Advanced Computing Seminar Data Mining and Its Industrial Applications — Chapter 7 — Fuzzy Sets

1. Advanced Computing Seminar Data Mining and Its Industrial Applications— Chapter 7 —Fuzzy Sets Zhongzhi Shi, Markus Stumptner, Yalei Hao, Gerald Quirchmayr Knowledge and Software Engineering Lab Advanced Computing Research Centre School of Computer and Information Science University of South Australia Chap7 Fuzzy Set and Logic Zhongzhi Shi

2. Outline • Introduction • Fuzzy Sets • Fuzzy Logic • Fuzzy Clustering • Fuzzy C-Means Clustering • Summary Chap7 Fuzzy Set and Logic Zhongzhi Shi

3. Introduction • Idea of Fuzzy Chap7 Fuzzy Set and Logic Zhongzhi Shi

4. Introduction • What does it offer? • generate precise solutions from certain or approximate information • While other approaches require accurate equations to model real-world behaviors, fuzzy design can accommodate the ambiguities of real-world human language and logic. • It provides both an intuitive method for describing systems in human terms and automates the conversion of those system specifications into effective models. Chap7 Fuzzy Set and Logic Zhongzhi Shi

5. Introduction • Aristotle • "Law of the Excluded Middle," • every proposition must either be True or False, A or not-A • Ex., rose is either red or not red • It cannot be red and not red • Plato • there was a third region (beyond True and False) where these opposites "tumbled about. Chap7 Fuzzy Set and Logic Zhongzhi Shi

6. Introduction • In the early 1900s, Lukasiewicz • described a three-valued logic, which can best be translated as the term `possible', and assigned it a numeric value between True and False. • Knuth, a former student of Lukasiewicz • proposed a three-valued logic apparently missed by Lukasiewicz, whiched used an integral range [-1, 0 +1] rather than [0, 1, 2]. Chap7 Fuzzy Set and Logic Zhongzhi Shi

7. Introduction • 1960‘s • Lotfi A. Zadeh, a professor of UC Berkeley • Observed that conventional computer logic was incapable of manipulating data representing subjective or vague human ideas such as "an atractive person" or "pretty hot". • Fuzzy logic, hence was designed to allow computers to determine the distinctions among data with shades of gray, similar to the process of human reasoning. Chap7 Fuzzy Set and Logic Zhongzhi Shi

8. Introduction • In 1965, • Zadeh published his seminal work "Fuzzy Sets“ • Described the mathematics of fuzzy set theory, and by extension fuzzy logic. • This theory proposed making the membership function (or the values False and True) operate over the range of real numbers [0.0, 1.0]. Chap7 Fuzzy Set and Logic Zhongzhi Shi

9. Introduction • US and certain parts of Europe ignored it, fuzzy logic was excepted with open arms in Japan, China and most Oriental countries. • The world's largest number of fuzzy researchers are in China with over 10,000 scientists. • The popularity of fuzzy logic in the Orient reflects the fact that Oriental thinking more easily accepts the concept of "fuzziness". Chap7 Fuzzy Set and Logic Zhongzhi Shi

10. Outline • Introduction • Fuzzy Sets • Fuzzy Logic • Fuzzy Clustering • Fuzzy C-Means Clustering • Summary Chap7 Fuzzy Set and Logic Zhongzhi Shi

11. Fuzzy Sets • Universal Set X – always a crisp set. • Crisp set assigns value {0,1} to members in X • Fuzzy set assigns value [0,1] to members in X • These values are called the membership functions m. • Membership function of a fuzzy set A is denoted by : A: X  [0,1] A: [x1/m1, x2/m2, …, xn/mn} Chap7 Fuzzy Set and Logic Zhongzhi Shi

12. Fuzzy Sets • Fuzzy sets • How fuzzy sets quantifying the same information can describe this natural drift. Chap7 Fuzzy Set and Logic Zhongzhi Shi

13. Fuzzy Sets • Let's talk about people and "youthness“ • Set S (the universe of discourse) is the set of people • Fuzzy subset YOUNG • "to what degree is person x young?" • To each person in the universe of discourse, we have to assign a degree of membership in the fuzzy subset YOUNG. • A membership function based on the person's age • young(x) = { 1, if age(x) <= 20,(30-age(x))/10, if 20 < age(x) <= 30,0, if age(x) > 30 } Chap7 Fuzzy Set and Logic Zhongzhi Shi

