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0 Figure 25 function. 0 Simulation results by the learning for determining the upper Umit of a nonlinear interval 26 Hisao Ishibuchi and Manahu Nii In the fuzzy regression analysis [69-73], the fuzzy-number coefficients of the fuzzy regression model in (34) are determined by the following linear programming problem: Minimize (45) ^^{[Y(Xp)]^), p=\ subject to yp e [Y(Xp)\, /? = 1, 2 , . . , m, (46) where [•]h is the /z-level set of a fuzzy number (see Fig. 26). Because the /z-level set of a fuzzy number is a closed interval, the linear programming problem in (45) and (46) for the fuzzy regression analysis is basically the same as the problem in (36) and (37) for the interval regression analysis.

The fuzzy classification can be done by slightly modifying the aforementioned procedure. In our fuzzy classification, the cost function is modified for determining the possibility area and the necessity area of each class. For determining the possibility area of Class 1, we use the following cost function: ^^ ^{(hOpfll. , epochs), and a>(w) is a monotonically decreasing function such that 0 <&)(«)< land(w(M) ^• 0 for M -^ oo. For example, we can use the following decreasing function: (W(M) = 1 / { 1 + ( M / 1 0 0 0 ) ^ } .

27 for the case of triangular fuzzy outputs and trapezoidal fuzzy outputs. Figure 27a is depicted from the nonlinear interval function in Fig. 0 = Yi^l > X Input (a) Figure 26 (b) Illustration of the /i-level set. 0 Figure 27 Two fuzzy functions derived from the interval function in Fig. 23b. where 7(x) is a symmetric triangular fuzzy number (see Fig. 27a). On the other hand, Fig. 27b is depicted from the same nonhnear interval function by the relation: (49) where 7 (x) is a symmetric trapezoidal fuzzy number (see Fig.

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