WORKSHOP Applications of Fuzzy Sets and Fuzzy Logic to Engineering Problems"' Pertisau, Tyrol, Austr

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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
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1. Propagation of uncertainty through
mathematical models in a decision support context
(Oberkampf et alia, 2002)
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Challenge Problem 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
• a is characterized by multiple intervals, b is
  specified by a probability distribution with
  imprecise parameters
• a is an interval, b is a precise probability
  distribution

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Challenge Problem B

• 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

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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)
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2. RANDOM SET THEORY Histograms of disjoint
subsets Ai ? X
Ø  if B ? Ai i k to l Pr (B) ?
m (Ai ) Ai ? B    else ? m
(Ai ) Ai ? B ?Pr (B) ? ? m (Ai ) Ai ?
B ? ?
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Histograms of not-disjoint subsets Ai ? X
Ø  Upper and Lower Probabilities from
multi-valued mapping (Dempster, 1967) Ø  Evidence
Theory (Shafer, 1976)
? m (Ai ) Ai ? B ? Pr (B) ? ? m (Ai )
Ai ? B ? ?   Belief Bel(B) ?
Probability ? Plausibility Pl(B)  
Bel(B) Pl(Bc) 1
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Distribution on the singletons of a focal element
Ai of the free probability m(Ai)
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Contour Function ? (x) Pl ( B ?x?)
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Consonant Random Sets Fuzzy Sets
Therefore ? B ? X , Pl ( B ) max ? (x) x ?
B Bel ( B ) 1 - max ? (x) x ? Bc
Ø   Possibility/Necessity Theory (Zadeh, 1978
Dubois Prade, 1986) Ø  (Normalized ) Fuzzy sets
(Zadeh, 1965) as consonant random sets Ø 
Probability Measures as non-consonant random sets
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Random Set from a Fuzzy Set
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3. Why Imprecise Probabilities in Engineering
Imprecise probabilities seem to be the natural
consequence of set-valued measurements  Ø 
directly in real-world observations (for example
geological or geo-mechanical surveys)  Ø  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. Ø  when lack of direct
experimental data forces us to resort to experts,
each one giving imprecise measures (consonant or
not-consonant)  Ø  Statistics from multi-choice
questionnaire
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4. Aggregation of different sources of
information Set uncertainty - case 1 AND
C(A,B) A?B ( A AND B)
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 .
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- case 2 OR
C(A,B) A?B ( A OR B)
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 .
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- case 3 Convolutive Averaging (X-Averaging)
If a distance d is defined in ? between points P
or subsets C(A,B) C d(A, C) d(C, B)
In a vectorial Euclidean space X
Notes 1 The rule in any case works and hides
the conflict to the decision maker
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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.
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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
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
(EX) hides the conflict
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Aggregating probabilistic assignements (Rule 2)
For two assigned relative frequencies of events
(focal elements)
For infinite number of realisations, simply
averaging
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Updating by means of Bayes Theorem (Rule 1)
Combining a probabilistic distribution m1(Xi)
and a deterministic event Xj (m2(Xj) 1)
Notes 1- Pro(Xj ) is a normalisation factor K 2-
If K??1, posterior probabilities increases
dramatically (reliability of m2(Xj) 1)
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Generalisation to random sets Dempsters
Rule(Shafers Evidence theory)
Combining two random sets ?1 (Ai , m1(Ai))
and ?2 (Bj , m2(Bj))
Notes 1- If Cij ?? for every i, j the rules
does not work 2- BayesRule is a particular
application of Dempsters Rule 3- Combining two
consonant random sets (two fuzzy sets) by means
of Dempsters Rule the resulting random sets is
generally not consonant.
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Criticism of Dempsters 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) )
Therefore Bel(brain tumor)Pro(brain
tumor)Pl(brain tumor) 1
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Yagers Modified Dempsters Rule (1987)
Therefore Bel(brain tumor) 10-4 lt Pro(brain
tumor) lt Pl(brain tumor) 1 Bel(meningitis)
0 lt Pro(meningitis) lt Pl(meningitis) 1- 10-4
Bel(concussion) 0 lt Pro(concussion) lt
Pl(concussion) 1- 10-4
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Fuzzy composition of consonant random sets(Rule
1)
Given two fuzzy sets A, B
Let Ai , Bi m1(Ai) m2(Bi) ?i-1 - ?i ,
i 2 to k their nested (strong) ?-cuts with
the same probabilistic assignement
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Normalization of Fuzzy composition Rule
Notes 1-If Ak?Bk ? the rule does not work 2-
If A2?B2 ? C is subnormal 3- K1-h(C) is the
probability assignement of the empty set ?
Therefore two alternative rules can be used for
normalization
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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 Yagers
modification of the Dempsters rule seems to be
reasonable in many cases