Ch02 fuzzyrelation

Fuzzy Relations
• A fuzzy relation R is a 2 D MF:
– R :{ ((x, y), µR(x, y)) | (x, y) ∈ X × Y}
Examples:
• x is close to y (x & y are real numbers)
• x depends on y (x & y are events)
• x and y look alike (x & y are persons or objects)
• Let X = Y = IR+
and R(x,y) = “y is much greater than x”
The MF of this fuzzy relation can be subjectively defined as:
if X = {3,4,5} & Y = {3,4,5,6,7}





≤
>
++
−
=µ
xyif,0
xyif,
2yx
xy
)y,x(R

Extension Principle
The image of a fuzzy set A on X
nnA22A11A
x/)x(x/)x(x/)x(A µ++µ+µ= 
under f(.) is a fuzzy set B:
nnB22B11B
y/)x(y/)x(y/)x(B µ++µ+µ= 
where yi = f(xi), i = 1 to n
If f(.) is a many-to-one mapping, then
)x(max)y( A
)y(fx
B 1
µ=µ −
=

Defintion: Extension Principle
• Suppose that function f is a mapping from an n-
dimensional Cartesian product space X1 × X2 × … × Xn
to a 1-dimensional universe Y s.t. y=f(x1, …, xn), and
suppose A1, …, An are n fuzzy sets in X1, …, Xn,
respectively.
• Then, the extension principle asserts that the fuzzy set B
induced by the mapping f is defined by

Example
– Application of the extension principle to fuzzy
sets with discrete universes
Let A = 0.1 / -2+0.4 / -1+0.8 / 0+0.9 / 1+0.3 / 2
and f(x) = x2 – 3
• Applying the extension principle, we obtain:
B = 0.1 / 1+0.4 / -2+0.8 / -3+0.9 / -2+0.3 /1
= 0.8 / -3+(0.4V0.9) / -2+(0.1V0.3) / 1
= 0.8 / -3+0.9 / -2+0.3 / 1
where “V” represents the “max” operator, Same
reasoning for continuous universes

Max-Min Composition
The max-min composition of two fuzzy relations
R1 (defined on X and Y) and R2 (defined on Y and
Z) is
Properties:
– Associativity:
– Distributivity over union:
– Week distributivity over intersection:
– Monotonicity:
µ µ µR R
y
R Rx z x y y z1 2 1 2 ( , ) [ ( , ) ( , )]= ∨ ∧
R S T R S R T    ( ) ( ) ( )=
R S T R S T   ( ) ( )=
R S T R S R T    ( ) ( ) ( )⊆
S T R S R T⊆ ⇒ ⊆( ) ( ) 

Max-Star Composition
• Max-product composition:
• In general, we have max-* composition:
where * is a T-norm operator.
µ µ µR R
y
R Rx z x y y z1 2 1 2 ( , ) [ ( , ) ( , )]= ∨
µ µ µR R
y
R Rx z x y y z1 2 1 2 ( , ) [ ( , )* ( , )]= ∨

Example of max-min & max-
product composition
Let R1 = “x is relevant to y”
R2 = “y is relevant to z”
be two fuzzy relations defined on X×Y and
Y×Z respectively :X = {1,2,3}, Y = {α,β,χ,δ}
and Z = {a,b}. Assume that:












=










=
0.20.7
0.60.5
0.30.2
0.19.0
R
0.20.30.80.6
0.90.80.20.4
0.70.50.31.0
R 21

R1oR2 may be interpreted as the derived fuzzy
relation “x is relevant to z” based on R1 & R2
Let’s assume that we want to compute the
degree of relevance between 2 ∈ X & a ∈ Z
Using max-min, we obtain:
{ }
{ }
0.7
5,0.70.4,0.2,0.max
7.09.0,5.08.0,2.02.0,9.04.0max)a,2(2R1R
=
=
∧∧∧∧=µ 
{ }
{ }
0.63
0.40,0.630.36,0.04,max
7.0*9.0,5.0*8.0,2.0*2.0,9.0*4.0max)a,2(2R1R
=
=
=µ 
Using max-product composition, we obtain:

