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EMU, COMPUTER ENGINEERING DEPARTMENT 1999/2000 ACADEMIC YEAR, SPRING SEMESTER

EMU, COMPUTER ENGINEERING DEPARTMENT 1999/2000 ACADEMIC YEAR, SPRING SEMESTER. C M P E 586 Software Implementation of Fuzzy Systems PREPARED BY DR. KONSTANTIN DEGTIAREV FEBRUARY/JUNE 2000 Slides use the material of books and journal papers.

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EMU, COMPUTER ENGINEERING DEPARTMENT 1999/2000 ACADEMIC YEAR, SPRING SEMESTER

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  1. EMU, COMPUTER ENGINEERING DEPARTMENT1999/2000 ACADEMIC YEAR, SPRING SEMESTER C M P E 586 SoftwareImplementationof FuzzySystems PREPARED BY DR. KONSTANTIN DEGTIAREV FEBRUARY/JUNE 2000 Slides use the material of books and journal papers

  2. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 1 CMPE 586 Software Implementationof FuzzySystems  Reference • G.J.Klir, U.H.St.Clair, Bo Yuan. Fuzzy Set Theory. Foundations & Applications, Prentice Hall PTR, 1997 • B.Kosko. Fuzzy Engineering, Prentice Hall, 1997 • T.J.Ross. Fuzzy Logic with Engineering Applications, McGraw-Hill, 1995 • J.Yen, R.Langari. Fuzzy Logic. Intelligence, Control, and Information, Prentice Hall, 1999 • L.-X.Wang. A Course in Fuzzy Systems and Control, Prentice Hall, 1997 • W.Pedrycz (ed.). Fuzzy Modelling. Paradigms and Practice (Int. Series in Intelligent Technologies), Kluwer Academic Publ., 1996 • J.Yen. Fuzzy Logic - A Modern Perspective // IEEE Transactions on Knowledge and Data Engineering, vol.11, #1, January/February 1999 • L.A.Zadeh. The Birth and Evolution of Fuzzy Logic // Int. Journalon General Systems, vol.17, 1990, pp.95-105 1

  3. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 2 CMPE 586 Software Implementationof FuzzySystems  Section1 Outline • Introduction. Uncertainty, Imprecission and Vagueness • Fuzzy Systems. Brief History of Fuzzy Logic. Foundation of Fuzzy Theory. • Fuzzy Sets and Systems. Fuzzy Systems in Commercial Products • Research fields in Fuzzy Theory [ the discussion of these topics takes approximately 4 lecture hours. One example is explained (CubiCalc and fuzzyTECHsoftware packages are used) ] 2

  4. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 3 CMPE 586 Software Implementationof FuzzySystems • Most of the phenomena we encounter everyday are imprecise - the imprecision may be associated with their shapes, position, color, texture, semantics that describe what they are • Fuzziness primarily describes uncertainty (partial truth) and imprecision • The key idea of fuzziness comes from the multivalued logic: Everything is a matter of degree • Imprecision raises in several faces, e.g. as a semantic ambiguity 3

  5. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 4 CMPE 586 Software Implementationof FuzzySystems • By fuzzifying crisp data obtained from measurements, FL enhances the robustness of a system • Imprecision raises in several faces - for example, as a semantic ambiguity the statement “the soup is HOT” is ambiguous, but not fuzzy e.g. [20º,80º] Definition of the domain of discourse Transaction to Fuzziness 4

  6. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 5 CMPE 586 Software Implementationof FuzzySystems • The word “fuzzy” can be defined as “imprecisely defined, confused, vague” • Humans represent and manage natural language terms (data) which are vague. Almost all answers to questions raised in everyday life are within some proximity of the absolute truth 5

  7. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 6 CMPE 586 Software Implementationof FuzzySystems • Probability theory is one of the most traditional theories for representing uncertainty in mathematical models • Nature of uncertainty in a problem is a point which should be clearly recognized by engineer - there is uncertainty that arises from chance, from imprecision, from a lack of knowledge, from vagueness, from randomness… • probability theory deals with the expectation of an event (future event, its outcome is not known yet), i.e. it is a theory of random events 6

