Fuzzy Chapter 04

0 z
2. Centroid method
z =
∗
∫ µ ( z ) ⋅ z dz
C
∫ µ ( z ) dz
C
3. Weighted average method:
z =
∗
∑ µ (z ) ⋅ z
C
∑ µ (z )
C
z × 0.5 + z × 0.3 z = 0.5 + 0.3
∗ 1 2
4. Mean max membership:
z = sup { z ∈ Z µ ( z ) = hgt (C )}
∗ z∈Z CK k
µ(x) 1 0.5 0 2 4 6 8 10
z
z∗ =
a+b 2
5. Center of sums:
z =
∗
∫ z ∑ µ ( z ) dz
z k =1 Ck
n
∫ ∑ µ ( z ) dz
Z k =1 Ck
n
µ(x) 1 0.5 0 2 4 6 8 10
z
defuzzification step
6. Center of largest area:
λ=1,
λ=0.9,
R 0 .9
1 0 = 0 0 0
λ=0.5,
1 1 R = 0 0 0
1
1 1 0 0 1
0 0 1 0 0
0 0 0 1 1
0 1 0 1 1
λ=0,
R0=E
Properties
1. ( R ∪ S )λ = Rλ ∪ Sλ 2. ( R ∩ S )λ = Rλ ∩ Sλ 3. ( R )λ ≠ Rλ 4.
~ ~
λ
λ
λ
2.(A∩B) = A ∩B
~ ~λλ来自λ3.(A) ≠ A except for a valueof λ = 0.5
~
λ
λ
4. For any λ ≤α, where0 ≤α ≤1,itistruethat A ⊆ A ,whereA = X
α λ
0
λ-cut(or α-cut) for fuzzy relation
µ A ( y ) ≥ min[ µ A ( x), µ A ( z )]
Nonconvex fuzzy: x,y,z in fuzzy A,
∃ y , if x<y<z implies
that µ A ( y ) < min[ µ A ( x ), µ A ( z )] If A and B are both convex, then A∩B is also convex Defuzzification to crisp sets λ-cut(or α-cut)
Rλ = {( x, y ) | µ R ( x, y ) ≥ λ}
1 0.8 R= 0 0.1 0.2 0.1 0.2 1 0.4 0 0.9 0.4 1 0 0 0 0 1 0.5 0.9 0 0.5 1 0.8 0
1 0 R1 = 0 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1
z =
∗
∫ µ ( z ) zdz
Cm
∫ µ ( z ) dz
Cm
7. First (or last) of maxima: First, the largest height in the union (denoted hgt(Ck)) is determined,
hgt (C ) = supµ ( z )
Aλ = {x | µ A ( x) > λ}
Ex.1 A: discrete fuzzy set, using Zadeh’s notation, defined on
universe X={a,b,c,d,e,f}.
1 0.9 0.6 0.3 0.01 0 A= + + + + + a b c d e f
For any λ ≤ α , 0 ≤ α ≤ 1, then Rα ⊆ Rλ .
Defuzzification to scalars
C = ∪C = C
k i =1 i
κ
1. Max membership principle:
µ ( z ) ≥ µ ( z)
∗ C C
for all z ∈Z
µ(x) 1
Then A1={a} A0.9={a,b} A0.6={a,b,c} A0.3={a,b,c,d} A0+={a,b,c,d,e} A0=X λ-cut(or α-cut) sets obey the following four very special properties:
1.(A∪B) = A ∪B
Chap.4 Membership Functions
Features of the membership function
Fig.4.1 Core, support, and boundary
0.5
Fig. 4.2 normal (a) and subnormal (b) Core: µ A ( x ) = 1 Support: µ A ( x) > 0 Boundary: 1 > µ A ( x) > 0 Normal fuzzy: µ A ( x) = 1, at least one element x in fuzzy A. Subnormal fuzzy: µ A ( x) < 1, for all x in A. Convex fuzzy: x,y,z in fuzzy A, x<y<z implies that
k z∈Z CK
Then the first of the maxima is found,
z = inf { z ∈ Z µ ( z) = hgt (C )}
∗ z∈Z CK k
An alternative to this method is called the last of maxima, and it is given by