Type:	Package
Title:	Stationary Subspace Analysis
Version:	0.1.1
Date:	2022-12-01
Maintainer:	Markus Matilainen <markus.matilainen@outlook.com>
Depends:	tsBSS (≥ 0.5.3), ICtest (≥ 0.3-4), JADE (≥ 2.0-2), BSSprep, ggplot2
Imports:	xts, zoo
Description:	Stationary subspace analysis (SSA) is a blind source separation (BSS) variant where stationary components are separated from non-stationary components. Several SSA methods for multivariate time series are provided here (Flumian et al. (2021); Hara et al. (2010) <doi:10.1007/978-3-642-17537-4_52>) along with functions to simulate time series with time-varying variance and autocovariance (Patilea and Raissi(2014) <doi:10.1080/01621459.2014.884504>).
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
NeedsCompilation:	no
Packaged:	2022-12-01 15:31:21 UTC; manmat
Author:	Markus Matilainen [cre, aut], Lea Flumian [aut], Klaus Nordhausen [aut], Sara Taskinen [aut]
Repository:	CRAN
Date/Publication:	2022-12-01 16:40:05 UTC

Stationary Subspace Analysis

Description

Stationary subspace analysis (SSA) is a blind source separation (BSS) variant where stationary components are separated from non-stationary components. Several SSA methods for multivariate time series are provided here (Flumian et al. (2021); Hara et al. (2010) <doi:10.1007/978-3-642-17537-4_52>) along with functions to simulate time series with time-varying variance and autocovariance (Patilea and Raïssi(2014) <doi:10.1080/01621459.2014.884504>).

Details

This package contains functions for identifying different types of nonstationarity

• SSAsir – SIR type function for mean non-stationarity identification

• SSAsave – SAVE type function for variance non-stationarity identification

• SSAcor – Function for identifying changes in autocorrelation

• ASSA – ASSA: Analytic SSA for identification of nonstationarity in mean and variance.

• SSAcomb – Combination of SSAsir, SSAsave, and SSAcor using joint diagonalization

The package also contains function rtvvar to simulate a time series with time-varying variance (TV-VAR), and function rtvAR1 to simulate a time series with time-varying autocovariance (TV-AR1).

Author(s)

Markus Matilainen, Léa Flumian, Klaus Nordhausen, Sara Taskinen

Maintainer: Markus Matilainen <markus.matilainen@outlook.com>

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

Hara S., Kawahara Y., Washio T. and von Bünau P. (2010). Stationary Subspace Analysis as a Generalized Eigenvalue Problem, Neural Information Processing. Theory and Algorithms, Part I, pp. 422-429.

Patilea V. and Raïssi H. (2014) Testing Second-Order Dynamics for Autoregressive Processes in Presence of Time-Varying Variance, Journal of the American Statistical Association, 109 (507), 1099-1111.

ASSA Method for Non-stationary Identification

Description

ASSA (Analytic Stationary Subspace Analysis) method for identifying non-stationary components of mean and variance.

Usage

ASSA(X, ...)

## Default S3 method:
ASSA(X, K, n.cuts = NULL, ...)
## S3 method for class 'ts'
ASSA(X, ...)

Arguments

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} = \frac{1}{T}\sum_{i = 1}^K \left({\bf m}_{T_i} {\bf m}_{T_i}^T + \frac{1}{2} {\bf S}_{T_i} {\bf S}_{T_i}^T\right) - \frac{1}{2} {\bf I},

where i = 1, \ldots, K, K is the number of breakpoints, {\bf I} is an identity matrix, and {\bf m}_{T_i} is the average of values of {\bf Y} 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 non-stationary components, and the last p-k rows to the subspace of stationary components.

Value

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

Author(s)

Markus Matilainen, Klaus Nordhausen

References

Hara S., Kawahara Y., Washio T. and von Bünau P. (2010). Stationary Subspace Analysis as a Generalized Eigenvalue Problem, Neural Information Processing. Theory and Algorithms, Part I, pp. 422-429.

