## 2 Materials and Methods
The original time series \(F=(f_{1}f_{2},\)\(\dots\)\(f_{\mathrm{N}})\) of length N is transformed into a sequence of multidimensional vectors (trajectory matrix X) during this stage. The matrix X has the dimension L \(\times\) K, where K is number of the lagged vectors selected:  
\[\mathrm{K=N-L+1} \tag{1}\]  
Therefore, K is the number of columns and L is the number of rows of the trajectory matrix X. Each column of this matrix is rolling window of length L, so 1 st column has elements of the time series F from the first till L, 2 nd column has elements from the second till L + 1, and so on. \[\mathrm{X= \begin{pmatrix}f_{1}&\cdots&f_{\mathrm{N-L+1}}\\ \vdots&\ddots&\vdots\\ f_{\mathrm{L}}&\cdots&f_{\mathrm{N}}\end{pmatrix} } \tag{2}\]  
X is matrix with equal elements on all diagonals (so it may be considered as a Hankel matrix if L = N - L + 1). The second stage of the decomposition phase is singular value decomposition (SVD) of trajectory matrix. The result of this stage is the decomposition of the trajectory matrix X into elementary parts, i.e., representation of the matrix X as a sum of elementary matrices. Let \(\lambda_{1},\lambda_{2},\dots,\lambda_{\mathrm{L}}\) denote the eigenvalues of the matrix \(\mathrm{S=X\times X^{T}}\) taken in non-decreasing order, and d is number of \(\lambda_{i}>0\). Let us also denote U\({}_{1}\), U\({}_{2}\), \(\dots\), U\({}_{L}\) the eigenvectors of the matrix S, corresponding to the ordered eigenvalues [19]. Then, the singular value decomposition for X is  
\[\mathrm{X=\Sigma X_{i}} \tag{3}\]  
X\({}_{i}\) is the elementary matrix with rank 1. These elementary matrices are calculated by the formula:  
\[\mathrm{X}_{i}=\sqrt{\lambda_{i}}\mathrm{U}_{i}\mathrm{V}_{i}^{T}, \tag{4}\]  
where elementary vectors V\({}_{i}\) can be found after we calculate U and \(\lambda_{i}\), using the formula:  
\[\mathrm{V}_{i}=\mathrm{X}^{T}\frac{\mathrm{U}_{i}}{\sqrt{\lambda_{i}}} \tag{5}\]  
After decomposition, we have reconstruction phase. To reconstruct time series, we use diagonal averaging. Each matrix X\({}_{i}\) we obtained after decomposition phase is translated into a series of length N. We suppose L* = min (L, K), K* = max (L, K). For each element x of matrix X\({}_{i}\), let us also denote x*ab = xab, if L \(<\) K, and x*ab = xba, if L \(>\) K. Diagonal averaging [18] transforms each resulting matrix Xi into a series according to the formula for each element \(\widetilde{f_{ik}}\) of vector \(\widetilde{F}_{i}\):  
\[\widetilde{f_{ik}}= \begin{cases}&\frac{1}{k+1}\sum_{n=-1}^{k+1}\mathrm{x}_{n,k-n+2}^{*},\ \forall\ 0\leq k<\mathrm{L}^{*}-1\\ &\frac{1}{\mathrm{L}^{*}}\sum_{n=-1}^{k}\mathrm{x}_{n,k-n+2}^{*},\ \forall\ \mathrm{L}^{*}-1\leq k< \mathrm{K}^{*}\\ &\frac{1}{\mathrm{N-K}}\sum_{n=-\mathrm{K}^{*}+2}^{*}\mathrm{x}_{n,k-n+2}^{*}, \ \forall\ \mathrm{L}^{*}-1\leq k<\mathrm{K}^{*}\end{cases}  \tag{6}\]  
This formula averages the elements along the diagonals. Therefore, applying diagonal averaging to the resulting matrices X\({}_{i}\), we obtain the series \(\widetilde{F}_{i}\). These series can be considered as additive components of the time series f that was under study. These components may be classified as a trend, cycle, or noise component [17], and then we can model it. In spite of its significant advantages, the SSA method requires fairly good expert analysis of the components of a time series in order to correctly assess their content. This is a rather labor-intensive process due to the large amount of work on expert evaluation in the comparative analysis of the dynamics of a large number of phenomena and processes,presumably related to each other (which is possible and can lead both to construction of quantitative models and to identification of fundamental features of the processes under study, as we showed in [7]). Expert assessments are also complicated by possible changes in trends, changes in the nature of fluctuations, as well as the complexity of expert separation based on visual assessment of short-term cyclical fluctuations and autocorrelation.