# Earthquake Magnitude and Frequency Forecasting in Northeastern Algeria using Time Series Analysis
## 1 Introduction
It is worth noting that several authors [25; 26; 27] suggested that the extreme value distribution is a more suitable analytical model and should be employed for the distribution of the magnitude set rather than the GR model. Stochastic processes, particularly Poisson processes, are also used to forecast earthquake frequencies by taking the earthquake occurrence time into account. The Poisson model is characterized by a constant hazard function and an exponential recurrence time distribution. This assumption leads to the unsatisfying result of only time-independent seismic hazard or seismic forecasting estimates, suggesting that the probability of an earthquake at any given time is independent of its magnitude and the time since the last one, respectively. Additionally, compared to the Poisson law, the distribution of earthquakes is over-dispersed, since the events are grouped in time and location. Various other models based on stochastic processes have also been investigated to predict the number of events, such as compound Poisson [28; 29], branching [15; 30] and stochastic point processes [31; 32]. These stochastic procedures made predicting the magnitude of the earthquake challenging. In addition, these approaches are ineffective for determining earthquake seasonality and patterns. Recently, a few studies attempted to use forecasting models based on time series, such as the Auto-Regressive Integrated Moving Average (ARIMA) and the generalized autoregressive conditional heteroscedasticity (GARCH) models [33; 34], to forecast earthquake magnitudes by considering the seasonality and trends of earthquake series. Another model, the singular spectrum analysis model, or SSA [35; 36; 37], is becoming more attractive in order to forecast earthquake magnitudes in a specific seismic zone. It is a reliable and advanced non-parametric time series analysis method combining dynamical systems, signal processing, multivariate statistics and traditional forecasting analysis. According to several authors [35; 36; 37], this approach is useful for determining the magnitude of earthquakes in a given area. The SSA model can identify significant seismic time series components exhibiting typical irregular behavior and provide accurate forecasts for them. Despite the fact that a few studies on earthquake magnitude forecasting using the SSA approach have been conducted [37], the primary goal of our study is to apply this method in Northeastern Algeria to predict earthquake magnitudes with time effects, which are frequently missed in estimates, while also capturing the dynamics of earthquake occurrences. Subsequently, the ARIMA and SSA models provided the annual maximum earthquake magnitudes for the first time in the studied area. The root mean square error approach, usually denoted as RMSE, is a common tool for comparing different models and/or methodologies in time series [38; 39]. In this study, the estimated RMSE values show that the SSA is the best model for describing the number of earthquakes and the annual maximum magnitude in this region. The seismicity in this area is continuous, and it is characterized by low-to-moderate seismic activity. Despite the interest of the scientific community in regional seismology and seismic hazards, the studied area remains of great interest due mainly to the tectonic complexity, which resulted in the identification of several active faults, the characteristics of which are still under investigation [40]. Then, this study is the first one in this region to address the issue of forecasting using time series, and it is part of the scientific community's efforts to manage and reduce seismic risk. It is worth noting that, previously, a model has been developed for Northern Morocco and Algeria to assess the probability of exceeding magnitudes Mw 5.0 and 6.0 in 10 years [41], but assuming a Poissonian process on a spatially smoothed seismicity model. This study enabled for the spatial variation of such results.