Integrated measures for rough sets based on general binary relations

Uncertainty measures are important for knowledge discovery and data mining. Rough set theory (RST) is an important tool for measuring and processing uncertain information. Although many RST-based methods for measuring system uncertainty have been investigated, the existing measures cannot adequately characterise the imprecision of a rough set. Moreover, these methods are suitable only for complete information systems, and it is difficult to generalise methods for complete information systems to incomplete information systems. To overcome these shortcomings, we present new uncertainty measures, integrated accuracy and integrated roughness, that are based on general binary relations, and we study important properties of these measures. A theoretical analysis and examples show that the proposed integrated measures are more precise than existing uncertainty measures, they are suitable for both complete and incomplete information systems, and they are logically consistent. Therefore, integrated accuracy and integrated roughness overcome the limitations of existing measures. This research not only develops the theory of uncertainty, it also expands the application domain of uncertainty measures and provides a theoretical basis for knowledge acquisition in information systems based on general binary relations.