## 2 Methods  
### Singular Spectrum Analysis  
The strategy of basic SSA is to decompose the observed time series into a sum of independent components. The procedure of basic SSA is in two isolated steps: decomposition and reconstruction. Each consists of another two steps separately. The decomposition step is composed of time lagged embedding and singular value decomposition (SVD), while the reconstruction step is composed of diagonal averaging and grouping. A flowchart of basic SSA, consisting of the sub-steps of decomposition and reconstruction, is shown in Figure 1.  
In the first step, an initial dataset \(\mathbf{Y}_{N}\),  
\[\mathbf{Y}_{N}=\left( \begin{array}{cccc}y_{0}&y_{1}&\cdots&y_{N-1}\\ \end{array} \right), \tag{1}\]  
consisting of \(N\) samples with an equal time interval is projected in a vector space via delayed coordinate mapping over a lagged window length \(L\), which is an integer satisfying \(2\leq L\leq N/2\). The variable \(L\) is called the embedding dimension of the lagged vector space and should be adequately large. Through delayed coordinate mapping, we take \(L\) consecutive samples to create a time delay vector, \(\mathbf{x}_{i}\),  
\[\mathbf{x}_{i}=\left( \begin{array}{cccc}y_{i}&y_{i+1}&\cdots&y_{i+L-1}\\ \end{array} \right)^{T}. \tag{2}\]  
Considering the relationship between the number of samples \(N\) and the lagging window length \(L\), we get \(N-L+1\) lagged vectors. Next, we get the trajectory matrix \(\mathbf{X}\) by connecting all the lagged vectors. Obviously, since the elements on the vice diagonal lines are equal, \(\mathbf{X}\) is a Hankel matrix,  
\[\mathbf{X}=\left( \begin{array}{cccc}y_{0}&y_{1}&y_{2}&\cdots&y_{N-L}\\ y_{1}&y_{2}&y_{3}&\cdots&y_{N-L+1}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ y_{L-1}&y_{L}&y_{L+1}&\cdots&y_{N-1}\\ \end{array} \right). \tag{3}\]  
In the second step, SVD is applied to decompose the trajectory matrix \(\mathbf{X}\) into the form as follows:  
\[\mathbf{X}=\mathbf{U}\mathbf{\Sigma}\mathbf{V}^{T}, \tag{4}\]  
where \(\mathbf{U}\) and \(\mathbf{V}\) are unitary matrices. Their corresponding column vectors \(\mathbf{u}_{i}\) and \(\mathbf{v}_{i}\) are the left-singular vectors and right-singular vectors of \(\mathbf{X}\), which are orthogonal eigenvectors of matrices \(\mathbf{X}\mathbf{X}^{T}\) and \(\mathbf{X}^{T}\mathbf{X}\), respectively. The vectors \(\mathbf{u}_{i}\) and \(\mathbf{v}_{i}\) are defined as the factor empirical orthogonal functions (EOFs) and principal components (PCs) of \(\mathbf{X}\), respectively [28]. \(\mathbf{\Sigma}\) is a rectangular diagonal matrix whose elements on the diagonal line are in descending order. The elements on the diagonal line \(\sigma\) are singular values of \(\mathbf{X}\), which are the positive square roots of the eigenvalues of matrix \(\mathbf{X}\mathbf{X}^{T}\). Due to the dimension of decomposition \(d\) being equal to the rank of the matrix \(\mathbf{X}\), \(\mathbf{X}\) can be written as:  
Figure 1: Flowchart of basic singular spectrum analysis (SSA). (**a**) Procedure of basic SSA; (**b**) sub-procedure: decomposition; (**c**) sub-procedure: reconstruction. \[\mathbf{X}=\sum_{i=1}^{d}\mathbf{X}_{i}=\sum_{i=1}^{d}\mathbf{u}_{i}\sigma_{i}\mathbf{v}_{i}^{T}, \tag{5}\]  
where \(\mathbf{X}_{i}\) is one of the decomposition matrices of \(\mathbf{X}\). For every decomposition \(\mathbf{X}_{i}\), the corresponding pairs \((\mathbf{u}_{i}\sigma_{i}\mathbf{v}_{i})\) are sorted by descending order of singular values. Note that the square of each singular value can describe the contribution of the corresponding component to the initial signal; thus, the ratio \(\sigma_{k}^{2}/\sum_{i=1}^{d}\sigma_{i}^{2}\) can be considered as the contribution ratio of the \(k\)-th component \(\mathbf{X}_{k}\).