Instance selection improves geometric mean accuracy: a study on imbalanced data classification


A natural way of handling imbalanced data is to attempt to equalise the class frequencies and train the classifier of choice on balanced data. For two-class imbalanced problems, the classification success is typically measured by the geometric mean (GM) of the true positive and true negative rates. Here we prove that GM can be improved upon by instance selection, and give the theoretical conditions for such an improvement. We demonstrate that GM is non-monotonic with respect to the number of retained instances, which discourages systematic instance selection. We also show that balancing the distribution frequencies is inferior to a direct maximisation of GM. To verify our theoretical findings, we carried out an experimental study of 12 instance selection methods for imbalanced data, using 66 standard benchmark data sets. The results reveal possible room for new instance selection methods for imbalanced data.

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  1. 1.

    We will use the terms “example”, “instance”, “object” and “prototype” interchangeably, meaning a data point in the feature space of interest, e.g. \(\mathbf {x}\in \mathbb {R}^n\).

  2. 2.

    We find it curious that no such methods, on this category, have yet been developed to maximise GM.

  3. 3.

    Available at

  4. 4.

    We noticed that, while the original OSS is defined by Kubat in [30] as CNN followed by TL, later on, Batista [5] defined it in reverse order and also independently proposed an equivalent to Kubat’s OSS. This misunderstanding has spread in subsequent works. However, we have maintained the original name OSS for CNN+TL, as used [30], and we use TL+CNN for Batista et al.’s method [5].

  5. 5.

    The random selection was performed by using the SpreadSubsample instance supervised filter.

  6. 6.

    Available in the KEEL GitHub repository:

  7. 7.

    Available in Google code:


  1. 1.

    Akbani, R., Kwek, S., Japkowicz, N.: Applying support vector machines to imbalanced datasets. In: Boulicaut, J.F., Esposito, F., Giannotti, F., Pedreschi, D. (eds.) Machine Learning: ECML 2004: 15th European Conference on Machine Learning, Pisa, Italy, 20-24 September, 2004. Proceedings, pp. 39–50. Springer Berlin Heidelberg, Berlin, Heidelberg (2004).

  2. 2.

    Alcala-Fdez, J., Fernández, A., Luengo, J., Derrac, J., García, S., Sánchez, L., Herrera, F.: KEEL data-mining software tool: data set repository and integration of algorithms and experimental analysis framework. J. Mult. Valued Logic Soft Comput. 17(2–3), 255–287 (2011)

    Google Scholar 

  3. 3.

    Barandela, R., Sánchez, J., García, V., Rangel, E.: Strategies for learning in class imbalance problems. Pattern Recognit. 36(3), 849–851 (2003)

    Article  Google Scholar 

  4. 4.

    Barandela, R., Valdovinos, R., Sánchez, J.: New applications of ensembles of classifiers. Pattern Anal. Appl. 6(3), 245–256 (2003)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Batista, G.E.A.P.A., Prati, R.C., Monard, M.C.: A study of the behavior of several methods for balancing machine learning training data. SIGKDD Explor. Newsl. 6(1), 20–29 (2004).

    Article  Google Scholar 

  6. 6.

    Batuwita, R., Palade, V.: FSVM-CIL: fuzzy support vector machines for class imbalance learning. IEEE Trans. Fuzzy Syst. 18(3), 558–571 (2010).

    Article  Google Scholar 

  7. 7.

    Chawla, N., Japkowicz, N., Kotcz, A.: Editorial: special issue on learning from imbalanced data sets. ACM SIGKDD Explor. Newsl. 6(1), 1–6 (2004)

    Article  Google Scholar 

  8. 8.

    Cieslak, D.A., Chawla, N.V., Striegel, A.: Combating imbalance in network intrusion datasets. In: GrC, pp. 732–737 (2006)

  9. 9.

    Cleofas-Sánchez, L., Sánchez, J.S., García, V.: Gene selection and disease prediction from gene expression data using a two-stage hetero-associative memory. Prog. Artif. Intell. (2018).

  10. 10.

