# Comprehensive Evaluation of Conveyor Belt Impact Resistance Using Canonical Correlation Analysis
## 4 Results and Discussion
### Canonical Correlation Analysis
Using the values of the standardized canonical coefficients in the first column of Table 11, the first canonical variable \(U_{1}\) (or variable \(V_{1}\) from Table 12) is determined using the following formula:  
\[U_{1}=-0.2456X_{1}-0.3481X_{2}+0.4818X_{3}+0.8645X_{4} \tag{8}\]  
\[V_{1}=-0.0831Y_{1}-0.5124Y_{2}-0.7457Y_{3}-0.1230Y_{4} \tag{9}\]  
The graphical depiction of the canonical correlation analysis for the first canonical pair is shown in Figure 13. On the left side, there are the original variables, which form the canonical variable \(U\), and on the right side, there are the variables forming the canonical variable \(V\). The pair \((U_{1},\ V_{1})\) represents the first pair of canonical variables, the correlation of which is determined by the canonical correlation coefficient \(R_{c1}\). In an analogous way, we can determine other pairs of canonical variables, \(U\) and \(V\). Standardized canonical coefficients are shown in Tables 11 and 12. \begin{table}
\begin{tabular}{l c c c c} \hline  & \(\mathbf{V_{1}}\) & \(\mathbf{V_{2}}\) & \(\mathbf{V_{3}}\) & \(\mathbf{V_{4}}\) \\ \hline \(F_{impact}\) (\(Y_{1}\)) & \(-0.0831\) & 0.9029 & \(-2.0751\) & \(-0.4810\) \\ \(\Delta F_{tension}\) (\(Y_{2}\)) & \(-0.5124\) & 0.1045 & 1.6507 & 0.9329 \\ \(E_{relat}\) (\(Y_{3}\)) & \(-0.7457\) & \(-0.0464\) & \(-1.7016\) & \(-0.0634\) \\ Damage (\(Y_{4}\)) & \(-0.1230\) & 0.0457 & 0.0636 & \(-1.1619\) \\ \hline \end{tabular}
\end{table} 
Table 12: Standardized canonical coefficients for variable \(V\). Figure 13: First pair of canonical variables. The values of the canonical variables \(U\) and \(V\) are calculated using canonical coefficients. A graphical representation of the course of dependence between individual pairs of canonical variables is shown in Figure 14. The graphs confirm previous findings that the first two pairs of canonical variables have a high canonical correlation and are the most important compared to the other pairs. A similar interpretation of variables is obtained by establishing a correlation between each variable and the corresponding canonical variable. In Table 13, there are the correlations (Pearson coefficient of correlation \(r\)) between the original variables determining the properties (or result) of the experiment and the canonical variables \(U\) and \(V\). For example, it follows (Table 13) that the first canonical variable \(U_{1}\) is dominated by the variable Type (\(r\) = 0.797). The canonical variable \(U_{2}\) is dominated by the variables Weight (\(r\) = 0.655) and Height (\(r\) = 0.661). The first canonical variable \(V_{1}\) is dominated by the variables \(E_{relat}\) (\(r\) = \(-\)0.769) and \(\Delta F_{tension}\) (\(r\) = \(-\)0.780). The canonical variable \(V_{2}\) is dominated by the variable \(F_{impact}\) (\(r\) = 0.996). Table 13 is also supplemented by the \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline
**Variables** & \(\mathbf{U_{1}}\) & \(\mathbf{U_{2}}\) & \(\mathbf{U_{3}}\) & \(\mathbf{U_{4}}\) & \(\mathbf{V_{1}}\) & \(\mathbf{V_{2}}\) & \(\mathbf{V_{3}}\) & \(\mathbf{V_{4}}\) \\ \hline Weight & \(-\)0.151 & 0.655 & 0.702 & \(-\)0.235 & \(-\)0.149 & 0.603 & 0.266 & \(-\)0.013 \\ Height & \(-\)0.248 & 0.661 & \(-\)0.643 & 0.294 & \(-\)0.245 & 0.609 & \(-\)0.244 & 0.016 \\ Strength & 0.389 & 0.263 & \(-\)0.352 & \(-\)0.809 & 0.384 & 0.242 & \(-\)0.133 & \(-\)0.045 \\ Type * & 0.797 & 0.306 & 0.136 & 0.503 & 0.786 & 0.282 & 0.052 & 0.028 \\ \hline \(F_{impact}\) & 0.067 & 0.917 & \(-\)0.022 & 0.002 & 0.068 & 0.996 & \(-\)0.028 & 0.029 \\ \(\Delta F_{tension}\) & \(-\)0.688 & 0.597 & 0.107 & 0.006 & \(-\)0.697 & 0.649 & 0.284 & 0.115 \\ \(E_{relat}\) & \(-\)0.769 & \(-\)0.534 & \(-\)0.088 & 0.003 & \(-\)0.780 & \(-\)0.580 & \(-\)0.232 & 0.047 \\ Damage & \(-\)0.538 & 0.129 & 0.100 & \(-\)0.044 & \(-\)0.545 & 0.140 & 0.264 & \(-\)0.783 \\ \hline \hline \end{tabular}  
* Reference category: Steel cord conveyor belt.  
\end{table} 
Table 13: Correlation between original variables and canonical variables.