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
Pedrycz and Gomide, FSE 2007
Fuzzy sets
Pedrycz and Gomide, FSE 2007
� 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
Pedrycz and Gomide, FSE 2007
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)
Pedrycz and Gomide, FSE 2007
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)
Pedrycz and Gomide, FSE 2007
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)
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)
Equality
Pedrycz and Gomide, FSE 2007
A = B iff A(x) = B(x) ∀x ∈ X
Inclusion
A ⊆ B iff A(x) ≤ B(x) ∀x ∈ X
Degree of inclusion
+≤
=⇒otherwise)()(1
)()(if1)()(
xBx-A
xBxAxBxA
∫ ⇒=⊂X
xxBxAxBxA d))()(()(Card
1)()(
X
Pedrycz and Gomide, FSE 2007
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
Energy measure of fuzziness
Pedrycz and Gomide, FSE 2007
∑=
=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
Pedrycz and Gomide, FSE 2007
Example
e(u) = u ∀u ∈ [0, 1] linear
∑=
==n
ii AxAAE
1)(Card)()(
),(|)()(|)()(11
φ=φ−== ∑∑==
AdxxAxAAEn
iii
n
ii
d = Hamming distance
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
Inclusion of probabilistic information
∑=
=n
iii xAepAE
1)]([)(
∫=X
xxAexpAE d)]([)()(
pi: probability of xi
p(x): probability density function
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
Specific fuzzy set Lack of specificity
1.0
x
A(x)
1.0
x
A(x)
xo
Specificity
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
∫α
αα= max
0d
)(1
)(ACard
ASpec
∑= α
α∆=m
ii
iACardASpec
1 )(1
)(
Yager (1993)
Geometric interpretation of sets and fuzzy sets
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
Motivation
� Need of granulation:
– abstract information– summarize information
� Purpose:
– comprehension– decision making– description
Discretization, quantization, granulation
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
⟨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
Frame of cognition
Pedrycz and Gomide, FSE 2007
� 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
Pedrycz and Gomide, FSE 2007
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
Pedrycz and Gomide, FSE 2007
• 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
Pedrycz and Gomide, FSE 2007
� 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
Pedrycz and Gomide, FSE 2007
� Focus of attention
1.0
x
AA
α
Regions of focus of attentionimplied by the correspondingfuzzy sets
Any fuzzy set can be viewed as a family of sets:
Pedrycz and Gomide, FSE 2007
)(sup)(]1,0[
]1,0[
xAxA
AA
α∈α
∈αα
α=
α= U 1.0
x
A
αj
αk
αi
αkAαk
Aαi
Example
Pedrycz and Gomide, FSE 2007
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]