# Earthquake Magnitude and Frequency Forecasting in Northeastern Algeria using Time Series Analysis
## 5 Validation of the Procedure
In 2016 and 2019, about four events were predicted, and three events were predicted in 2017. On the other hand, values found using the SSA (24,2) model are of the order of 3 for each year, and hence closer to the observed values. Regarding the annual maximum magnitude, similar values are obtained using the ARIMA (2,1,2) and SSA (24,2) models, of the order of Mw 5.0 for 2016, 2017 and 2019. These forecasted values are in agreement with the observed ones, which are equal to Mw 5.0, 4.5 and 4.7 for 2016, 2017 and 2019, respectively. Thus, clearly the SSA (24,2) model makes it possible to find realistic results corroborating the observed data. A statistical test is used to assess the consistency of the results obtained using the SSA algorithm in the period from 2016 to 2019. According to [PERSON] et al. [19], to take into account the uncertainty, the likelihood test, named L-test, can be conducted by simulating the observed events. In the current study, we focus on the statistical N-test [42], which consists to test the rate forecast. The N-test is intended to measure, in a probabilistic manner, how the forecasted number of events will match the observed number of earthquakes. Then, the N-test [19; 42; 106] is implemented. The N-test takes into account two linked quantile scores, \(\delta_{1}\) and \(\delta_{2}\), reflecting whether the produced sequences produced forecasted event numbers \(N_{fore}\) that were higher or lower than the observed values \(N_{obs}\), as given in the equations  
\[\delta_{1}=1-P\Big{(}(N_{obs}-1)\Big{|}N_{fore}\Big{)} \tag{15}\]  
\[\delta_{2}=P\Big{(}N_{obs}\Big{|}N_{fore}\Big{)} \tag{16}\]  
bearing in mind that \(P(\omega|\lambda)=\frac{\lambda^{\omega}}{\omega!}exp(-\lambda)\). The quantity \(\delta_{1}\) is the probability of observing at least \(N_{obs}\), and \(\delta_{2}\) is the probability of observing at most \(N_{obs}\). Both the overall forecast rate and observed number of events are assumed to be Poissonian and described by \(N_{fore}\) and \(N_{obs}\), respectively. The quantity \(\delta_{2}\) describes the right-continuous Poisson cumulative distribution with the expectation \(N_{fore}\) at corresponding \(N_{obs}\) at the times evaluated. This score describes the fraction of forecast expectations smaller than the observed events. The probability that more than \(N_{obs}\) events are forecasted is given by \((1-\delta_{2})\). The problem with this approach is addressed in [42]. Instead, the \(\delta_{1}\) probability was added in addition to the original N-test to describe at least \(N_{obs}\), in which the user only needs to be concerned about low probability values [19; 42]. The quantile scores in this statistical N-test assess if the number of forecasted occurrences is inconsistent with \(N_{obs}\). A small \(\delta_{1}\) indicates that the forecast underestimates the observed sequence, whereas a small \(\delta_{2}\) suggests that the forecast overestimates the number of occurrences. The forecast can thus be rejected if the probabilities of \(\delta_{1}\) and \(\delta_{2}\) are less than the effective significance level [42]. The effective significance threshold for the one-sided N-tests was set at 2.5% to coincide with a single quantile score, which corresponds to a 5% error rate for the test [42,107]. Table 5 presents the obtained results, the observed \(N_{obs}\) and the forecasted number of earthquakes \(N_{fore}\), and the quantile scores \(\delta_{1}\) and \(\delta_{2}\). The N-test performance can be interpreted by observing whether the quantile scores \(\delta_{1}\) and \(\delta_{2}\) fall the region between 0.025 and 1. In Figure 11, we can observe that high \(\delta_{1}\) scores tend to correspond to low \(\delta_{2}\) scores.