  8. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 7 CMPE 586 Software Implementationof FuzzySystems • Fuzziness deals with the impression of meaning of concepts expressed in natural language - it is not concerned with events at all • Fuzzy theory handles nonrandom uncertainty 7

  9. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 8 CMPE 586 Software Implementationof FuzzySystems • As it is stated by L.Zadeh, “in many cases there is more to be gained from cooperation than from arguments over which methodology is best…” • Many situations cover both kinds of uncertainty: assume the weather forecast - “tomorrow slight rains are highly probable” 8

  10. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 9 CMPE 586 Software Implementationof FuzzySystems • The principle of incompatibility (L.Zadeh, 1973): “As the complexity of a system increases, our ability to make precise and yet significant statements about its behavior diminishes until a threshold is reached beyond which precision and significance (or relevance) become almost mutually exclusive characteristics” 9

  11. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 10 CMPE 586 Software Implementationof FuzzySystems • Intimateconnectionbetweenfuzzinessandcomplexity (L.A.Zadeh) • anew approach to system analysis: approximate and yet effective means of describing the behavior of systems which aretoo complexor tooill-definedto admit ofprecise mathematical analysis 10

  12. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 11 CMPE 586 Software Implementationof FuzzySystems • A new approach to system analysis: a departure from the conventional quantitative techniques of system analysis • A new paradigm: to develop approximate solutions that are both cost-effective and highly useful • a Fuzzy System (FS) is defined as a system with operating principles based on fuzzy information processing and decision making • There are several ways to represent knowledge, but the most commonly used has a form of rules: IF (premise)A THEN (conclusion)B 11

  13. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 12 CMPE 586 Software Implementationof FuzzySystems • From a knowledge representation viewpoint, a fuzzy IF-THEN rule is a scheme for capturing knowledge that involves imprecision - if we know a premise (fact), then we can infer another fact (conclusion) • A fuzzy system (FS) is constructed from a collection of fuzzy IF-THEN rules • Acquisition of knowledge captured in IF-THEN rules is NOT a trivial task (expert knowledge, systems measurements, etc.) • The building blocks for fuzzy IF-THEN rules are FUZZY SETS 12

  14. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 13 CMPE 586 Software Implementationof FuzzySystems • The rule “IF the air is coolTHEN set the motor speed to slow” has a form: IFx is ATHENy is B, where fuzzy sets “cool” and “slow” are labeled by A and B, correspondingly • A and B characterize fuzzy propositions about variables x and y • Most of the information involved in human communication uses natural language terms that are often vague, imprecise, ambiguous by their nature, and fuzzy sets can serve as the mathematical foundation of natural language 13

  15. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 14 CMPE 586 Software Implementationof FuzzySystems • A Fuzzy Set is a set with a smooth boundaries • Fuzzy Set Theory generalizes classical set theory to allow partial membership • Fuzzy Set A is a universal set U is determined by a membership functionA(x) that assigns to each element xU a number A(x) in the unit interval [0,1] • Universal set U (Universe of Discourse) contains all possible elements of concern for a particular application • Fuzzy set has a one-to-one correspondence with its membership function 14

  16. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 15 CMPE 586 Software Implementationof FuzzySystems • Fuzzy set A is defined as A = { (x, A(x)) }, xU, A(x)[0,1] • A(x) = Degree(xA) is a grade of membership of element xU in set A 15

  17. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 16 CMPE 586 Software Implementationof FuzzySystems • The membership functions themselves are NOT fuzzy - they are precise mathematical functions; once a fuzzy property is represented by a membership function, nothing is fuzzy anymore • Suppose U is the interval [0,85] representing the age of ordinary human beings, and the linguistic term “young” as a function of age (value of the variable age) can be defined as [see the graphical representation on the next slide] [ !! pay attention to the usage of the symbol “ / “ ] 16

  18. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 17 CMPE 586 Software Implementationof FuzzySystems Universe of discourse U is continuos • If U is a set of integers from 1 to 10 ( U={1,2,…,10} ), then “small” is a fuzzy subset of U, and it can be defined using enumeration (summation notation): A = “small” = 1/1+1/2+0.85/3+0.75/4+0.5/5+0.3/6+0.1/7 17

