monte carlo and possibilistic methods for uncertainty analysis...example 1: uncertainty in event...
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Monte Carlo and Possibilistic Methods for
Uncertainty AnalysisDr. P. Baraldi
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Piero Baraldi
2‘The Uncertainty Factor’
OUTPUT
…Model parameters:
α1, α2,..
Uncertainty representation
Uncertainty propagation
u
y1
y2
yn
Model
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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)
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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
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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:
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Piero Baraldi
6Example 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
2~ (0, )N
hmax: “between 9 and 11,
with a preference for 10”
h = actual crack depth
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
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Piero Baraldi
7In This Lecture
Uncertainty representation
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
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Piero Baraldi
8
PART I: Uncertainty Representation
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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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Piero Baraldi
10
• Aleatory uncertainty
probability distributions
• Epistemic uncertainty
Probability distributions
Possibility distributions
Uncertainty representation
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Piero Baraldi
11
Epistemic Uncertainty Representation
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Piero Baraldi
12
Sufficient informative data:
e.g. N experiments (y1,y2,…, yN)
Probability distributions
1
y y
f(y) F (y)
Epistemic Uncertainty Representation
(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
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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)
Epistemic Uncertainty Representation
(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
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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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Piero Baraldi
15Example 1
π(y)
9 10 11 y
For any interval I=[y1,y2]
1
I
Possibility Measure:
Necessity Measure
0.5
y1 y2
? I
?IN
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Piero Baraldi
16Example 2
I=[9.8, 10.5]
π(y)
9 10 10.5 11 y
1
Possibility Measure:
Necessity Measure
? I
?IN
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Piero Baraldi
17Interpretation of the Possibility Distribution
Lower limiting probability value ≤ ≤ Upper limiting probability value IYP
π(y)
9 10 11 y
For any interval I=[y1,y2]
1
I
Possibility Measure, Π(I)Necessity Measure, N(I)
0.5
y1 y2
yIYPyIyIy
maxmax1
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Piero Baraldi
18Example 2: Interpretation
I=[9.8, 10.5]
π(y)
9 10 10.5 11 y
1
18.01
maxmax1
IYP
yIYPyIyIy
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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
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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
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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
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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”
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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
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Piero Baraldi
24Possibility distribution: Core
π(y)
1
yA1
10 ( ) 1P y A
π possibility distribution
A1 core
Y
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Piero Baraldi
25Possibility Distribution: Support
π(y)
1
yA0
1)( 0 AYP
π possibility distribution
A0 support
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Piero Baraldi
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
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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
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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!!!
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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
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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
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Piero Baraldi
31
PART II: Uncertainty propagation
u =g(y1,…,yn)
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Piero Baraldi
32Uncertainty propagation: objective
y1
y2
MODEL
g(y1,…,yn)
f(u)
f(y1)
u
yn
f(y2)
f(yn)
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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)
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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
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Piero Baraldi
35Level 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)
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
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Piero Baraldi
36Level 2 uncertainty propagation
• Level 2:
• y1,…,yn are uncertain parameters described by the probability
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
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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
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Piero Baraldi
38
Uncertainty propagation – Level 1
u =g(y1,…,yn)
1. Purely probabilistic method• y1,…,yn are uncertain parameters
described by the probability distributions
f(y1),…, f(yn)
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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 ),(
)(
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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
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41
Uncertainty propagation – Level 1
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 ,)(
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• 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
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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)
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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
)
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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
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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 .
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49
Uncertainty propagation – Level 1
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)
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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)
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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
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52
...
(yk+1)α
yk+1
α=0
α=α+Δα
0
1
u
(u)
Outer loop: probabilistic parameters
Inner loop: possibilistic parameters
α=1?
Details of the Hybrid Monte Carlo and
Possibilistic method
y1
Monte Carlo sampling of:
from f1(y1), ..., yk~ fk(yk) )(uj
𝑢𝛼 = 𝑖𝑛𝑓𝑦𝑘+1∈𝐴𝛼
𝑌𝑘+1 ,…,𝑦𝑛∈𝐴𝛼𝑌𝑛
𝑔 𝑦1, … , 𝑦𝑛
j =N?
𝑢𝛼
input
output
j=0
j =j+1
j
k
j yy ,...,1
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53
...
(yk+1)α
yk+1
α=0
α=α+Δα
0
1
u
(u)
Outer loop: probabilistic parameters
Inner loop: possibilistic parameters
α=1?
Details of the Hybrid Monte Carlo and
Possibilistic method
y1
Monte Carlo sampling of:
from f1(y1), ..., yk~ fk(yk) )(uj
𝑢𝛼 = 𝑠𝑢𝑝𝑦𝑘+1∈𝐴𝛼
𝑌𝑘+1 ,….,𝑦𝑛∈𝐴𝛼𝑌𝑛
𝑔 𝑦1, … , 𝑦𝑛
j =N?
input
output
𝑢𝛼
j=0
j =j+1
j
k
j yy ,...,1
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54
...
(y)
α
yk+1
α=0
α=α+Δα
0
1
u
(u)
Outer loop: probabilistic parameters
Inner loop: possibilistic parameters
α=1?
