# Earthquake Magnitude and Frequency Forecasting in Northeastern Algeria using Time Series Analysis
## 5 Validation of the Procedure  
After a certain earthquake forecast method has been developed using training data and set up by best fit on real data, and before any practical implementation, it must be validated [105]. The SSA method is used to perform forecasting analysis of the earthquake data file from 1910 to 2015, with the purpose of generating forecast values and comparing them to observed values from 2016 to 2019. As previously stated, the time series are divided into two sets, the first of which includes 80% of the data used as training data [74] and encompasses the years 1910 to 1997, while the remaining 20% covers the years 1998 to 2015, considered as testing data. The stationarity and the normality of the studied series in the considered training period are tested again using the ADF and the SW tests. The derived ADF values are 0.34 and 0.077 for the annual earthquake number and maximum magnitude, respectively, whereas the obtained SW values are 2.0 \(\times\) 10\({}^{-7}\) and 0.035 for the two investigated series, respectively. The ADF test results for the two series are higher than 5%, showing that the two series are non-stationary during the training period. Moreover, the two series are not normally distributed and exhibit a non-stationary behaviour as a result of the computed SW values. Therefore, as proceeded initially, and in order to achieve the stationarity in the data, the differentiated series of \(\{N_{k};k\geq 0\}\) and \(\{m_{k},k\geq 0\}\) are computed. Considering the d parameter of the ARIMA model equal to 1, the ACF and PACF functions of the resultant series are used to select the \(p\) and \(q\) parameters. Thus, ARIMA (2,1,2) and ARIMA (2,1,3) are the model-candidates for the time series \(\{N_{k};k\geq 0\}\), and ARIMA (2,1,2) and ARIMA (3,1,2) for the time series \(\{m_{k},k\geq 0\}\), derived during the training period 1910-1997. Analyzing the annual earthquake number series, AIC values equal to 3.99 and 4.00 are obtained for the ARIMA (2,1,2) and ARIMA (2,1,3) models, respectively, whereas BIC values equal to 4.20 and 4.24 are obtained for the same model candidates, respectively. In addition, analyzing the annual maximum magnitude time series, the obtained AIC values are equal to 2.21 and 2.23 for the ARIMA (2,1,2) and ARIMA (3,1,2) models, respectively. For the same model-candidates, the BIC values are equal to 2.41 and 2.16. Thus, on the base of AIC and BIC criteria, the ARIMA (2,1,2) model is considered the  
Figure 10: Observed and forecasted (in red) values for the number of earthquakes (top) and the annual maximum magnitude (bottom) using the SSA model. best one to describe the annual earthquake number and the annual maximum magnitude during the training period. The parameters of the SSA model are then obtained, as previously proceeded, by testing several window lengths (12, 24 and 36 samples). The best model is also derived using the RMSE criterion. Analyzing the annual earthquake number time series, the lower RMSE value equal to 3.62 is derived for the model SSA (24,2), whereas for the models SSA (12, 2) and SSA (36,2), the values are equal to 4.40 and 3.89, respectively. Concerning the annual maximum magnitude time series, the lower RMSE is derived for the model SSA (36,2), whereas for the models SSA (12,2) and SSA (36,2), the values are equal to 0.379 and 0.375, respectively. Finally, for the two considered time series, SSA (24,2) is considered the best model. Then, using the previously described procedure, the chosen models ARIMA (2,1,2) and SSA (24,2) are used to forecast the annual number of earthquakes and the annual maximum magnitude from 2016 to 2019. Uncommonly, no events with magnitudes greater than Mw 4.0 occurred in the studied region in 2018, whereas two events with magnitudes greater than Mw 4.0 occurred in 2016, 2017 and 2019. Then, the period from 2016-2019 appears as a low seismicity epoch. That being the case, the forecasted values using the model ARIMA (2,1,2) are greater than real ones.