ssaBSS (version 0.1.1)

SSAsave: Identification of Non-stationarity in Variance

Description

SSAsave method for identifying non-stationarity in variance

Usage

SSAsave(X, ...)
# S3 method for default
SSAsave(X, K, n.cuts = NULL, ...)
# S3 method for ts
SSAsave(X, ...)

Value

A list of class 'ssabss', inheriting from class 'bss', containing the following components:

W

The estimated unmixing matrix.

S

The estimated sources as time series object standardized to have mean 0 and unit variances.

M

Used separation matrix.

K

Number of intervals the time series is split into.

D

Eigenvalues of M.

MU

The mean vector of X.

n.cut

Used K+1 vector of values that correspond to the breaks which are used for splitting the data.

method

Name of the method ("SSAsave"), to be used in e.g. screeplot.

Arguments

X

A numeric matrix or a multivariate time series object of class ts, xts or zoo. Missing values are not allowed.

K

Number of intervals the time series is split into.

n.cuts

A K+1 vector of values that correspond to the breaks which are used for splitting the data. Default is intervals of equal length.

...

Further arguments to be passed to or from methods.

Author

Markus Matilainen, Klaus Nordhausen

Details

Assume that a \(p\)-variate \({\bf Y}\) with \(T\) observations is whitened, i.e. \({\bf Y}={\bf S}^{-1/2}({\bf X}_t - \frac{1}{T}\sum_{t=1}^T {\bf X}_{t})\), for \(t = 1, \ldots, T,\) where \({\bf S}\) is the sample covariance matrix of \({\bf X}\).

The values of \({\bf Y}\) are then split into \(K\) disjoint intervals \(T_i\). Algorithm first calculates matrix $${\bf M} = \sum_{i = 1}^K \frac{T_i}{T}({\bf I} - {\bf S}_{T_i}) ({\bf I} - {\bf S}_{T_i})^T,$$ where \(i = 1, \ldots, K\), \(K\) is the number of breakpoints, \({\bf I}\) is an identity matrix, and \({\bf S}_{T_i}\) is the sample variance of values of \({\bf Y}\) which belong to a disjoint interval \(T_i\).

The algorithm finds an orthogonal matrix \({\bf U}\) via eigendecomposition $${\bf M} = {\bf UDU}^T.$$ The final unmixing matrix is then \({\bf W} = {\bf U S}^{-1/2}\). The first \(k\) rows of \({\bf U}\) are the eigenvectors corresponding to the non-zero eigenvalues and the rest correspond to the zero eigenvalues. In the same way, the first \(k\) rows of \({\bf W}\) project the observed time series to the subspace of components with non-stationary variance, and the last \(p-k\) rows to the subspace of components with stationary variance.

References

Flumian L., Matilainen M., Nordhausen K. and Taskinen S. (2021) Stationary subspace analysis based on second-order statistics. Submitted. Available on arXiv: https://arxiv.org/abs/2103.06148

See Also

JADE

Examples

n <- 5000
A <- rorth(4)

z1 <- rtvvar(n, alpha = 0.2, beta = 0.5)
z2 <- rtvvar(n, alpha = 0.1, beta = 1)
z3 <- arima.sim(n, model = list(ma = c(0.72, 0.24))) 
z4 <- arima.sim(n, model = list(ar = c(0.34, 0.27, 0.18)))

Z <- cbind(z1, z2, z3, z4)
X <- as.ts(tcrossprod(Z, A)) # An mts object

res <- SSAsave(X, K = 6)
res$D # Two non-zero eigenvalues
screeplot(res, type = "lines") # This can also be seen in screeplot
ggscreeplot(res, type = "lines") # ggplot version of screeplot

# Plotting the components as an mts object
plot(res) # The first two are nonstationary in variance