## 2 Literature Review
### Ssa  
SSA is a component of the more inclusive classification of methods called Principal Component Analysis (PCA), is a powerful technique used to analyze and predict non-parametric time series that include non-stationarity data elements with complex seasonal components and is superior to classical techniques. On the other hand, SSA is a technique used for the analysis of time series that are observed, and it is capable of exposing the main features of the predictability of time series. The traditional models, namely ARMA, ARIMA, Box-Jenkins and artificial intelligence, such as fuzzy models, etc., demand certain assumptions as restrictive distributional, structural ones and adequate historical data. A new method inspired by genetic colonial theory is introduced to forecast and filter time series [17] This method has many similarity with SSA. However, SSA does not use any statistical assumptions, namely the series stationarity or residuals normality, and it need not rely on any parametric model to find out the trend or oscillations. Moreover, contrary to many other methods, it is independent of sample size, even small sample sizes. The theoretical underpinnings and practical background of the SSA technique were introduced by [PERSON] et al. in 2001 [18] and further by [PERSON] [19; 20] and [PERSON] for common time series [21]. The applications of this technique are many and various, for example, in meteorology, physics, economics, and financial mathematics. In recent years, SSA has been designed and employed to various practical problems, for instance, in industrial production [22], by [PERSON] et al. [23] to load forecasting in the electricity market as well as other markets, for forecasting CO\({}_{2}\) emissions [24], and for signal extraction in a genetics related application [25]. Moreover, a forecasting method was presented with Artificial Neural Network and SSA [26] and a hybrid model Coupled with SSA was used to forecast rainfall [27]. Finally, SSA is introduced and reviewed the separability techniques and window length [28]. The SSA technique is composed of two steps that are complementary to each other, decomposition and reconstruction. Figure 1 presents, the standard SSA, which includes two main stages and four subsets. By reconstructing the original series after decomposition, it is possible to remove noise. The first stage is the decomposition stage, which has two steps:  
Step 1, embedding:  
Embedding is similar to mapping and can be regarded as a form of mapping. This form of mapping, i.e., embedding, turns a time series that is one-dimensional, \(Y_{T}=(y_{1},\,y_{2},\,\ldots,\,y_{T})\), into series that are multi-dimensional, \(X_{1},X_{2},\ldots,X_{k}\), using vectors, \(X_{i}=(y_{i},\ldots,y_{i+L-2})\in R^{L}\). \[X=\left( \begin{array}{ccc}y_{0}&\cdots&y_{K-1}\\ \vdots&\ddots&\vdots\\ y_{L-1}&\cdots&y_{T-1}\end{array} \right)\]  
Embedding has certain assumptions: K = T \(-\) L + 1; the window length, L, as the single parameter or an integer that should be 2 \(\leq\) L \(\leq\) T. Besides, the L or window length must be big enough. This step results in a matrix called a Hankel matrix, \(H=(X_{1},\ldots,X_{K})\), which has equal elements on the diagonal, and the linear space that is spanned by columns of H is called the trajectory. Step 2, SVD:  
The second step is called the SVD step, which sums up elementary matrices of rank-one bi-orthogonal to obtain the singular value decomposition of the trajectory, \(X=E_{1}+\ldots+E_{d}\), where \(E_{i}=\sqrt{\lambda_{i}}U_{i}\hat{V}_{i}\) and \(\lambda_{i}\) and \(U_{i}\) are eigenvalues and eigenvectors of matrix, \(X\hat{X}\), \(V_{i}=\hat{X}U_{i}/\sqrt{\lambda_{i}}\) and \(\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{L}\geq 0\).