## 5 Conclusions  
The paper describes a hybrid PSO-IDW algorithm for the modeling of data; it is applied on maximum precipitation data in the experimental part. The algorithm is based on a well-known spatial interpolation method (IDW). Since the application of IDW makes use of the power parameter \(\beta\), we proposed the use of the PSO algorithm to identify a suitable value for this parameter. Empirical results proved that the IDW performance was maintained, while the new method offered lower computational costs. Results of the IDW and OIDW were further compared to Kriging.  
The optimization of IDW with PSO described in this paper offered an alternative to an exhaustive search for the problem of tuning the power parameter in the IDW method. In cases when the estimations provided by the traditional application of IDW are good, the results obtained with OIDW should preserve the quality of the prediction and be better in terms of computational effort.  
Kriging is a powerful geostatistical method, suitable to be applied in certain cases, and its application assumes profound knowledge of spatial statistics. When the spatial correlation is strong and the variogram can depict well the spatial variability, the KG algorithm predicts very well. Even so, for our precipitation data, the results are not substantially better by KG, as it was the case for other environmental variables studied in [17]. In cases where the nature of the input data does not support KG, instead of using IDW for prediction it is wiser to choose OIDW since it automates the process of identifying the appropriate power parameter, with smaller computational effort. Our approach also benefits from the advantages of the PSO algorithm: a simple implementation, robustness, small number of parameters to adjust and a high probability and efficiency in finding the global optima, fast convergence, short computational time and modeling accuracy. OIDW is easy to use by people that do not have statistical knowledge [29; 42].  
The limitations of IDW are preserved by OIDW: in locations with sparse neighbors (such as those near the borders of the study region), the errors for IDW simulated data are larger. While providing a solution for optimizing the IDW parameter, OIDW uses PSO, which has several parameters of its own that need to be set (e.g., swarm size, the weights in the Equations (7) and (8)). In our experiments, setting these parameters to values commonly used in the literature led to very good results.  
The contribution of our paper was two-fold. For hydrologists, we introduced an optimization of a well-known and widely used spatial interpolation method--namely IDW. For the artificial intelligence community, we proved the utility of the application of the PSO metaheuristic on a parameter optimization problem.  
The results obtained in our case study using time series data from 51 meteorological locations in the Dobrogea region (Romania) encouraged us to consider OIDW as a good choice for the spatial interpolation of precipitation data. Still, more experiments will be performed, using various types of environmental data and various measures of performance of the models, in order to decide the appropriateness of OIDW as a general spatial interpolator.  
Data curation: [PERSON]. ([PERSON]); Investigation: [PERSON]. ([PERSON]) and [PERSON] ([PERSON]); Methodology; E.B.; Software: [PERSON]. ([PERSON]); Supervision: [PERSON]. ([PERSON]); Writing original draft: [PERSON]. ([PERSON]) and E.B.; Writing--review and editing: [PERSON]. ([PERSON]) and E.B. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
We thank the anonymous reviewers for the very useful comments and insights offered in the reviews. Our paper was significantly improved by taking into account their observations.  
The authors declare no conflict of interest.