# On the Correlation of Symbols' Vibrational Behavior and Manufacturing Processes
## 2 Theoretical Background
### Spectral Correlation Index  
Concerning cymphals, the CFDAC can be used to monitor and classify unfinished cymphals during manufacturing stages based on alterations in the CFDAC pattern. This requires two CFDACs to be compared. The first CFDAC is the correlation of the FRFs of a cymphal, C1 (denoted by \(\text{CFDAC}_{h(\text{C1}),h(\text{C1})}\)). The second CFDAC is the correlation of the C1 FRFs to a set of FRFs of another cymphal, C2 (denoted by \(\text{CFDAC}_{h(\text{C1}),h(\text{C2})}\)). For assessing the similarity between \(\text{CFDAC}_{h(\text{C1}),h(\text{C1})}\) and \(\text{CFDAC}_{h(\text{C1}),h(\text{C2})}\), quantitative indicators are required, which are derived from image structure similarity indicators. The Pearson correlation coefficient (PCC) is used to quantify the degree of similarity of the compared CFDACs [26]. It is described by  
\[\text{PCC}_{A,B}=\frac{\sum_{f=1}^{N_{f}}\sum_{g=1}^{N_{g}}A_{fg}B_{fg}{-N_{f}N _{g}\overline{AB}}}{\Big{(}N_{f}N_{g}-1\Big{)}\sigma_{A}\sigma_{B}} \tag{2}\]  
where \(N_{f}\) and \(N_{g}\) are the FRFs' frequency lines, as stated above ( \(N_{f}=N_{g}\)). The terms \(A_{fg}\) and \(B_{fg}\) correspond to \(\text{CFDAC}_{h(\text{C1}),h(\text{C1})}\) and \(\text{CFDAC}_{h(\text{C1}),h(\text{C2})}\), respectively. As it may be assumed, in Equation (2), either the real parts, i.e., \(A_{fg}^{\text{Re}}=\text{Re}\Big{\{}\text{CFDAC}_{h(\text{C1}),h(\text{C1})} \Big{\}}\) and \(B_{fg}^{\text{Re}}=\text{Re}\Big{\{}\text{CFDAC}_{h(\text{C1}),h(\text{C2})} \Big{\}}\), or the imaginary parts, i.e., \(A_{fg}^{\text{Im}}=\text{Im}\Big{\{}\text{CFDAC}_{h(\text{C1}),h(\text{C1})} \Big{\}}\) and \(B_{fg}^{\text{Im}}=\text{Im}\Big{\{}\text{CFDAC}_{h(\text{C1}),h(\text{C2})} \Big{\}}\), can be used. It must be noted that for the calculation of PCC, the CFDAC matrices must be reshaped to \(1\times N_{f}N_{g}\) arrays.  
The term \(\overline{A}\) corresponds to mean value of the array according to  
\[\overline{A}=\frac{1}{N_{f}N_{g}}{\sum_{f=1}^{N_{f}}{\sum_{g=1}^{N_{g}}{A_{fg}}}} \tag{3}\]  
The same equation is used for the term \(\overline{B}\).  
The term \(\sigma_{A}\) corresponds to the standard deviation of the array according to  
\[\sigma_{A}=\sqrt{\frac{1}{N_{f}N_{g}-1}{\sum_{f=1}^{N_{f}}{\sum_{g=1}^{N_{g}} {\Big{(}A_{fg}-\overline{A}\Big{)}^{2}}}}} \tag{4}\]  
The same equation can be used for the term \(\sigma_{B}\).  
The PCC values are real and fall within the range of \([-1,\ 1]\). A value of 0 indicates no correlation, 1 represents perfect linear correlation, and \(-1\) indicates an inverse or negative correlation. However, the latter case, which represents inverse frequency correlation, is not applicable to the present study. Therefore, the range is modified to \([0,1]\).  
The spectral correlation index (SCI) is determined as  
\[\text{SCI}_{A,B}=1-|\text{PCC}_{A,B}|\in\mathbb{R} \tag{5}\]  
This transformation was adopted by [PERSON] et al. [26] for convenience, as they consider the value 0 to be a better descriptor for fully correlated states and 1 for completely uncorrelated states. Additionally, the SCI condenses all the information from the CFDAC matrices into a single value. Furthermore, the SCI can be calculated either from the real or the imaginary parts of the considered CFDACs, as described in [26].