Abstract：The knowledge representation and the knowledge acquisition of multi-scale data are crucial research directions in multi-granularity computing. While analyzing multi-scale data, a key issue is the selection of the optimal scale combination, with the aim of choosing a suitable subsystem for the final decision. To solve the problem of knowledge acquisition from multi-scale multiset-valued data, at first, similarity relations determined by the set of objects under different scale combinations are constructed based on Hellinger distance in generalized multi-scale multiset-valued decision systems, and the information granule representation is provided. Second, the concepts of optimal scale reducts and entropy optimal scale reducts are defined in consistent generalized multi-scale multiset-valued decision systems, and the equivalence between the optimal scale reducts and the entropy optimal scale reducts is proven. In inconsistent generalized multi-scale multiset-valued decision systems, the definition of generalized decision optimal scale reducts is proposed by introducing generalized decision functions. Furthermore, by employing conditional entropies and generalized decision functions, search algorithms of entropy optimal scale reducts and generalized decision optimal scale reducts are designed. Finally, a method of constructing generalized multi-scale multiset-valued decision systems is proposed, and the experiments demonstrate the validity and rationality of the proposed algorithms.
[1] PEDRYCZ W.Granular Computing for Data Analytics: A Manifesto of Human-Centric Computing. IEEE/CAA Journal of Automatica Sinica, 2018, 5(6): 1025-1034. [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] 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. [4] PAWLAK Z.Rough Sets: Theoretical Aspects of Reasoning about Data. Boston, USA: Kluwer Academic Publishers, 1991. [5] WANG W J, ZHAN J M, ZHANG C, et al. A Regret-Theory-Based Three-Way Decision Method with a Priori Probability Tolerance Dominance Relation in Fuzzy Incomplete Information Systems. Information Fusion, 2023, 89: 382-396. [6] XU W H, PAN Y Z, CHEN X W, et al. A Novel Dynamic Fusion Approach Using Information Entropy for Interval-Valued Ordered Datasets. IEEE Transactions on Big Data, 2023, 9(3): 845-859. [7] ZHANG X Y, HOU J L.A Novel Rough Set Method Based on Adjustable-Perspective Dominance Relations in Intuitionistic Fuzzy Ordered Decision Tables. International Journal of Approximate Reasoning, 2023, 154: 218-241. [8] CHEN B W, ZHANG X Y, YANG J L.Feature Selections Based on Three Improved Condition Entropies and One New Similarity Degree in Interval-Valued Decision Systems. Engineering Applications of Artificial Intelligence, 2023, 126. DOI:10.1016/j.engappai.2023.107165. [9] LUO L H, YANG J L, ZHANG X Y, et al. Tri-level Attribute Reduction Based on Neighborhood Rough Sets. Applied Intelligence, 2024, 54: 3786-3807. [10] YE J, ZHAN J M, DING W P, et al. A Novel Three-Way Decision Approach in Decision Information Systems. Information Sciences, 2022, 584: 1-30. [11] WU W Z, LEUNG Y.Theory and Applications of Granular Labelled Partitions in Multi-scale Decision Tables. Information Sciences, 2011, 181(18): 3878-3897. [12] WU W Z, LEUNG Y.Optimal Scale Selection for Multi-scale Decision Tables. International Journal of Approximate Reasoning, 2013, 54(8): 1107-1129. [13] 胡军,陈艳,张清华,等.广义多尺度集值决策系统最优尺度选择.计算机研究与发展, 2022, 59(9): 2027-2038. (HU J, CHEN Y, ZHANG Q H, et al. Optimal Scale Selection for Generalized Multi-scale Set-Valued Decision Systems. Journal of Computer Research and Development, 2022, 59(9): 2027-2038.) [14] 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. [15] WANG W J, HUANG B, WANG T X.Optimal Scale Selection Based on Multi-scale Single-Valued Neutrosophic Decision-Theoretic Rough Set with Cost-Sensitivity. International Journal of Approximate Reasoning, 2023, 155: 132-144. [16] XIAO Y B, ZHAN J M, ZHANG C, et al. A Preference Group Consensus Method with Three-Way Decisions and Regret Theory under Multi-scale Information Systems. Information Sciences, 2024, 660. DOI: 10.1016/j.ins.2024.120126. [17] 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. [18] ZHAN J M, DENG J, XU Z S, et al. A Three-Way Decision Methodology with Regret Theory via Triangular Fuzzy Numbers in Incomplete Multi-scale Decision Information Systems. IEEE Tran-sactions on Fuzzy Systems, 2023, 31(8): 2773-2787. [19] 张庐婧,林国平,林艺东,等.