kernels like the polynomial, Gaussian (or radial basis function, rbf) and sigmoid functions are popular due to their simplicity. These have the forms $$k_{poly}(\mathbf{X}_i,\mathbf{X}_j) = (\mathbf{X}_i \dot \mathbf{X}_j)^m$$ where $m$ is the degree of the polynomial, $$k_{rbf}(\mathbf{X}_i,\mathbf{X}_j) = \exp(-\gamma ||\mathbf{X}_i - 
\mathbf{X}_j||^2)$$ and $$k_{sigmoid}(\mathbf{X}_i,\mathbf{X}_j) = \tanh(\mathbf{X}_i \cdot 
\mathbf{X}_j + r),$$ where $\gamma$ and $r$ are parameters of the kernel.

The simple binary SVM described above can be generalized to support vector regression (SVR) [@Dea97] which is the algorithm we employ in this study. For the regression problem, we seek hyperplanes satisfying the equations $$\begin{aligned}
Y_i - (\mathbf{W} \cdot \mathbf{X}_i - b) &\leqslant& \epsilon + \xi_i \\
(\mathbf{W} \cdot \mathbf{X}_i - b) - Y_i &\leqslant& \epsilon + \xi_i^*\end{aligned}$$ Here, $\epsilon$ is a tolerance parameter, i.e. there is no penalty assigned to predictions that fall within $\epsilon$ of the true value. $\xi_i>0 $ and $\xi_i^*>0$ are slack variables corresponding to upper and lower constraints on the system output.

The quantity the SVR must minimize is $$\frac{1}{2}||\mathbf{W}||^2 + C\sum_{i=1}^{N}(\xi_i+\xi_i^*).
\label{eqn:svr}$$ which closely resembles Equation (\[eqn:nlsvm\]). The Lagrangian takes the form $$\begin{aligned}
L(\alpha_i,\alpha_i^*) = && 
\frac{1}{2}||\mathbf{W}||^2 + C\sum_{i=1}^{N}(\xi_i + \xi_i^*) \\ \nonumber
&& - \sum_{i=1}^{N} \alpha_i [\epsilon + \xi_i-y_i + (\mathbf{W} \cdot 
\mathbf{X}_i + 