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Fuzzy Sets

A set is a collection of objects. A special kind of set. Fuzzy Sets. Jan Jantzen www.inference.dk 2013. Summary: A set . This object is not a member of the set. This object is a member of the set. A classical set has a sharp boundary. ... and a fuzzy set.

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Fuzzy Sets

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  1. A set is a collection of objects A special kind of set Fuzzy Sets Jan Jantzen www.inference.dk 2013

  2. Summary: A set ... This object is not a member of the set This object is a member of the set A classical set has a sharp boundary

  3. ... and a fuzzy set This object is not a member of the set This object is a member of the set to a degree, for instance 0.8. The membership is between 0 and 1. A fuzzy set has a graded boundary

  4. Example: High and low pressures

  5. Example: "Find books from around1980" This could include 1978, 1979, 1980, 1981, and 1982

  6. Maybe even 41 or 42 could be all right?

  7. Example: A fuzzywashingmachine • If youfill it withonly a fewclothes, it willuseshorter time and thus save electricity and water. There is a computer inside that makes decisions depending on how full the machine is and other information from sensors. Samsung J1045AV capacity 7 kg

  8. A rule (implication) • IF the machine is full THEN wash long time Action. Condition The internal computer is able to execute an if—then rule even when the condition is only partially fulfilled.

  9. IF the machine is full … true • Example: Load = 3.5 kg clothes false Classical set Linear fuzzy set Nonlinear fuzzy set Three examples of functions that define ‘full’. The horizontal axis is the weight of the clothes, and the vertical axis is the degree of truth of the statement ‘the machine is full’.

  10. … THEN wash long time • long time couldbet = 120 minutes The duration depends on the washing program that the user selects.

  11. Decision Making (inference) • Rule. IF the machine is full THEN wash long time • Measurement. Load = 3.5 kg • Conclusion. Full(Load) × t = 0.5 × 120 = 60 mins The machine is only half full, so it washes half the time. Analogy The dots mean 'therefore'

  12. A rule base withfourrules • IF machine is full AND clothesaredirty THEN wash long time • IF machine is full AND clothesarenotdirty THEN wash medium time • IF machine is notfull AND clothesaredirty THEN wash medium time • IF machine is notfull AND clothesarenotdirty THEN wash short time There are two inputs that are combined with a logical 'and'.

  13. Theory of Fuzzy Sets

  14. Lotfi Zadeh’s Challenge Clearly, the “class of all real numbers which are much greater than 1,” or “the class of beautiful women,” or “the class of tall men,” do not constitute classes or sets in the usual mathematical sense of these terms (Zadeh1965).

  15. Sets The set of positive integers The set of belonging to for which {Live dinosaurs in British Museum} =  The empty set

  16. Fuzzy Sets {nice days} {adults} Much greater than Membership function

  17. 1 0.8 fuzzy 0.6 Membership 0.4 crisp 0.2 0 150 160 170 180 190 200 Height [cm] Tall Persons Degree of membership Membership function Universe

  18. Fuzzy(http://www.m-w.com) adjective Synonyms: faint, bleary, dim, ill-defined, indistinct, obscure, shadowy, unclear, undefined, vague Unfortunately, they all carry a negative connotation.

  19. 'Aroundnoon' Triangular Trapezoidal Smooth versions of the same sets.

  20. Spring Summer Autumn Winter 1 0.5 Membership 0 Time of the year we are here The 4 Seasons Seasons have overlap; the transition is fuzzy.

  21. Summary A set A fuzzy set

  22. Operations onFuzzy Sets

  23. Set Operations Union Intersection Negation Classical Fuzzy

  24. A B Fuzzy Set Operations Union Intersection Negation

  25. Example: Age The negation of 'very young' Primary term Primary term The square root of Old The square of Young

  26. Operations Here is a whole vocabulary of seven words. Each operates on a membership function and returns a membership function. They can be combined serially, one after the other, and the result will be a membership function.

  27. CartesianProduct The AND composition of all possible combinations of memberships from A and B The curves correspond to a cut by a horizontal plane at different levels

  28. Example: Donald Duck's family • Suppose, • nephew Huey resembles nephew Dewey • nephew Huey resembles nephew Louie • nephew Dewey resembles uncle Donald • nephew Louie resembles uncle Donald • Question: How much does Huey resemble Donald?

  29. Donald Dewey Louie 0.5 Donald Huey Dewey Louie 0.8 0.9 = Huey 0.6 Solution: Fuzzy Composition of Relations Relation Composition Relation Relation ? ? 0.8 0.5 Huey Dewey Donald Huey Louie Donald 0.6 0.9 ?

  30. If—Then Rules Rule 2 Rule 1 If x is Neg then y is Neg If x is Pos then y is Pos approximately equal

  31. Key Concepts • Universe • Membership function • Fuzzy variables • Set operations • Fuzzy relations • All of the above are parallels to classical set theory

  32. Application examples • Database and WWW searches • Matching of buyers and sellers • Rule bases in expert systems

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