## 2 Materials and Methods
### Support Vector Machines  
The SVMs are widely applied in the research related to classification, regression and novelty detection. They can be used to estimate the regression combined with different kernel functions. In the problems of classification, the SVMs act as the binary classification, defined as the largest linear classifier in the feature space [14]. Learning strategy of the SVMs is interval maximization, which can be converted into a solution of a convex quadratic programming problem. A kernel function is used to transform the training data from non-linear decision surface to the linear equation in a higher number of dimensions. Linear discriminant functions provide efficient 2-class classifiers if class features can be separated by a linear decision surface [20]. In the prediction process, frost events can be seen as a typical binary classification problem: frost occurs and frost does not occur. In case of a set of dataset with \(k\) samples are linearly inseparable, a maximum margin hyperplane can be calculated by the SVMs with the following equation. \[\min\Phi(w)=\frac{1}{2}w^{T}\cdot w+C\sum_{i=1}^{k}\xi_{i}, \tag{1}\]  
where, \(C\) is positive, a penalty factor for controlling the tradeoff between minimization error of the classification and maximization of the margin. \(w\) and \(\xi_{i}\) is the weight vector and slack variable. In this research, \(C\) is 1.0. \(w\) and \(\xi_{i}\) will be optimized in the training processing. Generally, if the training point is no longer satisfies the constraint of \(y_{i}\) (\((w\cdot x_{i})+b)\geq 1\) and \(\xi_{i}\geq 1\) will be introduced to make the decision of the constraint for:  
\[y_{i}(w\cdot x_{i}+b)\geq 1-\xi_{i},\ \xi_{i}\geq 0,\ i=1,\ldots\ldots,k, \tag{2}\]  
where, \(x_{i}\) and \(b\) is the input vector and the bias; \(w\cdot x_{i}+b\) is the distance between \(x_{i}\) and the hyperplane. The Lagrange multipliers is used to determine the optimal decision surface and finally the classification function is shown as,  
\[L=\frac{1}{2}w^{T}\cdot w+C\sum_{i=1}^{k}\xi_{i}-\sum_{i=1}^{n}\alpha_{i}[y_{ i}(w\cdot x_{i}+b)+\xi_{i}-1]-\sum_{i=1}^{k}\beta_{i}\xi_{i}, \tag{3}\]  
where, \(\alpha\) and \(\beta\) is Lagrange multipliers. Since \(\frac{\partial L}{\partial b}=0\), \(\frac{\partial L}{\partial w}=0\) and \(\frac{\partial L}{\partial\xi}=0\), the Equation (3) can be derived into the Equations (4) and (5):  
\[\sum_{i=1}^{k}y_{i}\alpha_{i}=0, \tag{4}\]  
\[w^{*}=\sum_{i=1}^{k}y_{i}\alpha_{i}x_{i}, \tag{5}\]where, \(w^{*}\) is the normal vector for the optimal hyperplane; \(C\geq\alpha_{i}\geq 0\), \(i=1\), \(2\), \(3\),..., \(k\). So, the classification function is shown as,  
\[f(x)=\sum_{i=1}^{k}y_{i}\alpha_{i}(x\cdot x_{i})+b. \tag{6}\]  
In case of a set of dataset with \(k\) samples are nonlinear separable, samples are mapped to the high-dimensional feature space and become a linear case through the nonlinear mapping function \(\Phi(x_{i})\). The \(w^{*}\) can be described as following:  
\[w^{*}=\sum_{i=1}^{k}y_{i}\alpha_{i}x_{i}\Phi(x_{i}), \tag{7}\]  
where, \(\Phi(x_{i})\) maps the eigenvectors to higher-dimensional eigenspaces. So, the classification function is shown as,  
\[f(x)=\sum_{i=1}^{n}\alpha_{i}^{*}y_{i}K(x_{i},x)+b^{*}, \tag{8}\]  
where, \(b^{*}\) and \(K(x_{i},x)\) is the bias and the kernel function, respectively. The kernel functions implicitly map the eigenvectors to the higher-dimensional eigenspaces by the nonlinear transformation to which it is associated, making the system learning samples linearly separable. The most common kernel functions are linear kernel function, polynomial kernel function and radial basis function kernel (BRF), which is defined as follows, respectively. \[K(x_{i},x)=(x_{i}\cdot x), \tag{9}\]  
\[K(x_{i},x)=(x_{i}\cdot x+1)^{d}, \tag{10}\]  
\[y_{i}(w\cdot x_{i}+b)\geq 1-\xi_{i},\ \xi_{i}\geq 0,\ i=1,\ldots\ldots,k, \tag{2}\]0  
where, \(d\) is the nature number. \(\sigma\) is a smoothing variate that determines the influence each of the points.