A hybrid method for increasing the speed of SVM training using belief function theory and boundary region

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The training of the support vector machine (SVM) classifier has high computational complexity and is not suitable for large data classification. Since the classification hyperplane is determined by the support vectors and the other instances do not have an effect on the classifier, a method is introduced that does not use all instances for training. Data set may include inappropriate instances such as noisy and outlier instances. In this paper, a novel method is introduced in which at the first step using the belief function theory, the instances uncertainty such as noisy and outlier instances are identified and discarded, at the second step using the geometric method, called, boundary region, the boundary instances are determined. Finally, at the last step, by using the obtained boundary instances, the training of the SVM classifier is done. In the proposed method BF–BR (Belief Function–Boundary Region), the computational cost of the classification training is reduced without losing classification accuracy. The performance has been evaluated on real world data sets from UCI repository by the tenfold cross validation method. The results of the experiments have been compared with the other methods, which indicate superiority of the proposed method in terms of the number of training instances and training time while good classification accuracy for SVM training is achieved.

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  1. 1.

    Vapnik V (1995) The nature of statistical learning theory. IEEE Trans Neural Netw 10(5):988–999

    MATH  Article  Google Scholar 

  2. 2.

    Yang L, Xu Z, Feature extraction by PCA and diagnosis of breast tumors using SVM with DE-based parameter tuning. Int J Mach Learn Cyber (2017) 1–11

  3. 3.

    Mao WT, Xu JC, Wang C et al (2014) A fast and robust model selection algorithm for multi-input multi-output support vector machine. Neurocomputing 130:10–19

    Article  Google Scholar 

  4. 4.

    Santhanama V, Morariua VI, Harwooda D, Davisa LS (2016) A non-parametric approach to extending generic binary classifiers for multi-classification. Pattern Recogn 58:149–158

    Article  Google Scholar 

  5. 5.

    Feng C, Liao S (2017) Scalable Gaussian kernel support vector machines with sublinear training time complexity. Inf Sci 419:480–494

    Article  Google Scholar 

  6. 6.

    Yan H, Ye Q, Zhang T, Yu DJ (2018) Least squares twin bounded support vector machines based on L1-norm distance metric for classification. Pattern Recogn 74:434–447

    Article  Google Scholar 

  7. 7.

    Abe S (2014) Fusing sequential minimal optimization and Newton’s method for support vector training. Int J Mach Learn Cyber 7:345–364

    Article  Google Scholar 

  8. 8.

    Colbert R, Bagnio S, Bagnio Y (2002) A parallel mixture of SVMs for very large scale problems. Neural Comput 14(5):1105–1114

    Article  Google Scholar 

  9. 9.

    Han D, Liu W, Dezert J, Yang Y (2016) A novel approach to pre-extracting support vectors based on the theory of belief functions. Knowl Based Syst 110:210–223

    Article  Google Scholar 

  10. 10.

    Chau AL, Li X, Yu W (2013) Convex and concave hulls for classification with support vector machine. Neurocomputing 122:198–209

    Article  Google Scholar 

  11. 11.

    Shafer G (1976) A mathematical theory of evidence. Princeton University Press, Princeton

    Google Scholar 

  12. 12.

    Jousselme AL, Liu CS, Grenier D (2006) Measuring ambiguity in the evidence theory. IEEE Trans Syst Man Cybern 36(5):890–903

    Article  Google Scholar 

  13. 13.

    Yager RR (2007) Entropy and specificity in a mathematical theory of evidence. Int J General Syst 9(4):249–260

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Karal O (2017) Maximum likelihood optimal and robust Support Vector Regression with lncosh loss function. Neural Networks 94:1–12

    Article  Google Scholar 

  15. 15.

    Zhou C, Lu X, Huang M (2016) Dempster–Shafer theory-based robust least squares support vector machine for stochastic modelling. Neurocomputing 182:145–153

    Article  Google Scholar 

  16. 16.

    Lu X, Liu W, Zhou C, Huang M (2017) Probabilistic weighted support vector machine for robust modeling with application to hydraulic actuator. IEEE Trans Ind Inform 13(4):1723–1733

    Article  Google Scholar 

  17. 17.

    Han DQ, Han CZ, Yang Y (2009) Approach for pre-extracting support vectors based on K-NN. Control Decis 24(4):494–498

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Yang X, Tan L, He L (2014) A robust least squares support vector machine for regression and classification with noise. Neurocomputing 140:41–52

    Article  Google Scholar 

  19. 19.

    Hu J, Zheng K (2015) A novel support vector regression for data set with outliers. Appl Soft Comput 31:405–411

    Article  Google Scholar 

  20. 20.

    Khosravani HR, Ruano AE, Ferreira PM (2016) A convex hull-based data selection method for data driven models. Appl Soft Comput 47:515–533

    Article  Google Scholar 

  21. 21.

