1 / 48

Fuzzy Rendszerek I.

1 / 1. Fuzzy Rendszerek I. Dr. K óczy László. An Example. 1 / 2. A class of students (E.G. M.Sc. Students taking „Fuzzy Theory”) The universe of discourse : X. “Who does have a driver’s licence?” A subset of X = A (Crisp) Set  (X) = CHARACTERISTIC FUNCTION.

talor
Download Presentation

Fuzzy Rendszerek I.

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 1 / 1 Fuzzy Rendszerek I. Dr. Kóczy László

  2. An Example 1 / 2 • A class of students(E.G. M.Sc. Students taking „Fuzzy Theory”) • The universe of discourse: X • “Who does have a driver’s licence?” • A subset of X = A (Crisp) Set • (X) = CHARACTERISTIC FUNCTION 1 0 1 1 0 1 1 • “Who can drive very well?” (X) = MEMBERSHIP FUNCTION 0.7 0 1.0 0.8 0 0.4 0.2 FUZZY SET

  3. 1 / 3 History of fuzzy theory • Fuzzy sets & logic: Zadeh 1964/1965- • Fuzzy algorithm: Zadeh 1968-(1973)- • Fuzzy control by linguistic rules: Mamdani & Al. ~1975- • Industrial applications: Japan 1987- (Fuzzy boom), KoreaHome electronicsVehicle controlProcess controlPattern recognition & image processingExpert systemsMilitary systems (USA ~1990-)Space research • Applications to very complex control problems: Japan 1991-E.G. helicopter autopilot

  4. 1 / 4 An application example One of the most interesting applications of fuzzy computing: “FOREX” system. 1989-1992, Laboratory for International Fuzzy Engineering Research (Yokohama, Japan) (Engineering – Financial Engineering) To predict the change of exchange rates (FOReign EXchange) ~5600 rules like:“IF the USA achieved military successes on the past day [E.G. in the Gulf War] THEN ¥/$ will slightly rise.” FOREX Inputs Prediction (Observations) Fuzzy Inference Engine

  5. ¥/$ 100 1993 1994 1995 Time 1 / 5 Another Example • What is fuzzy here? • What is the tendency of the ¥/$ exchange rate?“It’s MORE OR LESS falling” (The general tendency is “falling”, there’s no big interval of rising, etc.) • What is the current rate?Approximately 88 ¥/$  Fuzzy number • When did it first cross the magic 100 ¥/$ rate? SOMEWHEN in mid 1995

  6. 1 / 6 A complex problem Many components, very complex system. Can AI system solve it? Not, as far as we know. But WE can. Our car, save fuel, save time, etc.

  7. c• • x aAbA • b • d • y • a B Crisp set A If xA then also xB. B • y • x AB A 1 / 7 Definitions • Crisp set: • Convex set:A is not convex as aA, cA, butd=a+(1-)c A, [0, 1].B is convex as for every x, yB and[0, 1] z=x+(1-)y B. • Subset:

  8. 1 / 8 Definitions • Equal sets:If AB and BA then A=B if not so AB. • Proper subset:If there is at least one yB such that yA then AB. • Empty set: No such x. • Characteristic function:A(x): X{0, 1}, where X the universe.0 value: x is not a member,1 value: x is a member.

  9. 1 / 9 Definitions A={1, 2, 3, 4, 5, 6} • Cardinality: |A|=6. • Power set of A:P (A)={{}=Ø, {1}, {2}, {3}, {4}, {5}, {6}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {2, 3}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 5}, {3, 6}, {4, 5}, {4, 6}, {5, 6}, {1, 2, 3}, {1, 2, 4}, {1, 2, 5}, {1, 2, 6}, {1, 3, 4}, {1, 3, 5}, {1, 3, 6}, {1, 4, 5}, {1, 4, 6}, {1, 5, 6}, {2, 3, 4}, {2, 3, 5}, {2, 3, 6}, {2, 4, 5}, {2, 4, 6}, {2, 5, 6}, {3, 4, 5}, {3, 4, 6}, {3, 5, 6}, {4, 5, 6}, {1, 2, 3, 4}, {1, 2, 3, 5}, {1, 2, 3, 6}, {1, 2, 4, 5}, {1, 2, 4, 6}, {1, 2, 5, 6}, {1, 3, 4, 5}, {1, 3, 4, 6}, {1, 3, 5, 6}, {1, 4, 5, 6}, {2, 3, 4, 5}, {2, 3, 4, 6}, {2, 3, 5, 6}, {2, 4, 5, 6}, {3, 4, 5, 6}, {1, 2, 3, 4, 5}, {1, 2, 3, 4, 6}, {1, 2, 4, 5, 6}, {1, 3, 4, 5, 6}, {2, 3, 4, 5, 6}, {1, 2, 3, 4, 5, 6}}. |P (A)|=2|6|=64.

