[go: up one dir, main page]
More Web Proxy on the site http://driver.im/
Skip to main content
Springer Nature Link
Log in

Rough Validity, Confidence, and Coverage of Rules in Approximation Spaces

  • Conference paper
Transactions on Rough Sets III

Part of the book series: Lecture Notes in Computer Science ((TRS,volume 3400))

Abstract

From the granular computing perspective, the existing notions of validity, confidence, and coverage of rules in approximation spaces may be viewed as too crisp since granularity of the space is not, in general, taken into account in their definitions. In this article, an extension of the classical approach to a general rough case is discussed. We introduce and investigate graded validity, confidence, and coverage of rules as examples of rough validity, confidence, and coverage, respectively. The graded notions are based on the concepts of graded meaning of formulas and sets of formulas, studied in our earlier works. Among others, the notions of graded validitity, confidence, and coverage refine and extend the classical forms by taking into account granules of information drawn toward objects of an approximation space.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. An, A., Cercone, N.: Rule quality measures for rule induction systems: Description and evaluation. J. Comput. Intelligence 17, 409–424 (2001)

    Article  Google Scholar 

  2. Bruha, I.: Quality of decision rules: Definitions and classification schemes for multiple rules. In: Nakhaeizadeh, G., Taylor, C.C. (eds.) Machine Learning and Statistics, The Interface, pp. 107–131. John Wiley & Sons, New York (1997)

    Google Scholar 

  3. Stepaniuk, J.: Knowledge discovery by application of rough set models. In: Polkowski, L., Tsumoto, S., Lin, T.Y. (eds.) Rough Set Methods and Applications: New Developments in Knowledge Discovery in Information Systems, pp. 137–233. Physica, Heidelberg (2001)

    Google Scholar 

  4. Skowron, A., Stepaniuk, J.: Tolerance approximation spaces. Fundamenta Informaticae 27, 245–253 (1996)

    MATH  MathSciNet  Google Scholar 

  5. Stepaniuk, J.: Approximation spaces, reducts and representatives. In: Polkowski, L., Skowron, A. (eds.) Rough Sets in Knowledge Discovery, vol. 2, pp. 109–126. Physica, Heidelberg (1998)

    Google Scholar 

  6. Polkowski, L., Skowron, A.: Rough sets: A perspective. In: Polkowski, L., Skowron, A. (eds.) Rough Sets in Knowledge Discovery, pp. 31–56. Physica, Heidelberg (1998)

    Google Scholar 

  7. Polkowski, L., Skowron, A.: Towards adaptive calculus of granules. In: Zadeh, L.A., Kacprzyk, J. (eds.) Computing with Words in Information/Intelligent Systems, vol. 1, pp. 201–228. Physica, Heidelberg (1999)

    Google Scholar 

  8. Polkowski, L., Skowron, A.: Rough mereological calculi of granules: A rough set approach to computation. J. Comput. Intelligence 17, 472–492 (2001)

    Article  MathSciNet  Google Scholar 

  9. Polkowski, L., Skowron, A.: Rough mereology. LNCS (LNAI), vol. 869, pp. 85–94. Springer, Heidelberg (1994)

    Google Scholar 

  10. Polkowski, L., Skowron, A.: Rough mereology: A new paradigm for approximate reasoning. Int. J. Approximated Reasoning 15, 333–365 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Polkowski, L., Skowron, A.: Rough mereology in information systems. A case study: Qualitative spatial reasoning. In: Polkowski, L., Tsumoto, S., Lin, T.Y. (eds.) Rough Set Methods and Applications: New Developments in Knowledge Discovery in Information Systems, pp. 89–135. Physica, Heidelberg (2001)

