# Damage Evaluation in Tempered Vacuum Glazing via Multivariate Statistical Methods
## 2 Materials and Methods
### Evaluation Model  
#### 2.3.1 Principal Component Analysis (PCA)  
PCA is the method for finding the potential reigning factors in different related variables [21]. As a systematic product, characteristic parameters of TVG, such as surface microcracks and vacant pillars, have a close relationship between each other, exerting a direct influence on the surface stress. High multi-collinearity exists in different parameters, leading to incorrect regression equation parameters and even making the model unusable. PCA recombines original parameters to several aggregative indicators which not only provide more information but also have a poor linear relation due to the less information loss. The standardized values of six characteristic parameters of TVG (\(Z_{1}\), \(Z_{2}\),..., \(Z_{6}\)) are potential factors \(z_{1}\), \(z_{2}\),..., \(z_{6}\). Assuming that six new coordinate axes \(p_{i}\) (\(i=1\), \(2\),..., \(6\)) have to be found in the original six-dimensional parameter space, the relationship between the original and new parameters can be expressed as follow:  
\[\left\{ \begin{array}{l}p_{1}=b_{11}z_{1}+b_{12}z_{2}+\cdots+b_{16}z_{6}\\ p_{2}=b_{21}z_{1}+b_{22}z_{2}+\cdots+b_{26}z_{6}\\ \cdots\\ p_{i}=b_{i1}z_{1}+b_{i2}z_{2}+\cdots+b_{i6}z_{6}\end{array} \right. \tag{1}\]  
Since these six parameters are related to each other, PCA allows one to find a lesser number of new variables to gather as much information as possible about the variance from the original variables. Therefore, these new variables are called principal components, and each of them is a linear combination of the original variables. The maximum variance that can be found in the six orthogonal directions is presented by a feature vector and an eigenvalue. Both can be determined from the characteristic equation \(\mathbf{A}\mathbf{Z}=\lambda\mathbf{Z}\), where \(\mathbf{A}\) is the covariance matrix of samples and \(\lambda\) represents the reassignment result for the total variance of the original variables in the principal components. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Evaluating Level** & **Evaluating Parameter** & **Maximum Depth of Microcrack (\(\mathbf{\mu}\)m)** \\ \hline Excellent & 8–10 & \textless{}0.1 \\ Well & 6–8 & 0.1–1 \\ Poor & 4–6 & 1–10 \\ Worst & 0–4 & \(\geq\)10 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Evaluation basis for surface microcrack depth. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Evaluating Level** & **Evaluating Parameter** & **Microcrack Quantity within** \\  & & **1 mm\({}^{2}\) (Pieces)** \\ \hline Excellent & 8–10 & \textless{}200 \\ Well & 6–8 & 200–400 \\ Poor & 4–6 & 400–600 \\ Worst & 0–4 & \(\geq\)600 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Evaluation basis for surface microcrack quantity. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Evaluating Level** & **Evaluating Parameter** & **Bubble Quantity within 10 cm (Pieces)** \\ \hline Excellent & 8–10 & \textless{}150 \\ Well & 6–8 & 150–300 \\ Poor & 4–6 & 300–450 \\ Worst & 0–4 & \(\geq\)450 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Evaluation basis for bubble quantity at the sealing edge. The distributive variance \(S_{p_{i}}\) of the \(i\)'th principal component depends on the \(i\)'th feature vector in the numerical data as follows:  
\[S_{p_{i}}=\frac{\sum\limits_{i=1}^{6}(p_{i}-\overline{p_{i}})^{2}}{n-1}=\lambda _{i} \tag{2}\]  
Given the contribution rate of each component, the total information can be expressed as follows:  
\[m=\sum\limits_{i=1}^{6}\lambda_{i} \tag{3}\]  
The feature vector is used to measure the information from the variables.