Abstract
In this paper, a new version of Support Vector Machine (SVM) is proposed which any of training samples are considered the random variables. Hence, in order to achieve robustness, the constraint in SVM must be replaced with probability of constraint. In this new model, by applying the nonparametric statistical methods, we obtain the optimal separating hyperplane by solving a quadratic optimization problem. Afterwards, we present the least squares model of our proposed method. The efficiency of our proposed method is shown by several examples for both cases (linear and nonlinear) with probabilistic constraints.
Similar content being viewed by others
References
AVIDAN, S. (2000), “Support Vector Tracking”, IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(8), 1064–1072.
BACHE, K., and LICHMAN, M. (2013), UCIMachine Learning Repository, http://archive.ics.uci.edu/ml/index.php.
BARNETT, N.S., and DRAGOMIR, S.S. (2002), “Some Inequalities for Probability, Expectation and Variance of Random Variables Defined Over a Finite Interval”, Computers and Mathematics with Applications, 43, 1319–1357.
BEN-TAL, A., BHADRA, S., BHATTACHARYYA, C., and NATH, J.S. (2011), “Chance Constrained Uncertain Classification via Robust Optimization”, Mathematical Programming, 127(1), 145–173.
BOSCH, P., LOPEZ, J., RAMIREZ, H., and ROBOTHAM, H. (2013), “Support Vector Machine Under Uncertainty: An Application For Hydroacoustic Classification of Fish-Schools in Chile”, Expert Systems with Applications, 40, 4029–4034.
CHEN, Y., and WANG, J.Z. (2003), “Support Vector Learning for Fuzzy Rule-Based Classification Systems”, IEEE Transactions on Fuzzy Systems, 11(6), 716–728.
CHEN, X., YANG, J., LIANG, J., and YE, Q. (2012), “Recursive Robust Least Squares Support Vector Regression Based on Maximum Correntropy Criterion”, Neurocomputing, 97, 63–73.
CHIANG, J.H., and HAO, P.Y. (2004), “Support Vector Learning Mechanism for Fuzzy Rule-Based Modeling: A New Approach”, IEEE Transactions on Fuzzy Systems, 12(1), 1–12.
CORTES, C., and VAPNIK, V. (1995), “Support Vector Networks”, Machine Learning, 20, 273–297.
EFRON, B. (1979), “Bootstrap Methods: Another Look at the Jackknife”, Annals of Statistics, 7, 1-26.
GAO, J.B., GUNN, S.R., HARRIS, C.J., and BROWN, M. (2002), “A Probabilistic Framework for SVM Regression and Error Bar Estimation”, Machine Learning, 46, 71–89.
HUANG, H.P., and LIU, Y.H. (2002), “Fuzzy Support Vector Machines for Pattern Recognition and Data Mining”, International Journal of Fuzzy Systems, 4, 826–835.
HUANG, G., SONG, S.,WU, C., and YOU, K. (2012a), “Robust Support Vector Regression for Uncertain Input and Output Data”, IEEE Transactions on Neural Networks and Learning Systems, 23(11), 1690–1700.
HUANG, G.B., ZHOU, H., DING, X.J., and ZHANG, R. (2012b), “Extreme Learning Machine for Regression and Multiclass Classification”, IEEE Transactions on Systems, Man, and Cybernetics, Part B (TSMC), 42(2), 513–529.
HUANG, G.B., ZHU, Q.Y., and SIEW, C.K. (2006), “Extreme Learning Machine: Theory and Applications”, Neurocomputing, 70, 489–501.
JINGLIN, Y., LI, H.X., and YONG, H. (2011), “A Probabilistic SVM Based Decision System for Pain Diagnosis”, Expert Systems with Applications, 38, 9346–9351.
JOACHIMS, T. (1998), “Text Categorization with Support Vector Machines: Learning with Many Relevant Features”, in Proceedings European Conference on Machine Learning, eds. C. Nedellec, and C. Rouveirol, London UK: Springer, pp. 137–142.
KARR, A.F. (1992), Probability, New York: Springer.
KIM, S., KAVURI, S., and LEE, M. (2013), “Deep Network with Support Vector Machines”, in Neural Information Processing, Springer, pp. 458–465.
KIM, S., YUA, Z., KIL, R.M., and LEE, M. (2015), “Deep Learning of Support Vector Machines with Class Probability Output Networks”, Neural Networks, Special Issue, pp. 19–28.
KWOK, J.T.Y. (2000), “The Evidence Framework Applied to Support Vector Machines”, IEEE Transactions on Neural Networks, 11, 1162–1173.
LANCKRIET, G.R.G., GHAOUI, L.E., BHATTACHARYYA, CH., and JORDAN, M.I. (2002), “A Robust Minimax Approach to Classification”, Journal of Machine Learning Research, 3, 555–582.
LECUN, Y., BOTOU, L., JACKEL, L., DRUCKER, H., CORTES, C., DENKER, J., GUYON, I., MULLER, U., SACKINGER, E., SIMARD, P., and VAPNIK, V. (1995), “Learning Algorithms for Classification: A Comparison on Handwritten Digit Recognition”, in Neural Networks: The Statistical Mechanics Perspective, World Scientific, pp. 261–276.
