Transcript Workshop - Pertisau 30 Sept 2002

WORKSHOP
“Applications of Fuzzy Sets and Fuzzy Logic to Engineering Problems".
Pertisau, Tyrol, Austria - September 29th, October 1st, 2002
Aggregation of Evidence from
Random and Fuzzy Sets
Alberto Bernardini
Associate Professor
Dipartimento di Costruzioni e Trasporti
University of Padova, Italy
1. Propagation of uncertainty through mathematical models in
a decision support context (Oberkampf et alia, 2002)
Challenge Problem A:
1.
2.
3.
4.
Y  (a  b)
a
a is an interval, b is an interval
a is an interval, b is characterized by multiple intervals
a and b are characterized by multiple intervals
a is an interval, b is specified by a probability distribution
with imprecise parameters
5. a is characterized by multiple intervals, b is specified by a
probability distribution with imprecise parameters
6. a is an interval, b is a precise probability distribution
Challenge Problem B:
X
Ds 

Y /k
k
k  m  c 
2
2
• m is given by a precise triangular probability distribution
• k is given by n independent, equally credible, sources of
information through triangular probability distributions with
parameters measured by closed intervals
• c is given by q independent, equally credible, sources of
information through closed intervals
•  is given by a triangular probability distribution with parameters
measured by closed intervals
Two Key problems
1 -Combination of random and set uncertainty

(random set uncertainty)
2 - Aggregation of different, eventually independent,
sources of uncertain information
Both random and set uncertainty could be Aleatory (objective) or
Epistemic (subjective)
2. RANDOM SET THEORY
Histograms of disjoint subsets Ai  X
m
m(Ai )
Ai
x
1
B
B(x)

if B =  Ai | i = k to l :
else
Pr (B) =  m (Ai ) | Ai  B
 m (Ai ) | Ai  B Pr (B)   m (Ai ) | Ai  B  
Histograms of not-disjoint subsets Ai  X
 Upper and Lower Probabilities from multi-valued mapping (Dempster, 1967)
 Evidence Theory (Shafer, 1976)
m(A2 )
A2
m(A1)
A1
Probabilistic assignment m(Ai)
1
B
B(x)
 m (Ai ) | Ai  B  Pr (B)   m (Ai ) | Ai  B  
Belief Bel(B)  Probability  Plausibility Pl(B)
Bel(B) + Pl(Bc) = 1
Distribution on the singletons of a focal element Ai of the
“free probability” m(Ai)
m(Ai )
Ai = [xL , xR ]
m(Ai )/| Ai |
xL
xR
Upper and Lower Cumulative Distribution Functions
x
FLOW(x) = Pl ( B(x) = tX| t x)
F(x)
1
FUPP(x) = Bel ( B(x) = tX| t x)=
1 - Pl ( tX| t > x)
FWHP = White Probabilities
x ELOW (f(x))  E (f(x))  EUPP (f(x))
Contour Function  (x) = Pl ( B = x)
 (x)
1
m2
m1
x
Pl ( B = x) =  (x)
Bel ( B = x) = 0 if | Aì| > 1 for i = 1 to n
Consonant Random Sets: Fuzzy Sets
(x)
 B1, B2  X ,
Pl ( B1  B2 ) = max ( Pl (B1), Pl (B2) )
1
m1
m2
x
B1
B2
(x)
Pl(B)
1
Bel(B)
Therefore:
 B  X , Pl ( B ) = max  (x) | x  B
Bel ( B ) = 1 - max  (x) | x  Bc
 Possibility/Necessity Theory (Zadeh, 1978; Dubois & Prade, 1986)
 (Normalized ) Fuzzy sets (Zadeh, 1965) as consonant random sets
 Probability Measures as non-consonant random sets
B
Random Set from a Fuzzy Set

 (x)
1 = 1
1
2
A1
3
2
3 = 0
A = x : (x) > 
A = 
A = A1
; m(A1) = 1 - 2 = 1 - 2
A = A2
;m(A2) = 2 - 3 = 3
……….
A2
x
i+1
A = Ai
; m(Ai) = i - i+1
3. Why Imprecise Probabilities in Engineering
Imprecise probabilities seem to be the natural consequence of set-valued
measurements:
 directly in real-world observations (for example geological or geo-mechanical
surveys);
 when we analyse statistical data trough histograms, even if the measurements are
point-valued: the bars are in fact nothing else but non-overlapping focal elements.
 when lack of direct experimental data forces us to resort to experts, each one
giving imprecise measures (consonant or not-consonant)
 Statistics from multi-choice questionnaire
4. Aggregation of different sources of information
Set uncertainty - case 1 :AND
C(A,B) = AB
A

B
( A AND B)
 C ( P)  min A ( P),  B ( P)
C ( P)  min A ( P),  B ( P)
Notes:
1 – Total conflict (AB = ) – Total loss of information
2 – Partial conflict (AB  ). Uncertainty decreases for the
decision maker
3 – The rules works very well if AB   and the sources of
information for (A, B) are very reliable .
- case 2 : OR
C(A,B) = AB
A

