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Data Level Preprocessing Methods

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Book cover Learning from Imbalanced Data Sets

Abstract

The first mechanism to address the problem of imbalanced learning was the use of sampling methods. They consists of modifying a set of imbalanced data using different procedures to provide a balanced or more adequate data distribution to the subsequent learning tasks. In the specialized literature, many studies have shown that, for several types of classifiers, rebalancing the dataset significantly improves the overall performance of the classification compared to a non-preprocessed data set. Over the years, this procedure has been common and the use of sampling methods for imbalanced learning has been standardized. Still, classifiers do not always have to use this kind of preprocessing because many of them are able to directly deal with imbalanced datasets. There is no clear rule that tells us which strategy is best, whether to adapt the behavior of learning algorithms or to use data preprocessing techniques. However, data sampling and preprocessing techniques are standard techniques in imbalanced learning, they are widely used in Data Science problems. They are simple and easily configurable and can be used in synergy with any learning algorithm. This chapter will review the techniques of sampling, undersampling (the classical ones in Sect. 5.2 and advanced approaches in Sect. 5.3) and oversampling such as SMOTE in Sect. 5.4, as well as the most-known algorithm SMOTE and its derivatives in Sect. 5.5. Some hybridizations of undersampling and oversampling are described in Sect. 5.6. Experiments with graphical illustrations will be carried out to show the behavior of these techniques.

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Notes

  1. 1.

    We would like to thank Sergio González for the development of a visualization module for this task. See more at https://github.com/sergiogvz/imbalanced_synthetic_data_plots

References

  1. Abdi, L., Hashemi, S.: To combat multi-class imbalanced problems by means of over-sampling techniques. IEEE Trans. Know. Data Eng. 28(1), 238–251 (2016)

    Article  Google Scholar 

  2. Almogahed, B.A., Kakadiaris, I.A.: NEATER: filtering of over-sampled data using non-cooperative game theory. Soft Comput. 19(11), 3301–3322 (2015)

    Article  Google Scholar 

  3. Anand, A., Pugalenthi, G., Fogel, G.B., Suganthan, P.N.: An approach for classification of highly imbalanced data using weighting and undersampling. Amino Acids 39(5), 1385–1391 (2010)

    Article  Google Scholar 

  4. Angiulli, F., Basta, S., Pizzuti, C.: Distance-based detection and prediction of outliers. IEEE Trans. Know. Data Eng. 18(2), 145–160 (2006)

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  6. Barella, V., Costa, E., Carvalho, A.C.P.L.F.: ClusterOSS: a new undersampling method for imbalanced learning. Technical report (2014)

    Google Scholar 

  7. Barua, S., Islam, M.M., Murase, K.: A novel synthetic minority oversampling technique for imbalanced data set learning. In: 18th International Conference on Neural Information Processing, ICONIP, Shanghai, pp. 735–744 (2011)

    Chapter  Google Scholar 

  8. Barua, S., Islam, M.M., Yao, X., Murase, K.: MWMOTE-majority weighted minority oversampling technique for imbalanced data set learning. IEEE Trans. Know. Data Eng. 26(2), 405–425 (2014)

    Article  Google Scholar 

  9. Basu, M., Ho, T.K. (ed.): Data Complexity in Pattern Recognition. Springer, London (2006)

    MATH  Google Scholar 

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

    Article  Google Scholar 

  11. Bellinger, C., Drummond, C., Japkowicz, N.: Beyond the boundaries of SMOTE – a framework for manifold-based synthetically oversampling. In: European Conference on Machine Learning and Knowledge Discovery in Databases (ECML PKDD), Riva del Garda, pp. 248–263 (2016)

    Chapter  Google Scholar 

  12. Błaszczyński, J., Deckert, M., Stefanowski, J., Wilk, S.: Integrating selective pre-processing of imbalanced data with ivotes ensemble. In: Szczuka, M., Kryszkiewicz, M., Ramanna, S., Jensen, R., Hu, Q. (eds.) Rough Sets and Current Trends in Computing. LNSC, vol. 6086, pp. 148–157. Springer, Berlin/Heidelberg (2010)

    Chapter  Google Scholar 

  13. Bradley, A.P.: The use of the area under the roc curve in the evaluation of machine learning algorithms. Pattern Recogn. 30(7), 1145–1159 (1997)

    Article  Google Scholar 

  14. Breiman, L., Friedman, J.H., Olshen, R.A., Stone, C.J.: Classification and Regression Trees. Chapman and Hall, New York/Wadsworth and Inc., Belmont (1984)

