Fuzzy Rules and Fuzzy Reasoning. Chapter 3, Neuro-Fuzzy and Soft Computing: Fuzzy Rules and Fuzzy Reasoning by Jang

Transcription

1 Chapter 3, Neuro-Fuzzy and Soft Computing: Fuzzy Rules and Fuzzy Reasoning by Jang

2 Outline Extension principle Fuzzy relations Fuzzy if-then rules Compositional rule of inference Fuzzy reasoning 2

3 Extension Principle A is a fuzzy set on X : A = µ ( x )/ x + µ ( x )/ x + + µ ( x )/ x A 1 1 A 2 2 L A n n The image of A under f( ) is a fuzzy set B: B = µ ( x )/ y + µ ( x )/ y + + µ ( x )/ y B 1 1 B 2 2 L B n n where yi = f(xi), i = 1 to n. If f( ) is a many-to-one mapping, then µ ( y) = max µ ( x) B x f ( y) = 1 A 3

4 Fuzzy Relations A fuzzy relation R is a 2D MF: R = {(( x, y), µ ( x, y)) ( x, y) X Y} R 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) 4

5 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 µ R o R ( x, z) = [ µ R ( x, y) µ R ( y, z)] Properties: Associativity: Distributivity over union: y R o ( S o T ) = ( R o S ) o T R o ( S U T ) = ( R o S ) U( R o T ) Weak distributivity over intersection: R o ( S I T ) ( R o S ) I( R o T ) Monotonicity: S T ( R o S ) ( R o T ) 5

6 Max-Star Composition Max-product composition: µ R o R ( x, z) = [ µ R ( x, y) µ R ( y, z)] In general, we have max-* composition: where * is a T-norm operator. y µ R o R ( x, z) = [ µ R ( x, y)* µ R ( y, z)] y

7 Linguistic 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,...} 7

8 Linguistic Values (Terms) 8 complv.m

9 Operations on Linguistic Values Concentration: CON ( A) = A 2 Dilation: Contrast intensification: DIL( A) = A 05. A, A( x) INT( A) = µ 0 2 2( A), 05. µ ( x) A 9 intensif.m

10 Fuzzy 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. 0

11 Fuzzy If-Then Rules Two ways to interpret If x is A then y is B : y A coupled with B y A entails B B B A x A x 1

12 Fuzzy If-Then Rules Two ways to interpret If x is A then y is B : A coupled with B: (A and B) R = A B = A B = µ ( A x ) ~ µ ( B y ) ( x, y ) A entails B: (not A or B) - Material implication - Propositional calculus - Extended propositional calculus - Generalization of modus ponens 2

13 Fuzzy If-Then Rules Fuzzy implication function: µ µ µ ( x, y) = f ( ( x), ( y)) = f ( a, b) R A B A coupled with B 3 fuzimp.m

14 Fuzzy If-Then Rules A entails B 4 fuzimp.m

15 Compositional Rule of Inference Derivation of y = b from x = a and y = f(x): y y b b y = f(x) y = f(x) 5 a a and b: points y = f(x) : a curve x a a and b: intervals y = f(x) : an interval-valued function x

16 Compositional Rule of Inference a is a fuzzy set and y = f(x) is a fuzzy relation: 6 cri.m

17 Fuzzy Reasoning Single rule with single antecedent Rule: if x is A then y is B Fact: x is A Conclusion: y is B Graphic Representation: A A B w A X B Y x is A X y is B Y 7

18 Fuzzy Reasoning Single 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 A B B T-norm C X A B x is A X y is B Y z is C Y w C Z Z 8

19 Fuzzy Reasoning C' = µ C' (A'*B') o (A * B C) (z) premise 1 = = = = x,y x,y [ µ (x) µ (y)] [ µ (x) µ (y) µ (z)] {[ µ (x) µ (y) µ (x) µ (y)]} { [ ]} µ (x) µ (x) [ µ (y) µ (y)] A' 14 x 44 2 w1 (w 1 A' A' w premise 2 2 A ) µ B' B' 4443 C (z) A A B w 2 B µ C C (z) B' B µ y C (z) 9

20 Fuzzy Reasoning Multiple rules with multiple antecedent 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 Fact: x is A and y is B Conclusion: z is C Graphic Representation: (next slide) 0

21 Fuzzy Reasoning Graphics representation: A A1 B B1 C1 X A A2 B B2 Y w1 C2 Z w2 X A B Y T-norm x is A X y is B Y z is C C Z Z 1

22 Fuzzy Reasoning: MATLAB Demo >> ruleview mam21 2

23 Other Variants Some terminology: Degrees of compatibility (match) Firing strength Qualified (induced) MFs Overall output MF 3