Attribute reduction and optimal decision rules acquisition for continuous valued information systems

For continuous valued information systems, the attribute values of objects for the same attribute represent not only their ordinal relationship but also their relative distances. Therefore, the classical rough set model is not suitable for deducing attribute reductions and optimal decision rules for continuous valued information systems. Though some discretization methods are proposed to transform the continuous valued information systems into discrete ones, but those methods are too categorical and may lead to loss of information in some cases. To solve such information loss problem, we propose a tolerance rough set model in this paper. With a given level, the proposed model can divide a universe into some maximal tolerance classes. Also two types of lower and upper approximations are defined accordingly. Then the reductions of the maximal tolerance class and optimal decision rules based on the proposed attribute descriptors are defined, and the approximate discernibility function for the maximal tolerance class is constructed and used to compute all the corresponding optimal decision rules via using Boolean reasoning techniques. Finally, the general reductions and consistent reductions for continuous valued information systems are discussed.