## 4 Results and Discussion  
The experimental results of the study are presented in Tables 1-4, which contain:  
* \(\beta\) values rounded up to 4 decimals (which are mean values for the \(\beta\)'s obtained over the 50 runs of the PSO algorithm);
* Standard deviations of the computed \(\beta\) values, denoted by st.dev. ;
* Mean squared errors (MSE) of the series values obtained in the experiments. The MSE is the average squared difference between the estimated values and the actual values; it is a measure of a model's quality (Formula (15)). The smaller the MSE, the better the model is. * The time (in seconds) for running an experiment in the case of IDW*;
* The average run time (in seconds) over all the 50 experiments and the standard deviation of the time in the OIDW experiments. PSO is a stochastic algorithm; hence, mean and standard deviations are reported for both the optimized parameter, and the time consumed by the algorithm. The standard deviation is a proof of the algorithm stability (i.e., when the standard deviation is very small it means that the algorithm converges in every run; hence, it is stable and reliable). The results obtained using only the series recorded at the ten main stations in the experiments done in the scenario A1 are displayed in Table 1, as follows: IDW* (columns 2-4), OIDW (columns 5-7) and KG (column 8). The values of the \(\beta\) parameter are equal up to the fourth decimal in IDW* and OIDW. The average run time for IDW* is 82.19 times greater than that for running OIDW (column 4-60 s and column 7-0.73 s). Table 2 displays the results of the experiments performed using the series recorded at the main meteorological station, in the scenario B1. We remarked that the PSO convergence was very good. The standard deviations of time and \(\beta\) in the OIDW algorithm are of the order of \(10^{-4}\), respectively \(10^{-6}\), hence they are not reported individually in the table. The values identified by OIDW coincide with those identified by IDW\({}^{\star}\), by the 4 th decimal. A significant difference appears in the computational time. In the scenario B1, the average computational time for an OIDW run is more than 60 times smaller than the time for a IDW* run. \begin{table}
\begin{tabular}{l c c c c c c} \hline  & \multicolumn{2}{c}{**IDW\({}^{\star}\)**} & \multicolumn{2}{c}{**OIDW**} & \multicolumn{2}{c}{**KG**} \\ \hline
**Station** & \(\beta\) & **MSE** & **Time (s)** & \(\beta\) **(std.dev)** & **MSE** & **Time (s)** \\ \hline Adamclisi & & 32.4184 & & & 32.4184 & & 32.73 \\ Cernavoda & & 22.9189 & & & 22.9189 & & 23.40 \\ Constanta & & 30.3249 & & & 30.3249 & & 30.22 \\ Corugea & & 22.4062 & & & 22.4062 & & 22.48 \\ Harsova & & 35.6281 & & 1.1933618 & 35.6281 & 0.73 (0.132) & 35.03 \\ Jurilovca & & 24.1018 & & (3.2533 \(\times 10^{-4}\)) & 24.1018 & & 23.04 \\ Mangalia & & 42.0429 & & & 42.0429 & & 42.73 \\ Medgidia & & 22.3809 & & & 22.3809 & & 22.58 \\ Sulina & & 44.0204 & & & 44.0204 & & 43.54 \\ Tulcea & & 33.0844 & & & 33.0844 & & 34.47 \\ \hline \end{tabular}
\end{table} 
Table 1: The values of the \(\beta\) parameter obtained in experiments performed using the series recorded at the ten main meteorological stations, in scenario A1. For OIDW, the values in the parenthesis are the standard deviation of the parameter identified, computed over the 50 independent runs. Comparison with KG.