Information Entropy and Optimal Scale Combination in Multi-Scale Covering Decision Systems

In the traditional covering information systems, each object has only one scale, that is, the single-scale covering information system. But in realistic applications, one often needs to process and analyze data in different scales. To solve this problem, a new multi-scale covering information system is first introduced in this paper, and the optimal scale combination selections based on information entropy in multi-scale coveing decision systems are discussed. Firstly, entropy consistent scale combination and entropy optimal scale combination are defined in multi-scale covering information system. Secondly, entropy optimal scale combination is defined in multi-scale covering decision system, which is proved to be equivalent to the optimal scale combination in consistent multi-scale coveing decision system. Moreover, lower approximation entropy optimal scale combination is defined in the inconsistent multi-scale covering decision system. Finally, the corresponding selection algorithms and specific examples are given by using conditional entropy and limitary conditional entropy, respectively.