## 2 Materials and Methods
### Limit-Data Generating Process
#### 2.2.1 Synthetic Minority Oversampling Technique-Nominal Continuous (SmoteNC)
A confusion matrix [29; 30; 31] for fault diagnosis evaluation metrics is presented in Table 2. The equations for calculating these indicators are presented in Equations (4)-(6) below:

\[\text{Recall rate}= \frac{\text{TP}}{\text{TP}+\text{FN}} \tag{4}\] \[\text{Accuracy}= \frac{\text{TP}+\text{TN}}{\text{TP}+\text{TN}+\text{FP}+\text{ FN}}\] (5) \[\text{Balanced accuracy}= (\frac{\text{TP}}{(\text{TP}+\text{FN})}+\frac{\text{TN}}{( \text{TN}+\text{FP})})/2 \tag{6}\]

## 3 Results

The results section details the fault data set; the design of relevant model parameters; and the experiment results, including those for recall, accuracy, and balanced accuracy. ### Dataset Description

In this study, the UCI AI4I 2020 Predictive Maintenance Dataset [32] was used to verify the performance of the proposed model [33]. This data set contains 10,000 data points, of which, 3.4% of points represent fault data. Each data point has 12 features, specifically, six equipment operation features and six equipment fault features. These features are detailed in the following text. The six equipment operation features of the data points are as follows:

1. Product ID: Product ID, which represents categorical data, is a key feature used to distinguish the type of product processed and consists of a letter Low (50%), medium (30%), High (20%) as product quality variants. 2. Air temperature: Air temperature, which represents numerical data, refers to the temperature of the environment (between 2 K and 300 K after normalization). 3. Process temperature (K): Process temperature, which represents numerical data, refers to the temperature of the production process. 4. Rotational speed (rpm): Rotational speed, which represents numerical data, refers to the rotational speed of the main shaft. 5. Torque (Nm): Torque represents a type of numerical data and is generally equal to 40 Nm where \(\varepsilon=10\) and no negative values. 6. Tool wear (min): Tool wear, which represents numerical data, refers to the tool operation time. The six equipment fault features of the data points are as follows:
7. Tool wear failure (TWF): Tool wear failure causes a process failure. 8. Heat dissipation failure (HDF): Heat dissipation causes a process failure. 9. Power failure (PWF): Power failure causes a process failure. 10. Overstrain failure (OSF): OSF refers to the failure caused by overstrain in the production process. 11. Random failures (RNF): RNFs are failures whose cause cannot be determined. Their occurrence probability in the production process is 0.1%. 12. Machine failure: The original two-category label (0 represents normal, and 1 represents failure) was changed into a multicategory label (0 represents normal, 1 represents TWF, 2 represents HDF, 3 represents PWF, 4 represents OSF, and 5 represents RNF) to verify the multicategory prediction accuracy of the proposed model. ### Experiment Setting

The experimental method adopts a three-fold cross-validation, and explains the average number of training and testing data sets after each round of segmentation. Please refer to the following, Table 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & & \multicolumn{2}{c}{**Actual Condition**} \\ \hline  & & Failure & Normal \\
**Prediction** & Failure & TP & FP \\
**Condition** & & (True Positive) & (False Positive) \\  & Normal & FN & TN \\  & & (False Negative) & (True Negative) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Confusion matrix for fault diagnosis dataset. ### Parameter Setting

All parameters that are applied to ctGAN and CatBoost are listed in Tables 2 and 3, respectively. ### Experiment Results

Equipment fault data exhibit two types of features: categorical features (related to the product information) and numerical features. Considerable imbalance was observed in the collected data, with the fault data accounting for a small proportion of the collected data (Figure 3, in which 1 and 0 represent fault and normal data, respectively). Figure 3: (**A**) Machine Failure Minority class 3.4%.