## 3 Results  
The denoising process consists of four steps, embedding, singular-value decomposition, grouping, and diagonal averaging [35]. According to the observed time-series wind-speed data and these four steps, the trajectory matrix was constructed. The trajectory matrix was decomposed and reconstructed to extract signals representing different components of the original time series, such as long-term trend signals, periodic signals, and noise signals. In order to obtain satisfying results, the noise signals were abandoned. The detailed process is as follows:  
(1) Embedding:  
Form the trajectory matrix of the series \(X\), which is the \(L\times K\) matrix:  
\[X= [x_{1},\cdots,x_{N}]=\left(x_{ij}\right)_{i,j=1}^{L,K}=\left[ \begin{array}[ ]{cccc}x_{1}&x_{2}&x_{3}&\cdots&x_{K}\\ x_{2}&x_{3}&x_{4}&\cdots&x_{K+1}\\ x_{3}&x_{4}&x_{5}&\cdots&x_{K+2}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ x_{L}&x_{L+1}&x_{L+2}&\cdots&x_{N}\\ \end{array} \right] \tag{1}\]  
where \(X= \left(x_{1},\ \cdots,\ x_{i+L-1}\right)^{T},\left(1<i<K\right)\) are lagged vectors of size \(L\). Matrix \(X\) is a Hankel matrix that means that \(X\) has equal elements \(x_{ij}\) on the anti-diagonals \(i+j=const\). (2) Singular-Value Decomposition:  
Perform the singular value decomposition (\(SVD\)) of trajectory matrix \(X\). Set \(S=XX^{T}\) and denote by \(\lambda_{1},\cdots,\lambda_{L}\) the eigenvalues of \(S\) taken in the decreasing order of magnitude \(\lambda_{1}\geq\cdots\geq\lambda_{L}\geq 0\), and by \(U_{1},\ldots,U_{L}\) the orthonormal system of the eigenvectors of matrix \(S\) corresponding to these eigenvalues. Set \(d=rankX=\max\{i\), such that \(\lambda_{i}>0\}\) (note that \(d=L\) for a typical real-life series) and \(V_{i}=X^{T}U/\sqrt{\lambda_{i}},(i=1,\ldots,d)\). In this notation, the SVD of trajectory matrix \(X\) can be written as \(X=X_{1}+\cdots+X_{N}\). Where \(X_{i}=\sqrt{\lambda_{i}}U_{i}V_{i}^{T}\) are matrices having rank 1; these are called elementary matrices. Collection \((\sqrt{\lambda_{i}},\ U_{i},\ V_{i}^{T})\) is the \(i\)th eigentriple (ET) of the SVD. Vectors \(U_{i}\) are the left singular vectors of matrix \(X\), and numbers \(\sqrt{\lambda_{i}}\) are the singular values and provide the singular spectrum of \(X\); this gives the name to SSA. Vectors \(V_{i}\sqrt{\lambda_{i}}=\ X^{T}U_{i}\) are the vectors of principal components (PCs). (3) Eigentriple grouping:  
Partition set of indices \(\{1,\ \ldots,\ d\}\) into m disjoint subsets \(I_{1},\cdots,I_{m}\). Let \(I=\left\{i_{1},\cdots,i_{p}\right\}\). Then, resultant matrix \(X_{I}\) corresponding to group \(I\) is defined as \(X=X_{I_{1}}+\cdots+X_{I_{m}}\). The resultant matrices are computed for the groups, and the grouped SVD expansion of \(X\) can now be written as \(X=X_{I_{1}}+\ldots+X_{I_{m}}\). (4) Diagonal averaging:  
Each matrix \(X_{I_{j}}\) of the grouped decomposition is hankelized, and then the obtained Hankel matrix is transformed into a new series of length \(N\) using the one-to-one correspondence between Hankel matrices and the time series. Diagonal averaging applied to a resultant matrix \(X_{I_{k}}\) produces a \begin{table}
\begin{tabular}{c c c} \hline
**Metric** & **Definition** & **Equation** \\ \hline
**AE** & The average forecast error of n times forecast results & \(\mathbf{AE}=\frac{1}{N}\sum_{n=1}^{N}\left(y_{n}-\hat{y}_{n}\right)\) \\
**MAE** & The average absolute forecast error of n times forecast results & \(\mathbf{MAE}=\frac{1}{N}\sum_{n=1}^{N}\left|y_{n}-\hat{y}_{n}\right|\) \\
**MSE** & The average of the prediction error squares & \(\mathbf{MSE}=\frac{1}{N}\sum_{n=1}^{N}\left(y_{n}-\hat{y}_{n}\right)^{2}\) \\
**MAPE** & The average of absolute percentage error & \(\mathbf{MAPE}=\frac{1}{N}\sum_{n=1}^{N}\left|\frac{y_{n}-\hat{y}_{n}}{y_{n}} \right|\times 100\%\) \\ \hline \end{tabular}
\end{table} 
Table 1: Four metrics. reconstructed series \(\widetilde{X}^{(k)}=(\widetilde{X}_{1}^{(k)},\cdots,\widetilde{X}_{N}^{(k)})\).