Extending characteristic relations on an incomplete data set by the three-way decision theory

The methods of mining incomplete data based on characteristic sets or characteristic relation have been intensively studied in recent years. With the development of related research, many modifications of the definition of characteristic relation have been proposed. However, few of them can be used for decision rule induction due to the so-called definability problem. In this paper, by using the wide sense of the three-way decision theory, we extend the notions of characteristic relation and characteristic set to the systems with four types of characteristic relations and characteristic sets, respectively. Then we study the probabilistic approximations based on the extended characteristic set system. Moreover, we extend the maximal characteristic neighborhood system into four types of maximal characteristic neighborhood systems, and investigate the probabilistic approximations based on them. Finally, we generalize the definition of local definability for incomplete data processing. In particular, several existing methods based on characteristic sets are integrated into our model, and the new types of characteristic relations proposed by us seem more practical than the classical one in some aspects. In this way, our model illustrates the effectiveness and comprehensiveness of thinking in threes. (C) 2020 Elsevier Inc. All rights reserved.