Skip to main content
SpringerLink
Log in

Information Entropy and Granulation Co–Entropy of Partitions and Coverings: A Summary

  • Chapter
Transactions on Rough Sets X

Part of the book series: Lecture Notes in Computer Science ((TRS,volume 5656))

Abstract

Some approaches to the covering information entropy and some definitions of orderings and quasi–orderings of coverings will be described, generalizing the case of the partition entropy and ordering. The aim is to extend to covering the general result of anti–tonicity (strictly decreasing monotonicity) of partition entropy. In particular an entropy in the case of incomplete information systems is discussed, with the expected anti-tonicity result, making use of a partial partition strategy in which the missing information is treated as a peculiar value of the system.

On the other side, an approach to generate a partition from a covering is illustrated. In particular, if we have a covering γ coarser than another covering δ with respect to a certain quasi order relation on coverings, the induced partition π(γ) results to be coarser than π(δ) with respect to the standard partial ordering on partitions. Thus, one can compare the two coverings through the entropies of the induced partitions.

The author’s work has been supported by MIUR\PRIN project “Automata and Formal languages: mathematical and application driven studies”and by “Funds of Sovvenzione Globale INGENIO allocated by Fondo Sociale Europeo, Ministero del Lavoro e della Previdenza Sociale, and Regione Lombardia”.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ash, R.B.: Information theory. Dover Publications, New York (1990) (originally published by John Wiley & Sons, New York (1965))

    MATH  Google Scholar 

  2. Bianucci, D., Cattaneo, G.: Monotonic behavior of entropies and co-entropies for coverings with respect to different quasi-orderings. In: Kryszkiewicz, M., Peters, J.F., Rybiński, H., Skowron, A. (eds.) RSEISP 2007. LNCS (LNAI), vol. 4585, pp. 584–593. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  3. Bianucci, D., Cattaneo, G.: On non-pointwise entropies of coverings: Relationship with anti-monotonicity. In: Wang, G., Li, T., Grzymala-Busse, J.W., Miao, D., Skowron, A., Yao, Y. (eds.) RSKT 2008. LNCS (LNAI), vol. 5009, pp. 387–394. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  4. Bianucci, D., Cattaneo, G., Ciucci, D.: Entropies and co–entropies for incomplete information systems. In: Yao, J., Lingras, P., Wu, W.-Z., Szczuka, M.S., Cercone, N.J., Ślȩzak, D. (eds.) RSKT 2007. LNCS (LNAI), vol. 4481, pp. 84–92. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  5. Bianucci, D., Cattaneo, G., Ciucci, D.: Entropies and co–entropies of coverings with application to incomplete information systems. Fundamenta Informaticae 75, 77–105 (2007)

    MathSciNet  MATH  Google Scholar 

  6. Bianucci, D., Cattaneo, G., Ciucci, D.: Information entropy and co–entropy of crisp and fuzzy granulations. In: Masulli, F., Mitra, S., Pasi, G. (eds.) WILF 2007. LNCS (LNAI), vol. 4578, pp. 9–19. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  7. Bianucci, D.: Rough entropies for complete and incomplete information systems. Ph.D. thesis (2007)

    Google Scholar 

  8. Birkhoff, G.: Lattice theory, 3rd edn. American Mathematical Society Colloquium Publication, vol. XXV. American Mathematical Society, Providence (1967)

    MATH  Google Scholar 

  9. 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)

    Article  Google Scholar 

  10. Cattaneo, G., Ciucci, D.: Algebraic structures for rough sets. In: Dubois, D., Gryzmala-Busse, J.W., Inuiguchi, M., Polkowski, L. (eds.) Transactions on Rough Sets II. LNCS, vol. 3135, pp. 208–252. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  11. 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)

    Chapter  Google Scholar 

  12. Cattaneo, G., Ciucci, D., Bianucci, D.: Entropy and co-entropy of partitions and coverings with applications to roughness theory. In: Bello, R., Falcon, R., Pedrycz, W., Kacprzyk, J. (eds.) Granular Computing: at the Junction of Fuzzy Sets and Rough Sets. Studies in Fuzziness and Soft Computing, vol. 224, pp. 55–77. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  13. Hamming, R.W.: Cading and information theory. Prentice–Hall, New Jersey (1986) (Second edition of the 1980 first edition)

