Variable precision fuzzy rough set based on relative cardinality

The fuzzy rough set approach (FRSA) is a theoretical framework that can deal with data analysis of possibilistic information systems. While a set of comprehensive rules can be induced from a possibilistic information system by using FRSA, generation of several intuitively justified rules is sometimes blocked by objects that only partially satisfy the antecedents of the rules. In this paper, we use the variable precision models of FRSA to cope with the problem. The models admit rules that are not satisfied by all objects. It is only required that the proportion of objects satisfying the rules must be above a threshold called a a precision level. In the presented models, the proportion of objects is represented as a relative cardinality of a fuzzy set with respect to another fuzzy set. We investigate three types of models based on different definitions of fuzzy cardinalities including Σ-counts, possibilistic cardinalities, and probabilistic cardinalities; and the precision levels corresponding to the three types of models are respectively scalars, fuzzy numbers, and random variables.