1. Fuzzy Measures & Measures of Fuzziness Presented by : Armin

2. Fuzzy measures & measures of Fuzziness 2 Contents  Introduction  Measure & Integral  Fuzzy Measure  Fuzzy Integral ( Choquet & Sugeno)  Measure of fuzziness

3. Fuzzy measures & measures of Fuzziness 3 Introduction The measure is one of the most important concepts in mathematics and so is the integral with respect to the measure. They have many applications in engineering, and their main characteristic is the additivity. This characteristic is very effective and convenient, but often too inflexible or too rigid. As a solution to the rigidness problem the fuzzy measure was proposed.

4. Fuzzy measures & measures of Fuzziness 4 Measure  Set functions : A function defined on a family of sets is called a set function.  Additive :  Monotone :  Normalized :

5. Fuzzy measures & measures of Fuzziness 5 Measure  Measure  A measure on X : a non-negative additive set function defined on 2 x  A normalized measure : Probability measure  An additive set function : Signed measure

6. Fuzzy measures & measures of Fuzziness 6 Measures A measure measures the size of sets.  Counting measure m c : |A|  Probability : tossing a die  δ x 0 : Dirac measure on X focused on x 0 Null set

7. Fuzzy measures & measures of Fuzziness 7 Measures & Integral A f This is a set! and the integral is measuring the “size” of this set! Integrals are like measures! They measure the size of a set. We just describe that set by a function. Therefore, integrals should satisfy the properties of measures.

8. Fuzzy measures & measures of Fuzziness 8 Measures & Integral Integral : Integral over A : X: finite set, m : signed measure on X, f : function on X

9. Fuzzy measures & measures of Fuzziness 9 Measures & Integral

10. Fuzzy measures & measures of Fuzziness 10 Measures & Integral

11. Fuzzy measures & measures of Fuzziness 11 Measures & Integral

12. Fuzzy measures & measures of Fuzziness 12 Fuzzy Measures [Sugeno 1977] Fuzzy measure is a set function g defined on that: 1. g( ∅ ) = 0, g(X) = 1. 3. If and Then

13. Fuzzy measures & measures of Fuzziness 13 Fuzzy measure Fuzzy measure is: Non-additive Non-monotonic

14. Fuzzy measures & measures of Fuzziness 14 Fuzzy measure Monotone and non-additive

15. Fuzzy measures & measures of Fuzziness 15 Fuzzy measure General Discussion  Fuzzy set: a value is assigned to each element of the universal set signifying its degree of membership in a particular set with unsharp boundaries.  Fuzzy measure: assign a value to each crisp set signifying the degree of evidence or belief that a particular element belongs in the set.

16. Fuzzy measures & measures of Fuzziness 16 Fuzzy Measures Fuzzy Set versus Fuzzy Measure Fuzzy SetFuzzy Measure Underlying Set Vague boundaryCrisp boundary Vague boundary: Probability of fuzzy set RepresentationMembership value of an element in X Degree of evidence or belief of an element that belongs to A in X ExampleSet of large number A degree of defection of a tree Degree of Evidence or Belief of an object that is tree

17. Fuzzy measures & measures of Fuzziness 17 Fuzzy integral (Choquet) An extension of the ordinary integral Each worker xi works f (xi ) hours a day. f (xi ) = f (x1) + [f (x2 )- f (x1 )] + … +[f (xi )- f (xi-1 )] Group A produces the amount μ(A) in one hour { { XX\{x 1 } XnXn {

18. Fuzzy measures & measures of Fuzziness 18 Fuzzy integral (Choquet) a4a4 a3a3 a2a2 a1a1 0

19. Fuzzy measures & measures of Fuzziness 19 Is defined only for normalized fuzzy measures μ : a normalized fuzzy measure f : a function with range {a 1, a 2, a 3,…a n } ∫ f o μ = Max{Min{a i, μ({x | f(x) >= a i })}} If μ is a 0-1 fuzzy measure ∫ f o μ = (c) ∫ f dμ min f(x) <= ∫ f o μ <= max f(x) Fuzzy integral (Sugeno) 19

20. Fuzzy measures & measures of Fuzziness 20 μ : a normalized fuzzy measure f : a function with range {a 1, a 2, a 3,…a n } 20 Fuzzy integral (Sugeno)

21. Fuzzy measures & measures of Fuzziness 21 Sugeno Integral & Choquet Integral Choquet Integral:

22. Fuzzy measures & measures of Fuzziness 22 Sugeno Integral & Choquet Integral Sugeno Integral : 22 Student A i=1 : min(0.5, 1) = 0.5 i=2 : min(0.8,0.5) = 0.5 i=3 : min(0.9, 0.45) = 0.45 Max(0.5, 0.5, 0.45) = 0.5 LitPhysMath 18 16 10

23. Fuzzy measures & measures of Fuzziness 23 Sugeno Integral & Choquet Integral Sugeno Integral for aggregation : Choquet Integral for aggregation : 23

24. Fuzzy measures & measures of Fuzziness 24 Measures of fuzziness Indicates the degree of fuzziness of a fuzzy set  De Luca and Termini : based on entropy of fuzzy set and Shannon function  Higashi and Klir + Yager: based on the degree of distinction between the fuzzy set and its complement

25. Fuzzy measures & measures of Fuzziness 25 Measures of fuzziness The measure of fuzziness d(A) : if A is a crisp set d(A) = 0 if μ(x) = ½ for every x d(A) = the maximum value if B is crisper than A d(A) ≥ d(B) if  is complement of A d(Â) = d(A)

26. Fuzzy measures & measures of Fuzziness 26 Measures of fuzziness [De Luca & Termini 1972] The entropy as a measure of fuzzy set A={(x, μ(x) )} is defined as : “μ(A)” is the membership function of the fuzzy set A n is the number of element in support of A and K is a positive constant

27. Fuzzy measures & measures of Fuzziness 27 Measures of fuzziness [De Luca & Termini 1972] Example:  Let A=“ Integers close to 10” A={(7,.1), (8,.5), (9,.8), (10, 1), (11,.8), (12,.5), (13,.1)}  Let k=1, so d(A) =.325 +.693 +.501 + 0 +.501 +.693 +.325 = 3.038  Let B=“ Integers quite close to 10” B={(6,.1), (7,.3), (8,.4), (9,.7), (10, 1), (11,.8), (12,.5), (13,.3), (14,.1)}  Let k=1, so d(A) =.325 +.611 +.673 +.611 + 0 +.501 +.693 +.611 +.325 = 4.35 A is crisper than B.

28. Fuzzy measures & measures of Fuzziness 28 Measures of fuzziness [Yager 1979] Requirement of distinction between A and its complement is not satisfied by fuzzy sets :  So,any measure of fuzziness should be measure of this lack the measure is based on the distance between a fuzzy set and its complement

29. Fuzzy measures & measures of Fuzziness 29 Measures of fuzziness [Yager 1979] Distance : Let S=supp(A) : A measure of fuzziness : p=1 : Hamming metric p=2 : Euclidean metric

30. Fuzzy measures & measures of Fuzziness 30 Measures of fuzziness [Yager 1979] Example (cont. slide 13) :  Let A=“ Integers close to 10”  Let B=“ Integers quite close to 10”  For p=1:  For p=2:

31. Fuzzy measures & measures of Fuzziness 31 Reference “Fuzzy Measures and Fuzzy Integrals”, Toshiaki Murofushi and Michio Sugeno. “Fuzzy measures and integrals”, Radko Mesiar. “Fuzzy set Theory and Its Applications”, H.J.Zimmermann. “The Evolution of the Concept of Fuzzy Measure”, Luis Garmendia

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