Linguistic variables
• The concept of linguistic variables introduced
by Zadeh is an alternative approach to
modeling human thinking.
• Information is expressed in terms of
fuzzy sets instead of crisp numbers

Example
• numerical values: Age = 65
*A linguistic variables takes linguistic values Age is old
*A linguistic value is a fuzzy set
*All linguistic values form a term set
T(age) = {young, not young, very young, ..., middle
aged, not middle aged, ..., old, not old, very old, more or
less old, ...,not very young and A numerical variable
takes not very old, ...}
Where each term T(age) is characterized by fuzzy
set of a universe of discourse X= = [0,100]

– Example:
• A numerical variable takes numerical values
Age = 65
A linguistic variables takes linguistic values
Age is old
A linguistic value is a fuzzy set
• All linguistic values form a term set
• T (age) = {young, not young, very young, ...,middle
aged, not middle aged, ..., old, not old, very old, more
or less old, …, not very young and not very old, ...}

• Where each term T(age) is
characterized by a fuzzy set of a
universe of discourse X= [0,100]

Operations on linguistic variables
– Let A be a linguistic value described by a fuzzy set
with membership function µA(.)
– is a modified version of the original linguistic value.
– A2 = CON(A) is called the concentration operation
√A = DIL(A) is called the dilation operation
– CON(A) & DIL(A) are useful in expression the hedges such as
“very” & “more or less” in the linguistic term A
– Other definitions for linguistic hedges are also possible
∫ µ=
X
k
A
k
x/)]x([A

– Composite linguistic terms
– Let’s define:
– where A, B are two linguistic values whose
semantics are respectively defined by
µA(.) & µB(.)
– Composite linguistic terms such as: “not very
young”, “not very old” & “young but not too young”
can be easily characterized
∫
∫
∫
µ∨µ=∪=
µ∧µ=∩=
µ−=¬=
X
BA
X
BA
X
A
x/)]x()x([BABorA
x/)]x()x([BABandA
,x/)]x(1[A)A(NOT

– Example: Construction of MFs for composite
linguistic terms. Let’s define
– Where x is the age of a person in the universe of
discourse [0,100]
• More or less = DIL(old) = √old =
6old
4young
30
100x
1
1
)100,3,30,x(bell)x(
20
x
1
1
)0,2,20,x(bell)x(





 −
+
==µ






+
==µ
x/
30
100x
1
1
X
6∫





 −
+
k2x
1
1
),k,,x(bell)x((...)






σ
µ−
+
=µσ=µ

• Not young and not old = ¬young ∩ ¬old =
• Young but not too young = young ∩ ¬young2 (too =
very) =
• Extremely old ≡ very very very old = CON
(CON(CON(old))) =
x/
30
100x
1
1
1
20
x
1
1
1 6
X
4

















 −
+
−∧


















+
−∫
∫


































+
−∧


















+x
2
44
x/
20
x
1
1
1
20
x
1
1
∫

















 −
+x
8
6
x/
30
100x
1
1

– Contrast intensification
– the operation of contrast intensification on a
linguistic value A is defined by
• INT increases the values of µA(x) which are greater
than 0.5 & decreases those which are less or equal
that 0.5
• Contrast intensification has effect of reducing the
fuzziness of the linguistic value A




≤µ≤¬¬
≤µ≤
=
1)x(0.5if)A(2
5.0)x(0ifA2
)A(INT
A
2
A
2

– Orthogonality
• A term set T = t1,…, tn of a linguistic variable x on the
universe X is orthogonal if:
• Where the ti’s are convex & normal fuzzy sets
defined on X.
∑
=
∈∀=µ
n
1i
it Xx,1)x(

Fuzzy if-then rules
• General format:
– If x is A then y is B
(where A & B are linguistic values defined by fuzzy sets
on universes of discourse X & Y).
• “x is A” is called the antecedent or premise
• “y is B” is called the consequence or conclusion
– Examples:
• If pressure is high, then volume is small.
• If the road is slippery, then driving is dangerous.
• If a tomato is red, then it is ripe.
• If the speed is high, then apply the brake a little.

