## 5 Rotor Temperature Modeling and Prediction
### Spearman Correlation Analysis of Rotor Temperature Dependencies in PMSMs  
In motor systems, the relationship between rotor temperature and electrical/mechanical parameters often exhibits nonlinear characteristics. Compared to [PERSON]'s linear assumption, Spearman's nonparametric nature, which relies on the ranks of variables, can effectively analyze monotonic nonlinear associations. Moreover, Spearman is less sensitive to outliers and the distribution shape of data, while motor sensor data frequently contain noise or pulse interference. Therefore, [PERSON]'s correlation coefficient is more appropriate than Pearson or other correlation coefficients. First, the analysis was conducted using Group A, and the results are shown in Figure 8.  
_stator_yoke_ (0.9777), _stator_tooth_ (0.9692), and _stator_winding_ (0.9436) exhibit an extremely strong positive correlation with rotor temperature (\(pm\)). This phenomenon stems from the internal thermal conduction paths within the motor. The current in the stator winding generates Joule heat, which is conducted through the tooth to the yoke. Ultimately, heat transfers to the permanent magnets via heat conduction across the air gap and limited \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & **Stage** & **Error** & **Average** & **Median** &
\begin{tabular}{c} **Standard** \\ **Deviation** \\ \end{tabular} \\ \hline \multirow{8}{*}{XGBoost-Alpha} & & RMSE & 0.4443 & 0.4456 & 0.0251 \\  & & MSE & 0.1980 & 0.1986 & 0.0223 \\  & Train & MAE & 0.3575 & 0.3636 & 0.0229 \\  & & MSE & 0.2488 & 0.2526 & 0.0185 \\  & & R-squared & 0.9748 & 0.9753 & 0.0032 \\  & & RMSE & 0.8240 & 0.7992 & 0.1017 \\  & & MSE & 0.6888 & 0.6387 & 0.1770 \\  & Test & MAE & 0.6192 & 0.6087 & 0.0553 \\  & & MSE & 0.2720 & 0.2871 & 0.0990 \\  & & R-squared & 0.8977 & 0.9047 & 0.0177 \\  & Total & Runtime & 0.3359 & 0.3276 & 0.0292 \\ \hline \multirow{8}{*}{XGBoost-Beta} & & RMSE & 1.0984 & 1.1127 & 0.0642 \\  & & MSE & 1.2104 & 1.2381 & 0.1367 \\  & Train & MAE & 0.9745 & 0.9871 & 0.0634 \\  & & MSE & 0.7655 & 0.7684 & 0.0599 \\  & & R-squared & 0.8373 & 0.8377 & 0.0184 \\  & & RMSE & 1.2480 & 1.2134 & 0.1456 \\  & & MSE & 1.5777 & 1.4723 & 0.3938 \\  & Test & MAE & 1.0448 & 1.0288 & 0.0958 \\  & & MSE & 0.7317 & 0.7591 & 0.1560 \\  & & R-squared & 0.7830 & 0.7918 & 0.0429 \\  & Total & Runtime & 0.4117 & 0.2716 & 0.6223 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: _Cont._convection. These three parameters are highly suitable as indirect indicators for monitoring rotor temperature. \(u\_d\) (\(-0.7541\)) and \(u\_q\) (\(-0.7147\)) show significant negative correlations. In the vector control of a PMSM, the d-axis voltage is used for field-weakening control to extend the motor's high-speed operating range and may indirectly affect iron losses. The q-axis voltage regulates the torque current. These negative correlations might reflect a mechanism where the controller actively increases voltage under high-temperature conditions to compensate for reduced counter electromotive force. It suppresses heat accumulation. _coolant_ and \(i\_q\) are moderately correlated variables. When coolant temperature rises, reduced heat dissipation capability accelerates heat accumulation, consistent with thermal equilibrium principles. However, their correlation coefficients are lower than those of stator components. This suggests a potential thermal response lag in the cooling system. The q-axis current is proportional to motor torque. Its relatively lower correlation may arise from current-limiting control strategies implemented by the controller during high-temperature operation. _motor_speed_ shows no significant correlation with rotor temperature, which contradicts intuitive expectations. This could be due to the compensation effect of closed-loop control strategies or the absence of high-speed, heavy-load operational data in the experimental dataset.