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Issue No.03 - March (2009 vol.21)

pp: 443-462

Witold Pedrycz , University of Alberta, Edmonton

Tianyou Chai , Northwestern University, Shenyang

Xiaodong Liu , Dalian University of Technology, Dalian

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TKDE.2008.147

ABSTRACT

The notion of a rough set was originally proposed by Pawlak underwent a number of extensions and generalizations. Dubois and Prade (1990) introduced fuzzy rough sets which involve the use of rough sets and fuzzy sets within a single framework. Radzikowska and Kerre (2002) proposed a broad family of fuzzy rough sets, referred to as ( t)-fuzzy rough sets which are determined by some implication operator (implicator), and a certain t-norm. In order to describe the linguistically represented concepts coming from data available in some information system, the concept of fuzzy rough sets are redefined and further studied in the setting of the Axiomatic Fuzzy Set (AFS) theory. Compared with the ( t)-fuzzy rough sets, the advantages of AFS fuzzy rough sets are twofold. They can be directly applied to data analysis present in any information system without resorting to the details concerning the choice of the implication, t-norm and a similarity relation S. Furthermore such rough approximations of fuzzy concepts come with a well-defined semantics and therefore offer a sound interpretation. Some examples are included to illustrate the effectiveness of the proposed construct. It is shown that the AFS fuzzy rough sets provide a far higher flexibility and effectiveness in comparison with rough sets and some of their generalizations.

INDEX TERMS

rough sets, fuzzy rough sets

CITATION

Witold Pedrycz, Tianyou Chai, Xiaodong Liu, "The Development of Fuzzy Rough Sets with the Use of Structures and Algebras of Axiomatic Fuzzy Sets",

*IEEE Transactions on Knowledge & Data Engineering*, vol.21, no. 3, pp. 443-462, March 2009, doi:10.1109/TKDE.2008.147REFERENCES

