## 2 Experiments and Methods
### t-SNE Algorithm for Dimensionality Reduction  
The t-SNE algorithm is a nonlinear dimensionality reduction technique that is particularly effective for visualizing high-dimensional data by embedding them in a lower-dimensional space. This algorithm excels in preserving both local and global structures of the original data through probability distribution matching [26; 27]. In this study, all three dimensionality reduction methods (PCA, SNE, and t-SNE) were used to reduce the  
Figure 2: MIC values between temperature and indicator gases. high-dimensional gas concentration data to 2 dimensions to facilitate intuitive visualization and enable effective comparison between the clustering results. In dimensionality reduction context, local data structure refers to the preservation of proximity relationships between neighboring data points, ensuring that points that are close in the high-dimensional space remain close in the reduced space. Global data structure refers to the maintenance of large-scale patterns and relationships across the entire dataset, such as clusters, general shapes, and relative distances between distant points. Unlike PCA, which prioritizes global variance preservation, t-SNE effectively balances both local and global structural preservation through its probability-based similarity measurements, making it particularly suitable for capturing the complex evolutionary patterns in coal spontaneous combustion data where both short-range similarities and long-range relationships are critical for accurate classification. In the high-dimensional space, t-SNE first computes the conditional probability \(p_{ij}\) that represents the similarity between data points \(x_{i}\) and \(x_{j}\) using a Gaussian distribution:  
\[p_{ij}=\frac{\exp\left(-\left\|x_{i}-x_{j}\right\|^{2}/2\sigma_{i}^{2}\right) }{\sum_{k\
eq 1}\exp\left(-\left\|x_{i}-x_{k}\right\|^{2}/2\sigma_{i}^{2} \right)}, \tag{1}\]  
where \(\left|\right.x_{i}-x_{j}\left|\right.\) denotes the Euclidean distance between points \(x_{i}\) and \(x_{j}\), and \(\sigma_{i}\) is the standard deviation of the Gaussian distribution centered at \(x_{i}\). This standard deviation is determined through binary search to achieve a predefined perplexity value, which reflects the effective number of neighbors for each data point. To ensure symmetry, the joint probability distribution \(p_{\bar{y}}\) in the high-dimensional space is computed as follows:  
\[p_{ij}=\frac{p_{j|i}+p_{i|j}}{2n}, \tag{2}\]  
where \(n\) represents the total number of data points. The joint probability \(p_{\bar{y}}\) characterizes the pairwise similarities between data points, with larger values indicating a higher likelihood of points being neighbors. In the low-dimensional space, t-SNE employs a student t-distribution with one degree of freedom (Cauchy distribution) to compute the similarities \(q_{\bar{y}}\) between mapped points \(y_{i}\) and \(y_{j}\):  
\[q_{i|j}=\frac{\left(1+\left\|y_{i}-y_{j}\right\|^{2}\right)^{-1}}{\sum_{k\
eq 1 }\left(1+\left\|y_{k}-y_{l}\right\|^{2}\right)^{-1}}, \tag{3}\]  
The Student t-distribution was specifically chosen over a Gaussian distribution due to its heavier tails, which helps mitigate the \"crowding problem\"--the tendency for moderate-distance points to be collapsed together in the low-dimensional embedding. The objective function of t-SNE minimizes the Kullback-Leibler divergence between the high-dimensional and low-dimensional probability distributions:  
\[C=KL(P||Q)=\sum_{i}\sum_{j}p_{\bar{y}}log\left(p_{\bar{y}}/q_{\bar{y}}\right), \tag{4}\]  
The KL divergence quantifies the dissimilarity between the two probability distributions, where a smaller value indicates better preservation of the original data structure.