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Hedge

Hedge. Young, middle aged, old → Primary term ； not → negation ； Very, more or less, quite, extremely → hedges ； And, or, either, neither → connection. Hedge … continued.

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Hedge

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  1. Hedge Young, middle aged, old → Primary term ； not → negation ； Very, more or less, quite, extremely → hedges ； And, or, either, neither → connection

  2. Hedge…continued • Concentration and dilation of linguistic values let A be a linguistic value characterized by a fuzzy set with membership function μA(•), then AK is interpreted as a modified version of the original linguistic value expressed in particular, the operation of concentration is defined as CON (A) = A2 while the operation of dilation is defined as DIL (A) = A0.5 other appropriate values of K are possible

  3. Hedge…continued • Ex.

  4. Hedge…continued • Negative and connectives • Composite linguistic term not young and not old

  5. Hedge…continued young but not too young

  6. Fuzzy If-Then Rules • Or called fuzzy rules • A fuzzy inference system consists of a set of fuzzy if-then rules • Have been applied to • Control systems, decision-making, pattern recognition, system modeling • a fuzzy if-then rule associates • A condition described using linguistic variables and fuzzy sets to a conclusion Ex.If a tomato is red, then it is ripe

  7. Fuzzy If-Then Rules • Fuzzy rule-based inference can be understood from several points of view • (i) a multi-expert decision making metaphor • (ii) mathematically, fuzzy rule-based inference can be viewed as an interpolation scheme • (iii) from a logic view point, fuzzy rule-based inference is a generalization of a logic reasoning scheme – modes ponens

  8. Fuzzy If-Then Rules • The elastic condition and the consequent of a fuzzy rule are often described by words (ie. Linguistic labels/ values/terms) Ex. Linguistic variable：〝age〞 Term set： T：{young, not young, very young, not very young, …. middle aged, not middle aged, …….. old, not old, very old, more or less old, not very old, not very young and very old, …….} Use 〝age is young〞to denote the assignment of the linguistic value 〝young〞to the linguistic variable 〝age〞

  9. Fuzzy If-Then Rules • Fuzzy if-then rules (fuzzy rules) • a fuzzy if-then assume the form if x is A, then y is B A, B：linguistic values defined by fuzzy sets on universes X and Y x, y： linguistic variables “xis A”： antecedent / premise “y is B”：consequence / conclusion Ex. If pressure is high, then volume is small if the speed is high, the apply the brake a little

  10. Fuzzy If-Then Rules • Abbreviated as A→B • Such a expression describes a relation between two variables x and y This suggests that a fuzzy if-then rule be defined as a binary fuzzy relation on the product space X * Y • Classical implication A→B

  11. Fuzzy If-Then Rules • Fuzzy implication (A coupled with B)

  12. Fuzzy If-Then Rules • Ex. If x is medium, then y is small X={2,3,4,5,6,7,8,9} Y={1,2,3,4,5,6} Apply Rm： Medium ≡ 0.1/2 + 0.3/3 + 0.7/4 + 1/5 + 1/6 + 0.7/7 + 0.5/8 + 0.2/9 small ≡ 1/1 + 1/2 + 0.9/3 + 0.6/4 + 0.3/5 + 0.1/6

  13. Fuzzy If-Then Rules Apply Rp： Apply Rbp： (2,1)：0 ν(0.1+1-1)=0.1 (2,2)：0 ν(0.1+1-1)=0.1 (2,3)：0 ν(0.1+0.9-1)=0 (2,4)：0 ν(0.1+0.6-1)=0.0 (4,3)：0 ν(0.7+0.9-1)=0.6 • Fuzzy implication (A entails B) • 1 Λ (1-a+b) • (1-a) ν b

  14. Approximate reasoning • Modus ponens (離斷律) • Basic rule of inference in traditional logic Premise 1 (fact) ： x is A Premise 2 (rule) ： if x is A then y is B Consequence (conclusion)：y is B • Ex. fact ： The tomato is red rule ： if the tomato is red then the tomato is ripe conclusion：the tomato is ripe

  15. Approximate reasoning • Generalized modus ponens (GMP) • How about〝the tomato is more or less red〞？ Premise 1 (fact) ： x is A’ Premise 2 (rule) ： if x is A then y is B Consequence (conclusion)：y is B’ Where A’ is dose to A,B’ is close to B. when A, B, A’ and B’ are fuzzy sets, such an inference procedure is called approximate reasoning (fuzzy reasoning)