14. Fuzzy Sets • Graph of membership function • Example: the degree of truth of the statement "Parthiban is YOUNG" is 0.50 Chap7 Fuzzy Set and Logic Zhongzhi Shi

15. Fuzzy Sets Crisp set A 1.0 5’10’’ Heights • Sets with fuzzy boundaries A = Set of tall people Fuzzyset A 1.0 .9 Membership function .5 5’10’’ 6’2’’ Heights

16. Membership Functions (MFs) • Characteristics of MFs: • Subjective measures • Not probability functions “tall” in Asia MFs .8 “tall” in the US .5 “tall” in NBA .1 5’10’’ Heights

17. Fuzzy Sets • Formal definition: A fuzzy set A in X is expressed as a set of ordered pairs: Membership function (MF) Universe or universe of discourse Fuzzy set A fuzzy set is totally characterized by a membership function (MF).

18. Fuzzy Sets with Discrete Universes • Fuzzy set C = “desirable city to live in” X = {SF, Boston, LA} (discrete and nonordered) C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)} • Fuzzy set A = “sensible number of children” X = {0, 1, 2, 3, 4, 5, 6} (discrete universe) A = {(0, .1), (1, .3), (2, .7), (3, 1), (4, .6), (5, .2), (6, .1)}

19. Fuzzy Sets with Cont. Universes • Fuzzy set B = “about 50 years old” X = Set of positive real numbers (continuous) B = {(x, mB(x)) | x in X}

20. Alternative Notation • A fuzzy set A can be alternatively denoted as follows: X is discrete X is continuous Note that S and integral signs stand for the union of membership grades; “/” stands for a marker and does not imply division.

21. Fuzzy Partition • Fuzzy partitions formed by the linguistic values “young”, “middle aged”, and “old”: lingmf.m

22. Support Core Normality Crossover points Fuzzy singleton a-cut, strong a-cut Convexity Fuzzy numbers Bandwidth Symmetricity Open left or right, closed More Definitions

23. MF Terminology MF 1 .5 a 0 Core X Crossover points a - cut Support

24. Convexity of Fuzzy Sets • A fuzzy set A is convex if for any l in [0, 1], Alternatively, A is convex is all its a-cuts are convex. convexmf.m

25. Set-Theoretic Operations • Subset: • Complement: • Union: • Intersection:

26. Set-Theoretic Operations subset.m fuzsetop.m

27. MF Formulation • Triangular MF: Trapezoidal MF: Gaussian MF: Generalized bell MF:

28. MF Formulation disp_mf.m

29. MF Formulation • Sigmoidal MF: Extensions: • Abs. difference • of two sig. MF • Product • of two sig. MF disp_sig.m

30. MF Formulation • L-R MF: Example: c=65 a=60 b=10 c=25 a=10 b=40 difflr.m

31. Cylindrical Extension Base set A Cylindrical Ext. of A cyl_ext.m

32. 2D MF Projection Two-dimensional MF Projection onto X Projection onto Y project.m

33. Fuzzy Complement • General requirements: • Boundary: N(0)=1 and N(1) = 0 • Monotonicity: N(a) > N(b) if a < b • Involution: N(N(a) = a • Two types of fuzzy complements: • Sugeno’s complement: • Yager’s complement:

34. Fuzzy Complement Sugeno’s complement: Yager’s complement: negation.m

35. Fuzzy Intersection: T-norm • Basic requirements: • Boundary: T(0, 0) = 0, T(a, 1) = T(1, a) = a • Monotonicity: T(a, b) < T(c, d) if a < c and b < d • Commutativity: T(a, b) = T(b, a) • Associativity: T(a, T(b, c)) = T(T(a, b), c) • Four examples (page 37): • Minimum: Tm(a, b) • Algebraic product: Ta(a, b) • Bounded product: Tb(a, b) • Drastic product: Td(a, b)

36. T-norm Operator Algebraic product: Ta(a, b) Bounded product: Tb(a, b) Drastic product: Td(a, b) Minimum: Tm(a, b) tnorm.m

