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**Unformatted text preview: **Fuzzy set definition Specification of
membership
functions and
Multi-dimensional
fuzzy sets Typical membership functions
! "# Typical membership functions
"# gaussmf x, c, s = e µ A : X → [0,1]
µ Typical membership functions
! triangularmf x, c, h = max min h − c + x , c + h − x , 0
h
h $ ⊂ − 2
x−c
2
2⋅s "# trapmf x, a, b, c, d = max min x − a , 1, d − x , 0
b−a
d −c Typical membership functions
$ "# gbellmf x, a, b, c = x
a 1 − c 2⋅b
a +1 1 Typical membership functions
$ "# gbellmf x, a, b, c = x
a % 1 − c 2⋅b
a Typical membership functions
sigmf x, a, c = +1 Typical membership functions
! "# "# 1 e −a x−c +1 Multi-dimensional fuzzy sets
'
() & "# µ *
∈ =µ
* ∈ "# Multi-dimensional fuzzy sets
Base set A Multi-dimensional fuzzy sets Cylindrical Ext. of A +, )
µR X i 1× X i 2 × × X ik ( xi1 , xi 2 , , xik ) = max X j 1 , X j 2 , , X jn µ R (x1 , x2 , , xn ) 2 Multi-dimensional fuzzy sets
.
0 1 / Multi-dimensional fuzzy sets
Two-dimensional Projection Projection MF onto X onto Y ) 23 ! - 1 3
RX = max µ R ( x, y) | x
y X RY = max µ R ( x, y) | y
Y 4
/ x Multi-dimensional fuzzy sets
6 7 ) 5 Multi-dimensional fuzzy sets
,8 * "# % - 9*
:; f x, y = e − y−4 2 ⋅e 2
− x−3
2 2 ,8 *
9*
:;
f x, y = e f x, y = e − y−4 2− x − 3 2
2 2 − y−4 2 2 fx x, y = e 2 2
− x−3
2 2
2 8
⋅e 9; fy x, y = e 2
− x−3
2 2 2
− y−4 8 Multi-dimensional fuzzy sets fx x, y = e − x−3 2
2 2
2 2 fy x, y = e 2
− y−4 f x, y = e "# − y−4 2 ⋅e 2 :; Multi-dimensional fuzzy sets − x−3 2
2 2
2 2 fx x, y = e − x−3 2
2 2
2 2 fy x, y = e 2
− y−4 f ( x, y ) = fx ( x, y ) ∪ fy ( x, y ) !
- - 3 Multi-dimensional fuzzy sets
!
* f x, y = x−3 ⋅ y−4 +1 − 2.5 4 ...

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- Winter '16
- ganeshan
- Membership functions, Fuzzy set, Multi-dimensional fuzzy sets, Typical membership functions