## 3 Case Study
### TEP Dataset  
#### 3.1.1 Datasets Description  
The Tennessee Eastman process (TEP), detailed by [PERSON] and [PERSON] [30; 31], is a comprehensive dataset widely studied in the diagnosis of chemical process faults. The process schematic is depicted in Figure 2. TEP comprises five main subsystems: a reactor, a condenser, a vapor-liquid separator, a recycle compressor, and a product stripper, among others. Each time-series sample in the TEP dataset contains 52 features, including 41 process measurements and 11 manipulated variables, recorded every three minutes. The TEP dataset includes a normal operation set with 500 samples and 21 fault conditions in the training subset, each with 480 samples. Correspondingly, the testing subset contains 960 samples per condition, with fault information being injected after 8 h of operation.  
Because of the scant descriptions of the last six fault types in the dataset, only the first 15 fault categories are applied in the GOPML fault diagnosis. Table 1 elaborates on the 15 fault categories. The 15 scrutinized faults in TEP are unique from each other. Among these, Figure 3 displays the visualization of certain variables from normal samples and IDV2 samples in the training subset. Conventional diagnostic methods face challenges when there are no samples available for training the model for certain faults. Thus, the proposed GOPML approach to fault diagnosis is notably significant and practically applicable.  
To align with the preprocessing requirements of the GOPML algorithm, the TEP dataset was methodically segmented using a sliding window methodology. The process utilized the windows of 40 consecutive data points, each reflecting a 2 h operational period, with a step size of 20 points to maintain temporal consistency. This arrangement ensured that each window adequately captured the complex dynamics of the process for a comprehensive learning experience. The GOPML algorithm then utilized this structured data to identify patterns and anomalies within the process. In total, each of the 15 fault categories was divided into 23 windows, yielding a collective 345 unique intervals for analysis. Additionally, for the testing subset, the initial 8 h of normal operational data were excluded, focusing the analysis on the subsequent samples where anomalies were introduced.  
Figure 2: Flow chart of the Tennessee Eastman Process.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable** & **Fault State** & **Disturb** \\ \hline IDV1 & A/C feed ratio, B composition & Step change \\  & constant & Step change \\ IDV2 & B composition, A/C ratio & Step change \\ IDV3 & D feed temperature & Step change \\ IDV4 & Reactor cooling water inlet & Step change \\  & temperature & Step change \\ IDV5 & Condenser cooling water & Step change \\ IDV6 & A feed loss & Step change \\ IDV7 & C header pressure loss & Step change \\ IDV8 & A, B, C feed composition & Random variants \\ IDV9 & D feed temperature & Random variants \\ IDV10 & C feed temperature & Random variants \\ IDV11 & Reactor cooling water inlet & Random variants \\  & temperature & Random variants \\ IDV12 & Condenser cooling water & Random variants \\  & valve & Random variants \\ IDV13 & Reaction kinetics & Slow drift \\ IDV14 & Reactor cooling water valve & Sticking \\ IDV15 & Condenser cooling water & Sticking \\ \hline \hline \end{tabular} This table lists the 15 fault types identified in the Tennessee Eastman Process (TEP) that were used to evaluate the GOPML algorithm.  
\end{table}
Table 1: Fault types in the Tennessee Eastman Process evaluated by GOPML.  
Figure 3: The visualization of certain variables from normal samples and IDV2 samples in the training subset.