Optimal scale selection for multi-scale decision tables

Human beings often observe objects or deal with data hierarchically structured at different levels of granulations. In this paper, we study optimal scale selection in multi-scale decision tables from the perspective of granular computation. A multi-scale information table is an attribute-value system in which each object under each attribute is represented by different scales at different levels of granulations having a granular information transformation from a finer to a coarser labelled value. The concept of multi-scale information tables in the context of rough sets is introduced. Lower and upper approximations with reference to different levels of granulations in multi-scale information tables are defined and their properties are examined. Optimal scale selection with various requirements in multi-scale decision tables with the standard rough set model and a dual probabilistic rough set model are discussed respectively. Relationships among different notions of optimal scales in multi-scale decision tables are further analyzed.