A weighted uncertainty measure of rough sets based on general binary relation

Uncertainty measure is one of the important aspects of rough set theory. A new kind of weighted uncertainty measure called α-entropy is presented under general binary relation by considering the sample data with different importance, and some existing uncertainty measures are a special case of α-entropy by adjusting the variable parameter α. Thus it unites the corresponding uncertainty measures of complete and incomplete information systems. In addition, a well-justified uncertainty measures, α-roughness (α-accuracy) is proposed based on α-entropy. It is proved that α-roughness (α-accuracy) decreases (increases) monotonously as the information granularities become smaller. The numerical example proves that the α-accuracy and α-roughness are more reasonable and accurate than the existing methods. Finally, under general binary relation, a new heuristic weighted attribute reduction algorithm is proposed based on α-accuracy. The experiments demonstrate that the weighted measures in this paper provide a method for combining the subjective preferences and prior knowledge in uncertainty measures, and the combination classifier based on the variable parameter α can improve the accuracy of classification. These investigations developed the uncertainty theory and provide theory basis for knowledge acquisition in information systems based on general binary relation.