## 3 Results and Discussion
### Performance Parameters (Merits) and Model Discrimination  
Two performance parameters (\(r\) and RMSE) are not sufficient. Eight to twelve parameters are commonly used in the recent literature [28; 29; 30]. The usage of performance parameters differently for training and test sets has not been mentioned, either. Training and testing were used in the context of artificial neural networks (ANN), which are superfluous there, as parameter estimation can be completed without them. The description of MLP and ANFIS is no less than breaking a butterfly in a wheel. Similarly, over- and underfitting have not even been mentioned, along with bias-variance trade-off, although online course(s) are also available [31]. It should be mentioned that many performance parameters would provide a more sophisticated evaluation, and they are recommended below:  
Mean absolute error (MAE) is a straightforward metric because it is simple and easily interpretable to estimate the accuracy of predictive models. Figure 6: (Figure 10 from ref. [2]): performance of all models of Figure 3 was compared for description of epidemic data of Germany by using GWO for minimum search. The Bayesian Information Criterion (BIC) is applied for model selection from among several rival models. It takes into account the degrees of freedom and penalizes more complex models with the number of parameters to avoid overfitting [32]. Akaike's Information Criterion (AIC) is also a measure of the goodness of fit. AIC (similarly to BIC) is widely used for model selection from among numerous models [33]. BIC penalizes complex models more heavily than AIC does. Instead of the number of point pairs (\(N\)) used in the denominator of the RMSE formula in ref. [2], the degrees of freedom would have been the proper choice: (\(N\)-\(p\)), where \(p\) is the number of parameters in the models. As the number of parameters in all models is known (_cf._, Figure 3), one can only wonder why the degrees of freedom were not considered, at all. There are some unorthodox behaviors of the simple correlation coefficient as defined between independent (input) variables and dependent (output) variables. It can measure linear relations only, but some of the models--Equations (11)-(13) in ref. [2]--are highly nonlinear. On the other hand, a simple quadratic relation (parabola \(y=x^{2}\)) would provide exactly \(r=0\) (if all plus minus pairs are involved). Naturally, a multiple correlation coefficient (index) can be defined, although differently, between the input \(y\) and output \(\hat{y}\) variables (\(\overline{y}\) is the mean of all \(y_{i}\)-s):  
\[R_{1}^{2}=\frac{\sum_{i}^{N}(\hat{y}_{i}-\overline{y})^{2}}{\sum_{i}^{N}(y_{i }-\overline{y})^{2}} \tag{3}\]  
Equation (3) was suggested by [PERSON] and [PERSON] [34] and it can have values larger than 1 for nonlinear equations (but not necessarily). \[R_{2}^{2}=1-\frac{\sum_{i}^{N}(y_{i}-\hat{y}_{i})^{2}}{\sum_{i}^{N}(y_{i}- \overline{y})^{2}} \tag{4}\]  
Equation (4) is often called determination coefficient in analytical chemistry textbooks. I found the first occurrence in ref. [35]. It cannot be larger than 1, but it can have negative values, if the model is worse than a simple constant, the average of all \(y_{i}\)-s.  
\[R_{3}^{2}=\frac{\left(\sum_{i}^{N}(\hat{y}_{i}-\overline{y})^{2}\right)^{2}}{ \sum_{i}^{N}(y_{i}-\overline{y})^{2}\sum_{i}^{N}(\hat{y}_{i}-\overline{y})^{2}} \tag{5}\]  
Equation (5) [36] also might suffer from being greater than 1 in the case of nonlinear models. \[R_{4}^{2}=\frac{\left[\sum_{i}^{N}(y_{i}-\overline{y})\sum_{i}^{N}(\hat{y}_{i }-\overline{y})\right]^{2}}{\sum_{i}^{N}(y_{i}-\overline{y})^{2}\sum_{i}^{N}( \hat{y}_{i}-\overline{y})^{2}} \tag{6}\]  
Equation (6) eliminates the inconsistent behavior detailed above for (\(R_{1}^{2}\), \(R_{2}^{2}\), \(R_{3}^{2}\)). To my knowledge, Prof. [PERSON] (University of Bergen) derived Equation (6) but never published it, and his equation is not used widely, if at all. None of the above four equations has been applied in ref. [2] for model discrimination.