Uncertainty measurement for incomplete set-valued data with application to attribute reduction

A set-valued information system (SVIS) is the generalization of a single-valued information system. A SVIS with missing information values is called an incomplete set-valued information system (ISVIS). This paper focuses on studying uncertainty measurement for an ISVIS with application to attribute reduction. First, the similarity degree between information values on each attribute is presented in an ISVIS. Then, the tolerance relation induced by each subsystem is given and rough approximations based on this relation is considered. Next, some tools to measure the uncertainty of an ISVIS are put forwarded. Moreover, the validity of the proposed measures is analyzed from the statistical point of view. Finally, information granulation and information entropy are applied to attribute reduction, the incomplete rate is adopted, and the effectiveness under different incomplete rates is analyzed and verified by k-means clustering algorithm and Mean Shift clustering algorithm.