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doi: 10.3934/naco.2021027
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Smooth augmented Lagrangian method for twin bounded support vector machine

1. 

Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran

2. 

Department of Mathematics, Faculty of Science, University of Bojnord, Bojnord, Iran

* Corresponding author: Saeed Ketabchi

Received  September 2020 Revised  June 2021 Early access July 2021

In this paper, we propose a method for solving the twin bounded support vector machine (TBSVM) for the binary classification. To do so, we use the augmented Lagrangian (AL) optimization method and smoothing technique, to obtain new unconstrained smooth minimization problems for TBSVM classifiers. At first, the augmented Lagrangian method is recruited to convert TBSVM into unconstrained minimization programming problems called as AL-TBSVM. We attempt to solve the primal programming problems of AL-TBSVM by converting them into smooth unconstrained minimization problems. Then, the smooth reformulations of AL-TBSVM, which we called AL-STBSVM, are solved by the well-known Newton's algorithm. Finally, experimental results on artificial and several University of California Irvine (UCI) benchmark data sets are provided along with the statistical analysis to show the superior performance of our method in terms of classification accuracy and learning speed.

Citation: Fatemeh Bazikar, Saeed Ketabchi, Hossein Moosaei. Smooth augmented Lagrangian method for twin bounded support vector machine. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2021027
References:
[1]

Z. ArabasadiR. AlizadehsaniM. RoshanzamirH. Moosaei and A. Yarifard, Computer aided decision making for heart disease detection using hybrid neural network-Genetic algorithm, Computer Methods and Programs in Biomedicine, 141 (2017), 19-26.   Google Scholar

[2]

F. Bazikar, S. Ketabchi and H. Moosaei, DC programming and DCA for parametric-margin $\nu-$support vector machine, Applied Intelligence, (2020), 1–12. Google Scholar

[3]

D. P. Bertsekas, Nonlinear Programming, Belmont, 1995.  Google Scholar

[4]

E. G. Birgin and J. M. Martinez, Practical Augmented Lagrangian Methods for Constrained Optimization, Society for Industrial and Applied Mathematics, 2014. doi: 10.1137/1.9781611973365.  Google Scholar

[5]

C. C. Chang and C. J. Lin, LIBSVM: A library for support vector machines, ACM transactions on intelligent systems and technology (TIST), 2 (2011), 1-27.   Google Scholar

[6]

C. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Computational Optimization and Applications, 5 (1996), 97-138.  doi: 10.1007/BF00249052.  Google Scholar

[7]

C. Chen and O. L. Mangasarian, Smoothing methods for convex inequalities and linear complementarity problems, Mathematical Programming, 71 (1995), 51-69.  doi: 10.1007/BF01592244.  Google Scholar

[8]

J. Demsar, Statistical comparisons of classifiers over multiple data sets, Journal of Machine Learning Research, (2006), 1–30.  Google Scholar

[9]

A. GoliE. MoeiniA. M. ShafieeM. Zamani and E. Touti, Application of improved artificial intelligence with runner-root meta-heuristic algorithm for dairy products industry: a case study, International Journal on Artificial Intelligence Tools, 29 (2020), 1-30.   Google Scholar

[10]

A. GoliH. K. ZareR. T. Moghaddam and A. Sadeghieh, An improved artificial intelligence based on gray wolf optimization and cultural algorithm to predict demand for dairy products: a case study, IJIMAI, 5 (2019), 15-22.   Google Scholar

[11]

A. GoliH. K. ZareR. Tavakkoli-Moghaddam and A. Sadeghieh, Application of robust optimization for a product portfolio problem using an invasive weed optimization algorithm, Numerical Algebra, Control & Optimization, 9 (2019), 187-209.  doi: 10.3934/naco.2019014.  Google Scholar

[12]

A. Goli, H. K. Zare, R. Tavakkoli-Moghaddam and A. Sadeghieh, Hybrid artificial intelligence and robust optimization for a multi-objective product portfolio problem Case study: The dairy products industry, Computers & Industrial Engineering, 9 (2019), ID: 106090. doi: 10.1016/j.cie.2019.106090.  Google Scholar

[13]

R. L. Iman and J. M. Davenport, Approximations of the critical region of the fbietkan statistic, Communications in Statistics-Theory and Methods, 9 (1980), 571-595.   Google Scholar

[14]

S. Jafarian-NaminA. GoliM. QolipourM. Mostafaeipour and A. M. Golmohammadi, Forecasting the wind power generation using Box-Jenkins and hybrid artificial intelligence, International Journal of Energy Sector Management, 13 (2019), 1038-1062.  doi: 10.1108/IJESM-06-2018-0002.  Google Scholar

[15]

H. Javadi, H. Moosaei and D. Ciuonzo, Learning wireless sensor networks for source localization, Sensors, 19 (2019), 635. Google Scholar

[16]

R. K. JayadevaR. Khemchandani and S. Chandra, Twin support vector machines for pattern classification, IEEE Transactions on Pattern Analysis and Machine Intelligence, 29 (2007), 905-910.  doi: 10.1142/9789812813220_0009.  Google Scholar

[17]

H. JuQ. Hou and L. Jing, Fuzzy and interval-valued fuzzy nonparallel support vector machine, Journal of Intelligent and Fuzzy Systems, 36 (2019), 2677-2690.   Google Scholar

[18]

S. Ketabchi and H. Moosaei, Minimum norm solution to the absolute value equation in the convex case, Journal of Optimization Theory and Applications, 154 (2012), 1080-1087.  doi: 10.1007/s10957-012-0044-3.  Google Scholar

[19]

S. KetabchiH. MoosaeiM. Razzaghi and P. Pardalos, An improvement on parametric $\nu$-support vector algorithm for classification, Ann. Oper. Res., 276 (2019), 155-168.  doi: 10.1007/s10479-017-2724-8.  Google Scholar

[20]

S. Ketabchi and M. Behboodi-Kahoo, Smoothing techniques and augmented Lagrangian method for recourse problem of two-stage stochastic linear programming, Journal of Applied Mathematics, (2013), Article ID: 735916. doi: 10.1155/2013/735916.  Google Scholar