    Google Scholar 

  14. Hartley, R.V.L.: Transmission of information. The Bell System Technical Journal 7, 535–563 (1928)

    Article  Google Scholar 

  15. Huang, B., He, X., Zhong, X.Z.: Rough entropy based on generalized rough sets covering reduction. Journal of Software 15, 215–220 (2004)

    MathSciNet  MATH  Google Scholar 

  16. 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); 1, 17–75 (1965)

    Google Scholar 

  17. 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)

    Google Scholar 

  18. Kryszkiewicz, M.: Rough set approach to incomplete information systems. Information Sciences 112, 39–49 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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)

    Article  MathSciNet  MATH  Google Scholar 

  20. Liang, J., Shi, Z., Li, D., Wierman, M.J.: Information entropy, rough entropy and knowledge granulation in incomplete information systems. International Journal of General Systems 35(6), 641–654 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liang, J., Xu, Z.: Uncertainty measure of randomness of knowledge and rough sets in incomplete information systems. Intelligent Control and Automata 4, 2526–2529 (2000); In: Proc. of the 3rd World Congress on Intelligent Control and Automata

    Google Scholar 

  22. Pawlak, Z.: Information systems - theoretical foundations. Information Systems 6, 205–218 (1981)

    Article  MATH  Google Scholar 

  23. Pawlak, Z.: Rough sets. Int. J. Inform. Comput. Sci. 11, 341–356 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  24. Pawlak, Z.: Rough sets: Theoretical aspects of reasoning about data. Kluwer Academic Publishers, Dordrecht (1991)

    Book  MATH  Google Scholar 

  25. Reza, F.M.: An introduction to information theory. Dover Publications, New York (1994) (originally published by Mc Graw-Hill, New York (1961))

    MATH  Google Scholar 

  26. Shannon, C.E.: A mathematical theory of communication. The Bell System Technical Journal 27, 379–423, 623–656 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  27. Taylor, A.E.: General theory of functions and integration. Dover Publications, New York (1985)

    Google Scholar 

  28. Wierman, M.J.: Measuring uncertainty in rough set theory. International Journal of General Systems 28, 283–297 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhao, Y., Luo, F., Wong, S.K.M., Yao, Y.Y.: A general definition of an attribute reduct. In: Yao, J., Lingras, P., Wu, W.-Z., Szczuka, M.S., Cercone, N.J., Ślȩzak, D. (eds.) RSKT 2007. LNCS, vol. 4481, pp. 101–108. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Informatica, Sistemistica e Comunicazione, Università di Milano – Bicocca, Viale Sarca 336/U14, I–20126, Milano, Italia

    Daniela Bianucci & Gianpiero Cattaneo

Authors
  1. Daniela Bianucci
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gianpiero Cattaneo
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Electrical and Computer Engineering, Winnipeg, University of Manitoba, R3T 5V6, Manitoba, Canada

    James F. Peters

  2. Institute of Mathematics, Warsaw University, Banacha 2, 02-097, Warsaw, Poland

    Andrzej Skowron

  3. Department of Logic and Philosophy of Science, Maria Curie-Skłodowska University, Marii Curie-Skłodowskiej 4, 20-031, Lublin, Poland

    Marcin Wolski

  4. University of Calcutta, Kolkata, India

    Mihir K. Chakraborty

  5. Zhejiang Ocean University, Zhejiang, P.R. China

    Wei-Zhi Wu

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Bianucci, D., Cattaneo, G. (2009). Information Entropy and Granulation Co–Entropy of Partitions and Coverings: A Summary. In: Peters, J.F., Skowron, A., Wolski, M., Chakraborty, M.K., Wu, WZ. (eds) Transactions on Rough Sets X. Lecture Notes in Computer Science, vol 5656. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03281-3_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-03281-3_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-03280-6

  • Online ISBN: 978-3-642-03281-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation