# Regression Approach to a Novel Lateral Flatness Leveling System for Smart Manufacturing
## 4 Results and Discussion
Table 7 below shows the regression results for each of the five backup sets, namely backup set 1 to set 5. The \(R^{2}\) and adjusted \({R_{a}}^{2}\) were all around 0.81-0.88, with slight variation between each of the backup sets. The values for the root mean square error (RMSE) and standard deviation error (Std. Error) were all within 30. Thus, it was considered as a good fit for the regression models. From the above MLR model, the machine learning system was tested with two different common coil defects. The defects were based on the edge wave and center buckle of the sheet metal coil. The system was tested and recorded with different samples with training outputs, which later were compared with those from an experienced machine operator. The experiment was tested with a defined success rate of 100% being a flatness of 60 \(\upmu\)m or lower, a flatness specification which is a common standard in the sheet metal leveling industry. After the flatness reading was recorded, it was calculated as a percentage against 60 \(\upmu\)m as a reference. The experiment was conducted with 1000 test samples during coil leveling processing. The regression model was input to the system to train and provide a progressive output as shown in Figures 6 and 7 for leveling edge waves and center buckle, respectively. During the leveling of edge waves, the backup sets need to provide slightly tight space in the middle and loose space at the edges so that the sheet can expand laterally toward its two edges and make it flat. The results, shown in Figure 6, indicate that as the number of samples increased, the regression model could offer a success rate over 96%, whereas the experienced machine operator apparently was capped by a ceiling. The experienced machine operator may have produced a relatively good success rate when the number of samples was small. However, when the number of samples increased, the operator's \"experience\" may not have been good enough to make the proper judgement in adjusting the backup sets. In general, the experienced machine operator was shown to provide a success rate of around 80-90% during all of the 1000 test samples, which apparently was not as good as the results generated by the proposed method. Additionally, in comparing the results of the regression model and the machine operator, the regression model was shown to be more consistent for the success rate. Due to the edge wave possibly not being the same for both edges, it was harder for the machine operator to use solely experience for trials and errors to keep the leveling results consistent for the entire process. As a result, the regression model was shown to have a higher success rate and a more consistent for the edge wave leveling application. \begin{table}
\begin{tabular}{c c c c} \hline
**Backup Set 5** & **Coefficient** & **Standard Error** & \(p\)**-Value** \\ Constant & 20.75 & 4.32 & 6.9 \(\times\) 10\({}^{-8}\) \\ Coil thickness & \(-\)19.67 & 1.89 & 5.8 \(\times\) 10\({}^{-13}\) \\ Coil width & 2.22 & 0.97 & 2.8 \(\times\) 10\({}^{-13}\) \\ Coil tensile strength & 7.62 & 0.62 & 5.4 \(\times\) 10\({}^{-10}\) \\ Coil yield strength & 7.15 & 0.45 & 2.2 \(\times\) 10\({}^{-12}\) \\ Coil outer diameter & 2.97 & 0.18 & 7.5 \(\times\) 10\({}^{-11}\) \\ Coil inner diameter & 4.08 & 0.43 & 4.5 \(\times\) 10\({}^{-12}\) \\ Line speed & 0.22 & 0.15 & 8.2 \(\times\) 10\({}^{-11}\) \\ Flatness reading & 20.97 & 1.98 & 1.9 \(\times\) 10\({}^{-12}\) \\ \hline \end{tabular}
\end{table} 
Table 6: Multiple linear regression model for backup set 5. \begin{table}
\begin{tabular}{c c c c c} \hline  & \(R^{2}\) & \({R_{a}}^{2}\) & **RMSE** & **Std. Error** \\ \hline Backup set 1 & 0.8856 & 0.8156 & 20.55 & 19.52 \\ Backup set 2 & 0.8951 & 0.8513 & 21.54 & 18.62 \\ Backup set 3 & 0.8125 & 0.8763 & 22.35 & 20.56 \\ Backup set 4 & 0.8698 & 0.8865 & 26.25 & 22.65 \\ Backup set 5 & 0.8725 & 0.8425 & 30.15 & 26.79 \\ \hline \end{tabular}
\end{table} 
Table 7: Regression results for each backup set.