[21]

S. Ketabchi and M. Behboodi-Kahoo, Augmented Lagrangian method for recourse problem of two-stage stochastic linear programming, Kybernetika, 49 (2013), 188-198.   Google Scholar

[22]

R. KhemchandaniP. Saigal and S. Chandra, Improvements on $ \nu$-twin support vector machine, Neural Networks, 79 (2016), 97-107.   Google Scholar

[23]

M. A. Kumar and M. Gopal, Application of smoothing technique on twin support vector machines, Pattern Recognition Letters, 29 (2008), 1842-1848.  doi: 10.1007/978-1-84996-098-4.  Google Scholar

[24]

Y. J. Lee and O. L. Mangasarian, SSVM: A smooth support vector machine for classification, Computational Optimization and Applications, 20 (2001), 5-22.  doi: 10.1023/A:1011215321374.  Google Scholar

[25]

H. MoosaeiS. KetabchiM. Razzaghi and and M. Tanveer, Generalized twin support vector machines, Neural Processing Letters, 53 (2021), 1545-1564.   Google Scholar

[26]

M. Lichman, UCI Machine Learning Repository, Irvine, CA: University of California, School of Information and Computer Science, 2013. Google Scholar

[27]

W. Noble, S.William and others, Support vector machine applications in computational biology, Kernel Methods in Computational Biology, 71 (2004), 92. Google Scholar

[28]

R. LotfiY. Z. MehrjerdiM. S. PishvaeeA. Sadeghieh and G. W. Weber, A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk, Numerical Algebra, Control and Optimization, 11 (2021), 221-253.  doi: 10.3934/naco.2020023.  Google Scholar

[29]

R. Lotfi, Z. Yadegari, S. H. Hosseini, A. H. Khameneh, E. B. Tirkolaee and G. W. Weber, A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project, Journal of Industrial and Management Optimization, Online. doi: 10.3934/jimo.2020158.  Google Scholar

[30]

O. L. Mangasarian and E. W. Wild, Multisurface proximal support vector machine classification via generalized eigenvalues, IEEE Trans. Pattern Anal. Mach. Intell., 28 (2005), 69-74.   Google Scholar

[31]

O. L. Mangasarian and E. W. Wild, Proximal support vector machine classifiers, In Proceedings KDD-2001: Knowledge Discovery and Data Mining, 2001.  Google Scholar

[32]

J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research, Springer-Verlag, New York, 1999. doi: 10.1007/b98874.  Google Scholar

[33]

X. Peng, A $\nu$-twin support vector machine ($\nu$-TSVM) classifier and its geometric algorithms, Information Sciences, 180 (2010), 3863-3875.  doi: 10.1016/j.ins.2010.06.039.  Google Scholar

[34]

C. Platt, Fast training of support vector machines using sequential minimal optimization, in Advances in Kernel Methods, (1999), 185–208. Google Scholar

[35]

Y. H. ShaoC. H. ZhangX. B. Wang and N. Y. Deng, Improvements on twin support vector machines, IEEE Transactions on Neural Networks, 22 (2011), 962-968.  doi: 10.1109/TNNLS.2014.2379930.  Google Scholar

[36]

Y. Tian and Z. Qi, Review on: twin support vector machines, Annals of Data Science, 1 (2014), 253-277.  doi: 10.1007/s40745-014-0018-4.  Google Scholar

[37]

Y. TianX. JuZ. Qi and Y. Shi, Improved twin support vector machine, Science China Mathematics, 57 (2014), 417-432.  doi: 10.1007/s11425-013-4718-6.  Google Scholar

[38]

H. WangZ. Zhou and Y. Xu, An improved $ \nu$-twin bounded support vector machine, Appl. Intell., 48 (2018), 1041-1053.   Google Scholar

[39]

Y. WangT. Wang and J. Bu, Color image segmentation using pixel wise support vector machine classification, Pattern Recognition, 44 (2011), 777-787.   Google Scholar

[40]

Y. Yan and Q. Li, An efficient augmented Lagrangian method for support vector machine, Optimization Methods and Software, 35 (2020), 855-883.  doi: 10.1080/10556788.2020.1734002.  Google Scholar

show all references

References:
[1]

Z. ArabasadiR. AlizadehsaniM. RoshanzamirH. Moosaei and A. Yarifard, Computer aided decision making for heart disease detection using hybrid neural network-Genetic algorithm, Computer Methods and Programs in Biomedicine, 141 (2017), 19-26.   Google Scholar

[2]

F. Bazikar, S. Ketabchi and H. Moosaei, DC programming and DCA for parametric-margin $\nu-$support vector machine, Applied Intelligence, (2020), 1–12. Google Scholar

[3]

D. P. Bertsekas, Nonlinear Programming, Belmont, 1995.  Google Scholar

[4]

E. G. Birgin and J. M. Martinez, Practical Augmented Lagrangian Methods for Constrained Optimization, Society for Industrial and Applied Mathematics, 2014. doi: 10.1137/1.9781611973365.  Google Scholar

[5]

C. C. Chang and C. J. Lin, LIBSVM: A library for support vector machines, ACM transactions on intelligent systems and technology (TIST), 2 (2011), 1-27.   Google Scholar

[6]

C. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Computational Optimization and Applications, 5 (1996), 97-138.  doi: 10.1007/BF00249052.  Google Scholar

[7]

C. Chen and O. L. Mangasarian, Smoothing methods for convex inequalities and linear complementarity problems, Mathematical Programming, 71 (1995), 51-69.  doi: 10.1007/BF01592244.  Google Scholar

[8]

J. Demsar, Statistical comparisons of classifiers over multiple data sets, Journal of Machine Learning Research, (2006), 1–30.  Google Scholar

[9]

A. GoliE. MoeiniA. M. ShafieeM. Zamani and E. Touti, Application of improved artificial intelligence with runner-root meta-heuristic algorithm for dairy products industry: a case study, International Journal on Artificial Intelligence Tools, 29 (2020), 1-30.   Google Scholar

[10]