    Google Scholar 

  15. Brodley, C.E., Friedl, M.A.: Identifying mislabeled training data. J. Artif. Intell. Res. 11, 131–167 (1999)

    Article  MATH  Google Scholar 

  16. Bunkhumpornpat, C., Sinapiromsaran, K., Lursinsap, C.: Safe–level–SMOTE: safe–level–synthetic minority over–sampling TEchnique for handling the class imbalanced problem. In: Proceedings of the 13th Pacific–Asia Conference on Advances in Knowledge Discovery and Data Mining PAKDD’09, Bangkok, pp. 475–482 (2009)

    Google Scholar 

  17. Bunkhumpornpat, C., Sinapiromsaran, K., Lursinsap, C.: DBSMOTE: density-based synthetic minority over-sampling TEchnique. Appl. Intell. 36(3), 664–684 (2012)

    Article  Google Scholar 

  18. Cano, J.R., Herrera, F., Lozano, M.: Using evolutionary algorithms as instance selection for data reduction in KDD: an experimental study. IEEE Trans. Evol. Comput. 7(6), 561–575 (2003)

    Article  Google Scholar 

  19. Chawla, N.V.: Data mining for imbalanced datasets: an overview. In: Maimon, O., Rokach, L. (eds.) Data Mining and Knowledge Discovery Handbook, pp. 853–867. Springer, New York (2005)

    Chapter  Google Scholar 

  20. Chawla, N.V., Bowyer, K.W., Hall, L.O., Kegelmeyer, W.P.: SMOTE: synthetic minority over–sampling technique. J. Artif. Intell. Res. 16, 321–357 (2002)

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  22. Chawla, N.V., Cieslak, D.A., Hall, L.O., Joshi, A.: Automatically countering imbalance and its empirical relationship to cost. Data Min. Knowl. Disc. 17(2), 225–252 (2008)

    Article  MathSciNet  Google Scholar 

  23. Chen, S., Guo, G., Chen, L.: A new over-sampling method based on cluster ensembles. In: 7th International Conference on Advanced Information Networking and Applications Workshops, Perth, pp. 599–604 (2010)

    Google Scholar 

  24. Cohen, G., Hilario, M., Sax, H., Hugonnet, S., Geissbuhler, A.: Learning from imbalanced data in surveillance of nosocomial infection. Artif. Intell. Med. 37, 7–18 (2006)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  26. de la Calleja, J., Fuentes, O.: A distance-based over-sampling method for learning from imbalanced data sets. In: Proceedings of the Twentieth International Florida Artificial Intelligence, pp. 634–635 (2007)

    Google Scholar 

  27. Das, B., Krishnan, N.C., Cook, D.J.: RACOG and wRACOG: two probabilistic oversampling techniques. IEEE Trans. Know. Data Eng. 27(1), 222–234 (2015)

    Article  Google Scholar 

  28. Dietterich, T.G.: An experimental comparison of three methods for constructing ensembles of decision trees: bagging, boosting, and randomization. Mach. Learn. 40, 139–157 (2000)

    Article  Google Scholar 

  29. 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 39(5), 1097–1107 (2009)

    Article  Google Scholar 

  30. Estabrooks, A., Jo, T., Japkowicz, N.: A multiple resampling method for learning from imbalanced data sets. Comput. Intell. 20(1), 18–36 (2004)

    Article  MathSciNet  Google Scholar 

  31. Fernández, A., García, S., Herrera, F., Chawla, N.V.: Smote for learning from imbalanced data: progress and challenges, marking the 15-year anniversary. J. Artif. Intell. Res. 61, 863–905 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. Fernández-Navarro, F., Hervás-Martínez, C., Gutiérrez, P.A.: A dynamic over-sampling procedure based on sensitivity for multi-class problems. Pattern Recognit. 44(8), 1821–1833 (2011)

    Article  MATH  Google Scholar 

  33. Galar, M., Fernández, A., Barrenechea, E., Bustince, H., Herrera, F.: A review on ensembles for 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 

  34. Gao, M., Hong, X., Chen, S., Harris, C.J., Khalaf, E.: PDFOS: PDF estimation based over-sampling for imbalanced two-class problems. Neurocomputing 138, 248–259 (2014)

    Article  Google Scholar 

  35. García, S., Herrera, F.: Evolutionary under-sampling for classification with imbalanced data sets: proposals and taxonomy. Evol. Comput. 17(3), 275–306 (2009)

