## 2 Materials and Methods
### Limit-Data Generating Process
#### 2.2.1 Synthetic Minority Oversampling Technique-Nominal Continuous (SmoteNC)
(**B**) Tool wear failure (TWF) Minority class 0.5%. (**C**) Heat dissipation failure (HDF) Minority class 0.9%. (**D**) Power failure (PWF) Minority class 1.2%. (**E**) Overstrain failure (OSF) Minority class 1.0%. (**F**) Random failures (RNF) Minority class 0.2%. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Round** & **Total** & **Training Set** & **Test Set** \\ \hline Each Round & 10,000 (100\%) & 6700 & 3300 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The number of training and testing data sets after each round of segmentation. The severe imbalance in the collected data set increased the difficulty of obtaining accurate model predictions. The prediction results obtained for the TWF, HDF, PWF, OSF, and RNF diagnoses with the confusion matrix and other methods for processing limited data were compared. According to the results presented in Tables 4-13, the experimental results of five failures are presented. The proposed method exhibited a higher recall rate and balanced accuracy than did the other methods for the diagnoses of all types of failure. Perfect recall was achieved with the proposed method in PWF diagnosis and OSF diagnosis, and balanced accuracy achieved the best result at 98.77% and 98.05%, respectively. Although some false positive results were obtained using the proposed method in the aforementioned fault diagnoses, its performance was acceptable. For the TWF diagnosis, a balanced accuracy of 91.02%, which was 17.08% higher than the second-performing diagnosis, was observed, and the accuracy was only behind by 2.45%. For the HDF diagnosis, a balanced accuracy of 97.61% was observed, and other recall and accuracy data were higher than the second-performing ctGAN + CatBoost. For the RNF diagnosis, a balanced accuracy, recall rate, and accuracy of 63.84, 85.71%, and 42.06%, respectively, were achieved using the proposed method. This kind of random failure has no exact failure type, which makes it difficult to grasp. In addition, SmoteNC + CatBoost exhibited excellent performance in the HDF and OSF diagnoses, but performed unsatisfactorily in the PWF and RNF diagnoses. ctGAN + CatBoost exhibited favorable performance in the HDF and OSF diagnoses, but performed unsatisfactorily in the PWF and RNF diagnoses. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Failure Mode** & **Total** & **Training Set** & **SMOTE-NC** & **ctGAN** \\ \hline TWF & 19,658 & 6700 & 6673 & 6700 \\  & (Contains 27 Failure) & (Failure) & (Failure) & (Failure) \\ HDF & 19,658 & 6700 & 6620 & 6700 \\  & (Contains 80 Failure) & (Failure) & (Failure) \\ OSF & 19,658 & 6700 & 6638 & 6700 \\  & (Contains 62 Failure) & (Failure) & (Failure) \\ PWF & 19,658 & 6700 & 6640 & 6700 \\  & (Contains 60 Failure) & (Failure) & (Failure) \\ RNF & 19,658 & 6700 & 6688 & 6700 \\  & (Contains 12 Failure) & (Failure) & (Failure) \\ \hline Machine Failure & 19,658 & 6700 & 6479 & 6700 \\  & (Contains 221 Failure) & (Failure) & (Failure) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The number of samples generated by SMOTE-NC and ctGAN. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline echo & 10 \\ Size of the output samples & Generator: (256,256) \\ Discriminator: (256,256) \\ Optimizer & Adam \\ Learning Rate & 0.0002 \\ Loss Function & lower-bound (ELBO) loss \\ Activation & ReLU \\ Number of generated failure data & 6700 (Same as the number of failures in the training set) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameter settings for the ctGAN. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Iterations & 50 \\ Depth & 6 \\ Learning rate & 0.18176 \\ Early stopping rounds & 10 \\ Bagging temperature & 0.8278 \\ Iterations & 50 \\ Depth & 6 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameter settings for CatBoost (results of parameter optimization based on the Optuna method).