Table 1.

Commonly used kernel functions for vector machine algorithms

Kernel function name	Expression
Linear kernel	$\kappa \left({\mathit{x}}_i,{\mathit{x}}_j\right)=\left({\mathit{x}}_i^{\mathrm{T}}{\mathit{x}}_j\right)$
Polynomial kernel	$\kappa \left({\mathit{x}}_i,{\mathit{x}}_j\right)={\left({\mathit{x}}_i^{\mathrm{T}}{\mathit{x}}_j\right)}^n$
Laplacian kernel	$\kappa \left({\mathit{x}}_i,{\mathit{x}}_j\right)=\exp \left(-\Vert {\mathit{x}}_i-{\mathit{x}}_j\Vert /\sigma \right)$
Radial basis kernel (Gaussian kernel)	$\kappa \left({\mathit{x}}_i,{\mathit{x}}_j\right)=\exp \left(-{\Vert {\mathit{x}}_i-{\mathit{x}}_j\Vert }^2/2{\sigma }^2\right)$
Sigmoid kernel	$\kappa \left({\mathit{x}}_i,{\mathit{x}}_j\right)=\tanh \left[a\left({\mathit{x}}_i^{\mathrm{T}}{\mathit{x}}_j\right)-b\right],a,b>\mathrm{0}$

Note: x_i and x_j denote two sample vectors, n, a, b, σ denote the parameters in the kernel function.