## 5 Conclusions  
This study addressed the challenges of fault detection in dynamic and nonlinear industrial processes by enhancing the DCCA framework. Through the combination of BLS-based incremental learning and the Pyraformer model, we proposed the Incremental Pyraformer-DCCA framework, which integrates efficient feature mapping with robust temporal modeling. The experimental results validated the framework's effectiveness through applications to the TE process, CSTR process, and injection molding process. To evaluate the performance of the proposed framework, we compared it with several other methods based on the DCCA framework, including models that incorporate LSTM, GRU, and Transformer architectures into the network design of DCCA. These comparative models share the same foundational principles of CCA and DCCA but employ different network structures to capture temporal and nonlinear relationships in process data. In the TE Fault IDV (6) process, Incremental Pyraformer-DCCA achieved an FDR of 100% and a FAR of 4.35%. This result was superior to those of LSTM-DCCA (FDR: 100%, FAR: 16.15%), GRU-DCCA (FDR: 100%, FAR: 9.94%), and the best-performing Transformer-based DCCA model (FDR: 100%, FAR: 24.87%) among its variations. Similarly, in the CSTR Fault 4 process, the framework demonstrated an FDR of 100% and a FAR of 3.48%, compared to LSTM-DCCA (FDR: 100%, FAR: 19.9%), GRU-DCCA (FDR: 100%, FAR: 8.46%), and Transformer-based DCCA (FDR: 100%, FAR: 13.43%). In the injection molding Fault 1 and Normal 1 processes, Incremental Pyraformer-DCCA continued to excel, with an FDR of 97.02% and a FAR of 0%, significantly surpassing LSTM-DCCA (FDR: 88.07%, FAR: 11.61%), GRU-DCCA (FDR: 74.44%, FAR: 11.18%), and Transformer-based DCCA (FDR: 100%, FAR: 8.63%). In the ablation experiments using the CSTR process as an example, we evaluated the contributions of each component by incrementally enhancing the baseline DCCA framework. The baseline DCCA served as the starting point and achieved an FDR of 91.69% and a FAR of 6.97%. Incremental DCCA, which incorporated Incremental Learning, improved adaptability and reached an FDR of 95.67% and a FAR of 3.98%. Pyraformer-DCCA introduced temporal modeling, leading to further improvements, with an FDR of 93.20% and a FAR of 0%. The fully integrated Incremental Pyraformer-DCCA delivered the best results, with an FDR of 100% and a FAR of 3.48%. These results demonstrated superior fault detection accuracy and reduced False Alarm Rates when compared to traditional and advanced methods. This method provides a feasible solution for monitoring dynamic and nonlinear processes and achieves robust and adaptive fault detection in complex industrial environments. This study has some limitations that necessitate further investigation. One limitation is the restricted domain applicability of the proposed framework. Although validated on the TE process, CSTR systems, and injection molding processes, its generalizability to other industrial scenarios has not yet been thoroughly examined, leaving the question of broader scalability open for future exploration. The other limitation is the trade-offs associated with incremental learning. Although incremental learning reduces the need for full retraining, it may encounter challenges such as knowledge forgetting or inadequate adaptation to new patterns when faced with frequent updates or significant changes in the data. Future research should focus on enhancing the adaptability of the framework to diverse industrial applications and improving the robustness of incremental learning to mitigate these challenges. Furthermore, the proposed Incremental Pyraformer-DCCA framework holds considerable promise for enhancing Failure Mode and Effects Analysis (FMEA) in industrial processes. In FMEA, the detection score guides the prioritization of corrective actions by gauging how likely it is to identify failures before they lead to major disruptions.