  19. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 18 CMPE 586 Software Implementationof FuzzySystems • In the previous example elements of U (universal set) with zero membership degrees are not included into enumeration • A notion of a fuzzy set provides a convenient way of defining abstraction - a process which plays a basic role in human thinking and communication • All theories that use the basic concept of fuzzy set can be called in a whole Fuzzy Theory • Rough classification of Fuzzy Theory can be depicted as follows [note that dependencies between the branches are not shown] : 18

  20. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 19 CMPE 586 Software Implementationof FuzzySystems 19

  21. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 20 CMPE 586 Software Implementationof FuzzySystems • The idea of Fuzzy Sets appeared in 1964 : L.A.Zadeh (Professor of the University of California at Berkeley): “We need a radically different kind of mathematics, the mathematics of fuzzy or cloudy quantities which are not described in terms of probability distributions…” • The paper “Fuzzy Sets” (Zadeh L.A., Information and Control, vol.8, pp.338-353, 1965) first used the word “fuzzy” to mean “vague” in technical literature • criticized by academic community the idea caused a development of fuzzy set theory foundation (1965-1980) • academic research work stimulates first industrial applications of fuzzy systems (1977-1990) - cement kiln controller (Denmark), train control system (Sendai subway, Japan), digital and analog fuzzy chips (USA, Japan) 20

  22. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 21 CMPE 586 Software Implementationof FuzzySystems • Currently, the application fields of fuzzy systems cover signal processing, communications, expert systems, medicine, business/finance, control (industrial processes and consumer electronics), … • widening of collaboration between universities and industry, “fuzzy boom” (1987-present): Japan Europe  USA • 1992: 1st IEEE International Conference on Fuzzy Systems • appearance of software companies (INFORM, Aptronix,etc.) • Fuzzy Logic Toolbox for MATLAB was released in 1994 • Courses on fuzzy sets and systems in Universities curricula “Engineering consists largely of recommending decisions based on insufficient information.... It is essential that these students be exposed to ways of treating uncertainty and vagueness. This also requires that existing faculty utilize these methods…” (Colin Brown, conference of NAFIPS) 21

  23. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 22 CMPE 586 Software Implementationof FuzzySystems • Appearance of the new computational paradigms and intensification of research in certain areas (genetic algorithms/evolutionary strategies, neural networks) • L.A.Zadeh introduced a term soft computing (1992) ------------EXAMPLE 1 ------------- Fuzzy Toolbox Demo (MATLAB) by Dr.R.Babuška (Delft University of Technology, The Netherlands) If-Then Rules. Fuzzy reasoning (example) Word 97 document (preliminary explanations) * * * * * end of the Section 1 * * * * * 22

  24. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 23 CMPE 586 Software Implementationof FuzzySystems  Section 2 Outline • Mathematical Background of Fuzzy Systems. Classical (crisp) vs. Fuzzy Sets. Representation of Fuzzy Sets • Types of Membership Functions. Basic concepts (support, singleton, height, -cut, convexity). Fuzzy Set Operations • S- and T-norms. Properties of Fuzzy Sets. Sets as points in Hypercubes. Cartesian Product. Crisp and Fuzzy Relations • Linguistic variables and hedges. Membership function design (shape analysis) [ the discussion of these topics takes approximately 10 lecture hours. Examples are explained using CubiCalc, fuzzyTECHand FL Toolbox packages] 23

  25. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 24 CMPE 586 Software Implementationof FuzzySystems • Fuzzy SystemsF:npusemrules to map vector inputxto vector or scalar outputsF(x) • Fuzzy (Rule-based) Systems make use oflinguistic variablesin their antecedents and consequents • Linguistic variables can be naturally represented byfuzzy setsand logical connectives of these sets input X 24

  26. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 25 CMPE 586 Software Implementationof FuzzySystems • A classical (crisp) set A in the universe of discourse U can be defined in three ways: - by enumerating (listing) elements (often called list or extensional definition) - by specifying the common properties of elements (intensional or rule definition) the notation A = {x | P(x)} means that set A is composed of elements x such that everyx has the propertyP(x) - by introducing a zero-one membership function (characteristic or indicator definition) 25

  27. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 26 CMPE 586 Software Implementationof FuzzySystems • Crisp set is a set with precise boundary, and classical set theory is founded on the idea that we can make crisp, exact distinctions between two groups, i.e. between those individuals (elements) that are definitely in the result set (group 1), and those that are definitely outside it (group 2) • The basic operations on classical sets (A and B are crisp sets in the universe of discourse U): • complement divides the universal set U into 2 (two) parts 26

