0.2

0.4

0.6

0.8

1

Xs

(Xs)

1

Extension principle

working failed

Piero Baraldi



h1(TMiss) = 13.7(Hmax)

9 10 11

inputs

otherwise1

)(if0)),(()(

max

max

HThHThfTX

miss

missmissS

output

0 10

0.2

0.4

0.6

0.8

1

Xs

(Xs)

0 10

0.2

0.4

0.6

0.8

1

Xs

(Xs)

0 10

0.2

0.4

0.6

0.8

1

Xs

(Xs)

0 10

0.2

0.4

0.6

0.8

1

Xs

(Xs)

1

working failed

Extension principle

00)()(

00)(0

missSmiss

missS

TXPTR

TXP

sX

ssX

XXPXss

00

max0max1

Interpretation

Piero Baraldi



h2(TMiss) = 10.2

(Hmax)

9 10 11

inputs

otherwise1

)(if0)),(()(

max

max

HThHThfTX

miss

missmissS

output

2

working failed

Extension principle

8.0)(0

8.00)(0

miss

missS

TR

TXP

0 10

0.2

0.4

0.6

0.8

1

Xs

(Xs)

0 10

0.2

0.4

0.6

0.8

1

Xs

(Xs)

0 10

0.2

0.4

0.6

0.8

1

Xs

(Xs)

0 10

0.2

0.4

0.6

0.8

1

Xs

(Xs)

sX

ssX

XXPXss

00

max0max1

Interpretation

Piero Baraldi

81

0 10

0.2

0.4

0.6

0.8

1

Xs

(Xs)

0 10

0.2

0.4

0.6

0.8

1

Xs

(Xs)

0 10

0.2

0.4

0.6

0.8

1

Xs

(Xs)

0 10

0.2

0.4

0.6

0.8

1

Xs

(Xs)

Example of realizations of Possibility

distributions of the component state

Piero Baraldi

82Results

•Hybrid Monte Carlo and possibilistic method: the component

reliability is between [0.896,0.927]

Piero Baraldi

83Conclusions

How to collect the information and input it in the proper mathematical

format?

Piero Baraldi

84Conclusions


format?

Sufficient information probability distribution

Scarce and qualitative information possibility distribution

does not force information in the epistemic uncertainty representation

Piero Baraldi

85Conclusions


format?

How to propagate the uncertainty through the model so as to obtain the

proper representation of the uncertainty in the output of the analysis?

Pure Probabilistic Approach

Extension Principle

Hybrid Monte Carlo and possibilistic method

Piero Baraldi

86Conclusions


format?

How to propagate the uncertainty through the model so as to obtain the

proper representation of the uncertainty in the output of the analysis?

How to interpret the uncertainty results in a manner that is

understandable and useful to decision makers?

Interval of probabilities, e.g. the component reliability is between

[0.896,0.927]

Piero Baraldi

References

Uncertainty representation & propagation techniques:

• T. Aven, P. Baraldi, R. Flage, E. Zio, “Uncertainty in Risk Assessment: The

Representation and Treatment of Uncertainties by Probabilistic and Non-probabilistic

Methods, Editore: Wiley, Anno edizione: 2014, ISBN: 978-1-118-48958-1.

• C. Baudrit, D. Dubois and D. Guyonnet, Joint Propagation of Probabilistic and

Possibilistic Information in Risk Assessment, IEEE Transactions on Fuzzy Systems,

Vol. 14, 2006, pp. 593-608.

Applications:

• P. Baraldi, I. C. Popescu, E. Zio, "Methods Of Uncertainty Analysis In Prognostics",

International Journal of Performability Engineering, Vol 6 (4), pp. 303-331, 2010.

• P. Baraldi, E. Zio, “A Combined Monte Carlo and Possibilistic Approach to Uncertainty

Propagation in Event Tree Analysis”, Risk Analysis, Vol. 28 (5), pp. 1309-1325, 2008.

87

Documents

Monte Carlo and Possibilistic Methods for Uncertainty Analysis...Example 1: Uncertainty in Event Tree Analysis Safety system 1 Safety system 2 Piero Baraldi Example 2: Reliability