泛化的统一切比雪夫多项式核函数

Generalized Uniform Chebyshev Polynomial Kernel

针对分布稀疏、特征不明显的小样本数据回归中的属性冗余问题,基于统一切比雪夫多项式,提出了一种向量形式输入的可变正交多项式核函数——泛化的统一切比雪夫多项式核函数.新的核函数通过利用统一切比雪夫多项式的正交性和可变性扩大了函数的搜索空间,通过调整多项式阶数有效地控制了特征空间维数,从而解决了稀疏数据回归中的属性冗余问题.另外,利用Mer-cer定理证明了该核函数的有效性.在多组标准数据集和实际工程数据集上对核函数的性能进行了实验对比,结果证明新的核函数预测精度较高,泛化能力较好,在大多数标准数据集上的性能优于其他切比雪夫多项式核函数.

Based on a group of unified Chebyshev polynomials （UCP）, a new kernel for vector inputs, named generalized uniform Chebyshev polynomial kernel （GUCK）, is proposed to solve the problem of redundant attributes in the regression analysis on small-scale data sets. The proposal kernel can extend the search space of optimal kernel function by the orthogonality and adaptivity of UCP and control the dimension of the feature space by adjusting the polynomial coefficient of UCP. The problem of redundant attributes is settled by this method. Moreover, the proposal kernel, GUCK, has been proved that it is a valid support vector machine （SVM） kernel. The simulation results and application results show that GUCK can lead to better generalization performance in comparison with other common kernels, and is well applicable to the practical dataset. The GUCK has an advantage over other Chebyshev kernels on the majority of benchmark data sets