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Relationships among generalized rough sets in six coverings and pure reflexive neighborhood system. (English) Zbl 1250.68261
Summary: Rough set is a useful tool to deal with partition related uncertainty, granularity, and incompleteness of knowledge. Although the classical rough set is constructed on the basis of an indiscernibility relation, it can also be generalized by using some weaker binary relations. In this paper, a systematic approach is used to study the generalized rough sets in six coverings and pure reflexive neighborhood system. After two steps, relationships among generalized rough sets in six coverings and pure reflexive neighborhood system are obtained. The first step is to study the generalized rough sets in six coverings, and to get relationships between every two covering rough set models. The second step is to study the relationships between generalized rough sets in each covering and in pure reflexive neighborhood system. The inclusion relations or equivalence relations among the seven upper/lower approximations could be acquired. Finally, the accuracy measures of generalized rough sets in six coverings and that in pure reflexive neighborhood system are compared. The relationships among seven accuracy measures are also obtained. Some illustrative examples are employed to demonstrate our arguments.

MSC:
68T37 Reasoning under uncertainty in the context of artificial intelligence
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[1] R.B. Barot, T.Y. Lin, Granular computing on covering from the aspects of knowledge theory, in: IEEE NAFIPS, 2008.
[2] Błlaszczyński, J.; Słlowiński, R.; Szeląg, M., Sequential covering rule induction algorithm for variable consistency rough set approaches, Inform. sci., 181, 5, 987-1002, (2011)
[3] Calegari, S.; Ciucci, D., Granular computing applied to ontologies, Int. J. approx. reason., 51, 4, 391-409, (2010) · Zbl 1205.68394
[4] Chen, Y.; Miao, D.Q.; Wang, R.Z., A rough set approach to feature selection based on ant colony optimization, Pattern recogn. lett., 31, 3, 226-233, (2010)
[5] Cheng, Y.; Miao, D.Q.; Feng, Q.R., Positive approximation and converse approximation in interval-valued fuzzy rough sets, Inform. sci., 181, 11, 2086-2110, (2011) · Zbl 1216.68215
[6] Das, N.R.; Das, P., Neighborhood systems in fuzzy topological group, Fuzzy set. syst., 116, 3, 401-408, (2000) · Zbl 1025.54002
[7] Deng, T.Q.; Chen, Y.M.; Xu, W.L.; Dai, Q.H., A novel approach to fuzzy rough sets based on a fuzzy covering, Inform. sci., 177, 11, 2308-2326, (2007) · Zbl 1119.03051
[8] Diker, M.; Uğur, A.A., Textures and covering based rough sets, Inform. sci., 184, 1, 44-63, (2012) · Zbl 1239.03032
[9] Du, Y.; Hu, Q.H.; Zhu, P.F.; Ma, P.J., Rule learning for classification based on neighborhood covering reduction, Inform. sci., 181, 24, 5457-5467, (2011)
[10] Hu, Q.H.; Yu, D.R.; Xie, Z.X.; Liu, J.F., Fuzzy probabilistic approximation spaces and their information measures, IEEE trans. fuzzy syst., 14, 2, 191-201, (2006)
[11] Hu, Q.H.; An, S.; Yu, D.R., Soft fuzzy rough sets for robust feature evaluation and selection, Inform. sci., 180, 22, 4384-4400, (2010)
[12] Kryszkiewicz, M., Rough set approach to incomplete information systems, Inform. sci., 112, 1-4, 39-49, (1998) · Zbl 0951.68548
[13] Li, T.J.; Zhang, W.X., Rough fuzzy approximations on two universes of discourse, Inform. sci., 178, 3, 892-906, (2008) · Zbl 1128.68099
[14] Lin, T.Y., Granular computing: practices, theories, and future directions, Encyclopedia complex. syst. sci., 4339-4355, (2009)
[15] Lin, T.Y., Granular computing on binary relations I: data mining and neighborhood systems, (), 107-121 · Zbl 0927.68089
[16] Lin, T.Y., Granular computing on binary relations II: rough set representations and belief functions, (), 122-140 · Zbl 0927.68090
[17] T.Y. Lin, Neighborhood systems and approximation in database and knowledge base systems, in: Proceedings of the Fourth International Symposium on Methodologies of Intelligent Systems, Poster Session, October 12-15, 1989, pp. 75-86.
[18] T.Y. Lin, K.J. Huang, Q. Liu, W. Chen, Rough sets, neighborhood systems and approximation, in: Proceedings of the Fifth International Symposium on Methodologies of Intelligent Systems, Selected Papers, Knoxville, Tennessee, October 25-27, 1990, pp. 130-141.
[19] Lin, T.Y., Neighborhood systems – a qualitative theory for fuzzy and rough sets, (), 132-155
[20] T.Y. Lin, Neighborhood systems and relational databases, in: ACM Conference on Computer Science, 1988, pp. 725.
