## 2 Materials and Methods
### Limit-Data Generating Process
#### 2.2.1 Synthetic Minority Oversampling Technique-Nominal Continuous (SmoteNC)
First, the minority class features exhibited by the generated minority data samples depend on the condition vector and sample training method. These data are different from those obtained solely by simulating the distance between minority class data. Second, in the proposed method, a discriminator is used to verify whether the simulated minority class samples are representative of the real minority class samples. #### 2.2.3 Data Combination

In this study, the simulated and actual fault data are integrated, with the aim of reaching a balance between fault and normal operation data. There are three sources of integration. The first is the real operation data, which contains the fault and normal operation data. Second, the fault data generated by SmoteNC simulation can greatly increase the data for a very small number of fault samples, but the authenticity may not be enough. Finally, the equipment failure data simulated by ctGAN are closer to the real data after being screened by the discriminator. However, the disadvantage is that this requires a variety of training data. Therefore, we merged the real data and the two simulated data, so that the fault and normal operation data are balanced (Algorithm 3). ```
Input: Training set \(T\), which contains failure dataset (minority class) \(S\) Output: Synthesized failure dataset \(T_{new}\)
1. \(S_{new}\leftarrow\textit{SmoteNC}(T)\)
2. \(G_{new}\gets\textit{ctGAN}(S,\textit{Snew})\)
3. \(T_{new}\gets\textit{T}\cup\textit{Snew}\cup\textit{Snew}\)
4. Return \(T_{new}\)
```

**Algorithm 3**SmoteNC\(\leftarrow\)tGAN (Pseudocode)

Finally, we present the synthetic training dataset (\(T_{new}\)):

\[T_{new}=T\ \cup\ S_{new}\ \cup\ G_{new} \tag{3}\]

### Model Learning and Applications

In the model learning and application stage, the effects of simulated and real failure data in equipment failure diagnosis are verified. Two classification tasks are performed in this stage. The first task involves classifying the failure diagnosis results as failure or nonfailure data. The second task involves multicategory classification, where the fault diagnosis results are further classified according to the type of failure, such as tool wear failure (TWF), heat dissipation failure (HDF), power failure (PWF), overstrain failure (OSF), and random failure (RNF). Since obtaining equipment operation data is difficult, insufficient data are collected for deep learning in most studies [4; 25]. Therefore, in this study, the CatBoost classifier [26], which is a popular classifier, was employed to classification tasks. The following section details this classifier and the Optuna hyperparameter learning method [27]. #### 2.3.1 CatBoost Classifier and Optuna

CatBoost, which is an ensemble-learning algorithm based on a gradient boosting decision tree, employs an ordered boosting algorithm and a greedy algorithm to solve problems related to iterative gradient descent [15] and to reduce the risk of overfitting. To increase model accuracy and enhance model performance, the Optuna hyperparameter learning method [16] was adopted in the present study. The training parameters in this method are detailed in Section 3. #### 2.3.2 Model Evaluation

In equipment fault diagnosis, the rate of true positives is considerably higher than that of true negatives. Furthermore, a high false positive rate results in the constant triggering of a failure alarm, which decreases user confidence in a fault diagnosis model. Therefore, in this study, recall rate, accuracy [28], and balanced accuracy [29] were selected to evaluate the proposed model. In addition, balanced accuracy is an overall indicator for a small number of fault data.