Families of the Granules for Association Rules and Their Properties

  • Hiroshi SakaiEmail author
  • Chenxi Liu
  • Michinori Nakata
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9436)


We employed the granule (or the equivalence class) defined by a descriptor in tables, and investigated rough set-based rule generation. In this paper, we consider the new granules defined by an implication, and propose a family of the granules defined by an implication in a table with exact data. Each family consists of the four granules, and we show that three criterion values, support, accuracy, and coverage, can easily be obtained by using the four granules. Then, we extend this framework to tables with non-deterministic data. In this case, each family consists of the nine granules, and the minimum and the maximum values of three criteria are also obtained by using the nine granules. We prove that there is a table causing support and accuracy the minimum, and generally there is no table causing support, accuracy, and coverage the minimum. Finally, we consider the application of these properties to Apriori-based rule generation from uncertain data. These properties will make Apriori-based rule generation more effective.


Association rules Rule generation Apriori algorithm Granularity Uncertainty 



The authors would be grateful for reviewers’ useful comments. This work is supported by JSPS (Japan Society for the Promotion of Science) KAKENHI Grant Number 26330277.


  1. 1.
    Agrawal, R., Srikant, R.: Fast algorithms for mining association rules in large databases. In: Proceedings of VLDB’94, pp. 487–499. Morgan Kaufmann (1994)Google Scholar
  2. 2.
    Agrawal, R., Mannila, H., Srikant, R., Toivonen, H., Verkamo, A.I.: Fast discovery of association rules. In: Advances in Knowledge Discovery and Data Mining, pp. 307–328. AAAI/MIT Press (1996)Google Scholar
  3. 3.
    Blackburn, P., et al.: Modal Logic. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  4. 4.
    Frank, A., Asuncion, A.: UCI machine learning repository. Irvine, CA: University of California, School of Information and Computer Science (2010).
  5. 5.
    Grzymała-Busse, J.W.: Data with missing attribute values: generalization of indiscernibility relation and rule induction. Trans. Rough Sets 1, 78–95 (2004)CrossRefGoogle Scholar
  6. 6.
    Lipski, W.: On semantic issues connected with incomplete information databases. ACM Trans. Database Syst. 4(3), 262–296 (1979)CrossRefGoogle Scholar
  7. 7.
    Lipski, W.: On databases with incomplete information. J. ACM 28(1), 41–70 (1981)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Nakata, M., Sakai, H.: Twofold rough approximations under incomplete information. Int. J. Gen. Syst. 42(6), 546–571 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Orłowska, E., Pawlak, Z.: Representation of nondeterministic information. Theor. Comput. Sci. 29(1–2), 27–39 (1984)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Pawlak, Z.: Information systems theoretical foundations. Inf. Syst. 6(3), 205–218 (1981)CrossRefGoogle Scholar
  11. 11.
    Pawlak, Z.: Systemy Informacyjne: Podstawy Teoretyczne (in Polish) WNT (1983)Google Scholar
  12. 12.
    Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Dordrecht (1991)CrossRefGoogle Scholar
  13. 13.
    Sakai, H., Ishibashi, R., Koba, K., Nakata, M.: Rules and apriori algorithm in non-deterministic information systems. Trans. Rough Sets 9, 328–350 (2008)zbMATHGoogle Scholar
  14. 14.
    Sakai, H., Wu, M., Nakata, M.: Division charts as granules and their merging algorithm for rule generation in nondeterministic data. Int. J. Intell. Syst. 28(9), 865–882 (2013)CrossRefGoogle Scholar
  15. 15.
    Sakai, H., Wu, M., Nakata, M.: Apriori-based rule generation in incomplete information databases and non-deterministic information systems. Fundam. Informaticae 130(3), 343–376 (2014)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Sakai, H., Wu, M., Nakata, M.: The completeness of NIS-Apriori algorithm and a software tool getRNIA. In: Proceedings of International Conference on AAI2014, pp. 115–121 (2014)Google Scholar
  17. 17.
    Skowron, A., Rauszer, C.: The discernibility matrices and functions in information systems. In: Słowiński, R. (ed.) Intelligent Decision Support - Handbook of Advances and Applications of the Rough Set Theory, pp. 331–362. Kluwer Academic Publishers, Dordrecht (1992)Google Scholar
  18. 18.
    Wu, M., Sakai, H.: getRNIA web software (2013).
  19. 19.
    Wu, M., Nakata, M., Sakai, H.: An overview of the getRNIA system for non-deterministic data. Procedia Comput. Sci. 22, 615–622 (2013)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Graduate School of EngineeringKyushu Institute of TechnologyKitakyushuJapan
  2. 2.Faculty of Management and Information ScienceJosai International UniversityToganeJapan

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