## 4 Application Results
### The Case of Injection Molding Process  
The injection molding equipment is equipped with multiple built-in sensors for real-time process monitoring, eliminating the need for additional manual sensor installation. Specifically, the machine is equipped with an internal displacement sensor to measure the screw position and thermocouple sensors to monitor mold temperatures. Additionally, a Kistler 4262A pressure sensor was employed to measure injection and hydraulic pressures. All sensor data were transmitted to a monitoring PC for recording and subsequent analysis. The dataset is divided into a training set of 273 normal injection cycles and a test set of 188 normal samples and 20 abnormal samples. Each injection cycle contains 968 consecutive time-series data points for the monitored variables. The injection molding process monitors eight variables, including three pressure parameters, one screw position parameter, and four temperature parameters. In this study, the three pressure parameters and the screw position parameter were designated as variable X, while the four temperature parameters were assigned to variable Y. Consequently, the training data dimensions for both X and Y were 968 \(\times\) 4. From the test set, five specific subsets of data were selected to evaluate the fault detection performance. Two of these subsets, referred to as Normal 1 and Normal 2, represent normal operating conditions without observable anomalies. The other three subsets, Fault 1, Fault 2, and Fault 3, capture different abnormal conditions in the process data. These five subsets together provide a representative sample of both normal and fault scenarios for robust model evaluation. Figure 12 presents the monitoring performances for all the methods applied to Fault 1 in the mold injection process. Table 6 summarizes the fault detection performances in terms of the FDR for Fault 1. Fedformer-DCCA achieves the highest FDR of 100%, indicating perfect fault detection performance. Incremental Pyraformer-DCCA follows closely with an FDR of 97.02%, showcasing competitive results and outperforming other methods. Models such as LSTM-DCCA (88.07%), Autoformer-DCCA (82.85%), and GRU-DCCA (74.44%) demonstrate moderate fault detection capabilities. In contrast, traditional CCA (70.97%) and Crossformer-DCCA (69.76%) exhibit relatively lower performances. Table 7 presents a detailed evaluation of the fault detection models applied to the injection molding process, considering two fault scenarios (Fault 2 and Fault 3) and two normal conditions (Normal 1 and Normal 2). The table assesses each model's performance based on the FDR and FAR, providing a comprehensive analysis of their ability to detect faults while minimizing false alarms. \begin{table}
\begin{tabular}{c c} \hline \hline
**Model** & **FDR\%** \\ \hline CCA & 70.97 \\ LSTM–DCCA & 88.07 \\ Feedformer–DCCA & 100 \\ Crossformer–DCCA & 69.76 \\ Autoformer–DCCA & 82.85 \\ GRU–DCCA & 74.44 \\
**Incremental Pyraformer–DCCA** & **97.02** \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Monitoring results for Fault 1 in the mold injection process. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Model**} & **Fault 2** & **Fault 3** & **Normal 1** & **Normal 2** \\ \cline{2-5}  & **FDR\%** & **FDR\%** & **FAR\%** & **FAR\%** \\ \hline CCA & 70.14 & 73.86 & 21.82 & 54.24 \\ LSTM-DCCA & 96.81 & 82.00 & 11.61 & 11.93 \\ Fedformer-DCCA & 100.00 & 100.00 & 8.63 & 9.27 \\ Crossformer-DCCA & 97.34 & 84.03 & 6.60 & 9.69 \\ Autoformer-DCCA & 99.04 & 89.14 & 3.41 & 14.27 \\ GRU-DCCA & 81.79 & 77.10 & 11.18 & 27.80 \\
**Incremental Pyraformer–DCCA** & **100** & **100** & **0.00** & **5.32** \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Monitoring results for Fault 2, Fault 3, Normal 1, and Normal 2 in the injection molding process. Figure 12: Monitoring results for Fault 1 in the mold injection process. **(a)** Linear CCA. **(b)** LSTM–DCCA. **(c)** Fedformer–DCCA. **(d)** Crossformer–DCCA. **(e)** Autoformer–DCCA. **(f)** GRU–DCCA. **(g)** Incremental Pyraformer–DCCA.