Fuzzy Sets and Rough Sets n Introduction! n History and definition ! n Fuzzy Sets ! n Membership function! n Fuzzy set operations! n Rough Sets! n Approximation! n Reduction! INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 “Fuzzifica(on is a kind of scien(fic permisiveness; it tends to result in socially appealing slogans unaccompanied by the discipline of hard work.” R. E. Kalman, 1972 INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Introduction “Number of things of the same kind, that belong together because they are similar or complementary to each other."! The Oxford English Dictionary ! Set! (collections) of various objects of interest ! Set Theory: George Cantor (1893)! completely new, elegant approach to vagueness! Fuzzy Set! an element can belong to a set to a degree k (0 ≤ k ≤ 1) ! Fuzzy Set theory: Lotfi Zadeh(1965) ! another approach to vagueness ! imprecision is expressed by a boundary region of Rough Set! a set! Rough Set Theory: Zdzisaw Pawlak(1982)! INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Lotfi Zadeh INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Introduction n Early computer science! • Not good at solving real problems! • The computer was unable to make accurate inferences! • Could not tell what would happen, give some preconditions! • Computer always seemed to need more information! INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Lotfi Zadeh n “Fuzzy Sets” paper published in 1965! n Comprehensive - contains everything needed to implement FL! n Key concept is that of membership values:! extent to which an object meets vague or imprecise properties! n Membership function: membership values over domain of interest! n Fuzzy set operations! n Awarded the IEEE Medal of Honor in 1995! INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 History for fuzzy sets and system n First fuzzy control system, work done in 1973 with Assilian (1975)! n Developed for boiler-engine steam plant! n 24 fuzzy rules! n Developed in a few days! n Laboratory-based! n Served as proof-of-concept! INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Early European Researchers Hans Zimmerman, Univ. of Aachen! •Founded first European FL working group in 1975! •First Editor of Fuzzy Sets and Systems! •First President of Int’l. Fuzzy Systems Association! ! Didier Dubois and Henri Prade in France! •Charter members of European working group! •Developed families of operators! •Co-authored a textbook (1980)! INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Early U. S. Researchers K. S. Fu (Purdue) and Azriel Rosenfeld (U. Md.) (1965-75)! ! Enrique Ruspini at SRI! •Theoretical FL foundations! •Developed fuzzy clustering! ! James Bezdek, Univ. of West Florida! •Developed fuzzy pattern recognition algorithms! •Proved fuzzy c-means clustering algorithm! •Combined fuzzy logic and neural networks! •Chaired 1st Fuzz/IEEE Conf. in 1992 and others! •President of IEEE NNC 1997-1999! INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 The Dark Age •Lasted most of 1980s! •Funding dried up, in US especially! $“...Fuzzy logic is based on fuzzy thinking. It fails to distinguish between the issues specifically addressed by the traditional "methods of logic, definition and statistical decision- making...” ! """"- J. Konieki (1991) in AI Expert! ! •Symbolics ruled: “fuzzy” label amounted to the ‘kiss of death’! INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Michio Sugeno •Secretary of Terano’s FL working group, est. in 1972! •1974 Ph.D. dissertation: fuzzy measures theory! •Worked in UK! •First commercial application of FL in Japan: control system for water purification plant (1983)! % ! INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Other Japanese Developements n 1st consumer product: shower head using FL circuitry to control temperature (1987)! n Fuzzy control system for Sendai subway (1987)! n 2d annual IFSA conference in Tokyo was turning point for FL (1987)! n Laboratory for Int’l. Fuzzy Engineering Research (LIFE) founded in Yokohama with Terano as Director, Sugeno as Leading Advisor in 1989.! INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Fuzzy Systems Theory and Paradigms n Variation on 2-valued logic that makes analysis and control of real (non-linear) systems possible! n Crisp “first order” logic is insufficient for many applications because $almost all human reasoning is imprecise! n fuzzy sets, approximate reasoning, and fuzzy logic issues