New results on granular variable precision fuzzy rough sets based on fuzzy (co)implications

Abstract

Some fuzzy rough sets only consider relative error limit and others are still sensitive to mislabeled samples. Considering the absolute error limit, Fang and Hu [11] proposed granular variable precision fuzzy rough sets based on fuzzy (co)implications to remedy these defects, which are suitable to the databases with errors in conditional attribute(s) and decision making. However, there are some faults in the characterizations of granular variable precision fuzzy rough sets based on fuzzy (co)implications presented by Fang and Hu, such as false conclusions, insufficient and redundant conditions. In this paper, we further discuss the equivalent expressions of granular variable precision σ-lower and θ-upper approximation operators with the proper semicontinuity of fuzzy (co)implications. Based on those new equivalent expressions of granular variable precision σ-lower and θ-upper approximation operators, the composition of granular variable precision fuzzy rough sets based on fuzzy (co)implications is studied with respect to a general fuzzy relation. Moreover, we further study granular variable precision fuzzy rough sets based on fuzzy (co)implications to rectify those faults mentioned above.