## 3 Results  
Initially, an analysis was carried out to ascertain the optimal activation function for the hidden layers. The effectiveness of the linear and \"softplus\" activation functions was assessed. The implementation of the \"softplus\" activation function yielded faster and superior convergence. The \"softplus\" activation function demonstrated a convergence rate of 92%, while the linear activation function exhibited a convergence rate of 74%. Furthermore, the loss function's final value was thrice higher when employing the linear activation function in contrast to the \"softplus\" activation function. The next step of the analysis was the assessment of the activation function employed in the output layer. The \"softplus\" and linear functions were evaluated as potential activation functions. The aim of the study was to assess the degree of similarity between the generated data and the bell-shaped distribution of the source data. The outcomes are depicted in Figure 3.  
As illustrated in Figure 3, the data generated for both R and T groups exhibited a bell-shaped distribution when a linear activation function was used for the output layer (Figure 3b). It was observed that the utilization of a linear activation function for the output layer resulted in a more optimal distribution as opposed to the \"softplus\" activation function, which led to a right-skewed distribution (Figure 3a). Overall, it appears that the utilization of the \"softplus\" activation function in the output layer yields an exponential  
Figure 2: Schematic representation of the analysis strategy in this study. Initially, two randomly generated datasets were generated for the test (T) and reference (R) groups. Then followed subsampling to draw parts of the original population. Finally, the variational autoencoder was applied to the subsampled data in order to produce the generated datasets. The aim of the generated datasets was to exhibit the same properties as the original data. In this study, comparisons were made among the three datasets (original vs. subsampled vs. generated), as well as between the T and R groups of all datasets. like distribution for the resultant data, whereas the adoption of a linear activation function generates data with a bell-shaped distribution. The process of optimizing the number of epochs for model training was also executed. The values of 100, 500, 1000, 5000, and 10,000 underwent testing. Each of the variables mentioned above was used to train a VAE model and, after that, the trained model was used to generate data for the R and T groups. The results are illustrated in Figure 4.  
Figure 4 presents a graphical representation that reveals a considerable degree of variance in the data over 100 epochs (Figure 4a). Moreover, the data were not centered around the actual mean of both the R and T groups, which was 100. Similarly, this pertained to the situation in which 500 epochs were employed (Figure 4b). On the other hand, it was observed that increasing the number of epochs beyond 1000 made a significant difference in how well data were centered around the mean (which was 100) and how well variance  
Figure 4: Distribution of the generated data for both the R and T groups using variational autoencoders with 100 (**a**), 500 (**b**), 1000 (**c**), 5000 (**d**), and 10,000 (**e**) epochs. Figure 3: Distribution of the generated data for both R and T groups using the “softplus” (**a**) and linear (**b**) activation functions for the output layer. was placed within the desired range (Figure 4c). A similar pattern was further observed when the number of epochs was increased to 5000 and 10,000 (Figure 4d,e, respectively). Finally, an investigation was carried out to ascertain the optimal number of hidden layers for both the encoder and decoder components. The investigation analyzed the values of 2, 3, and 4 as the number of hidden layers in both the encoder and decoder. As a consequence, overall numbers of 4, 6, and 8 hidden layers were evaluated, correspondingly.