Theoretical Study on a New Information Entropy and its Use in Attribute Reduction Ping Luo 1,2 , Qing He 1 , Zhongzhi Shi 1 1 Key Laboratory of Intelligent Information Processing, Institute of Computing Technology, Chinese Academy of Sciences, 100080, Beijing, China 2 Graduate School of the Chinese Academy of Sciences, 100080, Beijing, China luop@ics.ict.ac.cn Abstract The positive region in rough set framework and Shan- non conditional entropy are two traditional uncertainty measurements, used usually as heuristic metrics in at- tribute reduction. In this paper first a new information entropy is systematically compared with Shannon entropy, which shows its competence of another new uncertainty measurement. Then given a decision system we theoret- ically analyze the variance of these three metrics under two reverse circumstances. Those are when condition (decision) granularities merge while decision (condition) granularities remain unchanged. The conditions that keep these measurements unchanged in the above different situations are also figured out. These results help us to give a new information view of attribute reduction and propose more clear understanding of the quantitative relations between these different views, defined by the above three uncertainty measurements. It shows that the requirement of reducing a condition attribute in new information view is more rigorous than the ones in the latter two views and these three views are equivalent in a consistent decision system. Keyword: Information entropy; conditional entropy; rough set; positive region. 1. Introduction Rough set is a valid mathematical tool that deals with imprecise, uncertain, vague or incomplete knowledge of a decision system. It has been applied successfully in data mining and machine learning field when we use attribute reduction in data preprocessing phase and value reduction in inductive learning phase. The original rough set theory is from algebra view since all its basic concepts, such as lower and upper approximation and positive region, are resulted from the indiscernibility relation between instances [1, 2]. However, some researchers introduce Shannon entropy to attribute reduction and adopt it as the heuristic metrics to select important attributes [3]. Then it gives the information view of attribute reduction. Paper [4, 5] also proposed a theoretical comparison of attribute reduction in algebra and Shannon information views. In this paper we aim to introduce a new information entropy [6, 7] to attribute reduction and analyze the quantitative relations between different attribute reductions including the new informa- tion view, Shannon information view and algebra view. Although this paper does not present any new attribute reduction algorithm, it is indeed about algorithms because it clarifies the different abilities of the three metrics to measure uncertainty and the inclusion relationship between the attribute importance in different views, and will result in more efficient heuristic reduction algorithms. We organize this paper as follows. Section 2 gives some basic notions in decision system and rough set theory. Section 3 presents the definition of new information entropy and the corresponding conditional entropy, and proposes some properties of these new definitions which imply those of Shannon entropy. These two sections are the basis for further analysis. In section 4 we theoretically analyze the variance of the three uncertainty measurements caused by the merging of condition granularities while decision granularities remain unchanged. These results help us to understand the quantitative relations between different attribute reductions in section 5. Additionally the change of uncertainty measurements along with the merging of decision granularities while condition granularities remain unchanged are also studied in section 4. We conclude this paper in section 6.