A Game-Theoretic Rough Set Approach for Handling Missing Data in Clustering

  • Nouman Azam
  • Mohammad Khan Afridi
  • JingTao Yao
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10868)


An important issue in clustering is how to deal with data containing missing values. A three-way decision making approach has recently been introduced for this purpose. It includes an added option of deferment which is exercised whenever it is not clear to include or exclude an object from a cluster. A critical issue in the three-way approach is how to decide the thresholds defining the three types of decisions. We examine the role of game-theoretic rough set model (GTRS) to address this issue. The GTRS model induces three-way decisions by implementing a game between multiple cooperative or competitive criteria. In particular, a game in GTRS is proposed which realizes the determination of thresholds from the viewpoint of tradeoff between accuracy and generality of clustering. Experimental results are reported for two datasets from UCI machine learning repository. The comparison of the GTRS results with another three-way model of (1, 0) suggests that the GTRS model significantly improves generality by upto 65% while maintaining similar levels of accuracy. In comparison to the (0.5, 0.5) model, the GTRS improves accuracy by upto 5% at a cost of some decrease in generality.


Clustering Missing data Three-way Game-theoretic rough sets 



This work was partially supported by a Discovery Grant from NSERC Canada and Indigenous Student Scholarship from HEC Pakistan.


  1. 1.
    Azam, N., Yao, J.T.: Formulating game strategies in game-theoretic rough sets. In: Lingras, P., Wolski, M., Cornelis, C., Mitra, S., Wasilewski, P. (eds.) RSKT 2013. LNCS (LNAI), vol. 8171, pp. 145–153. Springer, Heidelberg (2013). Scholar
  2. 2.
    Bugnet, M., Kula, A., Niewczas, M., Botton, G.A.: Segregation and clustering of solutes at grain boundaries in mgrare earth solid solutions. Acta Mater. 79, 66–73 (2014)CrossRefGoogle Scholar
  3. 3.
    Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Stat. Soc.: Ser. B (Methodol.) 39(1), 1–38 (1977)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Eye, A.V.: Statistical Methods in Longitudinal Research: Principles and Structuring Change. Statistical Modeling and Decision Science, vol. 1 (2014)Google Scholar
  5. 5.
    Haitovsky, V.: Missing data in regression analysis. J. Roy. Stat. Soc. 30, 67–82 (1968)zbMATHGoogle Scholar
  6. 6.
    Herbert, J.P., Yao, J.T.: Game-theoretic rough sets. Fundam. Inf. 108(3–4), 267–286 (2011)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Hu, J., Li, T., Wang, H., Fujita, H.: Hierarchical cluster ensemble model based on knowledge granulation. Knowl.-Based Syst. 91, 179–188 (2016)CrossRefGoogle Scholar
  8. 8.
    Iam-On, N., Boongeon, T., Garrett, S., Price, C.: A link-based cluster ensemble approach for categorical data clustering. IEEE Trans. Knowl. Data Eng. 24(3), 413–425 (2012)CrossRefGoogle Scholar
  9. 9.
    Li, J.H., Song, S.J., Zhang, Y.L., Zhou, Z.: Robust k-median and k-means clustering algorithms for incomplete data. Mathe. Probl. Eng. 2016, 1–8 (2016)MathSciNetGoogle Scholar
  10. 10.
    Lichman, M.: UCI machine learning repository (2013). Accessed 9 Feb 2017
  11. 11.
    Little, T.D., Lang, K.M., Wu, W., Rhemtulla, M.: Missing Data. Wiley, New York (2016)CrossRefGoogle Scholar
  12. 12.
    Timm, H., Dring, C., Kruse, R.: Different approaches to fuzzy clustering of incomplete datasets. Int. J. Approximate Reasoning 35(3), 239–249 (2004)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Xu, D., Tian, Y.: A comprehensive survey of clustering algorithms. Ann. Data Sci. 2(2), 165–193 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Xu, R., Wunsch, D.: Survey of clustering algorithms. IEEE Trans. Neural Netw. 16(3), 645–678 (2005)CrossRefGoogle Scholar
  15. 15.
    Yang, X., Li, T., Fujita, H., Liu, D., Yao, Y.: A unified model of sequential three-way decisions and multilevel incremental processing. Knowl.-Based Syst. 134, 172–188 (2017)CrossRefGoogle Scholar
  16. 16.
    Yao, Y.: Rough sets and three-way decisions. In: Ciucci, D., Wang, G., Mitra, S., Wu, W.-Z. (eds.) RSKT 2015. LNCS (LNAI), vol. 9436, pp. 62–73. Springer, Cham (2015). Scholar
  17. 17.
    Yu, H.: A framework of three-way cluster analysis. In: Polkowski, L., Yao, Y., Artiemjew, P., Ciucci, D., Liu, D., Ślęzak, D., Zielosko, B. (eds.) IJCRS 2017. LNCS (LNAI), vol. 10314, pp. 300–312. Springer, Cham (2017). Scholar
  18. 18.
    Yu, H., Su, T., Zeng, X.: A three-way decisions clustering algorithm for incomplete data. In: Miao, D., Pedrycz, W., Ślȩzak, D., Peters, G., Hu, Q., Wang, R. (eds.) RSKT 2014. LNCS (LNAI), vol. 8818, pp. 765–776. Springer, Cham (2014). Scholar
  19. 19.
    Yu, H., Zhang, C., Wang, G.: A tree-based incremental overlapping clustering method using the three-way decision theory. Knowl.-Based Syst. 91, 189–203 (2016)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Nouman Azam
    • 1
  • Mohammad Khan Afridi
    • 1
  • JingTao Yao
    • 2
  1. 1.National University of Computer and Emerging SciencesPeshawarPakistan
  2. 2.Department of Computer ScienceUniversity of ReginaReginaCanada

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