# Earthquake Magnitude and Frequency Forecasting in Northeastern Algeria using Time Series Analysis
## 3 Methodology Outline
The Akaike information criterion [78] and the Bayesian information criterion [79], denoted as _AIC_ and BIC, respectively, are used to select the best model, the one with minimum information criterion values and white noise error series. According to [PERSON] et al. [80], it is worth noting that the _AIC_ criterion represents the relationship between the Kullback-Leibler measure [81] and the maximum likelihood estimation method. The Kullback-Leibler measure is developed to capture the lost information in the estimation, which means that this measure selects the good model minimizing the loss of information. Usually, the AIC criterion is given by the equation  
\[AIC=-2 logL\Big{(}\widetilde{\theta}\Big{)}+\ 2K \tag{7}\]  
where \(\theta\) is the set of model parameters, \(L(\widetilde{\theta})\) is the likelihood of the candidate model given the data when evaluated at the maximum likelihood estimate of \(\theta\), and \(K\) is the number of the estimated parameters in the candidate model. For small samples, the corrected Akaike information criterion, denoted as \(AIC_{c}\), should be used instead of the \(AIC\) criterion described previously. The relation giving the \(AIC_{c}\) is  
\[AIC_{c}\ =-2\ logL(\widetilde{\theta})+\ 2K+\ \frac{2K+1}{N-K-1} \tag{8}\]  
where \(N\) is the number of observations. [PERSON] and [PERSON] [82] define a small sample size as one that is less than 40. It is noteworthy that, as N increases, the third term in \(AIC_{c}\) approaches zero, producing the same result as the \(AIC\) criterion. The Bayesian information criterion [79], denoted as \(BIC\), is another model selection criterion based on the information theory but set within a Bayesian context. The difference between the \(BIC\) and the \(AIC\) criteria is that the former puts a greater penalty for the number of parameters than the latter. It is computed using the following relation  
\[BIC=-2\ logL(\widetilde{\theta})+KlogN \tag{9}\]  
where the terms are the same as described in the definition of the \(AIC\) criterion. As in the previous criterion, the best model is the one providing the minimum \(BIC\) value. The Single Spectrum Analysis (SSA) model is a time series forecasting method commonly used to analyze time series with periodic oscillations. The application of the SSA method is advantageous because it encompasses decomposing a time series into simpler components, such as a gradually changing trend, oscillations and noise. Various disciplines have implemented the SSA method, including signal processing [83], nonlinear dynamics [84; 85], climate data [86; 87; 88], medical science [89] and mathematical statistics [90]. Furthermore, when combined with neural networks or other comparable techniques, it is a powerful pre-processing tool for time series forecasting [91; 92]. Decomposition and reconstruction are the two complementary processes that make up the SSA implementation algorithm. According to [PERSON] and [PERSON] [93], we consider a real-valued time series \(X=(x_{1},\,\ldots,\,x_{N})\) with length \(N\) (\(N>2\)), satisfying \(x_{j}\
eq 0\) for at least one \(j\). The first step in the decomposition stage, named the embedding step, consists in transforming the original time series into a sequence of \(L\)-dimensional vectors, \(X_{i}=(x_{i},\,\ldots,\,x_{j})^{T}\), where \(J=N-L+1\), and \(L\) is the window length. The \(J\)-formed vectors are called \(L\)-lagged vectors and present the columns of the trajectory matrix (or \(L\)-trajectory matrix) of the series \(X\) (\(L\times J\)) given by  
\[X=\left( \begin{array}{cccc}x_{1}&x_{2}&\cdots&x_{J}\\ x_{2}&x_{3}&\cdots&x_{J+1}\\ \vdots&\vdots&\ddots&\vdots\\ x_{L}&x_{L+1}&\cdots&x_{N}\end{array}