A class of fuzzy numbers induced by probability density functions and their arithmetic operations

In this paper we are interested in a class of fuzzy numbers which is uniquely identified by their membership functions. The function space, denoted by Xh,p , will be constructed by combining a class of nonlinear mappings h (subjective perception) and a class of probability density functions (PDF) p (objective entity), respectively. Under our assumptions, we prove that there always exists a class of h to fulfill the observed outcome for a given class of p. Especially, we prove that the common triangular number can be interpreted by a function pair (h, p). As an example, we consider a sample function space Xh,p where h is the tangent function and p is chosen as the Gaussian kernel with free variable & mu;. By means of the free variable & mu; (which is also the expectation of p(x; & mu;)), we define the addition, scalar multiplication and subtraction on Xh,p. We claim that, under our definitions, Xh,p has a linear algebra. Some numerical examples are provided to illustrate the proposed approach.& COPY; 2023 Elsevier B.V. All rights reserved.

Automated summary

\ This paper focuses on a specific class of fuzzy numbers that can be uniquely identified by their membership functions. The authors construct a function space, denoted as Xh,p, by combining a set of nonlinear mappings h that represents subjective perception, and a set of probability density functions p that represent objective entities. Under their assumptions, they prove the existence of a class of h functions that can accurately predict the observed outcomes based on a given class of p functions. They also demonstrate that the commonly used triangular number can be interpreted using a function pair (h, p). They provide a numerical example in which h is the tangent function and p is the Gaussian kernel with a free variable μ, and show that Xh,p exhibits linear algebra properties under their defined operations.