nsigma(x)
nu(x)
nv(x)
## S3 method for class 'ssa':
nspecial(x)
## S3 method for class 'ssa':
summary(object, digits = max(3, getOption("digits") - 3), ...)
## S3 method for class 'ssa':
$(x, name)nu, nv, nsigma,
nspecial routines, matrix or vector for $ operator.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 $\mathbf{X}$, while $V_i$ belongs to the row space of $\mathbf{X}$. Therefore, let us consider such decompositions called consistent.
Note that $(*)$ is a decomposition of $\mathbf{X}$ 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 $\mathbf{X}$. Minimal decomposition has exactly $r$ addends. Introduce the Frobenius-inner product as $\langle \mathbf{Z}, \mathbf{Y} \rangle _\mathrm{F} = \sum_{i, j} z_{i, j} \cdot y_{i, j}$. Thus, we can say about F-orthogonality and F-orthogonal decompositions if $\mathbf{X}_i$ are F-orthogonal. For F-orthogonality, left or right orthogonality is sufficient.
Generally, $\|\mathbf{X}\|^2$ can be not equal to $\sum_i \|\mathbf{X}_i\|^2$. For F-orthogonal decompositions, $\|\mathbf{X}\|^2 = \sum_i \|\mathbf{X}_i\|^2$.
The contribution of k-th matrix component is defined as ${\|\mathbf{X}_k\|^2} / {\|\mathbf{X}\|^2} = {\sigma_k^2} / (\sum_i \sigma_i^2)$.
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.
However, it is possible to extract some fields out of it using the
operator $, in particular, the following values of argument
name are supported:
[object Object],[object Object],[object Object]
If SSA with projections is being performed, then the eigentriples are ordered as follows:
at first, row projection triples, then column projection triples and then
SVD-triples. Non-SVD triples (like projection triples) are called `special triples'.
The number of special triples can be obtained by calling nspecial method.
Also, one can use the following fields of the SSA object:
[object Object],[object Object]
Rssa for an overview of the package, as well as,
ssa,
calc.v,
iossa,
fossa,# Decompose 'co2' series with default parameters
s <- ssa(co2, neig = 20)
# Show the number of eigentriples saved in the 's'
print(nsigma(s))
# Show the summary
summary(s)
# Show the set of singular values
print(s$sigma)
# Show the first eigenvector
print(s$U[, 1])
# Decompose `co2' series with polynomial projections
s <- ssa(co2, row.projector = 1, column.projector = 2)
print(nspecial(s))
print(c(s$nPL, s$nPR))
# Reconstruct a polynomial trend
plot(reconstruct(s, groups = list(trend = seq_len(nspecial(s)))))Run the code above in your browser using DataLab