tFOBI: FOBI for Tensor-Valued Observations
In tensorBSS: Blind Source Separation Methods for Tensor-Valued Observations

Description Usage Arguments Details Value Author(s) References See Also Examples

Computes the tensorial FOBI in an independent component model.

1	tFOBI(x, norm = NULL)

`x`	Numeric array of an order at least two. It is assumed that the last dimension corresponds to the sampling units.
`norm`	A Boolean vector with number of entries equal to the number of modes in a single observation. The elements tell which modes use the “normed” version of tensorial FOBI. If `NULL` then all modes use the non-normed version.

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.

tFOBI recovers then based on x the underlying independent components S by estimating the r unmixing matrices W_1, ..., W_r using fourth joint moments.

The unmixing can in each mode be done in two ways, using a “non-normed” or “normed” method and this is controlled by the argument norm. The authors advocate the general use of non-normed version, see the reference below for their comparison.

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

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

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.
`norm`	The vector indicating which modes used the “normed” version.
`Xmu`	The data location.
`datatype`	Character string with value "iid". Relevant for `plot.tbss`.

Joni Virta

Virta, J., Li, B., Nordhausen, K. and Oja, H., (2017), Independent component analysis for tensor-valued data, Journal of Multivariate Analysis, doi: 10.1016/j.jmva.2017.09.008

FOBI, tgFOBI

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)

tfobi <- tFOBI(X)

MD(tfobi$W[[1]], A1)
MD(tfobi$W[[2]], A2) 
tMD(tfobi$W, list(A1, A2))

# Digit data example

data(zip.train)
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)

tfobi <- tFOBI(x0)
plot(tfobi, col=y0)