How Good Are Probabilistic Approximations for Rule Induction from Data with Missing Attribute Values?

  • Patrick G. Clark
  • Jerzy W. Grzymala-Busse
  • Zdzislaw S. Hippe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7413)


The main objective of our research was to test whether the probabilistic approximations should be used in rule induction from incomplete data. Probabilistic approximations, well known for many years, are used in variable precision rough set models and similar approaches to uncertainty.

For our experiments we used five standard data sets. Three data sets were incomplete to begin with and two data sets had missing attribute values that were randomly inserted. We used two interpretations of missing attribute values: lost values and “do not care” conditions. Among these ten combinations of a data set and a type of missing attribute values, in one combination the error rate (the result of ten-fold cross validation) was smaller than for ordinary approximations; for other two combinations, the error rate was larger than for ordinary approximations.


Image Segmentation Probabilistic Approximation Decision Table Rule Induction Indiscernibility Relation 
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  1. 1.
    Wong, S.K.M., Ziarko, W.: INFER—an adaptive decision support system based on the probabilistic approximate classification. In: Proceedings of the 6th International Workshop on Expert Systems and their Applications, pp. 713–726 (1986)Google Scholar
  2. 2.
    Grzymala-Busse, J.W., Ziarko, W.: Data mining based on rough sets. In: Wang, J. (ed.) Data Mining: Opportunities and Challenges, pp. 142–173. Idea Group Publ., Hershey (2003)Google Scholar
  3. 3.
    Pawlak, Z., Skowron, A.: Rough sets: Some extensions. Information Sciences 177, 28–40 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Pawlak, Z., Wong, S.K.M., Ziarko, W.: Rough sets: probabilistic versus deterministic approach. International Journal of Man-Machine Studies 29, 81–95 (1988)zbMATHCrossRefGoogle Scholar
  5. 5.
    Ślęzak, D., Ziarko, W.: The investigation of the bayesian rough set model. International Journal of Approximate Reasoning 40, 81–91 (2005)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Yao, Y.Y.: Probabilistic rough set approximations. International Journal of Approximate Reasoning 49, 255–271 (2008)zbMATHCrossRefGoogle Scholar
  7. 7.
    Yao, Y.Y., Wong, S.K.M.: A decision theoretic framework for approximate concepts. International Journal of Man-Machine Studies 37, 793–809 (1992)CrossRefGoogle Scholar
  8. 8.
    Yao, Y.Y., Wong, S.K.M., Lingras, P.: A decision-theoretic rough set model. In: Proceedings of the 5th International Symposium on Methodologies for Intelligent Systems, pp. 388–395 (1990)Google Scholar
  9. 9.
    Ziarko, W.: Variable precision rough set model. Journal of Computer and System Sciences 46(1), 39–59 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Ziarko, W.: Probabilistic approach to rough sets. International Journal of Approximate Reasoning 49, 272–284 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Clark, P.G., Grzymala-Busse, J.W.: Experiments on probabilistic approximations. In: Proceedings of the 2011 IEEE International Conference on Granular Computing, pp. 144–149 (2011)Google Scholar
  12. 12.
    Grzymala-Busse, J.W., Wang, A.Y.: Modified algorithms LEM1 and LEM2 for rule induction from data with missing attribute values. In: Proceedings of the Fifth International Workshop on Rough Sets and Soft Computing (RSSC 1997) at the Third Joint Conference on Information Sciences (JCIS 1997), pp. 69–72 (1997)Google Scholar
  13. 13.
    Stefanowski, J., Tsoukias, A.: Incomplete information tables and rough classification. Computational Intelligence 17(3), 545–566 (2001)CrossRefGoogle Scholar
  14. 14.
    Grzymala-Busse, J.W.: On the unknown attribute values in learning from examples. In: Proceedings of the ISMIS 1991, 6th International Symposium on Methodologies for Intelligent Systems, pp. 368–377 (1991)Google Scholar
  15. 15.
    Kryszkiewicz, M.: Rough set approach to incomplete information systems. In: Proceedings of the Second Annual Joint Conference on Information Sciences, pp. 194–197 (1995)Google Scholar
  16. 16.
    Grzymala-Busse, J.W.: Rough set strategies to data with missing attribute values. In: Workshop Notes, Foundations and New Directions of Data Mining, in conjunction with the 3rd International Conference on Data Mining, pp. 56–63 (2003)Google Scholar
  17. 17.
    Grzymała-Busse, J.W.: Generalized Parameterized Approximations. In: Yao, J., Ramanna, S., Wang, G., Suraj, Z. (eds.) RSKT 2011. LNCS, vol. 6954, pp. 136–145. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  18. 18.
    Pawlak, Z.: Rough sets. International Journal of Computer and Information Sciences 11, 341–356 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Pawlak, Z.: Rough Sets. Theoretical Aspects of Reasoning about Data. Kluwer Academic Publishers, Dordrecht (1991)zbMATHCrossRefGoogle Scholar
  20. 20.
    Grzymala-Busse, J.W.: LERS—a system for learning from examples based on rough sets. In: Slowinski, R. (ed.) Intelligent Decision Support. Handbook of Applications and Advances of the Rough Set Theory, pp. 3–18. Handbook of Applications and Advances of the Rough Set Theory. Kluwer Academic Publishers, Dordrecht (1992)Google Scholar
  21. 21.
    Grzymala-Busse, J.W.: MLEM2: A new algorithm for rule induction from imperfect data. In: Proceedings of the 9th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, pp. 243–250 (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Patrick G. Clark
    • 1
  • Jerzy W. Grzymala-Busse
    • 1
    • 2
  • Zdzislaw S. Hippe
    • 3
  1. 1.Department of Electrical Engineering and Computer ScienceUniversity of KansasLawrenceUSA
  2. 2.Institute of Computer SciencePolish Academy of SciencesWarsawPoland
  3. 3.Department of Expert Systems and Artificial IntelligenceUniversity of Information Technology and ManagementRzeszowPoland

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