# A Review of Thermal Error Modeling Methods for Machine Tools
## 2 Thermal Error Modeling Method for Machine Tools
### Multivariable Regression Analysis (MRA)  
Multiple regression is another common method to establish a linear or non-linear mathematical model between one dependent variable and one or more independent variables through statistical analysis of the experimental sample data [22; 23]. Compared with the least square method, the multiple regression model has more independent variables and is more coordinate to the actual situation. Therefore, the thermal error model established by this method has higher accuracy and robustness.  
The basic model of linear multivariate regression thermal error [24] is shown in Equation (2),  
\[Y=\beta X+\varepsilon \tag{2}\]  
where \(Y\) represents the thermal error; \(X\) is the temperature; \(\beta\) is the coefficient; and \(\varepsilon\) is the residual. Multivariate regression modeling is used to find the right coefficient \(\beta\) to ensure that the residual is equal at each data point and the objective function value \(\sum\limits_{i=1}^{n}{(y_{i}-x_{i}^{T}\beta)}^{2}\) is minimized.  
In 2011, [PERSON] and [PERSON] [25] established an analytical model for the spindle based on the multiple regression model. When the model was applied for thermal error compensation of the spindle, the thermal error of reduced from 73 \(\upmu\)m to 13 \(\upmu\)m. By replacing the original objective function with the new objective function \(\sum\limits_{i=1}^{n}{w_{i}(y_{i}-x_{i}^{T}\beta)}^{2}\), an improved linear multiple regression model was proposed by [PERSON] et al. [26; 27] from 2011 to 2012. With this method, the thermal error model of the spindle was established and used to reduce the maximum residual error of the spindle in the \(Z\) and \(Y\) directions from 25 \(\upmu\)m and 7 \(\upmu\)m to 1.8 \(\upmu\)m and 0.7 \(\upmu\)m. In addition, in the paper of [PERSON] et al. [28] published in 2013, a position-independent thermal error model of multiple linear regression for the linear axis was constructed. After compensation based on the model, the maximum fitting residuals of the three errors of \(Y\) axis \(Sy\), \(Sz\), and \(R\alpha\) were reduced to 9.6 \(\upmu\)m, 2.4 \(\upmu\)m, and 7.1 \(\upmu\)rad, respectively. Similarly, [PERSON] et al. [29], [PERSON] et al. [30], and [PERSON] et al. [31] also established the linear axis thermal error models based on regression analysis. Among them, based on the homogeneous transformation, [PERSON] et al. [31] proposed a data-driven thermal error compensation method of high-speed and precision five-axis machine tools. The thermal error of the liner axis was expressed as the polynomial function of the time and the position.  
In this model, the thermal error and geometric error were separated. Compared with the uncompensated and traditional error compensation situations, the machining errors were reduced 85% and 37%, respectively. The fitting performance of the model is shown in Figure 2.  
Moreover, based on the linear regression analysis, [PERSON] et al. [32] presented a thermal deformation compensation model for lathe machining. The experimental results showed that the thermal errors were reduced to 19\(\upmu\)9 when temperature was rising and to 15 \(\upmu\)m when it reached to a stable state.  
However, it is worth noting that when there are more variables, the calculation time of MRA is too long. Moreover, the correlation between the thermal error and the temperature is only considered by the MRA, which will lead to the temperature variable coupling and reducing the accuracy of the model [33; 34; 35].