Rough set approach to incomplete numerical data

Rough set theory has been applied successfully in many fields. However, classical rough set model can only deal with complete and symbolic data sets. Some researchers have proposed extended rough set models to handle incomplete data while others proposed extensions of classical rough set models to deal with numerical data. However, there exist many data which contain numerical attributes and missing values simultaneously. We propose, in this paper, an extended rough set model, i.e. tolerance-fuzzy rough set model to deal with this type of data characterized with numerical attributes and missing values, that is, incomplete numerical data. Discernibility matrices and discernibility functions for incomplete numerical information systems and incomplete numerical decision systems are defined to compute reducts or relative reducts. Meanwhile, the relationship between the proposed tolerance-fuzzy rough set model and the tolerance rough set model is also examined. It is shown that the tolerance-fuzzy rough set model is an extension of the tolerance rough set model. Finally, uncertainty measurement is also investigated. It is suggested that the proposed tolerance-fuzzy rough set model provide an optional approach to incomplete numerical data.