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Towards Improved Trapezoidal Approximation to Intersection (Fusion) of Trapezoidal Fuzzy Numbers: Specific Procedure and General Non-Associativity Theorem

Gang Xiang1 and Vladik Kreinovich2

1Philips Healthcare, El Paso, TX 79902, USA 2University of Texas, El Paso, TX 79968, USA, vladik@utep.edu

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1. Data Fusion: Main Idea and Case of Interval Un- certainty

• Situation: often, we have several pieces of information

about the same quantity x.

• Data fusion: we combine (“fuse”) this information.
• Frequent case: we have estimates

xi with upper bounds ∆i on their uncertainty: | xi − x| ≤ ∆i.

• Interval uncertainty: xi ∈ Xi

def

= [xi, xi], where xi = xi − ∆i and xi = xi + ∆i.

• Fusion of intervals: if we know that x ∈ X1, . . . , and

x ∈ Xn, then x ∈ X = [x, x] = X1 ∩ . . . ∩ Xn, where: x = max(x1, . . . , xn) and x = min(x1, . . . , xn).

• Natural properties: X1∩X2 = X2∩X1 and associativity

X1 ∩ (X2 ∩ X3) = (X1 ∩ X2) ∩ X3.

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2. Data Fusion for Fuzzy Numbers

• Frequent situation: we have several piece of imprecise

(“fuzzy”) expert knowledge Xi about a quantity x.

• Natural description: in terms of fuzzy numbers µXi(x).
• Natural fusion operation: intersection

X = X1 ∩ . . . ∩ Xn, defined as µX(x) = min(µX1(x), . . . , µXn(x)).

• Towards an algorithm: a fuzzy set X can be described

by its α-cuts X(α)

def

= {x : µ(x) ≥ α}.

• Useful result: the α-cut of the intersection X is equal

to the intersection of α-cuts: X(α) = X1(α) ∩ . . . ∩ Xn(α).

• Resulting algorithm: use the above formula to find the

α-cut of the intersection for α = 0, 0.1, . . . , 1.0.

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3. Computer Representation of Fuzzy numbers: Trapezoidal Numbers

• Ideally: to adequately describe a fuzzy number, we

need to store α-cuts for all α ∈ [0, 1].

• In practice: we can only store finitely many α-cuts.
• Usually: we store the lower (α = 0) and upper (α = 1)

α-cuts and use linear interpolation.

✲ ✻

• ❅

❅ ❅ ❅ ❅

1 a a

• Resulting (approximate) fuzzy numbers are called

trapezoidal.

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4. Intersection of Two Trapezoidal Fuzzy Numbers Desirable: find the intersection µi(x) of the trapezoidal membership functions

✲ ✻ ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏

a a and

✲ ✻

• b

b

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5. Problem: Intersection of Two Trapezoidal Fuzzy Numbers Is, In General, Not Trapezoidal

• We are interested in: intersection (fusion) of member-

ship functions.

• We limit ourselves to: trapezoidal membership func-

tions.

• Problem: the intersection of two trapezoidal member-

ship functions, in general, has the non-trapezoidal form:

✲ ✻

• ✏✏✏✏✏✏✏

✏ ✏ ✏ ✏ ✏ ✏

• a

a b b

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6. Usual Solution to the Above Problem

• Usual solution to the above problem is simply:

– take an intersection of lower and alpha α-cuts, and then – as the result of the fusion, take the trapezoidal number corresponding to this intersection:

✲ ✻ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏

• ✟✟✟✟✟✟✟✟✟✟

a a b b

• Main limitation: this approach underestimates the re-

sulting membership function.

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7. Under-Estimation Explained: Mathematical Com- ment

• Reminder: trapezoidal membership functions are (lo-

cally) linear.

• Reminder: intersection is the minimum of two func-

tions.

• Fact: a minimum µX(x) of two linear functions is al-

ways concave.

• Meaning: the values µX(x) are always above the straight

line µt(x) connecting the endpoints:

✲ ✻

• ✏✏✏✏✏✏✏

✏ r r r r r r r r r ✏ ✏ ✏ ✏ ✏

• ✟✟✟✟✟✟✟✟✟✟

r r r r r r

µt µX µX a a b b

r r r r

µt µX

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8. How to Improve the Traditional Trapezoidal Ap- proximation to Intersection: A New Procedure

• Traditional approximation:

– starts with the α-cuts corresponding to α = 0 and α = 1, and – uses the linear interpolation to reconstruct other α-cuts.

• Problem: for 0 < α < 1, the reconstructed α-cuts are

biased (under-estimated).

• Natural idea:

– First, compute the actual intersection. – Second, approximate the non-0 and non-1 parts of this intersection by linear functions.

