## 4 Verification of the Method  
The effectiveness of the above method was verified by an experiment. The selected temperature sensors were used to establish the model to predict the thermal error of the machine tool and compared to other temperature sensor selection methods. In order to verify whether the selected temperature sensors can be used at different spindle rotation speeds, firstly the machine tool was operated at 1200 rpm and 2000 rpm, respectively, to generate the data for establishing the thermal error model, and then verify its prediction accuracy at other rotation speeds. According to Table 3, temperature sensor No. 10 (oil of the gear box), No. 14 (hydrostatic oil of the spindle), and No. 21 (near the spindle motor) were selected after optimization. Multiple regression analysis (MRA) [32; 33; 34] was used to establish the model. Since the thermal error along the \(Z\)-axis contributes most of the thermal error, the thermal error along \(Z\)-axis was considered in the model. Through MRA, the thermal error model was derived as follows:  
\[\mathbf{e}_{z}=0.0310\mathbf{x}_{10}+0.0152\mathbf{x}_{14}+0.0101\mathbf{x}_{21} \tag{16}\]  
in which, the value of temperature rise was used, so the constant term of the model is zero. Then, four comparison groups with different temperature sensor selection methods were established, in which the models were all established by MRA to make sure the differences between groups are mainly caused by the temperature selection method. As for the first comparison group, temperature sensors were clustered and selected only considering the Euclidean distance between the temperature sensors. Three temperature sensors (No. 4 (upper surface on the rear of the ram), No. 18 (cooling water of the spindle motor), and No. 21 (near the spindle motor)) were selected. Then the thermal error model was established:  
Figure 8: Clusters of the sensors after clustering: (**a**) Cluster 1; (**b**) Cluster 2; (**c**) Cluster 3. \begin{table}
\begin{tabular}{c c c} \hline
**Temperature Sensor 1** & **Temperature Sensor 2** & **Temperature Sensor 3** \\ \hline No. 14 (hydrostatic oil of the spindle) & No. 10 (oil of gear box) & No. 21 (near the spindle motor) \\ \hline \end{tabular}
\end{table} 
Table 3: Selection of temperature sensors from each cluster. \[\mathbf{e}_{z}=0.0423\mathbf{x}_{4}+0.0069\mathbf{x}_{18}+0.0156\mathbf{x}_{21} \tag{17}\]  
As for the second comparison group, temperature sensors were clustered and selected only according to the correlation coefficient between temperature sensors. Three temperature sensors (No. 10 (oil of the gear box), No. 16 (inlet of the cooling box), and No. 21 (near the spindle motor)) were selected to establish the thermal error model:  
\[\mathbf{e}_{z}=0.0638\mathbf{x}_{10}-0.0033\mathbf{x}_{16}+0.0161\mathbf{x}_{21} \tag{18}\]  
In the third comparison group, only one temperature sensor was selected. Considering temperature sensor No. 21 (near the spindle motor) has the minimum model prediction error among all sensors, temperature sensor No. 21 was selected to establish the proportional model of thermal error:  
\[\mathbf{e}_{z}=0.0198\mathbf{x}_{21} \tag{19}\]  
In the fourth comparison group, temperature sensors were selected according to the \"sensors-near-heat source\" method, in which the temperature sensors that were placed near the heat source of the machine tool, as well as the sensors reflecting the temperature of the environment, were selected. Thus, four sensors (No. 21 (near the spindle motor), No. 15 (surface of the worktable), No. 22 (near the hydrostatic oil in the ram), and No. 23 (environment temperature near the pillar)) were selected to establish the thermal error model:  
\[\mathbf{e}_{z}=0.0196\mathbf{x}_{21}-0.1144\mathbf{x}_{15}-0.0015\mathbf{x}_{22}+0.1360\mathbf{x}_ {23} \tag{20}\]  
After the establishment of the models, the machine tool was operated at 1000 rpm to verify the accuracy of the model.