Fuzzy Measures Measures of Fuzziness Presented by Armin

Fuzzy Measures & Measures of Fuzziness Presented by : Armin

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

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. Fuzzy measures & measures of Fuzziness 3

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

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 Fuzzy measures & measures of Fuzziness 5

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

Measures & Integral 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. A Fuzzy measures & measures of Fuzziness 7

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

Measures & Integral Fuzzy measures & measures of Fuzziness 9

Measures & Integral Fuzzy measures & measures of Fuzziness 10

Measures & Integral Fuzzy measures & measures of Fuzziness 11

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

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

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

Fuzzy measure n 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. Fuzzy measures & measures of Fuzziness 15

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

Fuzzy integral (Choquet) n n An extension of the ordinary integral Each worker xi works f (xi ) hours a day. n { X X{x 1} { { f (xi ) = f (x 1) + [f (x 2 )- f (x 1 )] + … +[f (xi )- f (xi-1 )] Xn Group A produces the amount μ(A) in one hour Fuzzy measures & measures of Fuzziness 17

Fuzzy integral (Choquet) a 4 a 3 0 a 2 a 1 Fuzzy measures & measures of Fuzziness 18

Fuzzy integral (Sugeno) n n n Is defined only for normalized fuzzy measures μ : a normalized fuzzy measure f : a function with range {a 1, a 2, a 3, …an} ∫ f o μ = Max{Min{a , μ({x | f(x) >= a })}} i n 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 measures & measures of Fuzziness 19 19

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

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

Sugeno Integral & Choquet Integral n Sugeno Integral : Student A 18 16 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 10 Lit Phys Math Max(0. 5 , 0. 45) = 0. 5 Fuzzy measures & measures of Fuzziness 22 22

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

Measures of fuzziness n 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 Fuzzy measures & measures of Fuzziness 24

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

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

Measures of fuzziness[De Luca & Termini 1972] n 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. Fuzzy measures & measures of Fuzziness 27

Measures of fuzziness [Yager 1979] n 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 Fuzzy measures & measures of Fuzziness 28

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

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

Reference n n “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 Fuzzy measures & measures of Fuzziness 31

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