1. CHAPTER 3 Selected Design and Processing Aspects of Fuzzy Sets

2. The Development of Fuzzy Sets: Elicitation of Membership Functions Elicitation of membership functions is of significant relevance to conceptual and algorithmic developments of fuzzy sets. A number of general approaches: – horizontal approach; – vertical approach; – Saaty priority method (analytical hierarchy process, AHP); – fuzzy clustering.

3. The Development of Fuzzy Sets: Elicitation of Membership Functions Semantics of fuzzy sets – some general observations: – Fuzzy sets as meaningful (semantically sound) constructs; – The number of fuzzy sets used to describe some variable (construct) limited to 7+/-2 terms (membership functions); – Fuzzy sets require calibration – adjustment of membership functions depending on the context in which fuzzy sets are used.

4. The Development of Fuzzy Sets: Elicitation of Membership Functions Fuzzy set as a descriptor of feasible solutions The intent is to describe a collection of solutions to a given optimization problem by characterizing then through degrees of feasibility as optimal solutions. – Determine maximum of F where F assumes positive values a collection of solutions and their global characterization as a fuzzy set

5. The Development of Fuzzy Sets: Elicitation of Membership Functions Fuzzy set as a descriptor of feasible solutions – Determine minimum of F a collection of solutions and their global characterization as a fuzzy set

6. The Development of Fuzzy Sets: Elicitation of Membership Functions Fuzzy set as a descriptor of feasible solutions If F assumes real numbers, then – For the maximization problem

7. The Development of Fuzzy Sets: Elicitation of Membership Functions Fuzzy set and a notion of typicality Issue of gradual typicality captured through membership degrees In geometry: ideal geometric figures (circle, ellipse, square...) Perception of geometry of ellipsoide: (a) higher differences |a-b|, less typicality of the figure (b) ratios a/b and the departure from an ideal shape where a/b=1

8. The Development of Fuzzy Sets: Elicitation of Membership Functions Horizontal scheme of membership function estimation – Identify a collection of elements of the universe of discourse X and query a panel of n experts: does x belong to concept A? – Count the number of ‘yes” responses (p) and calculate the ratio of p/n.

9. The Development of Fuzzy Sets: Elicitation of Membership Functions Horizontal scheme of membership function estimation – Membership value – the ratio of p/n. – Standard deviation of the membership estimate and associated confidence interval determined as [p-σ, p+σ].

10. The Development of Fuzzy Sets: Elicitation of Membership Functions Example: Horizontal scheme of membership function estimation

11. The Development of Fuzzy Sets: Elicitation of Membership Functions Vertical scheme of membership function estimation Determination of successive α-cuts and a formation of fuzzy set – Query a panel of n experts: what are the elements of X which belong to fuzzy set A at a degree not lower than α?

12. The Development of Fuzzy Sets: Elicitation of Membership Functions Saaty’s priority approach to membership function estimation Determination of membership function through a series of pairwise comparisons of elements of X with regard to their preference vis-a-vis a given concept – fuzzy set. Consider that for elements X 1, X 2, …, X n, we have the membership grades A(X 1 ), A(X 2 ), …, A(X n ). Organize them in a form of a reciprocal matrix:

13. The Development of Fuzzy Sets: Elicitation of Membership Functions Saaty’s priority approach to membership function estimation Properties of reciprocal matrices: – The diagonal values are equal to 1; – Entries symmetrically positioned with respect to the diagonal satisfy condition if multiplicative reciprocality that is M(X k,X l )=1/ M(X l,X k ). – Transitive property: M(X k,X l ) M(X k,X l )= M(X k,X l ) for all indexes i, j, k.

14. The Development of Fuzzy Sets: Elicitation of Membership Functions Saaty’s priority approach to membership function estimation Eigenvectors of reciprocal matrix – The i-th element of the above vector is equal to nA(X i ). – Overall MA=nA – A is the eigenvector of M associated with the largest eigenvalue of M equal to “n”.

15. The Development of Fuzzy Sets: Elicitation of Membership Functions Saaty’s priority approach to membership function estimation From reciprocal matrix to fuzzy set: – Construct a reciprocal matrix based on expert's pairwise comparisons – Use of scale of relative importance

16. The Development of Fuzzy Sets: Elicitation of Membership Functions Saaty’s priority approach to membership function estimation Experimentally constructed reciprocal matrix may not satisfy transitivity condition resulting in some level of inconsistency. Inconsistency index – = 0, if max =n; – >0, lack of consistency.

17. The Development of Fuzzy Sets: Elicitation of Membership Functions Principle of justifiable granularity Experimental data captured in a form of a certain information granule – fuzzy sets. The resulting fuzzy set is required to satisfy: – Sufficient level of experimental evidence – to be as high as possible. – Sufficient specificity – to be as high as possible. These two requirements are in conflict.

18. The Development of Fuzzy Sets: Elicitation of Membership Functions Principle of justifiable granularity: development strategy Reconciling conflicting criteria: Cover most data Make fuzzy set specific enough – minimize support of A, |m-a|

19. The Development of Fuzzy Sets: Elicitation of Membership Functions Design of fuzzy sets through fuzzy clustering: Fuzzy C-Means Grouping n-dimensional data located in R n into c clusters – fuzzy sets so that an objective function becomes minimized. Notation: – U- partition matrix – v 1, v 2, …, v c – prototypes – ||.|| - distance function – m- fuzzification coefficient

20. The Development of Fuzzy Sets: Elicitation of Membership Functions Design of fuzzy sets through fuzzy clustering: Fuzzy C-Means Partition matrix – two properties

21. The Development of Fuzzy Sets: Elicitation of Membership Functions FCM – minimization problem Minimize objective function with respect to: – prototypes; – partition matrix. Use of Lagrange multipliers in the optimization with respect to partition matrix. Prototypes easily determined when using Euclidean distance function.

