## 4 Application Results
### The Case of Continuous Stirred-Tank Reactor Process
Table 4 summarizes the fault detection performance of each model in terms of the FDR and FAR. Among the evaluated methods, Incremental Pyraformer-DCCA demonstrates the most favorable overall performance, achieving a perfect FDR of 100% while maintaining the lowest FAR of 3.48%. In contrast, Fedformer-DCCA and Crossformer-DCCA exhibit higher FARs of 25.87% and 25.37%, respectively, despite Fedformer-DCCA achieving a perfect FDR and Crossformer-DCCA showing a slightly reduced FDR of 96.19%. This indicates that while Transformer-based approaches generally maintain high detection accuracy, their False Alarm Rates vary significantly. GRU-DCCA also achieves an FDR of 100%, with a moderate FAR of 8.46%, demonstrating a balanced performance compared to other methods. Similarly, Autoformer-DCCA achieves a perfect FDR but records a FAR of 13.43%, which is notably lower than Fedformer-DCCA and Crossformer-DCCA but higher than GRU-DCCA and Incremental Pyraformer-DCCA. The traditional CCA method records the lowest FDR of 92.89%, accompanied by a FAR of 11.94%, indicating its limitations in both fault detection accuracy and false alarm suppression. \begin{table}
\begin{tabular}{c c c c} \hline
**Fault ID** & **Description**\({}^{\ddagger}\) & **Value of \(\delta\)\({}^{\ddagger}\) & **Type** \\ \hline
1 & \(a=a_{0}exp(-\delta t)\) & 0.0005 & Multiplicative \\
2 & \(b=b_{0}exp(-\delta t)\) & 0.001 & Multiplicative \\
3 & Simultaneous Faults 1 and 2 & & Multiplicative \\
4 & \(C_{i}=C_{i,0}+\delta t\) & 0.001 & Additive \\
5 & \(T_{i}=T_{i,0}+\delta t\) & 0.05 & Additive \\
6 & \(T_{ci}=T_{ci,0}+\delta t\) & 0.05 & Additive \\
7 & \(C=C_{0}+\delta t\) & 0.001 & Additive \\
8 & \(T=T_{0}+\delta t\) & 0.05 & Additive \\
9 & \(T_{C}=T_{C,0}+\delta t\) & 0.05 & Additive \\
10 & \(Q_{C}=Q_{C,0}+\delta t\) & \(-0.1\) & Additive \\ \hline \end{tabular} \(x^{\ddagger}\) All t in minutes. Subscript 0 refers to nominal value. \(x^{\ddagger}\) The \(\delta\) commands the rate of fault progression.  
\end{table} 
Table 3: Incipient fault scenarios in the CSTR [32]. Figure 11: Monitoring results for Fault 4 in the CSTR process. (**a**) Linear CCA. (**b**) LSTM–DCCA. (**c**) Fedformer–DCCA. (**d**) Crossformer–DCCA. (**e**) Autoformer–DCCA. (**f**) GRU–DCCA. (**g**) Incremental Pyraformer–DCCA. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **FDR\%** & **FAR\%** \\ \hline CCA & 92.89 & 11.94 \\ LSTM–DCCA & 100 & 19.9 \\ Fedformer–DCCA & 100 & 25.87 \\ Crossformer–DCCA & 96.19 & 25.37 \\ Autoformer–DCCA & 100 & 13.43 \\ GRU–DCCA & 100 & 8.46 \\
**Incremental Pyraformer–DCCA** & **100** & **3.48** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Monitoring results for Fault 4 in the CSTR process. Table 5 presents a detailed comparison of fault detection methods across four fault scenarios (Fault 3, Fault 5, Fault 6, and Fault 10) in the CSTR process. The methods are evaluated based on their FDR and FAR, providing insights into their detection accuracy and the frequency of false positives. For Fault 3 (Multiplicative faults), Incremental Pyraformer-DCCA achieves an outstanding FDR of 100% with a minimal FAR of 2.49%, demonstrating its superior capability to detect faults with very few false alarms. In comparison, other methods such as LSTM-DCCA and GRU-DCCA also achieve perfect FDRs but exhibit slightly higher FARs (7.46% and 6.97%, respectively), indicating a greater propensity for false positives. Fedformer-DCCA and Autoformer-DCCA achieve 100% FDRs but at the cost of higher FARs (14.30% and 8.96%, respectively), while Crossformer-DCCA shows a similar FDR but a significantly higher FAR of 19.90%. The traditional CCA method records a lower FDR of 87.99%, but it has the advantage of no false alarms (FAR = 0%). For Fault 5 (Additive faults), Incremental Pyraformer-DCCA, again, leads with an FDR of 100% and the lowest FAR of 3.98%, outperforming other models in minimizing false alarms.