Model-Based Clustering of Inhomogeneous Paired Comparison Data

  • Ludwig M. Busse
  • Joachim M. Buhmann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7005)


This paper demonstrates the derivation of a clustering model for paired comparison data. Similarities for non-Euclidean, ordinal data are handled in the model such that it is capable of performing an integrated analysis on real-world data with different patterns of missings.

Rank-based pairwise comparison matrices with missing entries can be described and compared by means of a probabilistic mixture model defined on the symmetric group. Our EM-method offers two advantages compared to models for pairwise comparison rank data available in the literature: (i) it identifies groups in the pairwise choices based on similarity (ii) it provides the ability to analyze a data set of heterogeneous character w.r.t. to the structural properties of individal data samples.

Furthermore, we devise an active learning strategy for selecting paired comparisons that are highly informative to extract the underlying ranking of the objects. The model can be employed to predict pairwise choice probabilities for individuals and, therefore, it can be used for preference modeling.


Pairwise Comparison Paired Comparison Linear Extension Collaborative Filter Pairwise Comparison Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ailon, N., Mohri, M.: An Efficient Reduction of Ranking to Classification, Technical Report, TR2007-903 (2007)Google Scholar
  2. 2.
    Ailon, N., Mohri, M.: Preference-Based Learning to Rank. Machine Learning 80, 189–211 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Barnes, S.H., Kaase, M.: Political Action: Mass Participation in Five Western Countries. Sage, Beverly Hills (1979)Google Scholar
  4. 4.
    Busse, L.M., Orbanz, P., Buhmann, J.M.: Cluster Analysis of Heterogeneous Rank Data. In: International Conference on Machine Learning (2007)Google Scholar
  5. 5.
    Busse, L.M., Buhmann, J.M.: Multicriteria Scaling for Utilities under Intransitivities (to appear, 2011)Google Scholar
  6. 6.
    Cao, Z., Qin, T., Liu, T.-Y., Tsai, M.-F., Li, H.: Learning to Rank: From Pairwise Approach to Listwise Approach, Microsoft Tech. Report (2007)Google Scholar
  7. 7.
    Cohen, W.W., Schapire, R.E., Singer, Y.: Learning to Order Things. In: Advances in Neural Information Processing Systems, vol. 10 (1998)Google Scholar
  8. 8.
    Diaconis, P.: Group Representations in Probability and Statistics, Institute of Mathematical Statistics (1988)Google Scholar
  9. 9.
    Fligner, M.A., Verducci, J.S.: Distance based rank models. Journal of the Royal Statistical Society B 48(3), 359–369 (1986)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Fürnkranz, J., Hüllermeier, E.: Pairwise Preference Learning and Ranking. In: Lavrač, N., Gamberger, D., Todorovski, L., Blockeel, H. (eds.) ECML 2003. LNCS (LNAI), vol. 2837, pp. 145–156. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  11. 11.
    Hofmann, T., Buhmann, J.: Pairwise Data Clustering by Deterministic Annealing. IEEE Transactions on Pattern Analysis and Machine Intelligence 19(1), 1–14 (1997)CrossRefGoogle Scholar
  12. 12.
    Kahn, J., Saks, M.: Every poset has a good comparison. In: Proc. 16-th Symposium on Theory of Computing, pp. 299–301 (1984)Google Scholar
  13. 13.
    Karzanov, A., Khachiyan, L.: On the Conductance of Order Markov Chains. Order 8, 7–15 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kendall, M.G., Babington Smith, B.: On the Method of Paired Comparisons. Biometrika 31, 324–345 (1940)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lebanon, G., Lafferty, J.D.: Cranking: Combining Rankings Using Conditional Probability Models on Permutations. In: International Conference on Machine Learning (2002)Google Scholar
  16. 16.
    Little, R.J.A., Rubin, D.B.: Statistical analysis with missing data. Wiley series in probability and mathematical statistics. Applied probability and statistics, NJ (2002)Google Scholar
  17. 17.
    Lu, T., Boutilier, C.: Learning Mallows Models with Pairwise Preferences. In: International Conference on Machine Learning (2011)Google Scholar
  18. 18.
    Mallows, C.L.: Non-null ranking models I. Biometrika 44, 114–130 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Marden, J.I.: Analyzing and Modeling Rank Data. Chapman & Hall, Boca Raton (1995)zbMATHGoogle Scholar
  20. 20.
    McLachlan, G.J., Krishnan, T.: The EM Algorithm and Extensions. John Wiley & Sons, Chichester (1997)zbMATHGoogle Scholar
  21. 21.
    Meila, M., Phadnis, K., Patterson, A., Bilmes, J.: Consensus ranking under the exponential model. In: Conference on Uncertainty in Artificial Intelligence, UAI (2007)Google Scholar
  22. 22.
    Saaty, T.L.: A scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology 15, 234–281 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Su, X., Khoshgoftaar, T.M.: A Survey of Collaborative Filtering Techniques. In: Advances in Artificial Intelligence (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Ludwig M. Busse
    • 1
  • Joachim M. Buhmann
    • 1
  1. 1.Department of Computer ScienceETH ZurichZurichSwitzerland

Personalised recommendations