chap 3: fuzzy rules and fuzzy reasoning j.-s. roger jang ( 張智星 ) cs dept., tsing hua univ.,...
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Chap 3: Fuzzy Rules and Fuzzy ReasoningChap 3: Fuzzy Rules and Fuzzy Reasoning
J.-S. Roger Jang (J.-S. Roger Jang (張智星張智星 ))
CS Dept., Tsing Hua Univ., TaiwanCS Dept., Tsing Hua Univ., Taiwanhttp://www.cs.nthu.edu.tw/~janghttp://www.cs.nthu.edu.tw/~jang
[email protected]@cs.nthu.edu.tw
Fuzzy Rules and Fuzzy Reasoning
Fuzzy Rules and Fuzzy Reasoning
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OutlineOutline
Extension principle
Fuzzy relations
Fuzzy if-then rules
Compositional rule of inference
Fuzzy reasoning
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Extension PrincipleExtension Principle
A is a fuzzy set on X :
A x x x x x xA A A n n ( ) / ( ) / ( ) /1 1 2 2
The image of A under f( ) is a fuzzy set B:
B x y x y x yB B B n n ( ) / ( ) / ( ) /1 1 2 2
where yi = f(xi), i = 1 to n.
If f( ) is a many-to-one mapping, then
Bx f y
Ay x( ) max ( )( )
1
Fuzzy Rules and Fuzzy Reasoning
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Fuzzy RelationsFuzzy Relations
A fuzzy relation R is a 2D MF:
Examples:• x is close to y (x and y are numbers)
• x depends on y (x and y are events)
• x and y look alike (x, and y are persons or objects)
• If x is large, then y is small (x is an observed reading and Y is a corresponding action)
R x y x y x y X YR {(( , ), ( , ))|( , ) }
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Max-Min CompositionMax-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: R Ry
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 ( ) ( )
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Max-Star CompositionMax-Star Composition
Max-product composition:
In general, we have max-* composition:
where * is a T-norm operator.
R Ry
R Rx z x y y z1 2 1 2 ( , ) [ ( , ) ( , )]
R Ry
R Rx z x y y z1 2 1 2 ( , ) [ ( , )* ( , )]
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Linguistic VariablesLinguistic Variables
A numerical variables takes numerical values:
Age = 65
A linguistic variables takes linguistic values:
Age is old
A linguistic values 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 yound and not very old, ...}
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Linguistic Values (Terms)Linguistic Values (Terms)
complv.m
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Operations on Linguistic ValuesOperations on Linguistic ValuesCON A A( ) 2
DIL A A( ) . 0 5
INT AA x
A xA
A
( ), ( ) .
( ) , . ( )
2 0 05
2 05 1
2
2
Concentration:
Dilation:
Contrast
intensification:
intensif.m
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Fuzzy If-Then RulesFuzzy If-Then Rules
General format:If x is A then y is B
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.
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Fuzzy If-Then RulesFuzzy If-Then Rules
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
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Fuzzy If-Then RulesFuzzy If-Then Rules
Two ways to interpret “If x is A then y is B”:• A coupled with B: (A and B)
• A entails B: (not A or B)
- Material implication
- Propositional calculus
- Extended propositional calculus
- Generalization of modus ponens
R A B A B x y x yA B ( ) ( )|( , )~
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Fuzzy If-Then RulesFuzzy If-Then RulesFuzzy implication function:
R A Bx y f x y f a b( , ) ( ( ), ( )) ( , )
fuzimp.m
A coupled with B
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Fuzzy If-Then RulesFuzzy If-Then Rules
A entails B
fuzimp.m
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Compositional Rule of InferenceCompositional Rule of Inference
Derivation of y = b from x = a and y = f(x):
a and b: points
y = f(x) : a curve
a
b
y
xx
y
a and b: intervals
y = f(x) : an interval-valued
function
a
b
y = f(x) y = f(x)
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Compositional Rule of InferenceCompositional Rule of Inference
a is a fuzzy set and y = f(x) is a fuzzy relation:
cri.m
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Fuzzy ReasoningFuzzy Reasoning
Single rule with single antecedentRule: if x is A then y is B
Fact: x is A’
Conclusion: y is B’
Graphic Representation:
A
X
w
A’ B
Y
x is A’
B’
Y
A’
Xy is B’
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Fuzzy ReasoningFuzzy ReasoningSingle rule with multiple antecedent
Rule: if x is A and y is B then z is C
Fact: x is A’ and y is B’
Conclusion: z is C’
Graphic Representation:A B T-norm
X Y
w
A’ B’ C2
Z
C’
ZX Y
A’ B’
x is A’ y is B’ z is C’
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Fuzzy ReasoningFuzzy Reasoning
Multiple rules with multiple antecedentRule 1: if x is A1 and y is B1 then z is C1
Rule 2: if x is A2 and y is B2 then z is C2
Fact: x is A’ and y is B’
Conclusion: z is C’
Graphic Representation: (next slide)
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Fuzzy ReasoningFuzzy Reasoning
Graphics representation: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’
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Fuzzy Reasoning: MATLAB DemoFuzzy Reasoning: MATLAB Demo
>> ruleview mam21
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Other VariantsOther Variants
Some terminology:• Degrees of compatibility (match)
• Firing strength
• Qualified (induced) MFs
• Overall output MF