LEE, Y.J., and HUANG, S.Y. (2007), “Reduced Support Vector Machines: A Statistical Theory”, IEEE Transactions on Neural Networks, 18, 1–13.
LI, H., YANG, J., ZHANG, G., and FAN, B. (2013), “Probabilistic Support Vector Machines for Classification of Noise Affected Data”, Information Sciences, 221, 60–71.
LIANG, N.Y., HUANG, G.B., SARATCHANDRAN, P., and SUNDARARAJAN, N. (2006), “A Fast and Accurate Online Sequential Learning Algorithm for Feed Forward Networks”, IEEE Transactions on Neural Networks, 17(6), 1411–1423.
LIN, C.F., and WANG, S.D. (2002), “Fuzzy Support Vector Machine”, IEEE Transactions on Neural Networks, 13, 464–471.
LIU, C., NAKASHIMA, K., SAKO, H., and FUJISAWA, H. (2003), “Handwritten Digit Recognition: Bench-Marking of State-of-the-Art Techniques”, Pattern Recognition, 36, 2271–2285.
LIU, W.Y., YUE, K., and GAO, M.H. (2011), “Constructing Probabilistic Graphical Model from Predicate Formulas for Fusing Logical and Probabilistic Knowledge”, Information Sciences, 181(18), 3828–3845.
LOBO, M., VANDENBERGHE, L., BOYD, S., and LEBRET, H. (1998), “Applications of Second-Order Cone Programming”, Linear Algebra Its Applications, 284, 193–228.
MEHROTRA, S. (1992), ”On the Implementation of a Primal-Dual Interior Point Method”, SIAM Journal on Optimization, 2(4), 575–601.
OSUNA, E., FREUND, R., and GIROSI, F. (1997), “Training Support Vector Machines: An Application to Face Detection”, in IEEE Conference Computer Vision Pattern Recognition, pp. 130–136.
PARK, W.J., and KIL, R.M. (2009), “Pattern Classification with Class Probability Output Network”, IEEE Transactions on Neural Networks, 20, 1659–1673.
PLATT, J.C. (1999), “Probabilistic Outputs for Support Vector Machines and Comparisons to Regularized Likelihood Methods”, in Advances in Large Margin Classifiers, MIT Press.
QI, Z., TIAN, Y., and SHI, Y. (2013), “Robust Twin Support Vector Machine for Pattern Classification”, Pattern Recognition, 46, 305–316.
RINALDI, F. (2009), “Mathematical Programming Methods forMinimizing the Zero Norm Over Polyhedral Sets,” Ph.D. thesis, Operations Research at Sapienza University of Rome.
SADOGHI YAZDI, H., EFFATI, S., and SABERI, Z. (2007), “The Probabilistic Constraints in the Support Vector Machine”, Applied Mathematics and Computation, 194, 467–479.
SEBALD, D.J., and BUCKLEW, J.A. (2000), “Support Vector Machine Techniques for Nonlinear Equalization”, IEEE Transactions on Signal Processing, 48(11), 3217–3226.
SOLLICH, P. (2002), “Bayesian Methods for Support Vector Machines: Evidence and Predictive Class Probabilities”, Machine Learning, 46, 21–52.
SUN, Y., YUAN, Y., and WANG, G. (2014), “Extreme Learning Machine for Classification Over Uncertain Data”, Neurocomputing, 128, 500–506.
SUYKENS, J.A.K., DE BRABANTER, J., LUKAS, L., and VDANEWALLE, J. (2002), “Weighted Least Squares Support Vector Machines: Robustness and Sparse Approximation”, Neurocomputing, 48, 85–105.
SUYKENS, J.A.K., and VANDEWALLE, J. (1999), “Least Squares Support Vector Machines Classifiers”, Neural Processing Letters, 9, 293–300.
THI, H.A.L., VO, X.T., and DINH, T.P. (2014), “Feature Selection for Linear SVMs Under Uncertain Data: Robust Optimization Based on Difference of Convex Functions Algorithms”, Neural Networks, 59, 36–50.
TRAFALIS, T.B., and GILBERT, R.C. (2006), “Robust Classification and Regression Using Support Vector Machines”, European Journal of Operational Research, 173(3), 893–909.
VAPNIK, V. (1995), The Nature of Statistical Learning Theory, New York: Springer-Verlag.
WANG, Y.Q.,WANG, S.Y., and LAI, K.K. (2005), “A New Fuzzy Support Vector Machine to Evaluate Credit Risk”, IEEE Transactions on Fuzzy Systems, 13, 820–831.
YANG, X., TAN, L., and HE, L. (2014), “A Robust Least Squares Support Vector Machine for Regression and Classification with Noise”, Neurocomputing, 140, 41–52.
YOU, L., JIZHEN, L., and YAXIN, Q. (2011), “A New Robust Least Squares Support Vector Machine for Regression with Outliers”, Procedia Engineering, 15, 1355–1360.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Abaszade, M., Effati, S. A New Method for Classifying Random Variables Based on Support Vector Machine. J Classif 36, 152–174 (2019). https://doi.org/10.1007/s00357-018-9282-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00357-018-9282-x