  10. 1 / 10 Definitions • Relative complement or difference:A–B={x | xA and xB}B={1, 3, 4, 5}, A–B={2, 6}.C={1, 3, 4, 5, 7, 8}, A–C={2, 6}! • Complement: where X is the universe.Complementation is involutive:Basic properties: • Union:AB={x | xA or xB} For (Law of excluded middle)

  11. 1 / 11 Definitions • Intersection:AB={x | xA and xB}. For • More properties:Commutativity: AB=BA, AB=BA.Associativity: ABC=(AB)C=A(BC), ABC=(AB)C=A(BC).Idempotence: AA=A, AA=A.Distributivity: A(BC)=(AB)(AC), A(BC)=(AB)(AC). (Law of contradiction)

  12. 1 / 12 Definitions • More properties (continued):DeMorgan’s laws: • Disjoint sets: AB=. • Partition of X:

  13. 1 / 13 Summarize properties

  14. 1 / 14 Membership function

  15. 1 / 15 Membership function (2) • More general Idea:L-Fuzzy set, A: XL, L has a linear (full) or partial ordering. • Vector valued fuzzy set:A: x[0, 1]n=[0,1]…[0,1] (n times). • Fuzzy measure:g: P (x)[0, 1]Degree of belonging to a crisp subset of the universe. Example:’teenager’=13..19 years old, ‘twen’=20..29 years old,‘in his thirties’=30..39 years old, etc.X={Years of men’s possible age}‘teenager’Xg(‘Susan is a twen’)=0.8,g(‘Susan is a teenager’)=0.2,g(‘Susan is in her thirties’)=0.1.Best guess has highest measure.

  16. mm 2A 1 m 1 0 X X1 X2 1 / 16 Definitions • Fuzzy set of type 2: • Level N:

  17. 1 / 17 Definitions • Interval valued fuzzy set: even • C(Y) is the family of convex subsets of Y, i.e. P(Y) • C([0,1]) is the family of all intervals • e.g.: m 1 0 X X

  18. L-Fuzzy Type N Type N-1 Type 2 Interval V’D Fuzzy Set Vector V’D Crisp Set 1 / 18 Definitions • Relation of various classes of sets discussed

  19. 1 / 19 Some basic concepts of fuzzy sets

  20. 1 / 20 Some basic concepts of fuzzy sets • Support: supp(A)={x | A(x)>0}.supp:Infant=0, so supp(Infant)=0.If |supp(A)|<, A can be definedA=1/x1+ 2/x2+…+ n/xn. • Kernel (Nucleus, Core):Kernel(A)={x | A(x)=1}.

  21. 1 / 21 Definitions • Height: • height(old)=1height(infant)=0 • If height(A)=1 A is normal • If height(A)<1 A is subnormal • height(0)=0 • a-cut: Strong Cut: • Kernel: • Support: • If A is subnormal, Kernel(A)=0

  22. 1 / 22 Definitions • Level set of A: • Convex fuzzy set: • A is convex if for every x,yX and [0,1]: • All sets on the previous figure are convex FS’s

  23. x2 Kernel =0.2 =0.4 =0.8 =0.6 x1 1 / 23 Definitions • Nonconvex fuzzy set: • Convex fuzzy set over R 2

  24. 1 / 24 Definitions • Fuzzy cardinality of FS A:Fuzzy numberfor all A

  25. N 1 0 r1 r* r2  N(r) 1 0  3 6 9 1 / 25 Definitions • Fuzzy number: Convex and normal fuzzy set of  • Example 1: height(N(r))=1 for any r1, r2: N(r*)= N(r1+(1- )r2)min(N(r1), N(r2)) • Example 2: “Approximately equal to 6” m

  26. 1 / 26 Definitions • Flat fuzzy number: There is a,b (ab, a,b) N(r)=1 IFF r[a,b] (Extension of ‘interval’) • Containment (inclusion) of fuzzy set AB IFF mA(x) mB(x) • Example:Old Adult • Equal fuzzy sets A=B IFF AB and ABIf it is not the case: AB • Proper subset AB IFF AB and A  B • Example:Old Adult

  27. 1 / 27 Definitions • Extension principle: How to generalize crisp concepts to fuzzy? Suppose that X and Y are crisp sets X={xi}, Y={yi} and f: XY If given a fuzzy set The latter is understood that if for

  28. 1 / 28 Definitions • Example X={a,b,c} Y={d,e,f} • Arithmetics with fuzzy numbers:Using extension principle a+b=c (a,b,c  ) A+B=C

  29. A B C 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12  1 / 29 Example

  30. 1 / 30 Definitions • Fuzzy set operations defined by L.A. Zadeh in 1964/1965 • Complement: • Intersection: • Union: (x):

  31. 1 / 31 Definitions This is really a generalization of crisp set op’s!