- [1] Z. Pawlak, “Rough Sets,”
Int'l J. Computer and Information Sciences, vol. 11, no. 5, pp. 341-356, 1982.- [2] Z. Pawlak, “Rough Probability,”
Bull. Polish Academy of Sciences, Math., vol. 32, pp. 607-612, 1984.- [3] Z. Pawlak, “Hard and Soft Sets,”
Rough Sets, Fuzzy Sets and Knowledge Discovery, W.P. Ziarko, ed., pp. 130-135, Springer, 1994.- [4] D. Dubois and H. Prade, “Rough Fuzzy Sets and Fuzzy Rough Sets,”
Int'l J. General Systems, vol. 17, nos. 2/3, pp. 191-209, 1990.- [5] A.M. Radzikowska and E.E. Kerre, “A Comparative Study of Fuzzy Rough Sets,”
Fuzzy Sets and Systems, vol. 126, pp. 137-155, 2002.- [6] J.M. Fernandez Salido and S. Murakami, “Rough Set Analysis of a General Type of Fuzzy Data Using Transitive Aggregations of Fuzzy Similarity Relations,”
Fuzzy Sets and Systems, vol. 139, pp.635-660, 2003.- [7] X.D. Liu, “The Fuzzy Theory Based on AFS Algebras and AFS Structure,”
J. Math. Analysis and Applications, vol. 217, pp.459-478, 1998.- [8] X.D. Liu, “The Topology on AFS Algebra and AFS Structure,”
J.Math. Analysis and Applications, vol. 217, pp. 479-489, 1998.- [9] X.D. Liu, “The Fuzzy Sets and Systems Based on AFS Structure, EIAlgebra and EII Algebra,”
Fuzzy Sets and Systems, vol. 95, pp.179-188, 1998.- [10] X.D. Liu, W. Pedrycz, and Q.L. Zhang, “Axiomatics Fuzzy Sets Logic,”
Proc. 12th IEEE Int'l Conf. Fuzzy Systems (FUZZ '03), vol. 1, pp. 55-60, 2003.- [11] X.D. Liu, T.Y. Chai, and W. Wang, “Approaches to the Representations and Logic Operations for Fuzzy Concepts in the Framework of Axiomatic Fuzzy Set Theory I, II,”
Information Sciences, vol. 177, pp. 1007-1026, pp. 1027-1045, 2007.- [12] X.D. Liu, W. Wang, and T.Y. Chai, “The Fuzzy Clustering Analysis Based on AFS Theory,”
IEEE Trans. Systems, Man and Cybernetics, Part B, vol. 35, no. 5, pp. 1013-1027, 2005.- [13] X.D. Liu and W. Pedrycz, “The Development of Fuzzy Decision Trees in the Framework of Axiomatic Fuzzy Set Logic,”
Applied Soft Computing, vol. 7, pp. 325-342, 2007.- [14] J.E. Graver and M.E. Watkins,
Combinatorics with Emphasis on the Theory of Graphs. Springer, 1977.- [15] G.J. Wang, “Theory of Topological Molecular Lattices,”
Fuzzy Sets and Systems, vol. 47, pp. 351-376, 1992.- [16] L.A. Zadeh, “Fuzzy Logic=Computing with Words,”
IEEE Trans. Fuzzy Systems, vol. 4, no. 2, pp. 103-111, 1996.- [17] P.R. Halmos,
Measure Theory. Springer, 1974.- [18] A. Skowron and C. Rauszer, “The Discernibility Matrices and Functions in Information Systems, Intelligent Decision Support,”
Handbook of Applications and Advances of the Rough Sets Theory, R.Slowinski, ed., pp. 331-362, Kluwer Academic Publishers, 1992.- [19] S.K. Pal and P. Mitra, “Case Generation Using Rough Sets with Fuzzy Representation,”
IEEE Trans. Knowledge and Data Eng., vol. 16, no. 3, pp. 292-300, Mar. 2004.- [20] Z. Bonikowski, “Algebraic Structures of Rough Sets in Representative Approximation Spaces,”
Electronic Notes in Theoretical Computer Science, vol. 82, no. 4, pp. 1-12, http://www.elsevier. nl/locate/entcsvolume82.html , Mar. 2003.- [21] K. Dembczynski, R. Pindur, and R. Susmag, “Generation of Exhaustive Set of Rules within Dominance-Based Rough Set Approach,”
Electronic Notes in Theoretical Computer Science, vol. 82, no. 4, pp. 1-12, http://www.elsevier.nl/locate/entcsvolume82.html , Mar. 2003.- [22] N. Zhong, J.Z. Dong, and S. Ohsuga, “Meningitis Data Mining by Cooperatively Using GDT-RS and RSBR,”
Pattern Recognition Letters, vol. 24, pp. 887-894, 2003.- [23] H.S. Nguyen, “On the Decision Table with Maximal Number of Reducts,”
Electronic Notes in Theoretical Computer Science, vol. 82, no. 4, pp. 1-8, http://www.elsevier.nl/locate/entcsvolume82.html , Mar. 2003.- [24] Q.H. Wang and J.R. Li, “A Rough Set-Based Fault Ranking Prototype System for Fault Diagnosis,”
Eng. Applications of Artificial Intelligence, vol. 17, pp. 909-917, 2004.- [25] S. Greco, B. Matarazzo, and R. Slowinski, “Axiomatic Characterization of a General Utility Function and Its Particular Cases in Terms of Conjoint Measurement and Rough-Set Decision Rules,”
European J. Operational Research, vol. 158, pp. 271-292, 2004.- [26] S. Greco, B. Matarazzo, and R. Slowinski, “Rough Sets Theory for Multicriteria Decision Analysis,”
European J. Operational Research, vol. 129, pp. 1-47, 2001.- [27] S. Greco, B. Matarazzo, and R. Slowinski, “Rough Approximation of a Preference Relation by Dominance Relations,”
European J.Operational Research, vol. 117, pp. 63-83, 1999.- [28] S. Greco, B. Matarazzo, and R. Slowinski, “Rough Sets Methodology for Sorting Problems in Presence of Multiple Attributes and Criteria,”
European J. Operational Research, vol. 138, pp. 247-259, 2002.- [29] K. Zaras, “Rough Approximation of a Preference Relation by a Multi-Attribute Stochastic Dominance for Determinist and Stochastic Evaluation Problems,”
European J. Operational Research, vol. 130, pp. 305-314, 2001.- [30] R. Slowinski and D. Vanderpooten, “A Generalized Definition of Rough Approximations Based on Similarity,”
IEEE Trans. Knowledge and Data Eng., vol. 12, no. 2, pp. 331-336, Mar./Apr. 2000.- [31] D. Kim and S.Y. Bang, “A Handwritten Numeral Character Classification Using Tolerant Rough Set,”
IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 22, no. 9, pp. 923-937, Sept. 2000.- [32] L. Geng and C.W. Chan, “An Algorithm for Case Generation from a Database,”
Applied Math. Letters, vol. 17, pp. 269-274, 2004.- [33] D. Kim, “Data Classification Based on Tolerant Rough Set,”
Pattern Recognition, vol. 34, pp. 1613-1624, 2001.- [34] C.J. Merz and P.M. Murphy,
UCI Repository for Machine Learning Data-Bases, Dept. of Information and Computer Science, Univ. of California, http://www.ics.uci.edu/~mlearnMLRepository. html , 1996.- [35] Z. Pawlak and A. Skowron, “Rudiments of Rough Sets,”
Information Sciences, vol. 177, pp. 3-27, 2007.- [36] K.H. Kim,
Boolean Matrix Theory and Applications. Marcel Dekker, 1982.- [37] X.D. Liu, K.J. Zhu, and H.Z. Huang, “The Representations of Fuzzy Concepts Based on the Fuzzy Matrix Theory and the AFS Theory,”
Proc. IEEE Int'l Symp. Intelligent Control (ISIC '03), pp.1006-1011, Oct. 2003.- [38] X.D. Liu, “The Structure of Fuzzy Matrices,”
J. Fuzzy Math., vol. 2, pp. 311-325, 1994.- [39] Z. Pawlak and A. Skowron, “Rough Sets: Some Extensions,”
Information Sciences, vol. 177, pp. 28-40, 2007.- [40] Z. Pawlak and A. Skowron, “Rough Sets and Boolean Reasoning,”
Information Sciences, vol. 177, pp. 41-70, 2007.- [41] L. Polkowski and A. Skowron, “Rough Mereology: A New Paradigm for Approximate Reasoning,”
Int'l J. Approximate Reasoning, vol. 15, pp. 333-365, 1996.- [42] S. Lesniewski,
Foundations of the General Theory of Sets. Polish Scientific Circle, 1916.- [43] X.D. Liu and W. Pedrycz,
Axiomatic Fuzzy Set Theory and Its Application. Springer-Verlag, in press. |