Abstract
The Luce model is one of the most popular ranking models used to estimate the ranks of items. In this study, we focus on grouping items with similar abilities and consider a new supervised clustering method by fusing specific parameters used in the Luce model. By modifying the penalty function conventionally used in grouping parameters, we obtain a new method of grouping items in the Luce model without pairwise comparison modeling and develop an efficient algorithm to estimate the parameters. Moreover, we give an application of the proposed algorithm to the Bradley-Terry model with ties. In the real data analysis, we confirm that the proposed estimator provides an easier interpretation of ranks and an improvement in the quality of prediction.
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References
Agresti A (2012) Categorical data analysis, 3rd edn. Springer, New York
Boyd S, Parikh N, Chu E, Peleato B, Eckstein J (2010) Distributed optimization and statistical learning via the alternating direction method of multipliers. Found Trend® Mach Learn 3:1–122
Bradley RA, Terry ME (1952) Rank analysis of incomplete block designs: I. The method of paired comparisons. Biometrika 39:324–345
Burges C, Shaked T, Renshaw E, Lazier A, Deeds M, Hamilton N, Hullender G (2005) Learning to rank using gradient descent. Proceed 22nd Int Conf Mach Learn 56:89–96
Cao Z, Qin T, Liu T-Y, Tsai M-F, Li H (2007) Learning to rank: from pairwise approach to listwise approach. In: Proceedings of the 24th International Conference on Machine Learning, pp 129–136
Davidson RR (1970) On extending the Bradley-Terry model to accommodate ties in paired comparison experiments. J Am Stat Assoc 65:317–328
Freund Y, Iyer R, Schapire R, Singer Y (2003) An efficient boosting algorithm for combining preferences. J Mach Learn Res 4:933–969
Gertheiss J, Tutz G (2010) Sparse modeling of categorial explanatory variables. Ann Appl Stat 4:2150–2180
Hunter DR, Lange K (2004) A tutorial on MM algorithms. Am Stat 58:30–37
Hunter DR (2004) MM algorithms for generalized Bradley-Terry models. Ann Stat 32:384–406
Jeon J, Kim Y (2013) Revisiting the Bradley-Terry model and its application to information retrieval. J Korean Data Inf Sci Soc 24:1089–1099
Joachims T (2002) Optimizing search engines using clickthrough data. In: Proceedings of the eighth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp 133–142
Ke T, Fan J, Wu Y (2015) Homogeneity pursuit. J Am Stat Assoc 110:175–194
Kendall M (1938) A new measure of rank correlation. Biometrika 30:81–93
Luce RD (1959) Individual choice behavior: A theoretical analysis. Wiley, New York
Masarotto G, Varin C (2012) The ranking lasso and its application to sport tournaments. Ann Appl Stat 6:1949–1970
McFadden D (1973) Conditional logit analysis of qualitative choice behavior. In: Frontiers in Econometrics. Academic Press, New York, pp 105–142
Mosteller F (1951) Remarks on the method of paired comparisons: I. The least squares solution assuming equal standard deviations and equal correlations. Psychometrika 16:3–9
Rao PV, Kupper LL (1967) Ties in paired-comparison experiments: a generalization of the Bradley-Terry model. J Am Stat Assoc 62:194–204
Schauberger G, Tutz G (2015) Modelling heterogeneity in paired comparison data - an l1 penalty approach with an application to party preference data. urn:nbn:de:bvb:19-epub-25175-7. Technical report
Shen X, Huang H -C (2010) Grouping pursuit through a regularization solution surface. J Am Stat Assoc 105:727–739
Shen X, Huang H -C, Pan W (2012) Simultaneous supervised clustering and feature selection over a graph. Biometrika 99:899–914
Thurstone L (1927) A law of comparative judgment. Psychol Rev 34:273–286
Tibshirani R, Saunders M, Rosset S, Zhu J, Knight K (2005) Sparsity and smoothness via the fused lasso. J Royal Stat Soc Ser B 67:91–108
Tibshirani R, Taylor J (2011) The solution path of the generalized lasso. Ann Stat 39:1335–1371
Tutz G (1986) Bradley-Terry-Luce models with an ordered response. J Math Psychol 30:306–316
Tutz G, Schauberger G (2015) Extended ordered paired comparison models with application to football data from German Bundesliga. AStA Adv Stat Anal 99:209–227
Varin C, Cattelan M, Firth D (2015) Statistical modelling of citation exchange between statistics journals. J Royal Stat Soc Ser A 179:1–33
Yuille AL, Rangarajan A (2003) The concave-convex procedure. Neural Comput 15:915–936
Zhou H (2006) The adaptive lasso and its oracle properties. J Am Stat Assoc 101:1418–1429
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This work was supported by Kyonggi University Research Grant 2014.
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Jeon, JJ., Choi, H. The sparse Luce model. Appl Intell 48, 1953–1964 (2018). https://doi.org/10.1007/s10489-016-0861-4
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DOI: https://doi.org/10.1007/s10489-016-0861-4