## 4 Results and Discussion
The MAPE is given by the formula:  
\[\text{MAPE}= \frac{1}{n}\sum_{i=1}^{n}\left|\frac{p_{i}-m_{i}}{m_{i}}\right| \tag{16}\]  
where \(n\) is number of stations used in the IDW, \(p_{i}\) is the value predicted by IDW for the precipitation at station \(i\) and \(m_{i}\) is the measured value for the precipitation at station \(i\). The KGE coefficient is introduced in [40]; it is defined by formula:  
\[\text{KGE}=1-\text{ED} \tag{17}\]  
where:  
\[\text{ED}=\sqrt{(r-1)^{2}+\left(\frac{\sigma_{s}}{\sigma_{0}}-1\right)^{2}+ \left(\frac{\mu_{s}}{\mu_{0}}-1\right)^{2}} \tag{18}\]  
where \(\mu_{s}\) is mean of the values resulted from the model, \(\mu_{0}\) is the mean of the recorded values, \(\sigma_{s}\) is the standard deviation of the values resulted from the model and \(\sigma_{0}\) standard deviation of the recorded values. MAPE is a scale-independent indicator that could be used in order to compare the performance of a given method on separate data sets. The lower MAPE is, the better the model is. The KGE coefficient is dimensionless as well and has an ideal value of unity. Although, in this study, the models were calibrated using the MSE as the model calibration criterion, the KGE and MAPE provide further validation of the obtained models. Table 8 provides the MAPE values together with the Root Mean Square Errors (RMSE = \(\sqrt{\text{MSE}}\)) and the KGE for the ten main stations. Figure 7: MSE in OIDW (scenario B2). Similar results for MAPE and KGE were obtained in the A2 and B2 scenarios, so we are not presenting them here. The highest MAPE value (so the worst data fit) was noticed for the Sulina station in both scenarios. It was expected, since Sulina is situated 12 km offshore, in the Danube Delta, and the climate presents different particularities compared with the rest of Dobrogea region. The same was true for the KGE coefficient: its lowest values were obtained for the Sulina and Mangalia stations (the most isolated stations). Since the values of both goodness-of-fit indicators, dimensional-RMSE and dimensionless-MAPE are small, and most of the KGE values are bigger that 0.5, it resulted in the OIDW performing well. The [PERSON]'s test could not reject the hypotheses that groups of MSE from Tables 1-4 (together), Tables 5 and 6 (together) and Tables 3-6 were homoskedastic, the corresponding \(p\)-values associated to this test were greater than the significance level of 5%. The Kruskal--Wallis test could not reject the null hypothesis, the corresponding \(p\)-values being 0.9356 (Tables 1 and 2), 0.8348 (Table 3 + Table 5), 0.6437 (Table 4 + Table 6) and 0.7519 (Tables 3-6 all the stations), respectively. The eqdist.etest test applied to the same groups of series as in the previous test could not reject the hypothesis that each group of series has the same distribution, the corresponding \(p\)-values being greater than the significance level of 5%. Concluding, the MSE's of the estimations obtained by OIDW, IDW* and KG were not significantly different. When KG is not very straightforward to be applied and searching for the best parameter \(\beta\) for IDW with grid search is computationally intensive, OIDW proves to be a convenient and rapid to use solution. Other optimization methods can be applied for the power parameter of IDW calibration, such as methods based on the Golden Section (GS) method, provided that the hypotheses of this method are satisfied. As an example, we tried to calibrate the parameter \(\beta\) by this additional optimization method, namely by the fminbnd Matlab function [41]. This function applies GS, followed by parabolic interpolation; the boundaries for the search are to be specified as parameters. In the experiments performed on the series from the ten main stations with the fminbnd method, we obtained an approximation of 1.1903\(f\) or \(\beta\) with a duration of 1 s, with MSE = 30.75. The results are comparable in this situation with those obtained by both OIDW and IDW.