A. GoliH. K. ZareR. T. Moghaddam and A. Sadeghieh, An improved artificial intelligence based on gray wolf optimization and cultural algorithm to predict demand for dairy products: a case study, IJIMAI, 5 (2019), 15-22.   Google Scholar

[11]

A. GoliH. K. ZareR. Tavakkoli-Moghaddam and A. Sadeghieh, Application of robust optimization for a product portfolio problem using an invasive weed optimization algorithm, Numerical Algebra, Control & Optimization, 9 (2019), 187-209.  doi: 10.3934/naco.2019014.  Google Scholar

[12]

A. Goli, H. K. Zare, R. Tavakkoli-Moghaddam and A. Sadeghieh, Hybrid artificial intelligence and robust optimization for a multi-objective product portfolio problem Case study: The dairy products industry, Computers & Industrial Engineering, 9 (2019), ID: 106090. doi: 10.1016/j.cie.2019.106090.  Google Scholar

[13]

R. L. Iman and J. M. Davenport, Approximations of the critical region of the fbietkan statistic, Communications in Statistics-Theory and Methods, 9 (1980), 571-595.   Google Scholar

[14]

S. Jafarian-NaminA. GoliM. QolipourM. Mostafaeipour and A. M. Golmohammadi, Forecasting the wind power generation using Box-Jenkins and hybrid artificial intelligence, International Journal of Energy Sector Management, 13 (2019), 1038-1062.  doi: 10.1108/IJESM-06-2018-0002.  Google Scholar

[15]

H. Javadi, H. Moosaei and D. Ciuonzo, Learning wireless sensor networks for source localization, Sensors, 19 (2019), 635. Google Scholar

[16]

R. K. JayadevaR. Khemchandani and S. Chandra, Twin support vector machines for pattern classification, IEEE Transactions on Pattern Analysis and Machine Intelligence, 29 (2007), 905-910.  doi: 10.1142/9789812813220_0009.  Google Scholar

[17]

H. JuQ. Hou and L. Jing, Fuzzy and interval-valued fuzzy nonparallel support vector machine, Journal of Intelligent and Fuzzy Systems, 36 (2019), 2677-2690.   Google Scholar

[18]

S. Ketabchi and H. Moosaei, Minimum norm solution to the absolute value equation in the convex case, Journal of Optimization Theory and Applications, 154 (2012), 1080-1087.  doi: 10.1007/s10957-012-0044-3.  Google Scholar

[19]

S. KetabchiH. MoosaeiM. Razzaghi and P. Pardalos, An improvement on parametric $\nu$-support vector algorithm for classification, Ann. Oper. Res., 276 (2019), 155-168.  doi: 10.1007/s10479-017-2724-8.  Google Scholar

[20]

S. Ketabchi and M. Behboodi-Kahoo, Smoothing techniques and augmented Lagrangian method for recourse problem of two-stage stochastic linear programming, Journal of Applied Mathematics, (2013), Article ID: 735916. doi: 10.1155/2013/735916.  Google Scholar

[21]

S. Ketabchi and M. Behboodi-Kahoo, Augmented Lagrangian method for recourse problem of two-stage stochastic linear programming, Kybernetika, 49 (2013), 188-198.   Google Scholar

[22]

R. KhemchandaniP. Saigal and S. Chandra, Improvements on $ \nu$-twin support vector machine, Neural Networks, 79 (2016), 97-107.   Google Scholar

[23]

M. A. Kumar and M. Gopal, Application of smoothing technique on twin support vector machines, Pattern Recognition Letters, 29 (2008), 1842-1848.  doi: 10.1007/978-1-84996-098-4.  Google Scholar

[24]

Y. J. Lee and O. L. Mangasarian, SSVM: A smooth support vector machine for classification, Computational Optimization and Applications, 20 (2001), 5-22.  doi: 10.1023/A:1011215321374.  Google Scholar

[25]

H. MoosaeiS. KetabchiM. Razzaghi and and M. Tanveer, Generalized twin support vector machines, Neural Processing Letters, 53 (2021), 1545-1564.   Google Scholar

[26]

M. Lichman, UCI Machine Learning Repository, Irvine, CA: University of California, School of Information and Computer Science, 2013. Google Scholar

[27]

W. Noble, S.William and others, Support vector machine applications in computational biology, Kernel Methods in Computational Biology, 71 (2004), 92. Google Scholar

[28]

R. LotfiY. Z. MehrjerdiM. S. PishvaeeA. Sadeghieh and G. W. Weber, A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk, Numerical Algebra, Control and Optimization, 11 (2021), 221-253.  doi: 10.3934/naco.2020023.  Google Scholar

[29]

R. Lotfi, Z. Yadegari, S. H. Hosseini, A. H. Khameneh, E. B. Tirkolaee and G. W. Weber, A robust time-cost-quality-energy-environment trade-off with resource-constrained in project management: A case study for a bridge construction project, Journal of Industrial and Management Optimization, Online. doi: 10.3934/jimo.2020158.  Google Scholar

[30]

O. L. Mangasarian and E. W. Wild, Multisurface proximal support vector machine classification via generalized eigenvalues, IEEE Trans. Pattern Anal. Mach. Intell., 28 (2005), 69-74.   Google Scholar

[31]

O. L. Mangasarian and E. W. Wild, Proximal support vector machine classifiers, In Proceedings KDD-2001: Knowledge Discovery and Data Mining, 2001.  Google Scholar

[32]

J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research, Springer-Verlag, New York, 1999. doi: 10.1007/b98874.  Google Scholar

[33]

X. Peng, A $\nu$-twin support vector machine ($\nu$-TSVM) classifier and its geometric algorithms, Information Sciences, 180 (2010), 3863-3875.  doi: 10.1016/j.ins.2010.06.039.  Google Scholar

[34]

C. Platt, Fast training of support vector machines using sequential minimal optimization, in Advances in Kernel Methods, (1999), 185–208. Google Scholar

[35]

Y. H. ShaoC. H. ZhangX. B. Wang and N. Y. Deng, Improvements on twin support vector machines, IEEE Transactions on Neural Networks, 22 (2011), 962-968.  doi: 10.1109/TNNLS.2014.2379930.  Google Scholar

