About decompositions
The result of Decomposition step of SSA and its modifications can be written down in the following form:
$$%
{(*)} \quad \mathbf{X} = \sum_i \mathbf{X}_i, \qquad \mathbf{X}_i = \sigma_i U_i V_i^\mathrm{T},$$
where $X$ is the trajectory matrix,
$U_i in R^L$, $V_i in R^K$, $\sigma_i$ are non-negative numbers.
Also, we assume that $||U_i|| = 1$, $||V_i|| = 1$. The Singular Value Decomposition is a particular case of $(*)$ and corresponds to orthonormal systems of
${U_i}$ and ${V_i}$.
We call $(\sigma_i, U_i, V_i)$ eigentriple, $\sigma_i$
are singular values, $U_i$ are left singular values or eigenvectors,
$V_i$ are right singular vectors or factor vectors, by analogy with the SVD. For the most of SSA decompositions, $U_i$ belongs to the column space of $X$,
while $V_i$ belongs to the row space of $X$.
Therefore, let us consider such decompositions called consistent. Note that $(*)$ is a decomposition of $X$ into a sum of rank-one matrices.
If the systems ${U_i}$ and ${V_i}$ are linearly-independent,
then the decomposition $(*)$ is minimal (has minimal possible number of addends). If at least one of the systems is not linear independent,
the decomposition $(*)$ is not minimal.
If both ${U_i}$ and ${V_i}$ are orthonormal,
then the decomposition $(*)$ is called bi-orthogonal.
If ${U_i}$ is orthonormal, the decomposition is called left-orthogonal;
If ${V_i}$ is orthonormal, the decomposition is called right-orthogonal. Let $r$ be rank of $X$. Minimal decomposition has exactly $r$ addends.
Introduce the Frobenius-inner product as
$_F = sum_{i, j} (z_{i, j} y_{i, j})$.
Thus, we can say about F-orthogonality and F-orthogonal decompositions if $X_i$
are F-orthogonal. For F-orthogonality, left or right orthogonality is sufficient. Generally, $||X||^2$ can be not equal to $sum_i ||X_i||^2$.
For F-orthogonal decompositions, $||X||^2 = sum_i ||X_i||^2$. The contribution of k-th matrix component is defined as
$||X_k||^2 / ||X||^2 = sigma_k^2 / (sum_i sigma_i^2)$. For F-orthogonal decompositions, the sum of component contributions is equal to 1.
Otherwise, this sum can considerably differ from 1
(e.g., the sum of component contributions can be 90% or 146%). Remark. If the system ${U_i}$ (or ${V_i}$) has vectors
that do not belong to the column (or row) spaces, then the decomposition can be not minimal even
if ${U_i}$ (or ${V_i}$) are linearly independent,
since these projections on the column (or row) space can be dependent.Decompositions for different SSA modifications
- Basic SSA
- the SVD, consistent, minimal, bi-orthogonal and
therefore F-orthogonal decomposition. Implemented in
ssa with kind='1d-ssa' - FOSSA
- consistent, minimal F-orthogonal
decomposition. Implemented in
fossa - IOSSA
- consistent, minimal oblique decomposition. Implemented
in
iossa - SSA with projections
- non-consistent if at least one basis
vector used for the projection does not belong to the column (row)
trajectory space, F-orthogonal decomposition. The components,
which are obtained by projections, are located at the beginning of
the decomposition and have numbers $1, \dots,
n_special$. Implemented in
ssa with
kind='1d-ssa' and non-NULL row.projector or
column.projector arguments - Toeplitz SSA
- generally, non-consistent, non-minimal
F-orthogonal decomposition. Implemented in
ssa with kind='toeplitz-ssa'