## 4 Proposed Methodology Description
### Mathematical Formulation of the Problem
Next, ANOM is used to find the variation caused by each factor in the overall mean and is plotted on appropriate graphs. The overall mean is indicated and the average of each factor for each level is also plotted [69]. Equation (2) is used for the calculation of means. \[\overline{SNR_{fp}}=\frac{1}{p}{\sum\limits_{i=1}^{\text{p}}SNR} \tag{2}\]  
where \(f\) is the key factor and \(p\) is the number of levels. The contribution of each factor is calculated with the use of Equations (3)-(6) [69]. \[\text{SS}={\sum\limits_{i=1}^{\text{fp}}(SNR_{i}-m)^{2}} \tag{3}\]  
where SS is the sum of the squares of each factor, fp is the number of experiments in the orthogonal matrix and \(m\) is the overall mean of SN ratio  
\[\text{SS}_{A}=3\big{(}\overline{SNR_{A1}}-m\big{)}^{2}+3\big{(}SNR_{A2}-m \big{)}^{2} \tag{4}\]  
where SS\({}_{A}\) is the sum of squares of factor \(A\)  
\[\text{MS}_{factor}=\frac{\text{SS}_{A}}{Degrees\ of\ Freedom} \tag{5}\]  
where MS\({}_{factor}\) is the mean square for each factor. \[CP=\frac{\text{SS}_{A}}{SS}\times 100\% \tag{6}\] \begin{table}
\begin{tabular}{c c c c} \hline
**Signal-to-Noise Ratio (SNR)** & **Goal of the Experiment** & **Data Characteristics** & **Signal-to-Noise Ratio Formulas** \\ \hline Larger is Better & Maximize the response & Positive & \(\text{SNR}=-10 log_{10}\left(\frac{1}{k}{\sum\limits_{i=1}^{\text{k}}}\frac{1}{ y_{i}^{2}}\right)\) \\ Smaller is Better & Minimize the response & Non-negative with a target value of zero & \(\text{SNR}=-10 log_{10}\left(s_{i}^{2}\right)\) \\  & Target the response and you & & s: variation \\ Nominal is Best & want to base the & Positive, zero, or negative & \(\text{SNR}=-10 log_{10}\left(\frac{1}{k}{\sum\limits_{i=1}^{\text{k}}}\frac{1}{ y_{i}^{2}}\right)\) \\  & standard deviations only & & \\  & Target the response and you & Non-negative with an & \\ Nominal is Best & want to base the & “absolute zero” in which the & \(\text{SNR}=10 log_{10}\left(\frac{y_{i}^{2}}{s^{2}}\right)\) \\ (default) & signal-to-noise ratio on means & standard deviation is zero & \(\overline{y^{2}}:\)_Average of Data_ \\  & and standard deviations & when the mean is zero & s: variation \\ \hline \end{tabular}
\end{table} 
Table 5: Types of Signal-to-noise Ratio. where \(CP\) presents the contribution percentage of each factor in the system performance. The parameters used in Equations (1)-(6) are summarized below:  
* \(f\rightarrow\) number of factors
* \(p\rightarrow\) number of levels per factor
* k \(\rightarrow\) number of trials
* \(y_{i}\rightarrow\) model response for experiment iteration \(i\)  
The number of control factors and the number of levels to be used affect the total number of experiments. The number (i.e., k) corresponds to the number of trials, and \(y_{i}\) refers to the response in each experiment and is used to calculate the signal-to-noise ratio (SNR). Therefore, the critical parameters are the control factor and the number of levels. Next, the \(y_{i}\) is calculated in order to proceed with Equations (1)-(6). As for these parameters, they are not fully related to the proposed framework, except '\(y_{i}\)'. The '\(y_{i}\)' parameter is measured using an ultra-sonic sensor, and the framework element corresponding to '\(y_{i}\)' is described during data acquisition. As regards the other parameters, (\(f,p,k\)) are not inserted into the Digital Shadow model, but they are required for DOE and ANOVA analysis.