  16. Approximate reasoning • Modus tolens (逆斷律) Premise 1 (fact) ： y is not B Premise 2 (rule) ： if x is A then y is B Consequence (conclusion)：x is not A • Ex. (P.12) Premise 1 (fact) ： x is small Premise 2 (rule) ： if x is medium then y is small conclusion：？ (y is B’)

  17. Approximate reasoning X={2,3,4,5,6,7,8,9} Y={1,2,3,4,5,6} smallx = 1/1 + 1/2 + 0.9/3 + 0.6/4 + 0.3/5 + 0.1/6 + 0/7 + 0/8 + 0/9 Mediumx = 0.1/2 + 0.3/3 + 0.7/4 + 1/5 + 1/6 + 0.7/7 + 0.5/8 + 0.2/9 (x is small) ∩ (x is medium) = 0.1/2 + 0.3/3 + 0.6/4 + 0.3/5 + 0.1/6 + 0/7 + 0/8 + 0/9 smally = 1/1 + 1/2 + 0.9/3 + 0.6/4 + 0.3/5 + 0.1/6 B’ = 0.6/1 + 0.6/2 + 0.6/3 + 0.6/4 + 0.3/5 + 0.1/6

  18. Approximate reasoning 2→1：0.1 (0.1 ν1) 3→1：0.3 (0.13ν1) 4→1：0.6 (0.6 ν0.9) 5→1：0.3 6→1：0.1 7→1：0 8→1：0 9→1：0 ν =>0.6

  19. Approximate reasoning • Let A, A’ and B be fuzzy sets of X, X and Y, respectively. Assume that the fuzzy implication A→B is expressed as a fuzzy relation R on X × Y. Then the fuzzy set B induced by〝x is A’〞and the fuzzy rule〝if x is A then y is B〞is defined by

  20. Approximate reasoning • Note：a fuzzy rule is represented mathematically as fuzzy relations formed by the Cartesian product of the variables referred in the rule’s if-part and then-part. • Ex. If x is A, then y is B =>μR(x,y) = μA×B (x,y) If we use〝min〞operator for the Cartesian product, R becomes μR(x,y) = min {μA(x), μB(y)}

  21. Approximate reasoning • Single rule with multiple antecedents • 〝if x is A and y is B then z is C〞 Premise 1 (fact) ： x is A’ and y is B’ Premise 2 (rule) ： if x is A and y is B then z is C Consequence (conclusion)：z is C’ ≡ μA×B→C (x,y,z) = μR(x,y,z) Firing strength

  22. Approximate reasoning Or, C’ = (A’ × B’)。(A × B→ C) W1, W2：degree of compatibility W1Λ W2：firing strength

  23. Approximate reasoning • Multiple rules with multiple antecedents Premise 1 (fact) ： x is A’ and y is B’ Premise 2 (rule 1) ： if x is A1 and y is B1then z is C1 Premise 3 (rule 2) ： if x is A2 and y is B2then z is C2 Consequence (conclusion)：z is C’ Let R1 = A1 × B1→C1 R2 = A2 × B2→C2 C’ = (A’ × B’)。(R1∪R2) = [(A’ × B’)。R1]∪[(A’ × B’)。R2] = C’1 ∪C’2

  24. Fuzzy inference systems • The basic fuzzy inference system can take either fuzzy inputs or crisp inputs, the outputs are almost always fuzzy sets • But sometimes it is necessary to have crisp output, especially in a situation where a fuzzy inference system is used as a controller => Need a method of defuzzification to extract a crisp value that best represents a fuzzy set

  25. Rule 1 (Fuzzy) W1 yis B1 Rule 2 (Fuzzy) (Crisp or Fuzzy) W2 yis B2 (Fuzzy) (Crisp) Aggregator Defuzzifier y ‧‧‧ ‧‧‧ ‧‧‧ Rule r (Fuzzy) Wr yis Br Fuzzy inference systems

  26. Mandani fuzzy models • If x is A1 and y is B1, then z is C1 • If x is A2 and y is B2, then z is C2 △ max：T-conorn mim：T-norn

  27. △ max：T-conorn product：T-norn

  28. Fuzzy inference systems • Defuzzification • The method a crisp value is extracted from a fuzzy set as a representative value • 5 methods • (1) COA (Centroid of area)

  29. Fuzzy inference systems • (2) BOA (Bisector of area) • (3) MOM • The average of the maximizing z at which the M.F. reach a maximum μx

  30. Fuzzy inference systems • (5) LOM (largest of maximum) • LOM = Zright • (4) SOM (smallest of maximum) • SOM = Zleft

  31. The Mamdani model Ri：if x1is Ai1 and ….xr is Air, then y is ci xj, j =1,….,r can be crisp or fuzzy

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