tPP: Projection pursuit 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

Applies mode-wise projection pursuit to tensorial data with respect to the chosen measure of interestingness.

1	tPP(x, nl = "pow3", eps = 1e-6, maxiter = 100)

`x`	Numeric array of an order at least three. It is assumed that the last dimension corresponds to the sampling units.
`nl`	The chosen measure of interestingness/objective function. Current choices include `pow3` (default) and `skew`, see the details below
`eps`	The convergence tolerance of the iterative algortihm.
`maxiter`	The maximum number of iterations.

The observed tensors (arrays) X of size p_1 x p_2 x ... x p_r measured on N units are standardized from each mode and then projected mode-wise onto the directions that maximize the L_2-norm of the vector of the values E[G(u_k^T X X^T u_k)] - E[G(c^2)], where G is the chosen objective function and c^2 obeys the chi-squared distribution with q degress of freedom. Currently the function allows the choices G(x) = x^2 (pow3) and G(x) = x √{x} (skew), which correspond roughly to the maximization of kurtosis and skewness, respectively. The algorithm is the multilinear extension of FastICA, where the names of the objective functions also come from.

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

`S`	Array of the same size as x containing the estimated components.
`W`	List containing all the unmixing matrices.
`iter`	The numbers of iteration used per mode.
`Xmu`	The data location.
`datatype`	Character string with value "iid". Relevant for `plot.tbss`.

Joni Virta

Nordhausen, K. and Virta, J. (2018), Tensorial projection pursuit, Manuscript in preparation.

Hyvarinen, A. (1999) Fast and robust fixed-point algorithms for independent component analysis, IEEE transactions on Neural Networks 10.3: 626-634.

fICA, NGPP

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)

tpp <- tPP(X)

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