## 3 Generating Synthetic Bearing Fault Data via cycleGAN
### Validation of the CycleGAN Generative Model  
The validation of the synthetic data generation methodology presented in this study involved the utilization of the MMD metric [83], the Frechet Inception Distance (FID) [84; 85] and a specifically designed TL methodology. \begin{table}
\begin{tabular}{c c} \hline Epochs & 200 \\ Mini-batch size & 1 \\ Optimizer & Adam \\ Learning rate & 0.0002 \\ Gradient decay factor & 0.5 \\ Squared gradient decay factor & 0.999 \\ Adversarial loss weight (\(\lambda\)) & 10 \\ \hline \end{tabular}
\end{table} 
Table 2: cycleGAN training hyperparameters. Figure 4: Images of CWT spectra @ 877 rpm: (**a**) simulated signal; (**b**) experimental signal; (**c**) image generated by cycleGAN. The MMD is a non-parametric statistical metric used to quantify the dissimilarity of two probability distributions. The empirical estimate of the MMD is calculated using Equation (10), where \(\phi(\cdot)\) represents the characteristic kernel of the nonlinear mapping function, and \(X=\{x_{i}\}_{i=1}^{N}\) and \(Y=\{y_{j}\}_{j=1}^{M}\) are two given datasets. In this study, a radial basis function (RBF) kernel [86] is employed. Similarly, the FID metric is widely employed in GANs to measure how distant the images produced by generative AI are from reality. Specifically, FID employs the components of the latent space formed by the Inception Net-V3 [87] for the purpose of computing the statistic specified in Equation (11), where \(\mu_{r}\) is the mean value of the real image, \(\mu_{g}\) is the mean value of the generated image, \(\Sigma_{r}\) is the covariance matrix of the real image, \(\Sigma_{g}\) is the covariance matrix of the generated image and \(Tr\) represents the trace. Within the framework of this research, the MMD and FID metrics were employed to assess the discrepancy between probability distributions pertaining to simulated data, real data, and synthetic data. This enables the measurement of the degree of similarity that arises in a qualitative manner (as shown in Figures 4-6) between the real and synthetic data. Consequently, decreased MMD and FID values signify a higher degree of similarity between the distributions, thereby implying that the synthetic data could potentially serve as valid alternatives to the experimental data. Table 3 presents the computed values of the MMD and FID for different machine conditions. The data generated by cycleGAN exhibit a higher degree of resemblance to the authentic data in comparison to the simulated data. This statement is true for both  
Figure 5: Images of CVT spectra @ 607 rpm: (**a**) simulated signal; (**b**) experimental signal; (**c**) image generated by cycleGAN. Figure 6: Images of CVT spectra @ 997 rpm: (**a**) simulated signal; (**b**) experimental signal; (**c**) image generated by cycleGAN. the explicit training condition of the generative algorithm (877 rpm) and the two distinct operational conditions (607 rpm and 997 rpm). \[M\MD^{2}(X,\ Y)=\left\|\frac{1}{N}{\sum_{t=1}^{N}\phi(x_{t})}-\frac{1}{M}{\sum_{ j=1}^{M}\phi(y_{t})}\right\|^{2}, \tag{10}\]  
\[FID=\left\|{\mu_{r}-\mu_{g}}\right\|^{2}+Tr(\Sigma_{r}+\Sigma_{g}-2\big{(} \Sigma_{r}\Sigma_{g}\big{)}^{\frac{1}{2}}). \tag{11}\]  
In order to evaluate the effectiveness of synthetic data as substitutes for real data, a series of training experiments were performed on three distinct diagnosis models based on the TL approach [49; 50]. Namely, the CNNs AlexNet [88], VGG16 [89], and ResNet18 [90], which were initially pre-trained for image recognition on ImageNet [38], were trained through the fine-tuning process. The initial and intermediate layers, which are responsible for extracting image features that can differentiate between different classes, were kept the same. However, the final layer was replaced, and its weights were adjusted through retraining to the specific diagnostic task. This methodology enables the utilization of the knowledge existing in models trained on image recognition, particularly on a vast dataset like ImageNet [38].