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SSA: algorithm

SSA (singular-spectrum analysis) is a novel technique of time series analysis incorporating the elements of classical time series analysis, multivariate statistics, multivariate geometry, dynamical systems and signal processing. The main idea of SSA is in performing a singular value decomposition (SVD) of the `trajectory matrix' obtained from the original time series with subsequent reconstruction of the series. The basic version of SSA consists of four steps, which are performed as follows (see Golyandina, Nekrutkin and Zhigljavsky (2001) for other versions of SSA).
Step 1.
Embedding. Let $x_1, \ldots, x_{N}$ be a time series of length $N$, and $M$ be an integer, which will be called the `lag parameter'. We set $K=N-M+1$ and define the$M$-lagged vectors $X_j = (x_{j}, \ldots, x_{j+M-1})^{\rm T},\,$$j=1,2,\ldots$, and the trajectory matrix
\begin{displaymath}{\bf X}=(x_{i+j-1})_{i,j=1}^{M,K}=[X_1:\ldots:X_{K}] \, .\end{displaymath}


Note that the trajectory matrix ${\bf X}$ is a Hankel matrix, which means that all the elements along the diagonal $i\!+\!j\!=$ const are equal.

Step 2.
SVD of the trajectory matrix. Compute the singular value decomposition (SVD) of the matrix ${\bf X}$. It can be obtained via eigenvalues and eigenvectors of the so-called lag-covariance matrix ${\bf S}= {\bf X}{\bf X}^{\rm T}$ of size $M\times M$. Denote by $\lambda_1,\ldots,\lambda_M$ the eigenvalues of ${\bfS}$ and assume that they are arranged in the decreasing order, so that $\lambda_1\geq \ldots \geq \lambda_M \geq 0.$ Let $U_1,\ldots,U_M$ be the corresponding orthonormal eigenvectors of${\bfS}$ (in SSA literature they are often called `empirical orthogonal functions' or simply EOFs) and denote $V_i={\bfX}^{\rm T}U_i$ for $i=1,\ldots, d$, where $d$ is the number of nonzero eigenvalues $\lambda_i$. The vectors $V_i$ are the eigenvectors of the matrix ${\bf X}^{\rm T}{\bf X}$; they are sometimes called `principal components'. As a result of the SVD we obtain a representation ${\bf X}\! =\! {\bf X}_1\!+ \!\ldots \!+ \!{\bf X}_d,$ where ${\bf X}_i=\sqrt{\lambda_i} U_iV_i^{\rmT}$ $(i=1,\ldots,d)$ are rank-one biorthogonal matrices.
Step 3.
Grouping. Split the set of indices $\{1,\ldots,d\}$ into two groups, namely $I =\{ i_1, \ldots, i_l\}$ and $\bar I= \{1,\ldots,d\}\setminus I;$ then sum the matrices${\bf X}_i$ within each group. The result of the step is the representation
\begin{displaymath}{\bf X}= {\bf X}_{I} + {\bf X}_{\bar I}\, , \;\; {\rmwh......and}\; {\bf X}_{{\bar I}} = \sum_{i\notin I} {\bf X}_i \, .\end{displaymath}
Step 4.
Reconstruction. Averaging over the diagonals$i\!+\!j\!=$ const of the matrices ${\bf X}_{{I}}$ and ${\bfX}_{{\bar I}}$ is performed. Applying then twice the one-to-one correspondence between the series of length $N$ and the Hankel matrices of size $M\times (N-M+1),$ we obtain two series (denote them by $z_t$ and $\varepsilon_t$) and the SSA decomposition, which is a decomposition of the original series into a sum of two series
\begin{displaymath}x_t = z_t+ \varepsilon_t\, , \;\;\;t=1,\ldots,N.\end{displaymath}
 (1)
Here the series $z_t$ (obtained from the diagonal averaging of${\bf X}_{{I}}$) can often be associated with signal and the residual series $\varepsilon_t$ with noise.
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