## 3 Results and Discussion
### Model Evaluation
#### 3.4.2 Model Evaluation on Artificially Generated Missing Data  
Since the previous evaluation could not capture all the actual missing gaps (due to the inconsistency of the data set), we thought to generalize the validation using another validation set that could capture all the missing patterns (which are and are not in the original data set) that could be found in an original data set of six rainfall gauging stations. In year 2016, for 6 months (from January to June), there were no missing observations. Therefore, we took the data of that period to create 56 subsets with different sequential missing patterns, as shown in Table 5. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Spatial Kriging** & **MLP** & **Hybrid** & **SPTK** \\ \hline RMSE & 12.76 & 11.68 & **10.72** & 15.97 \\ MAE & 7.12 & **4.01** & 5.12 & 10.25 \\ R\({}^{2}\) & **13\%** & 3\% & **13\%** & 0\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of prediction error of all models. Figure 8: The model evaluation criterion. When two stations have missing observations in the same period (consider one incident out of 15 combinations in case 02), each station was modeled separately. Thus, it resulted in two trials. Finally, all combinations of two stations with missing observations (i.e., 15) generated 30 trials (15 \(\times\) 2). This was the same for the rest of the cases where there were at least two stations with missing data. The case where the number of stations with missing observations, M = 5, was not considered because there should be at least two stations without missing observations in the spatiotemporal grid to model those gauging stations with missing values using spatial kriging (a model requirement). In each subset, missingness was created with the last 20 observations (without breaking the time sequence and allocating enough data points for training) of the relevant sub-station. The rest of the data were used to train and test the models and, hence, identify the best-fit model for a selected sample of data. The estimated values from each model were compared with the actual values and tabularized in Table S3. In terms of the lowest values of RMSE and MAE and the highest values of R\({}^{2}\), the hybrid model has outperformed the other approaches. Moreover, it is important to point out that, when kriging performs better than other individual methods in terms of three validation measurements, the hybrid model has performed approximately equally in most of the cases. The outperformance with respect to the total (where lowest RMSE and MAE values and highest R\({}^{2}\) value occur) shows the hybrid method could gain the score of 41% (see Table 6). In addition to the above outcome, all the cases (in Table S3) were taken to obtain the graphs shown in Figure 9.  
Figure 9 illustrates that kriging produced most cases within the lowest value range of MAE and within the highest value range of R\({}^{2}\), whereas the MLP gained the majority of cases within the lowest value range of RMSE. In all these occurrences, the hybrid model has shown approximately the same performance as the optimal model. This study demonstrated that the spatial and temporal variations should not be neglected in missing value imputation in spatiotemporal data. Further, considering the two-way evaluation, we could confirm that the hybrid model had the ability to outperform other pure approaches or almost equally performed compared to the best approach whenever it was selected as the second-best approach. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Method** & **Kriging** & **MLP** & **Hybrid** & **SPTK** \\ \hline No. of times that each approach produced best results (as a percentage with respect to the sum of trials) & **30** & **20** & **41** & **9** \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Final comparison of model performance.