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# Lecture 29 fuzzy systems

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### Lecture 29 fuzzy systems

1. 1. Soft Computing Fuzzy Logic
2. 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. 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. 4. FUZZY LOGICUniverse of DiscourseRange of possible values of a fuzzy variableExample: Speed: 0 to 100 mph
5. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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