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An Algorithm for Ordinal Classification Based on Pairwise Comparison

  • Yunli Yang
  • Baiyu Chen
  • Zhouwang YangEmail author
Article
  • 24 Downloads

Abstract

Ordinal classification problems are applied in many fields. In the field of multivariate statistical analysis, these tasks are referred to as ordinal regression problems. In the field of management decision-making, they are known as multi-criteria decision analyses or sorting problems. This paper introduces the PairCode algorithm for ordinal classification with small sample sizes, which is based on a pairwise comparison strategy. In addition, this work outlines how to use pairwise comparisons to transform ordinal classifications into disordered regressions and how to transform the results of disordered regressions back to their original ordinal categories. Some effective strategies have been put forward, such as designing a class-label encoding matrix for the pairwise comparison, balancing samples, training classifiers, and predicting new samples. In numerical experiments, our algorithm (PairCode) is compared with the ordinal logistic regressions (LogisticOP) (Hu et al., IEEE Transactions on Knowledge and Data Engineering, 24(11), 2052–2064, 2012; Harrell 2015b), SVMOP (Gutiérrez et al., IEEE Transactions on Knowledge and Data Engineering, 28(1), 127–146, 2016; Leathart et al. 2016), SVORIM (Chu and Sathiya Keerthi, Neural Computation, 19(3), 792–815 2007; Gutiérrez et al., IEEE Transactions on Knowledge and Data Engineering, 28(1), 127–146, 2016), SVOREX (Chu and Sathiya Keerthi, Neural Computation, 19(3), 792–815 2007; Gutiérrez et al., IEEE Transactions on Knowledge and Data Engineering, 28(1), 127–146, 2016), and ELMOP (Deng et al., Neurocomputing, 74(1), 447–456, 2010; Gutiérrez et al., IEEE Transactions on Knowledge and Data Engineering, 28(1), 127–146, 2016). The results show that the PairCode algorithm performs better and is relatively stable as reflected by the correct classification rate (CCR), the mean absolute error (MAE), and the maximum MAE value (MMAE). However, the computing speed of the PairCode algorithm for classification is slightly slow and therefore warrants further study to improve the speed.

Keywords

Ordinal classification Pairwise comparison Balanced samples Angle distance Mean absolute error 

Notes

Acknowledgements

We would like to thank the anonymous reviewers for their comments and suggestions which greatly improve the manuscript. The work is supported by the NSF of China (No. 11871447), and Anhui Initiative in Quantum Information Technologies (AHY150200).

