Entropy Based Optimal Scale Reducts for Consistent Generalized Multi-scale Interval-Set Decision Systems
HAO Tingting1, WU Weizhi1, TAN Anhui2
1. School of Information Engineering, Zhejiang Ocean University, Zhoushan 316022; 2. School of Mathematical Sciences, Huaqiao University, Quan-zhou 362021
Abstract：To solve the problem of knowledge acquisition in generalized multi-scale interval-set decision systems, an optimal scale reduction method based on interval-set decision entropy is proposed. First, the concept of scale combinations containing zero scales is defined in the generalized multi-scale interval-set decision systems. Then, similarity relations of the object sets are constructed via conditional attribute sets generated by different scale combinations to obtain the representations of corresponding information granules. Next, with the given scale combination, the concepts of interval lower approximations, interval upper approximations, interval approximation accuracy, and interval roughness of the decision classes with respect to the conditional attribute sets are defined. Furthermore, by integrating conditional entropy with interval roughness,the concept of interval-set decision entropy is introduced along with its properties. Finally, in a consistent generalized multi-scale interval-set decision system, the concepts of optimal scale combination, entropy optimal scale combination and corresponding optimal scale reducts are defined. The equivalence between the two concepts of optimal scale combinations and the equivalence between the concepts of optimal scale reduct and entropy optimal scale reduct are proved. The calculation of an optimal scale reduct is illustrated with an example.
[1] YAO Y Y.Information Granulation and Rough Set Approximation. International Journal of Intelligent Systems, 2001, 16(1): 87-104. [2] LIN T Y.Granular Computing: Structures, Representations, and Applications // Proc of the 9th International Conference on Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing. Berlin, Germany: Springer, 2003: 16-24. [3] YAO J T, VASILAKOS A V, PEDRYCZ W.Granular Computing: Perspectives and Challenges. IEEE Transactions on Cybernetics, 2013, 43(6): 1977-1989. [4] ZADEH L A.Toward a Theory of Fuzzy Information Granulation and Its Centrality in Human Reasoning and Fuzzy Logic. Fuzzy Sets and Systems, 1997, 90(2): 111-127. [5] 张清华,王宇泰,赵凡,等.复杂问题求解的多粒度计算框架.中国科学(信息科学), 2025, 55(5): 1122-1139. (ZHANG Q H, WANG Y T, ZHAO F, et al. Multi-granularity Computing Framework for Complex Problem Solving. Scientia Sinica Informationis, 2025, 55(5): 1122-1139.) [6] 梁吉业,钱宇华,李德玉,等.大数据挖掘的粒计算理论与方法.中国科学(信息科学), 2015, 45(11): 1355-1369. (LIANG J Y, QIAN Y H, LI D Y, et al. Theory and Method of Granular Computing for Big Data Mining. Scientia Sinica Informa-tionis, 2015, 45(11): 1355-1369.) [7] 徐计,王国胤,于洪.基于粒计算的大数据处理.计算机学报, 2015, 38(8): 1497-1517. (XU J, WANG G Y, YU H.Review of Big Data Processing Based on Granular Computing. Chinese Journal of Computers, 2015, 38(8): 1497-1517.) [8] 庞继芳,宋鹏,梁吉业.面向决策分析的多粒度计算模型与方法综述.模式识别与人工智能, 2021, 34(12): 1120-1130. (PANG J F, SONG P, LIANG J Y.Review on Multi-granulation Computing Models and Methods for Decision Analysis. Pattern Re-cognition and Artificial Intelligence, 2021, 34(12): 1120-1130.) [9] QIN J D, MARTÍNEZ L, PEDRYCZ W, et al. An Overview of Granular Computing in Decision-Making: Extensions, Applications, and Challenges. Information Fusion, 2023, 98. DOI: 10.1016/j.inffus.2023.101833. [10] 苗夺谦,张清华,钱宇华,等.从人类智能到机器实现模型——粒计算理论与方法.