Entropies and Co-Entropies of Coverings with Application to Incomplete Information Systems

• Autorzy: Bianucci D. , Cattaneo G. , Ciucci D.
• Języki publikacji: EN

Abstrakty

EN

Different generalizations to the case of coverings of the standard approach to entropy applied to partitions of a finite universe X are explored. In the first approach any covering is represented by an identity resolution of fuzzy sets on X and a corresponding probability distribution with associated entropy is defined. A second approach is based on a probability distribution generated by the covering normalizing the standard counting measure. Finally, the extension to a generic covering of the Liang-Xu approach to entropy is investigated, both from the "global" and the "local" point of view. For each of these three possible entropies the complementary entropy (or co-entropy) is defined showing in particular that the Liang-Xu entropy is a co-entropy.

Słowa kluczowe

EN

rough set theory

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2007
• Tom: Vol. 75, nr 1-4
• Strony: 77--105
• Opis fizyczny: bibliogr. 24 poz., tab.

Twórcy

autor

Bianucci D.

autor

Cattaneo G.

autor

Ciucci D.

• Dipartimento di Informatica, Sistemistica e Communicazione, Universita di Milano-Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy,

Bibliografia

• [1] Ash, R. B.: Information Theory, Dover Publications, New York, 1990, (originally published by John Wiley & Sons, New York, 1965).
• [2] Beaubouef, T., Petry, F. E., Arora, G.: Information-theoretic measures of uncertainty for rough sets and rough relational databases, Journal of Information Sciences, 109, 1998, 185-195.
• [3] Birkhoff, G.: Lattice Theory, vol. XXV of AmericanMathematical Society Colloquium Publication, Third edition, American Mathematical Society, Providence, Rhode Island, 1967.
• [4] Cattaneo, G.: Abstract approximation spaces for rough theories, in: Rough Sets in Knowledge Discovery 1 (L. Polkowski, A. Skowron, Eds.), Physica-Verlag, Heidelberg, New York, 1998, 59-98.
• [5] Cattaneo, G., Ciucci, D.: Some Methodological Remarks about Categorical Equivalence in the Abstract Approach to Roughness. Part I, Lecture Notes in Artificial Intelligence, vol. 4062 of Lecture Notes in Artificial Intelligence, Springer-Verlag, 2006, 277-283.
• [6] Hartley, R. V. L.: Transmission of information, The Bell System Technical Journal, 7, 1928, 535-563.
• [7] Huang, B., He, X., Zhong, X.: Rough entropy based on generalized rough sets covering reduction, Journal of Software, 15, 2004, 215-220.
• [8] Khinchin, A. I.: Mathematical Foundations of Information Theory, Dover Publications, New York, 1957, (translation of two papers appeared in Russian in Uspekhi Matematicheskikh Nauk, 3 1953, 3-20 and 1, 1965, 17-75).
• [9] Klir, G. J., Wierman, M. J.: Uncertainty Based Information, Physica-Verlag, New York, 1998.
• [10] Komorowski, J., Pawlak, Z., Polkowski, L., Skowron, A.: Rough sets: A tutorial, in: Rough Fuzzy Hybridization (S. Pal, A. Skowron, Eds.), Springer-Verlag, Singapore, 1999, 3-98.
• [11] Liang, J., Shi, Z.: The information entropy, rough entropy and knowledge granulation in rough set theory, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 12, 2004, 37-46.
• [12] Liang, J., Xu, Z.: Uncertainty measure of Randomness of knowledge and rough sets in incomplete information systems, Intelligent Control and Automata, 4, 2000, 2526-2529, Proc. of the 3rd World Congress on Intelligent Control and Automata.
• [13] Orlowska, E.: A logic of indiscernibility relations, Lecture Notes in Computer Sciences, 208, 1985, 177-186.
• [14] Pawlak, Z.: Information Systems - Theoretical Foundations, Information Systems, 6, 1981, 205-218.
• [15] Pawlak, Z.: Rough Sets, Int. J. Inform. Comput. Sci., 11, 1982, 341-356.
• [16] Pawlak, Z.: Rough sets: Theoretical Aspects of Reasoning about Data, Kluwer Academic Publishers, Dordrecht, 1991.
• [17] Reza, F. M.: An Introduction to Information theory, Dover Publications, New York, 1994, (originally published by Mc Graw-Hill, New York, 1961).
• [18] Shannon, C. E.: A Mathematical Theory of Communication, The Bell System Technical Journal, 27, 1948, 379-423, 623-656.
• [19] Slezak, D.: Approximate Entropy Reducts, Fundamenta Informaticae, 53, 2002, 365-390.
• [20] Slezak, D., Wroblewski, J.: Applications of Normalized Decision Measures to the New Case Classification, Lecture Notes in Artificial Intelligence, vol. 2005 of Lecture Notes in Artificial Intelligence, Springer-Verlag, Berlin, 2001, 553-560.
• [21] Vakarelov, D.: A modal logic for similarity relations in Pawlak knowledge representation systems, Fundamenta Informaticae, XV, 1991, 61-79.
• [22] Wierman, M.: Measuring uncertainty in Rough Set Theory, International Journal of General Systems, 28, 1999, 283-297.
• [23] Yao, Y., Li, X., Lin, T., Liu, Q.: Representation and Classification of Rough Set Models, Conference Proceeding of Third International Workshop on Rough Sets and Soft Computing, San Jose, California, November 10-12 1994, 630-637.
• [24] Zakowski, W.: Approximations in the space (U,Π), Demonstratio Mathematica, XVI, 1983, 761-769.