Support vector regression with random output variable and probabilistic constraints

[1] A. R. Arabpour and M. Tata, Estimating the parameters of a fuzzy linear regression model,
Iranian Journal of Fuzzy Systems, 5(2) (2008), 1–19.
[2] K. Bache and M. Lichman, UCI machine learning repository, Available on-line at:
http://archive.ics.uci.edu/ml/machine-learning-databases, 2013.
[3] A. Ben-Tal, S. Bhadra, C. Bhattacharyya and J. S. Nath, Chance constrained uncertain
classification via robust optimization, Math. Program., 127(1) (2011), 145–173.
[4] P. Bosch, J. Lopez, H. Ramirez and H. Robotham, Support vector machine under uncertainty:
an application for hydroacoustic classification of fish-schools in Chile, Expert Systems with
Applications, 40 (2013), 4029–4034.
[5] K. D. Brabanter, J. D. Brabanter, J. A. K. Suykens and B. D. Moor, Approximate confidence
and prediction intervals for least squares support vector regression, IEEE Transactions on
Neural Networks, 22 (2011), 110–120.
[6] E. Carrizosa, J. E. Gordillo and F. Plastria, Kernel support vector regression with imprecise
output, Dept. MOSI. Vrije Univ. Brussel. Belgium. Tech. Rep., Available on-line at:
http://www.optimization-online.org/DB_FILE/2008/01/1896.pdf, 2008.
[7] E. Carrizosa, J. E. Gordillo and F. Plastria, Support vector regression for imprecise
data, Dept. MOSI. Vrije Univ. Brussel. Belgium. Tech. Rep., Available on-line at:
http://www.optimization-online.org/DB_HTML/2007/11/1826.html, 2007.
[8] J. H. Chiang and P. Y. Hao, Support vector learning mechanism for fuzzy rule-based modeling:
a new approach, IEEE Trans. Fuzzy Syst., 12(1) (2004), 1–12.
[9] H. Drucker, Ch. J. C. Burges, L. Kaufman, A. Smola and V. Vapnik, Support vector regression
machines, Adv. Neural Inform. Process. Syst., 9 (1997), 155–161.
[10] B. Efron, Bootstrap methods: Another look at the jackknife, Annals of Statistics, 7 (1979)
1–26.
[11] A. Farag and R. M. Mohamed, Classification of multispectral data using support vector machines
approach for density estimation, IEEE Seventh International Conference on Intell.
Eng. Syst., (2003), 4–6.
[12] J. B. Gao, S. R. Gunn, C. J. Harris and M. Brown, A probabilistic framework for SVM
regression and error bar estimation, Machine Learning, 46 (2002), 71–89.
[13] P. Y. Hao and J. H. Chiang, A fuzzy model of support vector regression machine, International
Journal of Fuzzy Systems, 9(1) (2007), 45–49.
[14] H. P. Huang and Y. H. Liu, Fuzzy support vector machines for pattern recognition and data
mining, International Journal of Fuzzy Systems, 4 (2002), 826–835.
[15] G. Huang, S. Song, C. Wu and K. You, Robust support vector regression for uncertain input
and output data, IEEE Transactions on Neural Networks and Learning Systems, 23(11)
(2012), 1690–1700.
[16] R. K. Jayadeva, R. Khemchandani and S. Chandra, Twin support vector machines for pattern
classification, IEEE Transactions on Pattern Analysis and Machine Intelligence, 29(5)
(2007), 905–910.
[17] Y. Jinglin, H. X. Li and H. Yong, A probabilistic SVM based decision system for pain diagnosis,
Expert Systems with Applications, 38 (2011), 9346–9351.
[18] A. F. Karr, probability, Springer, New york, (1993), 52–74.
[19] M. A. Kumar and M. Gopal, Least squares twin support vector machines for pattern classification,
Expert Systems with Applications, 36(4) (2009), 7535–7543.
[20] J. T. Y. Kwok, The evidence framework applied to support vector machines, IEEE Transactions
on Neural Networks, 11 (2000), 1162–1173.
