Extension Principle Adriano Cruz ©2002 NCE e IM/UFRJ
[email protected]
Fuzzy Numbers
A fuzzy number is fuzzy subset of the universe of a numerical number. – A fuzzy real number is a fuzzy subset of the domain of real numbers. – A fuzzy integer number is a fuzzy subset of the domain of integers.
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 2
Fuzzy Numbers - Example u(x) Fuzzy real number 10
5
10
15
x
u(x) Fuzzy integer number 10
5 @2002 Adriano Cruz
10
15 NCE e IM - UFRJ
x No. 3
Functions with Fuzzy Arguments
A crisp function maps its crisp input argument to its image.
Fuzzy arguments have membership degrees.
When computing a fuzzy mapping it is necessary to compute the image and its membership value.
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 4
Crisp Mappings X
@2002 Adriano Cruz
f(X)
NCE e IM - UFRJ
Y
No. 5
Functions applied to intervals
Compute the image of the interval.
An interval is a crisp set. y y=f(I)
I @2002 Adriano Cruz
x NCE e IM - UFRJ
No. 6
Mappings f(X) Y X
Fuzzy argument?
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 7
Extension Principle
Suppose that f is a function from X to Y and A is a fuzzy set on X defined as A = µA(x1)/x1 + µA(x2)/x2 + ... + µA(xn)/xn
The extension principle states that the image of fuzzy set A under the mapping f(.) can be expressed as a fuzzy set B.
B = f(A) = µA(x1)/y1 + µA(x2)/y2 + ... + µA(xn)/yn where yi=f(xi) @2002 Adriano Cruz
NCE e IM - UFRJ
No. 8
Extension Principle
If f(.) is a many-to-one mapping, mapping then there exist x1, x2 ∈X, x1 ≠ x2, such that f(x1)=f(x2)=y*, y*∈ ∈Y.
The membership grade at y=y* is the maximum of the membership grades at x1 and x2
more generally, we have
µ B ( y ) = max µ A ( x) x = f −1 ( y )
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 9
Monotonic Continuous Functions
For each point in the interval – Compute the image of the interval. – The membership degrees are carried through.
I @2002 Adriano Cruz
NCE e IM - UFRJ
No. 10
Monotonic Continuous Functions y
y
x u(y) u(x)
x @2002 Adriano Cruz
NCE e IM - UFRJ
No. 11
Monotonic Continuous Ex.
Function: y=f(x)=0.6*x+4
Input: Fuzzy number - around-5 – Around-5 = 0.3 / 3 + 1.0 / 5 + 0.3 / 7
f(around-5) = 0.3/f(3) + 1/f(5) + 0.3/f(7)
f(around-5) = 0.3/0.6*3+4 + 1/ 0.6*5+4 + 0.3/ 0.6*7+4
f(around-5) = 0.3/5.8 + 1.0/7 + 0.3/8.2 I
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 12
Monotonic Continuous Ex.
1
0.3
8.2
f(x) 10
5.8
4 5
10
x
u(x) 1 0.3 3 5
@2002 Adriano Cruz
NCE e IM - UFRJ
7
x No. 13
Nonmonotonic Continuous Functions
For each point in the interval – Compute the image of the interval. – The membership degrees are carried through. – When different inputs map to the same value, combine the membership degrees.
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 14
Nonmonotonic Continuous Functions y
y
x u(y) u(x)
x @2002 Adriano Cruz
NCE e IM - UFRJ
No. 15
Nonmonotonic Continuous Ex.
