part 5 fuzzy relations
DESCRIPTION
FUZZY SETS AND FUZZY LOGIC Theory and Applications. PART 5 Fuzzy Relations. 1. Crisp and fuzzy relations 2. Projections/Cylindric Ext. 3. Binary fuzzy relations 4. Relations on a single set 5. Equivalence relations. FUZZY SETS AND FUZZY LOGIC Theory and Applications. - PowerPoint PPT PresentationTRANSCRIPT
PART 5Fuzzy Relations
1. Crisp and fuzzy relations2. Projections/Cylindric Ext.3. Binary fuzzy relations4. Relations on a single set5. Equivalence relations
FUZZY SETS AND
FUZZY LOGICTheory and Applications
6. Compatibility relations7. Ordering relations8. Fuzzy morphisms9. Sup-i compositions10. Inf-ωi compositions
FUZZY SETS AND
FUZZY LOGICTheory and Applications
3
Crisp/fuzzy relations
• Crisp Relation
A crisp relation represents the presence or absence of association, interaction or interconnectedness between the elements of two or more sets.
A relation among crisp sets X1, X2, ..., Xn is a subset of the Cartesian product It is denoted by R(X1, X2, ..., Xn).
.ii
Xn
N
form) ed(abbreviat )|(
) , , ,( 2121
ii
ni
nn
XiXR
XXXXXXR
nNN
4
Crisp/fuzzy relations
Using a characteristic function defines the crisp relation R :
A relation can be written as a set of ordered tuples. Another convenient way of representing a relation R(X1, X2, ..., Xn) involves an n-dimensional membership array:
otherwise
,, , , if
0
1), , ,( 21
21
RxxxxxxR n
n
otherwise
,, , , iff
0
1 21 , , , 21
Rxxxr n
iii n
].[ , , , 21 niiir R
5
Crisp/fuzzy relations
• Fuzzy relation
A fuzzy relation is a fuzzy set defined on the Cartesian product of crisp sets X1, X2, ..., Xn 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.
6
Projections/Cylindric Ext.
• Projection
Given a relation R(X1, X2, ..., Xn), let [R↓Y] denote the projection of R on Y that disregards all sets in X except those in the family
Then, [R↓Y] is a fuzzy relation whose membership function is defined on the Cartesian product of sets in Y by the equation
). ,|| ,( nrrJ|iXJj|X ninj NN XY
).(max)]([ xRyRyx
Y
7
Projections/Cylindric Ext.
8
Projections/Cylindric Ext.
• Cylindric Extension
This operation on relations in some sense is an inverse to the projection.
Let R be a relation defined on the Cartesian product of sets in Y, and let [ R↑ X - Y ] denote the cylindric extension of R into
that are in X but are not in Y. Then,
[ R↑ X - Y ](x) = R(y)
for each x such that x y.
)( ni iX N
9
Projections/Cylindric Ext.
• Cylindric closure
The resulting relation of a relation which be exactly reconstructed from several of its projections by taking the set intersection of their cylindric extensions.
When projections are determined by the max operator, the min operator is normally used for the set intersection.
10
Projections/Cylindric Ext.
Given a set of projections of a relation on X, the cylindric closure, cyl{ Pi }, base on these projections is defined by the equation
for each where Yi denotes the family of sets on Pi is defined.
Xx
}|{ IiPi
)]([min)}({cyl xPxP iiIi
i YX
11
Projections/Cylindric Ext.
12
Binary fuzzy relations
• Domain : • Range :• Height :• Membership matrices :
) ,(max)( dom yxRxRYy
) ,(max)(ran yxRyR
Xx
) ,(maxmax)( yxRyRhXxYy
). ,( where],[R yxRrr xyxy
13
Binary fuzzy relations
• Sagittal diagram
14
Binary fuzzy relations
• Inverse relation:
The inverse of R(X, Y) is denoted R-1(Y, X).
by a membership matrix • Max-min composition
• Relational join
) ,() ,(1 yxRxyR
.)( ],[ 1111 RRR --yxr
)] ,( ), ,(min[max) ,]([) ,( zyQyxPzxQPzxRYy
)] ,( ), ,(min[) , ,]([) , ,( zyQyxPzyxQPzyxR
15
Binary fuzzy relations
16
Relations on a single set
• For crisp relations• R(X, X) is reflexive iff <x, x> R for each x X.
Otherwise, R(X, X) is called irreflexive.
