1 Fuzzy Relations & Operations on Fuzzy Relations •Fuzzy Relation Consider two universes: A crisp set consisting of a subset of ordered points is a crisp relation in the Cartesian product 1 1 2 2 { } and { } X x X x = = 1 2 ( , ) x x 1 2 X X × 1 X 2 X 1 2 ( , ) 0 f x x =
46
Embed
Fuzzy Relations & Operations on Fuzzy Relationspami.uwaterloo.ca/~sd625/Files/fuzzylecture_3_4.pdf · 1 Fuzzy Relations & Operations on Fuzzy Relations •Fuzzy Relation Consider
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
1
Fuzzy Relations & Operations on Fuzzy Relations
•Fuzzy RelationConsider two universes:A crisp set consisting of a subset of ordered points
is a crisp relation in the Cartesian product
1 1 2 2{ } and { }X x X x= =
1 2( , )x x
1 2X X×1X
2X
1 2( , ) 0f x x =
2
The notation of relation in crisp sets is also extendable to fuzzy sets. Let us consider the sets of numbers in x and y that are simultaneously close to 0. This relation could be expressed using the Gaussian membership function:
2 2( )( , ) /( , ) /( , )x yRX Y X Y
R x y x y e x yµ − +
× ×= =∫ ∫
3
Let R be a binary fuzzy relation defined on .Similar to fuzzy sets, R can be defined as
Alternatively, in a discrete Cartesian product, R can be expressed as
In a Continuous one
X Y×
( ){ }( , ), ( , )RR x y x yµ=
( , )
( , ) ( , )i j
R i j i jx y X Y
R x y x yµ∈ ×
= ∑
for ( , ) ( )x y X Y∈ ×
( , ) ( , )R i j i jX YR x y x yµ
×= ∫
4
The membership matrix of an mxn binary fuzzy relation has the general form
1 1 1 2 1
2 1 2 2 2
1 2
( , ) ( , ) ( , )( , ) ( , ) ( , )
( , ) ( , ) ( , )
R R R n
R R R n
R m R m R m n
x y x y x yx y x y x y
R
x y x y x y
µ µ µµ µ µ
µ µ µ
=
LL
M M M ML
5
Example: (Representing a Fuzzy Relation)Let and be two discrete sets. The fuzzy relation R = “x is similar toy” may be represented in five different ways:
1. Linguistically, such as by the statement “x is similar to y”
2. By listing (or taking the union of) all fuzzy singletons
3. As a directed graph4. In a tabular form5. As a membership matrix
1 2 3 4{ , , , }X x x x x= 1 2 3 4{ , , , }Y y y y y=
6
Basic Operations with Fuzzy Relations:
Suppose that we have two fuzzy relations R1 and R2. Their union is a new relation
Where the membership function of is
1 21 2 ( , ) ( , ) ( , )R RX YR R x y x y x yµ µ
× ∪ = ∨ ∫
1 2R R∪
1 2 1 2( , ) ( , ) ( , )R R R Rx y x y x yµ µ µ∪ = ∨
7
The intersection of the two fuzzy relations R1 and R2is a new relation
Where the membership function of is
1 21 2 ( , ) ( , ) ( , )R RX YR R x y x y x yµ µ
× ∩ = ∧ ∫
1 2R R∪
1 2 1 2( , ) ( , ) ( , )R R R Rx y x y x yµ µ µ∩ = ∧
8
Example: (Union and Intersection of Fuzzy Relations)Consider the following two fuzzy relations
1
0.0 0.0 0.1 0.8
0.0 0.8 0.0 0.00.1 0.8 1.0 0.8
R =
R1 = “x is larger than y”
2
0.4 0.4 0.2 0.1
0.5 0.0 1.0 1.00.5 0.1 0.2 0.6
R =
R2 = “y is much bigger than x”
9
1 2
0.4 0.4 0.2 0.8
0.5 0.8 1.0 1.00.5 0.8 1.0 0.8
R R ∪ =
The union of the two relations is formed by taking the maximum (t-norm) of the two grades of membership for the corresponding elements of the two matrices.The new relation is defined by
10
For the intersection, we take the minimum (s-norm) of the two grades of membership for the corresponding elements of the two matrices.The new relation is defined by
1 2
0.0 0.0 0.1 0.1
0.0 0.0 0.0 0.00.1 0.1 0.2 0.6
R R ∩ =
11
•Projection
The first projection is a fuzzy set that results by eliminating the second set Y of X×Y by projecting the relation on X.
