Fuzzy Inference and Reasoning

```Fuzzy Inference and Reasoning
Proposition
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Logic variable
3
Basic connectives for logic variables
(1)Negation
(2)Conjunction
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Basic connectives for logic variables
(3) Disjunction
(4)Implication
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Logical function
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Logic Formula
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Tautology
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Tautology
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Predicate logic
11
Fuzzy Propositions
• Assuming that truthand falsity are expressed
by values 1 and 0, respectively, the degree of
truth of each fuzzy proposition is expressed by
a number in the unit interval [0, 1].
Fuzzy Propositions
p : temperature (V) is high (F).
Fuzzy Propositions
p : V is F is S
•
•
•
•
V is a variable that takes values v from some universal set V
F is a fuzzy set onV that represents a fuzzy predicate
S is a fuzzy truth qualifier
In general, the degree of truth, T(p), of any truth-qualified
proposition p is given for each v e V by the equation
T(p) = S(F(v)).
p : Age (V) is very(S) young (F).
Representation of Fuzzy Rule
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Representation of Fuzzy Rule
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Fuzzy rule as a relation
If x is A, then y is B
x is A, y is B fuzzy predicatesA(x), B( y )
If A(x), thenB( y )
can be represented by relation
R( x, y ) : A(x)  B( y )
R( x, y ) can be considereda fuzzy set with 2 - dim membershipfunction
 R ( x, y )  f (  A ( x),  B ( y ))
wheref is " fuzzy implication function", performsthe task
of transforming themembershipdegrees of x in A and y in B
into thoseof ( x, y ) in A  B
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Fuzzy implications
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Example of Fuzzy implications
Let T and H be universeof temperatu
re and humidity,
and define variablest  T and h  H.
A " high", A  T
B " fairly high", B  H
t hen t herule can be rewrit t enas
R(t ,h) : If t is A, thenh is B
R(t ): t is A, R(h) : h is B
R(t ,h) : R(t )  R(h)
R(t ,h)  A  B    A (t )   B (h) /(t , h)
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Example of Fuzzy implications
R(t, h)  A  B    A (t )   B (h) /(t , h)
h
t
20
30
40
20
50
70
90
0.1
0.2
0.2
0.1
0.5
0.6
0.1
0.5
0.7
0.1
0.5
0.9
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Example of Fuzzy implications
When" temperature is fairlyhigh"or t is A' , A'  T
We can use composition of fuzzy relations to find R(h)
R( h)  R(t ' )  R C (t , h)
h
t
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30
40
20
50
70
90
0.1
0.2
0.2
0.1
0.5
0.6
0.1
0.5
0.7
0.1
0.5
0.9
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Representation of Fuzzy Rule
Single input and single output
Fact:
u is A '
: R(u)
Rule: If u is A then w is C : R(u, w)
Result: w is C '
: R( w)  R(u) R(u, w)
Multiple inputs and single output
Fact:
u1 is A1' ' and u2 is A2' ' and ... and un is An' '
Rule: If u1 is A1 and u2 is A2 and ... and un is An then w is C
Result: w is C '
Multiple inputs and Multiple outputs
Fact:
u1 is A1' and u2 is A2' and ... and un is An'
Rule: If u1 is A1 and u2 is A2 and ... and un is An then w1 is C1 , w2 is C2 ,..., wm is Cm
Result: w1 is C1' , w2 is C2' ,..., wm is Cm'
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Representation of Fuzzy Rule
Multiple rules
Fact : u1 is A' 1 and u2 is A' 2 and ... and un is A' n
Rule j : If u1 is A' 1 j and u2 is A' 2 j and ... and un is A' nj , then w1j is C'1j , w 2j is C' 2j , ..., w mj is C' mj
Result : w1 is C'1 , w 2 is C' 2 , ..., w m is C' m
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Compositional rule of inference
The inference procedure is called as the “compositional rule of inference”. The
inference is determined by two factors : “implication operator” and
“composition operator”.
