## 3 Results and Discussion
### The Hybrid Model Formation  
The structure of the performed hybrid model with MLP and spatial kriging can be depicted as in Figure 5. In the notation, \(\widehat{y}_{\mathbf{k}}\) and \(\widehat{y}_{\mathbf{m}}\) refers to the spatial kriging and the MLP estimates corresponding to the missing rainfall value of the \(i^{th}\) station, for instance, St2 (there are situations where the rainfall values of more than one station are missing on the same day).  
\begin{table}
\begin{tabular}{l c c} \hline \hline \multicolumn{1}{c}{**Sub Station**} & \multicolumn{1}{c}{**Missing Count**} & \multicolumn{1}{c}{**Missing Percentage**} \\ \hline St1\_Bal & 32 & 1.75 \\ St2\_Det & 1 & 0.05 \\ St3\_Els & 519 & 28.42 \\ St4\_Ill & 491 & 26.89 \\ St5\_Ker & 123 & 6.74 \\ St6\_Mor & 183 & 10.02 \\ St7\_Niv & 1249 & **68.4** \\ St8\_Uli & 1343 & **73.55** \\ \hline \hline \end{tabular}
\end{table}
Table 1: Missing percentages of sub-stations’ rainfall data.  
Figure 3: Missing patterns of sub-stations’ rainfall data.  
Note that MLP (for multivariate time series data) was trained using the parameter values tuned with grid search approach (selected from a range of values) tabularized in Table 2.  
Figure 4: The heatmap of correlation between rainfall gauging stations.  
Figure 5: The hybrid model structure.  
The regularization parameters and number of training/testing samples were modified to obtain a convergent model in each MLP model. As depicted in Figure 5, in order to integrate the two estimates obtained from spatial kriging and MLP, the optimal weights from each estimate should be found.  
For this, a data set without missing values needs to be used. Therefore, the following procedure was applied:  
1. From the original data set, three complete subsets (with no missing cases for all the stations) were selected. Consequently, subsets from the years 2015, 2016, and 2019 were selected.
2. Consecutive missing periods were artificially created randomly within the three subsets to capture the missing pattern of the original data set.
3. The spatial kriging and MLP were applied to estimate the missing cases.
4. Those estimated rainfall values were combined to form a single data set of 120 estimated rainfall values.
5. Then, from the final data set mentioned in (d), 100 simple random samples (with replacement) with a moderate sample size of 60 were drawn.
6. For each sample, the optimal values of two weights, \(w_{k}\) and \(w_{m}\), were calculated by applying Equations (11) and (12) as described in Section 2.1.3. The procedure resulted in 100 optimal pairs of weights.
7. Then, the first optimal weight pair was used to calculate the weighted estimates (\(\widehat{y}_{f}\)) of the rest of the samples (99). For each sample, the RMSE values (including the first sample considered) were calculated. Using these RMSE values, the average RMSE was computed.
8. Likewise, steps (a) to (g) were repeated using the second optimal weight pair, then third optimal weight pair, and so on.  
Finally, there were 100 average RMSE values. Out of them, the pair of weights that produced the lowest average RMSE value was selected as the best optimal pair of weights for spatial kriging and MLP (Table S2). The results indicated that the 11 th random sample produced the final optimal weights, \(w_{k}=0.50013\) and \(w_{m}=0.49987\). Those weights were taken to obtain optimal weighted estimates for the missing values.