tJADE for Tensor-Valued Observations

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Description

Computes the tensorial JADE in an independent component model.

Usage

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tJADE(x, 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 sampling units.

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 with mutually independent elements and measured on N units. The tensor independent component model further assumes that the tensors S are mixed from each mode m by the mixing matrix A_m, m= 1, ..., r, yielding the observed data X. In R the sample of X is saved as an array of dimensions p_1, p_2, ..., p_r, N.

tJADE recovers then based on x the underlying independent components S by estimating the r unmixing matrices W_1, ..., W_r using fourth joint moments in a more efficient way than tFOBI.

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

For a generalization for tensor-valued time series see tgJADE.

Value

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

S

Array of the same size as x containing the independent components.

W

List containing all the unmixing matrices

Xmu

The data location.

datatype

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

Author(s)

Joni Virta

References

Virta, J., Li, B., Nordhausen, K. and Oja, H., (2016), JADE for Tensor-Valued Observation, submitted, ???, ???–???. Preprint available on ArXiv http://arxiv.org/abs/1603.05406.

See Also

JADE, tgJADE

Examples

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n <- 1000
S <- t(cbind(rexp(n)-1,
             rnorm(n),
             runif(n, -sqrt(3), sqrt(3)),
             rt(n,5)*sqrt(0.6),
             (rchisq(n,1)-1)/sqrt(2),
             (rchisq(n,2)-2)/sqrt(4)))
             
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)

tjade <- tJADE(X)

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

## Not run: 
# Digit data example
# Running will take a few minutes

library(ElemStatLearn)
x <- zip.train

rows <- which(x[, 1] == 0 | x[, 1] == 1)
x0 <- x[rows, 2:257]
y0 <- x[rows, 1] + 1

x0 <- t(x0)
dim(x0) <- c(16, 16, 2199)

tjade <- tJADE(x0)
plot(tjade, col=y0)

## End(Not run)