## 2 Materials and Methods
### Validation Apparatus
This back-validation approach tested whether the generator could accurately reproduce the distributional and correlational properties of the source datasets, effectively simulating real populations from aggregate-level data. Three cohorts were selected to test generalizability across different populations and clinical profiles. A general population cohort was drawn from NHANES (2021-2023) to represent a heterogeneous patient pool and was used to test baseline performance in a broad population and included individuals aged 8 to 80 with continuous variables only [18]. A cardiovascular cohort was drawn from the Framingham Heart Study to assess a pool of patients with a specific comorbidity rather than the general population and included patients with diagnoses such as hypertension, angina, or myocardial infarction. This dataset contained both continuous and categorical variables [19]. A third cohort, referred to as the low-dimensional group, consisted of NHANES patients with only four input variables: age, sex, body mass index, and systolic blood pressure. This group was used to test the algorithm with a minimal number of individual characteristics. To evaluate performance under real-world limitations in data availability, each cohort was simulated under five data-availability conditions. The first scenario used all available parameters from real data, including means, bounds, and a complete correlation matrix. This served as the reference condition. The second scenario replaced the correlation matrix with estimates made by clinical experts who followed the specifically prepared reference guide (Appendix D). Two experts independently reviewed anonymized summaries of each cohort and provided estimated pairwise correlations. Their average formed the correlation matrix. In the third scenario, the correlation structure was derived from published literature or internal reference libraries [17]. The fourth scenario combined literature-based correlations with physician input to fill missing or inconsistent values. The fifth scenario omitted the correlation matrix entirely, allowing the model to generate uncorrelated variables mimicking the situation of not considering correlations while generating synthetic cohorts. Each simulated dataset was evaluated using standard descriptive statistics assessing the central tendency and dispersion. For each parameter, the mean and standard deviation were calculated for numerical variables, while the count and percentage were determined for categorical variables; these values were then compared to their counterparts in the original dataset (Appendix E). In addition, a correlation comparison plot was generated to visualize the structure of the synthetic data relative to the original cohort. Validation focused on the agreement between the simulated and real correlation matrices. Five quantitative metrics were used. Mean absolute error (MAE) measured the average unsigned difference between simulated and target correlation coefficients. Mean squared error (MSE) measured the average of the squared differences, penalizing larger deviations. Root mean squared error (RMSE) was the square root of MSE and preserved the original units of measurement. The coefficient of determination denoted \(R^{2}\), quantified the proportion of variance in the target correlations that was captured by the simulated ones. Sum of squared errors (SSE) represented the total squared deviation across all correlation pairs. Below are the equations for the metrics:  
\[\text{MAE}=\frac{1}{n}\underset{i=1}{\overset{n}{\sum}}|\hat{\rho_{i}}\ -\ \rho_{i}|\]  
\[\text{MSE}=\frac{1}{n}\underset{i=1}{\overset{n}{\sum}}(\hat{\rho_{i}}-\rho_{ i})^{2}\]  
\[\text{RMSE}=\sqrt{MSE}\]  
\[\text{R}^{2}=1-\frac{\underset{i=1}{\overset{n}{\sum}}(\rho_{i}-\hat{\rho_{i }})^{2}}{\underset{i=1}{\overset{n}{\sum}}(\rho_{i}-\overline{\rho}\ )^{2}}\]  
\[\text{SSE}=\underset{i=1}{\overset{n}{\sum}}(\hat{\rho_{i}}-\rho_{i})^{2}\]  
where \(\overline{\rho}\) is the mean of reference correlations.