Abstract
Graded rough sets (GRSs) act as a bidirectional quantitative model of three-way decision, but their approximation operators cannot preserve union, intersection and complement operations. Aiming at GRSs, this paper mines binary boundaries and constructs the power set space, so the corresponding ECG (electrocardiogram) data analysis is eventually performed. Based on union and intersection inequalities of approximation operators, four types of binary boundaries and their operators are first proposed to generate fundamental union and intersection equations, and both their quantitative semantics regarding dual membership grades and their degenerate properties on quantitative parameters are revealed. Then, union, intersection and complement operations of approximation sets are redefined by boundaries to acquire the set operation preservation of approximation operators, so the power set space of GRSs is established to induce homomorphisms regarding the classical power set space. Finally, the binary boundaries in power set space are utilized for ECG dataset analysis, and experimental results demonstrate the effectiveness of theoretical structures and in-depth properties. This study adopts double viewpoints of operator theory and set theory to enrich GRSs, its quantitative extension underlies uncertainty modeling and granular computing, while its mathematical structures facilitate data mining in terms of parameter optimization.
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References
Yao, Y.Y.: Two views of the theory of rough sets in finite universes. Int. Approx. Reason. 15, 291–317 (1996)
Pawlak, Z.: Rough sets. Int. J. Comput. Inform. Sci. 11(5), 341–356 (1982)
Ma, X.A.: Fuzzy entropies for class-specific and classification-based attribute reducts in three-way probabilistic rough set models. Int. J. Mach. Learn. Cybern. 12(2), 433–457 (2020).
Sun, B.Z., Chen, X.T., Zhang, L.Y., Ma, W.M.: Three-way decision making approach to conflict analysis and resolution using probabilistic rough set over two universes. Inf. Sci. 507, 809–822 (2020)
Maldonado, S., Peters, G., Weber, R.: Credit scoring using three-way decisions with probabilistic rough sets. Inf. Sci. 507, 700–714 (2020)
Yao, Y.Y., Lin, T.Y.: Generalization of rough sets using modal logics. Intell. Autom. Soft Comput. 2(2), 103–119 (1996)
Liu, C.H., Miao, D.Q., Zhang, N.: Graded rough set model based on two universes and its properties. Knowl. Based Syst. 33, 65–72 (2012)
Huang, B., Guo, C.X., Li, H.X., Feng, G.F., Zhou, X.Z.: An intuitionistic fuzzy graded covering rough set. Knowl. Based Syst. 107, 155–178 (2016)
Xue, Z.A., Lv, M.J., Han, D.J., Xin, X.W.: Multi-granulation graded rough intuitionistic fuzzy sets models based on dominance relation. Symmetry 10(10), 446 (2018)
Zhang, X.Y., Mo, Z.W., Xiong, F., Cheng, W.: Comparative study of variable precision rough set model and graded rough set model. Int. J. Approx. Reason. 53(1), 104–116 (2012)
Fang, B.W., Hu, B.Q.: Probabilistic graded rough set and double relative quantitative decision-theoretic rough set. Int. J. Approx. Reason. 74, 1–12 (2016)
Zhang, X.Y., Mo, Z.W., Shu, L.: Operators of approximation and approximate power sets space. J. Electron. Sci. Technol. China 2(2), 91–96 (2004)
Li, F.H., Chen, K.H., Lie, J., Zhan, Y.W., Manogaran, G.: Automatic diagnosis of cardiac arrhythmia in electrocardiograms via multigranulation computing. Appl. Soft Comput. 80, 400–413 (2019)
Zhang, X.Y., Wang, X.X., Mo, Z.W., Wang, Q., Tang, X.: Three-level and three-way vagueness similarity measurements for electrocardiogram (ECG) data analysis. In: 2021 16th International Conference on Intelligent Systems and Knowledge Engineering (ISKE), p. 471−478. IEEE (2021)
Zhang, X.Y., Xie, S.C., Mo, Z.W.: Graded rough set. J. Sichuan Normal Univ. (Nat. Sci.) 1, 12–16 (2010)
Acknowledgments
The authors thank the reviewers for their valuable suggestions. The work was supported by Sichuan Science and Technology Program of China (2021YJ0085), and Joint Research Project of Laurent Mathematics Center of Sichuan Normal University and National-Local Joint Engineering Laboratory of System Credibility Automatic Verification (ZD20220101).
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Wang, Q., Wang, X., Zhang, X. (2022). Binary Boundaries and Power Set Space of Graded Rough Sets and Their Correlative ECG (Electrocardiogram) Data Analysis. In: Yao, J., Fujita, H., Yue, X., Miao, D., Grzymala-Busse, J., Li, F. (eds) Rough Sets. IJCRS 2022. Lecture Notes in Computer Science(), vol 13633. Springer, Cham. https://doi.org/10.1007/978-3-031-21244-4_7
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DOI: https://doi.org/10.1007/978-3-031-21244-4_7
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