Abstract
A large part of qualitative data analysis is concerned with approximations of sets on the basis of relational information. In this paper, we present various forms of set approximations via the unifying concept of modal–style operators. Two examples indicate the usefulness of the approach.
Co-operation for this paper was supported by EU COST Action 274 ”Theory and Applications of Relational Structures as Knowledge Instruments” (TARSKI), www.tarski.org. Ivo Düntsch gratefully acknowledges support from the National Sciences and Engineering Research Council of Canada.
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References
Düntsch, I., Gediga, G.: Roughian – Rough Information Analysis. International Journal of Intelligent Systems 16, 121–147 (2001)
Jónsson, B., Tarski, A.: Boolean algebras with operators I. American Journal of Mathematics 73, 891–939 (1951)
Tarski, A.: Sur quelque propriét’es charactéristiques des images d’ensembles. Annales de la Societé Polonaise de Mathématique 6, 127–128 (1927)
Pawlak, Z.: Rough sets. Internat. J. Comput. Inform. Sci. 11, 341–356 (1982)
Wong, S., Wang, L., Yao, Y.: On modeling uncertainty with interval structures. Computational Intelligence 11, 406–426 (1995)
Yao, Y.Y.: On generalizing Pawlak approximation operators. In: Polkowski, L., Skowron, A. (eds.) RSCTC 1998. LNCS (LNAI), vol. 1424, pp. 298–307. Springer, Heidelberg (1998)
Słowiński, R., Vanderpooten, D.: Similarity relations as a basis for rough approximations. ICS Research Report 53, Polish Academy of Sciences (1995)
Słowiński, R., Vanderpooten, D.: A generalized definition of rough approximations based on similarity. IEEE Transactions on Knowledge and Data Engineering 12, 331–336 (2000)
Wille, R.: Restructuring lattice theory: An approach based on hierarchies of concepts. In: Rival, I. (ed.) Ordered sets. NATO Advanced Studies Institute, vol. 83, pp. 445–470. Reidel, Dordrecht (1982)
Düntsch, I., Gediga, G.: Modal–style operators in qualitative data analysis. In: Proceedings of the 2nd IEEE International Conference on Data Mining (ICDM 2002), pp. 155–162 (2002)
McKinsey, J.C.C., Tarski, A.: The algebra of topology. Annals of Mathematics 45, 141–191 (1944)
McKinsey, J.C.C., Tarski, A.: On closed elements in closure algebras. Annals of Mathematics 47, 122–162 (1946)
Birkhoff, G.: Lattice Theory, 2nd edn. Am. Math. Soc. Colloquium Publications, vol. 25. AMS, Providence (1948)
Ore, O.: Theory of Graphs. Am. Math. Soc. Colloquium Publications, vol. 38. AMS, Providence (1962)
Jónsson, B.: A survey of Boolean algebras with operators. In: Algebras and Orders. NATO Adv. Sci. Inst. Ser. C, Math. Phys. Sci, vol. 389, pp. 239–286. Kluwer, Dordrecht (1993)
Yao, Y.Y.: Generalization of rough sets using modal logic. Intelligent Automation and Soft Computing 2, 103–120 (1996)
van Benthem, J.: Correspondence theory. In: Gabbay, D.M., Guenthner, F. (eds.) Extensions of classical logic. Handbook of Philosophical Logic, vol. 2, pp. 167–247. Reidel, Dordrecht (1984)
Yao, Y.Y.: Constructive and algebraic methods of the theory of rough sets. Information Sciences 109, 21–47 (1998)
Yao, Y.Y.: Relational interpretations of neighborhood operators and rough set approximation operators. Information Sciences 111, 239–259 (1998)
Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1990)
Koppelberg, S.: General Theory of Boolean Algebras. Handbook on Boolean Algebras, vol. 1. North Holland, Amsterdam (1989)
Caspard, N., Monjardet, B.: The lattices of closure systems, closure operators, and implicational systems on a finite set: a survey. Discrete Applied Mathematics 127, 241–269 (2003)
Pagliani, P.: Concrete neighbourhood systems and formal pretopological spaces (2002) (preprint)
Lin, T.Y.: Granular computing on binary relations. I: Data mining and neighborhood systems. In: Polkowski, L., Skowron, A. (eds.) Rough sets in knowledge discovery, vol. 1, pp. 107–121. Physica–Verlag, Heidelberg (1998)
Fitting, M.: Basic modal logic. In: Gabbay, D.M., Hogger, C.J., Robinson, J.A. (eds.) Logical foundations. Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 1, pp. 368–448. Clarendon Press, Oxford (1993)
Orłowska, E.: Information algebras. In: Alagar, V.S., Nivat, M. (eds.) AMAST 1995. LNCS, vol. 936, pp. 50–65. Springer, Heidelberg (1995)
Gediga, G., Düntsch, I.: Skill set analysis in knowledge structures. British Journal of. Mathematical and Statistical Psychology 55, 361–384 (2002)
Steiner, A.: The lattice of topologies: Structure and complementation. Trans. Amer. Math. Soc. 122, 379–398 (1966)
Huebener, J.: Complementation in the lattice of regular topologies. Pacific J. Math. 41, 139–149 (1972)
Orlowska, E.: Semantics of vague concepts. In: Dorn, G., Weingartner, P. (eds.) Foundations of Logic and Linguistics. Problems and Solutions. Selected contributions to the 7th Internat. Congress of Logic, Methodology, and Philosophy of Science, Salzburg 1983, pp. 465–482. Plenum Press, New York (1983)
Yao, Y.Y., Lin, T.Y.: Generalization of rough sets using modal logic. Intelligent Automation and Soft Computing 2, 103–120 (1996)
Demri, S., Orłowska, E.: Incomplete Information: Structure, Inference, Complexity. EATCS Monographs in Theoretical Computer Science. Springer, Heidelberg (2002)
Tversky, A.: Features of similarity. Psychological Review 84, 327–352 (1977)
Johannesson, M.: Modelling asymmetric similarity with prominence. Cognitive Studies 55, Lund University (1997), Available from http://www.lucs.lu.se/Abstracts/LUCS_Studies/LUCS55.html (December 27, 2002)
Järvinen, J.: On the structure of rough approximations. TUCS Technical Report 447, Turku Centre for Computer Science, University of Turku (2002)
Rothkopf, E.Z.: A measure of stimulus similarity and errors in some paired-associate learning tasks. Journal of Experimental Psychology 53, 94–101 (1957)
Yao, Y.: Two views of the theory of rough sets in finite universes. International Journal of Approximate Reasoning 15, 291–317 (1996)
Pagliani, P.: From concept lattices to approximation spaces: Algebraic structures of some spaces of partial objects. Fundamenta Informaticae 18 (1993)
Shepard, R.N.: Analysis of proximities as a technique for the study of information processing in man. Human Factors 5, 33–48 (1963)
Buja, A., Swayne, D.F.: Visualization methodology for multidimensional scaling (2001) (preprint)
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Düntsch, I., Gediga, G. (2003). Approximation Operators in Qualitative Data Analysis. In: de Swart, H., Orłowska, E., Schmidt, G., Roubens, M. (eds) Theory and Applications of Relational Structures as Knowledge Instruments. Lecture Notes in Computer Science, vol 2929. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24615-2_10
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DOI: https://doi.org/10.1007/978-3-540-24615-2_10
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