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As an extension of Pawlak’s rough sets, rough fuzzy sets are proposed to deal with fuzzy target concept. As we know, the uncertainty of Pawlak’s rough sets is rooted in the objects contained in the boundary region, while the uncertainty of rough fuzzy sets comes from three regions (positive region, boundary region, and negative region). In addition, in the view of traditional uncertainty measures, the two rough approximation spaces with the same uncertainty are not necessarily equivalent, and they cannot be distinguished. In this paper, firstly, a fuzziness-based uncertainty measure is proposed. Meanwhile, the essence of the uncertainty for rough fuzzy sets and its three regions in a hierarchical granular structure is revealed. Then, from the perspective of fuzzy distance, we introduce a modified uncertainty measure based on the fuzziness-based uncertainty measure and present that our method not only is strictly monotonic with finer approximation spaces, but also can distinguish the two rough approximation spaces with the same uncertainty. Finally, a case study is introduced to demonstrate that the modified uncertainty measure is more suitable for evaluating the significance of attributes. These works are useful for further study on rough sets theory and promote the development of uncertain artificial intelligence. [ABSTRACT FROM AUTHOR]

Abstract As an extension of Pawlak's rough sets, rough fuzzy sets is proposed to deal with the target concept which is typically fuzzy or uncertain. It is worthwhile to introduce cost-sensitive learning into granular computing, as the granularity with the optimal cost can be selected. In the terms of decision making, test cost and decision cost are the most popular cost types. In the sequential three-way decisions (S3WD) models, the granularity is sensitive to the test cost. Meanwhile, the accuracy is sensitive to the decision cost. Selecting an optimal cost-sensitive granularity for problem solving is helpful for achieving optimal results at the lowest total cost in S3WD. However, it is difficult to evaluate test cost precisely and objectively in real-life applications, and existing works only focus on searching for the total cost as the objective function. In this paper, we firstly present a sequential three-way decisions model with rough fuzzy sets (S3WDRFS). Then, for S3WDRFS and its three regions, the changing rules of their decision cost are revealed in a hierarchical granular structure. By considering user requirements, we propose an optimization mechanism to achieve the optimal cost-sensitive granularity selection based on S3WDRFS model. Finally, the experimental results demonstrate that exemplary optimal granularities can be obtained for high quality decision-making under certain constraints. [ABSTRACT FROM AUTHOR]