    Cover, T., Hart, P.: Nearest neighbor pattern classification. IEEE Trans. Inf. Theory 13(1), 21–27 (1967)

    Article  MATH  Google Scholar 

  11. 11.

    Dal Pozzolo, A., Caelen, O., Le Borgne, Y.A., Waterschoot, S., Bontempi, G.: Learned lessons in credit card fraud detection from a practitioner perspective. Expert Syst. Appl. 41(10), 4915–4928 (2014)

    Article  Google Scholar 

  12. 12.

    Dasarathy, B.V.: Nearest Neighbor (NN) Norms: NN Pattern Classification Techniques. IEEE Computer Society Press, Los Alamitos, California (1990)

    Google Scholar 

  13. 13.

    Dietterich, T.: Approximate statistical tests for comparing supervised classification learning algorithms. Neural Comput. 10(7), 1895–1923 (1998)

    Article  Google Scholar 

  14. 14.

    Díez-Pastor, J.F., Rodríguez, J.J., García-Osorio, C., Kuncheva, L.I.: Random balance: ensembles of variable priors classifiers for imbalanced data. Knowl. Based Syst. 85, 96–111 (2015)

    Article  Google Scholar 

  15. 15.

    Díez-Pastor, J.F., Rodríguez, J.J., García-Osorio, C.I., Kuncheva, L.I.: Diversity techniques improve the performance of the best imbalance learning ensembles. Inf. Sci. 325, 98–117 (2015)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Drown, D.J., Khoshgoftaar, T.M., Seliya, N.: Evolutionary sampling and software quality modeling of high-assurance systems. IEEE Trans. Syst. Man Cybern. Part A Syst. Hum. 39(5), 1097–1107 (2009)

    Article  Google Scholar 

  17. 17.

    Eskildsen, S.F., Coupé, P., Fonov, V., Collins, D.L.: Detecting alzheimer’s disease by morphological MRI using hippocampal grading and cortical thickness. In: Proceedings of the MICCAI Workshop Challenge on Computer-Aided Diagnosis of Dementia Based on Structural MRI Data, pp. 38–47 (2014)

  18. 18.

    Fernández, A., García, S., Galar, M., Prati, R., Krawczyk, B., Herrera, F.: Learning from imbalanced data sets. Springer International PU (2018). Accessed 4 Feb 2019

  19. 19.

    Fix, E., Hodges, J.L.: Discriminatory analysis: non parametric discrimination: small sample performance. Technical report project 21-49-004 (11), USAF School of Aviation Medicine, Randolph Field, Texas (1952)

  20. 20.

    Freund, Y., Schapire, R.E.: A decision-theoretic generalization of on-line learning and an application to boosting. J. Comput. Syst. Sci. 55(1), 119–139 (1997).

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Galar, M., Fernandez, A., Barrenechea, E., Bustince, H., Herrera, F.: A review on ensembles for the class imbalance problem: Bagging-, boosting-, and hybrid-based approaches. IEEE Trans. Syst. Man Cybern. Part C Appl. Rev. 42(4), 463–484 (2012)

    Article  Google Scholar 

  22. 22.

    Galar, M., Fernández, A., Barrenechea, E., Herrera, F.: Eusboost: enhancing ensembles for highly imbalanced data-sets by evolutionary undersampling. Pattern Recognit. 46(12), 3460–3471 (2013)

    Article  Google Scholar 

  23. 23.

    Garcia, S., Derrac, J., Cano, J., Herrera, F.: Prototype selection for nearest neighbor classification: taxonomy and empirical study. IEEE Trans. Pattern Anal. Mach. Intell. 34(3), 417–435 (2012).

    Article  Google Scholar 

  24. 24.

    García-Pedrajas, N., Pérez-Rodríguez, J., García-Pedrajas, M.D., Ortiz-Boyer, D., Fyfe, C.: Class imbalance methods for translation initiation site recognition in DNA sequences. Knowl. Based Syst. 25(1), 22–34 (2012)

    Article  Google Scholar 

  25. 25.

    García, S., Herrera, F.: Evolutionary undersampling for classification with imbalanced datasets: proposals and taxonomy. Evolut. Comput. 17(3), 275–306 (2009).