    Article  MathSciNet  Google Scholar 

  36. García, V., Mollineda, R.A., Sánchez, J.S.: On the k–NN performance in a challenging scenario of imbalance and overlapping. Pattern Anal. Appl. 11(3–4), 269–280 (2008)

    Article  MathSciNet  Google Scholar 

  37. García, S., Fernández, A., Herrera, F.: Enhancing the effectiveness and interpretability of decision tree and rule induction classifiers with evolutionary training set selection over imbalanced problems. Appl. Soft Comput. 9, 1304–1314 (2009)

    Article  Google Scholar 

  38. García, S., Derrac, J., Triguero, I., Carmona, C.J., Herrera, F.: Evolutionary-based selection of generalized instances for imbalanced classification. Know. Based Syst. 25(1), 3–12 (2012)

    Article  Google Scholar 

  39. García, V., Sánchez, J.S., Mollineda, R.A.: On the effectiveness of preprocessing methods when dealing with different levels of class imbalance. Knowl. Based Syst. 25(1), 13–21 (2012)

    Article  Google Scholar 

  40. García-Pedrajas, N., Pérez-Rodríguez, J., de Haro-García, A.: Oligois: scalable instance selection for class-imbalanced data sets. IEEE Trans. Cybern 43(1), 332–346 (2013)

    Article  Google Scholar 

  41. Gazzah, S., Amara, N.E.B.: New oversampling approaches based on polynomial fitting for imbalanced data sets. In: The Eighth IAPR International Workshop on Document Analysis Systems, Nara, pp. 677–684 (2008)

    Google Scholar 

  42. Han, H., Wang, W.Y., Mao, B.H.: Borderline–SMOTE: a new over–sampling method in imbalanced data sets learning. In: Proceedings of the 2005 International Conference on Intelligent Computing (ICIC’05), Hefei. Lecture Notes in Computer Science, vol. 3644, pp. 878–887 (2005)

    Article  Google Scholar 

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

    Article  Google Scholar 

  44. He, H., Garcia, E.A.: Learning from imbalanced data. IEEE Trans. Know. Data Eng. 21(9), 1263–1284 (2009)

    Article  Google Scholar 

  45. He, H., Bai, Y., Garcia, E.A., Li, S.: ADASYN: adaptive synthetic sampling approach for imbalanced learning. In: Proceedings of the 2008 IEEE International Joint Conference Neural Networks (IJCNN’08), Hong Kong, pp. 1322–1328 (2008)

    Google Scholar 

  46. Hu, F., Li, H.: A novel boundary oversampling algorithm based on neighborhood rough set model: NRSBoundary-SMOTE. Math. Probl. Eng. Article ID 694809, 10 (2013)

    Google Scholar 

  47. Huang, J., Ling, C.X.: Using AUC and accuracy in evaluating learning algorithms. IEEE Trans. Knowl. Data Eng. 17(3), 299–310 (2005)

    Article  Google Scholar 

  48. Kang, Y.I., Won, S.: Weight decision algorithm for oversampling technique on class-imbalanced learning. In: ICCAS, Gyeonggi-do, pp. 182–186 (2010)

    Google Scholar 

  49. Kim, H., Jo, N., Shin, K.: Optimization of cluster-based evolutionary undersampling for the artificial neural networks in corporate bankruptcy prediction. Expert Syst. Appl. 59, 226–234 (2016)

    Article  Google Scholar 

  50. Kubat, M., Holte, R.C., Matwin, S.: Learning when negative examples abound. In: van Someren, M., Widmer, G. (eds.) Proceedings of the 9th European Conference on Machine Learning (ECML’97). Lecture Notes in Computer Science, vol. 1224, pp. 146–153. Springer, Berlin/New York (1997)

    Google Scholar 

  51. Laurikkala, J.: Improving identification of difficult small classes by balancing class distribution. In: AIME’01: Proceedings of the 8th Conference on AI in Medicine in Europe, Cascais, pp. 63–66 (2001)

    Google Scholar 

  52. Lemaitre, G., Nogueira, F., Aridas, C.K.: Imbalanced-learn: a python toolbox to tackle the curse of imbalanced datasets in machine learning. J. Mach. Learn. Res. 18(17), 1–5 (2017)

    MathSciNet  MATH  Google Scholar 

  53. Liang, Y., Hu, S., Ma, L., He, Y.: MSMOTE: improving classification performance when training data is imbalanced. In: International Workshop on Computer Science and Engineering, Qingdao, vol. 2, pp. 13–17 (2009)