  32. 1 / 32 Classical (Two valued) logic • Classical (two valued logic):Any logic – Means of a reasoning propositions: true (1) or false (0). Propositional logic uses logic variables, every variable stands for a proposition. When combining LV’s new variables (new propositions) can be constricted.Logical function:f: v1, v2, …, vn vn+1 • different logic functions of n variables exist. E.G. if n=2, 16 different logic functions (see next page).

  33. 1 / 33 Logic functions of two variables

  34. 1 / 34 Definitions • Important concern:How to express all logic fn’s by a few logic primitives (fn’s of one or two lv’s)? A system of lp’s is (functionally) complete if all LF’s can be expressed by the FNs in the system. A system is a base system if it is functionally complete and when omitting any of its elements the new system isn’t functionally complete.

  35. 1 / 35 Definitions • A system of LFN’s is functionally complete if: • At least one doesn’t preserve 0 • At least one doesn’t preserve 1 • At least one isn’t monotonic • At least one isn’t self dual • At least one isn’t linear • Example: AND , NOT

  36. 1 / 36 Definitions • Importance of base systems, and functionally complete systems.Digital engineering:Which set of primitive digital circuits is suitable to construct an arbitrary circuit? • Very usual: AND, OR, NOT (Not easy from the point of view of technology!) • NAND (Sheffer stroke) • NOR (Pierce function) • NOT, IMPLICATION (Very popular among logicians.)

  37. 0 and 1 are LFs • If v is a LF is a LF 1 / 37 Definitions • Logic formulas: • E.G.: Let’s adopt +,-,* as a complete system. Then: • Similarly with -,, etc. • There are infinitely many ways to describe a LF in an equivalent way • E.G.: A, , A+A, A*A, A+A+A, etc. • Canonical formulas, normal form • DCF • CCF • Logic formulas with identical truth values are named equivalent • If A and B are LFs, then A+B, A*B are LFs • There are no other LFs a=b

  38. 1 / 38 Definitions • Always true logic formulas: Tautology: AA+A • Always false logic formulas: Contradiction: A • Various forms of tautologies are used for didactive inference = inference rules • Modus Ponens:(a*(ab))b • Modus Tollens: • Hypothetical Syllogism:((a b)*(b c)) (a c) • Rule of Substitution:If in a tautology any variable isreplaced by a LF then it remains a tautology.

  39. 1 / 39 Definitions • These inference rules form the base of expert systems and related systems (even fuzzy control!) • Abstract algebraic model:Boolean AlgebraB=(B,+,*,-)B has at least two different elements (bounds): 0 and 1+ some properties of binary OPs “+” and “*”, and unary OP “-”.

  40. 1 / 40 Properties of boolean algebras Lattice

  41. 1 / 41 Definitions • Correspondences defining isomorphisms between set theory, boolean algebra and propositional logic

  42. Lattice B.A. 1 / 42 Definitions • Isomorphic structure of crisp set and logic operations = boolean algebra • Structure of propositions: x is P • x2=Dr.Kim P(x2)=FALSE!

  43. 1 / 43 Definitions • Quantifiers: • (x) P(x): There exists an x such that x is P • (x) P(x): For all x, x is P • (!x) P(x): x and only one x such that x is P

  44. 1 / 44 Definitions • Two valued logic questioned since B.C.Three valued logic includes indeterminate value: ½Negation: 1-a, differ in these logics. • Examples:

  45. 1 / 45 Definitions • No difference from classical logic for 0 and 1. But: are not true! • Quasi tautology: doesn’t assume 0. Quasi contradiction: doesn’t assume 1. • Next step?

  46. 1 / 46 N-valued logic • N-valued logic: Degrees of truth (Lukasiewicz, ~1933) LOGIC PRIMITIVES

  47. 1 / 47 Definitions • Ln n=2, … ,   (0) Classical LogicIf T is extended to [0,1] we obtain (T1) L1 with continuum truth degrees • L1 is isomorphic with fuzzy setIt is enough to study one of them, it will reveal all the facts above the other. • Fuzzy logic must be the foundator of approximate reasoning, based on natural language!  Rational truth values

  48. 1 / 48 Fuzzy Proportion • Fuzzy proportion: X is P‘Tina is young’, where:‘Tina’: Crispage, ‘young’: fuzzy predicate. Fuzzy sets expressing linguistic terms for ages Truth claims – Fuzzy sets over [0, 1] • Fuzzy logic based approximate reasoning • is most important for applications!

More Related