  28. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 27 CMPE 586 Software Implementationof FuzzySystems • Fundamental properties of the basic operations (these properties are also encountered in propositional logic): 27

  29. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 28 CMPE 586 Software Implementationof FuzzySystems • Fundamental properties satisfy to a principle of duality: replacing of empty set, U, ,  with U, empty set, , , respectively, brings again valid property • The notion of membership in fuzzy sets becomes a matter of degree (number in the closed interval [0,1]) • Membership of an element from the universe in fuzzy set is measured by a function that attempts to describe vaguenessand ambiguity 28

  30. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 29 CMPE 586 Software Implementationof FuzzySystems • Membership functions can be represented (a)graphically, (b) in a tabular or list form, (c )analytically and (d)geometrically (as a points in the unit cube) • Geometrical representation for two-element universal set U= ({x1,x2}) has a following vizualization: 29

  31. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 30 CMPE 586 Software Implementationof FuzzySystems • [see the previous figure] Vertices (0,0), (0,1), (1,0) and (1,1) represent all crisp sets that can be defined for the universal set U, e.g. the point (1,0) corresponds to the crisp set {x1} (element x2 has no membership) • Membership functions can be symmetrical or asymmetrical, and the most commonly used forms are triangular, trapezoidal, Gaussian and bell (the first two dominate in applications due to simplicity and computational efficiency) • Membership functions are typically defined on one-dimensional universes, and in most cases, the membership function appears in the continuos form Fuzzy Toolbox Demo (MATLAB) by Dr.R.Babuška (Delft University of Technology) FuzzyTECHand CubiCalc (explanations) 30

  32. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 31 CMPE 586 Software Implementationof FuzzySystems • The height of a fuzzy set A is the highest (maximum) value of its membership function, i.e. height(A) = • If a fuzzy set has a height 1, then it is called a normal fuzzy set; in contrast, if height(A) < 1, the fuzzy set is said to be subnormal • A subnormal set is a fuzzy set that contains only elements with partial (<1) membership • In most of applications fuzzy sets are normal, and during the reasoning process usually subnormal fuzzy sets are generated • A set of all elements of the universal set U whose degree of membership in a fuzzy set A is nonzero is called the support of a fuzzy set A, i.e. supp(A) = 31

  33. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 32 CMPE 586 Software Implementationof FuzzySystems • A set of all elements x of the universal set U with a property A(x) = 1 (A is a fuzzy set) is called the core of a fuzzy set A (core(A)) 32

  34. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 33 CMPE 586 Software Implementationof FuzzySystems • A fuzzy set whose support is a single point in the universe of discourse U is called a fuzzy singleton • Each fuzzy set A is associated with a family of crisp subsets of A - their elements have such membership degrees that they are restricted to a crisp subset of [0,1] • A crisp set A that contains those xU for which is called an -cutof a fuzzy set A • The general property of -cuts: for any fuzzy set A and two values 1, 2 [0,1] that satisfy to the condition 1< 2 the following is true: 33

  35. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 34 CMPE 586 Software Implementationof FuzzySystems • Fuzzy sets may be completely characterized by their -cuts: (decomposition theorem of fuzzy sets) Example(Lecture hours) 34

  36. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 35 CMPE 586 Software Implementationof FuzzySystems • Consider a fuzzy set A which is represented analytically in the universe of discourse U = [5,15] as follows: Example 35

  37. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 36 CMPE 586 Software Implementationof FuzzySystems • step 1: Several particular values of  are chosen from the unit interval [0,1] - they are 0.1, 0.3, 0.5, 0.7 and 0.9 • step 2: converting each of the -cutsA to fuzzy sets for each xU using the formula: (fuz_set) = A(x) Sometimes the theorem is referred as resolutionprinciple (approximate representation of membership function): Example 36

  38. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 37 CMPE 586 Software Implementationof FuzzySystems • Fuzzy set A is convex if for any elements x1, x2 and x3 from the universal set U, the relation x1< x2< x3 implies that • General property: the intersection of two convex sets produces a convex set  Convexity and -cuts: 37

