Transcript fuzzy relation

Chapter 3
FUZZY RELATION AND
COMPOSITION
Chi-Yuan Yeh
Outline
•
•
•
•
•
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Product set
Crisp / fuzzy relations
Composition / decomposition
Projection / cylindrical extension
Extension of fuzzy set / fuzzy relation
Fuzzy distance between fuzzy sets
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Product set
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Product set
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Product set
• A={a1,a2} B={b1,b2} C={c1,c2}
• AxBxC =
{(a1,b1,c1),(a1,b1,c2),(a1,b2,c1),(a1,b2,c2),(a
2,b1,c1),(a2,b1,c2),(a2,b2,c1), (a2,b2,c2)}
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Crisp relation
• A relation among crisp sets A1 , A2 , , An is a
subset of the Cartesian product. It is denoted
by R .
R  A1  A2 
 An
• Using the membership function defines the
crisp relation R :
1 iff (x1 , x2 , ..., xn )  R,
 R ( x1 , x2 , ,xn )  
0 otherwise
where x1  A1 , x2  A2 ,..., xn  An
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Fuzzy relation
• A fuzzy relation is a fuzzy set defined on the
Cartesian product of crisp sets A1, A2, ..., An
where tuples (x1, x2, ..., xn) may have varying
degrees of membership within the relation.
• The membership grade indicates the strength
of the relation present between the elements of
the tuple.
R : A1  A2  ...  An  [0,1]
R  (( x1 , x2 ,..., xn ), R ) | R ( x1 , x2 ,..., xn )  0, x1  A1 , x2  A2 ,..., xn  An 
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Representation methods
• Bipartigraph
(Crisp)
(Fuzzy)
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Representation methods
• Matrix
B
y1
y2
y3
B
y4
x1
x1
x2
x2
x3
x3
x4
x4
(Crisp)
y1
y2
y3
y4
(Fuzzy)
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Representation methods
• Digraph
(Crisp)
(Fuzzy)
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Domain and range of fuzzy relation
 R ( x, y )
• Domain: dom ( R ) ( x)  max
yB
 R ( x, y )
• Range : ran ( R ) ( y )  max
xA
domain
range
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Domain and range of fuzzy relation
• Fuzzy matrix
dom ( R ) ( x1 )  1.0
dom ( R ) ( x2 )  0.4
dom ( R ) ( x3 )  1.0
dom ( R ) ( x4 )  1.0
dom ( R ) ( x5 )  0.5
dom ( R ) ( x6 )  0.2
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Operations on fuzzy matrices
• Sum: A  B  max[aij , bij ]
• Example
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Operations on fuzzy matrices
• Max product: C = A・B=AB=
• Example
C12  ?
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Max product
• Example
C12  0.1
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Max product
• Example
C13  0.5
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Max product
• Example
C
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Operations on fuzzy matrices
• Scalar product:  A where 0    1
• Example
a
0.5 A 
b
c
a
b
0.1 0.25 0.0
0.2 0.5 0.05
c
0.0
0.5
0.0
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Operations on fuzzy relations
• Union relation
( x, y)  A  B
RS ( x, y)  max( R ( x, y), s ( x, y))
 R ( x, y)  s ( x, y)
• For n relations
( x, y )  A  B
 R  R ... R ( x, y )    R ( x, y )
1
2
n
Ri
i
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Union relation
• Example
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Operations on fuzzy relations
• Intersection relation
( x, y)  A  B
 RS ( x, y)  min(  R ( x, y), s ( x, y))
  R ( x, y)  s ( x, y)
• For n relations
( x, y )  A  B
 R  R ... R ( x, y )    R ( x, y )
1
2
n
Ri
i
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Intersection relation
• Example
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Operations on fuzzy relations
• Complement relation:
( x, y)  A  B
R ( x, y)  1  R ( x, y)
• Example
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Composition of fuzzy relations
• Max-min composition
( x, y)  A  B, ( y, z )  B  C
S R ( x, z)  max[min(R ( x, y), S ( y, z))]
y
 [R ( x, y)  S ( y, z)]
• Example
y
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Composition of fuzzy relations
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Composition of fuzzy relations
• Example
S R (1,  )  max[min(0.1,0.9), min(0.2,0.2), min(0.0,0.8), min(1.0,0.4)]
 max[0.1,0.2,0.0,0.4]  0.4
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Composition of fuzzy relations
• Example
S R (1,  )  max[min(0.1,0.0), min(0.2,1.0), min(0.0,0.0), min(1.0,0.2)]
 max[0.0,0.2,0.0,0.2]  0.2
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Composition of fuzzy relations
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α-cut of fuzzy relation
•
• Example
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α-cut of fuzzy relation
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Decomposition of relation
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Decomposition of relation
0
0.9
0.0
MR 
0.0
0.4
0.4
1.0
0.7
0.0
0.0
0.4
1.0
0.0
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Decomposition of relation
0.9
0.0
MR 
0.0
0.4
0.4
1.0
0.7
0.0
0.0
0.4
1.0
0.0

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Projection / cylindrical extension
