An adaptive support vector regression based on a new sequence of unified orthogonal polynomials

In practical engineering, small-scale data sets are usually sparse and contaminated by noise. In this paper, we propose a new sequence of orthogonal polynomials varying with their coefficient, unified Chebyshev polynomials (UCP), which has two important properties, namely, orthogonality and adaptivity. Based on these new polynomials, a new kernel function, the unified Chebyshev kernel (UCK), is constructed, which has been proven to be a valid SVM kernel. To find the optimal polynomial coefficient and the optimal kernel, we propose an adaptive algorithm based on the evaluation criterion for adaptive ability of UCK. To evaluate the performance of the new method, we applied it to learning some benchmark data sets for regression, and compared it with other three algorithms. The experiment results show that the proposed adaptive algorithm has excellent generalization performance and prediction accuracy, and does not cost more time compared with other SVMs. Therefore, this method is suitable for practical engineering application.

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► We propose a new sequence of unified Chebyshev polynomials.
► The unified Chebyshev kernel (UCK) is constructed.
► The evaluation criterion for the adaptive ability of UCK is deduced based on Riemannian metric.
► The adaptive SVM based on UCK (ASVM) is proposed for finding the optimal UCK.