2 Notions and Concepts of Fuzzy Sets Fuzzy Systems Engineering Toward Human-Centric Computing
Contents 2.1 Sets and fuzzy sets: A departure from the principle of dichotomy 2.2 Interpretation of fuzzy sets 2.3 Membership functions and their motivation 2.4 Fuzzy numbers and intervals 2.5 Linguistic variables Pedrycz and Gomide, FSE 2007
2.1 Sets and fuzzy sets: A departure from the principle of dichotomy Pedrycz and Gomide, FSE 2007
Dichotomy threshold tall short tall short X X (a) (b) Set and the principle Relaxation of complete of dichotomy inclusion and exclusion Pedrycz and Gomide, FSE 2007
Inherent problems of dichotomization “One seed does not constitute a pile nor two or three. From the other side, everybody will agree that 100 million seeds constitutes a pile. What is therefore the appropriate limit?” E. Borel, 1950 Pedrycz and Gomide, FSE 2007
Sets τ S T X 0 1.8 3.0 x 1 x 2 Threshold τ = 1.8 Dichotomy S = { x ∈ X | 0 ≤ x ≤ 1.8 } x 1 ∈ S , x 1 ∉ T T = { x ∈ X | 1.8 < x ≤ 3.0 } x 2 ∈ T , x 2 ∉ S Pedrycz and Gomide, FSE 2007
A : X → {0,1} Characteristic function 1 , ∈ if x A ( ) = A x 0 , ∉ if x A threshold S T X 1 if [ 1 . 8 , 3 . 0 ] ∈ x ( ) = T x S ( x ) T ( x ) 0 if [ 1 . 8 , 3 . 0 ] ∉ x 1.0 1.0 0 1.8 3 X 0 1.8 3 X Pedrycz and Gomide, FSE 2007
Fuzzy set: Membership function A : X → [0,1] tall short X T ( x ) S ( x ) short tall 1.0 1.0 1.5 0 1.5 3 0 3 X X Pedrycz and Gomide, FSE 2007
Fuzzy sets in discrete universes A ( x ) A 1.0 0 1 2 3 4 5 6 7 8 9 10 X X = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = {( A ( x ), x )} A = {0/0, 0/1, 0/2, 0.2/3, 0.5/4, 1.0/5, 0.5/6, 0.2/7, 0/8, 0/9, 0/10} A = [0, 0, 0, 0.2, 0.5, 1.0, 0.5, 0.2, 0, 0, 0] Pedrycz and Gomide, FSE 2007
2.2 Interpretation of fuzzy sets Pedrycz and Gomide, FSE 2007
Fuzziness ≠ Probability John is tall Head or tail ? Height of people Pedrycz and Gomide, FSE 2007
A : X → [0,1] Fuzziness X : universe (set) A : membership function P ( A ) : F → [0,1] P : probability (set) function Probability A : set X : universe (set) F : σ - algebra, a set of subsets of X Pedrycz and Gomide, FSE 2007
Membership grades: semantics � Similarity: degree of compatibility (data analysis and processing) � Uncertainty: possibility (reasoning under uncertainty) � Preference: degree of satisfaction (decision-making, optimization) Pedrycz and Gomide, FSE 2007
2.3 Membership functions and their motivation Pedrycz and Gomide, FSE 2007
Choosing membership functions Criteria should reflect: � Nature of the problem at hand � Perception of the concept to represent � Level of details to be captured � Context of application � Suitability for design and optimization Pedrycz and Gomide, FSE 2007
Triangular membership function 1 A(x) a = -1 0.9 m = 2 b = 5 0.8 0 if ≤ x a 0.7 − x a 0.6 f [ ] ∈ i x a,m − m a 0.5 ( ) = A x − b x 0.4 if [ ] ∈ x m,b − b m 0.3 0 if ≥ x b 0.2 0.1 0 x -5 -1 0 5 10 2 ( , , , ) max{min[( } /( ), ( ) /( )], 0 } = − − − − A x a m b x a m a b x b m Pedrycz and Gomide, FSE 2007
Trapezoidal membership function 1 A(x) a = -2.5 0.9 m = 0 n = 2.5 0 f < i x a 0.8 b = 5.0 − x a 0.7 if [ , ) ∈ x a m 0.6 − m a ( ) = 1 if [ , ) A x ∈ x m n 0.5 − b x 0.4 if [ , ] ∈ x n b − 0.3 b n 0 if > x b 0.2 0.1 0 x -5 -2.5 0 2.5 5 10 ( , , , , ) = max{min[( − } /( − ), 1 , ( − ) /( − )], 0 } A x a m n b x a m a b x b n Pedrycz and Gomide, FSE 2007
Γ -membership function 1 A(x) 0.9 a = 1 k = 5 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -5 0 5 10 x 0 if ≤ x a 0 if ≤ x a 2 ( ) ( ) = or ( ) = − k x a A x A x 2 if x a − ( − ) > k x a 1 if − > e x a 2 1 ( ) + − k x a Pedrycz and Gomide, FSE 2007
S-membership function 1 A(x) 0.9 a = -1 b = 3 0 if ≤ x a 0.8 2 0.7 − x a 2 if [ , ) ∈ x a m 0.6 − b a ( ) = A x 0.5 2 − x b 0.4 1 2 if ( , ] − ∈ x m b − b a 0.3 1 f > i x b 0.2 0.1 0 -5 0 5 x 10 Pedrycz and Gomide, FSE 2007
Gaussian membership function 1 A(x) 0.9 k = 0.5 σ = 0.5 m = 2 m = 2.0 0.8 0.7 0.6 2 ( ) − x m ( ) exp( ) = − A x 0.5 2 σ 0.4 0.3 0.2 0.1 0 x -5 0 5 10 Pedrycz and Gomide, FSE 2007
Exponential-like membership function 1 A(x) 0.9 k = 1 m = 2 0.8 0.7 1 0.6 ( ) 0 = > A x k 2 0.5 1 ( ) + − k x m 0.4 0.3 0.2 0.1 0 -5 0 5 x 10 Pedrycz and Gomide, FSE 2007
2.4 Fuzzy numbers and intervals Pedrycz and Gomide, FSE 2007
A B 1.0 1.0 R R A is a fuzzy number B is not a fuzzy number Pedrycz and Gomide, FSE 2007
A 2.5 A about fuzzy number real number about 2.5 2.5 1 1 2.5 3.0 2.2 2.5 R R A around A [2.2, 3.0] fuzzy interval real interval around [2.2, 3.0] [2.2, 3.0] 1 1 2.2 3.0 R 2.2 2.5 3.0 R Pedrycz and Gomide, FSE 2007
2.5 Linguistic variables Pedrycz and Gomide, FSE 2007
Linguistic variables � A certain variable (attribute) can be quantified in terms of a small number of information granules – temperature is { low , high } – speed is { low , medium , high , very high } � Each information granule comes with a well-defined meaning (semantics) Pedrycz and Gomide, FSE 2007
Linguistic variables: A definition 〈 X , T ( X ), X , G , M 〉 X : is the name of the variable T ( X ): is term set of X ; elements of T are labels L of linguistic values of X X : universe G : grammar that generates the names of X M : semantic rule that assigns to each label L ∈ T ( X ) a meaning whose realization is a fuzzy set on X with base variable x Pedrycz and Gomide, FSE 2007
Example 〈 X , T ( X ), X , G , M 〉 X : temperature X : [0, 40] T ( X ): { cold , comfortable , warm } G : only terminal symbols, the terms of T ( X ) M ( cold ) → C M ( comfortable ) → F M (warm) → W C , F and W are fuzzy sets in [0, 40] Pedrycz and Gomide, FSE 2007
C ( x ) cold 1.0 cold X x universe F ( x ) comfortable Linguistic 1.0 temperature comfortable terms semantic L name rule X M ( L ) X x W ( x ) warm 1.0 warm Term set X x T ( X ) Base variable 〈 X , T( X ), X , G, M 〉 Pedrycz and Gomide, FSE 2007
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