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Probabilistic rough set approximations. (English) Zbl 1191.68702
Summary: Probabilistic approaches have been applied to the theory of rough set in several forms, including decision-theoretic analysis, variable precision analysis, and information-theoretic analysis. Based on rough membership functions and rough inclusion functions, we revisit probabilistic rough set approximation operators and present a critical review of existing studies. Intuitively, they are defined based on a pair of thresholds representing the desired levels of precision. Formally, the Bayesian decision-theoretic analysis is adopted to provide a systematic method for determining the precision parameters by using more familiar notions of costs and risks. Results from existing studies are reviewed, synthesized and critically analyzed, and new results on the decision-theoretic rough set model are reported.

MSC:
68T37 Reasoning under uncertainty in the context of artificial intelligence
68T30 Knowledge representation
Software:
LERS; RSBR_
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[1] M.M.E. Abd El-Monsef, N.M. Kilany, Decision analysis via granulation based on general binary relation, International Journal of Mathematics and Mathematical Sciences 2007 (2007), doi:10.1155/2007/12714. Available at <http://www.hindawi.com/GetArticle.aspx?>. · Zbl 1138.62003
[2] R. Agrawal, T. Imielinski, A. Swami, Mining association rules between sets of items in large databases, in: Proceedings of ACM Special Interest Group on Management of Data, 1993, pp. 207-216.
[3] Beauboef, T.; Petry, F.E.; Arora, G., Information-theoretic measures of uncertainty for rough sets and rough relational databases, Information sciences, 109, 185-195, (1998)
[4] Bryniarski, E.; Wybraniec-Skardowska, U., Generalized rough sets in contextual space, (), 339-354
[5] Deogun, J.S.; Raghavan, V.V.; Sarkar, A.; Sever, H., Data mining: trends in research and development, (), 9-45
[6] J.S. Deogun, V.V. Raghavan, H. Sever, Rough set based classification methods and extended decision tables, in: Proceedings of the International Workshop on Rough Sets and Soft Computing, RSSC’04, 1994, pp. 302-309.
[7] Duda, R.O.; Hart, P.E., Pattern classification and scene analysis, (1973), Wiley New York · Zbl 0277.68056
[8] Düntsch, I.; Gediga, G., Roughian: rough information analysis, International journal of intelligent systems, 16, 121-147, (2001) · Zbl 0969.68147
[9] Eells, E.; Fitelson, B., Symmetries and asymmetries in evidential support, Philosophical studies, 107, 129-142, (2002)
[10] Fattorosi-Barnaba, M.; Amati, G., Modal operators with probabilistic interpretations I, Studia logica XLVI, 383-393, (1987) · Zbl 0645.03016
[11] S. Greco, B. Matarazzo, R. Sowiński, Rough membership and Bayesian confirmation measures for parameterized rough sets, in: Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing, Proceedings of RSFDGrC’05, LNAI, vol. 3641, 2005, pp. 314-324. · Zbl 1134.68531
[12] S. Greco, B. Matarazzo, R. Słowiński, Parameterized rough set model using rough membership and Bayesian confirmation measures, International Journal of Approximate Reasoning, (2007), in press, doi:10.1016/j.ijar.2007.05.018. · Zbl 1191.68678
[13] Grzymala-Busse, J.W., LERS - a system for learning from example based on rough sets, (), 3-18
[14] Katzberg, J.D.; Ziarko, W., Variable precision rough sets with asymmetric bounds, (), 167-177 · Zbl 0819.68041
[15] Klir, G.J.; Yuan, B., Fuzzy sets and fuzzy logic: theory and applications, (1995), Prentice-Hall New Jersey · Zbl 0915.03001
[16] W.D. Lee, S.R. Ray, Probabilistic inference for variable certainty decisions, in: Proceedings of the 14th ACM Annual Conference on Computer Science, 1986, pp. 482.
[17] Y. Li, C. Zhang, J.R. Swanb, Rough set based model in information retrieval and filtering, in: Proceedings of the 5th International Conference on Information Systems Analysis and Synthesis, 1999, pp. 398-403.
