## 3 Results and Discussion
### Mathematical Model Validation  
Overfitting can become a prevalent issue while training machine learning models. It happens when a model becomes too complex and starts to fit the training data too closely, resulting in poor generalization performance on new data. Validation is performed to ensure optimal reliability and prevent overfitting of the models that are selected for further analysis. In the current study, validation was conducted on several different models, but the two more relevant models are outlined in Table 3. The validation results revealed that the exponential GPR model was the least robust for this system, and so the model could not generalize to new data and likely overfitted the training set. It is important to note that the validation process is a crucial step in developing machine learning models. Selecting the most robust model can ensure that results are reliable and can be applied to new data confidently. Overall, the validation process is a vital component of any machine learning workflow and should be given careful consideration when developing new models [54].  
Considering the model with the best validation metrics, it can be inferred that ANN represents a forecasting performance with high accuracy (Figure 4). It can be justified because it models complex and nonlinear processes based on a data set using high self-adaptive learning without requiring a normal distribution like conventional methods [55]. Exponential GPR can face challenges in normative modelling due to its assumptions and complexities. These limitations can impact the accuracy and reliability of predictions, especially in scenarios like turbidity prediction, where precise modelling is crucial [56]. While both ANN and GPR have advantages and disadvantages, ANN represents a highly accurate and adaptable approach to complex forecasting tasks. This technology will likely provide valuable insights and predictions that can drive improved decision-making and performance.  
Good model reliability is observed when the estimated values are on the diagonal line of the performance graph [57]; the only one that complies, as shown in Figure 4, is the neural network because all the data are scattered close to the solid line. Therefore, it is considered reliable for estimating turbidity in artisanal wines and efficiently represents  
Figure 4: Validation performance of the selected models: (**a**) neural network and (**b**) exponential GPR.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **R2** & **MSE** & **NRSME** & **AIC** \\ \hline ANN & 0.985 & 0.004 & 6.01 & \(-\)160.12 \\ Exponential & 0.943 & 0.008 & 8.14 & \(-\)283.02 \\ GPR & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Validation metrics of the selected models during training.  
the estimation of turbidity in artisanal wines. Applying the significance criterion to the selected model, it was found that it is representative with an F of 57.38 and a critical F of 1.44, and so it can be concluded that this model represents the studied system [33]. Even though statistical indicators and error measures indicate that the model obtained in this research can predict wine turbidity, it is highly recommended to continue studying this system so it can be applied to other varieties of grapes or a greater range of variable values, even including more features.