## 3 Results  
In the first test performed, only the past matrix of the dataset was considered both for the training phase and for the testing phase. However, this option proved disadvantage in terms of results, which are summarized in Tables 4-6. In fact, with this approach KPCA was completely unable to distinguish between the state of health and the state of failure of the machinery, ending up labeling everything as a state of failure. As a consequence, a 0% rate of faults and a 100% rate of false alarms were identified. In conclusion, it is possible to state that this approach was completely unenforceable. In this case, the response of the two indices was the same, in contrast to the results obtained from the previous analysis (Table 1). The results obtained from this analysis cannot be considered acceptable, even considering the benchmark situation, i.e., the situation in which the entire dataset without any manipulation is used as an input for the KPCA. This particular benchmark situation was chosen since it represented the outcome that needed to be overcome. Therefore, it was considered appropriate to implement new tests. In this case, due to the need to combine the information in the matrix of past measurements and of future measurements, it was considered appropriate to examine some arguments regarding the canonical variate dissimilarity analysis (CVDA), which can be defined as an of evolution of the CVA. The main difference between CVA and CVDA lies in the fact that the latter extends CVA with a new index--which in this paper will be referred to as \(D\)--based on the difference between the past-projected and future-projected canonical variables [30]. The idea behind this technique is based on CVA's ability to find correlations between past and future datasets in light of the possibility to identify small deviations from the correct trend of the parameters considering how well one parameter can predict future canonical variables based on the previous ones [30]. In this technique, the difference between the matrix of past measurements and that of future measurements is taken into account in order to calculate the health indices \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Accuracy** & **Precision** & **Sensitivity** \\ \hline \(T^{2}\) & 16.15\% & 16.15\% & 100\% \\ \(Q\) & 16.15\% & 16.15\% & 100\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Accuracy, precision, and sensitivity only with past matrix. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **FAR** & **MDR** \\ \hline \(T^{2}\) & 100\% & 0\% \\ \(Q\) & 100\% & 0\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: KPCA only with past matrix. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **TP** & **TN** & **FP** & **FN** \\ \hline \(T^{2}\) & 111 & 0 & 576 & 0 \\ \(Q\) & 111 & 0 & 576 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Results analysis only with past matrix. during the process monitoring phase. Obviously, in CVDA a simple difference between the two matrices is not applied. However, considering the purpose of the paper, the fault detection model was trained by giving it as input only the difference between the matrix of future measurements and that of past measurements. In the first case, only the matrix of past measurements will be given for testing and therefore for actual monitoring. In the second case, what is proposed for the training will also be reproduced for testing, thus computing the difference between the matrix of future measurements and that of past measurements. For the sake of clarity, the difference between the matrix of future measurements and the matrix of past measurements is defined by the following procedure. Firstly, the difference between the matrix of past measurements is represented by \(\hat{Y}_{p}\), while the matrix of future measurements is represented by \(\hat{Y}_{f}\). Subsequently, considering, for example, a matrix \(\hat{Y}_{p}\) of size \(2\times 2\) (6) and a matrix \(\hat{Y}_{f}\) of size \(2\times 2\) (7) asfollows:  
\[\hat{Y}_{p} =\left(