    MathSciNet  Article  Google Scholar 

  26. 26.

    Hart, P.: The condensed nearest neighbor rule (corresp.). IEEE Trans. Inf. Theory 14(3), 515–516 (1968)

    Article  Google Scholar 

  27. 27.

    Jain, A.K., Zongker, D.: Feature selection: evaluation, application, and small sample performance. IEEE Trans. Pattern Anal. Mach. Intell. 19(2), 153–158 (1997).

    Article  Google Scholar 

  28. 28.

    Krawczyk, B.: Learning from imbalanced data: open challenges and future directions. Prog. Artif. Intell. 5(4), 221–232 (2016).

    Article  Google Scholar 

  29. 29.

    Krawczyk, B., Galar, M., Jeleń, Ł., Herrera, F.: Evolutionary undersampling boosting for imbalanced classification of breast cancer malignancy. Appl. Soft Comput. 38, 714–726 (2016)

    Article  Google Scholar 

  30. 30.

    Kubat, M., Matwin, S.: Addressing the curse of imbalanced training sets: one-sided selection. In: Proceedings of the Fourteenth International Conference on Machine Learning, pp. 179–186. Morgan Kaufmann (1997)

  31. 31.

    Kuncheva, L.: Combining Pattern Classifiers: Methods and Algorithms. Wiley, New York (2014). Accessed 4 Feb 2019

  32. 32.

    Laurikkala, J.: Improving identification of difficult small classes by balancing class distribution. In: Quaglini, S., Barahona, P., Andreassen, S. (eds.) Artificial Intelligence in Medicine: 8th Conference on Artificial Intelligence in Medicine in Europe, AIME 2001 Cascais, Portugal, 1–4 July, 2001, Proceedings, pp. 63–66. Springer Berlin Heidelberg, Berlin, Heidelberg (2001).

  33. 33.

    López, V., Triguero, I., Carmona, C.J., García, S., Herrera, F.: Addressing imbalanced classification with instance generation techniques: IPADE-ID. Neurocomputing 126, 15–28 (2014)

    Article  Google Scholar 

  34. 34.

    Phua, C., Alahakoon, D., Lee, V.: Minority report in fraud detection: classification of skewed data. ACM SIGKDD Explor. Newsl. 6(1), 50–59 (2004)

    Article  Google Scholar 

  35. 35.

    Pudil, P., Novovičová, J., Kittler, J.: Floating search methods in feature selection. Pattern Recognit. Lett. 15(11), 1119–1125 (1994).

    Article  Google Scholar 

  36. 36.

    Saeys, Y., Inza, I., naga, P.L.: A review of feature selection techniques in bioinformatics. Bioinformatics 23(19), 2507–2517 (2007)

    Article  Google Scholar 

  37. 37.

    Sáez, J.A., Krawczyk, B., Woźniak, M.: Analyzing the oversampling of different classes and types of examples in multi-class imbalanced datasets. Pattern Recognit. (2016).

  38. 38.

    Sanz, J.A., Bernardo, D., Herrera, F., Bustince, H., Hagras, H.: A compact evolutionary interval-valued fuzzy rule-based classification system for the modeling and prediction of real-world financial applications with imbalanced data. IEEE Trans. Fuzzy Syst. 23(4), 973–990 (2015)

    Article  Google Scholar 

  39. 39.

    Seiffert, C., Khoshgoftaar, T., Van Hulse, J., Napolitano, A.: RUSBoost: a hybrid approach to alleviating class imbalance. IEEE Trans. Syst. Man Cybern. Part A Syst. Hum. 40(1), 185–197 (2010)

    Article  Google Scholar 

  40. 40.

    Seiffert, C., Khoshgoftaar, T.M., Van Hulse, J.: Improving software-quality predictions with data sampling and boosting. IEEE Trans. Syst. Man Cybern. Part A Syst. Hum. 39(6), 1283–1294 (2009)

    Article  Google Scholar 

  41. 41.

