## 4 Application Results
### The Case of Injection Molding Process
For Fault 2, Incremental Pyraformer-DCCA achieves the highest performance with an FDR of 100%, ensuring perfect fault detection, and a FAR of 0.00% for Normal 1, which indicates its excellent ability to avoid false alarms. In comparison, LSTM-DCCA and Fedformer-DCCA show solid detection capabilities, with FDRs of 96.81% and 100%, respectively, but both models exhibit higher FARs of 11.61% and 8.63% for Normal 1, highlighting a slight trade-off between detection accuracy and false positives. Autoformer-DCCA and Crossformer-DCCA achieve high FDRs of 99.04% and 97.34%, respectively, but their FARs are somewhat higher--3.41% for Normal 1 and 6.60% for Normal 1--showing that while they maintain good Fault Detection Rates, they are more prone to false alarms. The CCA method, although exhibiting a lower FDR of 70.14% for Fault 2, manages to maintain a FAR of 0%, indicating its conservative nature in detecting faults but also suggesting a limitation in terms of sensitivity to faults. For Fault 3, Incremental Pyraformer-DCCA, again, delivers the best performance, with an FDR of 100% and a FAR of 0.00% for Normal 1, ensuring high accuracy in fault detection and effectively suppressing false alarms. LSTM-DCCA and Autoformer-DCCA show good performances with FDRs of 82.00% and 89.14%, respectively, but they have relatively high FARs of 11.93% and 14.27%, indicating a higher rate of false alarms compared to Incremental Pyraformer-DCCA. Crossformer-DCCA achieves an FDR of 84.03% but with a FAR of 9.69%, reflecting its tendency to flag more anomalies. Fedformer-DCCA maintains an FDR of 100% but with FARs of 8.63% and 9.27% for Normal 1 and Normal 2, respectively, indicating that while it is excellent at detecting faults, it struggles more with false alarms than Incremental Pyraformer-DCCA. The CCA model has the lowest FDR of 73.86% for Fault 3, with FARs of 21.82% and 54.24% for Normal 1 and Normal 2, respectively, showing that it has significant difficulty in fault detection and a high tendency to produce false positives. To evaluate the computational efficiency of our proposed method, we conducted all experiments on a GeForce RTX 3090 GPU with 24 GB of memory. The training and test times were measured across different datasets and model configurations to assess feasibility in real-world applications. As summarized in Table 8, Incremental Pyraformer-DCCA demonstrates competitive training efficiency compared to the baseline models. In the injection molding scenario, our proposed Incremental Pyraformer-DCCA method demonstrated competitive training efficiency, requiring approximately 3.53 s to complete a training iteration. This performance was compared against several baseline models: LSTM-DCCA (3.48 s), Fedformer-DCCA (7.14 s), Crossformer-DCCA (5.09 s), Autoformer-DCCA (4.21 s), and GRU-DCCA (3.21 s). These results indicate that our method maintains computational efficiency while incorporating a multi-scale attention mechanism and incremental learning strategies. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{2}{c}{**TE Process**} & \multicolumn{2}{c}{**CSTR Process**} & \multicolumn{2}{c}{**Injection Molding Process**} \\ \cline{2-7}  & **Train(s)** & **Test(s)** & **Train(s)** & **Test(s)** & **Train(s)** & **Test(s)** \\ \hline LSTM–DCCA & 1.64 & 0.05 & 5.30 & 0.08 & 3.48 & 0.06 \\ Feedformer–DCCA & 4.31 & 0.12 & 9.00 & 0.14 & 7.41 & 0.12 \\ Crossformer–DCCA & 1.98 & 0.06 & 5.68 & 0.07 & 5.09 & 0.06 \\ Autoformer–DCCA & 2.21 & 0.04 & 4.75 & 0.05 & 4.21 & 0.05 \\ GRU–DCCA & 1.90 & 0.03 & 4.20 & 0.04 & 3.21 & 0.03 \\
**Incremental** & **1.41** & **0.02** & **4.19** & **0.04** & **3.53** & **0.03** \\
**Pyraformer–DCCA** & \multicolumn{1}{c}{} & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Training and test times (in seconds) for different models across three datasets. To further validate its efficiency, we analyzed the computational cost on additional benchmark datasets.