1. PART 7 Constructing Fuzzy Sets 1. Direct/one-expert 2. Direct/multi-expert 3. Indirect/one-expert 4. Indirect/multi-expert 5. Construction from samples FUZZY SETS AND FUZZY LOGIC Theory and Applications

2. 22 Direct/one-expert An expert is expected to assign to each given element x a membership grade A(x) that, according to his or her opinion, best captures the meaning of the linguistic term represented by the fuzzy set A. It can be done by either 1.defining the membership function completely in terms of a justifiable mathematical formula, 2.exemplifying it for some selected elements of X.

3. 3 Direct/multi-expert When a direct method is extended from one expert to multiple experts, the opinions of individual experts must be appropriately aggregated. One of the most common methods is based on a probabilistic interpretation of membership functions. 3

4. 4 Direct/multi-expert "x belongs to A" is either true or false, where A is a fuzzy set on X that represent a linguistic term associated with a given linguistic variable. let a i (x) denote the answer of expert i. Assume that a i (x) = 1 when the proposition is valued by expert i as true, and a i (x) = 0 when it is valued as false. n: number of expects

5. 5 Direct/multi-expert Generalize the interpretation A(x) by allowing one to distinguish degrees of competence, c i, of the individual experts. where

6. 6 Direct/multi-expert Let A and B are two fuzzy sets, defined on the same universal set X. We can calculate A(x) and B(x) for each x X, and then choose appropriate fuzzy operators to calculate,, A U B, A ∩ B, and so forth. Let

7. 7 Direct/multi-expert and

8. 8 Indirect/one-expert Given a linguistic term in a particular context, let A denote a fuzzy set that is supposed to capture the meaning of this term. Let x 1, …,n be elements of the universal set X for which we want to estimate the grades of membership in A. 8

9. 9 Indirect/one-expert Our problem is to determine the values a i = A(x i ). Instead of asking the expert to estimate values a i directly. We ask him or her to compare elements x 1, …,n in pairs according to their relative weights of belonging to A.

10. 10 Indirect/one-expert pairwise comparisons A square matrix P = [p ij ], i,j N n, which has positive entries everywhere. Assume first that it is possible to obtain perfect values p ij. In this case, p ij = a i / a j ; and matrix P is consistent in the sense that for all i, j, k N n, which implies that p ii = 1 and p ij = 1/ p ji.

11. 11 Indirect/one-expert Furthermore, for all i N n or, in matrix form, where

12. 12 Indirect/one-expert Pa = na means that n is an eigenvalue of P and a is the corresponding eigenvector. It also can be rewritten in the form where I is the identity matrix.

13. 13 Indirect/one-expert If we assume that then a j for any j N n can be determined by the following simple procedure: hence,

14. 14 Indirect/one-expert The problem of estimating vector a from matrix P now becomes the problem of finding the largest eigenvalue λ max and the associated eigenvector. That is, the estimated vector a must satisfy the equation Pa = λ max a, where λ max is usually close to n.

15. 15 Indirect/multi-expert Let us illustrate methods in this category by describing an interesting method, which enables us to determine degrees of competence of participating experts. It is based on the assumption that, in general, the concept in question is n-dimensional (based on n distinct features), each defined on R. Hence, the universal set on which the concept is defined is R n. 15

16. 16 Indirect/multi-expert The full opinion of expert i regarding the relevance of elements (n-tuples) of R n to the concept is expressed by the hyperparallelepiped Where, denote the interval of values of feature ; that, in the opinion of expert i, relate to the concept in question (i N m, j N n ).

17. 17 Indirect/multi-expert We obtain m hyperparallelepipeds of this form for m experts. Membership function of the fuzzy set by which the concept is to be represented is then constructed by the following algorithmic procedure:

18. 18 Indirect/multi-expert

19. 19 Indirect/multi-expert

20. 20 Indirect/multi-expert

21. 21 Indirect/multi-expert

22. 22 Construction from samples Lagrange Interpretation A curve-fitting method in which the constructed function is assumed to be expressed by a suitable polynomial form. The function f employed for the interpolation of given sample data for all x R has the form 22

23. 23 Construction from samples for all i N n. Since values f (x) need not be in [0,1] for some x R, function f cannot be directly considered as the sought membership function A. We may convert f to A for each x by the formula

24. 24 Construction from samples An advantage of this method is that the membership function matches the sample data exactly. Its disadvantage is that the complexity of the resulting function (expressed by the degree of the polynomial involved) increases with the number of data samples.

25. 25 Construction from samples

26. 26 Construction from samples

27. 27 Construction from samples

28. 28 Construction from samples Least-square curve fitting The method of least-square curve fitting selects that function f (x: α 0, β 0, ) from the class for which reaches its minimum. Then, for all x R. 28

29. 29 Construction from samples An example of the class of bell-shaped functions is frequently used for this purpose. where α controls the position of the center of the bell, (β / 2 ) -2 defines the inflection points, and γ control the height of the bell (Fig. 10.4a). 29

30. 30 Construction from samples Given sample data, we determine (by any effective optimization method) values α 0, β 0, γ 0 of parameters α, β, γ, respectively, for which reaches its minimum. Then, according to A(x), the bell-shape membership function A that best conforms to the sample data is given by the formula for all x R. 30

31. 31 Construction from samples Another class of functions that is frequently used for representing linguistic terms is the class of trapezoidal-shaped functions, The meaning of the five parameters is illustrated in Fig. 10.4b. 31

32. 32 Construction from samples 32

33. 33 Construction from samples 33

34. 34 Construction from samples 34

35. 35 Construction from samples

36. 36 Construction from samples 36

37. 37 Construction from samples 37

38. 38 Construction from samples Neural networks 38

39. 39 Construction from samples Following the backpropagation learning algorithm, we first initialize the weights in the network. This means that we assign a small random number to each weight. Then, we apply pairs 〈 x p, t p 〉 of the training set to the learning algorithm in some order. 39

40. 40 Construction from samples For each x p, we calculate the actual output y p and calculate the square error Using E p, we update the weights in the network according to the backpropagation algorithm described in Appendix A. We also calculate a cumulative cycle error, 40

41. 41 Construction from samples At the end, we compare the cumulative error with the largest acceptable error, E max, specified by the user. – E ≦ E max : the neural network represents the desired membership function. – E ＞ E max : we initiate a new cycle. The algorithm is terminated when either we obtain a solution or the number of cycles exceeds a number specified by the user. 41

42. 42 Construction from samples 42

43. 43 Construction from samples 43

44. 44 Construction from samples 44

45. 45 Construction from samples

46. 46 Construction from samples

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48. 48 Construction from samples

49. 49 Construction from samples

50. 50 Construction from samples

51. 51 Construction from samples

52. 52 Construction from samples

53. 53 Exercise 7 7.1 7.2 7.3 7.4 53