[18] Li, Y.; Zhang, C.; Swanb, J.R., An information filtering model on the web and its application in jobagent, Knowledge-based systems, 13, 285-296, (2000)
[19] ()
[20] Mi, J.S.; Wu, W.Z.; Zhang, W.X., Approaches to knowledge reduction based on variable precision rough set model, Information sciences, 159, 255-272, (2004) · Zbl 1076.68089
[21] Miao, D.; Hou, L., A comparison of rough set methods and representative inductive learning algorithms, Fundamenta informaticae, 59, 203-219, (2004) · Zbl 1098.68129
[22] Michalski, R.S.; Winston, P.H., Variable precision logic, Artificial intelligence, 29, 121-146, (1986) · Zbl 0623.68077
[23] Mitchell, T.M., Machine learning, (1997), McGraw-Hill New York · Zbl 0913.68167
[24] Nakamura, A.; Gao, J.M., On a KTB-modal fuzzy logic, Fuzzy sets and systems, 45, 327-334, (1992) · Zbl 0754.03014
[25] Nguyen, S.H.; Skowron, A.; Stepaniuk, J., Granular computing: a rough set approach, Computational intelligence, 17, 514-544, (2001)
[26] Pawlak, Z., Rough sets, International journal of computer and information sciences, 11, 341-356, (1982) · Zbl 0501.68053
[27] Pawlak, Z., Rough sets: theoretical aspects of reasoning about data, (1991), Kluwer Academic Publishers Boston · Zbl 0758.68054
[28] Pawlak, Z.; Skowron, A., Rough membership functions, (), 251-271 · Zbl 0794.03045
[29] Pawlak, Z.; Wong, S.K.M.; Ziarko, W., Rough sets: probabilistic versus deterministic approach, International journal of man – machine studies, 29, 81-95, (1988) · Zbl 0663.68094
[30] Polkowski, L.; Skowron, A., Rough mereology: a new paradigm for approximate reasoning, International journal of approximate reasoning, 15, 333-365, (1996) · Zbl 0938.68860
[31] Polkowski, L.; Skowron, A., Rough sets in knowledge discovery 1, vols. 1-2, (1998), Physica-Verlag Heidelberg
[32] Skowron, A.; Polkowski, L., Rough mereology and analytical morphology, (), 399-437
[33] Skowron, A.; Stepaniuk, J., Tolerance approximation spaces, Fundamenta informaticae, 27, 245-253, (1996) · Zbl 0868.68103
[34] Śle¸zak, D.; sets, Rough; factor, Bayes, LNCS transactions on rough sets III, Lncs, 3400, 202-229, (2005)
[35] Śle¸zak, D.; Ziarko, W., The investigation of the Bayesian rough set model, International journal of approximate reasoning, 40, 81-91, (2005) · Zbl 1099.68089
[36] Srinivasan, P.; Ruiz, M.E.; Kraft, D.H.; Chen, J., Vocabulary mining for information retrieval: rough sets and fuzzy sets, Information processing and management, 37, 15-38, (2001) · Zbl 1011.68589
[37] Stefanowski, J., On rough set based approaches to induction of decision rules, () · Zbl 0927.68094
[38] Tsumoto, S., Automated extraction of medical expert system rules from clinical databases on rough set theory, Information sciences, 112, 67-84, (1998)
[39] S. Tsumoto, Accuracy and coverage in rough set rule induction, in: Rough Sets and Current Trends in Computing, Proceedings of RSCTC’02, LNAI, vol. 2475, 2002, pp. 373-380. · Zbl 1013.68567
[40] van Rijsbergen, C.J., Information retrieval, (1979), Butterworths London · Zbl 0227.68052
[41] Wang, G.Y., Rough reduction in algebra view and information view, International journal of intelligent systems, 18, 679-688, (2003) · Zbl 1037.68138
[42] Wei, L.L.; Zhang, W.X., Probabilistic rough sets characterized by fuzzy sets, International journal of uncertainty, fuzziness and knowledge-based systems, 12, 47-60, (2004) · Zbl 1074.68073
[43] Wong, S.K.M.; Ziarko, W., Comparison of the probabilistic approximate classification and the fuzzy set model, Fuzzy sets and systems, 21, 357-362, (1987) · Zbl 0618.60002
[44] W.Z. Wu, Upper and lower probabilities of fuzzy events induced by a fuzzy set-valued mapping, in: Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing, Proceedings of RSFDGrC’05, LNAI, vol. 3641, 2005, pp. 345-353. · Zbl 1134.68563
[45] Wu, W.Z.; Leung, Y.; Zhang, W.X., On generalized rough fuzzy approximation operators, LNCS transactions on rough sets V, Lncs, 4100, 263-284, (2006)
[46] Xu, Z.B.; Liang, J.Y.; Dang, C.Y.; Chin, K.S., Inclusion degree: a perspective on measures for rough set data analysis, Information sciences, 141, 227-236, (2002) · Zbl 1008.68134
[47] J.T. Yao, J.P. Herbert, Web-based support systems based on rough set analysis, in: Rough Sets and Emerging Intelligent Systems Paradigms, Proceedings of RSEISP’07, LNAI, vol. 4585, 2007, pp. 360-370.