[21] Lin, T.Y., Topological and fuzzy rough sets, decision support by experience- application of the rough sets theory, (), 287-304
[22] Lin, T.Y., Granular computing on partitions, covering and neighborhood system, J. nanchang inst. technol., 25, 2, 1-7, (2006)
[23] Lin, T.Y.; Barot, R.; Tsumoto, S., Some remarks on the concept of approximations from the view of knowledge engineering, Int. J. cogni. inform. natural intell., 4, 2, 1-11, (2010)
[24] Liu, G.L.; Sai, Y., A comparison of two types of rough sets induced by coverings, Int. J. approx. reason., 50, 3, 521-528, (2009) · Zbl 1191.68689
[25] Liu, G.L.; Zhu, W., The algebraic structures of generalized rough set theory, Inform. sci., 178, 21, 4105-4113, (2008) · Zbl 1162.68667
[26] Michael, J.B.; Lin, T.Y., Neighborhoods, rough sets, and query relaxation in cooperative answering, Rough sets and data mining: analysis of imprecise data, (1997), Kluwer Academic Publisher, pp. 229-238
[27] Pawlak, Z., Rough sets: theoretical aspects of reasoning about data, (1991), Kluwer Academic Publishers. Boston · Zbl 0758.68054
[28] Pawlak, Z.; Skowron, A., Rudiments of rough sets, Inform. sci., 177, 1, 3-27, (2007) · Zbl 1142.68549
[29] Pawlak, Z.; Skowron, A., Rough sets: some extensions, Inform. sci., 177, 1, 28-40, (2007) · Zbl 1142.68550
[30] Pawlak, Z.; Skowron, A., Rough sets and Boolean reasoning, Inform. sci., 177, 1, 41-73, (2007) · Zbl 1142.68551
[31] Pomykala, J.A., Approximation operations in approximation space, Bull. Pol. acad. sci., 35, 9-10, 653-662, (1987) · Zbl 0642.54002
[32] Qian, Y.H.; Liang, J.Y.; Dang, C.Y., Converse approximation and rule extraction from decision tables in rough set theory, Comput. math. appl., 55, 8, 1754-1765, (2008) · Zbl 1147.68736
[33] Qian, Y.H.; Liang, J.Y.; Li, D.Y.; Zhang, H.Y.; Dang, C.Y., Measures for evaluating the decision performance of a decision table in rough set theory, Inform. sci., 178, 1, 181-202, (2008) · Zbl 1128.68102
[34] Qian, Y.H.; Liang, J.Y.; Wu, W.Z.; Dang, C.Y., Information granularity in fuzzy binary grc model, IEEE trans. fuzzy syst., 19, 2, 253-264, (2011)
[35] Qian, Y.H.; Liang, J.Y.; Pedrycz, W.; Dang, C.Y., Positive approximation: an accelerator for attribute reduction in rough set theory, Artif. intell., 174, 9-10, 597-618, (2010) · Zbl 1205.68310
[36] Qian, Y.H.; Liang, J.Y.; Yao, Y.Y.; Dang, C.Y., MGRS: a multi-granulation rough set, Inform. sci., 180, 949-970, (2010) · Zbl 1185.68695
[37] P. Samanta, M.K. Chakraborty, Covering based approaches to rough sets and implication lattices, in: RSFDGrC 2009, 2009, pp. 127-134.
[38] Shi, Z.H.; Gong, Z.T., The further investigation of covering-based rough sets: uncertainty characterization, similarity measure and generalized models, Inform. sci., 180, 19, 3745-3763, (2010) · Zbl 1205.68430
[39] Skowron, A.; Stepaniuk, J., Tolerance approximation spaces, Fundam. inform., 27, 245-253, (1996) · Zbl 0868.68103
[40] Skowron, A.; Stepaniuk, J.; Swiniarski, R., Modeling rough granular computing based on approximation spaces, Inform. sci., 184, 1, 20-43, (2012)
[41] Sun, B.Z.; Gong, Z.T.; Chen, D.G., Fuzzy rough set theory for the interval-valued fuzzy information systems, Inform. sci., 178, 13, 2794-2815, (2008) · Zbl 1149.68434
[42] E. Tsang, D. Cheng, J. Lee, D. Yeung, On the upper approximations of covering generalized rough sets, in: Proceedings of the 3rd International Conference Machine Learning and Cybernetics, 2004, pp. 4200-4203.
