gFOBI for Tensor-Valued Time Series

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Description

Computes the tensorial gFOBI for time series where at each time point a tensor of order r is observed.

Usage

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tgFOBI(x, lags = 0:12, maxiter = 100, eps = 1e-06)

Arguments

x

Numeric array of an order at least two. It is assumed that the last dimension corresponds to the time.

lags

Vector of integers. Defines the lags used for the computations of the autocovariances.

maxiter

Maximum number of iterations. Passed on to frjd.

eps

Convergence tolerance. Passed on to frjd.

Details

It is assumed that S is a tensor (array) of size p_1 x p_2 x ... x p_r measured at time points 1, ..., T. The assumption is that the elements of S are mutually independent, centered and weakly stationary time series and are mixed from each mode m by the mixing matrix A_m, m= 1, ..., r, yielding the observed time series X. In R the sample of X is saved as an array of dimensions p_1, p_2, ..., p_r, T.

tgFOBI recovers then based on x the underlying independent time series S by estimating the r unmixing matrices W_1, ..., W_r using the lagged fourth joint moments specified by lags. This reliance on higher order moments makes the method especially suited for stochastic volatility models.

If x is a matrix, that is, r = 1, the method reduces to gFOBI and the function calls gFOBI.

If lags = 0 the method reduces to tFOBI.

Value

A list with class 'tbss', inheriting from class 'bss', containing the following components:

S

Array of the same size as x containing the estimated uncorrelated sources.

W

List containing all the unmixing matrices

Xmu

The data location.

datatype

Character string with value "ts". Relevant for plot.tbss.

Author(s)

Joni Virta

References

Virta, J. and Nordhausen, K., (2016), Blind source separation of tensor-valued time series, submitted, ???, ???–???.

See Also

gFOBI, frjd, tFOBI

Examples

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library("stochvol")
n <- 1000
S <- t(cbind(svsim(n, mu = -10, phi = 0.98, sigma = 0.2, nu = Inf)$y,
             svsim(n, mu = -5, phi = -0.98, sigma = 0.2, nu = 10)$y,
             svsim(n, mu = -10, phi = 0.70, sigma = 0.7, nu = Inf)$y,
             svsim(n, mu = -5, phi = -0.70, sigma = 0.7, nu = 10)$y,
             svsim(n, mu = -9, phi = 0.20, sigma = 0.01, nu = Inf)$y,
             svsim(n, mu = -9, phi = -0.20, sigma = 0.01, nu = 10)$y))
dim(S) <- c(3, 2, n)

A1 <- matrix(rnorm(9), 3, 3)
A2 <- matrix(rnorm(4), 2, 2)

X <- tensorTransform(S, A1, 1)
X <- tensorTransform(X, A2, 2)

tgfobi <- tgFOBI(X)

MD(tgfobi$W[[1]], A1)
MD(tgfobi$W[[2]], A2) 
MD(tgfobi$W[[2]] %x% tgfobi$W[[1]], A2 %x% A1)