ssaBSS (version 0.1.1)

SSAcor: Identification of Non-stationarity in the Covariance Structure

Description

SSAcor method for identifying non-stationarity in the covariance structure.

Usage

SSAcor(X, ...)
# S3 method for default
SSAcor(X, K, n.cuts = NULL, tau = 1, eps = 1e-6, maxiter = 2000, ...)
# S3 method for ts
SSAcor(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

The sums of pseudo eigenvalues.

DTable

The peudo eigenvalues of size ntau*p to see which type of nonstationarity there exists in each component.

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.

k

The used lag.

method

Name of the method ("SSAcor"), 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.

tau

The lag as a scalar or a vector. Default is 1.

eps

Convergence tolerance.

maxiter

The maximum number of iterations.

...

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 \(T_i\). For all lags \(j=1, ..., ntau\), algorithm first calculates matrices $${\bf M_j} = \sum_{i = 1}^K \frac{T_i}{T}({\bf S}_{j,T} - {\bf S}_{j,T_i})({\bf S}_{j,T} - {\bf S}_{j,T_i})^T,$$ where \(i = 1, \ldots, K\), \(K\) is the number of breakpoints, \({\bf S}_{J,T}\) is the global sample covariance for lag \(j\), and \({\bf S}_{\tau,T_i}\) is the sample covariance of values of \({\bf Y}\) which belong to a disjoint interval \(T_i\).

The algorithm finds an orthogonal matrix \({\bf U}\) by maximizing $$\sum_{j = 1}^{ntau} ||\textrm{diag}({\bf}{\bf U}{\bf M}_j {\bf U}')||^2.$$ where \(j = 1, \ldots, ntau\).

The final unmixing matrix is then \({\bf W} = {\bf U S}^{-1/2}\). Then the pseudo eigenvalues \({\bf D}_i = \textrm{diag}({\bf}{\bf U}{\bf M}_i {\bf U}') = \textrm{diag}(d_{i,1}, \ldots, d_{i,p})\) are obtained and the value of \(d_{i,j}\) tells if the \(j\)th component is nonstationary with respect to \({\bf M}_i\). The first \(k\) rows of \({\bf W}\) project the observed time series to the subspace of components with non-stationary covariance, and the last \(p-k\) rows to the subspace of components with stationary covariance.

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 <- rtvAR1(n)
z2a <- arima.sim(floor(n/3), model = list(ar = c(0.5),
        innov = c(rnorm(floor(n/3), 0, 1))))
z2b <- arima.sim(floor(n/3), model = list(ar = c(0.2),
        innov = c(rnorm(floor(n/3), 0, 1.28))))
z2c <- arima.sim(n - 2*floor(n/3), model = list(ar = c(0.8),
        innov = c(rnorm(n - 2*floor(n/3), 0, 0.48))))
z2 <- c(z2a, z2b, z2c)
z3 <- arima.sim(n, model = list(ma = c(0.72, 0.24), ar = c(0.14, 0.45)))
z4 <- arima.sim(n, model = list(ar = c(0.34, 0.27, 0.18))) 

Z <- cbind(z1, z2, z3, z4)
library(zoo)
X <- as.zoo(tcrossprod(Z, A)) # A zoo object

res <- SSAcor(X, K = 6, tau = 1)
ggscreeplot(res, type = "barplot", color = "gray") # Two non-zero eigenvalues

# Plotting the components as a zoo object
plot(res) # The first two are nonstationary in autocovariance