[36]

Y. Tian and Z. Qi, Review on: twin support vector machines, Annals of Data Science, 1 (2014), 253-277.  doi: 10.1007/s40745-014-0018-4.  Google Scholar

[37]

Y. TianX. JuZ. Qi and Y. Shi, Improved twin support vector machine, Science China Mathematics, 57 (2014), 417-432.  doi: 10.1007/s11425-013-4718-6.  Google Scholar

[38]

H. WangZ. Zhou and Y. Xu, An improved $ \nu$-twin bounded support vector machine, Appl. Intell., 48 (2018), 1041-1053.   Google Scholar

[39]

Y. WangT. Wang and J. Bu, Color image segmentation using pixel wise support vector machine classification, Pattern Recognition, 44 (2011), 777-787.   Google Scholar

[40]

Y. Yan and Q. Li, An efficient augmented Lagrangian method for support vector machine, Optimization Methods and Software, 35 (2020), 855-883.  doi: 10.1080/10556788.2020.1734002.  Google Scholar

Figure 1.  Illustration of TWSVM
Figure 2.  Results of linear TWSVM, TBSVM, I$ \nu $-TBSVM and AL-STBSVM on generated data set
Table 1.  Descriptions of the data sets from the UCI repository
Data set $ \# $ Cases $ \# $ Features $ \# $ Classes Source
Sonar 208 60 2 UCI
Cancer 699 9 2 UCI
Diabet 768 8 2 UCI
Wdbc 569 30 2 UCI
Ionosphere 351 34 2 UCI
Australian 690 14 2 UCI
Heart 270 14 2 UCI
Haberman 306 3 2 UCI
German 1000 24 2 UCI
House Votes 435 16 2 UCI
Spect 237 22 2 UCI
Splice 1000 60 2 UCI
Lung-cancer 32 56 2 UCI
F-diagnosis 100 9 2 UCI
Breast-cancer 116 9 2 UCI
Bupa 345 6 2 UCI
Pima 768 9 2 UCI
Housing 506 14 2 UCI
Data set $ \# $ Cases $ \# $ Features $ \# $ Classes Source
Sonar 208 60 2 UCI
Cancer 699 9 2 UCI
Diabet 768 8 2 UCI
Wdbc 569 30 2 UCI
Ionosphere 351 34 2 UCI
Australian 690 14 2 UCI
Heart 270 14 2 UCI
Haberman 306 3 2 UCI
German 1000 24 2 UCI
House Votes 435 16 2 UCI
Spect 237 22 2 UCI
Splice 1000 60 2 UCI
Lung-cancer 32 56 2 UCI
F-diagnosis 100 9 2 UCI
Breast-cancer 116 9 2 UCI
Bupa 345 6 2 UCI
Pima 768 9 2 UCI
Housing 506 14 2 UCI
Table 2.  Comparison of linear TWSVM, TBSVM, I$ \nu $-TBSVM and proposed model (AL-STBSVM) on UCI benchmark data sets
Data set TWSVM TBSVM I$ \nu $-TBSVM AL-STBSVM
size Acc($ \% $), Time(s) $ c_{1}=c_{2} $ Acc($ \% $), Time(s) $ c_{3}=c_{4} $, $ c_{1}=c_{2} $ Acc($ \% $), Time(s) $ c_{1}=c_{2} $, $ \nu $ Acc($ \% $), Time(s) $ c_{3}=c_{4} $, $ c_{1}=c_{2} $
Sonar 76.88, 0.73 77.45, 1.53 70.29, 1.53 79.14, 0.52
208$ \times $ 60 $ 2^{6} $ $ 2^{6} $, $ 2^{-3} $ $ 2^{6} $, 0.2 1, $ 2^{-8} $
Cancer 96.13, 1.63 96.28, 2.63 90.13, 2.28 96.15, 1.99
699$ \times $9 $ 2^{-2} $ $ 2^{-5},2^{-3} $ 1, 0.1 $ 2^{-5},2^{-8} $
Diabet 69.13, 1.90 73.58, 2.68 62.24, 3.39 71.74, 27.35
768$ \times $8 $ 2^{3} $ $ 2^{5},2^{-4} $ $ 2^{4} $, 0.5 $ 2^{3},2^{-7} $
Wdbc 93.66, 1.6 94.74, 2.37 90.87, 2 95.62, 6.16
569$ \times $30 $ 2^{6} $ $ 2^{3},2^{-3} $ $ 2^{5} $, 0.8 $ 2^{-5},2^{-4} $
Ionosphere 84.89, 0.86 86.30, 1.71 84.03, 1.67 85.21, 0.62
351$ \times $34 $ 2^{-3} $ $ 2^{-2},2^{-2} $ $ 2^{-2} $, 0.7 $ 2^{3},2^{-7} $
Australian 83.05, 1.62 84.05, 2.68 67.66, 3.30 83.90, 2.93
690 $ \times $ 14 $ 2^{5} $ $ 2^{5} $, 1 $ 2^{3} $, 0.6 $ 2,2^{4} $
Heart 84.44, 0.98 84.44, 1.57 83.70, 1.64 85.19, 0.63
