3 Characterization of Fuzzy Sets - dca.fee.unicamp.br · 3.1 Generic characterization of fuzzy sets: fundamental descriptors 3.2 Equality and inclusion relationships in fuzzy sets

3 Characterizationof Fuzzy Sets

Fuzzy Systems EngineeringToward Human-Centric Computing

3.1 Generic characterization of fuzzy sets: fundamental descriptors

3.2 Equality and inclusion relationships in fuzzy sets

3.3 Energy and entropy measures of fuzziness

3.4 Specificity of fuzzy sets

3.5 Geometric interpretation of sets and fuzzy sets

3.6 Granulation of information

3.7 Characterization of the families of fuzzy sets

3.8 Fuzzy sets, sets, and the representation theorem

Contents

Pedrycz and Gomide, FSE 2007

3.1 Generic characterization of fuzzy sets:

Some fundamental descriptors


Fuzzy sets


� Fuzzy sets are membership functions

� In principle: any function is “eligible” to describe fuzzy sets

� In practice it is important to consider:– type, shape, and properties of the function– nature of the underlying phenomena– semantic soundness

A: X → [0, 1]

Normality


hgt(A) = 1

Normal Subnormal

hgt(A) < 1)(sup)(hgt xAA

x X∈=

1.0

x

A(x)

1.0

x

A

A

A(x)


Normalization

hgt(A) < 1

Subnormal Normal

hgt(Norm(A)) =1

)(hgt)(

))((NormA

xAxA =

1.0

x

1.0

x

Norm(A)

A

A(x) A(x)


Support

}0)(|{)(Supp >∈= xAxA X

}0)(|{)(CSupp >∈= xAxclosureA X

Open set

Closed set

1.0

x

A(x)

A

Supp(A)

1.0

x

A(x)

A

CSupp(A)


Core

1.0

x

A(x)

A

Core(A)

}1)(|{)(Core =∈= xAxA X

α−cut

1.0

x

A(x)

A

Aα+

α

1.0

x

A(x)

A

Aα

α

})(|{ α≥∈=α xAxA X })(|{ α>∈=α xAxA X

Stronger condition

Convexity

1.0

x

A(x)

Aα

Nonconvex

1.0

x

A(x)A[λx1 + (1−λ)x2)]

x1 x2

x = λx1 + (1−λ)x2

0 ≤ λ ≤ 1 Convex fuzzy set

})(|{ α>∈=α xAxA X

)](),(min[])1([ 2121 xAxAxxA ≥λ−+λ

Cardinality

∑∈

=Xx

xAA )()(Card

∫=X

xxAA d)()(Card

X finite or countable

Card(A) = | A | sigma count (σ–count)


3.2 Equality and inclusion relationships for fuzzy sets

Equality


A = B iff A(x) = B(x) ∀x ∈ X

Inclusion

A ⊆ B iff A(x) ≤ B(x) ∀x ∈ X


1.0

x

A(x)A B

1.0

x

A(x)A B

A ⊆ B A ⊄ B

Sets

Degree of inclusion

+≤

=⇒otherwise)()(1

)()(if1)()(

xBx-A

xBxAxBxA

∫ ⇒=⊂X

xxBxAxBxA d))()(()(Card

1)()(

X

Degree of equality

∫ ⇒⇒==X

xxAxBxBxAxBxA d))]()(),()([min()(Card

1)()(

X


Example

Examples of fuzzy sets A and B along with their degrees of inclusion:

(a) a = 0, n = 2, b =3; m = 4 σ = 2; ||A = B|| = 0.637

(b) b = 7 ||A = B|| = 0.864

(c) a = 0, n = 2, b = 9, m = 4, σ = 0.5 || A =B || = 0.987


3.3 Energy and entropymeasures of fuzziness

Energy measure of fuzziness


∑=

=n

iixAeAE

1)]([)( Card (X) = n

∫=X

xxAeAE d)]([)(

e : [0, 1] → [0, 1] such that

e(0) = 0

e(1) = 1

e: monotonically increasing


Example

e(u) = u ∀u ∈ [0, 1] linear

∑=

==n

ii AxAAE

1)(Card)()(

),(|)()(|)()(11

φ=φ−== ∑∑==

AdxxAxAAEn

iii

n

ii

d = Hamming distance


e(u) non-linear

1.0

u

e(u)

1.0

u

e(u)

1.0 1.0

Emphasis on highmembership values

Emphasis on lowmembership values


Inclusion of probabilistic information

∑=

=n

iii xAepAE

1)]([)(

∫=X

xxAexpAE d)]([)()(

pi: probability of xi

p(x): probability density function


Entropy measure of fuzziness

∑=

=n

iixAhAH

1)]([)(

∫=X

xxAhA d))(()(H

h : [0,1] → [0,1]

