Outline Classical Set Fuzzy Set
Computational Intelligence
Lecture 10:Fuzzy Sets
Farzaneh Abdollahi
Department of Electrical Engineering
Amirkabir University of Technology
Fall 2011
Farzaneh Abdollahi Computational Intelligence Lecture 10 1/26
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Este documento aborda os conceitos básicos de conjuntos difusos, incluindo a definição de conjunto difuso, a função de pertinência, suporte, corte α e operações como união, intersecção e complemento. Também são apresentadas diferentes classes de s-norm e t-norm, como as classes de dombi, dubois-prade e yager, e as leis de de morgan para conjuntos difusos.
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Computational Intelligence
Lecture 10:Fuzzy Sets
Farzaneh Abdollahi
Department of Electrical Engineering
Amirkabir University of Technology
Fall 2011
Classical Set
Fuzzy Set Basic Concepts in Fuzzy Sets Operations on Fuzzy Sets Fuzzy Complement Fuzzy Union Fuzzy Intersection Averaging Operator
Example: cars in Tehran
I (^) The universe of discourse U. I (^) Set A is the cars with 4 cylinders:
A = {x ∈ U|xhas 4 cylinders} OR
μA =
1 ifx ∈ U&x has 4 cylinders 0 ifx ∈ U&x does not have 4 cylinders
I (^) Set D is the car made in Iran I (^) BUT the distinction between an Iranian car and a non-Iranian a car is not crisp:( I (^) Most of them are not completely made in Iran I (^) So what should we do??!!
Fuzzy Set
I (^) some sets do not have clear boundaries. I (^) Fuzzy set: in a universe of discourse U is characterized by a membership function μA(x) that takes values in the interval [0, 1]. I (^) In classical sets the membership function of a classical set can only take zero and one I (^) In fuzzy set the membership function is a continuous function with range [0, 1]. I (^) A fuzzy set A in U is represented by: I (^) a set of ordered pairs of a generic element x and its membership value: A = {(x, μA(x))|x ∈ U} I (^) for continuous U: A = ∫ U μA(x)/x. I (^) for discrete U: μA(x): A = ∑ U μA(x)/x I ∫ and ∑ do not represent integral and summation. I (^) They denote collection of all points x ∈ U with the associated membership function μA(x)
I (^) Different membership functions can be defined to characterize the same description.
I (^) The membership functions are not fuzzy, themselves.
I (^) They are precise mathematical functions.
I (^) Fuzzy sets are used to defuzzify the world. I (^) How to determine the membership functions? I (^) Formulate human knowledge I (^) Usually, gives a rough formula of the membership function I (^) fine-tuning is required. I (^) Data collected from various sensors I (^) specify the structures of the membership functions and then fine-tune the parameters based on the data. I (^) A fuzzy set has a one-to-one correspondence with its membership function
Example: Old and Young [?]
I (^) U is in the interval of [0, 100] I (^) young∫ = 25 0 1 /x^ +^
25 (1 + (^
x− 25 5 )
(^2) )− (^1) /x
I (^) old =
50 (1 + (^
x− 50 5 )
− (^2) )− (^1) /x
Basic Concepts in Fuzzy Sets
I (^) Support of a fuzzy set A in the universe of discourse U is a crisp set that contains all the elements of U that have nonzero membership values in A: suppA = {x ∈ U|μA > 0 } I (^) In the digital thermometer example: suppA = [21, 30] I (^) empty fuzzy set: support is empty I (^) fuzzy singleton: support is a single point I (^) Center of a fuzzy set: I (^) If the mean value of all points at which the membership function of the fuzzy set achieves its maximum value is finite, then this mean value is the center I (^) If the mean value equals positive (negative) infinite, then the center is the smallest (largest) among all points that achieve the maximum membership value.
I (^) Crossover point of a fuzzy set: the point in U whose membership value in A equals 0.5. I (^) Height of a fuzzy set: the largest membership value attained by any point. I (^) Normal fuzzy set: the height of fuzzy set equals to one (digital thermometer). I (^) α-cut of a fuzzy set A a crisp set Aα contains all the elements in U that have membership values in A greater than or equal to α: Aα = {x ∈ U|μA(x) ≥ α} I (^) In digital thermometer for α = 0. 7 , Tα = [23, 24 , 25 , 26 , 27] I (^) A fuzzy set A is convex iff its α-cut is a convex set for ∀α ∈ (0, 1].