  39. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 38 CMPE 586 Software Implementationof FuzzySystems • Generalization of set operations to fuzzy sets is not obvious • Operations on fuzzy sets are crucial to the fuzzy inference process • In the rule IF(A or B)THENC the true value of C is the true value of the disjunction (operation or) • Assume two fuzzy setsA and B are defined on the universe of discourse U - three basic operations can be represented as follows: • Fuzzy set A is equal to fuzzy set B if and only if A(x) = B(x), xU 38

  40. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 39 CMPE 586 Software Implementationof FuzzySystems • Fuzzy sets overlap with their complements (an element may partially belong to both fuzzy set and set’s complement) • In contrast, classical (crisp) sets never overlap with their complements 39

  41. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 40 CMPE 586 Software Implementationof FuzzySystems • Two fundamental laws of Classical Set theory - law of Excluded Middle and law of Contradiction  are violated in Fuzzy Set Theory(!!) • Standard fuzzy operations are quite adequate in many practical applications of FS, but they do not utilize the real expressive power of fuzzy sets (what are the other possibilities that may satisfy the requirements of practice?) • In practice, algebraic sum(1’) and algebraic product(2’) are used for a definition of union and intersection of two fuzzy sets, respectively: 40

  42. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 41 CMPE 586 Software Implementationof FuzzySystems • General notation: • Operator s is called an s-norm if it satisfies to the following axioms for any x, y, z and w[0,1]: • Some of the operators (s-norms) that “model” (i.e. extend) fuzzy union: see the next slide... 41

  43. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 42 CMPE 586 Software Implementationof FuzzySystems (S1)Drastic sum: (S2)Hamacher sum: (S3)Dubois-Prade class: (S4)Yager class: • Note: for arbitrary fuzzy setsA and B membership values x and y stand for A(x) and B(x), correspondingly Continuation: t-norms (triangular norms) 42

  44. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 43 CMPE 586 Software Implementationof FuzzySystems • Operator t is called an t-norm (triangular norm) if it satisfies to the following axioms for any x, y, z and w[0,1]: • Some of the operators (t-norms) that “model” (extend) fuzzy intersection: (T1)Drastic product: (T2)Hamacher product: More still to come... 43

  45. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 44 CMPE 586 Software Implementationof FuzzySystems (T3)Dubois-Prade class: (T4)Yager class: Self-studying exercise: Prove that the Yager t-norm (class T4) converges to the min operator when the parameter  is in the infinite limit : • An important properties of s-norms and t-norms can be summarized as follows: 44

  46. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 45 CMPE 586 Software Implementationof FuzzySystems • s-norms are bounded below by max (standard fuzzy union) and bounded above by drastic sum (S1): • t-norms are bounded below by drastic product (T1) and bounded above by min (standard fuzzy intersection): • s-norms (a set of fuzzy disjunction operators) are often called triangular conorms or shortly, t-conorms • The alternative forms of operators AND and OR are called compensatory operators (they compensate the strictness of min and max operators proposed by L.A.Zadeh)   45

  47. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 46 CMPE 586 Software Implementationof FuzzySystems • Operator c is called a fuzzy complement if it satisfies to the following axioms for any x and y[0,1]: • Some of the operators that “model” (extend) fuzzy complement: (C1)Sugeno’s complement: (C2)Yager’s complement: Demonstration (MATLAB environment) 46

  48. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 47 CMPE 586 Software Implementationof FuzzySystems • Main types of membership functions (MF): (a)Triangular MF is specified by 3 parameters {a,b,c}: (b)Trapezoidal MF is specified by 4 parameters {a,b,c,d}: More to come... 47

  49. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 48 CMPE 586 Software Implementationof FuzzySystems (c)Gaussian MF is specified by 2 parameters {a,}: (d)Bell-shaped MF is specified by 3 parameters {a,b,}: (e)Sigmoidal MF is specified by 2 parameters {a,b}: 48

  50. Prepared Dr.Konstantin Degtiarev, February-June 2000 Slide 49 CMPE 586 Software Implementationof FuzzySystems Image form “Neuro-Fuzzy and Soft Computing” (J.-S.R.Jang, C.-T.Sun, E.Mizutanani - supplementary slides 49

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