270$ \times $14 $ 2^{2} $ $ 2,2^{3} $ $ 2^{2} $, 0.9 $ 2^{5},2^{-7} $
Haberman 74.51, 0.86 75.91, 1.64 72.91, 1.66 77.05, 0.55
306$ \times $3 $ 2^{-2} $ $ 2^{3},2^{2} $ 1, 0.6 $ 2^{4},2^{4} $
German 73.9, 1.78 75.4, 3.94 61.8, 6.59 75.1, 11.42
1000$ \times $24 $ 2^{-2} $ $ 2^{-5},2^{-3} $ $ 2^{-3} $, 0.4 $ 2^{10},2^{2} $
House Votes 95.62, 0.85 95.85, 1.90 93.11, 1.67 96.08, 0.65
435$ \times $16 $ 2^{8} $ $ 2,2^{-3} $ 1, 0.1 $ 2^{2},2^{2} $
Spect 68.60, 0.75 70.44, 1.59 73.57, 1.6 71.24, 0.55
237$ \times $22 $ 2^{-5} $ $ 2^{-5},2^{-6} $ $ 2^{9} $, 0.9 $ 2^{3},2^{4} $
Splice 75.40, 2.44 79.70, 3.77 78.20, 5.63 79.18, 10.19
1000$ \times $60 $ 2^{-5} $ $ 2^{-5},2^{-6} $ $ 2^{-5} $, 0.7 $ 2^{5},2^{4} $
Lung-cancer 82.5, 0.62 82.5, 1.40 85, 1.42 85.83, 0.42
32$ \times $56 $ 2^{-1} $ $ 2^{-2},2^{-3} $ $ 2^{-3} $, 0.5 $ 2^{6},2^{-8} $
F-diagnosis 76.49, 0.62 82.34, 1.40 68.35, 1.48 75.01, 0.49
100$ \times $9 $ 2^{5} $ $ 2^{8},2^{5} $ $ 2^{10} $, 0.2 $ 2^{4},2^{-7} $
Breast-cancer 72.52, 0.61 71.84, 1.38 57.04, 1.52 73.81, 0.45
116$ \times $9 $ 2^{6} $ $ 2^{3},2^{-2} $ $ 2^{6} $, 0.9 $ 2^{5},2^{-7} $
Bupa 64.35, 0.73 65.23, 1.69 69.29, 1.72 66.68, 0.58
345$ \times $6 $ 2^{-4} $ $ 2^{-3} $, 2 $ 2^{-4} $, 0.2 $ 2^{2},2^{4} $
Pima 71.36, 0.86 73.30, 1.64 64.72, 2.65 73.62, 2.04
768$ \times $9 $ 2^{-7} $ $ 2^{-5},2^{-3} $ $ 2^{-7} $, 0.2 $ 2^{2},2^{4} $
Housing 80.08, 0.96 80.24, 2.89 61.11, 2.01 93.09, 1.5
506$ \times $14 $ 2^{-8} $ $ 2^{-7},2^{-6} $ $ 2^{-7} $, 0.2 $ 2^{6},2^{-8} $
Avg.acc 79.08 80.53 74.11 83.31
Data set TWSVM TBSVM I$ \nu $-TBSVM AL-STBSVM
size Acc($ \% $), Time(s) $ c_{1}=c_{2} $ Acc($ \% $), Time(s) $ c_{3}=c_{4} $, $ c_{1}=c_{2} $ Acc($ \% $), Time(s) $ c_{1}=c_{2} $, $ \nu $ Acc($ \% $), Time(s) $ c_{3}=c_{4} $, $ c_{1}=c_{2} $
Sonar 76.88, 0.73 77.45, 1.53 70.29, 1.53 79.14, 0.52
208$ \times $ 60 $ 2^{6} $ $ 2^{6} $, $ 2^{-3} $ $ 2^{6} $, 0.2 1, $ 2^{-8} $
Cancer 96.13, 1.63 96.28, 2.63 90.13, 2.28 96.15, 1.99
699$ \times $9 $ 2^{-2} $ $ 2^{-5},2^{-3} $ 1, 0.1 $ 2^{-5},2^{-8} $
Diabet 69.13, 1.90 73.58, 2.68 62.24, 3.39 71.74, 27.35
768$ \times $8 $ 2^{3} $ $ 2^{5},2^{-4} $ $ 2^{4} $, 0.5 $ 2^{3},2^{-7} $
Wdbc 93.66, 1.6 94.74, 2.37 90.87, 2 95.62, 6.16
569$ \times $30 $ 2^{6} $ $ 2^{3},2^{-3} $ $ 2^{5} $, 0.8 $ 2^{-5},2^{-4} $
Ionosphere 84.89, 0.86 86.30, 1.71 84.03, 1.67 85.21, 0.62
351$ \times $34 $ 2^{-3} $ $ 2^{-2},2^{-2} $ $ 2^{-2} $, 0.7 $ 2^{3},2^{-7} $
Australian 83.05, 1.62 84.05, 2.68 67.66, 3.30 83.90, 2.93
690 $ \times $ 14 $ 2^{5} $ $ 2^{5} $, 1 $ 2^{3} $, 0.6 $ 2,2^{4} $
Heart 84.44, 0.98 84.44, 1.57 83.70, 1.64 85.19, 0.63
270$ \times $14 $ 2^{2} $ $ 2,2^{3} $ $ 2^{2} $, 0.9 $ 2^{5},2^{-7} $
Haberman 74.51, 0.86 75.91, 1.64 72.91, 1.66 77.05, 0.55
306$ \times $3 $ 2^{-2} $ $ 2^{3},2^{2} $ 1, 0.6 $ 2^{4},2^{4} $
German 73.9, 1.78 75.4, 3.94 61.8, 6.59 75.1, 11.42
1000$ \times $24 $ 2^{-2} $ $ 2^{-5},2^{-3} $ $ 2^{-3} $, 0.4 $ 2^{10},2^{2} $
House Votes 95.62, 0.85 95.85, 1.90 93.11, 1.67 96.08, 0.65
435$ \times $16 $ 2^{8} $ $ 2,2^{-3} $ 1, 0.1 $ 2^{2},2^{2} $
Spect 68.60, 0.75 70.44, 1.59 73.57, 1.6 71.24, 0.55
237$ \times $22 $ 2^{-5} $ $ 2^{-5},2^{-6} $ $ 2^{9} $, 0.9 $ 2^{3},2^{4} $
