## 3 Results
\begin{array}{cc}a_{11}&a_{12}\\ a_{21}&a_{22}\end{array} \right) \tag{6}\] \[\hat{Y}_{f} =\left( \begin{array}{cc}b_{11}&b_{12}\\ b_{21}&b_{22}\end{array} \right) \tag{7}\]  
It is possible to define the matrix \(C\), obtained by subtracting the matrices \(\hat{Y}_{f}\) and \(\hat{Y}_{p}\), as (8):  
\[C=\left( \begin{array}{cc}b_{11}-a_{11}&b_{12}-a_{12}\\ b_{21}-a_{21}&b_{22}-a_{22}\end{array} \right) \tag{8}\]  
Moving on to the analysis of the obtained results, the second investigated approach analyzed the difference between future and past matrices in both training and test datasets. The achieved results are presented in Tables 7-9. It is possible to note that the results obtained from this analysis are better that those presented in Table 4. Consequently, this improvement indicates that considering both the matrix of past and future events increases the information available, and it also increases the readiness of the model. However, the results are still worse than the ones presented in Table 1, and it is therefore necessary to investigate other solutions. In this regard, \begin{table}
\begin{tabular}{c c c} \hline \hline  & **FAR** & **MDR** \\ \hline \(T^{2}\) & 17.8\% & 59.6\% \\ \(Q\) & **4.8\%** & **7.6\%** \\ \hline \hline \end{tabular}
\end{table} 
Table 7: KPCA with the difference between the past and future matrix for both training and test. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **TP** & **TN** & **FP** & **FN** \\ \hline \(T^{2}\) & 45 & 473 & 103 & 66 \\ \(Q\) & 103 & 548 & 28 & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Results analysis with the difference between the past and future matrix for both training and test. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Accuracy** & **Precision** & **Sensitivity** \\ \hline \(T^{2}\) & 75.4\% & 30.4\% & 40.5\% \\ \(Q\) & 94.7\% & 78.6\% & 92.8\% \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Accuracy, precision, and sensitivity with the difference between the past and future matrix for both training and test. two considerations should be made. The first one concerns the overall error rate of this model, which is still unacceptable and requires further investigation about the achievable results. The second one concerns the sensitivity analysis of the two indices. In fact, \(T^{2}\) usually provides significantly better results than \(Q\), while in the presented case the results are inverted. In the third approach analyzed, the input of the training phase was the difference between the matrix of future measurements and that of past measurements, while for the test phase only the matrix of past measurements was given as the input to the model. The obtained results are presented in Tables 10-12. It is interesting to note that this combination of past and future matrices has significantly increased the achievable results, making them highly performant. Moreover, this is the first combination of matrices that gives better result than those presented in Table 1. However, even in this case \(Q\) performed better than \(T^{2}\), which is unusual. Despite that, the results obtained from this approach are excellent, and the graphs of \(T^{2}\) and \(Q\) are therefore presented below to provide a visual oversight of the success of monitoring with this technique. Specifically, Figure 4 shows the monitoring of healthy data and Figure 5 shows the monitoring of faulty data. The good results obtained with the combination of the past and future matrices, specifically their difference, allow us to make some interesting considerations. For instance, it would be interesting to see the results of the analysis by calculating the sum and not the subtraction of the matrix of past measurements and that of future measurements. The tests presented hereafter provide an answer to this question, thus finally leading to the definition of the best model. However, it was already stated that both approaches tested with the sum of the matrix of past and future measurements guaranteed better results than the benchmark situation (Table 1).