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Lecture 29 fuzzy systems

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  • 1. Soft Computing Fuzzy Logic
  • 2. FUZZY LOGICMotivation• Modeling of imprecise conceptsE.g. age, weight, height,…• Modeling of imprecise dependencies (e.g. rules), e.g. ifTemperature is high and Oil is cheap then I will turn-on thegenerator• Origin of information - Modeling of expert knowledge -Representation of information extracted from inherentedly imprecise data
  • 3. FUZZY LOGICTo quantify and reason about fuzzy or vague terms of naturallanguageExample: hot, cold temperature small, medium, tall height creeping, slow, fast speedFuzzy VariableA concept that usually has vague (or fuzzy) valuesExample: age, temperature, height, speed
  • 4. FUZZY LOGICUniverse of DiscourseRange of possible values of a fuzzy variableExample: Speed: 0 to 100 mph
  • 5. FUZZY LOGICFuzzy Set (Value)Let X be a universe of discourse of a fuzzy variable and x be its elementsOne or more fuzzy sets (or values) Ai can be defined over XExample: Fuzzy variable: Age Universe of discourse: 0 – 120 years Fuzzy values: Child, Young, OldA fuzzy set A is characterized by a membership function µA(x) that associates each element x with a degree of membership value in AThe value of membership is between 0 and 1 and it represents the degree to which an element x belongs to the fuzzy set A
  • 6. FUZZY LOGICFuzzy Set (Value)In traditional set theory, an object is either in a set or not in aset (0 or 1), and there are no partial membershipsSuch sets are called “crisp sets”
  • 7. FUZZY LOGICFuzzy Set RepresentationFuzzy Set A = (a1, a2, … an) ai = µA(xi) xi = an element of X X = universe of discourseFor clearer representation A = (a1/x1, a2/x2, …, an/xn)Example: Tall = (0/5’, 0.25/5.5’, 0.9/5.75’, 1/6’, 1/7’, …)
  • 8. FUZZY LOGICFuzzy Set RepresentationFor a continuous set of elements, we need some function tomap the elements to their membership valuesTypical functions: sigmoid, gaussian
  • 9. FUZZY LOGICFormation of Fuzzy Sets • Opinion of a single person • Average of opinion of a set of persons • Other methods (e.g. function approximation from data by neural networks) • Modification of existing fuzzy sets - Hedges - Application of Fuzzy set operators
  • 10. FUZZY LOGICFormation of Fuzzy SetsHedges: Modification of existing fuzzy sets to account for some added adverbsTypes:Concentration (very) Square of memberships Conc(µA(x)) = [µA(x)]2 reduces small memberships values 0.1 changes to 0.01 (10 times reduction) 0.9 changes to 0.81 (0.1 times reduction) Example: very tall
  • 11. FUZZY LOGICFormation of Fuzzy SetsDilation (somewhat) Square root of memberships Dil(µA(x)) = [µA(x)]1/2 increases small memberships values 0.09 changes to 0.3 0.81 changes to 0.9 Example: somewhat tall
  • 12. FUZZY LOGICFuzzy Sets OperationsIntersection (A  B)In classical set theory the intersection of two sets containsthose elements that are common to bothIn fuzzy set theory, the value of those elements in theintersection: µA  B(x) = min [µA(x), µB(x)]e.g. Tall = (0/5, 0.1/5.25, 0.5/5.5, 0.8/5.75, 1/6) Short = (1/5, 0.8/5.25, 0.5/5.5, 0.1/5.75, 0/6) Tall  Short = (0/5, 0.1/5.25, 0.5/5.5, 0.1/5.75, 0/6) = Medium
  • 13. FUZZY LOGICFuzzy Sets OperationsUnion (A  B)In classical set theory the union of two sets contains thoseelements that are in any one of the two setsIn fuzzy set theory, the value of those elements in the union: µA  B(x) = max [µA(x), µB(x)]e.g. Tall = (0/5, 0.1/5.25, 0.5/5.5, 0.8/5.75, 1/6) Short = (1/5, 0.8/5.25, 0.5/5.5, 0.1/5.75) Tall  Short = (1/5, 0.8/5.25, 0.5/5.5, 0.8/5.75, 1/6) = not Medium
  • 14. FUZZY LOGICFuzzy Sets OperationsComplement (A)In fuzzy set theory, the value of complement of A is: µ  A(x) = 1 - µA(x)e.g. Tall = (0/5, 0.1/5.25, 0.5/5.5, 0.8/5.75, 1/6)  Tall = (1/5, 0.9/5.25, 0.5/5.5, 0.2/5.75, 0/6)
  • 15. FUZZY RULESFuzzy RulesRelates two or more fuzzy propositions If X is A then Y is Be.g. if height is tall then weight is heavyX and Y are fuzzy variablesA and B are fuzzy sets
  • 16. FUZZY LOGICFuzzy RelationsClassical relation between two universesU = {1, 2} and V = {a, b, c} is defined as: a b c R=UxV= 1 1 1 1 2 1 1 1 Example: U = Weight (normal, over) V = Height (short, med, tall)
  • 17. FUZZY LOGICFuzzy RelationsFuzzy relation between two universes U and V is defined as: µR (u, v) = µAxB (u, v) = min [µA (u), µB (v)]i.e. we take the minimum of the memberships of the twoelements which are to be related
  • 18. FUZZY LOGICFuzzy RelationsExample:Determine fuzzy relation between A1 and A2A1 = 0.2/x1 + 0.9/x2A2 = 0.3/y1 + 0.5/y2 + 1/y3The fuzzy relation R isR = A1 x A2 = 0.2 x 0.3 0.5 1 0.9
  • 19. FUZZY LOGICFuzzy RelationsExample:R = A1 x A2 = 0.2 x 0.3 0.5 1 0.9 = min(0.2, 0.3) min(0.2, 0.5) min(0.2, 1) min(0.9, 0.3) min(0.9, 0.5) min(0.9, 1) = 0.2 0.2 0.2 0.3 0.5 0.9
  • 20. FUZZY LOGICFuzzy RelationsR = R(A1, A2) A2 a23 = 0.2 0.2 0.2 (1.0 0.3 0.5 0.9 ) 0.2 a22 0.9 (0.5 ) 0.2 0.5 a21 (0.3 ) 0.2 0.3 a11 a12 A1 (0.2) (0.9)
  • 21. FUZZY RULESFuzzy Associative MatrixSo for the fuzzy rule: If X is A then Y is BWe can define a fuzzy matrix M(nxp) which relates A to B M=Ax BIt maps fuzzy set A to fuzzy set B and is used in the fuzzyinference process
  • 22. FUZZY RULESFuzzy Associative MatrixConcept behind M a1  b1 a1  b2 … a2  b1 … . . .If a1 is true then b1 is true; and so on
  • 23. FUZZY RULESApproximate ReasoningExample: Let there be a fuzzy associative matrix M for therule: if A then Be.g. If Temperature is normal then Speed is mediumLet A = [0/100, 0.5/125, 1/150, 0.5/175, 0/200] B = [0/10, 0.6/20, 1/30, 0.6/40, 0/50]
  • 24. FUZZY RULESApproximate Reasoning: Max-Min InferenceLet A = [0/100, 0.5/125, 1/150, 0.5/175, 0/200] B = [0/10, 0.6/20, 1/30, 0.6/40, 0/50]then M= (0, 0) (0, 0.6) . . . (0.5, 0) . . . . . . = 0 0 0 0 0 0 0.5 0.5 0.5 0 by taking the minimum 0 0.6 1 0.6 0 of each pair 0 0.5 0.5 0.5 0 0 0 0 0 0