智能系统学报, 2016, 11(6): 743-757. (MIAO D Q, ZHANG Q H, QIAN Y H, et al. From Human Inte-lligence to Machine Implementation Model: Theories and Applications Based on Granular Computing. CAAI Transactions on Intelligent Systems, 2016, 11(6): 743-757.) [11] PAWLAK Z.Rough Sets.International Journal of Computer and Information Sciences, 1982, 11(5): 341-356. [12] WU W Z, LEUNG Y.Theory and Applications of Granular Labelled Partitions in Multi-scale Decision Tables. Information Sciences, 2011, 181(18): 3878-3897. [13] WU W Z, LEUNG Y.Optimal Scale Selection for Multi-scale Decision Tables. International Journal of Approximate Reasoning, 2013, 54(8): 1107-1129. [14] ZHANG H Y, LEUNG Y, ZHOU L.Variable-Precision-Dominan-ce-Based Rough Set Approach to Interval-Valued Information Systems. Information Sciences, 2013, 244: 75-91. [15] CHEN X W, LUO M K.Incremental Information Fusion in the Presence of Object Variations for Incomplete Interval-Valued Data Based on Information Entropy. Information Sciences, 2024, 667. DOI: 10.1016/j.ins.2024.120479. [16] LI R K, YANG J L, ZHANG X Y.Optimal Scale Selection Based on Three-way Decisions with Decision-Theoretic Rough Sets in Multi-scale Set-Valued Decision Tables. International Journal of Machine Learning and Cybernetics, 2023, 14: 3719-3736. [17] 陈应生,李进金,林荣德,等.多尺度集值决策信息系统.控制与决策, 2022, 37(2): 455-463. (CHEN Y S, LI J J, LIN R D, et al. Multi-scale Set Value Decision Information System. Control and Decision, 2022, 37(2): 455-463.) [18] LI W K, LI J J, HUANG J X, et al. A New Rough Set Model Based on Multi-scale Covering. International Journal of Machine Learning and Cybernetics, 2021, 12(1): 243-256. [19] 施虹艺,马周明.多尺度决策系统的覆盖粗糙模糊集及其最优尺度选择.山东大学学报(理学版), 2024, 59(5): 114-130. (SHI H Y, MA Z M.Covering Rough Fuzzy Sets and Optimal Scale Selection in Multi-scale Decision Systems. Journal of Shandong University(Natural Science), 2024, 59(5): 114-130.) [20] LI R, ZHANG C, LI D Y, et al. Improved Evidential Three-Way Decisions in Incomplete Multi-scale Information Systems. International Journal of Approximate Reasoning, 2025, 181. DOI: 10.1016/j.ijar.2025.109417. [21] HUANG X F, ZHAN J M, XU Z S, et al. A Prospect-Regret Theo-ry-Based Three-Way Decision Model with Intuitionistic Fuzzy Numbers under Incomplete Multi-scale Decision Information Systems. Expert Systems with Applications, 2023, 214. DOI: 10.1016/j.eswa.2022.119144. [22] HE X L, ZHAO L, SHE Y H.On Generalization Reducts in Incomplete Multi-scale Decision Tables. International Journal of Machine Learning and Cybernetics, 2024, 15: 253-266. [23] LI F, HU B Q.A New Approach of Optimal Scale Selection to Multi-scale Decision Tables. Information Sciences, 2017, 381: 193-208. [24] WANG T Y, YANG B.Optimal Scale Selection of Dynamic Incomplete Generalized Multi-scale Fuzzy Ordered Decision Systems Based on Rough Fuzzy Sets. Fuzzy Sets and Systems, 2025, 515. DOI: 10.1016/j.fss.2025.109420. [25] WANG L X, WU W Z, XIE Z H, et al. Optimal Scale Combination Selection in Generalized Multi-scale Hybrid Decision Systems. Information Sciences, 2025, 689. DOI: 10.1016/j.ins.2024.121429. [26] YANG Y, ZHANG Q H, ZHAO F, et al. Optimal Scale Combination Selection Based on Genetic Algorithm in Generalized Multi-scale Decision Systems for Classification. Information Sciences, 2025, 693. DOI: 10.1016/j.ins.2024.121685. [27] ZHANG X Y, HUANG Y Y.Optimal Scale Selection and Know-ledge Discovery in Generalized Multi-scale Decision Tables. International Journal of Approximate Reasoning, 2023, 161. DOI: 10.1016/j.ijar.2023.108983. [28] 朱康,吴伟志,刘梦欣.协调广义多尺度序模糊决策系统的最优尺度组合与属性约简.