Learning fuzzy rules from fuzzy samples based on rough set technique. (English) Zbl 1129.68069
Summary: Although the traditional rough set theory has been a powerful mathematical tool for modeling incompleteness and vagueness, its performance in dealing with initial fuzzy data is usually poor. This paper makes an attempt to improve its performance by extending the traditional rough set approach to the fuzzy environment. The extension is twofold. One is knowledge representation and the other is knowledge reduction. First, we provide new definitions of fuzzy lower and upper approximations by considering the similarity between the two objects. Second, we extend a number of underlying concepts of knowledge reduction (such as the reduct and core) to the fuzzy environment and use these extensions to propose a heuristic algorithm to learn fuzzy rules from initial fuzzy data. Finally, we provide some numerical experiments to demonstrate the feasibility of the proposed algorithm. One of the main contributions of this paper is that the fundamental relationship between the reducts and core of rough sets is still pertinent after the proposed extension.
MSC:
| 68T05 | Learning and adaptive systems in artificial intelligence |
References:
| [1] | Bhatt, R. B.; Gopal, M., On fuzzy rough sets approach to feature selection, Pattern Recognition Letters, 26, 1632-1640 (2005) |
| [2] | Chakrabarty, K.; Biswas, R.; Nanda, S., Fuzziness in rough sets, Fuzzy Sets and Systems, 110, 247-251 (2000) · Zbl 0943.03040 |
| [3] | Chen, D. G.; Wang, X. Z.; Yeung, D. S.; Tsang, E. C.C., Rough approximations on a complete completely distributive lattice with applications to generalized rough sets, Information Sciences, 176, 1829-1848 (2006) · Zbl 1104.03053 |
| [4] | Dubois, D.; Prade, H., Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems, 17, 191-208 (1990) · Zbl 0715.04006 |
| [5] | Dubois, D.; Prade, H., Putting rough sets and fuzzy sets together, (Slowinski, R., Intelligent Decision Support, Handbook of Applications and Advances of the Rough Sets Theory (1992), Kluwer Academic Publishers) · Zbl 0607.00019 |
| [6] | Greco, S.; Matarazzo, B.; Slowinski, R., Rough set processing of vague information using fuzzy similarity relations, (Calude, C. S.; Paun, G., Finite Versus Infinite-Contributions to an Eternal Dilemma (2002), Springer-Verlag: Springer-Verlag London), 149-173 |
| [7] | Guan, J. W.; Bell, D. A.; Guan, Z., Matrix computation for information system, Information Sciences, 131, 129-156 (2001) · Zbl 0976.68149 |
| [8] | T.P. Hong, Learning approximate fuzzy rules from training examples, in: The Tenth IEEE International Conference on Fuzzy Systems, 2001.; T.P. Hong, Learning approximate fuzzy rules from training examples, in: The Tenth IEEE International Conference on Fuzzy Systems, 2001. |
| [9] | Hong, T. P.; Wang, T. T.; Wang, S. L.; Chien, B. C., Learning a coverage set of maximally general fuzzy rules by rough sets, Expert Systems with Applications, 19, 97-103 (2000) |
| [10] | http://www.ics.uci.edu/ mlearn/MLRepository.html; http://www.ics.uci.edu/ mlearn/MLRepository.html |
| [11] | Inuiguchi, M.; Greco, S.; Slowinski, R.; Tanino, T., Possibility and necessity measure specification using modifiers for decision making under fuzziness, Fuzzy Sets and Systems, 137, 151-175 (2003) · Zbl 1076.91011 |
| [12] | Jensen, R.; Shen, Q., Fuzzy-rough attributes reduction with application to web categorization, Fuzzy Sets and Systems, 141, 469-485 (2004) · Zbl 1069.68609 |
| [13] | Kohonen, T., Fuzzy entropy and conditioning, Information Sciences, 30, 165-174 (1986) · Zbl 0623.94005 |
| [14] | Mi, J. S.; Zhang, W. X., An axiomatic characterization of a fuzzy generalization of rough sets, Information Sciences, 160, 235-249 (2004) · Zbl 1041.03038 |
| [15] | Morsi, N. N.; Yakout, M. M., Axiomatics for fuzzy rough sets, Fuzzy Sets and Systems, 100, 327-342 (1998) · Zbl 0938.03085 |
| [16] | Pawlak, Z., Rough sets, International Journal of Computer and Information Sciences, 11, 341-356 (1982) · Zbl 0501.68053 |
| [17] | Pawlak, Z., Rough Sets: Theoretical Aspects of Reasoning about Data [M] (1991), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0758.68054 |
| [18] | Pawlak, Z.; Skowron, A., Rough sets: some extensions, Information Sciences, 177, 28-40 (2007) · Zbl 1142.68550 |
| [19] | Quafafou, M., a-RST: A generalization of rough set theory, Information Sciences, 124, 301-316 (2000) · Zbl 0957.68114 |
| [20] | Quinlan, J. R., Induction of decision trees, Machine Learning, 1, 81-106 (1986) |
