A threshold-based similarity relation under incomplete information

The conventional rough set theory based on complete information systems stems from the observation that objects with the same characteristics are indiscernible according to available information. Although rough sets theory has been applied in many fields, the use of the indiscernibility relation may be too rigid in some real situations. Therefore, several generalizations of the rough set theory have been proposed some of which extend the indiscernibility relation using more general similarity or tolerance relations. In this paper, after discussing several extension models based on rough sets for incomplete information, a novel relation based on thresholds is introduced as a new extension of the rough set theory, the upper-approximation and the lower approximation defined on this relation are proposed as well. Furthermore, we present the properties of this extended relation. The experiments show that this relation works effectively, in incomplete information and generates rational object classification.