## 5 Conclusions  
Aiming at the thermal error problem of motorized spindles, this study puts forward a thermal error prediction modeling method, and the main work is as follows:  
1. Firstly, the H-K-P algorithm is employed to cluster the temperature-measuring locations near the spindle. The hierarchical clustering algorithm is applied to determine the clustering number \(K\), and then all the pre-selected temperature sensor locations are clustered into \(K\) classes by using the K-medoids algorithm. After the clustering is completed, the Pearson correlation coefficient is utilized to evaluate the association degree between temperature data and thermal error in each category, and the point with the largest association degree is determined as the critical temperature point of this group. In the end, 12 temperature measuring points are successfully reduced to four crucial temperature points, thereby resolving the issue of collinearity among temperature measurement locations.
2. Using temperature-sensitive point data in conjunction with thermal error data, a predictive model for thermal errors in CNC machine tool motorized spindles is trained and developed. Aiming at the special property of thermal error having both time-varying characteristics and spatial characteristics, the POA-CNN-LSTMNN model is suggested, which incorporates LSTMNN for capturing temporal characteristics, utilizes CNN for extracting spatial features, and employs POA to optimize the parameters of the CNN-LSTM hybrid neural network, thus improving its predictive performance.
3. To verify the model s generalization and validity, the experimental analysis is carried out under different rotating speeds of the motorized spindle. The CSOA-CNN-LSTMNN, CNN-LSTMNN, and LSTMNN models are used to compare with the model proposed in this study. The experimental findings reveal significant improvements achieved by the POA-CNN-LSTMNN model in comparison to the other three models. On average, the POA-CNN-LSTMNN model exhibited a reduction of 51% and 49.9% in the _RMSE_ and _MAE_, respectively. Additionally, the \(R^{2}\) was 2.6% higher on average compared to the other models, and _MSE_ was reduced by 75.7% on average. These results strongly support the superiority and effectiveness of the POA-CNN-LSTMNN model in thermal error prediction for motorized spindles. The model demonstrates promising applicability and potential in the field of thermal error forecasting for motorized spindles in CNC machine tools. During the experimental process, we encountered various challenges. For instance, due to environmental conditions and equipment limitations, there were constraints in data acquisition. Extensive data collection efforts were required over a long period of time to ensure data quality and model accuracy. In future research, we will need to further explore efficient methods for data acquisition. Additionally, we aim to optimize the model to enhance its prediction accuracy and further expand the performance and application domains of thermal error prediction models.  
Y.G. ([PERSON]): writing--original draft preparation, conceptualization, validation, formal analysis; X.X.: methodology, project administration, supervision; Y.G. ([PERSON]): investigation, writing--review and editing. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the Liaoning Province Intelligent Assembly and the Inspection Technology Engineering Research Center (P6-0343).  
Not applicable.  
The data presented in this study are available on request from the corresponding author. They are restricted to the experimental results.  
The authors declare no conflicts of interest.