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

TL;DR

This paper introduces a new class of fuzzy numbers derived from probability density functions and nonlinear mappings, establishing their algebraic operations and demonstrating their interpretability with numerical examples.

Contribution

It constructs a novel fuzzy number class based on combining subjective perception functions with objective PDFs, and defines their arithmetic operations within a linear algebra framework.

Findings

Existence of a function pair (h, p) for observed outcomes

Triangular fuzzy numbers can be represented by specific (h, p) pairs

Defined arithmetic operations create a linear algebra structure

Abstract

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 $X_{h, p}$, 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 $X_{h, p}$ 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 $X_{h, p}$. We claim that, under our definitions, $X_{h, p}$ has a linear algebra. Some numerical examples are provided to illustrate the proposed approach.