    Google Scholar 

  54. Liu, X.Y., Wu, J., Zhou, Z.H.: Exploratory undersampling for class-imbalance learning. IEEE Trans. Syst. Man Cybern. B 39(2), 539–550 (2009)

    Article  Google Scholar 

  55. López, V., Fernández, A., García, S., Palade, V., Herrera, F.: An insight into classification with imbalanced data: empirical results and current trends on using data intrinsic characteristics. Inf. Sci. 250, 113–141 (2013)

    Article  Google Scholar 

  56. 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 

  57. Luengo, J., Fernández, A., García, S., Herrera, F.: Addressing data complexity for imbalanced data sets: analysis of SMOTE–based oversampling and evolutionary undersampling. Soft Comput. 15(10), 1909–1936 (2011)

    Article  Google Scholar 

  58. Ma, L., Fan, S.: CURE-SMOTE algorithm and hybrid algorithm for feature selection and parameter optimization based on random forests. BMC Bioinf. 18, 169 (2017)

    Article  Google Scholar 

  59. Mahalanobis, P.: On the generalized distance in statistics. Proc. Nat. Inst. Sci. (Calcutta) 2, 49–55 (1936)

    Google Scholar 

  60. Menardi, G., Torelli, N.: Training and assessing classification rules with imbalanced data. Data Min. Knowl. Disc. 28(1), 92–122 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  61. Nakamura, M., Kajiwara, Y., Otsuka, A., Kimura, H.: LVQ-SMOTE – learning vector quantization based synthetic minority over-sampling technique for biomedical data. BioData Min. 6, 16 (2013)

    Article  Google Scholar 

  62. Ng, W.W.Y., Hu, J., Yeung, D.S., Yin, S., Roli, F.: Diversified sensitivity-based undersampling for imbalance classification problems. IEEE Trans. Cybern. 45(11), 2402–2412 (2015)

    Article  Google Scholar 

  63. Pérez-Ortiz, M., Gutiérrez, P.A., Hervás-Martínez, C.: Borderline kernel based over-sampling. In: 8th International Conference on Hybrid Artificial Intelligent Systems (HAIS), Salamanca, pp. 472–481 (2013)

    Google Scholar 

  64. Pérez-Ortiz, M., Gutiérrez, P.A., Tiño, P., Hervás-Martínez, C.: Oversampling the minority class in the feature space. IEEE Trans. Neural Netw. Learn. Syst. 27(9), 1947–1961 (2016)

    Article  MathSciNet  Google Scholar 

  65. Prati, R.C., Batista, G.E.A.P.A., Monard, M.C.: A survey on graphical methods for classification predictive performance evaluation. IEEE Trans. Know. Data Eng. 23(11), 1601–1618 (2011)

    Article  Google Scholar 

  66. Puntumapon, K., Waiyamai, K.: A pruning-based approach for searching precise and generalized region for synthetic minority over-sampling. In: 16th Pacific-Asia Conference Advances in Knowledge Discovery and Data Mining (PAKDD), Kuala Lumpur, pp. 371–382 (2012)

    Chapter  Google Scholar 

  67. Quinlan, J.R.: C4.5: Programs for Machine Learning. Morgan Kauffman, San Mateo (1993)

    Google Scholar 

  68. Ramentol, E., Caballero, Y., Bello, R., Herrera, F.: SMOTE-RSB*: a hybrid preprocessing approach based on oversampling and undersampling for high imbalanced data-sets using smote and rough sets theory. Know. Inf. Syst. 33(2), 245–265 (2012)

    Article  Google Scholar 

  69. Ramentol, E., Gondres, I., Lajes, S., Bello, R., Caballero, Y., Cornelis, C., Herrera, F.: Fuzzy-rough imbalanced learning for the diagnosis of high voltage circuit breaker maintenance: the SMOTE-FRST-2T algorithm. Eng. Appl. AI 48, 134–139 (2016)

    Google Scholar 

  70. Rivera, W.A., Xanthopoulos, P.: A priori synthetic over-sampling methods for increasing classification sensitivity in imbalanced data sets. Expert Syst. Appl. 66, 124–135 (2016)

    Article  Google Scholar 

  71. Rokach, L.: Ensemble-based classifiers. Artif. Intell. Rev. 33(1), 1–39 (2010)

    Article  MathSciNet  Google Scholar 

  72. Sáez, J.A., Luengo, J., Stefanowski, J., Herrera, F.: SMOTE-IPF: addressing the noisy and borderline examples problem in imbalanced classification by a re-sampling method with filtering. Inf. Sci. 291, 184–203 (2015)