If <x, x> R, R(X, X) is called antireflexive.• R(X, X) is symmetric iff <x, y>, <y, x> R for each
x, y X. Otherwise, R(X, X) is called asymmetric.
If <x, y> and <y, x> R implies x=y, then R(X, X) is called antisymmetric.
(strictly antisymmetric: If <x, y> or <y, x> R
implies )
yx
17
Relations on a single set
• R(X, X) is transitive iff <x, z> R whenever both <x, y>, <y, z> R for at least y X. Otherwise, R(X, X) is called nontransitive.
If <x, z> R whenever both <x, y>, <y, z> R, R(X, X) is called antitransitive.
18
Relations on a single set
• For fuzzy relations
19
Relations on a single set
• For fuzzy relations
20
Relations on a single set
For fuzzy relation R(X, X) • Reflexive:
irreflexive: if not for some x X.
antireflexive:
ε-reflexive:• Symmetric:
asymmetric: if not for some x, y X.
antisymmetric:
. allfor ,1) ,( XxxxR
. allfor ,0) ,( XxxxR
.10 where,) ,( xxR. , allfor ), ,() ,( XyxxyRyxR
. allfor that imples
0) ,( and 0) ,(when
Xx, yyx
xyRyxR
21
Relations on a single set
• Transitive
(or, more specifically max-min transitive):
nontransitive: if not for some members of X.
antitransitive:
.pair each for
)] ,( ), ,(min[max) ,(
2Xx, z
zyRyxRzxRYy
. allfor
)] ,( ), ,(min[max) ,(
2Xx, z
zyRyxRzxRYy
22
Relations on a single set
• Transitive closure - RT(X, X)
– For crisp relations: RT(X, X) is defined as the relation that is transitive, contains R(X, X), and has the fewest possible members.
– For fuzzy relations: the elements of the transitive closure have the smallest possible membership grades that still allow the first two requirements to be met.
23
Relations on a single set
• Algorithm of finding transitive closure
.' :Stop .3
.1 step togo and ' make ,' If .2
).(' .1
TRR
RRRR
RRRR
24
Relations on a single set
Example 5.8• Determine max-min closure RT(X, X) for a fuzzy
relation R (X, X)
08.00
004.0
1000
005.7.
R
25
Relations on a single set
Stop. .'
4.8.4.0
4.4.4.0
18.4.0
5.5.5.7.
1. step back to go '' Since
'.
4.8.4.0
4.4.4.0
18.4.0
5.5.5.7.
4.4.4.0
4.4.4.0
4.8.4.0
5.5.5.7.
1. step back to go '' Since
'.
08.4.0
4.04.0
18.00
5.05.7.
004.0
4.000
08.00
5.05.7.
1. Step
TRR
RR ,R R
RR)(RRRR
RR ,R R
RR)(RRRR
26
Equivalence relations
• Equivalence relation: A crisp binary relation R(X, X) that is reflexive,
symmetric, and transitive.
• Equivalence class: Ax is a crisp subset of X, where R(X, X) is a
equivalence relation. Ax is referred to as a equivalence class of R(X, X) with respect to x.
27
Equivalence relations
• Similarity relation:
A fuzzy binary relation R(X, X) that is reflexive, symmetric, and transitive.
• Similarity class :
A fuzzy set in which the membership grade of any particular element represents the similarity of that element to the element x.
28
Equivalence relations
Example 5.10• Let X = {1, 2 , . . . , 10}. The Cartesian product X
X ×Y contains 100 members: 〈 1, 1 〉 , 〈 1, 2 〉 , 〈 1, 3 〉 ,… , 〈 10, 10 〉 . Let R(X, X) = { 〈 x, y 〉 |x and y have the same remainder when divided by 3 }. The relation is easily shown to be reflexive, symmetric, and transitive and is therefore an equivalence relation on X. The three equivalence classes denned by this relation are:
29
Equivalence relations
A1 = A4 = A7 = A10 = {1, 4, 7, 10 },
A2 = A5 = A8 = {2, 5, 8 },
A3 = A6 = A9 = {3, 6, 9}.
Hence, in this example, X / R = { {1, 4, 7,10 }, {2, 5, 8 }, {3, 6, 9 } }.
30
Equivalence relations
• Example 5.10• The fuzzy relation R(X, X) represented by the
membership matrix is a similarity relation on
X =(a, b, c, d, e, f, g).
