Essay: Polynomial and RBF functions

Sample Essay

Generally, there are four SVM kernel functions that are commonly used: linear, polynomial, radial basis function (RBF), and sigmoid.  Previous studies showed that the polynomial and RBF functions can achieve better results than linear and sigmoid kernels for land cover classification (Huang et al. 2002; Shafri and Ramie 2009).

To select the optimal kernel for our dataset, we have compared the performance of the polynomial and RBF kernel functions with their optimal parameters obtained from a 10-fold cross-validation test, on a small sample size.  The comparison indicates the performance of RBF is better than the polynomial kernel.  Therefore, RBF has been chosen for SVM classification in this research. RBF has two advantages: 1) the RBF kernel can classify samples that do not have linear relationships in high dimensional space, and 2) RBF has less numerical computation difficulties (Su et al. 2007). The RBF kernel is shown below.

where, xi and xj are two samples, γ is a Gaussian parameter.  Penalty parameter C is another parameter, which determines the width of the margin.  Generally, a large C value generates a small margin. In this study, SVM classification was carried out in an open source package LIBSVM (Chang and Lin 2001), and the parameters C=100 and γ =1 were determined from a 10-fold cross-validation test.

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