Entropy based optimal scale selection and attribute reduction in multi-scale interval-set decision tables

One of the current research gaps in multi-scale data analysis is studying information systems characterized by attributes with interval sets as attribute values and multiple scales. To address this gap, we first introduce the concepts of a multi-scale interval-set information system (MISIS) and a multi-scale interval-set decision table (MISDT). We then define the similarity relation between objects in an MISIS and the corresponding rough approximations. We further propose the positive region optimal scale, the modified conditional entropy optimal scale, and the positive complementary conditional entropy optimal scale in an MISDT. We examine the relationships among these optimal scales in consistent and inconsistent MISDTs and show that the positive region optimal scale and the modified conditional entropy optimal scale are equivalent in a consistent MISDT, while in an inconsistent MISDT, the positive region optimal scale is the same as the positive complementary conditional entropy optimal scale, and the modified conditional entropy optimal scale is not greater than the positive complementary conditional entropy optimal scale. Based on the optimal scale, we also develop attribute reduction approaches in MISDTs. Finally, through experimental analysis of data on the UCI dataset, we verify the effectiveness and reasonableness of our proposed methods.