    Google Scholar 

  12. Gomolińska, A.: A graded meaning of formulas in approximation spaces. Fundamenta Informaticae 60, 159–172 (2004)

    MATH  MathSciNet  Google Scholar 

  13. Pawlak, Z.: Mathematical Foundations of Information Retrieval. CC PAS Report, Warsaw, vol. 101 (1973)

    Google Scholar 

  14. Pawlak, Z.: Information systems – Theoretical foundations. Information Systems 6, 205–218 (1981)

    Article  MATH  Google Scholar 

  15. Pawlak, Z.: Information Systems. In: Theoretical Foundations. Wydawnictwo Naukowo-Techniczne, Warsaw (1983) (In Polish)

    Google Scholar 

  16. Pawlak, Z.: Rough sets. Int. J. Computer and Information Sciences 11, 341–356 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  17. Pawlak, Z.: Rough Sets. In: Theoretical Aspects of Reasoning About Data, Kluwer, Dordrecht (1991)

    Google Scholar 

  18. Pawlak, Z.: Rough sets and decision tables. LNCS, vol. 208, pp. 186–196. Springer, Berlin (1985)

    Google Scholar 

  19. Pawlak, Z.: On decision tables. Bull. Polish Acad. Sci. Tech. 34, 553–572 (1986)

    Google Scholar 

  20. Pawlak, Z.: Decision tables – A rough set approach. Bull. EATCS 33, 85–96 (1987)

    MATH  Google Scholar 

  21. Pawlak, Z.: Bayes’ theorem – The rough set perspective. In: Inuiguchi, M., Hirano, S., Tsumoto, S. (eds.) Rough Set Theory and Granular Computing. Studies in Fuzziness and Soft Computing, vol. 125, pp. 1–12. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  22. Skowron, A., Stepaniuk, J.: Generalized approximation spaces. In: Proc. 3rd Int. Workshop on Rough Sets and Soft Computing, San Jose, November 10-12, pp. 156–163 (1994)

    Google Scholar 

  23. Peters, J.F.: Approximation space for intelligent system design patterns. Engineering Applications of Artificial Intelligence 17, 1–8 (2004)

    Article  Google Scholar 

  24. Skowron, A., Stepaniuk, J.: Generalized approximation spaces. In: Lin, T.Y., Wildberger, A.M. (eds.) Soft Computing, Simulation Councils, San Diego, pp. 18–21 (1995)

    Google Scholar 

  25. Skowron, A., Stepaniuk, J.: Information granules and approximation spaces. In: Proc. 7th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-based Systems (IPMU 1998), Paris, France, July 8-10, pp. 354–361 (1998)

    Google Scholar 

  26. Zadeh, L.A.: Outline of a new approach to the analysis of complex system and decision processes. IEEE Trans. on Systems, Man, and Cybernetics 3, 28–44 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  27. 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, Heidelberg (1998)

    Google Scholar 

  28. Lukasiewicz, J.: Die logischen Grundlagen der Wahrscheinlichkeitsrechnung. In: Borkowski, L. (ed.) Jan Lukasiewicz – Selected Works, Amsterdam, London Warsaw, pp. 16–63. North-Holland, Polish Scientific Publ. (1970); First published in Kraków, (1913)

    Google Scholar 

  29. Gomolińska, A.: A comparative study of some generalized rough approximations. Fundamenta Informaticae 51, 103–119 (2002)

    MATH  MathSciNet  Google Scholar 

  30. Barwise, J.: Information Flow. In: The Logic of Distributed Systems, Cambridge University Press, UK (1997)

    Google Scholar 

  31. Devlin, K.: Logic and Information. Cambridge University Press, UK (1991)

    MATH  Google Scholar 

  32. Kripke, S.A.: Semantical analysis of intuitionistic logic. In: Crossley, J.N., Dummet, M.A.E. (eds.) Formal Systems and Recursive Functions, pp. 92–130. North- Holland, Amsterdam (1965)

    Chapter  Google Scholar 

  33. Skowron, A., Stepaniuk, J., Peters, J.F.: Rough sets and infomorphisms: Towards approximation of relations in distributed environments. Fundamenta Informaticae 54, 263–277 (2003)

