2 Notions and Concepts of Fuzzy Sets Fuzzy Systems Engineering - - PowerPoint PPT Presentation

2 Notions and Concepts

• f Fuzzy Sets

Fuzzy Systems Engineering Toward Human-Centric Computing

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

Contents

Pedrycz and Gomide, FSE 2007

2.1 Sets and fuzzy sets: A departure from the principle

• f dichotomy

Pedrycz and Gomide, FSE 2007

Dichotomy

Pedrycz and Gomide, FSE 2007

(a) (b) short tall short tall

threshold

X X

Set and the principle

• f dichotomy

Relaxation of complete inclusion and exclusion

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

Pedrycz and Gomide, FSE 2007

3.0 1.8

τ S T

X x1 x2

S = {x ∈X | 0 ≤ x ≤ 1.8 } T = {x ∈X | 1.8 < x ≤ 3.0 } Threshold τ = 1.8 x1 ∈ S , x1 ∉ T x2 ∈ T , x2 ∉ S Dichotomy

Pedrycz and Gomide, FSE 2007

Characteristic function

S T

threshold

1.8 1.8 3 3 1.0 1.0

T(x) S(x) X

X X

   ∉ ∈ = A x if A x if x A , , 1 ) (

A : X → {0,1}

   ∉ ∈ = ] . 3 , 8 . 1 [ if ] . 3 , 8 . 1 [ if 1 ) ( x x x T

Pedrycz and Gomide, FSE 2007

Fuzzy set: Membership function

X X 1.5

1.5

3 3 1.0 1.0

T(x) S(x) short tall short tall X

A : X → [0,1]

Pedrycz and Gomide, FSE 2007

X 1 1.0

A(x)

2 3 4 5 6 8 9 10 7

X = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 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]

Fuzzy sets in discrete universes

A

A = {(A(x),x)}

Pedrycz and Gomide, FSE 2007

2.2 Interpretation of fuzzy sets

Fuzziness ≠ Probability

John is tall

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Height of people Head or tail ?

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Fuzziness

A : X → [0,1] X: universe (set) A: membership function

Probability

P(A) : F → [0,1] P: probability (set) function A: set X: universe (set) F: σ-algebra, a set of subsets of X

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

Pedrycz and Gomide, FSE 2007

2.3 Membership functions and their motivation

Choosing membership functions

Pedrycz and Gomide, FSE 2007

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

Triangular membership function

Pedrycz and Gomide, FSE 2007

• 5

5 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 A(x) x a = -1 m = 2 b = 5

         ≥ ∈ − − ∈ − − ≤ = b x m,b x m b x b a,m x i a m a x a x x A if ] [ if ] [ f if ) ( } )], /( ) ( ), /( } max{min[( ) , , , ( m b x b a m a x b m a x A − − − − =

• 1

2

Pedrycz and Gomide, FSE 2007

• 5

5 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 A(x) x a = -2.5 m = 0 n = 2.5 b = 5.0

Trapezoidal membership function

         > ∈ − − ∈ ∈ − − < = b x b n x n b x b n m x m a x a m a x a x i x A if ] , [ if ) , [ if 1 ) , [ if f ) ( } )], /( ) ( , 1 ), /( } max{min[( ) , , , , ( n b x b a m a x b n m a x A − − − − =

2.5

• 2.5

Pedrycz and Gomide, FSE 2007

Γ-membership function

     > − + − ≤ =      > − ≤ =

− −

a x if ) ( 1 ) ( if ) (

• r

if 1 if ) (

2 2 ) (

2

a x k a x k a x x A a x e a x x A

a x k

• 5

5 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 A(x) x a = 1 k = 5

Pedrycz and Gomide, FSE 2007

S-membership function

         > ∈       − − − ∈       − − ≤ = b x i b m x a b b x m a x a b a x a x x A f 1 ] , ( if 2 1 ) , [ if 2 if ) (

2 2

• 5

5 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 A(x) x a = -1 b = 3

Pedrycz and Gomide, FSE 2007

Gaussian membership function

• 5

5 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 A(x) x k = 0.5 m = 2

) ) ( exp( ) (

2 2

σ m x x A − − =

σ = 0.5 m = 2.0

Pedrycz and Gomide, FSE 2007

Exponential-like membership function

) ( 1 1 ) (

2

> − + = k m x k x A

• 5

5 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 A(x) x k = 1 m = 2

Pedrycz and Gomide, FSE 2007

2.4 Fuzzy numbers and intervals

Pedrycz and Gomide, FSE 2007

1.0 R A 1.0 R B

A is a fuzzy number B is not a fuzzy number

Pedrycz and Gomide, FSE 2007

1 1 2.5 2.5 1 1 2.2 2.2 3.0 2.2 3.0

real number

2.5

fuzzy number about 2.5 real interval [2.2, 3.0] fuzzy interval around [2.2, 3.0]

2.5

A2.5 A[2.2, 3.0] Aabout Aaround

R R R R 3.0

Pedrycz and Gomide, FSE 2007

2.5 Linguistic variables

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

〈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]

Example

Pedrycz and Gomide, FSE 2007

Pedrycz and Gomide, FSE 2007

comfortable

x X

name X Linguistic terms L Term set

T(X)

semantic rule M(L) Base variable universe temperature cold warm

1.0 1.0 1.0 cold comfortable warm X X x x C(x) F(x) W(x)

〈X, T(X), X, G, M 〉