## 4 Verification of the Method
Since the rotation speed was not within the range of the rotation speed that was used to build the thermal error model, the universality of the selected temperature sensors can be verified. Comparisons of the accuracy of these models are shown in Figure 9, and the RMSEs and maximum deviations are shown in Table 4. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **New Method** & \begin{tabular}{c} **Comparison** \\ **Group 1** \\ \end{tabular} & \begin{tabular}{c} **Comparison** \\ **Group 2** \\ \end{tabular} & \
\begin{tabular}{c} **Comparison** \\ **Group 3** \\ \end{tabular} &
\begin{tabular}{c} **Comparison** \\ **Group 4** \\ \end{tabular} \\ \hline RMSE & 0.0118 mm & 0.0192 mm & 0.0261 mm & 0.0164 mm & 0.0528 mm \\ Maximum deviation & 0.0227 mm & 0.0318 mm & 0.0498 mm & 0.0329 mm & 0.0785 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparisons of the root mean square errors (RMSEs) and the maximum deviations. Figure 9: Comparisons of the accuracy of the models. According to Figure 9 and Table 4, the conclusions can be drawn that the method in this paper can derive the best model accuracy, and the predicted thermal error can depict the general trend of the thermal error, as well as the small variation of the thermal error caused by the cooling water. For the maximum deviation between the predicted thermal error and the measured thermal error, the method in this paper still has the lowest value among all the compared methods. As for the first comparison group, only the Euclidian distance of the temperature sensors were considered, which would cause there to be one dominated temperature sensor among the selected sensors because its variation amplitude is much larger than the others. Thus, some information would be lost and the accuracy of the prediction is poor. The small variation caused by cooling water cannot be predicted either. As for the second comparison group, only the correlation coefficient of the sensors were considered. Thus, some temperature sensors that only have very small amplitudes would be clustered with temperature sensors that have large amplitudes, and probably be selected. Thus, although they can predict the variation caused by the cooling water, they failed to derive an accurate amplitude of the thermal error. As for the third comparison group, only one temperature sensor was included. This is similar to the first comparison group. Thus, the accuracy of the prediction would be lower, and the variation caused by cooling water cannot be predicted either. As for the fourth comparison group, temperature sensors near the heat sources and temperature sensors of the environment were selected. It seems that this method contains the main information of the heat sources. However, most of the heat sources have very similar temperature variation when the machine tool is working, so there is very high collinearity when establishing the thermal error model, resulting in instability. The model might have high accuracy for the same rotation speed as the measurement. However, when predicting the thermal error at another rotation speed, the accuracy is much lower. In order to further verify the effectiveness of the temperature selection method in a more real working condition, the machine tool operated according to a schedule that simulated the working process of the machine tool, as shown in Table 5, in which the spindle of the machine tool rotated at different rotation speeds and also paused for a period of time during working. Then the thermal error prediction accuracy of the method in this paper was also compared to other comparison groups. The results are shown in Figure 10 and the RMSEs and maximum deviations are shown in Table 6.  
According to the result, although the rotation speed in the verification is far less than the speed when establishing the model (1200 rpm and 2000 rpm), the method in this paper can still derive a predicted thermal error that has better accuracy.