Abstract
Rough sets are applied to information tables containing imprecise values that are expressed in a probability distribution. A family of weighted equivalence classes is obtained where each equivalence class is accompanied by the probability to which it is an actual one. By using the family of weighted equivalence classes, we derive lower and upper approximations. The lower and upper approximations coincide with ones obtained from methods of possible worlds. Therefore, the method of weighted equivalence classes is justified. In addition, this method is applied to missing values interpreted probabilistically. Using weighted equivalence classes correctly derives a lower approximation, even in the case where the method of Kryszkiewicz does not derive any lower approximation.
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Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases. Addison-Wesley, London, UK (1995)
Greco, S., Matarazzo, B., Slowinski, R.: Handling Missing Values in Rough Set Analysis of Multi-attribute and Multi-criteria Decision Problem. In: Zhong, N., Skowron, A., Ohsuga, S. (eds.) New Directions in Rough Sets, Data Mining, and Granular-Soft Computing. LNCS (LNAI), vol. 1711, pp. 146–157. Springer, Heidelberg (1999)
Grzymala-Busse, J.W.: On the Unknown Attribute Values in Learning from Examples. In: Raś, Z.W., Zemankova, M. (eds.) ISMIS 1991. LNCS(LNAI), vol. 542, pp. 368–377. Springer, Heidelberg (1991)
Grzymala-Busse, J.W: Data with Missing Attribute Values: Generalization of Indiscernibility Relation and Rule Induction, Transactions on Rough Sets I, pp. 78–95 (2004)
Grzymala-Busse, J.W.: Characteristic Relations for Incomplete Data: A Generalization of the Indiscernibility Relation, Transactions on Rough Sets IV, pp. 58–68 (2005)
Guan, Y.-Y., Wang, H.-K.: Set-valued Information Systems. Information Sciences 176, 2507–2525 (2006)
Imielinski, T., Lipski, W.: Incomplete Information in Relational Databases. Journal of the ACM 31(4), 761–791 (1984)
Kryszkiewicz, M.: Rough Set Approach to Incomplete Information Systems. Information Sciences 112, 39–49 (1998)
Kryszkiewicz, M.: Properties of Incomplete Information Systems in the framework of Rough Sets. In: Polkowski, L., Skowron, A. (eds.) Rough Set in Knowledge Discovery 1: Methodology and Applications. Studies in Fuzziness and Soft Computing, vol. 18, pp. 422–450. Physica Verlag, Heidelberg (1998)
Kryszkiewicz, M.: Rules in Incomplete Information Systems. Information Sciences 113, 271–292 (1999)
Latkowski, R.: On Decomposition for Incomplete Data. Fundamenta Informaticae 54, 1–16 (2003)
Leung, Y., Li, D.: Maximum Consistent Techniques for Rule Acquisition in Incomplete Information Systems. Information Sciences 153, 85–106 (2003)
Nakata, N., Sakai, H.: Rough-set-based approaches to data containing incomplete information: possibility-based cases, pp. 234–241. IOS Press, Amsterdam, Trento, Italy (2005)
Nakata, N., Sakai, H.: Checking Whether or Not Rough-Set-Based Methods to Incomplete Data Satisfy a Correctness Criterion. In: Torra, V., Narukawa, Y., Miyamoto, S. (eds.) MDAI 2005. LNCS (LNAI), vol. 3558, pp. 227–239. Springer, Heidelberg (2005)
Nakata, N., Sakai, H.: Rough Sets Handling Missing Values Probabilistically Interpreted. In: Ślęzak, D., Wang, G., Szczuka, M., Düntsch, I., Yao, Y. (eds.) RSFDGrC 2005. LNCS (LNAI), vol. 3641, pp. 325–334. Springer, Heidelberg (2005)
Nakata, N., Sakai, H.: Lower and Upper Approximations in Data Tables Containing Possibilistic Information, Transactions on Rough Sets VII, 170–189 (2007)
Orłowska, E., Pawlak, Z.: Representation of Nondeterministic Information. Theoretical Computer Science 29, 313–324 (1984)
Parsons, S.: Current Approaches to Handling Imperfect Information in Data and Knowledge Bases. IEEE Transactions on Knowledge and Data Engineering 8(3), 353–372 (1996)
Pawlak, Z.: Rough Sets. International Journal of Computer and Information Sciences 11, 341–356 (1982)
Sakai, H.: Effective Procedures for Handling Possible Equivalence Relation in Non-deterministic Information Systems. Fundamenta Informaticae 48, 343–362 (2001)
Sakai, H., Nakata, M.: An Application of Discernibility Functions to Generating Minimal Rules in Non-deterministic Information Systems. Journal of Advanced Computational Intelligence and Intelligent Informatics 10, 695–702 (2006)
Sakai, H., Okuma, A.: Basic Algorithms and Tools for Rough Non-deterministic Information Systems, Transactions on Rough Sets I, pp. 209–231 (2004)
Słowiński, R., Stefanowski, J.: Rough Classification in Incomplete Information Systems. Mathematical and Computer Modelling 12(10/11), 1347–1357 (1989)
Stefanowski, J., Tsoukiàs, A.: On the Extension of Rough Sets under Incomplete Information. In: Zhong, N., Skowron, A., Ohsuga, S. (eds.) New Directions in Rough Sets, Data Mining, and Granular-Soft Computing. LNCS (LNAI), vol. 1711, pp. 212–219. Springer, Heidelberg (1999)
Stefanowski, J., Tsoukiàs, A.: Incomplete Information Tables and Rough Classification. Computational Intelligence 17(3), 545–566 (2001)
Ziarko, W.: Probabilistic Rough Sets. In: Ślęzak, D., Wang, G., Szczuka, M., Düntsch, I., Yao, Y. (eds.) RSFDGrC 2005. LNCS (LNAI), vol. 3641, pp. 283–293. Springer, Heidelberg (2005)
Ziarko, W.: Stochastic Approach to Rough Set Theory. In: Greco, S., Hata, Y., Hirano, S., Inuiguchi, M., Miyamoto, S., Nguyen, H.S., Słowiński, R. (eds.) RSCTC 2006. LNCS (LNAI), vol. 4259, pp. 38–48. Springer, Heidelberg (2006)
Zimányi, E., Pirotte, A.: Imperfect Information in Relational Databases. In: Motro, A., Smets, P. (eds.) Uncertainty Management in Information Systems: From Needs to Solutions, pp. 35–87. Kluwer Academic Publishers, Boston, MA (1997)
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Nakata, M., Sakai, H. (2007). Applying Rough Sets to Information Tables Containing Probabilistic Values . In: Torra, V., Narukawa, Y., Yoshida, Y. (eds) Modeling Decisions for Artificial Intelligence. MDAI 2007. Lecture Notes in Computer Science(), vol 4617. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73729-2_27
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DOI: https://doi.org/10.1007/978-3-540-73729-2_27
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