ch03
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- 1. Chapter 3: Fuzzy Rulesand Fuzzy Reasoning
- J.-S. Roger Jang ( )
- CS Dept., Tsing Hua Univ., Taiwan
- Modified by Dan Simon
- Cleveland State University
Fuzzy Rules and Fuzzy Reasoning 2. Outline
- Extension principle
- Fuzzy relations
- Fuzzy if-then rules
- Compositional rule of inference
- Fuzzy reasoning
3. Extension Principle Ais a fuzzy set onX: The image ofAunderf(.)is a fuzzy setB: wherey i= f(x i ) ,fori = 1ton . Iff(.)is a many-to-one mapping, then 4. Example: Extension Principle 0 1 2 3 0 1 4 9 0 1 2 3 0 1 4 9 -1 y=x 2 (x) (x) (y) (y) Example 1 Example 2 5. Fuzzy Relations
- A fuzzy relationRis a 2D MF:
- Examples:
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- x is close to y (x and y are numbers)
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- x depends on y (x and y are events)
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- x and y look alike (x and y are persons or objects)
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- If x is large, then y is small (x is an observed instrument reading and y is a corresponding control action)
6. Example: x is close to y 7. Example: X is close to Y 8. Max-Min Composition
- The max-min composition of two fuzzy relationsR 1(defined onXandY ) andR 2(defined onYandZ ):
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- Associativity:
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- Distributivity over union:
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- Weak distributivity over intersection:
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- Monotonicity:
(max) (min) 9. Max-Star Composition
- Max-product composition:
- In general, we have max * compositions:
- where * is a T-norm operator.
10. Example 3.4 Max * Compositions R 1 : x is relevant to y R 2 : y is relevant to z How relevant is x=2 to z=a?y= y= y= y= x=1 0.1 0.3 0.5 0.7 x=2 0.4 0.2 0.8 0.9 x=3 0.6 0.8 0.3 0.2 z=a z=b y= 0.9 0.1 y= 0.2 0.3 y= 0.5 0.6 y= 0.7 0.2 11. Example 3.4 (contd.) 1 2 3 a b 0.4 0.2 0.8 0.9 0.9 0.2 0.5 0.7 x y z 12. Linguistic Variables
- 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 aterm set(set of terms):
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- T(age) = {young, not young, very young, ...
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- middle aged, not middle aged, ...
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- old, not old, very old, more or less old, ...
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- not very young and not very old, ...}
13. Operations on Linguistic Values Concentration: Dilation: Contrast intensification: intensif.m (very) (more or less) 14. Linguistic Values (Terms) complv.m How are these derived from the above MFs? 15. Fuzzy If-Then Rules
- General format:
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- If x is A then y is B
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- This is interpreted as a fuzzy set
- Examples:
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- If pressure is high, then volume is small.
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- If the road is slippery, then driving is dangerous.
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- If a tomato is red, then it is ripe.
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- If the speed is high, then apply the brake a little.
16. Fuzzy If-Then Rules A is coupled with B: (x is A)(y is B) A A B B A entails B: (x is not A)(y is B) Two ways to interpret If x is A then y is B y x x y 17. Fuzzy If-Then Rules
- Example:
- if (profession is athlete) then (fitness is high)
- Coupling:Athletes, and only athletes, have high fitness.
- The if statement (antecedent) is a necessary and sufficient condition.
- Entailing:Athletes have high fitness, and non-athletes may or may not have high fitness.
- The if statement (antecedent) is a sufficient butnotnecessary condition.
18. Fuzzy If-Then Rules
- Two ways to interpret If x is A then y is B:
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- A coupled with B:( A and B T-norm)
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- A entails B:( not A or B )
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- Material implication
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- Propositional calculus
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- Extended propositional calculus
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- Generalization of modus ponens
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19. Fuzzy If-Then Rules
- Fuzzy implication
fuzimp.m Acoupledwith B (bell-shaped MFs, T-norm operators) Example: only fit athletes satisfy the rule 20. Fuzzy If-Then Rules AentailsB (bell-shaped MFs) Arithmetic rule: (x is not A)(y is B) (1 x) + y Example: everyone except non-fit athletes satisfies the rule fuzimp.m 21. Compositional Rule of Inference
- Derivation ofy = bfromx = aandy = f(x) :
aandb: points y = f(x): a curve Crisp : if x = a, then y=b a b y x x y aandb: intervals y = f(x): interval-valued function Fuzzy : if (x is a) then (y is b) a b y = f(x) y = f(x) 22. Compositional Rule of Inference
- Ais a fuzzy set of x andy = f(x)is a fuzzy relation:
cri.m 23. Fuzzy Reasoning
- Single rule with single antecedent
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- Rule:if x is A then y is B
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- Premise:x is A, where A is close to A
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- Conclusion:y is B
- Use max of intersection between A and A to get B
A X w A B Y x is A B Y A X y is B 24. Fuzzy Reasoning
- Single rule with multiple antecedents
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- Rule:if x is A and y is B then z is C
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- Premise:x is A and y is B
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- Conclusion:z is C
- Use min of (AA) and (BB) to get C
A B X Y w A B C Z C Z X Y A B x is A y is B z is C 25. Fuzzy Reasoning
- Multiple rules with multiple antecedents
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- Rule 1:if x is A1 and y is B1 then z is C1
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- Rule 2:if x is A2 and y is B2 then z is C2
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- Premise:x is A and y is B
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- Conclusion:z is C
- Use previous slide to get C 1 and C 2
- Use max of C 1 and C 2 to get C (next slide)
26. Fuzzy Reasoning
- Multiple rules with multiple antecedents
A 1 B 1 A 2 B 2 X X Y Y w 1 w 2 A A B B C 1 C 2 Z Z C Z X Y A B x is A y is B z is C 27. Fuzzy Reasoning: MATLAB Demo
- >> ruleview mam21 (Matlab Fuzzy Logic Toolbox)
28. Other Variants
- Some terminology:
- Degrees of compatibility (match between input variables and fuzzy input MFs)
- Firing strength calculation (we used MIN)
- Qualified (induced) MFs (combine firing strength with fuzzy outputs)
- Overall output MF (we used MAX)