## 4 Results and Discussion
The CGO-ANN model demonstrates substantially higher R\({}^{2}\) values compared to the other models, indicating its superior ability to capture the underlying relationships between input variables and PPV. For instance, in the testing dataset, the CGO-ANN model achieves an RMSE of 0.425 in the testing dataset, outperforming all other models, outperforming all other models. Moreover, the R\({}^{2}\) values further reinforce the superiority of the CGO-ANN model in explaining the variance in PPV. The CGO-ANN model demonstrates substantially higher R\({}^{2}\) values compared to the other models, indicating its superior ability to capture the underlying relationships between input variables and PPV. For instance, in the testing dataset, the CGO-ANN model achieves an RMSE of 0. ANN, single ANN, and USBM models record R\({}^{2}\) values of 0.777, 0.748, 0.731, and 0.520, respectively. Based on the results depicted in Figure 10, the reliability range of the proposed CGO-ANN model, determined by the significance level of the estimates, falls within the range of 89% to 91%. In contrast, other models exhibited reliability ranges between 50% and 80%. Figure 10: Regression analysis of the outcome predictions on the testing dataset, with a 95% confidence interval. These numerical results provided strong evidence supporting the efficacy of the CGO-ANN model in accurately predicting PPV. The consistently lower prediction errors and higher R\({}^{2}\) values demonstrated the superior performance of the CGO-ANN model compared to traditional ANN models and the empirical USBM model. Therefore, based on the numerical evidence, it can be concluded that the CGO-ANN model is the most effective predictive model for PPV estimation in this study. In Figure 11, we evaluate the developed models through the distribution of residuals. The distribution of residuals provides crucial insights into the accuracy and reliability of the predictive models. Ideally, the residuals should follow a normal distribution, indicating that the model effectively captures the underlying patterns in the data. A normal distribution of residuals suggests that the model accurately accounts for the variability in PPV, with the errors evenly distributed around zero. This implies that the model predicts PPV values with minimal bias and consistent accuracy across the range of observed PPV values. From Figure 11, we can see that the CGO-ANN model follows a normal distribution, indicating that this model effectively captures the underlying patterns in the data. Meanwhile, the other models, such as GA-ANN, PSO-ANN, single ANN, and the USBM model, yielded deviations from a normal distribution of residuals, and this could indicate potential issues with the predictive models, such as the model systematically underestimating or overestimating PPV values in certain ranges. In the GA-ANN, PSO-ANN, single ANN, and  
Figure 11: Distribution of residuals for each PPV predictive model developed. USBM models, all of their residuals are positively skewed, indicating that the models tend to underestimate PPV in higher ranges, whereas negatively skewed residuals suggest underestimation in lower ranges. In addition, heteroscedastic residuals imply that the spread of errors varies across different levels of PPV. This suggests that the model's predictions are less reliable for certain ranges of PPV values, leading to inconsistent prediction accuracy. To directly assess the normality of the residuals, a Q-Q plot was employed, providing a clearer indication of deviations from the expected normal distribution. By comparing the observed quantiles of residuals to those of a theoretical normal distribution, as demonstrated in Figure 12, we obtain a more nuanced understanding of residual normality for each PPV predictive model. This integrated approach strengthens the evaluation process, ensuring a thorough examination and resolution of any deviations from normality.