# Comprehensive Evaluation of Conveyor Belt Impact Resistance Using Canonical Correlation Analysis
## 3 Theory/Calculation
### Canonical Correlation Analysis  
Canonical correlation analysis (CCA) is a multidimensional statistical method that aims to describe as simply and concisely as possible the dependence between two groups of original variables using linear combinations [39; 40]. The method is a certain generalization of multiple regression and correlation analysis. In this method, we transform a system of mutually correlated variables into a system of new variables. The method assumes a linear dependence between variables and between groups of variables. It does not require the assumption of normality of variables. Let the variables \(X_{1},X_{2},\ X_{3},\ \cdots X_{k}\) form the first group of variables (variable group 1), and let the variables \(Y_{1},Y_{2},\ Y_{3},\ \cdots Y_{m}\) form the second group of variables (variable group 2). The goal of CCA is to find linear combinations of original variables from each group, thereby creating new artificial variables, canonical variables (similar to the principal component analysis method). Let the new canonical variable \(U\) be a linear combination of the original variables \(X_{1},X_{2},\ X_{3},\ \cdots X_{k}\) and let the canonical variable \(V\) be a linear combination of the original variables \(Y_{1},Y_{2},\ Y_{3},\ \cdots Y_{m}\). The goal of CCA is to find such pairs of canonical variables so that the dependence (correlation) between them is maximum. Initially, the first pair of canonical variables \(U_{1}\) and \(V_{1}\) is found. We gradually repeat the process by searching for other pairs of canonical variables (\(U_{2},\ V_{2}\)) up to (\(U_{q},\ V_{q}\)), whereby for the pair (\(U_{l},\ V_{h}\)) it holds that the following occurs:  
\[U_{h}=a_{1h}X_{1}+a_{2h}X_{2}+a_{3h}X_{3}+\cdots+a_{kh}X_{k}, \tag{3}\]  
\[V_{h}=b_{1h}Y_{1}+b_{2h}Y_{2}+b_{3h}Y_{3}+\cdots+b_{mh}Y_{m}, \tag{4}\]  
for \(h=1,\cdots,q\), where \(a_{h}=\{a_{1h},a_{2h},a_{3h},\cdots,a_{kh}\}\), \(b_{h}=\{b_{1h},b_{2h},b_{3h},\cdots,b_{mh}\}\), are canonical coefficients. In general, the number of pairs of canonical variables is equal to the lowest number of variables in each set, i.e., \(q=\min(k,m)\). The intensity of dependence between a pair of canonical variables is determined using a point estimate of the canonical correlation efficient \(R_{c}\). The canonical correlation coefficient has similar properties to the Pearson correlation coefficient \(r\). The squared of the canonical correlation coefficient \(R_{c2}\) is the canonical coefficient of determination. \(R_{c2}\) takes values from 0 to 1, where 0 means low correlation. The closer the value of the canonical coefficient of determination to 1, the stronger the correlation of the canonical variables. A graphical representation of the canonical correlation analysis, in general, is in Figure 6. On the left side are the original variables \(X_{1},X_{2},\ X_{3},\ \cdots X_{k}\) (variable group 1), which form the canonical variable \(U\); on the right side are the variables \(Y_{1},Y_{2},\ Y_{3},\ \cdots Y_{m}\) (variable group 2), which form the canonical variable \(V\). The variable \(U_{1}\) is a linear combination of the variables \(X_{1},X_{2},\ X_{3},\ \cdots X_{k}\) and the variable \(V_{1}\) is a linear combination of the variables \(Y_{1},Y_{2},\ Y_{3},\ \cdots Y_{m}\). The pair \((U_{1},\ V_{1})\) represents the first pair of canonical variables, the correlation of which is determined by the canonical correlation coefficient \(R_{c1}\). To verify the statistical significance of the canonical variables, an approximate F-test is used, which is based on [PERSON] lambda statistics. [PERSON] lambda statistics are interpreted in exactly the opposite direction to the canonical coefficient of determination \(R_{c2}\). A value near 0 means high correlation, while a value near 1 means a low correlation between canonical variables [41].