Noise-Tolerant Discrimination Indexes for Fuzzy γ Covering and Feature Subset Selection

Fuzzy
$\beta $
covering (FBC) has attracted considerable attention in recent years. Nevertheless, as the basic information granularity of FBC, fuzzy
$\beta $
neighborhood does not satisfy reflexivity, which may lead to instability in classification learning and decision-making. Although a few studies have involved reflexive fuzzy
$\beta $
neighborhoods, they only focus on a single fuzzy covering and cannot effectively deal with the information representation and information fusion of multiple fuzzy coverings. Moreover, there is a lack of investigation on noise-tolerant uncertainty measures for FBC, as well as their application in feature selection. Motivated by these issues, we investigate a noise-tolerant variable precision discrimination index (VPDI) by means of a new reflexive fuzzy covering neighborhood. To this end, fuzzy
$\gamma $
neighborhood with reflexivity is introduced to characterize the information fusion of a fuzzy covering family. An uncertainty measure called fuzzy
$\gamma $
neighborhood discrimination index is then presented to reflect the discriminatory power of fuzzy covering families. Some variants of the uncertainty measure, such as variable precision joint discrimination index, variable precision conditional discrimination index, and variable precision mutual discrimination index, are then put forth by means of fuzzy decision. These VPDIs can be used as an evaluation metric for a family of fuzzy coverings. Finally, the knowledge reduction of fuzzy covering decision systems is addressed from the point of keeping the discriminatory power, and a heuristic feature selection algorithm is designed by means of the variable precision conditional discrimination index. The experiments on 16 public datasets exhibit that the proposed algorithm can effectively reduce redundant features and achieve competitive results compared with six state-of-the-art feature selection algorithms. Moreover, it demonstrates strong robustness to the interference of random noise.