| [21] | Shen, Q.; Jensen, R., Selecting informative features with fuzzy-rough sets and its application for complex systems monitoring, Pattern Recognition, 37, 1351-1363 (2004) · Zbl 1070.68600 |
| [22] | R. Slowinski, D. Vanderpooten, Similarity relation as a basis for rough approximations, in: P.P. Wang (Ed.), Advances in Machine Intelligence and Soft-Computing, Department of Electrical Engineering, Duke University, Durham, NC, USA, 1997, pp. 17-33.; R. Slowinski, D. Vanderpooten, Similarity relation as a basis for rough approximations, in: P.P. Wang (Ed.), Advances in Machine Intelligence and Soft-Computing, Department of Electrical Engineering, Duke University, Durham, NC, USA, 1997, pp. 17-33. |
| [23] | H. Thiele, Fuzzy rough sets versus rough fuzzy sets—an interpretation and a comparative study using concepts of modal logics, University of Dortmund, Tech. Report No. CI-30=98, 1998, also in Proc. EUFIT’1997, pp. 159-167.; H. Thiele, Fuzzy rough sets versus rough fuzzy sets—an interpretation and a comparative study using concepts of modal logics, University of Dortmund, Tech. Report No. CI-30=98, 1998, also in Proc. EUFIT’1997, pp. 159-167. |
| [24] | H. Thiele, On axiomatic characterization of fuzzy approximation operators I, the fuzzy rough set based case, in: RSCTC 2000 Conference Proceedings, Banff Park Lodge, Bariff, Canada, 19 October 2000, pp. 239-247.; H. Thiele, On axiomatic characterization of fuzzy approximation operators I, the fuzzy rough set based case, in: RSCTC 2000 Conference Proceedings, Banff Park Lodge, Bariff, Canada, 19 October 2000, pp. 239-247. |
| [25] | H. Thiele, On axiomatic characterization of fuzzy approximation operators II, the rough fuzzy set based case, in: Proceedings of the 31st IEEE International Symposium on Multiple-Valued Logic, 2001, pp. 330-335.; H. Thiele, On axiomatic characterization of fuzzy approximation operators II, the rough fuzzy set based case, in: Proceedings of the 31st IEEE International Symposium on Multiple-Valued Logic, 2001, pp. 330-335. |
| [26] | E.C.C. Tsang, D.G. Chen, D.S. Yeung, X.Z. Wang, J.W.T. Lee, Attributes reduction using fuzzy rough sets, IEEE Transaction on Fuzzy System, in press.; E.C.C. Tsang, D.G. Chen, D.S. Yeung, X.Z. Wang, J.W.T. Lee, Attributes reduction using fuzzy rough sets, IEEE Transaction on Fuzzy System, in press. |
| [27] | Wang, X. Z.; Hong, J. R., Learning optimization in simplifying fuzzy rules, Fuzzy Sets and Systems, 106, 349-356 (1999) · Zbl 0978.68560 |
| [28] | Wang, X. Z.; Yeung, D. S.; Tsang, E. C.C., A comparative study on heuristic algorithms for generating fuzzy decision trees, IEEE Transactions on Systems, Man, and Cybernetics—Part B: Cybernetics, 31, 215-236 (2001) |
| [29] | Wu, W. Z.; Mi, J. S.; Zhang, W. X., Generalized fuzzy rough sets, Information Sciences, 151, 263-282 (2003) · Zbl 1019.03037 |
| [30] | Wu, W. Z.; Zhang, W. X., Constructive and axiomatic approaches of fuzzy approximation operators, Information Sciences, 159, 233-254 (2004) · Zbl 1071.68095 |
| [31] | Yao, Y. Y., Generalized rough set model, (Polkowski, L.; Skowron, A., Rough Sets in Knowledge Discovery. 1. Methodology and Applications (1998), Physica-Verlag: Physica-Verlag Heidelberg), 286-318 · Zbl 0946.68137 |
| [32] | Yao, Y. Y., Combination of rough and fuzzy sets based an a-level sets, (Lin, T. Y.; Cercone, N., Rough Sets and Data Mining: Analysis for Imprecise Data (1997), Kluwer Academic Publishers: Kluwer Academic Publishers Boston), 301-321 · Zbl 0859.04005 |
| [33] | Yeung, D. S.; Chen, D. G.; Tsang, E. C.C.; Lee, J. W.T.; Wang, X. Z., On the generalization of fuzzy rough sets, IEEE Transactions on Fuzzy Systems, 13, 343-361 (2005) |
| [34] | Yuan, Y. F.; Shaw, M. J., Introduction of fuzzy decision tree, Fuzzy Sets and Systems, 69, 125-139 (1995) |
| [35] | Zhang, W. X.; Wu, W. Z.; Liang, J. Y.; Li, D. Y., The Theory and Methodology of Rough Sets (2001), The Science Press: The Science Press Beijing, (in Chinese) |
| [36] | Zhu, W., Topological approaches to covering rough sets, Information Sciences, 177, 1499-1508 (2007) · Zbl 1109.68121 |
| [37] | Zhu, W.; Wang, F. Y., Some results on the covering generalized rough sets, International Journal of Pattern Recognition and Artificial Intelligence, 5, 6-13 (2002) |
| [38] | Zhu, W.; Wang, F. Y., Reduction and axiomization of covering generalized rough sets, Information Sciences, 152, 217-230 (2003) · Zbl 1069.68613 |
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