[43] Wei, W.; Liang, J.Y.; Qian, Y.H., A comparative study of rough sets for hybrid data, Inform. sci., 190, 1-12, (2012)
[44] Wu, W.Z.; Leung, Y., Theory and applications of granular labelled partitions in multi-scale decision tables, Inform. sci., 181, 18, 3878-3897, (2011) · Zbl 1242.68258
[45] Yang, T.; Li, Q.G., Reduction about approximation spaces of covering generalized rough sets, Int. J. approx. reason., 51, 3, 335-345, (2010) · Zbl 1205.68433
[46] Yang, X.B.; Yang, J.Y.; Wu, C.; Yu, D.J., Dominance-based rough set approach and knowledge reductions in incomplete ordered information system, Inform. sci., 178, 4, 1219-1234, (2008) · Zbl 1134.68057
[47] Yang, X.B.; Zhang, M.; Dou, H.L.; Yang, J.Y., Neighborhood systems-based rough sets in incomplete information system, Knowl. based syst., 24, 6, 858-867, (2011)
[48] Yang, X.B.; Zhang, M., Dominance-based fuzzy rough approach to an interval-valued decision system, Front. comput. sci. China, 5, 2, 195-204, (2011) · Zbl 1267.68300
[49] Yang, X.B.; Lin, T.Y.; Yang, J.Y.; Li, Y.; Yu, D.J., Combination of interval-valued fuzzy set and soft set, Comput. math. appl., 58, 3, 521-527, (2009) · Zbl 1189.03064
[50] Yang, X.B.; Yu, D.J.; Yang, J.Y.; Wei, L.H., Dominance-based rough set approach to incomplete interval-valued information system, Data knowl. eng., 68, 11, 1331-1347, (2009)
[51] Yang, X.B.; Yu, D.J.; Yang, J.Y.; Song, X.N., Difference relation-based rough set and negative rules in incomplete information system, Int. J. uncertain. fuzz., 17, 5, 649-665, (2009) · Zbl 1185.68696
[52] Yang, X.B.; Xie, J.; Song, X.N.; Yang, J.Y., Credible rules in incomplete decision system based on descriptors, Knowl. based syst., 22, 1, 8-17, (2009)
[53] Yao, Y.Y., Neighborhood systems and approximate retrieval, Inform. sci., 176, 23, 3431-3452, (2006) · Zbl 1119.68074
[54] Zadeh, L.A., Fuzzy set and information granularity, (), 3-18 · Zbl 0377.04002
[55] Zakowski, W., Approximations in the space, Demonst. math., 16, 761-769, (1983) · Zbl 0553.04002
[56] Ziarko, W., Variable precision rough sets model, J. comput. syst. sci., 46, 1, 39-59, (1993) · Zbl 0764.68162
[57] Zhao, J.; Liu, L., Construction of concept granule based on rough set and representation of knowledge-based complex system, Knowl. based syst., 24, 6, 809-815, (2011)
[58] Zhang, Y.L.; Luo, M.K., On minimization of axiom sets characterizing covering-based approximation operators, Inform. sci., 181, 14, 3032-3042, (2011) · Zbl 1216.68300
[59] Zhang, Y.L.; Li, J.J.; Wu, W.Z., On axiomatic characterizations of three pairs of covering based approximation operators, Inform. sci., 180, 2, 274-287, (2010) · Zbl 1186.68470
[60] Zhu, P., Covering rough sets based on neighborhoods: an approach without using neighborhoods, Int. J. approx. reason., 52, 3, 461-472, (2011) · Zbl 1229.03047
[61] W. Zhu, F.Y. Wang, A new type of covering rough sets, in: IEEE IS 2006, London, 4-6 September, 2006, pp. 444-449.
[62] Zhu, W., Topological approaches to covering rough sets, Inform. sci., 177, 6, 1499-1508, (2007) · Zbl 1109.68121
[63] Zhu, W., Relationship between generalized rough sets based on binary relation and covering, Inform. sci., 179, 3, 210-225, (2009) · Zbl 1163.68339
[64] Zhu, W., Relationship among basic concepts in covering-based rough sets, Inform. sci., 179, 14, 2478-2486, (2009) · Zbl 1178.68579
[65] Zhu, W., Generalized rough sets based on relations, Inform. sci., 177, 22, 4997-5011, (2007) · Zbl 1129.68088
[66] Zhu, W.; Wang, F.Y., On three types of covering rough sets, IEEE trans. knowl. data eng., 19, 8, 1131-1144, (2007)
[67] W. Zhu, F.Y. Wang, Properties of the first type of covering-based rough sets, in: Proceedings of DM Workshop 06, ICDM 06, Hong Kong, China, December 18, 2006, pp. 407-411.
[68] W. Zhu, Properties of the second type of covering-based rough sets, in: Workshop Proceedings of GrC& BI 06, IEEE WI 06, Hong Kong, China, December 18, 2006, pp. 494-497.
[69] W. Zhu, Properties of the fourth type of covering-based rough sets, in: HIS’06, AUT Technology Park, Auckland, New Zealand, December 13-15, 2006, pp. 43.
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