## 2 Accuracy Metrics  
### Correlation-Based Metrics  
The correlation coefficient (R) and coefficient of determination (R\({}^{2}\)) are widely used for the evaluation of the goodness of linear fit of regression models in ANNs [28]. Pearson correlation coefficient, Spearman's rank correlation coefficient and Kendall rank correlation coefficient are commonly used correlation coefficients in statistics. The R refer to the Pearson correlation coefficient in this study. The R interprets the degree of correlation between the actual and predicted variables [29; 30]. The calculation of R is illustrated in the Equation (1), the numerator is the sum of squares of residuals also called the residual sum of squares, and the denominator is the total sum of squares that is proportional to the variance of the data.  
The magnitude of R ranges from \(-1\) to +1 [31]. The strength of correlation of two variables can be described in five degrees, as illustrated in Figure 2. A value of +1 (or \(-1\)) indicates the perfect correlation between two variables, +1 is positive correlation and \(-1\) is inverse correlation between two variables; a range from 0.8 to 1 (or from \(-0.8\) to \(-1\)) indicates very strong correlation; a range from 0.6 to 0.8 (or from \(-0.6\) to \(-0.8\)) indicates strong correlation; a range from 0.4 to 0.6 (or from \(-0.4\) to \(-0.6\)) indicates moderate correlation; a range from 0.2 to 0.4 (or from \(-0.2\) to \(-0.4\)) indicates week correlation; and a range from 0 to 0.2 (or from 0 to \(-0.2\)) indicates very week correlation.  
Figure 1: Step-by-step block diagram of the proposed research. Step 1: theoretical study; step 2: application of accuracy metrics; step 3: result analysis; step 4: discussion and proposal of a novel selection method. ANN: artificial neural network; MSE: mean square error; RMSE: root mean square error; MAE: mean absolute error; MAPE: mean absolute percentage error; SMAPE: symmetric mean absolute percentage error.  
The R\({}^{2}\) is the ratio of the predicted variable that is explained by a regression model [32]. In other word, it is the ratio of explained variable from the total variable. R\({}^{2}\) is the square of correlation between the actual variable and predicted variable [33]. Thus, R\({}^{2}\) ranges from 0 to 1. A value of 0 indicates that the regression model explains none of the predicted variable, which means that there is no correlation between the two variables. A value of 1 indicates that the regression model explains all of the predicted variables, which means the that the correlation between the two variables is perfect. The explanations of other values between 0 and 1 can be found in Figure 2.  
The calculation of R and R\({}^{2}\) are defined as [28; 29; 30; 31; 32; 33]  
\[\begin{array}{c}\text{R}=\frac{\frac{1}{n}\sum\limits_{1}^{n}\left(\text{D}_ {\text{act}}-\overline{\text{D}}_{\text{act}}\right)\left(\text{D}_{\text{pre }}-\overline{\text{D}}_{\text{pre}}\right)}{\sqrt{\frac{1}{n}\frac{n}{1}\left( \text{D}_{\text{act}}-\overline{\text{D}}_{\text{act}}\right)^{2}}\sqrt{ \frac{1}{n}\sum\limits_{1}^{n}\left(\text{D}_{\text{pre}}-\overline{\text{D}}_ {\text{pre}}\right)^{2}}}\\ \text{R}^{2}=1-\frac{\frac{\sum\limits_{1}^{n}\left(\text{D}_{\text{act}}- \text{D}_{\text{pre}}\right)^{2}}{\sum\limits_{1}^{n}\left(\text{D}_{\text{ act}}-\overline{\text{D}}_{\text{act}}\right)^{2}}}{\text{D}_{\text{act}}^{2}} \end{array} \tag{1}\]  
where D\({}_{\text{act}}\) is the actual variable, D\({}_{\text{pre}}\) is the predicted variable, \(\overline{\text{D}}_{\text{act}}\) is the mean value of the actual variable, \(\overline{\text{D}}_{\text{pre}}\) is the mean value of the predicted variable and n is the amount of collected data; variables refer to the distance from ground level in this case study.