The Uncertainty Analysis of Vague Sets in Rough Approximation Spaces

Abstract

Vague sets, as well as intuitionistic fuzzy sets, are extensions of fuzzy sets. Based on fuzzy sets, vague sets generalize the membership degree from a single value to an interval value. Vague sets have a more powerful ability to process fuzzy information than fuzzy sets to some degree. In addition as we all know, human cognition is usually a gradual process. As a result, in a given multi-granularity space, how to characterize a vague concept and further measure its uncertainty, become a hot issue worth studying. However, the uncertainty of vague sets in rough approximation spaces is still lacking relative studies. Therefore, in order to more effectively excavate knowledge from vague sets, this paper focuses on the uncertainty of vague sets and reveals its hidden rules. First, change rules of the average fuzziness of the vague value with changing its truth membership degree and false membership degree are discussed. Second, in rough approximation spaces, the uncertainty of vague sets, i.e., the uncertainty of average-step-vague sets are analyzed. Then, its change rules with changing granularity are analyzed and discussed. Next, to better approximate a vague concept, change rules of uncertainty for approximation sets of vague sets with changing knowledge granularity are discussed. Finally, several illustration examples are listed to verify the obtained conclusions. These rules are in accordance with human cognitive mechanisms in multi-granularity knowledge spaces.