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Value-at-risk support vector machine: stability to outliers

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Abstract

A support vector machine (SVM) stable to data outliers is proposed in three closely related formulations, and relationships between those formulations are established. The SVM is based on the value-at-risk (VaR) measure, which discards a specified percentage of data viewed as outliers (extreme samples), and is referred to as \(\mathrm{VaR}\)-SVM. Computational experiments show that compared to the \(\nu \)-SVM, the VaR-SVM has a superior out-of-sample performance on datasets with outliers.

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Notes

  1. VaR with a confidence level \(\alpha \) is the \(\alpha \)-percentile of the distribution of loss and is widely used to monitor and control the market risk of financial instruments (Jorion 1997; Duffie and Pan 1997).

  2. A statistically robust SVM should not be confused with an SVM relying on methods of robust optimization (Ben-Tal et al. 2009): the former is unaffected if a certain percentage of the training data is changed, whereas the latter finds the separation hyperplane under the assumption that the training data are not clearly specified but rather are known to be from some set.

  3. The approach can be readily extended to the case of arbitrary probabilities of outcomes: \(\mathrm{Pr}(\omega _i)=p_i, i=1,\dots ,l\).

  4. The dataset was taken from UCI Machine Learning Repository (http://archive.ics.uci.edu/ml/datasets.html).

  5. http://www.aorda.com/aod/welcome.action/psg.action

  6. http://archive.ics.uci.edu/ml/datasets.html

  7. http://www.ise.ufl.edu/uryasev/research/testproblems/advanced-statistics/case-study-nu-support-vector-machine-based-on-tail-risk-measures/

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Acknowledgments

We are grateful to the referees for their comments and suggestions, which helped to improve the quality of the paper. This research was supported by AFOSR Grant FA9550-11-1-0258, New Developments in Uncertainty: Linking Risk Management, Reliability, Statistics and Stochastic Optimization.

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Authors and Affiliations

  1. Department of Industrial and Systems Engineering, University of Florida, 303 Weil Hall, PO Box 116595, Gainesville, FL, 32611-6595, USA

    Peter Tsyurmasto & Stan Uryasev

  2. Department of Mathematical Sciences, Stevens Institute of Technology, Castle Point on Hudson, Hoboken, NJ, 07030, USA

    Michael Zabarankin

Authors
  1. Peter Tsyurmasto
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  2. Michael Zabarankin
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  3. Stan Uryasev
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Correspondence to Peter Tsyurmasto.

Appendix 1: A PSG meta-code for VaR-SVM

Appendix 1: A PSG meta-code for VaR-SVM

The PSG meta-code, data, and solutions for the optimization problem (13) are available at the University of Florida Optimization Test Problems webpage,Footnote 7 see Problem 1b. For convenience, the PSG meta-code is presented below.

\(1\)       Problem: problem_var_svm, type = minimize

\(2\)       objective: objective_svm

\(3\)       quadratic_matrix_quadratic(matrix_quadratic)

\(4\)       var_risk_1(0.5,matrix_prior_scenarios)

\(5\)       box_of_variables: upperbounds =1,000, lowerbounds = \(-1{,}000\)

\(6\)       Solver: VAN, precision = 6, stages = 6

Command minimize instructs the solver that (13) is a minimization problem, whereas objective is a declaration of the objective function stated in lines 3 and 4: commands quadratic and var_risk_1 refer to the quadratic and VaR terms in (13), respectively, and matrix_quadratic and matrix_prior_scenarios are the names of text files (*.txt) that store corresponding data matrices. The coefficients \(C\) and \(\alpha \) are set to 0.5.

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Tsyurmasto, P., Zabarankin, M. & Uryasev, S. Value-at-risk support vector machine: stability to outliers. J Comb Optim 28, 218–232 (2014). https://doi.org/10.1007/s10878-013-9678-9

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