[48] J.T. Yao, M. Zhang, Feature selection with adjustable criteria, in: Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing, Proceedings of RSFDGrC’05, LNAI, vol. 3641, 2005, pp. 204-213.
[49] Yao, Y.Y., Two views of the theory of rough sets in finite universes, International journal of approximation reasoning, 15, 291-317, (1996) · Zbl 0935.03063
[50] Y.Y. Yao, On generalizing Pawlak approximation operators, in: Rough Sets and Current Trends in Computing, Proceedings of RSCTC’98, LNAI, vol. 1424, 1998, pp. 298-307. · Zbl 0955.68505
[51] Yao, Y.Y., Information granulation and rough set approximation, International journal of intelligent systems, 16, 87-104, (2001) · Zbl 0969.68079
[52] Yao, Y.Y., Information granulation and approximation in a decision-theoretical model of rough sets, (), 491-518
[53] Y.Y. Yao, On generalizing rough set theory, in: Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing, Proceedings of RSFDGrC’03, LNAI, vol. 2639, 2003, pp. 44-51.
[54] Yao, Y.Y., Probabilistic approaches to rough sets, Expert systems, 20, 287-297, (2003)
[55] Yao, Y.Y., Semantics of fuzzy sets in rough set theory, LNCS transactions on rough sets II, Lncs, 3135, 297-318, (2004)
[56] Yao, Y.Y.; Wong, S.K.M., A decision theoretic framework for approximating concepts, International journal of man – machine studies, 37, 793-809, (1992)
[57] Yao, Y.Y.; Wong, S.K.M.; Lin, T.Y., A review of rough set models, (), 47-75 · Zbl 0861.68101
[58] Yao, Y.Y.; Wong, S.K.M.; Lingras, P., A decision-theoretic rough set model, (), 17-24
[59] Zadeh, L.A., Fuzzy sets, Information and control, 8, 338-353, (1976) · Zbl 0139.24606
[60] Zhang, M.; Xu, L.D.; Zhang, W.X.; Li, H.Z., A rough set approach to knowledge reduction based on inclusion degree and evidence reasoning theory, Expert systems, 20, 298-304, (2003)
[61] W.X. Zhang, Y. Leung, Theory of including degrees and its applications to uncertainty inference, in: Soft Computing in Intelligent Systems and Information Processing, Proceedings of 1996 Asian Fuzzy System Symposium, 1996, pp. 496-501.
[62] Zhang, W.X.; Wu, W.Z.; Liang, J.Y.; Li, D.Y., Rough set theory and methodology, (2001), Xi’an Jiaotong University Press Xi’an, China, (in Chinese)
[63] W.Q. Zhao, Y.L. Zhu, An email classification scheme based on decision-theoretic rough set theory and analysis of email security, in: Proceedings of 2005 IEEE Region 10 TENCON, doi: 10.1109/TENCON.2005.301121.
[64] Zhong, N.; Dong, J.Z.; Ohsuga, S., Data mining: a probabilistic rough set approach, (), 127-146
[65] Zhong, N.; Skowron, A., A rough set-based knowledge discovery process, International journal of mathematics and computer science, 11, 603-619, (2001) · Zbl 0990.68139
[66] Ziarko, W., Variable precision rough set model, Journal of computer and system science, 46, 39-59, (1993) · Zbl 0764.68162
[67] Ziarko, W., Acquisition of hierarchy-structured probabilistic decision tables and rule from data, Expert systems, 20, 305-310, (2003)
[68] W. Ziarko, Probabilistic rough sets, in: Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing, Proceedings of RSFDGrC’05, LNAI, vol. 3641, 2005, pp. 283-293. · Zbl 1134.68567
[69] W. Ziarko, Probabilistic approach to rough sets, International Journal of Approximate Reasoning (2007), in press, doi:10.1016/j.ijar.2007.06.014. · Zbl 1191.68705
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