# Rank aggregation using latent-scale distance-based models

- 69 Downloads

## Abstract

Rank aggregation aims at combining rankings of a set of items assigned by a sample of rankers to generate a consensus ranking. A typical solution is to adopt a distance-based approach to minimize the sum of the distances to the observed rankings. However, this simple sum may not be appropriate when the quality of rankers varies. This happens when rankers with different backgrounds may have different cognitive levels of examining the items. In this paper, we develop a new distance-based model by allowing different weights for different rankers. Under this model, the weight associated with a ranker is used to measure his/her cognitive level of ranking of the items, and these weights are unobserved and exponentially distributed. Maximum likelihood method is used for model estimation. Extensions to the cases of incomplete rankings and mixture modeling are also discussed. Empirical applications demonstrate that the proposed model produces better rank aggregation than those generated by Borda and the unweighted distance-based models.

## Keywords

Ranking data Latent-scale distance-based model Rank aggregration Incomplete ranking## References

- Aledo, J.A., Gámez, J.A., Molina, D.: Tackling the rank aggregation problem with evolutionary algorithms. Appl. Math. Comput.
**222**, 632–644 (2013)MathSciNetzbMATHGoogle Scholar - Ali, A., Meilă, M.: Experiments with Kemeny ranking: what works when? Math. Soc. Sci.
**64**(1), 28–40 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - Aslam, J.A., Montague, M.: Models for metasearch. In: Proceedings of the 24th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval, pp. 276–284 (2001)Google Scholar
- Borg, I., Groenen, P.J.: Modern Multidimensional Scaling: Theory and Applications. Springer, Berlin (2005)zbMATHGoogle Scholar
- Busse, L.M., Orbanz, P., Buhmann, J.M.: Cluster analysis of heterogeneous rank data. In: Proceedings of the 24th International Conference on Machine Learning, pp. 113–120 (2007)Google Scholar
- Ceberio, J., Irurozki, E., Mendiburu, A., Lozano, J.A.: Extending distance-based ranking models in estimation of distribution algorithms. In: Evolutionary Computation (CEC), 2014 IEEE Congress on, pp. 2459–2466. IEEE (2014)Google Scholar
- de Borda, J.: Mémoire sur les élections au scrutin. Mémoires de l’Académie Royale des Sciences Année, pp. 657–664 (1781)Google Scholar
- DeConde, R.P., Hawley, S., Falcon, S., Clegg, N., Knudsen, B., Etzioni, R.: Combining results of microarray experiments: a rank aggregation approach. Stat. Appl. Genet. Mol. Biol. 5(1):Article 15 (2006)Google Scholar
- Deng, K., Han, S., Li, K.J., Liu, J.S.: Bayesian aggregation of order-based rank data. J. Am. Stat. Assoc.
**109**(507), 1023–1039 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - Dwork, C., Kumar, R., Naor, M., Sivakumar, D.: Rank aggregation methods for the web. In: Proceedings of the 10th International Conference on World Wide Web, pp. 613–622 (2001)Google Scholar
- Fligner, M.A., Verducci, J.S.: Distance-based ranking models. J. R. Stat. Soc. B
**48**(3), 359–369 (1986)MathSciNetzbMATHGoogle Scholar - Fligner, M.A., Verducci, J.S.: Posterior probabilities for a consensus ordering. Psychometrika
**55**(1), 53–63 (1990)MathSciNetCrossRefGoogle Scholar - Irurozki, E., Calvo, B., Lozano, J.A.: Sampling and learning the Mallows model under the Ulam distance. Technical report, Department of Computer Science and Artificial Intelligence, University of the Basque Country (2014)Google Scholar
- Irurozki, E., Calvo, B., Lozano, J.A.: Sampling and learning the Mallows and generalized Mallows models under the Cayley distance. Methodol. Comput. Appl. Probab.
**20**(1), 1–35 (2018)MathSciNetCrossRefzbMATHGoogle Scholar - Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P., et al.: Optimization by simulated annealing. Science
**220**(4598), 671–680 (1983)MathSciNetCrossRefzbMATHGoogle Scholar - Lee, M.D., Steyvers, M., Miller, B.: A cognitive model for aggregating people’s rankings. PLoS One
**9**(5), e96431 (2014)CrossRefGoogle Scholar - Lin, S., Ding, J.: Integration of ranked lists via cross entropy Monte Carlo with applications to mRNA and microRNA studies. Biometrics
**65**(1), 9–18 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - Mallows, C.L.: Non-null ranking models. I. Biometrika
**44**(1–2), 114–130 (1957)MathSciNetCrossRefzbMATHGoogle Scholar - Murphy, T.B., Martin, D.: Mixtures of distance-based models for ranking data. Comput. Stat. Data Anal.
**41**(3), 645–655 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - Naume, B., Zhao, X., Synnestvedt, M., Borgen, E., Russnes, H.G., Lingjærde, O.C., Strømberg, M., Wiedswang, G., Kvalheim, G., Kåresen, R., et al.: Presence of bone marrow micrometastasis is associated with different recurrence risk within molecular subtypes of breast cancer. Mol. Oncol.
**1**(2), 160–171 (2007)CrossRefGoogle Scholar - Niederreiter, H.: Quasi-Monte Carlo Methods. Wiley Online Library (2010)Google Scholar
- Černý, V.: Thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm. J. Optim. Theory Appl.
**45**(1), 41–51 (1985)MathSciNetCrossRefzbMATHGoogle Scholar