Fuzzy If-Then Rules
A coupled with B A entails B
Two ways to interpret “If x is A then y is B”:

Fuzzy if-then rules
• Note that R can be viewed as a fuzzy set with a
two-dimensional MF
µR(x, y) = f(µA(x), µB(y)) = f(a, b)
• With a = µA(x), b = µB(y) and f called the fuzzy
implication function provides the membership
value of (x, y)

Fuzzy if-then rules
– Case of “A coupled with B” (A and B)
(minimum operator proposed by Mamdani, 1975)
(product proposed by Larsen, 1980)
(bounded product operator)
∫ ∧==
YX
BAm yxyxBAR
*
),/()()(* µµ
)y,x/()y()x(B*AR
Y*X
BAp ∫ µµ==
)y,x/()1)y()x((0
)y,x/()y()x(B*AR
BA
Y*X
Y*X
BAbp
−µ+µ∨=
µ⊗µ==
∫
∫

Fuzzy if-then rules
• Fuzzy implication function:
A coupled with B
µA(x)=bell(x;4,3,10) and µB(y)=bell(y;4,3,10)
µ µ µR A Bx y f x y f a b( , ) ( ( ), ( )) ( , )= =

Fuzzy if-then rules
– Case of “A entails B” (not A or B)
(Zadeh’s arithmetic rule by using bounded sum
operator for union)
(Zadeh’s max-min rule)
)ba1(1)b,a(f:where
)y,x/()y()x(1(1BAR
a
Y*X
BAa
+−∧=
µ+µ−∧=∪¬= ∫
)ba()a1()b,a(f:where
)y,x/())y()x(())x(1()BA(AR
m
Y*X
BAAmm
∧∨−=
µ∧µ∨µ−=∩∪¬= ∫

Fuzzy if-then rules
(Boolean fuzzy implication with max for union)
(Goguen’s fuzzy implication with algebraic product for T-norm)
b)a1()b,a(f:where
)y,x/()y())x(1(BAR
s
Y*X
BAs
∨−=
µ∨µ−=∪¬= ∫


 ≤
=<
µ<µ= ∫∆
otherwisea/b
baif1
b~a:where
)y,x/())y(~)x((R
Y*X
BA

Fuzzy if-then rules
A entails B

Fuzzy Reasoning
– Known also as approximate reasoning
– It is an inference procedure that derives
conclusions from a set of fuzzy if-then-rules &
known facts
– The compositional rule of inference plays a key
role in F.R.
– Using the compositional rule of inference, an
inference procedure upon a set of fuzzy if-
then rules are formalized.

Compositional Rule of Inference
Derivation of y = b from x = a and y = f(x):
a and b: points
y = f(x) : a curve
x
y
a
b
y = f(x)
a
b
y
x
a and b: intervals
y = f(x) : an interval-valued
function
y = f(x)

– The extension principle is a special case of the
compositional rule of inference
• F is a fuzzy relation on X*Y, A is a fuzzy set
of X & the goal is to determine the resulting
fuzzy set B
– Construct a cylindrical extension c(A) with
base A
– Determine c(A) ∧ F (using minimum operator)
– Project c(A) ∧ F onto the y-axis which
provides B

• a is a fuzzy set and y = f(x) is a fuzzy relation:
B?
(a) F: X × Y
(b) c(A)
(c) c(A)∩F
(d) Y as a
fuzzy
set B
on the
y-axis.

