## 5 Experimental Results and Discussion
### Benchmark Simulation Data: Tennessee Eastman Process  
TEP is an industrial process simulation dataset that is widely used in the field of fault detection [38]. These benchmark data are essential because obtaining fault cases from real large-scale processes is challenging. Developed by Eastman Chemical Company, TEP simulates data collected at specific times of failure, closely resembling those of real industrial processes. For patent-related reasons, TEP data are based on simulations of real industrial processes, including their components, dynamics, and operating conditions. The primary equipment of process includes reactors, coolers, and compressors, and the TEP control structure is illustrated in Figure 6.  
Figure 5: Comparison of nearest neighbors between kNN and SNN for faulty data: (**a**) when neighbors are located at the center; (**b**) when neighbors are close to the center of cluster; (**c**) when many neighbors are located on the outskirts. The TEP comprises 52 process variables, all of which are contaminated with Gaussian noise. The dataset includes 21 types of predefined faults, of which 16 have known causes, while the remaining 5 are of unknown origin. For instance, faults from 8 to 12 result from increased variability in process variables, and fault 13 is attributed to an issue in the reactor. As described above, TEP simulations represent cases where a fault arises from an individual process or a combination of multiple processes. For each fault scenario, the TEP simulation confirms whether the fault detection method can effectively identify the fault. The training and validation data for TEP are available at [[https://doi.org/10.7910/DVN/6C3](https://doi.org/10.7910/DVN/6C3) JR1]([https://doi.org/10.7910/DVN/6C3](https://doi.org/10.7910/DVN/6C3) JR1) (accessed on 16 December 2024). The stepwise faults (TEP faults 1-6) involve staircase function faults in specific variables, making them easier to detect compared to other cases. These faults alter the distribution of related variables such as material balance, levels, and pressures. In Figure 7c, the distance-based method, LOF, showed a higher type I error in the normal section than other models. Despite the simplicity of this case, such false alarms can confuse operators, reducing model reliability. As shown in Figure 7, all fault detection models successfully detect faults, as indicated by a type II error of zero in all six models. Among them, AASK demonstrates lower false alarms and no miss detections, outperforming the conventional AAKR. When comparing the average type I and II errors for faults 1 to 6, kNN (type I = 8.75, type II = 18.44) and LOF (type I = 5.73, type II = 28.08) need improvement in both false alarm and miss detection rates. The performance of AAKR (type I = 2.09, type II = 26.63) and AASKR-NP (type I = 1.57, type II = 26.75) is similar to that of the proposed method. However, the proposed method outperforms the others, with an average type I error of 0.84 and type II error of 26.48, which are the lowest among the models. Figure 6: Control structures of TEP [38]. The TEP fault cases from 1 to 6 involve irregularly shaped faults with random variation. These faults result from changes in flow rate, reactor, and condenser cooler temperatures. Accordingly, Figure 8 shows greater variability in the faults compared to Figure 7. In Figure 8a, PCA calculates the largest type II error among all models. Although kNN and LOF calculate low type II errors, their high type I errors make them unsuitable as reliable fault detection models. On the other hand, AASKR, as shown in Figure 8f, calculates both lower type II errors and fewer false alarms compared to the other models. Discussing the average type I and II errors for faults 8 to 12, PCA calculates a high miss detection rate, with an average type II error of 38.23, the largest among all models. This performance degradation is primarily due to the smearing effect in complex industrial processes, where fault signatures are distributed across multiple principal components.