## 3 Experimental Settings  
The experiments were run using an improved deterministic PSO variant from [30], which is among the most popular versions of the PSO algorithm [31]. We are interested in assessing the quality of predictions for OIDW and compare them with the traditional grid search IDW (denoted, in the following. by IDW*). Figure 4: The flowchart of the Optimizing Inverse Distance Weighting (OIDW) algorithm. The experiments were run in 2 scenarios:  
(Scenario A)  
First, we used PSO and grid search to identify a single \(\beta\) that minimizes the sum of prediction error over all stations. The \(\beta\) value identified is characteristic of the entire Dobrogea region, hence it can be useful to infer values at points where no recordings are available, for example. When the predicted values for a given station are computed, the weights in Formula (1) corresponding to that station are set to 0. (Scenario B)  
We used PSO and grid search to search a \(\beta\) for each station that minimizes the prediction error when the series from that station is estimated using the other series from the other stations. For example, if we have ten stations, to estimate the precipitation at the station I, the series recorded at the stations II-X are used as input, while the series recorded at the station I are hold out as control values. Thus, they are used for comparison with the values computed by applying Formula (2) with the parameter \(\beta_{1}\) identified by OIDW. The \(\beta_{1}\) obtained minimizes the MSE for the station I. Another example: to estimate the values of the series from the station V, the series recorded at station I-IV and VI-X are used as an input, and a \(\beta_{5}\) is computed, corresponding to the minimum MSE for the station V, and so on. Further, we refined the experimental settings by performing experiments in two stages. First, we employed a reduced dataset comprising the series from ten main meteorological stations. We will refer to these scenarios as (A1) and (B1). The same experiments were repeated with the extended dataset that contains all 51 series. We will refer to these scenarios as (A2) and (B2). Hence we obtained four scenarios. We took a special interest in type (A) experiments because they should yield a unique empirically determined \(\beta\) to be characteristic for the entire region. For each scenario, 50 different runs of the PSO algorithm were performed. The default Mersenne twister randomizer with an initial seed of 0 was used in the experiments reported in the tables below. Setting the PSO parameters to optimal values is a very difficult optimization problem by itself [30; 32]. In our experiments, the PSO parameters are chosen to match those widely used in the literature [30]:  
* The swarm size is 24,
* Personal best influence is 2,
* Global best influence is 2,
* The inertia weight, decreasing during the iterations of the algorithm from 0.9 to 0.4,
* The number of epochs (i.e., PSO iterations) is 100,
* \(\beta\) is searched in the interval (1, 5). The results are compared with those obtained by applying IDW and KG. For IDW, we identified the parameter \(\beta\) performing a grid search with the step-size of \(10^{-4}\), in the interval [1.001, 5]. The grid search identifies the parameter \(\beta\) that yields the minimum prediction error (that will be listed in the results tables). The experiments for OIDW and IDW* were performed in the Matlab environment, in Matlab R2012, on a computer with Intel Core i5 quad-core processor at 2.30 GHz with 8 GB RAM, using the PSO toolbox, available online [33]. Matlab was also used for performing the IDW* computation (IDW with the grid search). For KG, we used Ordinary Kriging and report the mean prediction error obtained. The Kriging algorithm was run using the gstat, sp, spacetime, automap and geoR libraries from the R software.