Approximations of fuzzy numbers by trapezoidal fuzzy numbers preserving the ambiguity and value
ABSTRACT Value and ambiguity are two parameters which were introduced to represent fuzzy numbers. In this paper, we find the nearest trapezoidal approximation and the nearest symmetric trapezoidal approximation to a given fuzzy number, with respect to the average Euclidean distance, preserving the value and ambiguity. To avoid the laborious calculus associated with the Karush–Kuhn–Tucker theorem, the working tool in some recent papers, a less sophisticated method is proposed. Algorithms for computing the approximations, many examples, proofs of continuity and two applications to ranking of fuzzy numbers and estimations of the defect of additivity for approximations are given.
- SourceAvailable from: Luciano Stefanini
- "where a ≤ c ≤ d ≤ b ∈ R, u L : [a, c] −→   is a nondecreasing upper semicontinuous function, u L (a) = 0, u L (c) = 1, called the left side of the fuzzy number and u R : [d, b] −→   is a nonincreasing upper semicontinuous function, u R (d) = 1, u R (b) = 0, called the right side of the fuzzy number. "
Conference Paper: A Note on Fuzzy-Transform Approximation of Fuzzy Numbers[Show abstract] [Hide abstract]
ABSTRACT: A possibilistic representation of fuzzy num-bers/intervals can be obtained in terms of a fuzzy distribution function (the average of the possibility and necessity functions). The fuzzy distribution function is monotonic non decreasing upper-semicontinuous and there exists a simple one-to-one correspondence between the space of such functions and the space of fuzzy numbers, i.e., with normal, upper semicontinuous quasi concave membership function. As a consequence, the monotonic F-transform approximation of the fuzzy distribution function produces an approximation of any fuzzy number. Properties and examples are given.NAFIPS2015; 08/2015
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- "Woxman  extends this canonical representation to discrete fuzzy numbers. A more recent paper that discusses such problems is given in . They obtain the nearest trapezoidal approximation and nearest symmetric trapezoidal approximation to a given fuzzy number using the average Euclidean distance, preserving the value and ambiguity. "
ABSTRACT: This paper focuses on modelling fuzzy numbers with meaningful membership functions. More precisely, it proposes a method to construct trapezoidal fuzzy number approximations from raw discrete data. In many applications, input information is numerical, and therefore, particular fuzzy sets, such as fuzzy numbers, hold great interest and relevance in managing data imprecision and vagueness. The proposed technique provides an efficient way to obtain trapezoidal numbers using linear regression. The technique is simple, fast, and effective. Preliminary tests are performed using different types of input data: a Gaussian function, a Sigmoidal function, three datasets of synthetic discrete data, and an histogram obtained from a colour satellite image.Information Sciences 06/2013; 233:1–14. DOI:10.1016/j.ins.2013.01.023 · 4.04 Impact Factor
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- "In  some connections between trapezoidal approximation and aggregation were discussed. Other aspects of the trapezoidal approximation and its generalizations could be found in        . "
ABSTRACT: The problem of the nearest approximation of fuzzy numbers by piecewise linear 1-knot fuzzy numbers is discussed. By using 1-knot fuzzy numbers one may obtain approximations which are simple enough and flexible to reconstruct the input fuzzy concepts under study. They might be also perceived as a generalization of the trapezoidal approximations. Moreover, these approximations possess some desirable properties. Apart from theoretical considerations approximation algorithms that can be applied in practice are also given.Fuzzy Sets and Systems 02/2013; 233:26-51. DOI:10.1016/j.fss.2013.02.005 · 1.99 Impact Factor