    Mavroforakis ME, Theodoridis S (2006) A geometric approach to support vector machine classification. IEEE Trans Neural Netw 17(3):671–682

    Article  Google Scholar 

  22. 22.

    Wu SJ, Pham VH, Nguyen TN (2017) Two-phase optimization for support vectors and parameter selection of support vector machines: two-class classification. Appl Soft Comput 59:129–142

    Article  Google Scholar 

  23. 23.

    Cai F, Cherkassky V (2012) Generalized SMO algorithm for SVM-based multitask learning. IEEE Trans Neural Netw Learn Syst 23:997–1003

    Article  Google Scholar 

  24. 24.

    Huang X, Shi L, Suykens JAK (2015)” Sequential minimal optimization for SVM with pinball loss. Neurocomputing 149:1596–1603

    Article  Google Scholar 

  25. 25.

    Nalepa J, Kawulok M (2016) Adaptive memetic algorithm enhanced with data geometry analysis to select training data for SVMs. Neurocomputing 185:113–132

    Article  Google Scholar 

  26. 26.

    Zeng M, Yang Y, Luo S, Cheng J (2016) One-class classification based on the convex hull for Bearing fault detection. Mech Syst Signal Process 81:274–293

    Article  Google Scholar 

  27. 27.

    Lucidi S, Palagi L, Risi A, Sciandrone M (2009) A convergent hybrid decomposition algorithm model for SVM training. IEEE Trans Neural Netw 20:1055–1060

    Article  Google Scholar 

  28. 28.

    Vanek J, Michalek J, Psutka J (2017) A GPU-architecture optimized hierarchical decomposition algorithm for support vector machine training. IEEE Trans Comput Soc 28:3330–3343

    Google Scholar 

  29. 29.

    Schleif FM, Tino P (2017) Indefinite core vector machine. Pattern Recogn 71:187–195

    Article  Google Scholar 

  30. 30.

    Tsang IW, Kwok JT, Zurada J (2006) Generalized core vector machines. IEEE Trans Neural Netw 17:1126–1140

    Article  Google Scholar 

  31. 31.

    Xu J (2013) Fast multi-label core vector machine. Pattern Recogn 46:885–898

    MATH  Article  Google Scholar 

  32. 32.

    Almeida AR, Almeida OM, Junior BFS, Barreto LHSC, Barros AK (2017) ICA feature extraction for the location and classification of faults in high-voltage transmission lines. Electr Power Syst Res 148:254–263

    Article  Google Scholar 

  33. 33.

    Shah S, Batool S, Khan I, Ashraf M, Feature extraction through parallel probabilistic principal component analysis for heart disease diagnosis. Phys A Stat Mech Appl (2017) 796–807

    Article  Google Scholar 

  34. 34.

    Lia K, Xiea J, Sunb X, Maa Y (2011)” Multi-class text categorization based on LDA and SVM. Procedia Engineering 15:1963–1967

    Article  Google Scholar 

  35. 35.

    Wang G, Zhang G, Choi KS, Lu J (2017) Deep additive least squares support vector machines for classification with model transfer. IEEE Trans Syst Man Cybern Soc 99:1–14

    Google Scholar 

  36. 36.

    Hamidzadeh J, Monsefi R, Sadoghi Yazdi H (2014) Large symmetric margin instance selection algorithm. Int J Mach Learn Cyber 7:25–45

    MATH  Article  Google Scholar 

  37. 37.

    Mavroforakis ME, Theodoridis S A geometric approach to support vector machine (SVM) classification IEEE Trans Neural Netw 17: 671–382 (2006)

  38. 38.

    Zeng M, Yang Y, Zheng J, Cheng J (2015) Maximum margin classification based on flexible convex hulls Neurocomputing 149:957–965

    Article  Google Scholar 

  39. 39.

    Han DQ, Dezert J, Duan ZS (2016) Evaluation of probability transformations of belief functions for decision making. IEEE Trans Syst Man Cybern 46(1):93–108

    Article  Google Scholar 

  40. 40.

    Liu ZG, Pan Q, Dezert J (2013) Evidential classifier for imprecise data based on belief functions. Knowl Based Syst 52:246–257

    Article  Google Scholar 

  41. 41.

    Liu ZG, Pan Q, Dezert J, Mercier G (2015)” Credal c-means clustering method based on belief functions. Knowl Based Syst 74:119–132

    Article  Google Scholar 

  42. 42.

    Smets P, Kennes R (1994) The transferable belief model. Artif Intell 66(2):191–234

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Denœux T, Kanjanatarakul O, Sriboonchitta S (2015) “EK-NNclus: a clustering procedure based on the evidential K-nearest neighbor rule. Knowl Based Syst 88:57–69

    Article  Google Scholar 

  44. 44.

