## 3 Results and Discussion
### The Spatiotemporal Kriging Model (SPTK Model)  
Figure 6 summarizes the steps followed in applying the spatiotemporal kriging model.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & **Value** \\ \hline Activation function & ReLU \\ Epochs & 80 \\ Number of hidden layers & 1 \\ Number of neurons in hidden layer & 20 \\ Batch size & 72 \\ Learning rate & 0.01 \\ Optimizer & Adam \\ Loss Function & MAE \\ \hline \hline \end{tabular}
\end{table}
Table 2: Common parameter settings in MLP models.  
Figure 6: The spatiotemporal kriging method.  
In this approach, a variogram for each missing sequence was fitted using the available rainfall values of stations. For example, according to Figure 6, the first gap of the above data set can be seen in station St2. Then, the data from the remaining stations were employed to fit a variogram. Similarly, the missing blocks of St4 and St6 were estimated using the available rainfall values of the other stations and so on.  
Four variogram types (metric, separable, sum metric, and product sum) were compared in terms of RMSE values. Using the variogram with the lowest RMSE, the spatiotemporal kriging was applied to estimate the missing values in a substation(s) that was/were in the created grid. Table 3 provides the outputs resulting from 11 blocks of missing values.  
According to Table 3, it can be noticed that, in most of the cases, the metric variogram has become the best fit and Figure 7 depicts a sample of figures (out of 11 figures), which illustrates the comparison between variograms. In each case, the sample variogram and fitted four types of variograms are given.  
In fitting the spatiotemporal model for the rainfall data, the empirical variogram surface (or sample variogram) based on space and time is computed and used as inputs for fitting different models: metric, separable, sum metric, and product sum models. Out of them, the best fitting model can be selected, comparing with the surface of the empirical variogram. When considering Missing Gap 01 in Figure 7, the metric-type variogram shows the most similar surface shape with respect to distance and time lags as that of the original  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Gap no** & **Metric** & **Separable** & **SumMetric** & **Product Sum** \\ \hline
1 & **13.79** & 23.16 & 64.75 & 23.33 \\
2 & 38.56 & 22.871 & 52.43 & **22.868** \\
3 & **15.46** & 16.19 & 31.14 & 16.25 \\
4 & **30.31** & 30.86 & 71.54 & 30.82 \\
5 & 136.15 & 132.79 & **128.77** & 132.79 \\
6 & 17.53 & 10.03 & 31.8 & **9.99** \\
7 & 11.02 & 9.67 & **9.13** & 9.66 \\
8 & **88.75** & 89.13 & 110.29 & 89.28 \\
9 & **21.58** & 21.95 & 41.33 & 22.13 \\
10 & **81.02** & 81.03 & 118.85 & 81.04 \\
11 & 34.95 & **13.54** & 58.8 & 13.63 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of prediction errors of variogram types.  
Figure 7: Differences between the sample and the best fitting spatiotemporal variogram of each family of the first two gaps. The half of the variogram function (\(\gamma\)—gamma) which gives measures of spatiotemporal correlation of the random field is plotted against each space–time co-ordinate.  
sample. Nevertheless, in Missing Gap 02, the product sum variogram is more similar to the pattern of the actual sample. Likewise, in 11 gaps, the optimal variogram type deferred with the missing pattern of the data.  
Through optimal variograms the covariances of the data were calculated and, thereby, the optimal weights to predict the missing value at the targeted location (see Section 2.1.4).