For the implication, the two operators are often used:
For the composition, the two operators are often used:
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Representation of Fuzzy Rule
Fact:
u is A '
: R(u)
Rule: If u is A then w is C : R(u, w)
Result: w is C '
: R( w)  R(u) R(u, w)
Max-min composition operator
R(u, w) : A  C
Mamdani: min operator for the implication
Larsen: product operator for the implication
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One singleton input and one fuzzy
output
Mamdani
Fact:
u is A '
: R(u)
Rule: If u is A then w is C : R(u, w)
Result: w is C '
: R( w)  R(u) R(u, w)
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One singleton input and one fuzzy
output
Mamdani
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One singleton input and one fuzzy
output
Larsen
Fact:
u is A '
: R(u)
Rule: If u is A then w is C : R(u, w)
Result: w is C '
: R( w)  R(u) R(u, w)
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One singleton input and one fuzzy
output
Larsen
31
One fuzzy input and one fuzzy output
Mamdani
Fact:
u is A '
: R(u)
Rule: If u is A then w is C : R(u, w)
Result: w is C '
: R( w)  R(u) R(u, w)
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One fuzzy input and one fuzzy output
Mamdani
33
Ri consists of R1 and R2
Rule i : If u is Ai and v is Bi , then wis Ci
C 'i  (A' , B' )  (Ai and Bi  Ci )
μ C'i  (μ A ' , μ B' )  (μ A i Bi  μ C )
 (μ A ' , μ B' )  (min(μ A i , μ Bi )  μ C )
 (μ A ' , μ B' )  min[(μ A i  μ C ), (μ Bi  μ C )]
 max min{(μ A ' , μ B' ), min[(μ A i  μ C ), (μ Bi  μ C )]}
u ,v
 max min{min[μ A ' , (μ A i  μ C )],min[μ B' , (μ Bi  μ C )]}
u ,v
 min{[μ A '  (μ A i  μ C )],[μ B'  (μ Bi  μ C )]}
C 'i  [A'  (Ai  Ci )]  [B'  (Bi  Ci )]
 [A'  R 1i ]  [A'  R i2 ]
 C1i  Ci2
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Example
R : if x is A and y is B, thenz is C
where A  (0,1,2), B  (1,2,3),C  (5,6,7)are triangular fuzzy sets.
If input x0  1 and y0  1.5 (Singleton) , thenoutput?
35
Two singleton inputs and one fuzzy
output
Mamdani
Fact:
u is A ' and v is B '
: R(u, v)
Rule: If u is A and v is B then w is C : R(u, v, w)
Result: w is C '
: R(w)  R(u, v) R(u, v, w)
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Two singleton inputs and one fuzzy
output
Mamdani
37
Example
R : if x is A and y is B, thenz is C
where A  (0,1,2), B  (1,2,3),C  (5,6,7)are triangular fuzzy sets.
If input x0  1 and y0  1.5 (Singleton) , thenoutput?
38
Two fuzzy inputs and one fuzzy output
Mamdani
Fact:
u is A ' and v is B '
: R(u, v)
Rule: If u is A and v is B then w is C : R(u, v, w)
Result: w is C '
: R(w)  R(u, v) R(u, v, w)
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Two fuzzy inputs and one fuzzy output
Mamdani
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Two fuzzy inputs and one fuzzy output
Mamdani
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Example
R : if x is A and y is B, thenz is C
where A  (0,1,2), B  (1,2,3),C  (5,6,7)are triangular fuzzy sets.
If input A'  (1,2,3)and B'  (1.5,2.5,3.5)(Fuzzy set) , thenoutput?
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Multiple rules
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Multiple rules
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Multiple rules
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Example
R 1 : if x is A1 , thenz is C1
R 2 : if x is A 2 , thenz is C 2
where A1  (0,1,2), C1  (1,2,3),A 2  (0.5,1.5,2.5),C 2  (2,3,4)
are triangular fuzzy sets.
If input x0  1 (Singleton) , thenoutput ?
46
Mamdani method
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Mamdani method
48
Mamdani method
49
Mamdani method
50
Larsen method
51
Larsen method
52
Larsen method
53
Larsen method
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Fuzzy Logic Controller
55
Inference
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Inference
57
Inference
58
Inference
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Defuzzification
• Mean of Maximum Method (MOM)
60
Defuzzification
• Center of Area Method (COA)
61
Defuzzification
• Bisector of Area (BOA)
62
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Logic

29 Cards

Conjectures

15 Cards

Hypotheses

16 Cards