Abstract
In this paper, we introduce a generic way to represent and manipulate pairwise information about partial orders (representing rankings, preferences, ...) with belief functions. We provide generic and practical tools to make inferences from this pairwise information and illustrate their use on the machine learning problems that are label ranking and multi-label prediction. Our approach differs from most other quantitative approaches handling complete or partial orders, in the sense that partial orders are here considered as primary objects and not as incomplete specifications of ideal but unknown complete orders.
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\((\lambda _i, \lambda _j),(\lambda _j,\lambda _k) \in R \Rightarrow (\lambda _i,\lambda _k) \in R\).
The data sets are available at http://www.uni-marburg.de/fb12/kebi/research/repository/labelrankingdata.
The software can be downloaded from https://www.hds.utc.fr/~tdenoeux/.
References
Bache K, Lichman M (2013) UCI machine learning repository. University of California, School of Information and Computer Science, Irvine. http://archive.ics.uci.edu/ml
Blockeel H, Raedt LD, Ramon J (1998) Top-down induction of clustering trees. Proceedings of the 15th international conference on machine learning. Morgan Kaufmann Publishers Inc., San Francisco, pp 55–63
Boutell MR, Luo J, Shen X, Brown CM (2004) Learning multi-label scene classification. Pattern Recogn 37(9):1757–1771
Boutilier C, Brafman RI, Domshlak C, Hoos HH, Poole D (2004) CP-nets: a tool for representing and reasoning with conditional ceteris paribus preference statements. J Artif Intell Res (JAIR) 21:135–191
Cheng W, Rademaker M, De Baets B, Hüllermeier E (2010) Predicting partial orders: ranking with abstention. Machine learning and knowledge discovery in databases. Springer, Berlin, Heidelberg, pp 215–230
Cheng W, Waegeman W, Welker V, Hüllermeier E (2012) Label ranking with partial abstention based on thresholded probabilistic models. In: Advances in neural information processing systems, pp 2510–2518.
Chow C (1970) On optimum recognition error and reject tradeoff. IEEE Trans Inf Theory 16(1):41–46
Clare A, King RD (2001) Knowledge discovery in multi-label phenotype data. Principles of data mining and knowledge discovery. Springer, New York, pp 42–53
Cobb BR, Shenoy PP (2006) On the plausibility transformation method for translating belief function models to probability models. Int J Approx Reason 41(3):314–330
Dekel O, Singer Y, Manning CD (2003) Log-linear models for label ranking. In: Advances in neural information processing systems, pp 497–504.
Denœux T (1995) A k-nearest neighbor classification rule based on Dempster-Shafer theory. IEEE Trans Syst Man Cybern 25(5):804–813
Denœux T, Masson M-H (2012) Evidential reasoning in large partially ordered sets. Ann Oper Res 195(1):135–161
Destercke S (2013) A pairwise label ranking method with imprecise scores and partial predictions. Machine learning and knowledge discovery in databases. Springer, New York, pp 112–127
El Zoghby N, Cherfaoui V, Denœux T (2013) Optimal object association from pairwise evidential mass functions. 16th international conference on information fusion (FUSION). IEEE, New York, pp 774–780
Elisseeff A, Weston J (2001) A kernel method for multi-labelled classification. In: Advances in neural information processing systems, pp 681–687.
Fürnkranz J, Hüllermeier E (2010) Preference learning. Springer, New York
Fürnkranz J, Hüllermeier E, Mencía EL, Brinker K (2008) Multilabel classification via calibrated label ranking. Mach Learn 73(2):133–153
Grabisch M, Labreuche C (2008) A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid. 4OR 6(1):1–44.
Greco S, Kadziński M, SŁowiński R (2011) Selection of a representative value function in robust multiple criteria sorting. Comput Oper Res 38(11):1620–1637
Har-Peled S, Roth D, Zimak D (2003) Constraint classification for multiclass classification and ranking. Adv Neural Inf Process Syst 809–816.
