## 5 Thermal Damage
### Sensitivity Analysis (Importance of the Input Parameters)  
To identify and rank the most influential input parameters affecting thermal damage (TD), a sensitivity analysis was conducted using the Relief algorithm implemented in MATLAB (R2024b). This step forms part of the broader machine learning workflow, specifically focusing on feature importance analysis--a critical phase used to interpret model behavior and prioritize variables for further investigation or simplification. The Relief method is a distance-based algorithm suitable for regression problems. It estimates  
Figure 5: The correlation between the actual and predicted TD of the best predictive models: (**a**) Ensemble Bagged Trees model; (**b**) Regression Tree (Fine tree) model; (**c**) Gaussian Process Regression model.  
the importance (or weight) of each predictor by repeatedly sampling instances and evaluating how well feature values distinguish between instances with similar or different target values. This approach captures not only individual feature effects but also potential interactions with other variables. In this study, the Relieff function was applied to the input dataset used in the machine learning models. The resulting feature weights indicated the relative importance of each predictor to TD. Figure 6 illustrates the ranked weights, showing the predictors in the following order of importance: P1, P10, P7, P9, P8, P6, P2, P5, P3, and P4 (see Figure 6). Among these, the three most important parameters were identified as (1) temperature, (2) tensile strength, and (3) porosity. These results suggest that mineralogical properties (such as P2 to P5), although expected to be among the top influencing parameters of thermal damage due to differential expansion and microcrack formation during heating, exhibited negative weights and are influenced by TD. Instead, TD mechanics cause changes in mineralogical properties. Interestingly, UCS showed a near-zero weight, implying a limited direct role in the observed TD variations. This may seem counterintuitive, as UCS often initiates cracking. However, it is possible that heating may initiate and close existing microcracks, thereby masking the effects of UCS alone. This outcome highlights the complex interactions among material properties under thermal stress and suggests that a broader set of mechanical and mineralogical features should be considered to fully understand TD behavior.  
Since mineralogical properties contribute in a reverse way in the TD process, the mineral content parameters were removed from Equation (2), and the equation was redefined as  
\[\mathrm{TD}_{\mathrm{emp2}}=-0.1115+0.0007\cdot\mathrm{P1}+0.018\cdot\mathrm{ P7}+0.0001\cdot(\mathrm{P9}/\mathrm{P10}) \tag{3}\]  
Similarly, Figure 7 shows the updated correlation between the actual (defined in Equation (1)) and predicted TD (defined in Equation (3)). The updated equation can still estimate the thermal damage very accurately, with an R\({}^{2}\) close to 0.94. Compared to the previous equation, a better coefficient of correlation, R = 0.97, was reached.  
Figure 6: The order of importance of the TD input parameters.  
Figure 7: The updated correlation between the actual and predicted thermal damage.  
Further removing the porosity term from Equation (3) leads to  
\[\text{TD}_{\text{emp3}}=-0.0991+0.0008\cdot\text{P1}-0.0001\cdot(\text{P9}/\text{ P10}) \tag{4}\]  
Now, Figure 8 shows the updated correlation between the actual (defined in Equation (1)) and predicted TD (defined in Equation (4)). The updated equation can still estimate the thermal damage very accurately, where R\({}^{2}\) is close to 0.93.