Domain-wise approaches for updating approximations with multi-dimensional variation of ordered information systems

Dominance-based Rough Set Approach (DRSA) is widely applied in processing the information and data with a preference-order relation. In many real-life Ordered Information Systems (OIS), the attribute values and objects may vary simultaneously with time. Existing incremental updating algorithms in OIS are all single-dimensional, i.e., they only consider the individual variation of attribute values or objects. In addition, all the objects need to be compared with each other during updating. In this paper, we focus on designing specific multi-dimensional algorithms to efficiently update approximations on the simultaneous variation of attribute values and objects in OIS. The properties of the P-generalized decision in DRSA are firstly presented. Then two novel notions e.g., the P-generalized decision lower and upper domains, are defined, respectively. These two notions indicate the practical dominance dependence between different objects in OIS, and provide a simplified definition of the P-generalized decision. The approach based on these two notions can greatly reduce the comparisons between different objects, as well as provide novel strategies which can efficiently obtain the updated P-generalized decision to obtain the updated approximations. Furthermore, we develop two domain-wise algorithms which correspond to two cases of the multi-dimensional variation of attribute values and objects in OIS, respectively. Moreover, in implementing the algorithms, the sorting strategies are integrated to solve the problem of time-consuming traversals. A series of experimental results illustrate that our proposed domain-wise algorithms are not only more efficient than both of the traditional static algorithm and the integrated single-dimensional algorithms for dealing with the multi-dimensional variation of attribute values and objects, but also more efficient than the single-dimensional algorithms for dealing with the individual variation of objects in OIS.