[21] G. R. G. Lanckriet, L. E. Ghaoui, Ch. Bhattacharyya and M. I. Jordan, A robust minimax
approach to classification, J. Mach. Learn. Res., 3 (2002), 555–582.

[22] Y. J. Lee and S. Y. Huang, Reduced support vector machines: a statistical theory, IEEE
Transactions on Neural Networks, 18 (2007), 1–13.
[23] H. Li, J. Yang, G. Zhang and B. Fan, Probabilistic support vector machines for classification
of noise affected data, Information Sciences, 221 (2013), 60–71.
[24] C. F. Lin and S. D. Wang, Fuzzy support vector machine, IEEE Transactions on Neural
Networks, 13 (2002), 464–471.
[25] W. Y. Liu, K. Yue and M. H. Gao, Constructing probabilistic graphical model from predicate
formulas for fusing logical and probabilistic knowledge, Information Sciences, 181(18) (2011),
3828–3845.
[26] M. Lobo, L. Vandenberghe, S. Boyd and H. Lebret, Applications of second-order cone programming,
Linear Algebra Its Appl., 284 (1998), 193–228.
[27] O. L. Mangasarian, Nonlinear Programming, McGraw-Hill, New York, (1969), 69–75.
[28] S. Mehrotra, On the implementation of a primal-dual interior point method, SIAM J. Optim.,
2 (1992), 575–601.
[29] X. Peng, TSVR: an efficient twin support vector machine for regression, Neural Networks.,
23(3) (2010), 365–372.
[30] J. C. Platt, Probabilistic outputs for support vector machines and comparisons to regularized
likelihood methods, Advances in Large Margin Classifiers, 10(3) (1999), 61–74.
[31] Z. Qi, Y. Tian and Y. Shi, Robust twin support vector machine for pattern classification,
Pattern Recognition, 46 (2013), 305–316.
[32] H. Sadoghi Yazdi, S. Effati and Z. Saberi, The probabilistic constraints in the support vector
machine, App. Math. Comput., 194 (2007), 467–479.
[33] P. K. Shivaswamy, Ch.Bhattacharyya and A.J.Smola, Second order cone programming approaches
for handling missing and uncertain data,J. Mach. Learn. Res.,7 (2006), 1283-1314.
[34] P. Sollich, Bayesian methods for support vector machines: evidence and predictive class
probabilities, Machine Learning, 46 (2002), 21–52.
[35] J. A. K. Suykens and J. Vandewalle, Least squares support vector machine classifiers, Neural
Processing Letters, 9(3) (1999), 293–300.
[36] T. B. Trafalis and S. A. Alwazzi, Support vector regression with noisy data: a second order
cone programming approach, Int. J. General Syst., 36 (2007), 237–250.
[37] T. B. Trafalis and R. C. Gilbert, Robust classification and regression using support vector
machines, Eur. J. Oper. Res., 173(3) (2006), 893–909.
[38] URL http://www.csie.ntu.edu.tw/ cjlin/libsvmtools/datasets/regression.html.
[39] URL http://www.dcc.fc.up.pt/ ltorgo/Regression/DataSets.html.
[40] URL http://www.itl.nist.gov/div898/strd/nls/nls_main.shtml.
[41] V. Vapnik, The nature of statistical learning theory, Springer-Verlag, New York, (1995),
123-146, 181-186.
[42] V. Vapnik, S. Golowich and A. Smola, Support vector method for multivariate density estimation,
Adv. Neural Inform. Process. Syst., 12 (1999), 659–665.
[43] Y. Xu and L. Wang, A weighted twin support vector regression, Knowledge-Based Syst., 33
(2012), 92–101.
[44] Y. Xu, W. Xi, X. Lv and R. Guo, An improved least squares twin support vector machine,
Journal of information and computational science, 9(4) (2012), 1063–1071.
[45] X. Yang, L. Tan and L. He, A robust least squares support vector machine for regression and
classification with noise, Neurocomputing, 140 (2014), 41–52.