## 4 Application Results
### The Case of the Tennessee Eastman Process
Additionally, the traditional CCA method records an FDR of 99.87% with a comparatively high FAR of 17.39%. These results demonstrate the varying trade-offs between fault detection accuracy and False Alarm Rates across different methodologies, highlighting the specific strengths and limitations inherent to each approach. Table 2 presents a comprehensive comparison of the fault detection performances across eight methods evaluated under four fault scenarios (IDV (12), IDV (13), IDV (14), and IDV (17)) within the TE process. The methods are assessed using the FDR and FAR to determine their detection capability and specificity. For IDV (12), representing random variation faults, Autoformer-DCCA achieves the highest FDR of 99.62%, but its FAR of 29.81% highlights a trade-off with false alarms. Crossformer-DCCA and GRU-DCCA also show decent FDR values of over 90%, but their FARs are considerably higher, at 45.96% and 34.16%, respectively, suggesting that these models may be prone to false positives. CCA and Fedformer-DCCA strike a balance between their FDR and FAR, with CCA offering a moderate FDR of 97.75% and a FAR of 11.18% and Fedformer-DCCA achieving an FDR of 93.03% with a FAR of 18.01%. Incremental Pyraformer-DCCA stands out by maintaining an excellent FDR of 99.22% while reducing its FAR to 15.53%, showing a more balanced detection-to-alarm ratio. In the case of IDV (13), representing slow-drift faults, LSTM-DCCA and Autoformer-DCCA perform well, with FDRs near 98%, although their FARs of 16.15% and 28.57%, respectively, leave room for improvement. Crossformer-DCCA achieves a high FDR of 98.23%, but its FAR of 32.30% suggests it flags many false alarms. Fedformer-DCCA scores in the midrange, with an FDR of 94.94% and FAR of 16.77%, while CCA balances a slightly lower detection accuracy with a minimal FAR of 6.21%. The Incremental Pyraformer-DCCA model emerges as the top performer, with an FDR of 98.83% and the lowest FAR of 3.73%, indicating superior robustness in detecting gradual process deviations. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **FDR\%** & **FAR\%** \\ \hline CCA & 99.87 & 17.39 \\ LSTM-DCCA & 100 & 16.15 \\ Feedformer-DCCA & 98.73 & 3.73 \\ Crossformer–DCCA & 100 & 24.84 \\ Autoformer–DCCA & 100 & 25.47 \\ GRU-DCCA & 100 & 9.94 \\
**Incremental Pyraformer–DCCA** & **100** & **4.35** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Monitoring results for Fault IDV (6) in the TE process. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Model** & **IDV (12)** & \multicolumn{2}{c}{**IDV (13)**} & \multicolumn{2}{c}{**IDV (14)**} & \multicolumn{2}{c}{**IDV (17)**} \\ \cline{2-9}  & **FDR\%** & **FAR\%** & **FDR\%** & **FAR\%** & **FDR\%** & **FAR\%** \\ \hline CCA & 97.75 & 11.18 & 95.87 & 6.21 & 90.24 & 6.83 & 85.73 & 8.07 \\ LSTM-DCCA & 97.94 & 26.71 & 98.31 & 16.15 & 80.63 & 9.94 & 95.95 & 19.25 \\ Fedformer-DCCA & 93.03 & 18.01 & 94.94 & 16.77 & 92.99 & 26.09 & 94.68 & 12.42 \\ Crossformer-DCCA & 97.85 & 45.96 & 98.23 & 32.30 & 98.86 & 43.48 & 99.87 & 44.10 \\ Autoformer-DCCA & 99.62 & 29.81 & 97.97 & 28.57 & 72.53 & 33.54 & 99.49 & 44.72 \\ GRU-DCCA & 90.71 & 34.16 & 96.46 & 19.25 & 97.97 & 23.60 & 97.72 & 23.60 \\
**Incremental Pyraformer–DCCA** & **99.22** & **15.53** & **98.83** & **3.73** & **97.47** & **7.45** & **97.22** & **2.48** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Monitoring results for Fault IDV (12), IDV (13), IDV (14), and IDV (17) in the TE process. For IDV (14), representing sticking faults, Crossformer-DCCA delivers the highest FDR at 98.86%, but with a significantly high FAR of 43.48%, indicating a trade-off between detection accuracy and false positives. Autoformer-DCCA, on the other hand, struggles with a low FDR of 72.53%, despite a moderate FAR of 33.54%. GRU-DCCA and Fedformer-DCCA offer intermediate performances, with FDRs of 97.97% and 92.99% and FARs of 23.60% and 26.09%, respectively, showcasing the challenge of balancing sensitivity and specificity.