See Also

JADE

Examples

n <- 5000
A <- rorth(4)

z1 <- arima.sim(n, model = list(ar = 0.7)) + rep(c(-1.52, 1.38), c(floor(n*0.5),
        n - floor(n*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 <- ASSA(X, K = 6)
screeplot(res, type = "lines") # Two non-zero eigenvalues

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

Combination Main SSA Methods

Description

SSAcomb method for identification for non-stationarity in mean, variance and covariance structure.

Usage

SSAcomb(X, ...)

## Default S3 method:
SSAcomb(X, K, n.cuts = NULL, tau = 1, eps = 1e-6, maxiter = 2000, ...)
## S3 method for class 'ts'
SSAcomb(X, ...)

Arguments

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 the {\bf M} matrices from SSAsir (matrix {\bf M}_1), SSAsave (matrix {\bf M}_2) and SSAcor (matrices {\bf M}_{j+2}).

The algorithm finds an orthogonal matrix {\bf U} by maximizing

\sum_{i = 1}^{ntau+2} ||\textrm{diag}({\bf}{\bf U}{\bf M}_i {\bf U}')||^2.

where i = 1, \ldots, ntau + 2. The final unmixing matrix is then {\bf W} = {\bf US}^{-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 jth component is nonstationary with respect to {\bf M}_i.

Value

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

Author(s)

Markus Matilainen, Klaus Nordhausen

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 frjd

Examples

n <- 10000
A <- rorth(6)

z1 <- arima.sim(n, model = list(ar = 0.7)) + rep(c(-1.52, 1.38), 
        c(floor(n*0.5), n - floor(n*0.5)))
z2 <- rtvAR1(n)
z3 <- rtvvar(n, alpha = 0.2, beta = 0.5)
z4 <- arima.sim(n, model = list(ma = c(0.72, 0.24), ar = c(0.14, 0.45)))
z5 <- arima.sim(n, model = list(ma = c(0.34))) 
z6 <- arima.sim(n, model = list(ma = c(0.72, 0.15)))

Z <- cbind(z1, z2, z3, z4, z5, z6)
library(xts)
X <- tcrossprod(Z, A)
X <- xts(X, order.by = as.Date(1:n)) # An xts object

res <- SSAcomb(X, K = 12, tau = 1)
ggscreeplot(res, type = "lines") # Three non-zero eigenvalues
res$DTable # Components have different kind of nonstationarities

# Plotting the components as an xts object
plot(res, multi.panel = TRUE) # The first three are nonstationary

Identification of Non-stationarity in the Covariance Structure

Description

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

Usage

SSAcor(X, ...)

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

Arguments

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 jth 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.

Value

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

Author(s)

Markus Matilainen, Klaus Nordhausen

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

Identification of Non-stationarity in Variance

Description

SSAsave method for identifying non-stationarity in variance

Usage

SSAsave(X, ...)

## Default S3 method:
SSAsave(X, K, n.cuts = NULL, ...)
## S3 method for class 'ts'
SSAsave(X, ...)

Arguments

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.

Value

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

Author(s)

Markus Matilainen, Klaus Nordhausen

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

Identification of Non-stationarity in Mean

Description

SSAsir method for identifying non-stationarity in mean.

Usage

SSAsir(X, ...)

## Default S3 method:
SSAsir(X, K, n.cuts = NULL, ...)
## S3 method for class 'ts'
SSAsir(X, ...)

Arguments

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 m}_{T_i} {\bf m}_{T_i}^T),

where i = 1, \ldots, K, K is the number of breakpoints, and {\bf m}_{T_i} is the average 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 mean, and the last p-k rows to the subspace of components with stationary mean.