Function: y=f(x)=x2-6x+11 Input: Fuzzy number - around-4 Around-4 = 0.3/2+0.6/3+1/4+0.6/5+0.3/6 y = 0.3/f(2)+0.6/f(3)+1/f(4)+0.6/f(5)+0.3/f(6) y = 0.3/3+0.6/2+1/3+0.6/6+0.3/11 y = 0.6/2+(0.3 v 1)/3+0.6/6+0.3/11 y = 0.6/2 + 1/3 + 0.6/6 + 0.3/11 I
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 16
Nonmonotonic Continuous Functions 1 v 0.3
y
y
x u(y) 1 0.6
0.3 u(x) 1 0.6 0.3
x 2 3 4 5 6
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 17
Function Example 1
Consider
2
x y = f ( x) = 1 − 4
Consider fuzzy set ~
A = ∫ [1 2 | x |] / x | − 2 ≤ x ≤ +2
Result
@2002 Adriano Cruz
~
~
B = f ( A) = ∫ µ B ( y ) / y NCE e IM - UFRJ
Y
No. 18
Function Example 2
Result according to the principle ~
~
B = f ( A) = ∫ µ B ( y ) / y = ∫ µ A ( x) / f ( x) x = 2 1− y
Y 2
Y
µ A ( x) = 1 2 | x | µ A ( x) =| 1 − y | 2
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 19
Function Example 3
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 20
Extension Principle
Let f be a function with n arguments that maps a point in X1xX2x...xXn to a point in Y such that y=f(x1,…,xn). Let A1x…xAn be fuzzy subsets of X1xX2x...xXn The image of A under f is a subset of Y defined by
−1 [ ⊗ µ ( x )] if f ( y) ≠ 0 i Ai i ⊕ µ B ( y ) = ( x1 ,Kxn ),( x1 ,K, xn ) = f −1 ( y ) 0 if f −1 ( y ) = 0
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 21
Arithmetic Operations
Applying the extension principle to arithmetic operations it is possible to define fuzzy arithmetic operations Let x and y be the operands, z the result. Let A and B denote the fuzzy sets that represent the operands x and y respectively.
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 22
Fuzzy addition
Using the extension principle fuzzy addition is defined as
µ A+ B ( z ) = ∨ ( µ A ( x) ∧ µ B ( y )) x, y x+ y= z
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 23
Fuzzy addition - Examples
A = (3~) = 0.3/1+0.6/2+1/3+0.6/4+0.3/5
B =(11~)= 0.5/10 + 1/11 + 0.5/12
A+B=(0.3^0.5)/(1+10) + (0.6^0.5)/(2+10) + (1^0.5)/(3+10) + (0.6^0.5)/(4+10) + (0.3^0.5)/(5+10) + (0.3^1)/(1+11) + (0.6^1)/(2+11) + (1^1)/(3+11) + (0.6^1)/(4+11) + (0.3^1)/(5+11) +( 0.3^0.5)/(1+12) + (0.6^0.5)/(2+12) + (1^0.5)/(3+12) + (0.6^0.5)/(4+12) + (0.3^0.5)/(5+12)
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 24
Fuzzy addition - Examples
A = (3~) = 0.3/1+0.6/2+1/3+0.6/4+0.3/5 B =(11~)= 0.5/10 + 1/11 + 0.5/12 Getting the minimum of the membership values A+B=0.3/11 + 0.5/12 + 0.5/13 + 0.5/14 + 0.3/15 + 0.3/12 + 0.6/13 + 1/14 + 0.6/15 + 0.3/16 + 0.3/13 + 0.5/14 + 0.5/15 + 0.5/16 + 0.3/17
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 25
Fuzzy addition - Examples
A = (3~) = 0.3/1+0.6/2+1/3+0.6/4+0.3/5
B =(11~)= 0.5/10 + 1/11 + 0.5/12
Getting the maximum of the duplicated values
A+B=0.3/11 + (0.5 V 0.3)/12 + (0.5 V 0.6 V 0.3)/13 + (0.5 V 1 V 0.5)/14 + (0.3 V 0.6 V 0.5)/15 + (0.3 V 0.5)/16 + 0.3/17
A+B=0.3 / 11 + 0.5 / 12 + 0.6 / 13 + 1 / 14 + 0.6 / 15 + 0.5 / 16 + 0.3 / 17
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 26
Fuzzy addition
A, x=3
B, y=11
C, x=14
0.6 0.5 0.3
@2002 Adriano Cruz
NCE e IM - UFRJ
No. 27
Fuzzy Arithmetic
Using the extension principle the remaining fuzzy arithmetic fuzzy operations are defined as: µ A− B ( z ) = ∨ µ A ( x ) ∧ µ B ( y ) x, y
x− y= z
µ A*B ( z ) = ∨ µ A ( x) ∧ µ B ( y ) x, y
x* y = z
µ A / ( z ) = ∨ µ A ( x) ∧ µ B ( y ) @2002 Adriano Cruz
x, y x / y= z NCE e IM - UFRJ
No. 28