## 2 Methodology
### Methodology for Machine Learning-Based Risk Prediction  
This methodology describes the process of modeling and analyzing risk probability prediction through different phases of dataset development and model optimization. The initial dataset consisted of 28 samples, each containing three input variables (complexity coefficients) and seven output variables (probabilities of different risk groups). Given the small dataset size, overfitting was identified as a primary concern, particularly for complex models. Additionally, the input and output values were continuous numerical variables with limited variance, which could restrict the ability of machine learning models to detect meaningful patterns. To better understand the relationships between inputs and outputs, a correlation analysis was performed using Pearson's correlation coefficient. A correlation matrix was generated, displaying values ranging from \(-1\) to \(1\). Higher values, close to \(1\), indicated a strong positive correlation, meaning that an increase in an input variable led to an increase in the corresponding output variable. Conversely, values close to \(-1\) signified a strong negative correlation, suggesting that higher complexity might reduce the probability of specific risks. Values near \(0\) indicated no significant linear relationship between the variables. Given the dataset's limited size, several challenges arose. One of the main challenges was the risk of overfitting, where more complex models might memorize the dataset instead of learning generalizable patterns. Another issue was model instability, meaning that predictions could vary significantly when applied to new data, leading to unreliable outcomes. Additionally, standard validation techniques, such as k-fold cross-validation, could not be effectively applied due to the small number of samples, limiting the ability to assess model performance accurately. To mitigate these challenges, a progressive modeling approach was adopted. Initially, simpler models were tested to establish baseline performance before introducing more complex techniques. Linear regression was the first model to assess whether a dominant linear relationship existed between inputs and outputs. If successful, this would indicate that more complex models were unnecessary. The next step involved testing the Random Forest model, which is well suited for capturing nonlinear relationships. However, due to the small dataset size, it was expected that this model might struggle to generate meaningful patterns. Finally, a basic single-hidden-layer neural network was tested. While neural networks excel at capturing complex relationships, they require substantial amounts of data to train effectively. Given the dataset constraints, it was anticipated that the neural network would underperform compared to simpler models. If linear regression demonstrated strong performance, it would suggest that the dataset predominantly contained linear relationships. If Random Forest showed improvement over linear regression, it would indicate the presence of significant nonlinear relationships. On the other hand, if the neural network performed poorly, it would confirm that the dataset was too small to support deep learning models effectively. To address dataset limitations, a data expansion technique using Multivariate Copula Models was employed. This method increased the dataset size from 28 to 201 samples while preserving the statistical distribution of the original dataset. After expansion, a distribution analysis was conducted to verify that the newly generated data maintained the same statistical properties as the original dataset. Following dataset expansion, all variables were standardized to ensure uniform scaling across features. The dataset was split into training and test sets to evaluate model performance on unseen data. The first step in model evaluation involved testing linear regression to determine whether a dominant linear relationship existed.