22. The Development of Fuzzy Sets: Elicitation of Membership Functions FCM – impact of fuzzification coefficient of geometry clusters By changing the values of m (>1) a shape of membership functions becomes affected.

23. The Development of Fuzzy Sets: Elicitation of Membership Functions Separation measure for fuzzy clusters By quantifying among clusters, an extent of their separation is expressed

24. The Development of Fuzzy Sets: Elicitation of Membership Functions Fuzzy equalization Select membership functions A 1, A 2, …, A c in such a way so that their expected values are made equal for all fuzzy sets, that is

25. The Development of Fuzzy Sets: Elicitation of Membership Functions Design of membership functions: main guidelines – Highly visible, well-articulated semantics, keep the number of terms in the range 7+/-2. – Different views at fuzzy sets associated with their underlying estimation techniques. – Fuzzy sets are context-dependent and require calibration (when applied to a certain problem). – Two main categories of estimation techniques - data–driven and user- driven. Also some hybrid approaches are anticipated.

26. Aggregation Operations Aggregation operator regarded as a mapping satisfying conditions: – monotonicity g(x 1, x 2,…, x n ) > g(y 1, y 2,…, y n ), if x i > y j – boundary conditions g(0, 0, ….., 0) = 0 and g(1, 1, ….., 1) = 1

27. Aggregation Operations Averaging operators - generalized mean A class of operators in the form:

28. Aggregation Operations Examples of generalized means

29. Transformations of Fuzzy Sets Extension principle Different ways of mapping input (number, set, fuzzy set) through a given function f

30. Transformations of Fuzzy Sets Transformations of numeric argument Transformation of a point through a function

31. Transformations of Fuzzy Sets Transformations of sets Transformation of a given set through function f B = f(A) = {y  Y| y = f(x),  x  A}

32. Transformations of Fuzzy Sets Transformations of fuzzy sets Transformation of a given fuzzy set through function f

33. Transformations of Fuzzy Sets Transformations of fuzzy sets: a multivariate case Transformation of a collection of fuzzy set through function f:

34. Transformations of Fuzzy Sets Fuzzy numbers and fuzzy arithmetic Fuzzy numbers and fuzzy intervals satisfy the conditions of: – normality; – unimodality; – continuity; – boundness of support.

35. Transformations of Fuzzy Sets Examples: Information granules of numbers, intervals, fuzzy intervals and fuzzy numbers

36. Transformations of Fuzzy Sets Examples: Consider that you traveled for 2 hours at speed of about 110 km/hr. What was the distance you traveled? The speed is described in the form of some fuzzy set S whose membership function is given. The next example is a more general version of the above problem. You traveled at speed of about 110 km/hr for about 3 hours. What was the distance traveled? We assume that both the speed and time of travel are described by fuzzy sets. In a certain manufacturing process, there are five operations completed in series. Given the nature of the manufacturing activities, the duration of each of them can be characterized by fuzzy sets T 1, T 2,…, and T 5. What is the time of realization of this process?

37. Transformations of Fuzzy Sets Interval arithmetic and  -cuts Basic arithmetic operations on intervals: –addition: [a,b] + [c,d] = [a + c, b + d] –subtraction: [a,b] - [c,d] = [a - d, b - c ] –multiplication: [a,b].[c,d] = [min(ac, ad, bc, bd), max(ac, ad, bc, bd)] –division:

38. Transformations of Fuzzy Sets Interval arithmetic and  -cuts Use α-cuts for operation A*B (A  B)  = A   B  and then combine the obtained results by taking a union of the obtained α- cuts

39. Transformations of Fuzzy Sets Fuzzy arithmetic and extension principle The membership function of A*B is given in the form Considering some t-norm, one has Depending on t-norm (minimum, drastic product) the following inequality holds

40. Transformations of Fuzzy Sets Examples: Fuzzy arithmetic Depending on t-norm used, different membership functions of the result A+B are obtained

41. Transformations of Fuzzy Sets Fuzzy arithmetic: a fundamental result

42. Transformations of Fuzzy Sets Computing with triangular fuzzy numbers Algebraic operations on triangular fuzzy numbers produce interesting and practically relevant results. Consider two triangular fuzzy numbers

43. Transformations of Fuzzy Sets Computing with triangular fuzzy numbers: addition In calculations, we consider separately increasing and decreasing segments of the membership functions of A and B This leads to an interesting result - the sum is a triangular fuzzy number with the membership function:

44. Transformations of Fuzzy Sets Computing with triangular fuzzy numbers: multiplication As before we consider separately increasing and decreasing segments of the membership functions of A and B. For the increasing parts of the membership functions: x=(m-a)α+ay=(n-c) α+c z=xy=[(m-a) α+a][(n-c) α+c] The result is not a triangular fuzzy number.

45. Transformations of Fuzzy Sets Computing with triangular fuzzy numbers: division As before we consider separately increasing and decreasing segments of the membership functions of A and B. For the increasing parts of the membership functions x=(m-a)α+ay=(n-c) α+c The result is not a triangular fuzzy number.