Abstract
Support vector machines have been widely applied in binary classification, which are constructed based on crisp data. However, the data obtained in practice are sometimes imprecise, in which classical support vector machines fail in these situations. In order to handle such cases, this paper employs uncertain variables to describe imprecise observations and further proposes a hard margin uncertain support vector machine for the problem with imprecise observations. Specifically, we first define the distance from an uncertain vector to a hyperplane and give the concept of a linearly α-separable data set. Then, based on maximum margin criterion, we propose an uncertain support vector machine for the linearly α-separable data set, and derive the corresponding crisp equivalent forms. New observations can be classified through the optimal hyperplane derived from the model. Finally, a numerical example is given to illustrate the uncertain support vector machine.
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References
Boser, B., Guyon, I., & Vapnik, V. (1992). A training algorithm for optimal margin classifiers. Fifth annual workshop on computational learning theory (pp. 144–152). Pittsburgh: ACM.
Burges, C. (1998). A tutorial on support vector machines for pattern recognition. Data Mining and Knowledge Discovery, 2(2), 121–167.
Cortes, C., & Vapnik, V. (1995). Support-vector networks. Machine Learning, 20(3), 273–297.
Fang, L., & Hong, Y. (2020). Uncertain revised regression analysis with responses of logarithmic, square root and reciprocal transformations. Soft Computing, 24, 2655–2670.
Hu, Z., & Gao, J. (2020). Uncertain Gompertz regression model with imprecise observations. Soft Computing, 24, 2543–2549.
Lio, W., & Liu, B. (2018). Residual and confidence interval for uncertain regression model with imprecise observations. Journal of Intelligent and Fuzzy Systems, 35(2), 2573–2583.
Liu, B. (2007). Uncertainty theory (2nd ed.). Berlin: Springer.
Liu, B. (2009). Some research problems in uncertainty theory. Journal of Uncertain Systems, 3, 3–10.
Liu, B. (2010). Uncertainty theory: A branch of mathematics for modeling human uncertainty. Berlin: Springer.
Liu, B. (2012). Why is there a need for uncertainty theory. Journal of Uncertain Systems, 6, 3–10.
Liu, B. (2015). Uncertainty theory (4th ed.). Berlin: Springer.
Liu, Y., & Liu, B. (2022). Residual analysis and parameter estimation of uncertain differential equations. Fuzzy Optimization and Decision Making, 21(4), 513–530.
Liu, Z., & Yang, Y. (2020). Least absolute deviations uncertain regression with imprecise observations. Fuzzy Optimization and Decision Making, 19, 33–52.
Provost, F., & Fawcett, T. (2001). Robust classification for imprecise environments. Machine Learning, 42(3), 203–231.
Qin, Z. (2015). Mean-variance model for portfolio optimization problem in the simultaneous presence of random and uncertain returns. European Journal of Operational Research, 245(2), 480–488.
Song, Y., & Fu, Z. (2018). Uncertain multivariable regression model. Soft Computing, 22, 5861–5866.
Vapnik, V. (1995). The nature of statistical learning theory. New York: Springer.
Vapnik, V. (1999). An overview of statistical learning theory. IEEE Transactions on Neural Networks, 10(5), 988–999.
Yang, X., & Liu, B. (2019). Uncertain time series analysis with imprecise observations. Fuzzy Optimization and Decision Making, 18(3), 263–278.
Yao, K. (2015). A formula to calculate the variance of uncertain variable. Soft Computing, 19(10), 2947–2953.
Yao, K. (2018). Uncertain statistical inference models with imprecise observations. IEEE Transactions on Fuzzy Systems, 26(2), 409–415.
Yao, K., & Liu, B. (2018). Uncertain regression analysis: An approach for imprecise observations. Soft Computing, 22(17), 5579–5582.
Ye, T., & Liu, B. (2022). Uncertain hypothesis test for uncertain differential equations. Fuzzy Optimization and Decision Making. https://doi.org/10.1007/s10700-022-09389-w.
Zhao, M., Liu, Y., Ralescu, D., & Zhou, J. (2018). The covariance of uncertain variables: Definition and calculation formulae. Fuzzy Optimization and Decision Making, 17, 211–232.
Acknowledgements
This work was supported in part by National Natural Science Foundations of China (Nos. 72071008 and 71771011).
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Qin, Z., Li, Q. An uncertain support vector machine with imprecise observations. Fuzzy Optim Decis Making 22, 611–629 (2023). https://doi.org/10.1007/s10700-022-09404-0
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DOI: https://doi.org/10.1007/s10700-022-09404-0