The Uncertainty Measure of Hierarchical Quotient Space Structure

Qinghua Zhang,Guoyin Wang

doi:10.1155/2011/513195

Abstract

In the application of fuzzy reasoning, researchers usually choose the membership function optionally in some degree. Even though the membership functions may be different for the same concept, they can generally get the same (or approximate) results. The robustness of the membership function optionally chosen has brought many researchers' attention. At present, many researchers pay attention to the structural interpretation (definition) of a fuzzy concept, and find that a hierarchical quotient space structure may be a better tool than a fuzzy set for characterizing the essential of fuzzy concept in some degree. In this paper, first the uncertainty of a hierarchical quotient space structure is defined, the information entropy sequence of a hierarchical quotient space structure is proposed, the concept of isomorphism between two hierarchical quotient space structures is defined, and the sufficient condition of isomorphism between two hierarchical quotient space structures is discovered and proved also. Then, the relationships among information entropy sequence, hierarchical quotient space structure, fuzzy equivalence relation, and fuzzy similarity relation are analyzed. Finally, a fast method for constructing a hierarchical quotient space structure is presented.

Highlights

Since the fuzzy set theory was proposed by Zadeh in 1965 1, it has been successfully applied to many application areas, such as fuzzy control, fuzzy reasoning, fuzzy clustering analysis, and fuzzy decision
It was a bridge from “fuzzy” granule world to “crisp” granule world and could better uncover the characteristics of human beings dealing with uncertain problems and better interpret the relationship between “fuzziness” and “crispness.” Many important conclusions about the fuzzy quotient space theory could be referred to 10, and the fuzzy quotient space theory for the cut relation of fuzzy equivalence relation with any threshold was discussed by Zhang et al 12
In fuzzy quotient space theory, a fuzzy equivalence relation and a hierarchical quotient space structure are one to one, and the hierarchical quotient space structure is a structural description of fuzzy equivalence relation

Summary

Introduction

Since the fuzzy set theory was proposed by Zadeh in 1965 1 , it has been successfully applied to many application areas, such as fuzzy control, fuzzy reasoning, fuzzy clustering analysis, and fuzzy decision. The structural description is more essential to a fuzzy concept than the membership function This structure is called hierarchical quotient space structure in quotient space theory developed by B. The isomorphic fuzzy equivalence relations have the same hierarchical quotient space structure 10 ; that is, they have the same classification ability to the objects in domain X. Each person has his/her own membership function for the same concept, and he/she may get the different fuzzy similarity relations, he/she may obtain the same or isomorphic fuzzy equivalent relation which can produce the same hierarchical quotient space structure and classification of objects in the domain X. What is the reason that the different fuzzy similarity relations can produce the same hierarchical quotient space structure and the same classification?

Preliminary Concepts

The Information Entropy Sequence of Hierarchical Quotient Space Structure

H I2 k k i1 al n ln al n ap n ln ap n aq n ln aq n ap n

A Fast Constructing Hierarchical Quotient Space Structure Method

Conclusions