# Comprehensive Evaluation of Conveyor Belt Impact Resistance Using Canonical Correlation Analysis
## 4 Results and Discussion
### Canonical Correlation Analysis
The second group of variables called Result (impact force, increase in tension force, relative amount of absorbed energy, degree of damage) represent the variables describing the result of the conducted experiment. The first group represents the new canonical variable \(U\) and the second group the new canonical variable V. Verification of the statistical significance of canonical correlation analysis is carried out by means of several approximations of F-tests based on multivariate statistics (Table 7). The tests confirmed that there is a statistically significant dependence (\(p\)-value \(<\alpha\)) between the two groups of variables. That is, we reject the null hypothesis about the independence of the correlation coefficient. In general, the number of canonical dimensions is equal to the number of variables in the smaller set. In our case, we have four pairs (roots) of canonical variables: \(U_{1}\) and \(V_{1}\), \(U_{2}\) and \(V_{2}\), \(U_{3}\) and \(V_{3}\), and \(U_{4}\) and \(V_{4}\). The number of significant pairs may be smaller. The next step is to verify the statistical significance of each pair of canonical variables by means of an approximative F-test and a Wilks' lambda test (Table 8). We test the null hypothesis that all pairs of canonical variables and uncorrelated, or \(H_{0}\) : \(R_{c1}=R_{c2}=R_{c3}=R_{c4}=0\), where \(R_{ci}\) is the canonical correlation coefficient for the \(i\)th pair of canonical variables, \(i=1,2,3,\ 4\). The estimated canonical correlations \(R_{c}\) and the canonical coefficient of determination \(R_{c}^{2}\) for all pairs are sorted by size. The first value of the F-test in [PERSON]'s lambda statistical significance test (row 1 to 4, Table 8) tests the significance of all four canonical correlations. Since \(p\)-value \(<\alpha\), we reject the null hypothesis and assume that the first pair of canonical variables is significantly \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Statistic** & **Value** & **Approx.F** & **Df** & **Error Df** & \(p\)**-Value** \\ \hline Wilks’ & 0.0035 & 127.460 & 16 & 379.46 & \textless{}0.0001 \\ Lambda & 1.967 & 30.734 & 16 & 508 & \textless{}0.0001 \\ Pillai’s & 42.068 & 322.086 & 16 & 490.00 & \textless{}0.0001 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Significance test of canonical correlation analysis (\(\alpha\) = 0.05). \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline  & \(R_{c}\) & \(R_{c2}\) & **F-Test** & **Df** &
\begin{tabular}{c} **Wilk’s** \\ **Lambda** \\ \end{tabular} & \(p\)**-Value** \\ \hline
1 to 4 & 0.9865 & 0.9731 & 127.460 & 16 & 0.0035 & 2.2 \(\times\) 10\({}^{-16}\) \\
2 to 4 & 0.9206 & 0.8475 & 44.337 & 9 & 0.1302 & 2.2 \(\times\) 10\({}^{-16}\) \\
3 to 4 & 0.3790 & 0.1436 & 5.186 & 4 & 0.8537 & 4.9 \(\times\) 10\({}^{-4}\) \\
4 to 4 & 0.0558 & 0.0031 & 0.396 & 1 & 0.9969 & 0.5302 * \\ \hline \hline \end{tabular} \({}^{*}p\)-value \(<\) alpha = 0.05.  
\end{table} 
Table 8: Test of canonical dimensions (\(\alpha\) = 0.05). correlated. The second value of the F-test (row 2 to 4, Table 8) is used to test the significance of the second, third, and fourth canonical correlations, and the third value tests the significance of only the third and fourth canonical correlations. Finally, the last test is used to test whether the fourth canonical correlation is significant by itself (it is not, because \(p\)-value \(>\alpha\)). It follows that only the first three pairs of canonical variables are statistically significant. The value of Wilks' lambda is interpreted in the opposite direction to the canonical coefficient of determination \(R_{c}^{2}\). A value close to zero indicates a high correlation, and vice versa, a value close to 1 indicates a low correlation. The largest paired correlation coefficient is for the first pair of canonical variables \(U_{1}\) and \(V_{1}\) (\(R_{c1}=0.9865\)) and the second largest is for the pair \(U_{2}\) and \(V_{2}\) (\(R_{c2}=0.9206\)). The \(R_{c}^{2}\) value can be interpreted similarly to the \(r^{2}\) value in the regression analysis.