# k_tJADE: k-tJADE for Tensor-Valued Observations In tensorBSS: Blind Source Separation Methods for Tensor-Valued Observations

## Description

Computes the faster “k”-version of tensorial JADE in an independent component model.

## Usage

 `1` ```k_tJADE(x, k = NULL, 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. `k` A vector with one less element than dimensions in `x`. The elements of `k` give upper bounds for cumulant matrix indices we diagonalize in each mode. Lower values mean faster computation times. The default value `NULL` puts k equal to 1 in each mode (the fastest choice). `maxiter` Maximum number of iterations. Passed on to `rjd`. `eps` Convergence tolerance. Passed on to `rjd`.

## 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.

`k_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 at the same time in a more efficient way than `tFOBI` but also in fewer numbers than `tJADE`. `k_tJADE` diagonalizes in each mode only those cumulant matrices C^ij for which |i - j| < k_m.

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

## 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. `k` The used vector of k-values. `datatype` Character string with value "iid". Relevant for `plot.tbss`.

Joni Virta

## References

Miettinen, J., Nordhausen, K., Oja, H. and Taskinen, S. (2013), Fast Equivariant JADE, In the Proceedings of 38th IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2013), 6153–6157, doi: 10.1109/ICASSP.2013.6638847

Virta J., Li B., Nordhausen K., Oja H. (2018): JADE for tensor-valued observations, Journal of Computational and Graphical Statistics, 27, 628-637, doi: 10.1080/10618600.2017.1407324

Virta J., Lietzen N., Ilmonen P., Nordhausen K. (2021): Fast tensorial JADE, Scandinavian Journal of Statistics, 48, 164-187, doi: 10.1111/sjos.12445

`k_JADE`, `tJADE`, `JADE`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29``` ```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) k_tjade <- k_tJADE(X) MD(k_tjade\$W[], A1) MD(k_tjade\$W[], A2) tMD(k_tjade\$W, list(A1, A2)) k_tjade <- k_tJADE(X, k = c(2, 1)) MD(k_tjade\$W[], A1) MD(k_tjade\$W[], A2) tMD(k_tjade\$W, list(A1, A2)) ```