Value

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

Author(s)

Markus Matilainen, Klaus Nordhausen

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 <- arima.sim(n, model = list(ar = 0.7)) + rep(c(-1.52, 1.38),
        c(floor(n*0.5), n - floor(n*0.5)))
z2 <- arima.sim(n, model = list(ar = 0.5)) + rep(c(-0.75, 0.84, -0.45),
        c(floor(n/3), floor(n/3), n - 2*floor(n/3)))
z3 <- arima.sim(n, model = list(ma = 0.72))
z4 <- arima.sim(n, model = list(ma = c(0.34)))

Z <- cbind(z1, z2, z3, z4)
X <- tcrossprod(Z, A)

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

# Plotting the components
plot(res) # The first two are nonstationary in mean

Simulation of Time Series with Time-varying Autocovariance

Description

Simulating time-varying variance based on TV-AR1 model

Usage

rtvAR1(n, sigma = 0.93)

Arguments

Details

Time varying autoregressive processes of order 1 (TV-AR1) is

x_t = a_t x_{t-1} + \epsilon_t,

with x_0=0, \epsilon_t is iid N(0, \sigma^2) and a_t = 0.5\cos(2\pi t/T).

Value

The simulated series as a ts object.

Author(s)

Sara Taskinen, Markus Matilainen

References

Patilea V. and Raïssi H. (2014) Testing Second-Order Dynamics for Autoregressive Processes in Presence of Time-Varying Variance, Journal of the American Statistical Association, 109 (507), 1099-1111.

Examples

n <- 5000
X <- rtvAR1(n, sigma = 0.93)
plot(X)

Simulation of Time Series with Time-varying Variance

Description

Simulating time-varying variance based on TV-VAR model

Usage

rtvvar(n, alpha, beta = 1, simple = FALSE)

Arguments

Details

Time varying variance (TV-VAR) process x_t with parameters \alpha and \beta is of the form

x_t = \tilde h_t \epsilon_t,

where, if simple = FALSE,

\tilde h_t^2 = h_t^2 + \alpha x_{t-1}^2,

where \epsilon are iid N(0,1), x_0 = 0 and h_t = 10 - 10 \sin(\beta \pi t/T + \pi/6) (1 + t/T),

and if simple = TRUE,

\tilde h_t = t/T.

Value

The simulated series as a ts object.

Author(s)

Sara Taskinen, Markus Matilainen

References

Patilea V. and Raïssi H. (2014) Testing Second-Order Dynamics for Autoregressive Processes in Presence of Time-Varying Variance, Journal of the American Statistical Association, 109 (507), 1099-1111.

Examples

n <- 5000
X <- rtvvar(n, alpha = 0.2, beta = 0.5, simple = FALSE)
plot(X)

Class: ssabss

Description

Class 'ssabss' (blind source separation in stationary subspace analysis) with methods plot, screeplot (prints a screeplot of an object of class 'ssabss') and ggscreeplot (prints a screeplot of an object of class 'ssabss' using package ggplot2).

The class 'ssabss' also inherits methods from the class 'bss' in package JADE: for extracting the components (bss.components), for plotting the components (plot.bss), for printing (print.bss), and for extracting the coefficients (coef.bss).

Usage

## S3 method for class 'ssabss'
plot(x, ...)

## S3 method for class 'ssabss'
screeplot(x, type = c("lines", "barplot"), xlab = "Number of components",
          ylab = NULL, main = paste("Screeplot for", x$method),
          pointsize = 4, breaks = 1:length(x$D), color = "red", ...)
                              
## S3 method for class 'ssabss'
ggscreeplot(x, type = c("lines", "barplot"), xlab = "Number of components",
            ylab = NULL, main = paste("Screeplot for", x$method),
            pointsize = 4, breaks = 1:length(x$D), color = "red", ...)

Arguments

Details

A screeplot can be used to determine the number of interesting components. For SSAcomb it plots the sum of pseudo eigenvalues and for other methods it plots the eigenvalues.

Author(s)

Markus Matilainen

See Also

ASSA, SSAsir, SSAsave, SSAcor, SSAcomb, JADE, ggplot2