    Sun, Z., Song, Q., Zhu, X.: Using coding-based ensemble learning to improve software defect prediction. IEEE Trans. Syst. Man Cybern. Part C Appl. Rev. 42(6), 1806–1817 (2012)

    Article  Google Scholar 

  42. 42.

    Tao, T.: An Introduction to Measure Theory. Graduate Studies in Mathematics. American Mathematical Society, Providence (2013). Accessed 4 Feb 2019

  43. 43.

    Tesfahun, A., Bhaskari, D.L.: Intrusion detection using random forests classifier with smote and feature reduction. In: 2013 International Conference on Cloud & Ubiquitous Computing & Emerging Technologies (CUBE), pp. 127–132. IEEE (2013)

  44. 44.

    Tomek, I.: Two modifications of CNN. IEEE Trans. Syst. Man Cybern. 6, 769–772 (1976)

    MathSciNet  MATH  Google Scholar 

  45. 45.

    Triguero, I., Derrac, J., García, S., Herrera, F.: A taxonomy and experimental study on prototype generation for nearest neighbor classification. IEEE Trans. Syst. Man Cybern. Part C Appl. Rev. 42(1), 86–100 (2012)

    Article  Google Scholar 

  46. 46.

    Visa, S., Ralescu, A.: Issues in mining imbalanced data sets-a review paper. In: Proceedings of the Sixteen Midwest Artificial Intelligence and Cognitive Science Conference, vol. 2005, pp. 67–73 (2005)

  47. 47.

    Wilson, D.L.: Asymptotic properties of nearest neighbor rules using edited data. IEEE Trans. Syst. Man Cybern. SMC–2(3), 408–421 (1972).

    MathSciNet  Article  MATH  Google Scholar 

  48. 48.

    Wilson, D.R., Martinez, T.R.: Reduction techniques for instance-based learning algorithms. Mach. Learn. 38(3), 257–286 (2000).

    Article  MATH  Google Scholar 

  49. 49.

    Witten, I.H., Frank, E., Hall, M.A.: Data Mining: Practical Machine Learning Tools and Techniques, 3rd edn. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA (2011)

    Google Scholar 

  50. 50.

    Wu, X., Kumar, V., Quinlan, J.R., Ghosh, J., Yang, Q., Motoda, H., McLachlan, G.J., Ng, A., Liu, B., Yu, P.S., Zhou, Z.H., Steinbach, M., Hand, D.J., Steinberg, D.: Top 10 algorithms in data mining. Knowl. Inf. Syst. 14(1), 1–37 (2008)

    Article  Google Scholar 

  51. 51.

    Yang, P., Xu, L., Zhou, B.B., Zhang, Z., Zomaya, A.Y.: A particle swarm based hybrid system for imbalanced medical data sampling. BMC Genom. 10(3), 1–14 (2009).

    Article  Google Scholar 

  52. 52.

    Zhang, J., Mani, I.: kNN approach to unbalanced data distributions: a case study involving information extraction. In: Proceedings of The Twentieth International Conference on Machine Learning (ICML-2003), Workshop on Learning from Imbalanced Data Sets (2003)

  53. 53.

    Zheng, B., Myint, S.W., Thenkabail, P.S., Aggarwal, R.M.: A support vector machine to identify irrigated crop types using time-series landsat NDVI data. Int. J. Appl. Earth Obs. Geoinf. 34, 103–112 (2015)

    Article  Google Scholar 

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This work was done under project RPG-2015-188 funded by the Leverhulme Trust, UK; the project TIN2015-67534-P funded by the Ministerio de Economía y Competitividad of the Spanish Government; and the BU085P17 funded by the Junta de Castilla y León. The authors gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan Xp GPU used for this research.

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Correspondence to José-Francisco Díez-Pastor.

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Kuncheva, L.I., Arnaiz-González, Á., Díez-Pastor, J. et al. Instance selection improves geometric mean accuracy: a study on imbalanced data classification. Prog Artif Intell 8, 215–228 (2019).

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  • Imbalanced data
  • Geometric mean (GM)
  • Instance/prototype selection
  • Nearest neighbour
  • Ensemble methods
  • Theoretical perspective