1-monotonically increasing [0, ½]

2-monotonically decreasing (½, 1]

3-boundary conditions:h(0) = h (1) = 0h(½) = 1

Specificity of fuzzy sets


Specific fuzzy set Lack of specificity

1.0

x

A(x)

1.0

x

A(x)

xo

Specificity


1-Spec(A) = 1 if and only if ∃x0 ∈ A(x0) = 1, A(x) = 0 ∀x≠ x0

2-Spec(A) = 0 if and only if A(x) = 0 ∀x ∈ X

3-Spec(A1) ≤ Spec(A2) if A1 ⊃ A2

1.0

x

A1

A2

Examples


∫α

αα= max

0d

)(1

)(ACard

ASpec

∑= α

α∆=m

ii

iACardASpec

1 )(1

)(

Yager (1993)

Geometric interpretation of sets and fuzzy sets


X = { x1, x2} P(X) = {∅, {x1}, { x2}, { x1, x2}}

1.01.0

x1

x2

1.0 1.0

x2

x1

A

F(X)

∅ ∅

{ x1, x2} { x1, x2}

Set Fuzzy set


3.4 Granulation of information


Motivation

� Need of granulation:

– abstract information– summarize information

� Purpose:

– comprehension– decision making– description

Discretization, quantization, granulation


1.0

x

A1 A2 A3 A4

x

1.0

x

C1 C2 C3 C4

1.0F1 F2 F3 F4

Discretatization Quantization Granulation

Formal mechanisms of granulation


⟨X, G, S, C ⟩

X : universe

G: formal framework of granulation

S: collection of information granules

C: transformation G

S1

S2

Sm

Fuzzy Interval Rough

S

C

3.5 Characterization of families of fuzzy sets

Frame of cognition


� Codebook of conceptual entities

– family of linguistic landmarks

Φ = {A1, A2, … ., Am}

Ai is a fuzzy set on X, i = 1, …, m

� Granulation that satisfies semantic constraints

– coverage

– semantic soundness

Coverage


Φ= {A1, A2, … ., Am} covers X if, for any x ∈ X

∃i ∈ I | Ai(x) > 0

∃i ∈ I | Ai(x) > δ (δ-level coverage) δ ∈ [0, 1]

Ai´s are fuzzy set on X, i ∈ I = {1, …, m}

Semantic soundness


• Each Ai, i ∈ I = {1, …, m} is unimodal and normal

• Fuzzy sets Ai are disjoint enough (λ-overlapping)

• Number of elements of Φ is low

1.0

x

Ai

λ

δ

Characteristics of frames of cognition


� Specificity: Φ1 more specific than Φ2 if Spec(A1i) > Spec(A2j)

� Granularity: Φ1 finer than Φ2 if |Φ1| > |Φ2|

1.0

x

A2A1 A3

Φ1

1.0

x

A3A2 A4A1

Φ2


� Focus of attention

1.0

x

AA

α

Regions of focus of attentionimplied by the correspondingfuzzy sets

• Information hiding

1.0

x

A

B

a1 a2 a3 a4

x ∈ [a2, a3] indistinguishable for A, but not for B

3.6 Fuzzy sets, sets andthe representationtheorem


Any fuzzy set can be viewed as a family of sets:


)(sup)(]1,0[

]1,0[

xAxA

AA

α∈α

∈αα

α=

α= U 1.0

x

A

αj

αk

αi

αkAαk

Aαi

Example


X = {1, 2, 3, 4}

A = {0/1, 0.1/2, 0.3/3, 1/4, 0.3/5} = [0, 0.1, 0.3, 1, 0.3]

A0.1 = {0/1, 1/2, 1/3, 1/4, 1/5} = [0, 1, 1, 1, 1] → 0.1A0.1 = [0, 0.1, 0.1, 0.1, 0.1]

A0.3 = {0/1, 0/2, 1/3, 1/4, 1/5} = [0, 0, 1, 1, 1] → 0.3A0.3 = [0, 0 , 0.3, 0.3, 0.3]

A1 = {0/1, 0/2, 0/3, 1/4, 0/5} = [0, 0, 0, 1, 0] → 1.0A1 = [0, 0, 0, 1, 0]

A = max (0.1A0.1 , 0.3A0.3 , 1A1 )

A = [max (0,0,0), max(0.1,0,0), max(0.1,0.3,0), max(0.1,0.3,1), max(0.1,0.3,0)]

A = [0, 0.1, 0.3, 1, 0.3]

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3 Characterization of Fuzzy Sets - dca.fee.unicamp.br · 3.1 Generic characterization of fuzzy sets: fundamental descriptors 3.2 Equality and inclusion relationships in fuzzy sets