Splice 75.40, 2.44 79.70, 3.77 78.20, 5.63 79.18, 10.19
1000$ \times $60 $ 2^{-5} $ $ 2^{-5},2^{-6} $ $ 2^{-5} $, 0.7 $ 2^{5},2^{4} $
Lung-cancer 82.5, 0.62 82.5, 1.40 85, 1.42 85.83, 0.42
32$ \times $56 $ 2^{-1} $ $ 2^{-2},2^{-3} $ $ 2^{-3} $, 0.5 $ 2^{6},2^{-8} $
F-diagnosis 76.49, 0.62 82.34, 1.40 68.35, 1.48 75.01, 0.49
100$ \times $9 $ 2^{5} $ $ 2^{8},2^{5} $ $ 2^{10} $, 0.2 $ 2^{4},2^{-7} $
Breast-cancer 72.52, 0.61 71.84, 1.38 57.04, 1.52 73.81, 0.45
116$ \times $9 $ 2^{6} $ $ 2^{3},2^{-2} $ $ 2^{6} $, 0.9 $ 2^{5},2^{-7} $
Bupa 64.35, 0.73 65.23, 1.69 69.29, 1.72 66.68, 0.58
345$ \times $6 $ 2^{-4} $ $ 2^{-3} $, 2 $ 2^{-4} $, 0.2 $ 2^{2},2^{4} $
Pima 71.36, 0.86 73.30, 1.64 64.72, 2.65 73.62, 2.04
768$ \times $9 $ 2^{-7} $ $ 2^{-5},2^{-3} $ $ 2^{-7} $, 0.2 $ 2^{2},2^{4} $
Housing 80.08, 0.96 80.24, 2.89 61.11, 2.01 93.09, 1.5
506$ \times $14 $ 2^{-8} $ $ 2^{-7},2^{-6} $ $ 2^{-7} $, 0.2 $ 2^{6},2^{-8} $
Avg.acc 79.08 80.53 74.11 83.31
Table 3.  Comparison of nonlinear TWSVM, TBSVM, I$ \nu $-TBSVM and proposed model (AL-STBSVM) on UCI benchmark data sets
Dataset TWSVM TBSVM I$ \nu $-TBSVM AL-STBSVM
size Acc($ \% $), Time(s) $ c_{1}=c_{2} $, $ \gamma $ Acc($ \% $), Time(s) $ c_{3}=c_{4} $, $ c_{1}=c_{2} $, $ \gamma $ Acc($ \% $), Time(s) $ c_{1}=c_{2} $, $ \nu $, $ \gamma $ Acc($ \% $), Time(s) $ c_{3}=c_{4} $, $ c_{1}=c_{2} $, $ \gamma $
Sonar 84.53, 0.80 86.14, 1.62 83.27, 1.67 87.54, 0.84
208$ \times $ 60 $ 2^{5},2^{-6} $ $ 2^{6},2^{-3},2^{-3} $ $ 2^{5} $, 0.2, $ 2^{-3} $ $ 2^{2},2^{-8},2^{-6} $
Cancer 96.42, 4.71 96.42, 6.59 96.02, 2.90 96.85, 3.85
699$ \times $9 $ 2^{-2},2^{-6} $ $ 2^{-3},2^{-3},2^{-6} $ $ 2 $, 0.1, $ 2^{-4} $ $ 2^{-5},2^{-8},2^{-3} $
Diabet 65.49, 10.18 66.28, 8.21 65.11, 5.33 69.14, 21.28
768$ \times $8 $ 2^{5},2^{-6} $ $ 2^{5},2^{-4},2^{-6} $ $ 2^{4} $, 0.4, $ 2^{-3} $ $ 2^{5},2^{-6},2^{-3} $
Wdbc 62.74, 2.64 62.74, 3.53 91.76, 2.17 87.18, 10.69
569$ \times $30 $ 2^{5},2^{-6} $ $ 2^{4},2^{-3},2^{-6} $ $ 2^{-9} $, 0.3, $ 2^{-8} $ $ 2^{-5},2^{-6},2^{-1} $
Ionosphere 94.89, 1.42 94.54, 2.15 87.17, 1.88 94.89, 2.96
351$ \times $34 $ 2^{3},2^{-6} $ $ 2^{-4},2^{-2},2^{-3} $ $ 2^{-2} $, 0.2, $ 2^{-2} $ $ 2^{4},2^{-6},2^{-6} $
Australian 55.65, 4.89 55.22, 5.47 55.51, 6.94 65.21, 17.95
690$ \times $14 $ 2^{3},2^{-3} $ $ 2^{5},2,2^{-6} $ $ 2^{3} $, 0.5, $ 2^{-3} $ $ 2^{2},2^{4},2^{-5} $
Heart 82.59, 0.92 83.33, 1.86 81.48, 1.76 83.33, 1.58
270$ \times $14 $ 2^{2},2^{-3} $ $ 2,2,2^{-6} $ $ 2^{2} $, 0.9, $ 2^{9} $ $ 2^{5},2^{-7},2^{-6} $
Haberman 73.85, 1.28 73.52, 2.01 73.19, 1.88 73.52, 2.67
306$ \times $3 $ 2^{-1},2^{-6} $ $ 2,2,2^{-6} $ $ 1 $, 0.6, $ 2^{9} $ $ 2^{3},2^{4},2^{-6} $
German 70.1, 14.69 70.2, 14.34 70, 7.33 71.5, 35.33
1000$ \times $24 $ 2^{-2},2^{-6} $ $ 2^{-4},2^{-4},2^{-6} $ $ 2^{-3} $, 0.4, $ 2^{7} $ $ 2^{9},2^{4},2^{-6} $
House Votes 92.64, 1.22 93.55, 2.20 91.70, 1.78 94.71, 4.12
435$ \times $16 $ 2^{8},2^{-6} $ $ 2,2^{-2},2^{-6} $ $ 1 $, 0.1, $ 2^{-5} $ $ 2^{3},2^{4},2^{-3} $
Spect 71.89, 0.86 73.76, 1.66 68.90, 1.66 74.17, 1.45
237$ \times $22 $ 2^{-5},2^{-6} $ $ 2^{-3},2^{-6},2^{-4} $ $ 2^{-7} $, 0.6, $ 2^{-5} $ $ 2^{3},2^{4},2^{-1} $
