## 2 Experiments and Methods  
### Experimental System and Data Acquisition  
To develop a universally applicable prediction method for coal spontaneous combustion in goaf, 21 coal samples were collected from various mining regions, including Shaanxi, Guizhou, Anhui, and Shanxi provinces. The raw coal samples were sealed and transported to the laboratory, where they were crushed and screened into five particle size fractions: \(\leq\)0.9 mm, \(>\)0.9-\(\leq\)3 mm, \(>\)3-\(\leq\)5 mm, \(>\)5-\(\leq\)7 mm, and \(>\)7-\(\leq\)10 mm. A 250 g mixed sample was prepared by taking 50 g from each size fraction for the programmed temperature experiment to simulate the oxidation and temperature rise processes of coal spontaneous combustion. The experimental system with programmable oil bath heating consists of three main components: a gas circuit, a temperature control chamber, and a data acquisition system, as shown in Figure 1. An SPB-3 automatic air pump supplied the gas flow, while a temperature control device regulated the heating of the oil bath chamber. The heating rate was set at 0.3 \({}^{\circ}\)C min\({}^{-1}\) with an air flow rate of 30 mL min\({}^{-1}\). Gas samples were collected at 10 \({}^{\circ}\)C intervals when the coal temperature ranged from 30 \({}^{\circ}\)C to 180 \({}^{\circ}\)C. The collected gas samples were analyzed using gas chromatography to determine their composition and concentration. Through the experiments, a comprehensive database of 336 sets of data was established. The database includes temperature measurements and corresponding gas concentrations (O\({}_{2}\), N\({}_{2}\), CO\({}_{2}\), CO, CH\({}_{4}\), C\({}_{2}\)H\({}_{2}\), C\({}_{2}\)H\({}_{4}\), and C\({}_{2}\)H\({}_{6}\)). Partial data from the programmed temperature experiment is presented in Appendix A. Given the complex nature of coal spontaneous combustion and the multiple gas indicators involved, it is essential to understand the relationships between these variables for effective monitoring and early warning. Therefore, a systematic analysis of the correlations between gas concentrations and temperature variations was conducted. To quantitatively analyze the complex relationships between gas indicators and temperature variation during coal spontaneous combustion, the Maximal Information Coefficient (MIC) was employed due to its ability to capture both linear and nonlinear associations. As illustrated in Figure 2, the analysis revealed distinct patterns in the relationship between temperature and various gases. CO exhibited the strongest correlation with temperature (MIC = 0.95), which can be attributed to its continuous production during the coal oxidation process. The high MIC value suggests that CO serves as a primary indicator for monitoring temperature evolution. O\({}_{2}\) showed the second highest correlation (MIC = 0.81), reflecting its crucial role in the oxidation reaction and its consistent consumption pattern as temperature increases. Figure 1: Schematic diagram of programmed oil bath experimental system. The varying MIC values among different gases reflect the dynamic nature of coal oxidation processes and provide insights into the evolution of gas compositions at different stages of spontaneous combustion. A detailed examination of these relationships reveals distinctive patterns in gas generation and consumption. Hydrocarbon gases demonstrated varying degrees of correlation: C\({}_{2}\)H\({}_{4}\) (MIC = 0.74) showed moderate correlation, primarily emerging during higher temperature stages, while C\({}_{2}\)H\({}_{6}\) (MIC = 0.42) exhibited weaker correlation, typically appearing in the later stages of spontaneous combustion. CO\({}_{2}\) (MIC = 0.70) displayed intermediate correlation, indicating its value as a supplementary indicator. The relatively low MIC value for N\({}_{2}\) (MIC = 0.47) reflects its role as an environmental background gas rather than a direct product of coal oxidation. This hierarchical correlation structure presents both opportunities and challenges for data analysis.