多尺度邻域决策信息系统的特征子集选择.模式识别与人工智能, 2023, 36(1): 49-59. (ZHANG L J, LIN G P, LIN Y D, et al. Feature Subset Selection for Multi-scale Neighborhood Decision Information System. Pattern Recognition and Artificial Intelligence, 2023, 36(1): 49-59. ) [20] LI F, HU B Q.A New Approach of Optimal Scale Selection to Multi-scale Decision Tables. Information Sciences, 2017, 381: 193-208. [21] 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. [22] 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 Simultaneous Selection of Attributes and Scales in Multi-scale Decision Sys-tems. Pattern Recognition and Artificial Intelligence, 2024, 37(4): 368-382. ) [23] 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.) [24] 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. [25] ZHANG L J, LIN G P, WEI L, et al. Feature Subset Selection for Multi-scale Neighborhood Decision Information System via Mutual Information. Artificial Intelligence Review, 2024, 57. DOI: 10.1007/s10462-023-10626-w. [26] ZHANG Q H, CHENG Y L, ZHAO F, et al. Optimal Scale Combination Selection Integrating Three-Way Decision with Hasse Diagram. IEEE Transactions on Neural Networks and Learning Systems, 2022, 33(8): 3675-3689. [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] SHANNON C E.A Mathematical Theory of Communication. The Bell System Technical Journal, 1948, 27(3): 379-423. [29] 苗夺谦,王珏.粗糙集理论中知识粗糙性与信息熵关系的讨论.模式识别与人工智能, 1998, 11(1): 34-40. (MIAO D Q, WANG J.On the Relationships between Information Entropy and Roughness of Knowledge in Rough Set Theory. Pattern Recognition and Artificial Intelligence, 1998, 11(1): 34-40.) [30] LIANG J Y, SHI Z Z. The Information Entropy, Rough Entropy and Knowledge Granulation in Rough Set Theory. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 2004, 12(1): 37-46. [31] LIANG J Y, SHI Z Z, LI D Y, et al. Information Entropy, Rough Entropy and Knowledge Granulation in Incomplete Information Systems. International Journal of General Systems, 2006, 35(6): 641-654. [32] DAI J H, DAI W T, XU Q.An Uncertainty Measure for Incomplete Decision Tables and Its Applications. IEEE Transactions on Cybernetics, 2013, 43(4): 1277-1289. [33] WU J M, LIU D Y, HUANG Z H, et al. Multi-scale Decision Systems with Test Cost and Applications to Three-Way Multi-attribute Decision-Making. Applied Intelligence, 2024, 54(4): 3591-3605. [34] WANG Z H, CHEN H M, YUAN Z, et al. Multiscale Fuzzy Entropy-Based Feature Selection. IEEE Transactions on Fuzzy Systems, 2023, 31(9): 3248-3262. [35] MIYAMOTO S.Multisets and Fuzzy Multisets//LIU Z Q, MIYAMOTO S, eds. Soft Computing and Human-Centered Machines. Berlin, Germany: Springer, 2000: 9-33. [36] 王维琼,谢琼,许豪杰,等.两方有理数多重集的保密计算.电子与信息学报, 2023, 45(5): 1722-1730. (WANG W Q, XIE Q, XU H J, et al. Secure Computation of Two-Party Multisets with Rational Numbers. Journal of Electronics and Information Technology, 2023, 45(5): 1722-1730.) [37] 赵前进,平昕瑞,苏树智,等.标签敏感的多重集正交相关特征融合方法.电子与信息学报, 2022, 44(10): 3458-3464. (ZHAO Q J, PING X R, SU S Z, et al. Feature Fusion Method Based on Label-Sensitive Multi-set Orthogonal Correlation. Journal of Electronics and Information Technology, 2022, 44(10): 3458-3464.) [38] DU D H, FENG Q J, CHEN W F, et al. Mix-Supervised Multiset Learning for Cancer Prognosis Analysis with High-Censoring Survi-val Data. Expert Systems with Applications, 2024, 239. DOI: 10.1016/j.eswa.2023.122430. [39] ZHAO X R, HU B Q.Three-Way Decisions with Decision-Theore-tic Rough Sets in Multiset-Valued Information Tables. Information Sciences, 2020, 507: 684-699. [40] HUANG D, LIN H, LI Z W.Information Structures in a Multiset-Valued Information System with Application to Uncertainty Mea-surement. Journal of Intelligent and Fuzzy Systems, 2022, 43(6): 7447-7469. [41] SONG Y, LIN H, LI Z W.Outlier Detection in a Multiset-Valued Information System Based on Rough Set Theory and Granular Computing. Information Sciences, 2024, 657. DOI: 10.1016/j.ins.2023.119950. [42] 王蕾晰,吴伟志,谢祯晃.基于熵的多尺度多重集值信息系统的最优尺度选择与属性约简.模式识别与人工智能, 2023, 36(6): 495-510. (WANG L X, WU W Z, XIE Z H.Optimal Scale Selection and Attribute Reduction of Multi-scale Multiset-Valued Information Systems Based on Entropy. Pattern Recognition and Artificial Intelligence, 2023, 36(6): 495-510.)
2025, Vol. 38 