Splice 76.71, 15.98 75.20, 16.58 74.41, 14.66 87.20, 42.10
1000$ \times $60 $ 2^{-6},2^{-6} $ $ 2^{-5},2^{-5},2^{-6} $ $ 2^{-4} $, 0.2, $ 2^{-3} $ $ 2^{5},2^{5},2^{-6} $
Lung-caner 80.83, 0.67 85.83, 1.41 84.16, 1.51 85, 0.35
32$ \times $56 $ 2,2^{-6} $ $ 2^{-2},2^{-3},2^{-6} $ $ 2^{-5} $, 0.1, $ 2^{-3} $ $ 2^{5},2^{-8},2^{-6} $
F-diagnosis 88.14, 0.64 88.14, 1.42 87.16, 1.59 89.25, 0.44
100$ \times $9 $ 2^{5},2^{-3} $ $ 2^{6},2^{5},2^{-3} $ $ 2^{10} $, 0.2, $ 2^{7} $ $ 2^{4},2^{-7},2^{-2} $
Breast-cancer 55.16, 0.65 55.16, 1.48 55.15, 1.56 60.31, 0.48
116$ \times $9 $ 2^{5},2^{-3} $ $ 2^{3},2^{-5},2^{-3} $ $ 2^{7} $, 0.9, $ 2^{-6} $ $ 2^{4},2^{-7},2^{-6} $
Bupa 64.37, 1.19 64.35, 2.02 65.80, 1.94 66.36, 2.62
345$ \times $6 $ 2^{-4},2^{-6} $ $ 2^{3},2,2^{-3} $ $ 2^{-5} $, 0.2, $ 2^{-3} $ $ 2^{3},2^{4},2^{-6} $
Pima 65.63, 9.66 66.15, 8.08 65.12, 4.74 69, 20.07
768$ \times $9 $ 2^{-3},2^{-6} $ $ 2^{-6},2^{-6},2^{-6} $ $ 2^{-3} $, 0.6, $ 2^{-6} $ $ 2^{2},2^{5},2^{-3} $
Housing 93.09, 2.12 93.09, 3.65 93.09, 4.30 93.09, 7.01
506$ \times $14 $ 2^{-6},2^{6} $ $ 2^{-5},2^{-3},2^{7} $ $ 2^{10} $, 0.7, $ 2^{7} $ $ 2^{7},2^{-8},2^{-6} $
Avg.acc 76.37 76.86 77.16 80.46
Dataset TWSVM TBSVM I$ \nu $-TBSVM AL-STBSVM
size Acc($ \% $), Time(s) $ c_{1}=c_{2} $, $ \gamma $ Acc($ \% $), Time(s) $ c_{3}=c_{4} $, $ c_{1}=c_{2} $, $ \gamma $ Acc($ \% $), Time(s) $ c_{1}=c_{2} $, $ \nu $, $ \gamma $ Acc($ \% $), Time(s) $ c_{3}=c_{4} $, $ c_{1}=c_{2} $, $ \gamma $
Sonar 84.53, 0.80 86.14, 1.62 83.27, 1.67 87.54, 0.84
208$ \times $ 60 $ 2^{5},2^{-6} $ $ 2^{6},2^{-3},2^{-3} $ $ 2^{5} $, 0.2, $ 2^{-3} $ $ 2^{2},2^{-8},2^{-6} $
Cancer 96.42, 4.71 96.42, 6.59 96.02, 2.90 96.85, 3.85
699$ \times $9 $ 2^{-2},2^{-6} $ $ 2^{-3},2^{-3},2^{-6} $ $ 2 $, 0.1, $ 2^{-4} $ $ 2^{-5},2^{-8},2^{-3} $
Diabet 65.49, 10.18 66.28, 8.21 65.11, 5.33 69.14, 21.28
768$ \times $8 $ 2^{5},2^{-6} $ $ 2^{5},2^{-4},2^{-6} $ $ 2^{4} $, 0.4, $ 2^{-3} $ $ 2^{5},2^{-6},2^{-3} $
Wdbc 62.74, 2.64 62.74, 3.53 91.76, 2.17 87.18, 10.69
569$ \times $30 $ 2^{5},2^{-6} $ $ 2^{4},2^{-3},2^{-6} $ $ 2^{-9} $, 0.3, $ 2^{-8} $ $ 2^{-5},2^{-6},2^{-1} $
Ionosphere 94.89, 1.42 94.54, 2.15 87.17, 1.88 94.89, 2.96
351$ \times $34 $ 2^{3},2^{-6} $ $ 2^{-4},2^{-2},2^{-3} $ $ 2^{-2} $, 0.2, $ 2^{-2} $ $ 2^{4},2^{-6},2^{-6} $
Australian 55.65, 4.89 55.22, 5.47 55.51, 6.94 65.21, 17.95
690$ \times $14 $ 2^{3},2^{-3} $ $ 2^{5},2,2^{-6} $ $ 2^{3} $, 0.5, $ 2^{-3} $ $ 2^{2},2^{4},2^{-5} $
Heart 82.59, 0.92 83.33, 1.86 81.48, 1.76 83.33, 1.58
270$ \times $14 $ 2^{2},2^{-3} $ $ 2,2,2^{-6} $ $ 2^{2} $, 0.9, $ 2^{9} $ $ 2^{5},2^{-7},2^{-6} $
Haberman 73.85, 1.28 73.52, 2.01 73.19, 1.88 73.52, 2.67
306$ \times $3 $ 2^{-1},2^{-6} $ $ 2,2,2^{-6} $ $ 1 $, 0.6, $ 2^{9} $ $ 2^{3},2^{4},2^{-6} $
German 70.1, 14.69 70.2, 14.34 70, 7.33 71.5, 35.33
1000$ \times $24 $ 2^{-2},2^{-6} $ $ 2^{-4},2^{-4},2^{-6} $ $ 2^{-3} $, 0.4, $ 2^{7} $ $ 2^{9},2^{4},2^{-6} $
House Votes 92.64, 1.22 93.55, 2.20 91.70, 1.78 94.71, 4.12
435$ \times $16 $ 2^{8},2^{-6} $ $ 2,2^{-2},2^{-6} $ $ 1 $, 0.1, $ 2^{-5} $ $ 2^{3},2^{4},2^{-3} $
