Reduction foundation with multigranulation rough sets using discernibility

When multiple granulated knowledge in multigranulation spaces are involved in decision making, protocol principles are adopted to arrive at the final consensus. Multigranulation rough set theory utilizes a voting principle to combine the decision options derived from individual granulated knowledge. Note that those knowledge may provide different degrees of support to the final results, some are key, some are of less importance and some are even of no use. Selecting valuable knowledge and reducing worthless one are thus necessary for data processing, which can alleviate the storage occupancy and facilitate the logical and statistical analysis. However, the basic reduction foundation of multigranulation spaces has been rarely touched by researchers, which brings in many difficulties in algorithmic and real applications. This work aims to disclose the principles of multiple knowledge reduction in multigranulation spaces from the viewpoint of discernibility. First, the notions of knowledge reduction of multigranulation spaces are defined based on multigranulation rough set theory. Second, a decision function mapping each object into the decision options of its neighborhood granule is introduced. Third, several pairs of discernibility matrices and discernibility functions are successively developed using the decision function. We claim that the valuable and worthless knowledge in multigranulation spaces can be explicitly chose and eliminated respectively by using the proposed discernibility matrices and discernibility functions. That is to say, these discernibility tools provide a precise criterion for the knowledge reduction of multigranulation spaces. As a theoretical extension, a multigranulation information entropy is proposed and an approximate algorithm is constructed to compute a suboptimal reduct of a multigranulation space based on this entropy. In the end, numerical experiments are performed on public data sets to verify the effectiveness of the proposed reduction methods. This study can get us a grasp of the foundational principle of knowledge reduction and may bring a new insight for the designation of substantial reduction algorithms of multigranulation knowledge.