37. Fuzzy Union: T-conorm or S-norm • Basic requirements: • Boundary: S(1, 1) = 1, S(a, 0) = S(0, a) = a • Monotonicity: S(a, b) < S(c, d) if a < c and b < d • Commutativity: S(a, b) = S(b, a) • Associativity: S(a, S(b, c)) = S(S(a, b), c) • Four examples (page 38): • Maximum: Sm(a, b) • Algebraic sum: Sa(a, b) • Bounded sum: Sb(a, b) • Drastic sum: Sd(a, b)

38. T-conorm or S-norm Algebraic sum: Sa(a, b) Bounded sum: Sb(a, b) Drastic sum: Sd(a, b) Maximum: Sm(a, b) tconorm.m

39. Fuzzy Variables • Variables whose states are defined by linguistic concepts like low, medium, high. • These linguistic concepts are fuzzy sets themselves. Very high Very Low Med ium Low High Membership Temperature Trapezoidal membership functions Chap7 Fuzzy Set and Logic Zhongzhi Shi

40. Outline • Introduction • Fuzzy Sets • Fuzzy Logic • Fuzzy Clustering • Fuzzy C-Means Clustering • Summary Chap7 Fuzzy Set and Logic Zhongzhi Shi

41. Fuzzy Logic • Fuzzy logic is a superset of conventional(Boolean) logic • It extended to handle the concept of partial truth- truth values between "completely true" and "completely false". • It is the logic underlying modes of reasoning which are approximate rather than exact. • The importance of fuzzy logic derives from the fact that most modes of human reasoning and especially common sense reasoning are approximate in nature. Chap7 Fuzzy Set and Logic Zhongzhi Shi

42. Fuzzy Logic • Essential characteristics of fuzzy logic: • In fuzzy logic, exact reasoning is viewed as a limiting case of approximate reasoning. • In fuzzy logic everything is a matter of degree. • Any logical system can be fuzzified (thus, define Boolean logic as a subset of Fuzzy logic ) • In fuzzy logic, knowledge is interpreted as a collection of elastic or, equivalently , fuzzy constraint on a collection of variables • Inference is viewed as a process of propagation of elastic constraints. Chap7 Fuzzy Set and Logic Zhongzhi Shi

43. Fuzzy Logic • Bivalent sets • To characterize the temperature of a room Chap7 Fuzzy Set and Logic Zhongzhi Shi

44. Fuzzy logic • Limitations of bivalent sets • mutually exclusive • it is not possible to have membership of more than one set ( opinion would widely vary as to whether 50 degrees Fahrenheit is 'cold' or 'cool' hence the expert knowledge we need to define our system is mathematically at odds with the humanistic world). • Clearly, it is not accurate to define a transition from a quantity such as 'warm' to 'hot' by the application of one degree Fahrenheit of heat. • In the real world a smooth drift from warm to hot would occur. Chap7 Fuzzy Set and Logic Zhongzhi Shi

45. Fuzzy Logic • Fuzzy Set Operations—Union • The membership function of the Union of two fuzzy sets A and B with membership functions A and B respectively is defined as the maximum of the two individual membership functions. This is called the maximum criterion. • Union operation in Fuzzy set theory is the equivalent of the OR operation in Boolean algebra Chap7 Fuzzy Set and Logic Zhongzhi Shi

46. Fuzzy logic • Union—maximum criterion Chap7 Fuzzy Set and Logic Zhongzhi Shi

47. Fuzzy Logic • Fuzzy Set Operations—Union • The membership function of the Union of two fuzzy sets A and B with membership functions A and B respectively is defined as the maximum of the two individual membership functions. This is called the maximum criterion. • Union operation in Fuzzy set theory is the equivalent of the OR operation in Boolean algebra Chap7 Fuzzy Set and Logic Zhongzhi Shi

48. Fuzzy Logic • Union—maximum criterion Chap7 Fuzzy Set and Logic Zhongzhi Shi

49. Fuzzy Loic • Fuzzy Set Operations—Intersection • The membership function of the Intersection of two fuzzy sets A and B with membership functions A and B is defined as the minimum of the two individual membership functions. This is called the minimum criterion. • The Intersection operation in Fuzzy set theory is the equivalent of the AND operation in Boolean algebra. Chap7 Fuzzy Set and Logic Zhongzhi Shi

50. Fuzzy logic • Intersection—minimum criterion Chap7 Fuzzy Set and Logic Zhongzhi Shi