I (^) Crossover point of a fuzzy set: the point in U whose membership value in A equals 0.5. I (^) Height of a fuzzy set: the largest membership value attained by any point. I (^) Normal fuzzy set: the height of fuzzy set equals to one (digital thermometer). I (^) α-cut of a fuzzy set A a crisp set Aα contains all the elements in U that have membership values in A greater than or equal to α: Aα = {x ∈ U|μA(x) ≥ α} I (^) In digital thermometer for α = 0. 7 , Tα = [23, 24 , 25 , 26 , 27] I (^) A fuzzy set A is convex iff its α-cut is a convex set for ∀α ∈ (0, 1]. I (^) Let C be a set in a real or complex vector space. C is convex if, ∀x, y ∈ C and all λ ∈ [0, 1] , λx + (1 − λ)y ∈ C
I (^) Crossover point of a fuzzy set: the point in U whose membership value in A equals 0.5. I (^) Height of a fuzzy set: the largest membership value attained by any point. I (^) Normal fuzzy set: the height of fuzzy set equals to one (digital thermometer). I (^) α-cut of a fuzzy set A a crisp set Aα contains all the elements in U that have membership values in A greater than or equal to α: Aα = {x ∈ U|μA(x) ≥ α} I (^) In digital thermometer for α = 0. 7 , Tα = [23, 24 , 25 , 26 , 27] I (^) A fuzzy set A is convex iff its α-cut is a convex set for ∀α ∈ (0, 1]. I (^) Lemma: A fuzzy set A ∈ Rn^ is convex iff μA[λx 1 + (1 − λ)x 2 ] ≥ min[μA(x 1 ), μA(x 2 )] ∀x 1 , x 2 ∈ Rn, λ ∈ [0, 1].
I (^) The De Morgan’s Laws are true for fuzzy sets:
F ∪ D = F¯ ∩ D¯ F ∩ D = F¯ ∪ D¯
I (^) For Iranian Cars example:
I (^) μF ∪D =
{ μD if 0 ≤ p(x) ≤ 0. 5 μF if 0. 5 ≤ p(x) ≤ 1 I (^) μF ∩B =
{ μF if 0 ≤ p(x) ≤ 0. 5 μD if 0. 5 ≤ p(x) ≤ 1
Further Operations
I (^) An other difference between fuzzy sets and crisp sets: I (^) for crisp sets only one type of operation is defined for complement, union, and intersection I (^) for fuzzy sets, we can define several operations for them based on the given axioms. I (^) Why do we need different type of operations? I (^) Some operations may not be satisfactory in some situations.
Fuzzy Complement
I (^) Let c : [0, 1] → [0, 1] be a mapping that transforms the membership function of fuzzy set A into the membership function of the complement of A: c[μA(x)] = μ¯A(x) I (^) It was defined: c[μA(x)] = 1 − μA I (^) Let a = μA(x 1 ) and b = μA(x 2 ) I (^) the function c is qualified as a complement if: I (^) Axiom c1: c(0) = 1 and c(1) = 0 (boundary condition) I (^) Axiom c2: ∀a, b ∈ [0, 1], if a < b, then c(a) ≥ c(b) (nonincreasing condition) I (^) an increase in membership value must result in a decrease or no change in membership value for the complement I (^) Some types of fuzzy complement: I (^) Sugeno class: cλ(a) = (^) 1+^1 −λaa , λ ∈ (− 1 , ∞) I (^) λ = 0 basic fuzzy complement I (^) Yager class: cw (a) = (1 − aw^ )^1 /w^ , w ∈ (0, ∞) I (^) w = 1 basic fuzzy complement
Fuzzy set-S Norm
I (^) Let s : [0, 1] × [0, 1] → [0, 1] be a mapping that transforms the membership functions of fuzzy sets A and B into the membership function of the union of A and B, that s[μA(x), μB (x)] = μA ⋃^ B. I (^) the function S to be qualified as an union I (^) Let a = μA(x) and b = μB (x) I (^) Axiom s1.s(1, 1) = 1, s(0, a) = s(a, 0) = a (boundary condition). I (^) Axiom s2. s(a, b) = s(b, a) (commutative condition). I (^) Axiom s3. If a ≤ a′^ and b ≤ b′, then s(a, b) ≤ s(a′, b′) (nondecreasing condition). I (^) Axiom s4. s(s(a, b), c) = s(a, s(b, c)) (associative condition). I (^) Popular types of s-norm I (^) Dombi class: sλ(a, b) = (^) 1+[( 1 1 a −1)−λ+(^1 b −1)−λ]−^1 /λ^ , λ ∈ (0, ∞) I (^) Dobios-Prade class: sα(a, b) = a+bmax(1−ab−−min(a, 1 −a,bb,α,^1 −) α), α ∈ [0, 1] I (^) Yager class: sw (a, b) = min[1, (aw^ + bw^ )^1 /w^ ], w ∈ (0, ∞)