On the Degree of Independence of a Contingency Matrix

  • Shoji Hirano
  • Shusaku Tsumoto
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3066)


A contingency table summarizes the conditional frequencies of two attributes and shows how these two attributes are dependent on each other. Thus, this table is a fundamental tool for pattern discovery with conditional probabilities, such as rule discovery. In this paper, a contingency table is interpreted from the viewpoint of statistical independence and granular computing. The first important observation is that a contingency table compares two attributes with respect to the number of equivalence classes. For example, a n × n table compares two attributes with the same granularity, while a m × n (mn) table compares two attributes with different granularities. The second important observation is that matrix algebra is a key point of analysis of this table. Especially, the degree of independence, rank plays a very important role in evaluating the degree of statistical independence. Relations between rank and the degree of dependence are also investigated.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Pawlak, Z.: Rough Sets. Kluwer Academic Publishers, Dordrecht (1991)zbMATHGoogle Scholar
  2. 2.
    Tsumoto, S.: Knowledge discovery in clinical databases and evaluation of discovered knowledge in outpatient clinic. Information Sciences, 125–137 (2000)Google Scholar
  3. 3.
    Tsumoto, S., Tanaka, H.: Automated discovery of medical expert system rules from clinical databases based on rough sets. In: Proceedings of the Second International Conference on Knowledge Discovery and Data Mining 1996, Palo Alto, pp. 63–69. AAAI Press, Menlo Park (1996)Google Scholar
  4. 4.
    Tsumoto, S.: Statistical independence as linear independence. In: Skowron, A., Szczuka, M. (eds.) Electronic Notes in Theoretical Computer Science, vol. 82, Elsevier, Amsterdam (2003)Google Scholar
  5. 5.
    Skowron, A., Grzymala-Busse, J.: From rough set theory to evidence theory. In: Yager, R., Fedrizzi, M., Kacprzyk, J. (eds.) Advances in the Dempster-Shafer Theory of Evidence, pp. 193–236. John Wiley & Sons, New York (1994)Google Scholar
  6. 6.
    Butz, C.: Exploiting contextual independencies in web search and user profiling. In: Proceedings of World Congress on Computational Intelligence (WCCI 2002), CDROM (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Shoji Hirano
    • 1
  • Shusaku Tsumoto
    • 1
  1. 1.Department of Medical InformaticsShimane University, School of MedicineEnya-cho Izumo CityJapan

Personalised recommendations