Table 5.

The Method for Selecting the Gaussian KDA Parameter.

Input: A reasonable candidate set $\mathrm{S}=\left\{{\mathrm{s}}_1,{\mathrm{s}}_2,\cdots ,{\mathrm{s}}_{\mathrm{m}}\right\}$ for Gaussian kernel parameter, $\mathrm{X}=\left\{{\mathrm{X}}_1,{\mathrm{X}}_2,\cdots ,{\mathrm{X}}_{\mathrm{C}}\right\}$, the training set ${\mathrm{X}}_{\mathrm{i}}=\left\{{\mathrm{x}}_1,{\mathrm{x}}_2,\cdots ,{\mathrm{x}}_{{\mathrm{N}}_{\mathrm{i}}}\right\}$ $\left(1\le \mathrm{i}\le \mathrm{C}\right)$, the number of retained eigenvectors $\mathrm{d}$.

1. Get the internal sample set ${\mathsf{\Omega }}_{\mathrm{i}\mathrm{n}}$ and the edge sample set ${\mathsf{\Omega }}_{\mathrm{e}\mathrm{d}}$ from the training set ${\mathrm{X}}_{\mathrm{i}}$ using Algorithm 1.

2. For each parameter ${\mathrm{s}}_{\mathrm{i}}\in \mathrm{S},\mathrm{i}=1,2,\cdots ,\mathrm{m}$ Calculate the kernel matrix $\mathrm{K}$ using Equation (7). Reduce dimension of the $\mathrm{K}$ using the Gaussian KDA algorithm. Calculate $\mathrm{R}\mathrm{E}\left({\mathsf{\Omega }}_{\mathrm{e}\mathrm{d}}\right)$ and $\mathrm{R}\mathrm{E}\left({\mathsf{\Omega }}_{\mathrm{i}\mathrm{n}}\right)$ using Equation (8). Calculate the value of objective function $\mathrm{f}\left({\mathrm{s}}_{\mathrm{i}}\right)$ using Equation (12).

3. Select the optimum parameter $\mathrm{s}=\mathrm{a}\mathrm{r}\mathrm{g}\underset{{\mathrm{s}}_{\mathrm{i}}\in \mathrm{S}}{\mathrm{m}\mathrm{a}\mathrm{x}}\mathrm{f}\left({\mathrm{s}}_{\mathrm{i}}\right)$

Output: the optimum Gaussian kernel parameter $\mathrm{S}$.