## 2 Literature Review  
Over the last two decades, many mathematical modeling methods have been utilized to establish thermal error models of machine tools, such as least square method, multiple regression, grey system, support vector machine, neural network and so on. In 2015, [PERSON] et al. [26] used the Chebyshev polynomial-based orthogonal least squares regression to fit the thermal drift error curve of the spindle. The obtained curve fit well with the measurement error curve. In 2018, [PERSON] et al. [27] used the weighted least square adaptive parameter updating method to adapt the model parameters and boundary condition changes, to process a random set of sample rates from any measurement of tool center point deviations. Thus, the optimum order of the model was determined and the thermal errors of the five-axis machine tool were predicted. During the 88 h experimental survey, the compensation reduced the occurrence of errors by more than 95%. In addition, with the use of the least square method in 2020, [PERSON] et al. [28] established the radial and axial thermal error model for the spindle of machine tool and carried out error compensation. As a result, the accuracy of workpiece contour was increased by 78.4% after compensation. In 2011, the multiple regression model was used by [PERSON] et al. [29] to construct an analytic model of the machine tool spindle. After it was applied for compensation, the thermal error of the spindle declined from 73 \(\upmu\)m to 13 \(\upmu\)m. In 2020, [PERSON] et al. [30] put forward a thermal error model of linear axis on the basis of regression analysis, which separated thermal error from geometric error. Based on the homogeneous coordinate transformation data, the thermal error compensation method of machine tool was proposed. In comparison with uncompensated conditions, the machining error of the machine tool declined by 85%. Furthermore, in the paper by [PERSON] et al. [31] published in 2022, the traditional multiple linear regression model was improved, and a modeling and compensation method based on the size error of machined parts was proposed. Accordingly, a generalized model of thermal error compensation for machine tool in X direction was established, which realizedreal-time thermal error compensation, and contributed to a decrease of nearly 52% in the machining error of machined parts. On the basis of the standard gray system model GM(1, 1), [PERSON] and his colleagues [32] used a genetic algorithm to optimize the dimensions and weights of the variables in 2010, so as to minimize the residuals of the optimized model. It was able to reflect the tendency of thermal errors systematically, as well as lessen the effect of the stochastic change in thermal error, which contributed to a large promotion of the predictive accuracy of the thermal error model. In order to make the solution of grey prediction model GM(1, N) more accurate, [PERSON] et al. [33] adopted gray control variables, which are more accurate due to their linear gray differential equation, and the corresponding solution based on the superposition principle. To have a higher predictive accuracy, a new gray GMC(1, N) model based on the cuckoo search algorithm, named as CS-GMC(1, N), was proposed for thermal error of machine tool by [PERSON] et al. [34] in 2020. Comparing with the gray model based on PSO, it had a higher predictive accuracy. Based on SVM, [PERSON] et al. [35] proposed the thermal error model for the spindle of a machining center in 2013, and used it for thermal error compensation, which indicated that the SVM model had good performance on accuracy and robustness. In order to improve the performance of the thermal error model based on SVM, the grid search method was coupled with the SVM model to optimize its penalty and kernel parameters in the article by [PERSON] et al. [36] published in 2017. By applying the improved SVM model to the CNC machine tool, the positioning errors of \(X\) axis and \(Z\) axis were decreased by 89.55% and 85.67% respectively. In 2022, [PERSON] et al.