## 2 Experiments and Methods
### Support Vector Classification Model  
Support Vector Classification (SVC) is a sophisticated machine learning algorithm founded on the structural risk minimization principle. It constructs an optimal hyperplane in a high-dimensional feature space to separate different classes while maximizing the margin between them. This approach effectively balances model complexity with classification accuracy [30; 31; 32].  
The fundamental concept of SVC is expressed through its decision function, which combines a weight vector, bias term, and feature mapping:  
\[f(x)=w\phi(x)+b, \tag{15}\]  
This function represents the classifier's decision boundary, where \(\mathrm{w}\) is the weight vector determining the orientation of the separating hyperplane, \(\phi(x)\) is a nonlinear mapping function that transforms input data into a higher dimensional feature space, and \(\mathrm{b}\) is the bias term adjusting the hyperplanes position.  
To find the optimal hyperplane, SVC formulates a primary optimization problem that minimizes both structural risk and classification errors:  
\[\min\left(\frac{1}{2}\right)\|w\|^{2}+C\sum(\xi_{i}+\xi_{i}^{*}), \tag{16}\]This optimization is subject to essential constraints that enforce proper classification while allowing for some margin violations:  
\[\begin{array}{l}y_{i}(w\phi(x_{i})+b)<\varepsilon+\xi_{i}\\ -(w\phi(x_{i})+b)-y_{i}\leq\varepsilon+{\xi_{i}}^{*}\\ \xi_{i},{\xi_{i}}^{*}\geq 0,1=1,2,\cdots,m\end{array} \tag{17}\]  
Through the application of Lagrangian multipliers and optimization theory, the problem transforms into its dual form, which is computationally more tractable. The kernel function \(K(x_{i},x_{j})\) plays a crucial role in enabling nonlinear classification by implicitly mapping data to a higher dimensional space. The linear kernel, being the simplest form, directly computes the dot product in input space:  
\[K\big{(}x_{i},x_{j}\big{)}=x_{i}\cdot x_{j}, \tag{18}\]  
The Radial Basis Function (RBF) kernel introduces nonlinearity by measuring the Gaussian distance between points:  
\[K\big{(}x_{i},x_{j}\big{)}=\exp\Big{(}-\gamma\big{\|}x_{i}-x_{j}\big{\|}^{2} \Big{)}, \tag{19}\]  
For problems requiring polynomial decision boundaries, the polynomial kernel offers an alternative approach:  
\[K\big{(}x_{i},x_{j}\big{)}=(\gamma x_{i}+r)^{d}, \tag{20}\]  
The final classification decision is made using the sign of the decision function, which incorporates the learned support vectors and their corresponding coefficients:  
\[f(x)=\text{sign}\big{(}\sum\big{(}\alpha_{i}-{\alpha_{i}}^{*}\big{)}K(x_{i},x )+b\big{)}, \tag{21}\]  
This model demonstrates remarkable capability in handling complex classification tasks through its kernel-based approach and margin maximization strategy. The solution's sparseness, achieved through support vector identification, ensures computational efficiency during prediction while maintaining robust generalization performance. The hyperparameter \(C\) and kernel-specific parameters provide flexible control over the model's behavior, allowing practitioners to adapt the classifier to specific problem requirements and data characteristics.  
For the SVC implementation in this study, parameter optimization was conducted to ensure optimal classification performance. The radial basis function (RBF) kernel was selected after comparative testing with linear and polynomial kernels, as it demonstrated superior ability in handling the nonlinear relationships in coal spontaneous combustion data. The key parameters were determined through grid search with 5-fold cross-validation: the penalty parameter C was set to 10.0 to balance model complexity and classification accuracy, while the kernel coefficient \(\gamma\) was optimized to 0.01 to control the influence radius of support vectors.