## 3 Methods
### Signal Preprocessing Method  
Singular spectrum analysis (SSA) is a set of techniques that are widely applicable to a multitude of real-world problems, with applications including, but not being limited  
Figure 1: Proposed architecture of MFFD. to, the fields of classical time series analysis, multivariate statistical analysis, multivariate geometry, dynamical systems research, and signal processing. The main purpose of SSA is to decompose the original time series into multiple component sequences, each of which can be classified into trend components, periodic components, quasi-periodic components, and noise components. The SSA method consists of two stages and four steps: the decomposition stage, which includes embedding analysis and singular value decomposition; and the reconstruction stage, which is divided into grouping and diagonal averaging. The four steps are as follows:  
(1) Embedding: Convert the one-dimensional time series \(x_{n}\) (n = 1, 2, \(\ldots\), N) into a multi-dimensional column vector \(x_{1}\), \(x_{2}\), \(\ldots\), \(x_{N}\), \(X_{l}\) = \((x_{1}\), \(\ldots\), \(x_{i+L-1})^{\mathrm{T}}\) (with T indicating the transpose) for i = 1, 2, \(\ldots\), k. Here, \(\mathrm{k}\) = N - L + 1, and the embedding parameter L is an integer that satisfies 2 \(\leq\) L \(\leq\) N.  
(2) Singular value decomposition (SVD) [37]: The trajectory matrix \(X\) is constructed as follows. \[X=[X_{1},X_{2},\cdots,X_{k}]= \begin{bmatrix}x_{1}&x_{2}&\cdots&x_{k}\\ x_{2}&x_{3}&\cdots&x_{k+1}\\ \vdots&\vdots&\ddots&\vdots\\ x_{L}&x_{L+1}&\cdots&x_{N}\end{bmatrix}  \tag{1}\]  
The matrix \(X\) has dimensions L \(\times\) (N - L + 1), representing the result of the embedding operation. Now, define the covariance matrix \(Cx\) = \(X^{\mathrm{T}}X\), which is a square matrix of dimensions (N - L + 1) \(\times\) (N - L + 1) and can be decomposed using eigenvalue decomposition. The eigenvalues are \(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{L}\geq 0\), and the corresponding eigenvectors form an orthogonal matrix \(U\) = (\(U_{1}\), \(U_{2}\), \(\ldots\), \(U_{L}\)), called the empirical orthogonal functions (EOFs). Using singular value decomposition, approximate the matrix \(X\) as a sum of rank-1 matrices as follows. \[X=X_{1}+X_{2}+\cdots+X_{d}=\sum\
olimits_{m=1}^{d}\sum\sqrt{\lambda_{m}}U_{ m}V_{m}^{\mathrm{T}} \tag{2}\]  
\(\lambda_{m}\) is the m-th singular value, and \(U_{m}\) and \(V_{m}\) are the corresponding eigenvectors. This step represents decomposing the matrix \(X\) into the sum of multiple low-rank matrices. (3) Grouping: In this step, partition the set {1, 2, \(\ldots\), d} into p non-overlapping subsets {1, \(\ldots\), Ip}. For each subset I, define the composite matrix \(X_{I}\) as follows. \[X_{I}=X_{i1}+X_{i2}+\cdots+X_{in} \tag{3}\]  
Thus, the matrix \(X\) can be represented as follows. \[X=X_{I1}+X_{I2}+\cdots+X_{Ip} \tag{4}\]  
(4) Diagonal averaging: Each \(X_{Ij}\) (j = 1, 2, \(\ldots\), p) can be transformed into a time series of length N using diagonal averaging. Let A be an L \(\times\) K matrix, with elements \(a_{ij}\), \(1\leq i\leq\mathrm{L},1\leq j\leq\mathrm{K},\mathrm{L}^{\ast}=\min\{\mathrm{L },\mathrm{K}\}\), \(\mathrm{K}^{\ast}=\max\{\mathrm{L},\mathrm{K}\}\), and N = L + K - 1. The matrix A can be transformed into the time series \(a  
Therefore, for each matrix \(X_{ij}\) using the method described above, a reconstructed sequence can be generated as follows. \[RC_{m}=\left(\widetilde{x}_{m,1},\cdots,\widetilde{x}_{m,N}\right) \tag{6}\]  
Finally, each reconstructed sequence \(RC_{m}\) is generated from each composite matrix \(X_{ij}\), and the original time series can be reconstructed using the following formula. \[x_{n}=\sum\
olimits_{m=1}^{p}RC_{m,n}=\sum\
olimits_{m=1}^{p}\widetilde{x}_ {m,n}n=1,\cdots,N \tag{7}\]  
This step represents the process of reconstructing the original time series by summing the weighted contributions from the individual components.