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Projection / cylindrical extension
dom ( R ) ( x)  max  R ( x, y)
yB
ran ( R ) ( y )  max  R ( x, y )
xA
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Projection in n dimension
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Projection
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Projection
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Projection
max(0.4,0.5)  0.5
max(0.2,0.1)  0.2
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Projection
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Projection / cylindrical extension
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Cylindrical extension
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Cylindrical extension
C ( R) ( x1, x2 , x3 )  R ( x3 )
3
C ( R ) (0, 0, 0)   R (0)  1.0
3
C ( R ) (0,1, 0)   R (0)  1.0
3
C ( R ) (1, 0, 0)   R (0)  1.0
3
C ( R ) (1,1, 0)   R (0)  1.0
3
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Cylindrical extension
• x1 = 0 : x，x1 = 1 : y
• x2 = 0 : a， x2 = 1 : b
• x3 = 0 : α， x3 = 1 : β
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Cylindrical extension
(x 1 x2 x3)
R123
R12
R23
R123'
R123''
R123'''
000
001
010
011
100
101
110
111
0.9
0.0
0.4
0.0
1.0
0.7
0.0
0.0
0.9
0.9
0.4
0.4
1.0
1.0
0.0
0.0
1.0
0.7
0.4
0.0
1.0
0.7
0.4
0.0
0.9
0.9
0.4
0.4
1.0
1.0
0.0
0.0
1.0
0.7
0.4
0.0
1.0
0.7
0.4
0.0
0.9
0.7
0.4
0.0
1.0
0.7
0.0
0.0
• Join(R123’，R123’’) = C(R123’)∩C(R123’’)
= Min(R123’，R123’’)
= R123’’’
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Extension of fuzzy set
A crisp function
f : X Y
f ( A)  { y | y  f ( x), x  A}
f ( B)  {x | f ( x)  B}
Let A  {a1, a 2, a3} B  {b1, b2} then
f : a1  b1
f : a 2  b2
f : a 3  b3
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Extension of fuzzy set
There are two universal sets
X  {x1, x 2,...,xn}
Y  { y1, y 2,..., yn}
And A  X , B  Y
A  {( x1,1), ( x2,1), ( x3,1)}
R  {((x1, y1),1), ((x2, y 2),1), ((x3, y 2),1)}
We can obtain B by A and R, use
 ( y)  Max[min( ( x), R( x, y))],x  A, y  B
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Extension of fuzzy set
By  ( y)  Max[min( ( x), R( x, y))],x  A, y  B
 ( y1)  Max[min( ( x1), R ( x1, y1))]  max[min(1,1)]  1
 ( y 2)  Max[min( ( x 2), R ( x 2, y 2)),min( ( x 3), R ( x 3, y 2))]
 max[min(1,1), min(1,1)]  1
get B  {( y1,1), ( y 2,1)}
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Extension of fuzzy set
If A is a fuzzy set A  {( x1,0.2), ( x2,0.7), ( x3,0.6)}
and R is R  {((x1, y1),1), ((x2, y 2),1), ((x3, y 2),1)}
We can also get B by A an R, use
 ( y)  Max[min( ( x), R( x, y))],x  X , y Y
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Extension of fuzzy set
By use
 ( y)  Max[min( ( x), R( x, y))],x  X , y Y
 ( y1)  Max[min( ( x1), R ( x1, y1))]  Max[min(0.2,1)]  0.2
 ( y 2)  Max[min( ( x 2), R ( x 2, y 2)),min( ( x3), R ( x3, y 2))]
 Max[min(0.7,1), min(0.6,1)]  0.7
so we can get B  {( y1,0.2), ( y 2,0.7)}
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Extension of fuzzy set
If A is a fuzzy set A  {( x1,0.2), ( x2,0.7), ( x3,0.6)}
and R is a fuzzy relation
R  {((x1, y1),0.5), ((x2, y 2),0.4), ((x3, y 2),0.9)}
We can get B by using
 ( y)  Max[min( ( x), R( x, y))],x  X , y Y
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Extension of fuzzy set
By  ( y)  Max[min( ( x), R( x, y))],x  X , y Y
 ( y1)  Max[min( ( x1), R ( x1, y1))]  Max[min(0.2,0.5)]  0.2
 ( y 2)  Max[min( ( x 2), R ( x 2, y 2)),min( ( x3), R ( x 3, y 2))]
 Max[min(0.7,0.4), min(0.6,0.9)]  0.6
so we can get B  {( y1,0.2), ( y 2,0.6)}
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Extension of fuzzy set
Extension of a crisp relation
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Extension of fuzzy set
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Extension by fuzzy relation
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Extension by fuzzy relation
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Extension by fuzzy relation
MR 2
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Extension by fuzzy relation
MR 2
B ' (b1 )  max[min(0.8, 0.3), min(0.3, 0.8)]  0.3
B ' (b2 )  max[min(0.8,1.0), min(0.3, 0.0)]  0.8
B ' (b3 )  max[min(0.8, 0.0), min(0.3, 0.0)]  0.0
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Extension by fuzzy relation
B '  {(b1,0.3),(b2 ,0.8),(b3 ,0.0)}
MR 2
C ' (c1 )  max[min(0.3,0.7), min(0.8,0.2), min(0.0, 0.0)]  0.3
C ' (c2 )  max[min(0.3,0.4), min(0.8,0.0), min(0.0, 0.3)]  0.3
C ' (c3 )  max[min(0.3,1.0), min(0.8,0.3), min(0.0, 0.9)]  0.8
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Extension by fuzzy relation
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Fuzzy distance between fuzzy sets
nonnegative
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Fuzzy distance between fuzzy sets
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Fuzzy distance between fuzzy sets
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Fuzzy distance between fuzzy sets
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Fuzzy distance between fuzzy sets
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Thanks for your attention!
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