## 2 Singular Spectrum Analysis  
SSA is a method to study the de-noising or prediction of one-dimensional time-series signals. It mainly includes two essential parts: decomposition and reconstruction. Decomposition includes embedding and SVD. Reconstruction includes regrouping and diagonal averaging. The specific steps are as follows. (1) Embedding  
One-dimensional TEM data \(y=[y(1),y(2),\ldots,y(N)]\) can be mapped into an \(M\times L\) trajectory matrix \(\mathbf{Y}\)  
\[\mathbf{Y}=\left[ \begin{array}{cccc}y(1)&y(2)&\cdots\cdots&y(L)\\ y(2)&y(3)&\cdots\cdots&y(L+1)\\ \vdots&\vdots&\cdots\cdots&\vdots\\ y(M)&y(M+1)&\cdots\cdots&y(N)\end{array} \right] \tag{1}\]  
where \(M\) is the window length and \(L=N-M+1\), the trajectory matrix \(\mathbf{Y}\) is called the delay matrix [24]. (2) Singular Value Decomposition  
First, calculate the covariance matrix \(\mathbf{C}=\mathbf{Y}\mathbf{Y}^{\mathrm{T}}\) of the trajectory matrix \(\mathbf{Y}\), which is expressed as  
\[\mathbf{C}= \begin{bmatrix}c(0)&c(1)&\ldots&c(M-1)\\ c(1)&c(0)&\ldots&c(M-2)\\ \vdots&\vdots&\ddots&\vdots\\ c(M-1)&c(M-2)&\ldots&c(0)\end{bmatrix}  \tag{2}\]  
**C** is a Toeplitz matrix and needs to obtain its eigenvalue \(\lambda\) and eigenvector matrix U. The eigenvalues are arranged in descending order and correspond to \(M\times 1\) eigenvector. The eigenvalue vector is expressed as  
\[\lambda=\left[\lambda_{1},\lambda_{2},\cdots,\lambda_{M}\right],\ \left|\lambda_{1}\ \right. \left|\geq\left|\lambda_{2}\right|\geq\cdots\geq\left|\lambda_{M}\right|\right. \tag{3}\]  
The singular value vector \(\sigma\) of the matrix **C** can be presented as  
\[\sigma=\left[\sigma_{1},\sigma_{2},\cdots,\sigma_{M}\ \right]=\left(\sqrt{ \left|\lambda_{1}\right|},\ \sqrt{\left|\lambda_{2}\right|}\cdots,\ \sqrt{\left|\lambda_{M}\right|}\right) \tag{4}\]  
In general, the singular value vector \(\sigma\) of matrix **C** is called the singular spectrum of **Y**. The effective singular value is the singular value that is greater than zero in the singular value vector. In the actual data processing, the elements in the singular spectrum vector \(\sigma\) of the real sampled signal vector are generally larger than zero. The part with the smaller singular value is close to zero. The part that is more significant than zero is usually regarded as an effective singular value. The trajectory matrix **Y** is composed of several elementary matrices:  
\[\textbf{Y}=\textbf{Y}_{\textbf{1}}+\textbf{Y}_{\textbf{2}}+\ldots+\textbf{Y}_ {\textbf{j}} \tag{5}\]  
Let \(=\textbf{U}_{j}/\sqrt{\lambda_{j}}\), where \(\textbf{U}_{j}\) and \(\textbf{V}_{j}\) are the left and right eigenvectors corresponding to the eigenvalues after the trajectory matrix **Y** is decomposed. Then the trajectory matrix of the \(j\)th component in **Y** can be expressed as  
\[\textbf{Y}_{j}=\sqrt{\lambda_{j}}\textbf{U}_{j}\textbf{V}_{j}^{T}\ \ \ \ \ j=1,2, \cdots,M \tag{6}\]  
Substituting \(\textbf{V}_{j}\) in (6) and merging similar terms, \(\textbf{Y}_{j}\) can be presented as  
\[\textbf{Y}_{j}=\textbf{U}_{j}\textbf{U}_{j}^{T}\textbf{Y}\ \ \ \ \ \ j=1,2, \cdots,M \tag{7}\]
3. Reconstruction Let \(J=\{j_{1},j_{2},\cdots,j_{d}\}\) be the indices corresponding to \(d\) eigenvalues, then the matrix \(\textbf{Y}_{J}\) will be synthesized as  
\[\textbf{Y}_{\textbf{j}}=\sum_{i=1}^{d}\textbf{Y}_{\textbf{j}_{\textbf{j}}} \tag{8}\]  
Divide the subscripts of the matrix \(\textbf{Y}_{j}\) into \(p\) disjoint subsets \(J_{1},J_{2},\cdots J_{p}\), then the original trajectory matrix **Y** can be expressed as  
\[\textbf{Y}=\sum_{j=1}^{p}\textbf{Y}_{J_{p}} \tag{9}\]  
(4) Diagonal averaging  
Finally, the trajectory matrix is converted into one-dimensional data with \(N\) time channels in the diagonal averaging operation. \(\hat{S}\) is the \(L\times K\) trajectory matrix. According to the diagonal averaging formula, \(\hat{S}(n)\) is converted into a one-dimensional time series. The diagonal averaging formula is as  
\[\hat{S}(n)=\left\{