    Article  Google Scholar 

  73. Smith, M.R., Martinez, T.R., Giraud-Carrier, C.G.: An instance level analysis of data complexity. Mach. Learn. 95(2), 225–256 (2014)

    Article  MathSciNet  Google Scholar 

  74. Stefanowski, J., Wilk, S.: Selective pre-processing of imbalanced data for improving classification performance. In: Proceedings of the 10th International Conference on Data Warehousing and Knowledge Discovery (DaWaK08), Turin, pp. 283–292 (2008)

    Google Scholar 

  75. Sun, Y., Wong, A.K.C., Kamel, M.S.: Classification of imbalanced data: a review. Int. J. Pattern Recogn. Artif. Intell. 23(4), 687–719 (2009)

    Article  Google Scholar 

  76. Sundarkumar, G.G., Ravi, V.: A novel hybrid undersampling method for mining unbalanced datasets in banking and insurance. Eng. Appl. Artif. Intell. 37, 368–377 (2015)

    Article  Google Scholar 

  77. Tahir, M.A., Kittler, J., Yan, F.: Inverse random under sampling for class imbalance problem and its application to multi-label classification. Pattern Recogn. 45(10), 3738–3750 (2012)

    Article  Google Scholar 

  78. Tang, S., Chen, S.: The generation mechanism of synthetic minority class examples. In: 5th International Conference on Information Technology and Applications in Biomedicine (ITAB), Shenzhen, pp. 444–447 (2008)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  80. Wang, J., Xu, M., Wang, H., Zhang, J.: Classification of imbalanced data by using the SMOTE algorithm and locally linear embedding. In: 8th International Conference on Signal Processing (ICSP), Beijing, vol. 3, pp. 1–6. IEEE (2006)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  82. Wu, X., Kumar, V. (eds.): The top ten algorithms in data mining. In: Data Mining and Knowledge Discovery Series. Chapman and Hall/CRC Press, London (2009)

    Book  Google Scholar 

  83. Xie, Z., Jiang, L., Ye, T., Li, X.: A synthetic minority oversampling method based on local densities in low-dimensional space for imbalanced learning. In: 20th International Conference on Database Systems for Advanced Applications (DASFAA), Hanoi, pp. 3–18 (2015)

    Google Scholar 

  84. Yen, S., Lee, Y.: Under-sampling approaches for improving prediction of the minority class in an imbalanced dataset. In: ICIC, Kunming. LNCIS, vol. 344, pp. 731–740 (2006)

    MATH  Google Scholar 

  85. Yen, S.J., Lee, Y.S.: Cluster-based under-sampling approaches for imbalanced data distributions. Expert Syst. Appl. 36(3), 5718–5727 (2009)

    Article  MathSciNet  Google Scholar 

  86. Yeung, D.S., Ng, W.W.Y., Wang, D., Tsang, E.C.C., Wang, X.: Localized generalization error model and its application to architecture selection for radial basis function neural network. IEEE Trans. Neural Netw. 18(5), 1294–1305 (2007)

    Article  Google Scholar 

  87. Yoon, K., Kwek, S.: An unsupervised learning approach to resolving the data imbalanced issue in supervised learning problems in functional genomics. In: HIS’05: Proceedings of the Fifth International Conference on Hybrid Intelligent Systems, Rio de Janeiro, pp. 303–308 (2005)

    Google Scholar 

  88. Yu, H., Ni, J., Zhao, J.: Acosampling: an ant colony optimization-based undersampling method for classifying imbalanced dna microarray data. Neurocomputing 101, 309–318 (2013)

    Article  Google Scholar 

  89. Zhang, H., Li, M.: RWO-Sampling: a random walk over-sampling approach to imbalanced data classification. Inf. Fusion 20, 99–116 (2014)

    Article  Google Scholar 

  90. Zhang, J., Mani, I.: KNN approach to unbalanced data distributions: a case study involving information extraction. In: Proceedings of the 20th International Conference on Machine Learning (ICML’03), Workshop Learning from Imbalanced Data Sets (2003)

    Google Scholar 

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Fernández, A., García, S., Galar, M., Prati, R.C., Krawczyk, B., Herrera, F. (2018). Data Level Preprocessing Methods. In: Learning from Imbalanced Data Sets. Springer, Cham. https://doi.org/10.1007/978-3-319-98074-4_5

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