31
Equivalence relations
• To verify that R is reflexive and symmetric is trivial. To verify its transitivity, we may employ the algorithm for calculating transitive closures introduced in Sec. 5.4.
• If the algorithm is applied to R and terminates after the first iteration, then, clearly, R is transitive. The level set of R is ΛR = {0, .4, .5, .8, .9,1}. Therefore, R is associated with a sequence of five nested partitions π (αR), for α ΛR and α > 0. Their refinement relationship can be conveniently diagrammed by a partition tree, as shown in Fig. 5.7.
32
Equivalence relations
33
Compatibility relations
R(X, X) is a reflexive and symmetric fuzzy relation, it is sometimes called a proximity relation.
• Compatibility class : Given a crisp compatibility relation R(X, X), a compatibility class is a subset A of X such that 〈 x, y 〉 R for all x, y A.
• Maximal compatibility class : a compatibility class that is not properly contained within any other compatibility class.
34
Compatibility relations
• Complete cover : The family consisting of all the maximal compatibles induced by R on X is called a complete cover of X with respect to R.
• α-compatibility class : A subset A of X such that R(x, y) ≧α for all x, y A.
35
Compatibility relations
Example 5.11• Consider a fuzzy relation R(X, X) defined on X =
N9 by the following membership matrix:
36
Compatibility relations
• Since the matrix is symmetric and all entries on the main diagonal are equal to 1, the relation represented is reflexive and symmetric; therefore, it is a compatibility relation.
• The graph of the relation is shown in Fig. 5.8; its complete α-covers for α > 0 and α ΛR =
{0, .4, .5, .7, .8,1} are depicted in Fig. 5.9.
37
Compatibility relations
38
Compatibility relations
39
Ordering relations
• Partial ordering : A crisp binary relation R(X, X) that is reflexive, antisymmetric, and transitive is called a partial ordering.
• Minimum, maximum, minimal, maximal
40
Ordering relations
• Lower bound, greatest lower bound: Let X be a set on which a partial ordering is
defined, and let A be a subset of X(A X). If x X and x y for every y A, then x is called a lower bound of A on X with respect to the partial ordering.
If a particular lower bound succeeds every other lower bound of A, then it is called the greatest lower bound, or infimum, of A.
41
Ordering relations
• Upper bound, least upper bound: If x X and y x for every y A, then x is called
an upper bound of A on X with respect to the relation.
If a particular upper bound precedes every other upper bound of A, then it is called the least upper bound, or supremum, of A.
• Lattice: A partial ordering on a set X that contains a
greatest lower bound and a least upper bound for every subset of two elements of X is called a lattice.
42
Ordering relations
• Hasse diagram: Each element of X is expressed by a single node
that is connected only to the nodes representing its immediate predecessors and immediate successors. The connections are directed in order to distinguish predecessors from successors; the arrow ← indicates the inequality
. Diagrams of this sort are called ≦ Hasse diagrams.
43
Ordering relations
Example 5.12• Three crisp partial orderings P, Q, and R on the
set X = {a, b, c, d, e} are defined by their membership matrices (crisp) and their Hasse diagrams in Fig. 5.10. The underlined entries in each matrix indicate the relationship of the immediate predecessor and successor employed in the corresponding Hasse diagram. P has no special properties, Q is a lattice, and R is an example of a lattice that represents a linear ordering.
44
Ordering relations
45
Ordering relations
• Fuzzy partial ordering : A fuzzy binary relation R on a set X is a fuzzy
partial ordering iff it is reflexive, antisymmetric, and transitive under some form of fuzzy transitivity.
• Dominating class:
• Dominated class:
. where) ,()(][ XyyxRyR x
. where) ,()(][ XyxyRyR x
46
Ordering relations
• Undominated:
• Undominating:. and allfor
.0) ,(
iff dundominate is element An
yxXy
yxR
Xx
. and allfor
.0) ,(
iff ngundominati is element An
xyXy
xyR
Xx
47
Ordering relations
• Fuzzy upper bound :
on.intersectifuzzy eappropriatan denotes where
,) (
by
defined and ) (by denotedset fuzzy theis for
thedefined, is ordering partial
fuzzy aon which set a of subset crisp aFor
][
xAx
RAR,U
AR,UA
r boundfuzzy uppeR
XA
48
Ordering relations
• Least upper bound :
).,( ofsupport in the elements allfor
0) ,( and 0))( ,(
such that ),(in element unique
theisit exists, set theof a If
ARUy
yxRxARU
ARUx
Ar boundleast uppe
49
Ordering relations
Example 5.13• The following membership matrix defines a fuzzy partial
ordering R on the set X = {a, b, c, d, e}:
• The columns of the matrix give the dominated class for each element. Under this ordering, the element d is undominated, and the element c is undominadng.
