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:

• 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 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 ):

• Associativity:

• Distributivity over union:

• Weak distributivity over intersection:

• 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):

• 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, ...}

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:

• If x is A then y is B

• This is interpreted as a fuzzy set

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.

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:

• A coupled with B:( A and B T-norm)

• A entails B:( not A or B )

• • Material implication

• • Propositional calculus

• • Extended propositional calculus

• • Generalization of modus ponens

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

• Rule:if x is A then y is B

• Premise:x is A, where A is close to A

• 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

• Rule:if x is A and y is B then z is C

• Premise:x is A and y is B

• 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

• Rule 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

• Premise:x is A and y is B

• 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)