General approximation of fuzzy numbers by F-transform

A note on fuzzy-transform approximation of fuzzy numbers

“…By (8) we easily get that F k > 0 and hence ∈ (d, b]. We will use a similar reasoning as in the proof of Theorem 12 (ii) in [6]. As h ≤ d−c 4 , there exist k 1 , k 2 ∈ {1, ..., n − 1}…”

“…We also note that in the above two theorems the order of uniform approximation is the same as in the case of the general approximation of fuzzy numbers by using the extended iF-transform proposed in [6].…”

“…We end this section by mentioning that using similar reasonings as in [6] it can be proved that if u is strictly increasing on [a, c] then u is strictly increasing on [a, c ] and if u is strictly decreasing on [d, b] then u is strictly decreasing on [d , b]. We also note that in the above two theorems the order of uniform approximation is the same as in the case of the general approximation of fuzzy numbers by using the extended iF-transform proposed in [6].…”

“…In a recent paper (see [6]), we have proposed a general procedure to approximate fuzzy numbers by using the extended monotonic F −transform setting proposed in [14] and we have seen that the extended inverse F -transform of a fuzzy number will always preserve the support and the quasiconcavity property; but in general, a normalization of the approximating function is required.…”

Comparison of T-Norms and S-Norms for Interval Type-2 Fuzzy Numbers in Weight Adjustment for Neural Networks

“…[25] outlined a method for approximating trapezoidal fuzzy numbers in comparison with other methods of approximation. Coroianu L. et al [26] described the inverse F-transform to accomplish optimal fuzzy numbers, maintaining the support and the convergence of the core. Yang D. et al [27] presented an interval and modified interval neuron perceptron with interval weights and biases, and modified the learning algorithm for this approach.…”