## 1 Introduction
In addition, the dataset must be standardized (centering and scaling to have zero mean and unit standard deviation) before training the neural network to ensure consistent scales for all variables [29; 30]. Proper training of the neural network requires the representation of patterns that evenly cover the entire range of variables. To ensure this, the value range of the variables must be determined, and their statistical distribution must be evaluated. The analysis emphasizes the importance of precisely defining the range of independent variables for neural models to avoid errors when extrapolating beyond the range of the training data, especially in multidimensional input domains. Selecting the optimal number of neurons in the hidden layer of an MLP network is about finding a balance between approximation and generalization, with overfitting being a well-studied problem. There are different approaches to determine the optimal number of neurons, often favoring the lowest error value, although the arbitrary application of this criterion may increase the risk of overfitting. The evaluation of neural models relies heavily on a comprehensive test set that adequately represents the full range of the model. Statistical values for the test set, such as the mean absolute error and the correlation coefficient, should match those of the training set and, thus, provide a crucial insight into the quality of the model [1; 31]. Splitting a dataset into different subsets for training and validation is a fundamental aspect of machine learning. It plays a crucial role in various tasks, such as model evaluation, model comparison, and hyperparameter tuning. Common methods, such as holdout, bootstrap, and cross-validation (CV), are often used [32]. In this method, the available dataset is divided into two different subsets: a training set, which is used to determine the model parameters, and a separate validation set (test set), also known as the hold-out set or development set. The training set is crucial for model parameterization, while the validation set serves as an independent dataset for evaluating the model's performance. The selection criterion for the final model is to select the one with the fewest errors in the validation set. This approach ensures that the selected model generalizes well to new, unseen data [24]. Cross-validation (CV) is a well-known resampling method that is widely used in statistical learning methods. It serves as an essential tool for estimating the test error associated with a particular statistical learning method and makes an important contribution to model evaluation and selection. Evaluating the overall performance of a model and selecting an appropriate degree of flexibility are integral aspects of cross-validation that are essential for refining and optimizing statistical learning models [33]. The selection of k in k-fold cross-validation involves a trade-off between bias and variance. The parameter k determines the number of folds into which the dataset is divided and influences the estimation of the model's test error. It is expected to use k = 5 or k = 10, as empirical evidence suggests that these values lead to a balance yielding test error rate estimates with moderate bias and variance. This choice helps to reduce the risk of underfitting (high bias) or overfitting (high variance) in the model evaluation process [33]. Furthermore, a comparison between 5-fold cross-validation and 10-fold cross-validation shows that 10-fold cross-validation generally performs better in regression scenarios [34]. Bayesian optimization (BO), known for its efficiency in optimizing expensive black-box functions, has attracted considerable attention, especially in the field of hyperparameter optimization. The basic concept involves approximating the unknown function, typically using Gaussian processes, with initial random estimates and iterative updates with each new observation.