Summary
The abstract notion of rough approximation space is applied to the concrete cases of topological spaces with the particular situation of clopen–topologies generated by partitions, according to the Pawlak approach to rough set theory. In this partition context of a finite universe, typical of complete information systems, the probability space generated by the counting measure is analyzed, with particular regard to a local notion of rough entropy linked to the Shannon approach to these arguments. In the context of partition the notion of entropy as measure of uncertainty is distinguished from the notion of co–entropy as measure of granularity.
The above considerations are extended to the case of covering, typical situation of incomplete information systems with the associated similarity relation.
This is a preview of subscription content, log in via an institution to check access.
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
Cattaneo, G.: Abstract approximation spaces for rough theories. In: Polkowski, L., Skowron, A. (eds.) Rough Sets in Knowledge Discovery 1, pp. 59–98. Physica–Verlag, Heidelberg (1998)
Pawlak, Z.: Rough sets. Int. J. Inform. Comput. Sci. 11, 341–356 (1982)
Cattaneo, G., Ciucci, D.: Investigation about Time Monotonicity of Similarity and Preclusive Rough Approximations in Incomplete Information Systems. In: Tsumoto, S., Słowiński, R., Komorowski, J., Grzymała-Busse, J.W. (eds.) RSCTC 2004. LNCS (LNAI), vol. 3066, pp. 38–48. Springer, Heidelberg (2004)
Pawlak, Z.: Rough sets: A new approach to vagueness. In: Zadeh, L.A., Kacprzyc, J. (eds.) Fuzzy Logic for the Management of Uncertainty, pp. 105–118. J. Wiley and Sons, New York (1992)
Taylor, A.: General Theory of Functions and Integration. Dover Publications, New York (1985)
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, 3–20 (1953) and 1, 17–75 (1965)
Hartley, R.V.L.: Transmission of information. The Bell System Technical Journal 7, 535–563 (1928)
Ash, R.B.: Information Theory. Dover Publications, New York (1990) (originally published by John Wiley & Sons, New York, 1965)
Reza, F.M.: An Introduction to Information theory. Dover Publications, New York (1994) (originally published by Mc Graw-Hill, New York, 1961)
Shannon, C.E.: A mathematical theory of communication. The Bell System Technical Journal 27, 379–423, 623–656 (1948)
Bianucci, D., Cattaneo, G., Ciucci, D.: Entropies and co–entropies of coverings with application to incomplete information systems. Fundamenta Informaticae 75, 77–105 (2007)
Wierman, M.: Measuring uncertainty in rough set theory. International Journal of General Systems 28, 283–297 (1999)
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, 37–46 (2004)
Beaubouef, T., Petry, F.E., Arora, G.: Information–theoretic measures of uncertainty for rough sets and rough relational databases. Journal of Information Sciences 109, 185–195 (1998)
Pawlak, Z.: Rough sets: Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Dordrecht (1991)
Pawlak, Z.: Information systems - theoretical foundations. Information Systems 6, 205–218 (1981)
Komorowski, J., Pawlak, Z., Polkowski, L., Skowron, A.: Rough sets: A tutorial. In: Pal, S., Skowron, A. (eds.) Rough Fuzzy Hybridization, pp. 3–98. Springer, Singapore (1999)
Birkhoff, G.: Lattice Theory. American Mathematical Society, Providence, Rhode Island. American Mathematical Society Colloquium Publication, 3rd edn., vol. XXV (1967)
Liang, J., Xu, Z.: Uncertainty measure of randomness of knowledge and rough sets in incomplete information systems. Proc. of the 3rd World Congress on Intelligent Control and Automata 4, 2526–2529 (2000)
Bianucci, D., Cattaneo, G.: Monotonic behavior of entropies and co-entropies for coverings with respect to different quasi-orderings. LNCS (LNAI), vol. 4585, pp. 584–593. Springer, Heidelberg (to appear, 2007)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Cattaneo, G., Ciucci, D., Bianucci, D. (2008). Entropy and Co–entropy of Partitions and Coverings with Applications to Roughness Theory. In: Bello, R., Falcón, R., Pedrycz, W., Kacprzyk, J. (eds) Granular Computing: At the Junction of Rough Sets and Fuzzy Sets. Studies in Fuzziness and Soft Computing, vol 224. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-76973-6_4
Download citation
DOI: https://doi.org/10.1007/978-3-540-76973-6_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-76972-9
Online ISBN: 978-3-540-76973-6
eBook Packages: EngineeringEngineering (R0)