Hüllermeier E, Furnkranz J, Cheng W, Brinker K (2008) Label ranking by learning pairwise preferences. Artif Intell 172(16–17):1897–1916
Kamishima T, Kazawa H, Akaho S (2011) A survey and empirical comparison of object ranking methods. Preference learning. Springer, New York, pp 181–201
Kocev D, Vens C, Struyf J, Džeroski S (2007) Ensembles of multi-objective decision trees. Mach Learn ECML 2007:624–631
Labreuche C (2010) On the robustness for the Choquet integral. Computational intelligence for knowledge-based systems design. Springer, New York, pp 484–493
Li T, Ogihara M (2006) Toward intelligent music information retrieval. IEEE Trans Multimed 8(3):564–574
Loza Mencía E, Park S-H, Fürnkranz J (2010) Efficient voting prediction for pairwise multilabel classification. Neurocomputing 73(7):1164–1176
Madjarov G, Kocev D, Gjorgjevikj D, Džeroski S (2012) An extensive experimental comparison of methods for multi-label learning. Pattern Recogn 45(9):3084–3104
Marden JI (1995) Analyzing and modeling rank data, vol 64. Chapman & Hall, London
Rademaker M, De Baets B (2010) A threshold for majority in the context of aggregating partial order relations. 2010 IEEE international conference on fuzzy systems (FUZZ). IEEE, New York, pp 1–4
Read J, Pfahringer B, Holmes G, Frank E (2011) Classifier chains for multi-label classification. Mach Learn 85(3):333–359
Shafer G (1976) A mathematical theory of evidence. Princeton University Press, New Jersey
Smets P (1993) Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem. Int J Approx Reason 9(1):1–35
Smets P, Kennes R (1994) The transferable belief model. Artif Intell 66:191–234
Tritchler D, Lockwood G (1991) Modelling the reliability of paired comparisons. J Math Psychol 35(3):277–293
Troffaes M (2007) Decision making under uncertainty using imprecise probabilities. Int J Approx Reason 45(1):17–29
Tsoumakas G, Katakis I (2007) Multi-label classification: an overview. Int J Data Wareh Min (IJDWM) 3(3):1–13
Tsoumakas G, Katakis I, Vlahavas I (2008) Effective and efficient multilabel classification in domains with large number of labels. In: Proceedings of the ECML/PKDD 2008 workshop on mining multidimensional data (MMD08), pp 30–44.
Tsoumakas G, Katakis I, Vlahavas I (2010) Mining multi-label data. Data mining and knowledge discovery handbook. Springer, New York, pp 667–685
Tsoumakas G, Vlahavas I (2007) Random k-labelsets: an ensemble method for multilabel classification. Machine learning: ECML 2007. Springer, New York, pp 406–417
Ueda N, Saito K (2002) Parametric mixture models for multi-labeled text. In: Advances in neural information processing systems, pp 721–728.
Utkin LV (2009) A new ranking procedure by incomplete pairwise comparisons using preference subsets. Intell Data Anal 13(2):229–241
Vembu S, Gärtner T (2011) Label ranking algorithms: a survey. In: Preference learning, pp 45–64. Springer, New York.
Zaffalon M (2002) The naive credal classifier. J Probab Plan Inference 105:105–122
Zhang M-L, Zhou Z-H (2007) ML-KNN: a lazy learning approach to multi-label learning. Pattern Recogn 40(7):2038–2048
Acknowledgments
This work was carried out in the framework of the Labex MS2T, which was funded by the French Government, through the program “Investments for the future” managed by the National Agency for Research (Reference ANR-11-IDEX-0004-02).
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Communicated by V. Loia.
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Masson, MH., Destercke, S. & Denoeux, T. Modelling and predicting partial orders from pairwise belief functions. Soft Comput 20, 939–950 (2016). https://doi.org/10.1007/s00500-014-1553-9
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DOI: https://doi.org/10.1007/s00500-014-1553-9