NL (A, B) = (Ao B) ∧(AoB) ˆ
c
格贴近度
is said a lattice measure of similarity of A and B on F (X), NL(A,B) is called a lattice measure function of similarity of A and B on F (X).
1.4.3 Crisp domain of fuzzy operator (p.33)
清晰点 A(x)=0 or A(x)=1—— x is a crisp point. A(x)∈(0,1) —— x is a fuzzy point. Definition 1.4.3 Let * be a fuzzy operator on [0,1] and
1 NH (A B) =1− (0.2+0.1+0.1+0.1+0.1+0.2) ≈ 0.867 , 6 1 2 2 2 2 2 2 1 NE (A B) =1− , (0.2 + 0.1 + 0.1 + 0.1 + 0.1 + 0.2 )2 ≈ 0.859 6
0.5+0.7 +0.9+0.9+0.6+0.3 NM (A B) = , ≈ 0.830 0.7 +0.8+1.0+1.0+0.7 +0.5
2 n d1(A) = ⋅ ∑ A(xi ) − A (xi )| | 0.5 n i=1
d2 (A) =
2 n
1 2
⋅ (∑ A xi ) − A (xi )| ) | ( 0.5
i=1
n
1 2 2
, ( 1 A xi ) ≥ 0.5 ln2 A (xi ) = 0.5 ( 0, A xi ) < 0.5
What degree of fuzziness is for a fuzzy set (Luca & Termini, 1972)
Definition 1.4.2 If a mapping d : F (X) → [0,1] satisfies the following conditions: (1) A∈P (X) ⇔ d(A)=0; (2) A(x)≡1/2 ⇔ d(A)=1; (4) ∀A∈F (X), d(A)=d(Ac).
对称性 最贴近与最不贴近的情形 离的越远贴近度越小
(3) A⊆B⊆C⇒N(A,C)≤N(A,B)∧N(B,C). Then N is said measure function of similarity.
贴近度
☺1 Hamming NH
1 n NH (A B) =1− ∑ A xi ) − B(xi )| , | ( n i=1
经典集不模糊 这样的集最模糊 越靠近1/2越模糊
(3) (∀x∈X)B(x)≤A(x)≤1/2 ⇒ d(B)≤d(A); ∀
对称性
Then d is said measure of fuzziness in F (X). 模糊度
Examples (p.32)
☞ Hamming ☞ Euclid Where ☞ Shannon
☺4 Min-average NA
1 A= B =∅ , n ∑ A(xi ) ∧ B(xi )) ( NA(A B) = i=1 , he wie , ot r s n 1 ∑ A(xi ) + B(xi )) ( 2 i=1
1 A= B =∅ , b NA(A B) = ∫a (AIB)(x)dx , , ot r s he wie 1 b ( 2 ∫a (A x) + B(x))dx
0.8 = b, 0.3
Algebra Bounded Einstein Hamacher Yager
ˆ a +b = a +b−ab
a ⊕b = m a +b,1 in( )
a ˆ b = ab ⋅
a b=m ax(0, a +b −1 )
see book
Theorem 1.3.3 For all T and S △≤T≤∧≤∨≤S≤▽ Theorem 1.3.8 Let a, b∈[0,1], then (1) (2) (3) (4)
Ao B = ∨ (A x) ∧ B(x)) (
x∈X
Ao B = ∧ (A(x) ∨ B(x)) ˆ
x∈X
ˆ Then Ao B and Ao B are respectively said inner product and outer product of fuzzy A and B.
Definition 1.4.8 Let A,B∈F (X), then
Definition 1.4.1 For finite universe X
| A|= ∑A x) (
x∈X
基数或势
—— Cardinality of fuzzy set A
| A| || A||= | X|
—— Relative cardinality of fuzzy set A
1.4.2 Measure of fuzziness on fuzzy sets (p.29)
0.5+0.7 +0.9+0.9+0.6+0.3 NA(A B) = 1 , ≈ 0.907 2 (1.2 +1.5+1.9 +1.9 +1.3+ 0.8)
1.4.6 Lattice measure of similarity of fuzzy sets (p.39)
Lattice degree of similarity is ቤተ መጻሕፍቲ ባይዱnother sort of measure for two sets on their degree of closeness. Definition 1.4.5 Let A, B∈F (X) and
0
1 n H(A) = ∑s(A(xi )) nln2 i=1
1/2
1
−xln x −(1− x)ln(1− x), Where s(x) = 0,
x∈(0,1 ) x = 0,1
e.g. 1.4.3 Let X={a, b, c, d} and
0.8 0.9 0.1 0.8 A= + + + a b c d
Name Zadeh
0.5 ˆ 0.8 = 0.4 ⋅
0.5
t-conorm (S) ∨
t-norm (T) ∧
a =1 b, a∆b = a, b =1 且 0, a ≠1 b ≠1
a =0 a∇ = a, b = 0 b Drastic ∧ 0.8 = 0.5 0.5 1 ab ≠ 0 ,
1 b NH (A B) =1− , ∫a | A(x) − B(x)| dx b −a
☺2 Euclid NE
1 1 n NE (A B) =1− , (∑ A(xi ) − B(xi ))2 )2 ( n i=1
NE (A B) =1− ,
1 b −a
(∫ (A(x) − B(x))2 dx)
a
tt§1.3 t-norm and t-conorm (p.10)
Problem ∪ and ∩ of fuzzy sets are defined by operators ∨ and ∧, But
?∨ 0.8 = 0.8
↑ [0,0.8]
Information is lost.
Triangle Norm (Menger, 1942) Definition 1.3.1/1.3.2 A mapping △:[0,1]×[0,1]→[0,1] is said a triangle norm, if it satisfies the following conditions (1) Commutativity △(a, b)=△(b, a); (2) Associativity △(△(a, b), c)=△(a, △(b, c)); (3) Monotony a≤c, b≤d ⇒ △(a, b)≤△(c, d)
格贴近度函 数
e.g. In Example 1.4.6, X={x1, x2, x3, x4, x5, x6} and
0.5 0.7 1.0 0.9 0.6 0.3 A= + + + + + x x2 x3 x4 x5 x6 1 0.7 0.8 0.9 1.0 0.7 0.5 B= + + + + + x x2 x3 x4 x5 x6 1
b
1 2
☺3 Max-min NM
1 A= B =∅ , n ∑ A(xi ) ∧ B(xi )) ( NM (A B) = i=1 , , ot r s he wie n ∑ A(xi ) ∨ B(xi )) ( i=1
1 A= B =∅ , b (AIB)(x)dx NM (A B) = ∫a , , ot r s he wie b ∫a (AUB)(x)dx
e.g. 1.4.6 Let X ={x1, x2, x3, x4, x5, x6} and
A= 0.5 0.7 1.0 0.9 0.6 0.3 + + + + + x x2 x3 x4 x5 x6 1
Then
0.7 0.8 0.9 1.0 0.7 0.5 B= + + + + + x x2 x3 x4 x5 x6 1
T is t-norm = triangle norm + T(1,a)=a S is t-conorm = triangle norm + S(a,0)=a T(a, b) and S(a, b) are written as aTb and aSb respectively.