# Earthquake Magnitude and Frequency Forecasting in Northeastern Algeria using Time Series Analysis
## 4 Earthquake Forecasting
Figure 5b shows the ACF residual plots, indicating that  
Figure 3: (**a**) Annual number of earthquakes (circles) and maximum annual magnitude (stars) in the period 1910–2019. (**b**) Differential time series for annual earthquake number (circles) and for maximum annual magnitude (stars) in the period 1910–2000. the residuals of the two models are uncorrelated. Finally, Figure 5c shows that the residuals are normally distributed according to the residuals Quantile-Quantile plots (Q-Q plot) statistical test [102,103]. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Series** & **ADF Test** & **SW Test** & **Model** & **AIC** & **BIC** \\ \hline  & & & ARIMA (2,1,2) & 4.061 & 4.255 \\  & & & ARIMA (2,1,1) & 3.995 & 4.175 \\  & & & SARIMA (2,1,2) & 4.068 & 4.294 \\  & & & (1,0,1)(s=12) & & \\ \hline  & & & ARIMA (2,1,2) & 2.169 & 2.364 \\ Max\_Mag & 0.063 & 0.029 & & ARIMA (2,1,1) & 2.191 & 2.425 \\  & & & SARIMA (2,1,2) & 2.198 & 2.425 \\ \hline  & & & & **(a)** & \\ \end{tabular}
\end{table} 
Table 1: Annual earthquake number and annual maximum magnitude time series stationarity (ADF) and normality (SW) tests results. ARIMA model selection using the AIC and BIC criterions are also showed. Figure 4: ACF (circles) and PACF (stars) values (**a**) for the differential annual earthquake number and (**b**) for the differential annual maximum magnitude. The second approach considered in our study is based on the SSA method. The first stage of the SSA method consists of decomposing the \(\{N_{k};k\geq 0\}\) and \(\{m_{k},\,k\geq 0\}\) time series into principal independent components, which are the trend, the seasonality and the noise. The second stage consists of reconstructing the original series using only the trend and the seasonality. Figure 6 indicates that the seasonality and trend components used together (blue line) describe the observed earthquake number and the annual maximum magnitude in the studied region better than the trend component alone (red line). The parameters of the SSA model were empirically chosen based on a visual presentation and clear separability of the main independent components. According to [PERSON] [104], a window length less than half of the sample size is considered adequate. According to [PERSON] et al. [94], for seasonal time series, this parameter must be a common multiple of the seasonality period. Several window lengths (12, 24 and 36 samples) were thus evaluated to select the best one, which has been chosen using the RMSE criterion. The reconstruction stage parameter (see Section 3), denoted \(r\), is determined using the eigenvalues plot of the \(\{N_{k};k\geq 0\}\) and \(\{m_{k},\,k\geq 0\}\) series presented in Figure 7, where the slow decrease in these plots suggests the beginning of the noise component. Then, according to Figure 7, two components are used both in the reconstruction process of the \(\{N_{k};k\geq 0\}\) series and in the reconstruction of the \(\{m_{k},\,k\geq 0\}\) series. These results can be confirmed using the w-correlation matrices shown in Figure 8. The w-correlation matrices  
Figure 5: Residual analysis of the selected ARIMA model for earthquakes number time series (circles) and for annual maximum magnitudes time series (stars). (**a**) residuals of the selected models; (**b**) ACF residual plots, indicating that the residuals of the two models are uncorrelated; (**c**) residuals are normally distributed according to the residuals Quantile-Quantile plots (Q-Q plot). indicate that the first three components are uncorrelated in the \(\{N_{k};k\geq 0\}\) series (Figure 8a), and two components in the \(\{m_{k},k\geq 0\}\) series (Figure 8b). Therefore, from the w-correlation matrices, the reconstruction parameter is fixed to 3 and 2 for the \(\{N_{k};k\geq 0\}\) and \(\{m_{k},k\geq 0\}\) time series, respectively. Figure 6: Fit of the recorded annual number and the annual maximum earthquake magnitude (grey line) by the seasonality and trend components combined (blue line) and by the trend component alone (red line).