(a) F: X × Y µF(x,y)
(b) c(A) µc(A)(x,y) = µA(x)
(c) c(A)∩F µc(A)∩F(x,y) = min[µc(A)(x,y), µF(x,y)]
= min[µA(x), µF(x,y)]
= ∧ [µA(x), µF(x,y)]
(d) Y as a fuzzy set B on the y-axis.
µB(y) = maxx min[µA(x), µF(x,y)]
= ∨x [µA(x) ∧ µF(x,y)]
B? B = A ° F
f) Extension principle is a special case of the compositional
rule of inference:
µ µB
x f y
Ay x( ) max ( )
( )
=
= −1

Fuzzy Reasoning
• Given A, A ⇒ B, infer B
– A = “today is sunny”
– A ⇒ B: day = sunny then sky = blue
infer: “sky is blue”
• illustration
– Premise 1 (fact): x is A
– Premise 2 (rule): if x is A then y is B
– Consequence: y is B

Fuzzy Reasoning
• Approximation
A’ = “ today is more or less sunny”
B’ = “ sky is more or less blue”
• illustration
Premise 1 (fact): x is A’
Premise 2 (rule): if x is A then y is B
Consequence: y is B’
(approximate reasoning or fuzzy reasoning!)

Fuzzy Reasoning (Approximate Reasoning)
Approximate Reasoning ⇔ Fuzzy Reasoning
Definition:
Let A, A’, and B be fuzzy sets of X, X, and Y,
respectively. Assume that the fuzzy implication
A→B is expressed as a fuzzy relation R on X×Y.
Then, the fuzzy set B induced by “x is A′ ” and the
fuzzy rule “if x is A then y is B” is defined by
µB′ (y) = maxx min [µA′ (x), µR (x,y)]
= ∨x [µA′ (x) ∧ µR (x,y)]
⇔ B′ = A′ ° R = A′ ° (A→B)
Mamdani’s fuzzy implication functions and max-min
composition for simplicity and their wide applicability.

Fuzzy Reasoning (Approximate Reasoning)
Y
x is A’
Y
A
X
w
A’ B
B’
A’
X
y is B’
• Single rule with single antecedent
– Rule: if x is A then y is B
– Fact: x is A’ µB′ (y) = [∨x (µA′ (x) ∧ µA (x))] ∧ µB (y)
– Conclusion: y is B’ = w ∧ µB (y)
• Graphic Representation:

Fuzzy Reasoning
– Single rule with multiple antecedents
• Premise 1 (fact): x is A’ and y is B’
• Premise 2 (rule): if x is A and y is B then z is C
• Conclusion: z is C’
• Premise 2: A*B  C
[ ] [ ]
[ ]{ }
[ ]{ } [ ]{ }
)z()w(w
)z()y()y()x()x(
)z()y()x()y()x(
)z()y()x()y()x()z(
)CB*A()'B'*A('C
)z,y,x/()z()y()x(C*)B*A()C,B,A(R
C21
C
2w
B'By
w
A'Ax
CBA'B'Ay,x
CBA'B'Ay,x'C
2premise1premise
CB
Z*Y*X
Amamdani
1
µ∧∧=
µ∧µ∧µ∨∧µ∧µ∨=
µ∧µ∧µ∧µ∧µ∨=
µ∧µ∧µ∧µ∧µ∨=µ
→=
µ∧µ∧µ== ∫
    




Fuzzy Reasoning
A B
T-norm
X Y
w
A’ B’ C2
Z
C’
Z
X Y
A’ B’
x is A’ y is B’ z is C’

Fuzzy Reasoning
– Multiple rules with multiple antecedents
Premise 1 (fact): x is A’ and y is B’
Premise 2 (rule 1): if x is A1 and y is B1 then z is C1
Premise 3 (rule 2): If x is A2 and y is B2 then z is C2
Consequence (conclusion): z is C’
R1 = A1 * B1  C1
R2 = A2 * B2  C2
Since the max-min composition operator o is distributive over
the union operator, it follows:
C’ = (A’ * B’) o (R1 ∪ R2) = [(A’ * B’) o R1] ∪ [(A’ * B’) o R2]
= C’1 ∪ C’2
Where C’1 & C’2 are the inferred fuzzy set for rules 1 & 2
respectively

Fuzzy Reasoning
A1 B1
A2 B2
T-norm
X
X
Y
Y
w1
w2
A’
A’ B’
B’ C1
C2
Z
Z
C’
Z
X Y
A’ B’
x is A’ y is B’ z is C’

Ch02 fuzzyrelation

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