## 3 Results
### UMAP Transformation and Feature Distribution  
The first phase of the methodology relies on transforming the input data into a lower-dimensional space using UMAP. This transformation provides a two-dimensional representation that facilitates the visualization of feature distributions within the datasets. Such visualizations are critical for validating the quality of the synthetic data generated. However, UMAP is known to be sensitive to its initialization and hyperparameters, which can significantly affect the resulting cluster structure and data distribution in low-dimensional space. To address this sensitivity, we performed a detailed sensitivity analysis using prostate, breast, and cardiovascular datasets. UMAP's key parameters include n_neighbors, min_dist, and random_state. The n_neighbors parameter controls the balance between local and global data structure, determining the size of the neighborhood UMAP considers when learning the data manifold. The min_dist parameter controls how closely UMAP packs points in the low-dimensional representation, effectively setting a lower bound on the distance between points. Finally, the random_state parameter affects the randomness of the initialization, allowing reproducibility when set to a fixed value. To assess the impact of these parameters, we performed 50 UMAP transformations for each dataset, varying the values of n_neighbors, min_dist, and random_state. Specifically:  
* n_neighbors parameter was randomly assigned integer values between 2 and 50. * min_dist parameter was sampled from a uniform distribution between 0.0 and 0.9. * random_state was randomly chosen as an integer between 0 and 10,000. Each combination of these parameters resulted in a unique UMAP transformation, producing different data distributions in two-dimensional space. To quantify the variability across these transformations, we computed the cluster centroids for each class in each dataset and calculated the variance in their coordinates across the 50 transformations. The results, presented in Table 2, show considerable variability in the centroid positions, with the smallest variance being 18.68. This significant variation highlights the sensitivity of UMAP to initialization and hyperparameters, which can lead to inconsistent cluster structures depending on the settings chosen. To further illustrate this variability, Figure 4 displays 10 random UMAP transformations for each dataset. These visualizations show that, while some transformations maintain clear cluster separation, others fail to preserve the underlying structure of the data. For example, in several transformations, the two clusters remain clearly separated, but their spatial arrangement varies significantly. In other cases, suboptimal initializations result in poor cluster definition, compromising the interpretability of the representation. This sensitivity analysis highlights the importance of carefully selecting UMAP parameters to ensure meaningful and reproducible results, especially when used for synthetic data validation and clustering tasks. Figure 5 illustrates the UMAP-transformed representations of three different datasets for the appropriate initialization. For these datasets, the chosen parameters are n_neighbors\(=15\) and min_dist\(=0.1\). The random_state values were set to 23 for the prostate cancer data, 80 for the breast cancer data, and 76 for the cardiovascular disease data to ensure reproducibility. Specifically, Figure 4(a) displays the UMAP transformation of the PI-CAI prostate cancer dataset, highlighting how samples are grouped according to the target variable, (case_csPCA). Similarly, Figure 4(b) presents the UMAP visualization of the BC-MLR breast cancer dataset, showing distinct clusters that correspond to the target feature, Class. Finally, Figure 4(c) depicts the UMAP-transformed Ind-CardiOB cardiovascular dataset, where samples are also distinctly grouped based on the target variable.