    MATH  MathSciNet  Google Scholar 

  34. Pawlak, Z., Polkowski, L., Skowron, A.: Rough sets and rough logic: A KDD perspective. In: Polkowski, L., Tsumoto, S., Lin, T.Y. (eds.) Rough Set Methods and Applications: New Developments in Knowledge Discovery in Information Systems, pp. 583–646. Physica, Heidelberg (2001)

    Google Scholar 

  35. Gomolińska, A.: Satisfiability and meaning in approximation spaces. In: Lindemann, G., et al. (eds.) Proc. Workshop Concurrency, Specification, and Programming (CS&P’2004), Caputh, Germany, September 24-26. Informatik-Berichte, vol. 170, pp. 229–240. Humboldt-Universität zu Berlin, Berlin (2004)

    Google Scholar 

  36. Gomolińska, A.: A graded applicability of rules. In: Tsumoto, S., Słowiński, R., Komorowski, J., Grzymała-Busse, J.W. (eds.) RSCTC 2004. LNCS (LNAI), vol. 3066, pp. 213–218. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  37. Pogorzelski, W.A.: Notions and Theorems of Elementary Formal Logic. Białystok Division of Warsaw University, Białystok (1994)

    Google Scholar 

  38. Tsumoto, S.: Modelling medical diagnostic rules based on rough sets. In: Polkowski, L., Skowron, A. (eds.) RSCTC 1998. LNCS (LNAI), vol. 1424, pp. 475–482. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  39. Mollestad, T.: A Rough Set Approach to Data Mining: Extracting a Logic of Default Rules from Data, Ph.D. Dissertation. NTNU, Trondheim (1997)

    Google Scholar 

  40. Reiter, R.: A logic for default reasoning. Artificial Intelligence J. 13, 81–132 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  41. Agrawal, R., Imieliński, T., Swami, A.: Mining association rules between sets of items in large databases. In: Proc. ACM SIGMOD Int. Conf. on Management of Data, Washington, D.C, pp. 207–216 (1993)

    Google Scholar 

  42. Agrawal, R., Srikant, R.: Fast algorithms for mining association rules. In: Proc. 20th Conf. on Very Large Databases, Santiago, Chile, pp. 487–499 (1994)

    Google Scholar 

  43. Kryszkiewicz, M.: Fast discovery of representative association rules. In: Polkowski, L., Skowron, A. (eds.) RSCTC 1998. LNCS (LNAI), vol. 1424, pp. 214–221. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  44. Lin, T.Y., Louie, E.: Association rules with additional semantics modeled by binary relations. In: Inuiguchi, M., Hirano, S., Tsumoto, S. (eds.) Rough Set Theory and Granular Computing. Studies in Fuzziness and Soft Computing, vol. 125, pp. 147–156. Springer, Heidelberg (2003)

    Google Scholar 

  45. Murai, T., Nakata, M., Sato, Y.: Association rules from a point of view of conditional logic. In: Inuiguchi, M., Hirano, S., Tsumoto, S. (eds.) Rough Set Theory and Granular Computing. Studies in Fuzziness and Soft Computing, vol. 125, pp. 137–145. Springer, Heidelberg (2003)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Białystok, Akademicka 2, 15267, Białystok, Poland

    Anna Gomolińska

Authors
  1. Anna Gomolińska
    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, University of Manitoba, R3T 5V6, Winnipeg, Manitoba, Canada

    James F. Peters

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

    Andrzej Skowron

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Gomolińska, A. (2005). Rough Validity, Confidence, and Coverage of Rules in Approximation Spaces. In: Peters, J.F., Skowron, A. (eds) Transactions on Rough Sets III. Lecture Notes in Computer Science, vol 3400. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11427834_3

Download citation

  • DOI: https://doi.org/10.1007/11427834_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-25998-5

  • Online ISBN: 978-3-540-31850-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation