Maximal consistent block based optimal scale selection for incomplete multi-scale information systems

Numerical incomplete data are common in many real-life applications, and they are often hierarchically structured at different levels of granularity. A numerical incomplete multi-scale information system (NIMIS) is a special hierarchical data set in which each object can take on different values at different scales. In such a data set, an important issue is to choose the optimal scale in order to maintain certain conditions for final decision. In this paper, by employing maximal consistent block technique, we study the optimal scale selection with various requirements in NIMISs and numerical incomplete multi-scale decision systems (NIMDSs). We first introduce the concept of scale in NIMISs and NIMDSs. We then define the optimal scale and the maximal consistent block based optimal scale. Finally, we examine the relationship between the maximal consistent block based optimal scale and the optimal scale. We show that the maximal consistent block based optimal scale and the optimal scale are equivalent for both NIMISs and consistent NIMDSs. And in inconsistent NIMDSs, there is no static relationship between notions of the maximal consistent block based lower-approximation optimal scale and the upper-approximation optimal scale.