模式识别与人工智能, 2024, 37(6): 538-556. (ZHU K, WU W Z, LIU M X.Optimal Scale Combinations and Attribute Reduction for Consistent Generalized Multi-scale Ordered Fuzzy Decision Systems. Pattern Recognition and Artificial Intelligence, 2024, 37(6): 538-556.) [29] WANG F, LI J H, YU C C.Optimal Scale Selection Approach for Classification Based on Generalized Multi-scale Formal Context. Applied Soft Computing, 2024, 152. DOI: 10.1016/j.asoc.2024.111277. [30] CHENG Y L, ZHANG Q H, WANG G Y, et al. Optimal Scale Selection and Attribute Reduction in Multi-scale Decision Tables Based on Three-Way Decision. Information Sciences, 2020, 541: 36-59. [31] 廖淑娇,吴迪,卢亚倩,等.多尺度决策系统中测试代价敏感的属性与尺度同步选择.模式识别与人工智能, 2024, 37(4): 368-382. (LIAO S J, WU D, LU Y Q, et al. Test Cost Sensitive Simulta-neous Selection of Attributes and Scales in Multi-scale Decision Systems. Pattern Recognition and Artificial Intelligence, 2024, 37(4): 368-382.) [32] SHANNON C E.A Mathematical Theory of Communication. The Bell System Technical Journal, 1948, 27(3): 379-423. [33] JI X, LI J, YAO S,et al. Attribute Reduction Based on Fusion Information Entropy. International Journal of Approximate Reaso-ning, 2023, 160. DOI: 10.1016/j.ijar.2023.108949. [34] QU L D, HE J L, ZHANG G Q, et al. Entropy Measure for a Fuzzy Relation and Its Application in Attribute Reduction for He-terogeneous Data. Applied Soft Computing, 2022, 118. DOI: 10.1016/j.asoc.2022.108455. [35] XU W H, YANG Y F.Matrix-Based Feature Selection Approach Using Conditional Entropy for Ordered Data Set with Time-Evolving Features. Knowledge-Based Systems, 2023, 279. DOI: 10.1016/j.knosys.2023.110947. [36] BAO H, WU W Z, ZHENG J W,et al. Entropy Based Optimal Scale Combination Selection for Generalized Multi-scale Information Tables. International Journal of Machine Learning and Cyberne-tics, 2021, 12: 1427-1437. [37] 刘梦欣,谢祯晃,吴伟志,等.广义多尺度多重集值决策系统的最优尺度约简.模式识别与人工智能, 2025, 38(1): 2-21. (LIU M X, XIE Z H, WU W Z, et al. Optimal Scale Reducts for Generalized Multi-scale Multiset-Valued Decision Systems. Pattern Recognition and Artificial Intelligence, 2025, 38(1): 2-21.) [38] SANG B B, YANG L, XU W H, et al. VCOS: Multi-scale Information Fusion to Feature Selection Using Fuzzy Rough Combination Entropy. Information Fusion, 2025, 117. DOI: 10.1016/j.inffus.2024.102901. [39] YAO Y Y.Interval-Set Algebra for Qualitative Knowledge Representation // Proc of the 5th International Conference on Computing and Information. Washington, USA: IEEE, 1993: 370-374. [40] XIE Z H, WU W Z, WANG L X, et al. Entropy Based Optimal Scale Selection and Attribute Reduction in Multi-scale Interval-Set Decision Tables. International Journal of Machine Learning and Cybernetics, 2024, 15: 3005-3026. [41] YAO Y Y, LIU Q.A Generalized Decision Logic in Interval-Set-Valued Information Tables // Proc of the 7th International Workshop on New Directions in Rough Sets, Data Mining, and Granular-Soft Computing.Berlin, Germany: Springer, 1999: 285-293. [42] XIE X, ZHANG X Y, ZHANG S Y.Rough Set Theory and Attri-bute Reduction in Interval-Set Information System. Journal of Inte-lligent & Fuzzy Systems, 2022, 42(6): 4919-4929. [43] 张倚萌,贾修一,唐振民.基于条件信息熵的区间集决策信息表不确定性度量.南京理工大学学报, 2019, 43(4): 393-401. (ZHANG Y M, JIA X Y, TANG Z M.Uncertainty Measurement for Interval Set Decision Information Tables Based on Conditional Information Entropy. Journal of Nanjing University of Science and Technology, 2019, 43(4): 393-401.) [44] LIANG D C, FU Y Y, XU Z S.Novel AQM Analysis Approach Based on Similarity and Dissimilarity Measures of Interval Set for Multi-expert Multi-criterion Decision Making. International Journal of Approximate Reasoning, 2022, 142: 266-289.
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