## 5 Experiments
### Comparison with Baselines  
As described in Section 2, [13] illustrate the results of four models representing the state of the art in synthetic sequential data generation. The models used to calculate these metrics are identical to those used in this study; however, the data refer to Google's actions observed over a different time interval, a 10-year period. In addition, the models were evaluated using 100% of the data for training, in contrast to the VAE-GRU model with MCMC sampling, which used 90%. Despite these minor differences, it is possible to make a meaningful comparison with the model developed in this work, which shows a very good placement in both the discriminative and predictive scores, being competitive and distant only from the TimeVAE and TimeGAN models, which represent two particularly sophisticated approaches in the field. The analysis of the results shows that the use of 90% of the data for training, compared to 100% as in TimeGAN, did not significantly affect the performance of the model. As per Tables 3 and 4's metrics, including discriminative score, predictive score, and the KS test, they remained almost unchanged. The visual representation using t-SNE as per Figure 1 also shows a similar overlap between synthetic and original data, regardless of the portion of data used for training. This emphasises that the model retains its robustness even with a reduction in the training set, thus confirming the quality of the synthetic data generated. Statistical tests provide a formal approach to assessing the quality of synthetic data, allowing a direct comparison of the statistical and temporal properties of the generated samples with those of real data. This enables the verification of whether the generative model has succeeded in preserving the essential characteristics of the time series. Two fundamental statistical tests were considered:  
* The KS test is a non-parametric statistical technique used to determine the maximum distance between the cumulative distribution functions (CDFs) of two datasets. In this case, the datasets are the real and synthetic data. The objective is to determine whether the synthetic data follow the same statistical distribution as the real one. * The autocorrelation function is a statistical measure that assesses the degree of dependence of a time series on its past values. The test allows the comparison of temporal patterns between real and synthetic data by analysing the consistency of sequential dependencies. The aim is therefore to determine whether the synthetic data preserve the inherent temporal relationships present in the real data. Table 5 presents the results of the KS test, which is used to ascertain the maximum distance between the cumulative distributions of the synthetic and real data for the three models under consideration. A lower value of the KS test indicates a greater similarity between the two distributions. The results for the models with convolutional and GRU layers demonstrate a notable divergence between the two distributions, suggesting a limitation in accurately capturing the underlying data distribution. In contrast, the model with MCMC sampling integration exhibits a significantly lower KS test value than the first two models, indicating a smaller discrepancy between the synthetic and real data distributions and a greater capacity to accurately represent the underlying characteristics. Figure 2 illustrates the difference of the autocorrelation function (ACF) between real and synthetic data for the three models across a range of lag values. The model with convolutional layers has been demonstrated to be effective for low lags but shows difficulties in capturing long-term correlations, highlighting a limited ability to model the extended temporal relationships present in the real data. In contrast, the VAE model with GRU exhibits difficulties in the short term but the results are more effective in representing long-term correlations.