Discernibility Measures for Fuzzy β Covering and Their Application

As a combination of fuzzy sets and covering rough sets, fuzzy $\beta $ covering has attracted much attention in recent years. The fuzzy $\beta $ neighborhood serves as the basic granulation unit of fuzzy $\beta $ covering. In this article, a new discernibility measure with respect to the fuzzy $\beta $ neighborhood is proposed to characterize the distinguishing ability of a fuzzy covering family. To this end, the parameterized fuzzy $\beta $ neighborhood is introduced to describe the similarity between samples, where the distinguishing ability of a given fuzzy covering family can be evaluated. Some variants of the discernibility measure, such as the joint discernibility measure, conditional discernibility measure, and mutual discernibility measure, are then presented to reflect the change of distinguishing ability caused by different fuzzy covering families. These measures have similar properties as the Shannon entropy. Finally, to deal with knowledge reduction with fuzzy $\beta $ covering, we formalize a new type of decision table, that is, fuzzy $\beta $ covering decision tables. The data reduction of fuzzy covering decision tables is addressed from the viewpoint of maintaining the distinguishing ability of a fuzzy covering family, and a forward attribute reduction algorithm is designed to reduce redundant fuzzy coverings. Extensive experiments show that the proposed method can effectively evaluate the uncertainty of different types of datasets and exhibit better performance in attribute reduction compared with some existing algorithms.