and applications! INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Fuzziness is not probability • Probability is used, for example, in weather forecasting! • Probability is a number between 0 and 1 that is the certainty that an event will occur! • The event occurrence is usually 0 or 1 in crisp logic, but fuzziness says that it happens to some degree! • Fuzziness is more than probability; probability is a subset of fuzziness! • Probability is only valid for future/unknown events! • Fuzzy set membership continues after the event ! INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Probability • Probability is based on a closed world model in which it is! $assumed that everything is known! ! • Probability is based on frequency; Bayesian on subjectivity! • Probability requires independence of variables! • In probability, absence of a fact implies knowledge! ! INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Fuzzy Sets INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Fuzzy Set Membership •In fuzzy logic, set membership occurs by degree! •Set membership values are between 0 and 1! •We can now reason by degree, and apply logical operations to fuzzy sets! ! We usually write ! . µA (x) = m or, the membership value of x in the fuzzy set A is m, where 0 ≤ m ≤1 INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Fuzzy Set Membership Functions • Fuzzy sets have “shapes”: the membership values plotted versus! $the variable! ! • Fuzzy membership function: the shape of the fuzzy set over the! $range of the numeric variable! $Can be any shape, including arbitrary or irregular! $Is normalized to values between 0 and 1! $Often uses triangular approximations to save computation time! INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Fuzzy Sets Are Membership Functions from Bezdek INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Representations of Membership Functions ⎧ 0 .50 1 ⎫ TAMPBP = ⎨ + + ⎬ ⎩1.75 1.95 2.15⎭ ⎧ 0 .5 1 .5 0 ⎫ Warm = ⎨ + + + + ⎬ ⎩50 60 70 80 90⎭ −( p−80)2 / 50 µ FAIR _ PRICE (p) = e INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Two Types of Fuzzy Membership Function INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Equality of Fuzzy Sets • In traditional logic, sets containing the same members are equal:! ${A,B,C} = {A,B,C}! ! • In fuzzy logic, however, two sets are equal if and only if all! $elements have identical membership values:! $${.1/A,.6/B,.8C} = {.1/A,.6/B,.8/C}! INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Fuzzy Union • In traditional logic, all elements in either (or both) set(s)! $are included! ! • In fuzzy logic, union is the maximum set membership value! $! If mA(x) = 0.7 and mB(x) = 0.9 then mA∪B(x) = 0.9 INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Fuzzy Relations and Operators INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Summary: FUZZY SETS Membership function and Fuzzy set operations% INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Examples on fuzzy concepts Tom is rather tall, but Judy is short.! ! If you are tall, than you are quite likely heavy.! INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Examples on fuzzy concepts • The description of a human characteristic such as healthy.! • The classification of patients as depressed.! • The classification of certain objects as large.! • The classification of people by age such as old.! • A rule for driving such as &if an obstacle is close, then brake immediately".! INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Concept and set intension (内涵 )： attributes of the object! concept! extension (外延 )： all of the objects defined by the concept（ set） G. Cantor (1887) AaPa= {| ()} INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 If a fuzzy concept can be rigidly described by Cantor’s notion of sets or the bivalent (true/false or two-valued)… INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 模糊概念能否用Cantor集合来刻画？ 秃头悖论 ⼀位已经谢顶的⽼教授与他的学⽣争论他是否为秃头问题 。 教授 ： 我是秃头吗 ？ 学⽣ ： 您的头顶上已经没有多少头发 ， 确实应该说是 。 教授 ： 你秀发稠密 ， 绝对不算秃头 ， 问你 ， 如果你头上脱落了⼀根头发之后 ， 能说变成了秃头了吗 ？ 学⽣ ： 我减少⼀根头发之后 ， 当然不会变成秃头 。 教授 ： 好了 ， 总结我们的讨论 ， 得出下⾯的命题 ： ‘如果⼀个⼈不是秃头 ， 那 么他减少⼀根头发仍不是秃头 ’， 你说对吗 ？ 学⽣ ： 对 ！ 教授 ： 我年轻时代也和你⼀样⼀头秀发 ， 当时没有⼈说我秃头 ， 后来随着年龄 的增⾼ ， 头发⼀根根减少到今天的样⼦ 。 但每掉⼀根头发 ， 根据我们刚 才的命题 ， 我都不应该称为秃头 ， 这样经有限次头发的减少 ， ⽤这⼀命 题有限次 ， 结论是 ： ‘我今天仍不是秃头 ’。 INTRODUCTION TO COMPUTATIONAL INTELLIGENCE, Nanjing University Spring 2015 Postulate: If a man with n (a nature number) hairs is baldheaded, then