Abstract
The rough sets theory has proved to be a very useful tool for analysis of information tables describing objects by means of disjoint subsets of condition and decision attributes. The key idea of rough sets is approximation of knowledge expressed by decision attributes using knowledge expressed by condition attributes. From a formal point of view, the rough sets theory was originally founded on the idea of approximating a given set represented by objects having the same description in terms of decision attributes, by means of an indiscernibility binary relation linking pairs of objects having the same description by condition attributes. The indiscernibility relation is an equivalence binary relation (reflexive, symmetric and transitive) and implies an impossibility to distinguish two objects having the same description in terms of the condition attributes. It produces crisp granules of knowledge that are used to build approximations. In reality, due to vagueness of the available information about objects, small differences are not considered significant. This situation may be formally modelled by similarity or tolerance relations instead of the indiscernibility relation. We are using a similarity relation which is only reflexive, relaxing therefore the properties of symmetry and transitivity.
Moreover, the credibility of our similarity relation is gradual, i.e., it is a fuzzy similarity relation. As a consequence, the granules of knowledge produced by this relation are fuzzy and their credibility can change gradually from any finite degree to an infinitely small degree, thus giving different credibility of rough approximations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980.
D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets, Int. J. of General Systems, 17 (1990), 191–200.
D. Dubois, H. Prade, Putting rough sets and fuzzy sets together, in Intelligent Decision Support, Handbook of Applications and Advances of the Rough Sets Theory (R. Slowinski, ed.), Kluwer, Dordrecht, 1992, 203–233.
J. Fodor, M. Roubens, Fuzzy Preference Modelling and Multicriteria Decision Support, Kluwer, Dordrecht, 1994.
S. Greco, B. Matarazzo, R. Slowinski, The use of rough sets and fuzzy sets in MCDM, Chapter 14 in Advances in Multiple Criteria Decision Making (T. Gal, T. Hanne, T. Stewart, eds.), Kluwer, Dordrecht, 1999, 14.1–14.59.
J. W. Grzymala-Busse, LERS — a system for learning from examples based on rough sets, in Intelligent Decision Support. Handbook of Applications and Advances of the Rough Sets Theory (R. Slowinski, ed.), Kluwer, Dordrecht, 1992, -3–18.
T. Lin, Neighborhood systems and approximation in database and knowledge base systems, in Proc. 4th International Symposium on Methodologies for Intelligent Systems, 1989.
S. Marcus, Tolerance rough sets, Cech topologies, learning processes, Bull. of the Polish Academy of Sciences, Technical Sciences, 42 (1994), 471–487.
K. Menger, Probabilistic theories of relations, Proc. Nat. Acad. Sci. (Math.), 37 (1951), 178–180.
R. Mienko, J. Stefanowski, K. Toumi, D. Vanderpooten, Discoveryoriented induction of decision rules, Cahier du LAMSADE 141, Université de Paris Dauphine, Paris, 1996.
Z. Pawlak, Rough sets, Intern. J. of Information and Computer Sciences, 11 (1982), 341–356.
Z. Pawlak, Rough Sets. Theoretical Aspects of Reasoning about Data, Kluwer, Dordrecht, 1991.
Z. Pawlak, Rough sets and fuzzy sets, Fuzzy Sets and Systems, 17 (1985), 99–102.
L. Polkowski, A. Skowron, J. Zytkow, Tolerance based rough sets, in Soft Computing: Rough Sets, Fuzzy Logic, Neural Networks, Uncertainty Management, Knowledge Discovery (T. Y. Lin, A. Wildberger, eds.), Simulation Councils, Inc., San Diego, CA, 1995, 55–58.
B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland, Amsterdam, 1983.
A. Skowron, J. Stepaniuk, Generalized approximation spaces, in Soft Computing: Rough Sets, Fuzzy Logic, Neural Networks, Uncertainty Management, Knowledge Discovery (T. Y. Lin, A. Wildberger, eds.), Simulation Councils, Inc., San Diego, CA, 1995, 18–21.
R. Slowinski, A generalization of the indiscernibility relation for rough set analysis of quantitative information, Rivista di matematica per le scienze economiche e sociali, 15 (1993), 65–78.
R. Slowinski, Rough set processing of fuzzy information, in Soft Computing: Rough Sets, Fuzzy Logic, Neural Networks, Uncertainty Management, Knowledge Discovery (T. Y. Lin, A. Wildberger, eds.), Simulation Councils, Inc., San Diego, CA, 1995, 142–145.
R. Slowinski, J. Stefanowski, Rough classification in incomplete in-formation systems, Math. Comput. Modelling, 2, 10/11 (1989), 1347–1357.
R. Slowinski, J. Stefanowski, Handling various types of uncertainty in the rough set approach, in Rough Sets, Fuzzy Sets and Knowledge Discovery (W. R Ziarko, ed.), Springer-Verlag, London, 1993.
R. Slowinski, J. Stefanowski, Rough set reasoning about uncertain data, Fundamenta Informaticae, 27 (1996), 229–243.
R. Slowinski, J. Stefanowski, Rough family — software implementation of the rough set theory, in Rough Sets in Knowledge Discovery (L. Polkowski, A. Skowron, eds.), vol. 2, Physica-Verlag, Heidelberg, New York, 1998, 581–586.
R. Slowinski, D. Vanderpooten, Similarity relation as a basis for rough approximations, in Advances in Machine Intelligence and Soft-Computing (P. P. Wang, ed.), vol.IV, Duke University Press, Durham, NC, 1997, 17–33.
R. Slowinski, D. Vanderpooten, A generalized definition of rough approximation based on similarity, IEEE Transactions on Data and Knowledge Engineering, 1999 (to appear).
Y. Yao, S. Wong, Generalization of rough sets using relationships between attribute values, Proc. 2nd Annual Joint Conference on Information Sciences, Wrightsville Beach, NC, 1995, 30–33.
S. Weber, A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms, Fuzzy Sets and Systems, 11 (1983), 115–134.
W. Ziarko, D. Golan, D. Edwards, An application of DATALOGIC/R knowledge discovery tool to identify strong predictive rules in stock market data, Proc. AAAI Workshop on Knowledge Discovery in Databases, Washington D.C., 1993, 89–101.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag London Limited
About this chapter
Cite this chapter
Greco, S., Matarazzo, B., Slowinski, R. (2000). Rough Set Processing of Vague Information Using Fuzzy Similarity Relations. In: Finite Versus Infinite. Discrete Mathematics and Theoretical Computer Science. Springer, London. https://doi.org/10.1007/978-1-4471-0751-4_10
Download citation
DOI: https://doi.org/10.1007/978-1-4471-0751-4_10
Publisher Name: Springer, London
Print ISBN: 978-1-85233-251-8
Online ISBN: 978-1-4471-0751-4
eBook Packages: Springer Book Archive