References

  1. Allwein, E.L., Schapire, R.E., Singer, Y. (2000). Reducing multiclass to binary: A unifying approach for margin classifiers. Journal of Machine Learning Research, 1 (Dec), 113–141.MathSciNetzbMATHGoogle Scholar
  2. Baccianella, S., Esuli, A., Sebastiani, F. (2009). Evaluation measures for ordinal regression. In: 2009. ISDA’09. Ninth International Conference on Intelligent Systems Design and Applications (pp. 283–287) IEEE.Google Scholar
  3. Cao-Van, K., & De Baets, B. (2003). Growing decision trees in an ordinal setting. International Journal of Intelligent Systems, 18(7), 733–750.CrossRefGoogle Scholar
  4. Carrizosa, E., & Morales, D.R. (2013). Supervised classification and mathematical optimization. Computers & Operations Research, 40(1), 150–165.MathSciNetCrossRefGoogle Scholar
  5. Chu, W., & Sathiya Keerthi, S. (2007). Support vector ordinal regression. Neural Computation, 19(3), 792–815.MathSciNetCrossRefGoogle Scholar
  6. Cruz-Ramírez, M., Hervás-Martínez, C., Sánchez-Monedero, J., Gutiérrez, P.A. (2011). A preliminary study of ordinal metrics to guide a multi-objective evolutionary algorithm. In: 2011 11th International Conference on Intelligent Systems Design and Applications (ISDA) (pp. 1176–1181) IEEE.Google Scholar
  7. Cruz-Ramírez, M., Hervás-Martínez, C., Sánchez-Monedero, J., Gutiérrez, P.A. (2014). Metrics to guide a multi-objective evolutionary algorithm for ordinal classification. Neurocomputing, 135, 21–31.CrossRefGoogle Scholar
  8. Deng, W.-Y., Zheng, Q.-H., Lian, S., Chen, L., Wang, X. (2010). Ordinal extreme learning machine. Neurocomputing, 74(1), 447–456.CrossRefGoogle Scholar
  9. Dietterich, T.G., & Bakiri, G. (1995). Solving multiclass learning problems via error-correcting output codes. Journal of Artificial Intelligence Research, 2, 263–286.CrossRefGoogle Scholar
  10. Fernández-Navarro, F., Riccardi, A., Carloni, S. (2014). Ordinal neural networks without iterative tuning. IEEE Transactions on Neural Networks and Learning Systems, 25(11), 2075–2085.CrossRefGoogle Scholar
  11. Gutiérrez, P A., Perez-Ortiz, M., Sanchez-Monedero, J., Fernandez-Navarro, F., Hervas-Martinez, C. (2016). Ordinal regression methods: survey and experimental study. IEEE Transactions on Knowledge and Data Engineering, 28(1), 127–146.CrossRefGoogle Scholar
  12. Harrell, F. (2015a). Regression Modeling Strategies: with Applications to Linear Models, Logistic and Ordinal Regression, and Survival Analysis. Berlin: Springer.CrossRefGoogle Scholar
  13. Harrell, F.E. Jr. (2015b). Ordinal logistic regression. In Regression Modeling Strategies (pp. 311–325). Berlin: Springer.Google Scholar
  14. Hastie, T., Tibshirani, R., et al. (1998). Classification by pairwise coupling. Annals of Statistics, 26(2), 451–471.MathSciNetCrossRefGoogle Scholar
  15. Hu, Q., Che, X., Zhang, L., Zhang, D., Guo, M., Yu, D. (2012). Rank entropy-based decision trees for monotonic classification. IEEE Transactions on Knowledge and Data Engineering, 24(11), 2052–2064.CrossRefGoogle Scholar
  16. Leathart, T., Pfahringer, B., Frank, E. (2016). Building ensembles of adaptive nested dichotomies with random-pair selection. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases (pp. 179–194). Berlin: Springer.Google Scholar
  17. López, V., Fernández, A., García, S., Palade, V., Herrera, F. (2013). An insight into classification with imbalanced data: Empirical results and current trends on using data intrinsic characteristics. Information Sciences, 250, 113–141.CrossRefGoogle Scholar
  18. López, V., Fernández, A., Moreno-Torres, J.G., Herrera, F. (2012). Analysis of preprocessing vs. cost-sensitive learning for imbalanced classification. open problems on intrinsic data characteristics. Expert Systems with Applications, 39(7), 6585–6608.CrossRefGoogle Scholar
  19. Potharst, R., & Bioch, J.C. (2000). Decision trees for ordinal classification. Intelligent Data Analysis, 4(2), 97–111.CrossRefGoogle Scholar
  20. Potharst, R., & Feelders, A.J. (2002). Classification trees for problems with monotonicity constraints. ACM SIGKDD Explorations Newsletter, 4(1), 1–10.CrossRefGoogle Scholar
  21. Sun, Y., Wong, A.K.C., Kamel, M.S. (2009). Classification of imbalanced data A review. International Journal of Pattern Recognition and Artificial Intelligence, 23 (04), 687–719.CrossRefGoogle Scholar
  22. Wu, T.-F., Lin, C.-J., Weng, R.C. (2004). Probability estimates for multi-class classification by pairwise coupling. Journal of Machine Learning Research, 5(Aug), 975–1005.MathSciNetzbMATHGoogle Scholar
  23. Xia, F., Zhang, W., Li, F., Yang, Y. (2008). Ranking with decision tree. Knowledge and Information Systems, 17(3), 381–395.CrossRefGoogle Scholar

Copyright information

© The Classification Society 2019

Authors and Affiliations

  1. 1.University of Science and Technology of ChinaHefeiChina

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