50
Ordering relations
• For the subset A ={a, b}, the upper bound is the fuzzy set produced by the intersection of the dominating classes for a and b. Employing the min operator for fuzzy intersection, we obtain
• The unique least upper bound for the set A is the element b. All distinct crisp orderings captured by the given fuzzy partial ordering R are shown in Fig. 5.11. We can see that the orderings became weaker with the increasing value of α.
51
Ordering relations
52
Fuzzy morphisms
• Homomorphism :
If two crisp binary relations R(X, X) and Q(Y, Y) are defined on sets X and Y, respectively, then a function h : X → Y is said to be a homomorphism from 〈 X, R 〉 to 〈 Y, Q 〉 if
for all x1, x2 X.
,)( ),( implies , 2121 QxhxhRxx
53
Fuzzy morphisms
• Strong homomorphism:
. )( and )( where, , allfor
, implies ,
and , , allfor
)( ),( implies ,
21
211
121
2121
21
2121
yhxyhxYyy
RxxQyy
Xxx
QxhxhRxx
54
Fuzzy morphisms
55
Fuzzy morphisms
56
Fuzzy morphisms
• Isomorphism :
If h : X → Y is a homomorphism from 〈 X, R 〉 to 〈 Y, Q 〉 , and if h is completely specified, one-to-one, and onto, then it is called an isomorphism.
57
Sup-i compositions
• Sup-i composition :
Given a particular t-norm i and two fuzzy relations P(X, Y) and Q(Y, Z), the sup- i composition of P and Q is a fuzzy relation on X x Z denned by
for all
)] ,( ), ,([sup) ,]([ zyQyxPizxQPYy
i
. , ZzXx
58
Sup-i compositions
• Basic properties
59
Sup-i compositions
60
Sup-i compositions
• i-transitive: We say that relation R on X2 is i-transitive iff
for all x, y, z X.
• i-transitive closure: When a relation R is not i-transitive, we define its
i-transitive closure as a relation RT(i) that is the smallest i-transitive relation containing R.
)] ,( ), ,([) ,( zyRyxRizxR
61
Sup-i compositions
• Theorem 5.1 For any fuzzy relation R on X2, the fuzzy relation
is the i-transitive closure of R.
• Theorem 5.2 Let R be a reflexive fuzzy relation on X2, where |
X| = n 2. Then,≧ RT(i)=R(n-1).
1
)()(
n
niT RR
62
Inf-ωi compositions
• Operation ωi :
Given a continuous t-norm i, let
for every a, b [0, 1].
}) ,(|]1 ,0[sup{) ,( bxaixbai
63
Inf-ωi compositions
• Inf- ωi composition
. , allfor
])( ),([inf) ,)( (
equation by the defined
is )( and )(relation fuzzy of , n,compositio
-inf the,operation associated theand norm- aGiven
ZzXx
y, zQx, yPzxQP
Y, ZQX, YPQP
it
iYy
ii
i
i
64
Inf-ωi compositions
• Theorem 5.3 For any a, aj, b, d [0, 1], where j takes values
from an Index set J, operation ωi has the following properties:
65
Inf-ωi compositions
66
Inf-ωi compositions
• Theorem 5.4
Let P(X, Y), Q(Y, Z), R(X, Z), and S(Z, V) be fuzzy relations. Then:
67
Inf-ωi compositions
• Theorem 5.5
Let P(X, Y), Pj(X, Y), Q(Y, Z), and Qj(Y, Z) be fuzzy relations, where j takes values in an index set J. Then,
68
Inf-ωi compositions
• Theorem 5.6 Let P(X, Y), Q1(Y, Z), Q2(Y, Z), and R(Z, V) be
fuzzy relations. If Q1 Q2, then
69
Inf-ωi compositions
• Theorem 5.7 Let P(X, Y), Q(Y, Z), and R(X, Z) be fuzzy
relations. Then,
70
Exercise 5
• 5.1
• 5.4
• 5.6
• 5.11
• 5.19
• 5.20