# Earthquake Magnitude and Frequency Forecasting in Northeastern Algeria using Time Series Analysis
## 4 Earthquake Forecasting
Figure 7: Eigenvalues plots of the first 12 components for the annual earthquake number series (circles) and for the annual maximum magnitudes series (stars). The first group is composed of solely the first component, which is uncorrelated to the other components. Because the second and third components are highly correlated, the pair (2,3) forms the second group. After linking the components from 4 to 8, it is shown that the fourth component has a correlation with the fifth, sixth and eighth components. In addition, the sixth component has a high correlation with the seventh component. Thus, the pairs (4,5) and (6,7), as well as the eighth component, configure the last group. Therefore, the rest of components in \(\{N_{k};k\geq 0\}\) series correspond to the noise. For \(\{m_{k},\,k\geq 0\}\) series, according to the w-correlation matrix in Figure 8b, two groups can be used in the reconstruction of the original series. The first group consists of mainly the first component, whereas the second group consists of Components 2-8 and Component 12, which are all correlated with one another. As a result, all of the other components form the noise in the \(\{m_{k},\,k\geq 0\}\) series. It is worth noting that the first component in both matrices that is uncorrelated with the other component corresponds to the trend component in both  
Figure 8: W-correlation matrix plots for the first 12 components of the earthquake number series (**a**) and the annual maximum magnitude series (**b**). series. Therefore, the reconstruction parameters in this case are fixed to 3 and 2 for the \(\{N_{k};k\geq 0\}\) and \(\{m_{k},\,k\geq 0\}\) time series, respectively. Table 2 presents the RMSE value of the candidate SSA models for the studied series. According to the results given in Table 2, the best SSA models for \(\{N_{k};k\geq 0\}\) and \(\{m_{k},\,k\geq 0\}\) are the SSA (24,3) and SSA (12,2) models, respectively. Figure 9 depicts the independent main component extracted from the \(\{N_{k};k\geq 0\}\) (a) and \(\{m_{k},\,k\geq 0\}\) (b) series, representing the trend, seasonality and noise components. Finally, the RMSE value is used to compare the different approaches. Table 3 displays the RMSE values of the ARIMA and SSA models for the \(\{N_{k};k\geq 0\}\) and \(\{m_{k},\,k\geq 0\}\) time series, with the smaller RMSE value indicating the better model. As a result, the SSA (24,3) and SSA (12,2) models are the best ones for describing the number of earthquakes and the maximum magnitude series, respectively. Table 4 also shows the forecasted values from 2020 to 2030 using the selected SSA models. Finally, Figure 10 depicts the original (grey dashed line) and forecasted (red dashed line) series, where a stationary behavior can be observed in both time series from 2020 to 2030. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Series**} & \multicolumn{2}{c}{**Series**} \\ \cline{2-3}  & **Earthq\_Num** & **Max\_Mag** \\ \hline
2020 & 5 & 4.9 \\
2021 & 6 & 5.1 \\
2022 & 5 & 4.8 \\
2023 & 5 & 4.9 \\
2024 & 5 & 5.1 \\
2025 & 4 & 5.0 \\
2026 & 5 & 4.9 \\
2027 & 6 & 5.0 \\
2028 & 4 & 5.1 \\
2029 & 4 & 5.0 \\
2030 & 5 & 5.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Forecasted annual earthquake number and annual maximum magnitude time series from 2020 to 2030 from the selected SSA model. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Series**} & **Model** & **RMSE** \\ \hline  & SSA (12,2) & 3.847 \\ Earthq\_Num & SSA (24,3) & 3.048 \\  & SSA (36,2) & 3.426 \\ \hline \multirow{2}{*}{Max\_Mag} & SSA (12,2) & 0.314 \\  & SSA (24,2) & 0.365 \\ \cline{1-1}  & SSA (36,2) & 0.395 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The SSA chosen model based on the RMSE of the analyzed time series.