Spect 71.89, 0.86 73.76, 1.66 68.90, 1.66 74.17, 1.45
237$ \times $22 $ 2^{-5},2^{-6} $ $ 2^{-3},2^{-6},2^{-4} $ $ 2^{-7} $, 0.6, $ 2^{-5} $ $ 2^{3},2^{4},2^{-1} $
Splice 76.71, 15.98 75.20, 16.58 74.41, 14.66 87.20, 42.10
1000$ \times $60 $ 2^{-6},2^{-6} $ $ 2^{-5},2^{-5},2^{-6} $ $ 2^{-4} $, 0.2, $ 2^{-3} $ $ 2^{5},2^{5},2^{-6} $
Lung-caner 80.83, 0.67 85.83, 1.41 84.16, 1.51 85, 0.35
32$ \times $56 $ 2,2^{-6} $ $ 2^{-2},2^{-3},2^{-6} $ $ 2^{-5} $, 0.1, $ 2^{-3} $ $ 2^{5},2^{-8},2^{-6} $
F-diagnosis 88.14, 0.64 88.14, 1.42 87.16, 1.59 89.25, 0.44
100$ \times $9 $ 2^{5},2^{-3} $ $ 2^{6},2^{5},2^{-3} $ $ 2^{10} $, 0.2, $ 2^{7} $ $ 2^{4},2^{-7},2^{-2} $
Breast-cancer 55.16, 0.65 55.16, 1.48 55.15, 1.56 60.31, 0.48
116$ \times $9 $ 2^{5},2^{-3} $ $ 2^{3},2^{-5},2^{-3} $ $ 2^{7} $, 0.9, $ 2^{-6} $ $ 2^{4},2^{-7},2^{-6} $
Bupa 64.37, 1.19 64.35, 2.02 65.80, 1.94 66.36, 2.62
345$ \times $6 $ 2^{-4},2^{-6} $ $ 2^{3},2,2^{-3} $ $ 2^{-5} $, 0.2, $ 2^{-3} $ $ 2^{3},2^{4},2^{-6} $
Pima 65.63, 9.66 66.15, 8.08 65.12, 4.74 69, 20.07
768$ \times $9 $ 2^{-3},2^{-6} $ $ 2^{-6},2^{-6},2^{-6} $ $ 2^{-3} $, 0.6, $ 2^{-6} $ $ 2^{2},2^{5},2^{-3} $
Housing 93.09, 2.12 93.09, 3.65 93.09, 4.30 93.09, 7.01
506$ \times $14 $ 2^{-6},2^{6} $ $ 2^{-5},2^{-3},2^{7} $ $ 2^{10} $, 0.7, $ 2^{7} $ $ 2^{7},2^{-8},2^{-6} $
Avg.acc 76.37 76.86 77.16 80.46
Table 4.  Rank of accuracy linear classifiers on UCI benchmark data sets
Data set TWSVM TBSVM I$ \nu $-TBSVM AL-STBSVM
Sonar 3 2 4 1
Cancer 3 1 4 2
Diabet 3 1 4 2
Wdbc 3 2 4 1
Ionosphere 3 1 4 2
Australian 3 1 4 2
Heart 2.5 2.5 4 1
Haberman 3 2 4 1
German 3 1 4 2
House Votes 3 2 4 1
Spect 4 3 1 2
Splice 4 1 3 2
Lung-cancer 3.5 3.5 2 1
F-diagnosis 2 1 4 3
Breast-cancer 2 3 4 1
Bupa 4 3 1 2
Pima 3 2 4 1
Housing 3 2 4 1
Average rank 3.06 1.89 3.5 1.56
Data set TWSVM TBSVM I$ \nu $-TBSVM AL-STBSVM
Sonar 3 2 4 1
Cancer 3 1 4 2
Diabet 3 1 4 2
Wdbc 3 2 4 1
Ionosphere 3 1 4 2
Australian 3 1 4 2
Heart 2.5 2.5 4 1
Haberman 3 2 4 1
German 3 1 4 2
House Votes 3 2 4 1
Spect 4 3 1 2
Splice 4 1 3 2
Lung-cancer 3.5 3.5 2 1
F-diagnosis 2 1 4 3
Breast-cancer 2 3 4 1
Bupa 4 3 1 2
Pima 3 2 4 1
Housing 3 2 4 1
Average rank 3.06 1.89 3.5 1.56
Table 5.  Rank of accuracy nonlinear classifiers on UCI benchmark data sets
Data set TWSVM TBSVM I$ \nu $-TBSVM AL-STBSVM
Sonar 3 2 4 1
Cancer 2.5 2.5 4 1
Diabet 3 2 4 1
Wdbc 3.5 3.5 1 2
Ionosphere 1.5 3 4 1.5
Australian 2 4 3 1
Heart 3 1.5 4 1.5
Haberman 1 2.5 4 2.5
German 3 2 4 1
House Votes 3 2 4 1
Spect 3 2 4 1
Splice 2 3 4 1
Lung-cancer 4 1 3 2
F-diagnosis 2.5 2.5 4 1
Breast-cancer 2.5 2.5 4 1
Bupa 3 4 2 1
Pima 2 3 4 1
Housing 2.5 2.5 2.5 2.5
Average rank 2.61 2.53 3.53 1.33
Data set TWSVM TBSVM I$ \nu $-TBSVM AL-STBSVM
Sonar 3 2 4 1
Cancer 2.5 2.5 4 1
Diabet 3 2 4 1
Wdbc 3.5 3.5 1 2
Ionosphere 1.5 3 4 1.5
Australian 2 4 3 1
Heart 3 1.5 4 1.5
Haberman 1 2.5 4 2.5
German 3 2 4 1
House Votes 3 2 4 1
Spect 3 2 4 1
Splice 2 3 4 1
Lung-cancer 4 1 3 2
F-diagnosis 2.5 2.5 4 1
Breast-cancer 2.5 2.5 4 1
Bupa 3 4 2 1
Pima 2 3 4 1
Housing 2.5 2.5 2.5 2.5
Average rank 2.61 2.53 3.53 1.33
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