## 4 Application Results  
### The Case of the Tennessee Eastman Process  
The Tennessee Eastman (TE) process is a widely used benchmark in the field of process control and fault detection research. It is a simulated chemical production process that replicates a realistic industrial environment with various complexities. Originally developed by Downs and Vogel in 1993 Downs and Vogel in 1993 [PERSON] and Vogel (1993), the TE process serves as a challenging dataset for evaluating different control strategies, fault detection methods, and data-driven approaches. The TE process simulates a chemical production plant that involves multiple reactions, including both exothermic and endothermic processes, as shown in Figure 8.  
Figure 7: Incremental Pyraformer–DCCA process monitoring framework. Different colors in the figure represent distinct process input variables. The plant has five major units: a reactor, a condenser, a vapor-liquid separator, a compressor, and a product stripper. These units work together to produce two final products from four raw materials. The process includes 11 operation variables, which are designated as \(X\) and can be adjusted to control the system, along with 41 measurement variables, with the first 22 designated as \(Y\). The TE process involves the simulation of multiple training and testing datasets. In this study, only one training dataset without faults was utilized, consisting of 500 samples. One of the testing datasets represents normal operational conditions, while the remaining datasets contain various types of faults. Therefore, the training data dimensions were 500 \(\times\) 11 for X and 500 \(\times\) 22 for Y. The faulty datasets consisted of 960 samples, with faults being introduced at the 161 st sample point. These faults can be categorized into five distinct types: step changes, random variations, slow drifts, sticking, and unknown faults. Moreover, as argued by [PERSON] et al. (2016), certain faults, such as IDV (3), IDV (9), IDV (15), and IDV (21), present significant challenges for detection using conventional data-driven monitoring techniques. Below, the approach employed for fault detection focuses on analyzing processes involving IDVs (6), (12), (13), (14), and (17). Specifically, IDV (6) is identified as a step change fault, IDV (12) represents random variation faults, IDV (13) is classified as a slow-drift fault, IDV (14) exemplifies sticking faults, and IDV (17) is categorized as an unknown fault. Figure 9 presents the monitoring performances for all the methods applied to Fault IDV (6). Figure 8: Flowchart of the TE process (Kal  
Table 1 summarizes the fault detection performance of each model in terms of the Fault Detection Rate (FDR) and False Alarm Rate (FAR) for Fault IDV (6). To evaluate the effectiveness of the proposed method, comparisons were made with various approaches that integrate DCCA. These include time-series models such as LSTM [27] and RNN-based variations like GRU [28] combined with DCCA. Additionally, Transformer-based models, including Fedformer [29], Autoformer [30], and Crossformer [31], were included in the comparison. Among the evaluated methods, Incremental Pyraformer-DCCA exhibits the most favorable overall performance, achieving an FDR of 100% alongside a FAR of 4.35%. In comparison, Fedformer-DCCA attains a lower FAR of 3.73%; however, this improvement  
Figure 9: Monitoring results for Fault IDV (6) in the TE process. (**a**) Linear CCA. (**b**) LSTM–DCCA. (**c**) Fedformer–DCCA. (**d**) Crossformer–DCCA. (**e**) Autoformer–DCCA. (**f**) GRU–DCCA. (**g**) Incremental Pyraformer–DCCA. is accompanied by a slight reduction in its FDR to 98.73%. While Transformer-based approaches generally maintain high FDR values, notable variations in the FARs illustrate the performance distinctions among these methods. Specifically, Crossformer-DCCA and Autoformer-DCCA both achieve a perfect FDR of 100%, but they exhibit relatively elevated FARs of 24.84% and 25.47%, respectively.