## 3 Results and Discussions  
The main purpose of this study is to examine the Weibull and Rayleigh distribution functions, which are statistical methods for the daily average wind speed and wind power potential estimation of Canakkale Province, as well as the hybrid artificial intelligence-based ANN-GA and ANN-PSO methods, and then to present a comparison of these four methods. The gap in the literature was filled by using the ANN-GA and ANN-PSO methods for the first time to estimate the wind speed and wind power density in Canakkale Province. The statistical error indices listed in Table 4 were used to evaluate the accuracy of the presented estimation techniques and to determine the error rates. The error rates were calculated with the formulas for the five most commonly used statistical error indices given in this table: R\({}^{2}\), MSE, RMSE, MAE, and MAPE. In these formulas, \(n\) is the number of observations, \(y_{i}\) and \(y_{i}\) are the number of observations, \(y_{i}\) and \(y_{i}\), respectively. Figure 5: Flowchart of the hybrid ANN-GA and ANN-PSO algorithms [53]. and \(x_{i}\) represent the estimated and measured test data, respectively, and \(z_{i}\) represents the mean of the measured data. Figure 6 depicts the mean, minimum, and best fitness values (MSE) in relation to the generation size of the GA. The best MSE for the testing data was reached during the GA's 52 nd iteration as 0.3653, while the correlation was computed as 0.94839. Figure 7 demonstrates the linear correlation between the actual and estimated outcomes for the training and testing data. As seen in that figure, the regression line of the test and the estimated data are provided as y = 0.81 \(\times\) T + 0.62. The error between the estimated result and actual data and a comparison of them are also shown in Figure 7. As can be observed from the figure, the error graph has a few peak points, but overall, the error rate is close to zero, which leads one to deduce that the proposed ANN-GA algorithm performs successfully. Figure 8 represents the mean, minimum, and best fitness values (MSE) in relation to the generation size of PSO. The best MSE for the testing data was 0.4721, while the correlation was obtained as 0.94042. Figure 9 illustrates the linear correlation between the actual and estimated values for the training and testing data. The regression line of the test and the predicted data are provided as y = 0.78 \(\times\) T + 0.81. The error between the predicted and actual results and a comparison between them is also shown in that figure. According to Figure 9, the test and training errors are both quite low, which implies that the suggested ANN-PSO technique works satisfactorily. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Error Metrics** & **Definitions** & **Formulas** \\ \hline \(R^{2}\) & Coefficient of Determination & \(R^{2}=1-\frac{\sum_{i=1}^{n}{(y_{i}-x_{i})^{2}}}{\sum_{i=1}^{n}{(y_{i}-x_{i})^ {2}}}\) \\ _MSE_ & Mean Square Error & \(MSE=\frac{\sum_{i=1}^{n}{(y_{i}-x_{i})^{2}}}{\frac{n}{n}}\) \\ _RMSE_ & Root Mean Square Error & \(RMSE=\sqrt{\frac{\sum_{i=1}^{n}{(y_{i}-x_{i})^{2}}}{n}}\) \\ _MAE_ & Mean Absolute Error & \(MAE=\frac{1}{n}\frac{n}{\sum_{i=1}^{n}{|y_{i}-x_{i}|}}\) \\ _MAPE_ & Absolute Percentage Error & \(MSE=\frac{\sum_{i=1}^{n}{(y_{i}-x_{i})^{2}}}{n}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The mathematical formulas of the statistical error indicators. Figure 6: The minimum and best values of fitness with respect to the generation size of GA.  
Figure 8: The minimum and best values of fitness with respect to the generation size of PSO. Figure 7: Correlation between target and predicted results for the training and test data (ANN-GA). The comparison graph for the prediction of the 2019 daily wind power density for the Weibull distribution, Rayleigh distribution, hybrid ANN-GA, and ANN-PSO algorithms is depicted in Figure 10. The estimated and measured (actual) wind power density results are shown and compared in the figure.