## 5 Experiments
### Robustness Analysis  
A further analysis was conducted to evaluate the robustness of the final model in the presence of missing data, which is a common condition in real datasets. This evaluation was conducted by simulating the introduction of increasing proportions of missing values into the dataset, with a uniform random distribution both over time and between variables. The objective of this test was to assess the impact of increasing missing data on the quality of the synthetic data generated, using the same evaluation metrics employed in the previous analyses. Therefore, the goal is to identify the operational limits of the model, establishing the threshold beyond which the considered architecture is no longer capable of generating data with an acceptable quality. Figures 6 and 7 show the performance of two main metrics, the KS statistic and the discriminative score, as a function of the variation in the percentage of missing values, from 0% to 60%. These are considered for the final model and the model with the GRU layers but employing traditional latent sampling. The results show a progressive deterioration in the performance of the final model as the percentage of missing values increases, highlighting the difficulty of preserving patterns and temporal relationships under conditions of incomplete data. However, the final model shows greater robustness than approaches based solely on traditional sampling techniques. In particular, the discriminative score of the model with MCMC sampling exhibits less deterioration, remaining up to 30% missing values at levels similar to those of the model with no missing values, and then converging to the scores observed in the basic GRU model. Regarding the KS test, the increase in the value of the statistic is less pronounced in the final model with MCMC sampling than in the GRU model with traditional sampling. Figure 5: T-SNE plot of the VAE-GRU with MCMC model. In addition to the robustness of the synthetic data generated, assessing the model's ability to handle the presence of missing data, it is also important to evaluate the capacity to augment the dimensionality of the original dataset. Data augmentation is a fundamental technique, especially when employing analysis and prediction models in scenarios with a limited availability of data for training. In this context, generative models can support  
Figure 6: KS test of the VAE-GRU and VAE-GRU with MCMC models, across a different percentage of missing values. Figure 7: Discriminative score of the VAE-GRU and VAE-GRU with MCMC models, across different percentages of missing values. the creation of significantly larger datasets than the original ones, thereby enabling more robust training of machine learning models. In detail, the final model developed, a VAE with GRU layers and MCMC sampling, has been used to progressively generate an increasing volume of synthetic data. This strategy has been supported by the examination of the advancement in predictive performance, quantified by the predictive score, which assesses the model's capacity to accurately predict real data when trained on synthetic data. In addition to the scenarios previously analysed, in which the number of synthetic samples was equal to that of the real data, further experiments have been conducted with the generation of synthetic datasets comprising 4000, 5000, and finally, 7000 observations, which is almost equal to twice the original dimensionality. Figure 8 illustrates the scenario mentioned above, showing the trend of the predictive score concerning the generation of synthetic data with progressively increasing dimensionality. From this, it can be observed that the predictive score remains relatively consistent in comparison to that obtained with a dimensionality equal to that of the original dataset. However, a reduction in the score is evident in the samples comprising 5000 and 7000 units.