模糊数学pptCh1_Sec3-4-5

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
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
a ∨b = a ⊕(a b) c a ∧b = a (a ⊕b)
c
a b = (a +b −1 ∨0 )
ˆ a ˆ b = a (a +b) ⋅ ˆ b = a⊕(ac ˆ b) a+ ⋅
c
Proof (1) a ⊕(ac
a ⊕b = (a +b) ∧1
) b) = (a +((1−a) +b−1 ∨0) ∧1
☺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
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).
0.3 0.3 B= + a c
Then
1 1 1 A = + + 0.5 a b d
B =∅ 0.5
.
2 d1(A) = (| 0.8−1| +|0.9−1| +| 0.1−0| +|0.8−1|) = 0.3 4 2 d1(B) = (| 0.3| +| 0.3|) = 0.3 4 2 2 2 2 2 1 d2 (A) = (|0.8−1| +|0.9 −1| +| 0.1−0| +|0.8−1| )2 = 0.1 ≈ 0.316 4 2 2 2 1 d2 (B) = (| 0.3| +| 0.3| )2 = 0.18 ≈ 0.424 4
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
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
b, b ≥ a = (a +(b − a) ∨0) ∧1= a, b < a
= a ∨b
§1.4 Measurement of Fuzziness (p.29)
1.4.1 Cardinality of fuzzy sets e.g. A={1,2,3,4} How many elements are there in crisp A Answer: 4 Cardinality of A is 4. 基数或势
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)
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)
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
σ(∗) ={(x, y) ∈[0,1] | x∗ y = 0 or 1 }
2
Then domain σ(*) is said the crisp domain of fuzzy operator *. 清晰域
e.g. σ(∧)={(x, y) | x∧y=0 or 1} ={(x, y) | x=0 or y=0}∪{( x, y) | x=y=1} σ(⊙)={(x, y) | max(0, x+y-1) = 0 or 1} ={(x, y) | x+y≤1}∪{1,1}
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
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.
Common operator
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
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
Name Zadeh
0.5 ˆ 0.8 = 0.4 ⋅
0.5
t-conorm (Swk.baidu.com ∨
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 ,
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
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 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
y
1
(1, 1)
0
1
x
1.4.5 Degree of similarity of fuzzy sets (p.36)
Degree of similarity is a sort of measure for two sets on their degree of closeness. Definition 1.4.5 Let A, B, C∈ F(X) and a mapping N: F(X) × F(X) →[0,1] satisfy the following conditions: (1) N(A,B)=N(B,A); (2) N(A, A)=1, N(X,Φ)=0;
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)
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).
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 another sort of measure for two sets on their degree of closeness. Definition 1.4.5 Let A, B∈F (X) and