## 2 Materials and Methods
### Hyperparameter Optimization  
Model hyperparameters are critical factors that directly influence the behavior and performance of machine learning models. The process of hyperparameter tuning plays a significant role in determining the success of a model. However, many studies in the literature have typically relied on the default parameter values of machine learning methods, with limited in-depth investigation into the effects of hyperparameter tuning on model performance. This study employs Bayesian optimization to determine the optimal model parameters. Bayesian optimization facilitates a more efficient search process by considering the results of previous trials, thereby enabling the identification of the best hyperparameter combinations. This method allows for more accurate results with fewer trials, making the optimization process more efficient. Additionally, narrowing the search space aims to reach the optimal solution more quickly. This approach has proved to be highly effective in enhancing the model's overall performance and reducing computational costs.  
In contrast, traditional methods such as manual search, grid search, and random search are commonly used. However, these methods can be computationally expensive and time-consuming when applied to models with many hyperparameters or when evaluating each hyperparameter set, which is costly. Bayesian optimization, on the other hand, offers a more efficient alternative by directing the search process through probabilistic models. This method bases the model's performance on a probabilistic function and iteratively improves the search space, allowing for the determination of optimal hyperparameters with fewer evaluations. As a result, the computational burden is reduced and the hyperparameter tuning process is expedited, making Bayesian optimization a preferred option, especially for complex and high-dimensional machine learning problems [81]. The optimization process follows these steps:  
1. Selection of Initial Points: Several random combinations of parameters are initially selected.
2. Model Training: The model is trained with the selected hyperparameters, and predictions are made on the test data.
3. Acquisition Function Maximization: The acquisition function selects new hyperparameters.
4. Iteration: These steps are repeated until the optimal set of parameters is found.  
In this study, hyperparameter optimization was conducted with two objectives. The first objective was to identify the most successful predictive model by determining the optimal combinations of hyperparameters yielding the highest R values under ideal conditions. The second objective was to identify the hyperparameter combinations in suboptimal conditions, where the models exhibit the poorest predictive performance, and to ensure that the model still achieves high accuracy in this scenario. In this way, the model's performance was tested for high accuracy, consistency, and stability in its results.  
The _BayesianOptimization_ function was used in the Python environment to develop this model, with default parameters applied for the acquisition function and kernel type. The parameters used for the models, the ranges of tested parameter values, and the hyperparameters determined for both optimal and suboptimal conditions are presented in Table 4. The parameter ranges of the models were kept broad to reduce the risk of overfitting and achieve more robust results. Broad parameter ranges allow the model to be effectively tested with diverse parameter combinations, thereby contributing to the most accurate results.