## 5 Discussion  
Figure 8 shows the mean residual errors \(Y_{1}\) and \(Y_{2}\) for the entire sequence of experiments. The plotted point named \"WC\" corresponds to a measurement carried out without compensation. It characterizes the mean machine tool error existing without compensation. The values recorded by the LI and the TL respectively turn at around 8 and 3 \(\upmu\)m. The TL measures are of the same order as the data provided by the machine tool manufacturer. Figure 8 highlights the relative similarity between the two sets of measures. The two sets of values are vertically shifted. As explained in Section 3.3.1, this is because the LI measurement truncates the real error vector. Table 4 shows the entire set of estimated parameters derived from the coefficient matrix \(\mathbf{\hat{B}}_{k}\) (\(k=1\) and \(2\)). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Factors**} & \multicolumn{4}{c}{**DOE**} \\ \cline{2-5}  & **Notation** & **Lower Value** & **Middle Value** & **Upper Value** \\ \hline Acquisition time (s) & \(X_{1}\) & 0.5 & 1 & 1.5 \\ Number of TL & \(X_{2}\) & 4 & 5 & 6 \\ positions & \(X_{3}\) & Small & Medium & Large \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Factors used in the design of experiment (DOE) process. To ease the comparison between the two quadratic models, a Student test (\(t\)-test) is applied to the set of coefficients \(\mathbf{\hat{B}}_{1}\) and \(\mathbf{\hat{B}}_{2}\) characterizing \(\hat{Y}_{1}\) and \(\hat{Y}_{2}\) to simplify the expressions of the response surfaces. Only the factors that significantly influence the response are kept. The result is summarized in Equations (10) and (11). \[\hat{Y}_{1} =\left(\hat{b}_{0}\right)_{1}+\left(\hat{b}_{1}\right)_{1}\cdot X _{1}+\left(\hat{b}_{12}\right)_{1}\cdot X_{1}\cdot X_{2}+\left(\hat{b}_{11} \right)_{1}\cdot X_{1}^{2}+\left(\hat{b}_{22}\right)_{1}\cdot X_{2}^{2}; \tag{10}\] \[\hat{Y}_{2} =\left(\hat{b}_{0}\right)_{2}+\left(\hat{b}_{1}\right)_{2}\cdot X _{1}+\left(\hat{b}_{12}\right)_{2}\cdot X_{1}\cdot X_{2}+\left(\hat{b}_{11} \right)_{2}\cdot X_{1}^{2}+\left(\hat{b}_{22}\right)_{2}\cdot X_{2}^{2}. \tag{11}\]  
The two models are not identical, but similar. In the case of a compact machine tool, the offset size \(X_{3}\) does not significantly influence the residual error map. Whatever the appliance used, only the acquisition time \(X_{1}\) and the number \(X_{2}\) of the TL positions matter in calibrating a machine tool using a TL. Therefore, the response surfaces, which represent \(\hat{Y}_{1}\) and \(\hat{Y}_{2}\), are plotted only in function of \(X_{1}\) and \(X_{2}\). The two responses and their respective SCBs are presented through a 3D graph, which is shown in Figure 9.  
The order of magnitude of the error bars computed respectively for \(\hat{Y}_{1}\) and \(\hat{Y}_{2}\) are 2 and 0.8 \(\upmu\)m. The lower value estimated for the results obtained with the TL is explained by more reliable measurements and data acquisition processes. The TL is, therefore, the appliance better able to check the calibration that it provided. Nevertheless, the two sets of optimized setup parameters are identical. On the other hand, since the residual error map provided by the LI is truncated, only the response \(\hat{Y}_{2}\) is used to provide the optimized calibration setup factors. The results are summarized in Table 5.