• Implementation: use the Least Squares Method for this

approximation.

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9. Least Squares Method

• Problem: find

p for which e(a, p ) ≈ 0 for all a.

• Solution: min
• a e2(a,

p )2 da = a

a (p + q · a− µi(a))2 da.

• Resulting formulas:

p · (a)2 − (a)2 2 + q · (a)3 − (a)3 3 = a

a

µi(a) · a da. p · (a)2 − (a)2 2 + q · (a)3 − (a)3 3 = a

a

µi(a) · a da.

✲ ✻ ✏ ✏ ✏ ✏ ✏

• ✟✟✟✟✟✟✟✟✟✟

a b µℓ µi

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10. Problem with the New Approximation Procedure: Non-Associativity

• Situation: we have three different pieces of knowledge

X1, X2, and X3.

• Options: we can combine them in two different ways:

– we can first combine X1 and X2 into X1 ⊗ X2, and then combine X1 ⊗ X2 with X3; – we can first combine X2 and X3 into X2 ⊗ X3, and then combine X1 with X2 ⊗ X3.

• Intuitively: we expect the results to be equal:

(X1 ⊗ X2) ⊗ X3 = X1 ⊗ (X2 ⊗ X3).

• In practice: the Least Squares-based operation ⊗ is

not associative.

• What we prove in this paper: non-associativity is in-

evitable.

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11. Properties of the General Approximation Opera- tion ⊗

• 1st idea: it is sufficient to separately consider the left

and the right sides of the membership functions.

• 2nd idea: left-hand side is uniquely described by the

interval a = [a, a].

✲ ✻

• a

a

• Conclusion: we consider operations on intervals.
• 1st property – commutativity: a ⊗ b = b ⊗ a
• 2nd property – idempotence: a ⊗ a = a.

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12. 3rd Property of the General Approximation Op- eration otimes

• Situation: a ≤ b

def

= b ≤ a and b ≤ a.

✲ ✻

• b

b

• a

a

• Fact: in this situation, a ∪ b = b.
• Resulting requirement: a ⊗ b = b.

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13. 4th Property of the General Approximation Op- eration ⊗

• Non-trivial case: a ≤ b and b ≤ a.
• 1st requirement: the upper endpoint of c = a ⊗ b is

strictly in between the upper endpoints of a and b.

• 2nd requirement: the lower endpoint of c must also be

strictly in between the corresponding endpoints:

✲ ✻ ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏

• ✟✟✟✟✟✟✟✟✟✟

a b b a c c

• Formally: a ≤ a ⊗ b, a ⊗ b ≤ a, b ≤ a ⊗ b, and

a ⊗ b ≤ b.

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14. General Non-Associativity Result

• Definition. By an improved trapezoidal approximation to

the intersection of trapezoidal fuzzy numbers, we mean an interval operation ⊗ which is:

• commutative, i.e., a ⊗ b = b ⊗ a for all a and b;
• idempotent, i.e., a ⊗ a for all intervals a;
• preserves trapezoidal intersection, i.e.,

– if a ≤ b, – then a ⊗ b = b;

• improves over the usual approximation, i.e.,

– if a ≤ b and b ≤ a, – then a ≤ a⊗b, a⊗b ≤ a, b ≤ a⊗b, and a⊗b ≤ b.

• Proposition. Every improved trapezoidal approximation

to the intersection of trapezoidal fuzzy numbers is non- associative.

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15. Proof by Contradiction: Main Idea

• Let us assume that associativity holds for a = [1, 4]

and b = c = [2, 3], i.e., (a ⊗ b) ⊗ b = a ⊗ (b ⊗ b).

• Due to idempotence b ⊗ b = b, so we have d ⊗ b = d

for d

def

= a ⊗ b.

• Here, a ≤ b and b ≤ a, so due to the improvement

property, b ≤ d = a ⊗ b and d = a ⊗ b ≤ b.

• Since ⊗ is an improving operation, b ≤ d and d ≤ b

imply that d ≤ d ⊗ b.

• On the other hand, we have d = d⊗b hence d ≤ d⊗b.
• The contradiction proves that the above associativity

equality cannot be true.

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16. Acknowledgments This work was supported in part

• by the National Science Foundation grants HRD-0734825

and DUE-0926721,

• by Grant 1 T36 GM078000-01 from the National Insti-

tutes of Health,

• by Grant MSM 6198898701 from Mˇ

SMT of Czech Re- public, and

• by Grant 5015 “Application of fuzzy logic with opera-

tors in the knowledge based systems” from the Science and Technology Centre in Ukraine (STCU), funded by European Union. The authors are greatly thankful to Ronald R. Yager for valuable discussions.