Information Structures and Uncertainty Measures in an Incomplete Probabilistic Set-Valued Information System

An information system is a database that represents relationships between objects and attributes. A set-valued information system is the generalized model of a single-valued information system. A set-value information system that contains probability distributions and missing values is called an incomplete probability set-value information system (IPSIS). Uncertainty measure is an effective tool for evaluation. This paper explores information structures and uncertainty measures in an IPSIS. According to the Bhattacharyya distance, the distance between two objects in a given subsystem of an IPSIS is first proposed. Then, the tolerance relation on an object set, induced by a probability set-valued information system by using this distance, is obtained. Next, the information structure of this subsystem is introduced by a set vector. Moreover, the dependence between two information structures is studied by using the inclusion degree. Finally, as an application for information structures, measures of uncertainty for an IPSIS are investigated, and to evaluate the performance of the proposed measures, effectiveness analysis is given from the angle of statistics. These results will be helpful for establishing a framework of granular computing and understanding the essence of uncertainty in an IPSIS.