# Regression Approach to a Novel Lateral Flatness Leveling System for Smart Manufacturing
## 3 Materials and Methods
### Regression Analysis  
In this paper, multiple linear regression (MLR) analysis was applied to determine the relationship between the dependent variables and independent variables [31; 32; 33; 34; 35]. As mentioned by [PERSON] et al. [22], this multiple regression analysis can be used to estimate the value of a dependent variable by substituting the values of the independent variables. The dependent variables (\(y_{i}\)) in this paper are defined as the output of each backup roll adjustment value. The independent variables (\(x_{i}\)) are the measured input data and the specifications of the sheet metal coil. The backup set adjustment output is estimated using the MLR model as follows [31; 32]:  
\[y_{i}=\beta_{0}+\beta_{1}x_{1i}+\beta_{2}x_{2i}+\cdots+\beta_{\rho}x_{pi}+e_{i} \tag{1}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Brand Measure Range (Z-Axis)** & **Baumer 100–150 mm** \\ \hline Field of View (\(X\)-axis) & 48–72 mm \\ Resolution & 2–4 \(\upmu\)m \\ Measuring Rate & \textless{}500 Hz \\ Ambient Light Immunity & \textless{}35 kLux \\ Interface & Analog, RS485 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Laser sensor specifications.  
Figure 4: (**a**) Schematic drawing of the sensors’ placement and (**b**) a photograph of the model lateral leveling sensor measurement system.  
where the \(\beta_{k}\) (\(k\) = 0, 1, 2,..., \(\rho\)) coefficients are unknown parameters and \(e_{i}\) is a set of random error terms. The subscript \(i\) indicates the observational unit from which the observations of \(y\) and the \(\rho\) independent variable were taken [34].  
In this case, the relationship between the target and predictor variables is unknown. Therefore, it is often required to estimate this relationship or to equivalently estimate the unknown parameters \(\beta_{k}\). To achieve this goal, least squares regression is used for estimation, in which the sum of the squares of the residuals is a measure of the discrepancy between the data and an estimation model mentioned by [PERSON] [31], such as linear regression. A small sum of the squared residuals (\(SSR\)) indicates a tight fit of the model to the data, which mathematically can be expressed as  
\[SSR=\sum_{i=1}^{n}{{e_{i}}^{2}}=\sum_{i=1}^{n}\left[{{y_{i}}-\left({\beta_{0}+ \beta_{1}x_{1i}+\cdots+\beta_{\rho}x_{\rho i}}\right)}\right]^{2} \tag{2}\]  
or simply as  
\[SSR=\sum_{i=1}^{n}{{e_{i}}^{2}}={{e_{1}}^{2}}+{{e_{2}}^{2}}+\cdots+{{e_{n}}^{2}} \tag{3}\]  
As demonstrated by [PERSON] et al. [32] and [PERSON] [35], to measure the strength of the regression relationship \(R^{2}\), this is expressed as the fraction of the regression squared residuals over the corrected total squared residuals as  
\[R^{2}=\frac{\sum_{i}\left({\hat{y}_{i}-\overline{Y}}\right)^{2}}{\sum_{i} \left({y_{i}-\overline{Y}}\right)^{2}}=\frac{Regression\ SS}{Corrrcted\ total\ SS} \tag{4}\]  
where \(\hat{y}_{i}\) is the fitted value and \(\overline{Y}\) is the sample mean of the target value. This \(R^{2}\) value (also called the coefficient of determination) estimates the population proportion of variability in \(y\) accounted for by the best linear combination of the predictors. Its value being closer to 1 indicates a good deal of predictive power from the predictors for the target variable, while a value closer to 0 indicates little predictive power.  
Multiple regression analysis has two or more independent variables. It is necessary to consider the adjusted coefficient of determination, denoted as \({{R_{a}}^{2}}\), as \(R^{2}\) is an estimate of the population proportion of variability accounted for by the regression. \({{R_{a}}^{2}}\), in fact, compensates for the characteristic of \(R^{2}\) and corrects this bias as expressed by the following equation [32]:  
\[{{R_{a}}^{2}}=R^{2}-\frac{p}{n-p-1}\left({1-R^{2}}\right) \tag{5}\]  
where (\(n-p-1\)) is the degree of freedom, \(n\) is the number of samples, and \(p\) is the number of independent variables.