decompSSAM: Singular System Analysis for time series with missing values
In simsalabim: A collection of methods for time series analysis and signal detection.

Description Usage Arguments Details Value Warning Author(s) References See Also Examples

Implementation of the SSA algorithm for time series with missing values following Golyandina and Osipov (2007).

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decompSSAM(x, L, tau = 0, toeplitz = FALSE, getFreq = TRUE)
reconSSAM(dSSAM, x, groups,
      method = list(alpha = "PI", beta = "simultaneous"))

`x`	A vector representing the time series.
`L`	Embedding dimension.
`tau`	A number ranging from 0 to 1, indicating the maximal proportion of missing values within columns of the trajectory matrix for the usage of a modification of the inner product. If tau is negative it is treated as if it is zero. Only applies if `toeplitz = FALSE`.
`toeplitz`	Whether to use the Toeplitz modification of SSA for stationary time series or not.
`getFreq`	Whether dominant frequencies of the eigenvectors shall be determined.
`dSSAM`	Output of decompSSAM
`groups`	A list of vectors. Each vector is representing a selection of eigenvalues and eigenvectors which shall be used to compute reconstructed components.
`method`	Method or combination of methods that shall be used to estimate the reconstructed components at the position of missing values. Currently one of `"PC"` or `list(alpha="PI",beta="simultaneous")`.

decompSSAM performs the SSA decomposition whereas reconSSAM performs the SSA reconstruction, which is a type of band pass filtering.

In general the application of SSA for time series with missing values follows the same principals as for standard SSA. For general comments on the application of these functions see the documentation of decompSSA.

SSA embeds lagged copies of a time series x into a augmented matrix X (trajectory matrix ). In a second step the orthonormal basis of X is found via singular value decomposition (SVD). One of the internal steps of SVD is the computation of XX^T. There are two strategies available to obtain this product if values of x are missing. The first possibility is to omit any column of X and to compute XX^T for that reduced matrix. The other possibility is to use a modification of the inner product for vectors containing missing values. The threshold parameter tau controls the computation of XX^T. It gives the acceptable proportion of missing values whithin a vector for the applicatoin of the modified inner product. If tau is <=0, all columns of X containing any missing value are ommited. If tau >=1 no column of X is ommited. For any value of tau in between all columns of X having a propotion of more or equal than tau are omitted. There are several possibilities to capture the reconstructed components. One is the recovery by means of principal component, which has first been introduced by Schoellhamer (2001). Golyandina and Osipov split the recovery of the reconstructed signals into two steps alpha and beta (see reference for more details). Currently only one method for each step is implemented, namely alpha="PI", denoting the usage of the Pi - projector and beta="simultaneous", denoting simultaneous filling in.

The output of decompSSAM is an object of class decompSSAM inheriting from decompSSA with following items:

`lambda`	The eigenvalues, ordered decreasing.
`U`	The eigenvectors (columns), ordered by decreasing eigenvalues.
`freq`	Dominant frequency of the eigenvectors, ordered by decreasing eigenvalues.
`rank`	Rank of the eigenvalues, ordered by decreasing eigenvalues.
`N`	Length of the input series.
`L`	Embedding dimension or window length.
`toeplitz`	Logical, indicates if Toeplitz modification has been used.
`numMisssing`	The number of missing entries in the input.
`tau`	The user specification of `tau`.
`seriesName`	Name of input series.
`call`	Call of generating function.

The output of reconSSAM is a matrix with length(groups) column and length(x) rows. Each columns represents the sum of the reconstructed components defined by the list entries of groups.

May cause extreme memory demands. reconSSAM is computionally expensive.

Lukas Gudmundsson

Golyandina, N. & Osipov, E. The "Caterpillar"-SSA method for analysis of time series with missing values. Journal of Statistical Planning and Inference, 5th St. Petersburg Workshop on Simulation, 2007, 137, 2642-2653 http://www.gistatgroup.com/cat/mvssa1en.pdf

Schoellhamer, D. Singular spectrum analysis for time series with missing data. Geophysical Research Letters, 2001, 28, 3187-3190

plot.decompSSAM, decompSSA, sdTest

x <- sin(seq(0,10*pi,len=200))
x <- x + rnorm(x)/2
x[100:120] <- NA
x.dc <- decompSSAM(x,L=40)
plot(x.dc,by="rank",log="")
x.rc1 <- reconSSAM(x.dc, x, groups=list(1:2),
  method=list(alpha = "PI", beta = "simultaneous"))
x.rc2 <- reconSSAM(x.dc, x, groups=list(1:2),
  method="PC")
# compare result with input signal
plot(x,type="l")
lines(x.rc1,col="red",lwd=2,lty=1)
lines(x.rc2,col="black",lwd=2, lty=2)
lines(sin(seq(0,10*pi,len=200)),col="blue",lwd=2)