    Moghaddam VH, Hamidzadeh J (2016) New Hermite orthogonal polynomial kernel and combined kernels in support vector machine classifier. Pattern Recogn 60:921–935

    MATH  Article  Google Scholar 

  45. 45.

    Hamidzadeh J, Monsefi R, Yazdi HS (2015) IRAHC: Instance reduction algorithm using hyperrectangle clustering. Pattern Recogn 48:1878–1889

    MATH  Article  Google Scholar 

  46. 46.

    Yuan F, Xia X, Shi J, Li H, Li G, Non-linear dimensionality reduction and gaussian process based classification method for smoke detection IEEE Access (2017) 5:6833–6841

    Article  Google Scholar 

  47. 47.

    Liu C, Wang W, Wang M, Lv F, Konan M (2017) An efficient instance selection algorithm to reconstruct training set for support vector machine. Knowl Based Syst 116:58–73

    Article  Google Scholar 

  48. 48.

    Hamidzadeh J, Sadeghi R, Namaei N (2017)” Weighted support vector data description based on chaotic bat algorithm. Appl Soft Comput 60:540–551

    Article  Google Scholar 

  49. 49.

    Berg M, Cheong O, Kreveld M, Overmars M, Computational geometry: algorithms and applications. Springer, New York (2008)

    Google Scholar 

  50. 50.

    Jarvis RA (1973) On the identification of the convex hull of a finite set of points in the plane. Inf Process Lett 2:18–21

    MATH  Article  Google Scholar 

  51. 51.

    Lichman M, UCI Machine Learning Repository (2013) http://archive.ics.uci.edu/ml

  52. 52.

    Demsar J (2006) Statistical comparisons of classifiers over multiple data sets. J Mach Learn Res 7:1–30

    MathSciNet  MATH  Google Scholar 

  53. 53.

    Denœux T (1995) A k-nearest neighbor classification rule based on Dempster–Shafer theory. IEEE Trans Syst 25:804–813

    Google Scholar 

  54. 54.

    Asaeedi s, Didehva F, Mohades M (2017) α-Concave hull, a generalization of convex hull. Theor Comput Sci 702:48–59

    MathSciNet  MATH  Article  Google Scholar 

  55. 55.

    Nandan M, Khargonekar PP, Talathi SS (2014) Fast SVM training using approximate extreme points. J Mach Learn Res 15(1):59–98

    MathSciNet  MATH  Google Scholar 

  56. 56.

    Vanir V (1999) An overview of statistical learning theory. IEEE Trans Neural Netw 10(5):988–999

    Article  Google Scholar 

  57. 57.

    Bang S, Kang J, Jhun M, Kim E (2016) Hierarchically penalized support vector machine with grouped variables. Int J Mach Learn Cyber 8:1211–1221

    Article  Google Scholar 

  58. 58.

    Cortes C, Vapnik V (1995) Support-vector networks. Mach Learn 20(3):273–297

    MATH  Google Scholar 

  59. 59.

    Xu P, Davoine F, Zha H, Denœux T (2016) Evidential calibration of binary SVM classifiers. Int J Approx Reason 72:55–70

    MathSciNet  MATH  Article  Google Scholar 

  60. 60.

    Gu B, Sheng VS, Wang Z, Ho D, Osman S, Li S (2015) Incremental learning for ν-support vector regression. Neural Netw 67:140–150

    MATH  Article  Google Scholar 

  61. 61.

    Gu B, Wang JD, Yu YC, Zheng GS, Huang YF, Xu T (2012) Accurate on-line ν-support vector learning. Neural Netw 27:51–59

    MATH  Article  Google Scholar 

  62. 62.

    Xiaa Sy, Xiong Zy, Luo Yg, Dong Lm (2015) A method to improve support vector machine based on distance to hyperplane. Optik Int J Light Electron Opt 126:2405–2410

    Article  Google Scholar 

  63. 63.

    Gu B, Quan X, Gu Y, Sheng VS, Zheng G (2018) Chunk incremental learning for cost-sensitive hinge loss support vector machine. Pattern Recogn 83:196–208

    Article  Google Scholar 

  64. 64.

    Triguero I, Peralta D, Bacardit J, García S, Herrera F (2015) MRPR: a MapReduce solution for prototype reduction in big data classification. Neurocomputing 150:331–345

    Article  Google Scholar 

  65. 65.

    Chang CC, Lin CJ (2011) Libsvm: a library for support vector machines. ACM Trans Intell Syst Technol (TIST) 2(3):27

    Google Scholar 

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Correspondence to Javad Hamidzadeh.

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Moslemnejad, S., Hamidzadeh, J. A hybrid method for increasing the speed of SVM training using belief function theory and boundary region. Int. J. Mach. Learn. & Cyber. 10, 3557–3574 (2019). https://doi.org/10.1007/s13042-019-00944-3

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  • Support vector machine
  • SVM classifier
  • Belief function theory
  • Boundary region
  • Noisy instances