## 4 Synthesis and Validation of ML Models for the GIC Diagnostics  
The estimation of the coefficient of determination \(R^{2}\) showed that, for the diagnostics of the GIC values \(\mid\!J_{\text{VKH}}\mid\), the use of the geomagnetic field variability from nearby magnetic stations is reasonable. Here, we apply the methods based on multiple linear regression and an artificial neural network (ANN). The regression relationship is as follows:  
Figure 3: Statistics of the distribution of the GIC magnitudes \(\mid\!J\!\mid\) at VKH station (**a**) and the rate of the change of the Y-component of geomagnetic variations recorded at IVA station (**b**) for 2015 (\(\Delta t=1\) min). Red solid and dashed lines correspond to PDF and SF of the log-normal distribution, green solid and dashed lines describe the tail of the distribution and the PDF and SF of the generalized Pareto distribution; the black solid line corresponds to ESF.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mid\!dY_{\text{LOZ}}/dt\!\mid\) & \(\mid\!dY_{\text{IVA}}/dt\!\mid\) & \(\mid\!dY_{\text{SOD}}/dt\!\mid\) & \(\mid\!dY_{\text{KEV}}/dt\!\mid\) & **IE-Index** \\ \hline \(r\) & 0.882 & 0.878 & 0.847 & 0.841 & 0.772 \\ \hline \hline \end{tabular} Note: When assessing the correlation coefficient between two time series, if the \(i\)-th value was absent in the first time series, then it was excluded along with the \(i\)-th value of the second time series and vice versa.  
\end{table}
Table 3: Correlation of the values \(\mid\!J_{\text{VKH}}\!\mid\) with values of geomagnetic variations (data averaged over 15 min intervals).  
\[f(x,\beta)=\beta_{1}x_{1}+\beta_{2}x_{2}+\ldots+\beta_{k}x_{k}=\sum_{j=1}^{k} \beta_{j}x_{j}=x^{T}\beta^{T}, \tag{4}\]  
here, \(x^{T}=(x_{1},\,x_{2},\,\ldots,\,x_{k})\) is a vector of regressors; \(\beta^{T}=(\beta_{1},\,\beta_{2},\,\ldots,\,\beta_{k})\) is a vector column of coefficients; and \(k\) is a number of model input variables.  
To increase the stability of the developed approach, the calculation of the values of the regression coefficients in expression (4) is performed by the Elastic Net method, which is a hybrid method of LASSO (\(\lambda_{1}=\lambda,\,\lambda_{2}=0\), where \(\lambda\) is a regularization parameter) and ridge regression (\(\lambda_{2}=\lambda,\,\lambda_{1}=0\)) [21]:  
\[\hat{\beta}_{\text{ElasticNet}}=\underset{\beta}{\text{argmin}}\,\{||\mathbf{y}- \mathbf{X}\beta||_{2}+\lambda_{2}||\beta||_{2}+\lambda_{1}||\beta\,||_{1}\}, \tag{5}\]  
where \(||\beta||_{2}=\sum_{j=1}^{k}\beta_{j}^{2}\); \(||\beta||_{1}=\sum_{j=1}^{k}|\beta_{j}|\); \(\mathbf{X}\) is the \(n\times k\) predictor matrix of standardized variables; \(\mathbf{y}\) is the response vector.  
The analysis of the quality of the model's feature objects by relief scoring method [22] revealed that the input variable \(|dY_{\text{SOD}}/dt|\) has the least quality at a sufficiently high correlation coefficient with the objective function (Table 3). This may indicate the multi-collinearity of the regressors and the necessity to exclude \(|dY/dt|\) from SOD analysis.  
Finally, we have:  
\[|J_{VKH}|=\beta_{0}+\beta_{1}\bigg{|}\frac{dY_{\text{LOZ}}}{dt}\bigg{|}+\beta_ {2}\bigg{|}\frac{dY_{\text{JVA}}}{dt}\bigg{|}+\beta_{3}\bigg{|}\frac{dY_{\text{ KEV}}}{dt}\bigg{|}+\beta_{4}\text{IE}, \tag{6}\]  
where \(\beta_{0}=0.1\); \(\beta_{1}=90.56\times 10^{-3}\); \(\beta_{2}=32.25\times 10^{-3}\); \(\beta_{3  
As follows from Figure 4 and Table 4, the diagnostics of the GIC by means of the ANN because of the quantity of neurons in the hidden layer is able to provide a slightly better result compared to the regression model (6). However, the ANN requires much more computational resources (computer time) and is less interpretable.