Fuzzy Rules & Fuzzy Reasoning

Transcription

1 Sistem Cerdas : PTK Pasca Sarjana - UNY Fuzzy Rules & Fuzzy Reasoning Pengampu: Fatchul rifin Referensi: Jyh-Shing Roger Jang et al., Neuro-Fuzzy and Soft Computing: Computational pproach to Learning and Machine Intelligence, First Edition, Prentice Hall, 1997

2 Fuzzy if-then rules (3.3) (cont.) 22 Orthogonality term set T = t 1,, t n of a linguistic variable x on the universe X is orthogonal if: n i1 ti (x) 1, x X Where the t i s are convex & normal fuzzy sets defined on X.

3 Fuzzy if-then rules (3.3) (cont.) 23 General format: If x is then y is (where & are linguistic values defined by fuzzy sets on universes of discourse X & Y). x is is called the antecedent or premise y is is called the consequence or conclusion 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.

4 Fuzzy if-then rules (3.3) (cont.) 24 Meaning of fuzzy if-then-rules ( ) It is a relation between two variables x & y; therefore it is a binary fuzzy relation R defined on X * Y There are two ways to interpret : coupled with entails if is coupled with then: R * ~ X*Y where * is a T - normoperator. ~ (x)* (y) /(x,y)

5 Fuzzy if-then rules (3.3) (cont.) 25 If entails then: R = = ( material implication) R = = ( ) (propositional calculus) R = = ( ) (extended propositional calculus) ~ R ( x, y) supc; (x)*c (y),0 c 1

6 Fuzzy if-then rules (3.3) (cont.) 26 Two ways to interpret If x is then y is : y coupled with y entails x x

7 Fuzzy if-then rules (3.3) (cont.) 27 Note that R can be viewed as a fuzzy set with a two-dimensional MF R (x, y) = f( (x), (y)) = f(a, b) With a = (x), b = (y) and f called the fuzzy implication function provides the membership value of (x, y)

8 Fuzzy if-then rules (3.3) (cont.) 28 Case of coupled with R m * ( x) ( y) /( x, y) X * Y (minimum operator proposed by Mamdani, 1975) Rp * (x) X*Y (product proposed by Larsen, 1980) R bp * X*Y 0 ( X*Y (x) (bounded product operator) (x) (y) /(x,y) (y) /(x,y) (y) 1) /(x,y)

9 Fuzzy if-then rules (3.3) (cont.) 29 Case of coupled with (cont.) R dp * X * Y ( x).ˆ ( y) /( x, y) a if b 1 where :f(a,b) a.ˆ b b if a 1 0 if otherwise Example for ( x) bell( x;4,3,10) and ( y) bell( y;4,3,10) (Drastic operator)

10 Fuzzy if-then rules (3.3) (cont.) 30 coupled with

11 Fuzzy if-then rules (3.3) (cont.) Case of entails 31 R a where : f a X*Y 1 (1 (a,b) 1 (1 a b) (x) (y) /(x, y) (Zadeh s arithmetic rule by using bounded sum operator for union) R mm where : ( ) f m X*Y (1 (a,b) (1 a) (a b) (x)) ( (x) (y)) /(x,y) (Zadeh s max-min rule)

12 Fuzzy if-then rules (3.3) (cont.) 32 Case of entails (cont.) R s where : f s X*Y (1 (a,b) (1 a) b (x)) (oolean fuzzy implication with max for union) (y) /(x, y) R X*Y ( (x) ~ (y)) /(x, y) 1 if a b where : a ~ b b / a otherwise (Goguen s fuzzy implication with algebraic product for T-norm)

13 33 entails

14 34 Fuzzy Reasoning (3.4) Definition Known also as approximate reasoning It is an inference procedure that derives conclusions from a set of fuzzy if-then-rules & known facts

15 Fuzzy Reasoning (3.4) (cont.) 35 Compositional rule of inference Idea of composition (cylindrical extension & projection) Computation of b given a & f is the goal of the composition Image of a point is a point Image of an interval is an interval

16 Fuzzy Reasoning (3.4) (cont.) Derivation of y = b from x = a and y = f(x): 36 y y b b y = f(x) y = f(x) a a and b: points y = f(x) : a curve x a a and b: intervals y = f(x) : an interval-valued function x

17 Fuzzy Reasoning (3.4) (cont.) 37 The extension principle is a special case of the compositional rule of inference F is a fuzzy relation on X*Y, is a fuzzy set of X & the goal is to determine the resulting fuzzy set Construct a cylindrical extension c() with base Determine c() F (using minimum operator) Project c() F onto the y-axis which provides

18 Fuzzy Reasoning (3.4) (cont.) 38 a is a fuzzy set and y = f(x) is a fuzzy relation: cri.m

19 39 Fuzzy Reasoning (3.4) (cont.) Given,, infer = today is sunny : day = sunny then sky = blue infer: sky is blue illustration Premise 1 (fact): x is Premise 2 (rule): if x is then y is Consequence: y is

20 40 Fuzzy Reasoning (3.4) (cont.) pproximation = today is more or less sunny = sky is more or less blue iilustration Premise 1 (fact): x is Premise 2 (rule): if x is then y is Consequence: y is (approximate reasoning or fuzzy reasoning!)

21 Fuzzy Reasoning (3.4) (cont.) Definition of fuzzy reasoning 41 Let, and be fuzzy sets of X, X, and Y, respectively. ssume that the fuzzy implication is expressed as a fuzzy relation R on X*Y. Then the fuzzy set induced by x is and the fuzzy rule if x is then y is is defined by: ' ( y) max min ' (x), R Single rule with single antecedent x Rule : if x is then y is Fact: x is Conclusion: y is ( (y) = [ x ( (x) (x)] (y)) (x, y)

22 Fuzzy Reasoning (3.4) (cont.) 42 w X Y x is X y is Y

23 43 R C' C' (z) Single rule with multiple antecedents Premise 1 (fact): x is and y is Premise 2 (rule): if x is and y is then z is C mamdani Conclusion: z is C Premise 2: * C (,,C) ('*') ( * C) premise 1 x,y x,y (x) (y) (x) (y) (z) (x) (x) (y) (y) (w 1 ' ' w (x) ' x w 1 premise 2 2 ) ( * )* C ' ' C (y) (z) X*Y*Z (x) w 2 (x) (y) (y) C C (z) ' y (z) /(x, y, z) (z) C C

24 44 T-norm C2 w 1 w w 2 Z X Y C Z x is X y is Y z is C

25 Fuzzy Reasoning (3.4) (cont.) 45 Multiple rules with multiple antecedents Premise 1 (fact): x is and y is Premise 2 (rule 1): if x is 1 and y is 1 then z is C 1 Premise 3 (rule 2): If x is 2 and y is 2 then z is C 2 Consequence (conclusion): z is C R 1 = 1 * 1 C 1 R 2 = 2 * 2 C 2 Since the max-min composition operator o is distributive over the union operator, it follows: C = ( * ) o (R 1 R 2 ) = [( * ) o R 1 ] [( * ) o R 2 ] = C 1 C 2 Where C 1 & C 2 are the inferred fuzzy set for rules 1 & 2 respectively

26 C1 X Y w1 Z 2 2 C2 w2 X Y T-norm Z Z x is X y is Y z is C C