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Practical selection of SVM parameters and noise estimation for SVM regression. (English) Zbl 1075.68632
Summary: We investigate practical selection of hyper-parameters for support vector machines (SVM) regression (that is, $\epsilon$-insensitive zone and regularization parameter $C$). The proposed methodology advocates analytic parameter selection directly from the training data, rather than re-sampling approaches commonly used in SVM applications. In particular, we describe a new analytical prescription for setting the value of insensitive zone $\epsilon$, as a function of training sample size. Good generalization performance of the proposed parameter selection is demonstrated empirically using several low- and high-dimensional regression problems. Further, we point out the importance of Vapnik’s $\epsilon$-insensitive loss for regression problems with finite samples. To this end, we compare generalization performance of SVM regression (using proposed selection of $\epsilon$-values) with regression using `least-modulus’ loss ($\epsilon=0$) and standard squared loss. These comparisons indicate superior generalization performance of SVM regression under sparse sample settings, for various types of additive noise.

68T05Learning and adaptive systems
Full Text: DOI
[1] Chapelle, O.; Vapnik, V.: Model selection for support vector machines. Advances in neural information processing systems 12 (1999)
[2] Cherkassky, V.; Ma, Y.: Comparison of model selection for regression. Neural computation 15, 1691-1714 (2003) · Zbl 1046.62001
[3] Cherkassky, V.; Mulier, F.: Learning from data: concepts, theory, and methods. (1998) · Zbl 0960.62002
[4] Cherkassky, V.; Shao, X.; Mulier, F.; Vapnik, V.: Model complexity control for regression using VC generalization bounds. IEEE transaction on neural networks 10, No. 5, 1075-1089 (1999)
[5] Drucker, H.; Burges, C.; Kaufman, L.; Smola, A.; Vapnik, V.: Support vector regression machines. Neural information processing systems 9, 155-161 (1997)
[6] Hastie, T.; Tibshirani, R.; Friedman, J.: The elements of statistical learning: data mining, inference and prediction. (2001) · Zbl 0973.62007
[7] Huber, P.: Robust estimation of a location parameter. Annals of mathematical statistics 35, 73-101 (1964) · Zbl 0136.39805
[8] Kwok, J. T.: Linear dependency between ${\epsilon}$ and the input noise in ${\epsilon}$-support vector regression. Lncs 2130, 405-410 (2001) · Zbl 1001.68888
[9] Mattera, D.; Haykin, S.: Support vector machines for dynamic reconstruction of a chaotic system. Advances in kernel methods: support vector machine (1999)
[10] Muller, K.; Smola, A.; Ratsch, G.; Schölkopf, B.; Kohlmorgen, J.; Vapnik, V.: Using support vector machines for time series prediction. Advances in kernel methods: support vector machine (1999)
[11] Schölkopf, B.; Bartlett, P.; Smola, A.; Williamson, R.: Support vector regression with automatic accuracy control. Perspectives in neural computing, 111-116 (1998)
[12] Schölkopf, B.; Burges, J.; Smola, A.: Advances in kernel methods: support vector machine. (1999) · Zbl 0935.68084
[13] Schölkopf, B.; Smola, A.: Learning with kernels: support vector machines, regularization, and beyond. (2002)
[14] Smola, A.; Murata, N.; Schölkopf, B.; Muller, K.: Asymptotically optimal choice of ${\epsilon}$-loss for support vector machines. Proceedings of ICANN 1998 (1998)
[15] Smola, A., & Schölkopf, B (1998). A tutorial on support vector regression. NeuroCOLT Technical Report NC-TR-98-030, Royal Holloway College, University of London, UK. · Zbl 0910.68189
[16] Vapnik, V.: Statistical learning theory. (1998) · Zbl 0935.62007
[17] Vapnik, V.: The nature of statistical learning theory. (1999) · Zbl 0928.68093
[18] Vapnik, V (2001). Personal communication.