## 4 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c} \hline  & \multicolumn{2}{c}{**IDW\({}^{\star}\)**} & \multicolumn{2}{c}{**OIDW**} & \multicolumn{2}{c}{**KG \({}^{\star}\)\({}^{\star}\)\({}^{\star}\)**} \\ \hline
**Scenario** & **A1 \({}^{\ddagger}\)** & **B1 \({}^{\ddagger}\)** & **A1 \({}^{\ddagger}\)** & **B1 \({}^{\ddagger}\)** & **B1 \({}^{\ddagger}\)** \\ \hline Main stations & 30.93269 & 30.3704 & 30.93269 & 30.37178 & 31.022 \\ \hline  & \multicolumn{2}{c}{**IDW\({}^{\star}\)**} & \multicolumn{2}{c}{**OIDW**} & \multicolumn{2}{c}{**KG**} \\ \hline
**Scenario** & **A2** & **B2** & **A2** & **B2** & \\ \hline Main stations & 30.64324 \({}^{\ddagger}\) & 29.22245 \({}^{\ddagger}\) & 30.64315 \({}^{\ddagger}\) & 29.22245 \({}^{\ddagger}\) & 30.876 \({}^{\ddagger}\) \\ Secondary stations & 31.0671 \({}^{\ddagger}\) & 30.30585 \({}^{\ddagger}\) & 31.0671 \({}^{\ddagger}\) & 30.30585 \({}^{\ddagger}\) & 29.5280 \({}^{\ddagger}\) \\ All stations & 30.9840 \({}^{\ddagger}\) & 30.0934 \({}^{\ddagger}\) & 30.984 With respect to computational effort: for detecting the \(\beta\) that minimizes the MSE for fitting the values of a series in IDW* using a grid search with a step \(r\), on an interval [a. b], the computation should be performed \([(b-a)/r]+1\) times (where [] is the integer part of the number inside the bracket). For example, for a grid search with a step of 0.1, IDW* was performed 50 times for each station, so 500 times when only the main stations were employed and 2550 times when all the 51 series were used. Once again, taking into account the consumed time in scenarios A1 or A2, B1 or B2 is more convenient to run OIDW. This idea is supported by the fact that the \(\beta\)'s identified with OIDW in all the runs were identical to optimal values of \(\beta\) found by grid search IDW* up to the third digits (so, OIDW converged to a global optimum, in all cases). OIDW performed better than KG in 60% of cases in scenario A1, in 80% of cases in scenario B1, and in 49.01% cases in scenarios A2 and B2. While in the scenarios A1 and B1, the OIDW MSE and KG MSE are comparable, in the A2 and B2 scenarios, there are some discrepancies between them, as for example, for Sulina, Cheia, Negureni, Pantelimon and Mihai Viteazu (in A2 and B2). This situation could be explained by the following reasons: (a) the low number of series used in A1 and B1 scenario, (b) the inhomogeneity of the stations on the in the Dobrogea region, (c) the climate differences on the different part of the region and (d) the possible anisotropy was not considered. The results of the statistical tests on the SE series for OIDW and IDW* are presented in the following. The normality test for the MSE series corresponding to the experiments in scenarios A1, B1, OIDW, KG and IDW* from Tables 1 and 2, A2, B2, IDW*, KG from Tables 3 and 5, KG from Table 3 + Table 4 and KG from Table 5 + Table 6 could not reject the normality hypothesis. The same test led to the normality rejection for the IDW* MSE and OIDW MSE series corresponding to the secondary stations from Tables 4 and 6, but not for OIDW MSE corresponding to all the series in Table 3 + Tables 4 and 5 + Table 6.  
Figure 5: MSE’s for Adamclisi and Cernavoda series in a grid search with a step of \(10^{-4}\) on the study interval. Figure 6: Average MSE computed as the average of the MSEs corresponding to all the series in a grid search with a step of \(10^{-4}\) on the study interval. A spatial map containing illustration the distribution of MSE in OIDW form B2 scenario (values from Table 5 + Table 6) is presented in Figure 7. The highest MSEs correspond to the station near the region border, where the stations had few close neighbors. To offer a different perspective of the OIDW performance, we computed the Mean Absolute Percentage Error (MAPE) and the Kling-Gupta efficiency (KGE).