Details of the Hybrid Monte Carlo and
Possibilistic method
y1
𝑢𝛼 = 𝑖𝑛𝑓𝑦𝑘+1∈𝐴𝛼
𝑌𝑘+1 ,…,𝑦𝑛∈𝐴𝛼𝑌𝑛
𝑔 𝑦1, … , 𝑦𝑛
𝑢𝛼 = 𝑠𝑢𝑝𝑦𝑘+1∈𝐴𝛼
𝑌𝑘+1 ,….,𝑦𝑛∈𝐴𝛼𝑌𝑛
𝑔 𝑦1, … , 𝑦𝑛
Monte Carlo sampling of:
from f1(y1), ..., yk~ fk(yk) )(uj
j =N?
j=0
j =j+1
j
k
j yy ,...,1
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55
...
0
1
u
(u)
α=0
α=α+Δα
0z
Outer loop: probabilistic parameters
Inner loop: possibilistic parameters
α=1?
Details of the Hybrid Monte Carlo and
Possibilistic method
y1
)(uj
j=0
j =j+1
Monte Carlo sampling of:
from f1(y1), ..., yk~ fk(yk)
j
k
j yy ,...,1
)(uj
j =N?
𝑢𝛼 = 𝑖𝑛𝑓𝑦𝑘+1∈𝐴𝛼
𝑌𝑘+1 ,…,𝑦𝑛∈𝐴𝛼𝑌𝑛
𝑔 𝑦1, … , 𝑦𝑛
𝑢𝛼 = 𝑠𝑢𝑝𝑦𝑘+1∈𝐴𝛼
𝑌𝑘+1 ,….,𝑦𝑛∈𝐴𝛼𝑌𝑛
𝑔 𝑦1, … , 𝑦𝑛
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56
...
0
1
z
(u)
α=0
α=α+Δα
0
Outer loop: probabilistic parameters
Inner loop: possibilistic parameters
α=1?
Details of the Hybrid Monte Carlo and
Possibilistic method
y1
𝑢𝛼 = 𝑖𝑛𝑓𝑦𝑘+1∈𝐴𝛼
𝑌𝑘+1 ,…,𝑦𝑛∈𝐴𝛼𝑌𝑛
𝑔 𝑦1, … , 𝑦𝑛
𝑢𝛼 = 𝑠𝑢𝑝𝑦𝑘+1∈𝐴𝛼
𝑌𝑘+1 ,….,𝑦𝑛∈𝐴𝛼𝑌𝑛
𝑔 𝑦1, … , 𝑦𝑛
Monte Carlo sampling of:
from f1(y1), ..., yk~ fk(yk) )(uj
j=0
j =j+1 j =N?
j
k
j yy ,...,1
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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
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58
Level 1 uncertainty propagation - applications:
1. Event tree analysis
2. Reliability Assessment of a Degrading Component
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59
Application 1)
Event tree analysis
(only epistemic uncertainty is considered)
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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
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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.
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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
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63Hardware-Failure-Dominated events
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64Human-Error-Dominated events
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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
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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
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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
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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
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69
Application 2:
Reliability Assessment of a Degrading Structure
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70Example 3: Reliability Assessment of a
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”
h = actual crack depth
C, , = constants related to the material properties( )dh
e C hdt
ASSUMPTIONS:
• h monitored parameter
•h>Hmax structure fails
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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
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72
Time
h
TTF=?
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
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73
Time
h
TTF=?
TTF=?
Hmax
1pt
1pth
aleatory with
epistemic Hmax: between 9 and 11, with preference for 10
2~ (0, )N ( )dh
e C hdL
11
10
9
Epistemic uncertainty representation in the
case study
![Page 74: 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](https://reader036.vdocument.in/reader036/viewer/2022081623/6144fa7534130627ed50b22a/html5/thumbnails/74.jpg)
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74
Application of the Hybrid Monte Carlo and possibilistic
method
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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
![Page 76: 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](https://reader036.vdocument.in/reader036/viewer/2022081623/6144fa7534130627ed50b22a/html5/thumbnails/76.jpg)
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76Uncertainty propagation
Probabilistic distributionsPossibilistic distributions
•
•
ω
working
failed
otherwise1
)(if0)),(()(
max
max
HThHThfTX
miss
missmissS
)0)(()( missSmiss TXPTR
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77
( ) ( 1) ( )th t h t e C K
Tmiss = 500
MC simulations of the degradation
evolution: h(Tmiss)
20,t N
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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
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Piero Baraldi
79Possibility 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
working failed
Extension principle
00)()(
00)(0
missSmiss
missS
TXPTR
TXP
sX
ssX
XXPXss
00
max0max1
Interpretation
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80Possibility distributions of the component
state: MC simulation 2
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
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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
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82Results
•Hybrid Monte Carlo and possibilistic method: the component
reliability is between [0.896,0.927]
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83Conclusions
How to collect the information and input it in the proper mathematical
format?
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84Conclusions
How to collect the information and input it in the proper mathematical
format?
Sufficient information probability distribution
Scarce and qualitative information possibility distribution
does not force information in the epistemic uncertainty representation
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Piero Baraldi
85Conclusions
How to collect the information and input it in the proper mathematical
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
![Page 86: 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](https://reader036.vdocument.in/reader036/viewer/2022081623/6144fa7534130627ed50b22a/html5/thumbnails/86.jpg)
Piero Baraldi
86Conclusions
How to collect the information and input it in the proper mathematical
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]
![Page 87: 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](https://reader036.vdocument.in/reader036/viewer/2022081623/6144fa7534130627ed50b22a/html5/thumbnails/87.jpg)
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