B
( A OR B)
 C ( P)  max A ( P),  B ( P)
C ( P)  max A ( P),  B ( P)
Notes:
1 – Total conflict (AB = ) – No loss of information
2 – Partial conflict (AB  ). Uncertainty increases for the
decision maker
3 – The rule is reasonable when the sources of information for
(A, B) are not very reliable .
- case 3 : Convolutive Averaging (X-Averaging)
If a distance d is defined in  between
points P or subsets:
B
A
C(A,B) = C | d(A, C) = d(C, B)
In a vectorial Euclidean space X:
 C ( x)  sup min A ( x A ),  B ( xB ) 

x
x A  xB
2
C ( x)  sup min A ( x A ),  B ( xB ) 
x
x A  xB
2
Notes:
1 – The rule in any case works and hides the conflict to the
decision maker
General properties of the rules and discussion
-Commutativity: C(A,B) = C(B,A)
-Associativity: C(A, C(B, D)= C(C(A,B), D)
-Idempotence : C(A, A) = A
Notes:
1 – Idempotence does not capture that our confidence in A
grows with the repetitions.
Statistical aggregation and probability theory
Our confidence grows linearly with the number of repetitions of
events (focal elements).
For n realisations of events in a finite space of events:
n  X  X 
ni
j
i
m X i    
n j 1
n
Notes:
1- Probabilities are obtained mixing (p-averaging)  functions
2- Probabilities disclose the conflict to the decision maker (rule 2)
3- c-averaging of probability distributions (E[X]) hides the conflict
Aggregating probabilistic assignements (Rule 2)
For two assigned relative frequencies of events (focal elements):
m1  X i   n1,i / n1 ; m2  X i   n2,i / n2
m12  X i  
n1,i  n2,i
n1  n2
For infinite number of realisations, simply averaging:
m1  X i   m2  X i 
m12  X i  
2
Updating by means of Bayes Theorem (Rule 1)
Combining: a probabilistic distribution m1(Xi) and a deterministic event
Xj (m2(Xj) = 1):
m12  X i   P roX i / X j  

m1  X i  m1 X j / X i 
P roX i  X j 
P roX j 
m1 X j 
Notes:
1- Pro(Xj ) is a normalisation factor K
2- If K1, posterior probabilities increases dramatically
(reliability of m2(Xj) = 1)

Generalisation to random sets: Dempster’s Rule
(Shafer’s Evidence theory)
Combining: two random sets 1 = (Ai , ; m1(Ai)) and
2 = (Bj , ; m2(Bj)) :
m1  Ai  m2 B j  


12   Cij  Ai  B j ; m12 Cij  
K


K  1   m1  Ai  m2 B j 
Cij 
Notes:
1- If Cij  for every i, j the rules does not work;
2- Bayes’Rule is a particular application of Dempster’s Rule
3- Combining two consonant random sets (two fuzzy sets) by means of
Dempster’s Rule the resulting random sets is generally not consonant.
Criticism of Dempster’s Rule (Zadeh, 1984)
Combining two diagnosis about neurological symptoms in a patient:
1 =
(A1 = {meningitis}; m1(A1) = 0.99),
(A2 = {brain tumor}; m1(A2) = 0.01) )
2 =
(B1 = {concussion}; m2(A1) = 0.99),
(B2 = {brain tumor}; m2(A2) = 0.01) )
0.01 0.01 

12   C22  A2  B2  brain tumor; m12 C22  
 1
K


K  1  0.99 0.99  2  0.99 0.01  0.01 0.01
Therefore:
Bel({brain tumor})=Pro({brain tumor})=Pl({brain tumor})= 1
Yager’s Modified Dempster’s Rule (1987)
 Cij  Ai  B j ; m12 Cij   m1  Ai  m2 B j ;



12    ; m   m  m  
m1  Ai  m2 B j 

12
1
2


C


ij


 C22  A2  B2  brain t umor; m12 C22   0.01 0.01  1104 

12  
   meningit is, brain t umor, concussion; m    1  110 4 
12


Therefore:
Bel({brain tumor})= 10-4 < Pro({brain tumor}) < Pl({brain tumor})= 1
Bel({meningitis})= 0 < Pro({meningitis}) < Pl({meningitis})= 1- 10-4
Bel({concussion})= 0 < Pro({concussion}) < Pl({concussion})= 1- 10-4
1=1
2
i
h(C)
k=0
Let:
Fuzzy composition of consonant random sets
(Rule 1)
Given two fuzzy sets A, B
C ( x)  min A ( x), B ( x)
x
Ai , Bi ; m1(Ai) = m2(Bi) = i-1 - i , i = 2 to k
their nested (strong) -cuts with the same probabilistic assignement:
 Ci  Ai  Bi ; m12 Ci   m1  Ai   m2 Bi    i 1   i 

12  
 for : i  2 t o k

Normalization of Fuzzy composition Rule
Notes:
1=1
2
1-If AkBk =  the rule does not
work
i
2- If A2B2 =  C is subnormal
h(C)
3- K=1-h(C) is the probability
x assignement of the empty set 
k=0
Therefore two alternative rules can be used for normalization:
1
Dempster
1
Yager
1-h(C)
h(C)
x
x
5. CONCLUSIONS
1) when information is affected by both randomness and imprecision, a
reliability analysis can be conducted, taking into account the whole spectrum of
uncertainty experienced in data collection. In this case imprecision leads to
upper and lower bounds on the probability of an event of interest;
2) imprecision on basic parameters heavily has repercussions on the prediction of
the behaviour of a construction, so that probabilistic analyses that ignore
imprecision are meaningless, especially when very low probability of failure are
calculated or required.
3) Three alternative basic rules has been identified for the aggregation of
imprecise data: the subjective choice of the decision maker depend on the
reliability of the available information and the aims of the analysis.
4) In the application of the “Intersection” rules attention should be given to the
normalisation of the obtained probabilistic assignement: Yager’s modification of
the Dempster’s rule seems to be reasonable in many cases