## 3 Results
### Evaluation Metrics
Conversely, the synthetic datasets produced by CTGAN and TVAE show slight deviations from the real cumulative function, characterized by a subtle sinusoidal pattern. The quantitative results of the fidelity assessment, as measured by the Kolmogorov-Smirnov (K-S) statistic, are presented in Table 4, with the lowest values highlighted in bold. For the prostate cancer dataset, the synthetic datasets generated by CopulaGAN and TVAE demonstrate the smallest differences between the real and synthetic distributions, indicating superior fidelity for this case. Conversely, for the breast cancer data, the UMAP-generated and CTGAN synthetic datasets yield better fidelity results. Notably, the UMAP-based synthetic data achieve the lowest K-S statistic for most attributes in the breast cancer dataset, emphasizing their ability to closely replicate the real data distributions in this scenario. The utility assessment evaluates the performance of three machine learning models--random forest classifier, logistic regression, and support vector machine--in predicting the target variable for each dataset (case_csPCa, Class, or target). Each model is trained on three types of datasets: real, synthetic, and augmented (a combination of real and synthetic data). For the synthetic and augmented cases, the training incorporates datasets generated using UMAP-based methods, CTGAN, CopulaGAN, and TVAE, resulting in nine different training datasets that will lead to different results. The results of the random forest classifier are presented in Figure 11. For the prostate cancer data (Figure 11a), the UMAP-generated and TVAE synthetic and augmented datasets show superior performance compared with training on real data alone or training with CTGAN and CopulaGAN synthetic and augmented datasets. For the breast cancer data (Figure 11b), the performance enhancement provided by TVAE is less pronounced. However, UMAP synthetic and augmented datasets exhibit better results than real data across all metrics, except for AUC. In this specific case, only the augmented UMAP and TVAE datasets match the AUC performance achieved by real data. In the case of the cardiovascular data (Figure 11c), none of the synthetic or augmented datasets surpass the performance of the real dataset. Additionally, the CTGAN-generated dataset demonstrates particularly poor performance metrics for this scenario. Figure 11: Cont. The performance metrics for the logistic regression models, as displayed in Figure 12, reveal distinct patterns across the datasets. In the prostate cancer case (Figure 11(a)), while UMAP- and TVAE-generated datasets outperform CTGAN and CopulaGAN, they do not consistently exceed the performance of models trained on real data. For breast cancer data (Figure 11(b)), CopulaGAN-generated datasets demonstrate improved performance compared with previous observations, particularly in the F1-score metric. Additionally, both CopulaGAN- and UMAP-generated datasets slightly outperform the real dataset in accuracy. In the cardiovascular diseases scenario (Figure 11(c)), similar to the random forest results (Figure 10(c)), none of the synthetic datasets outperform the model trained with real data. Support vector machine (SVM) results, presented in Figure 13, provide further insights. For prostate cancer data (Figure 12(a)), synthetic and augmented datasets perform similarly, slightly exceeding the performance of real data across most metrics. Notably, the F1-score demonstrates a significant improvement with synthetic datasets. In the breast cancer case (Figure 12(b)), no synthetic dataset clearly outperforms the real data model across all metrics. However, subtle improvements in the F1-score are observed for some synthetic and augmented datasets, and augmented TVAE data show a marked enhancement in AUC performance.