( , )( , )R
X Y
x yR
x yµ
×
= ∫
1
11
maximize over all
( )( ) [ ( , )]R
R RyX y
xR x x y
x
µµ µ= = ∨∫
12
2
22
( )( ) [ ( , )]R
R RxY
yR y x y
y
µµ µ= = ∨∫
The second projection is a fuzzy set that results by eliminating the first set X of X×Y by projecting the relation on Y.
13
14
15
•Total ProjectionThis is a combined projection over the space X and Y. It is represented by the following:
is strongly related to and weakly related tois strongly related to and weakly related tois strongly related to and weakly related tois strongly related to and weakly related tois strongly related to and weakly related to
1x 1y 2 4y y…2x 2y 1 4y y…3x 3y 1 4y y…4x
4y 1 4y y…5x 5y 1 4y y…
34
[ ][ ] [ ]
1
2
3
4
1 2 3 4 11 0.3 0.9 0 1
00.3 1 0.3 0 0
0.050.9 0.8 1 0.8 0
0.10 1 0.8 1 0
0
1 0 0 0 0 5 4 1 0.1 0.05 0
y y y y Fxx
Rxx
S
= =
× =
35
•ExampleThree relations involved in max-min composition
When two of the components in the above equation are given and the other is unknown, we have a set of equations known as fuzzy equations:
.P Q R=o
[ ]
[ ]1 2 3
1 2 3
1 2 3
1 2 3
0.9 0.6 10.8 0.3 0.5 0.6 0.6 0.50.6 0.4 0.6
( 0.9) ( 0.8) ( 0.6) 0.6
( 0.6) ( 0.3) ( 0.4) 0.6( 1) ( 0.5) ( 0.6) 0.5
P
P p p p
p p p
p p pp p p
=
=
∧ ∨ ∧ ∨ ∧ =
∧ ∨ ∧ ∨ ∧ =∧ ∨ ∧ ∨ ∧ =
o
36
Generalized DeMorgan’s Law
T-norm and T-conorms are dual operators. They satisfy DeMorgan’s laws.
( )( )( )( )
( , ) ( ), ( ) ( , )
( , ) ( ), ( )
( , ) ( , )
( , ) ( , )
( , ) ( , )
( , ) ( , )
m m
a a
b b
d d
T a b N S N a N b S a b
S a b N T N a N b
T a b S a b
T a b S a b
T a b S a b
T a b S a b
= =
=
↔↔
↔↔
37
Fuzzy Reasoning
Definition: A linguistic variable is characterized by a quintuplet
is the name of the variableis the term set of ≡ set of linguistic values
is the universe of discourseis a syntactic rule which generates the terms in is a semantic rule, it associates with each fuzzy
set A, its meaning
( ), ( ), , ,x T x X G Mx
x( )T xXG ( )T x
M( )M A
38
Example
If age is interpreted as a linguistic variable; then its term set: •T(age)={young, not young, middle age, old, not old, very old}•T(x) is a term set over X=[0,100]•The syntactic refers to the way the linguistic values are represented in T(x)•The semantic rules define the membership function of each linguistic value of the term set T(x)
39
40
Fuzzy If-Then Rules
A fuzzy if-then rule (fuzzy rule ≡ fuzzy implication) assumes the form of: if x is A then y is B.•A and B represent fuzzy sets•x and y are two fuzzy linguistic variables•“x is A” is called antecedent•“y is B” is called consequent
41
Example:
•If pressure is high, then the volume is small.•If tomato is red, then it is ripe.
A fuzzy if-then rule is a binary relation R .
IF A an B are two fuzzy sets over X and Y respectively, thentwo ways for representing the implication
§ A is coupled with B§ A entails B
:R X Y→
42
A coupled with B
AA
B B
A entails B
Two ways to interpret “If x is A then y is B”:
y
xx
y
43
A is coupled with B
( ) ( ):
( , )A B
X Y
x yR X Y A B
x yµ µ
×
∗→ = × = ∫
44
A entails B
1 (1 ( ) ( )): :
( , )A B
aX Y
x yR A B
x yµ µ
×
∧ − +∪ ∫
“Zadeh” arithmetic operator:(1 ( )) ( ( ) ( ))
: ( ):( , )
A A Bmn
X Y
x x yR A A B
x yµ µ µ
×
− ∨ ∧∪ ∩ ∫
45
Presume that
f is called the fuzzy implication. It transforms elements of A to and elements of B in
For A entails B we have the two following Oprts:•Material implication•Extended proposition calculus
For A Coupled with B
Minimum operator:
Mamdani operator:
:R X Y A B→ = ∪: ( )R X Y A B B→ = ∩ ∪
( )( , ) ( ), ( ) ( , )R A Bx y f x y f a bµ µ µ= =A B→