Tensor Envelope Partial Least Squares Regression (TEPLS)

Fits the tensor-predictor partial least squares regression of Zhang & Li (2017). A tensor predictor X (order m) is reduced to a low-dimensional latent tensor via per-mode envelope factor matrices estimated with a SIMPLS iteration (their Algorithm 4), and the response is regressed on the latent tensor. The coefficient tensor is reconstructed in the original predictor space (their Lemma 2), giving the B_PLS estimator.

Usage

tepls(X, Y, u = NULL, ridge = 1e-08)

Arguments

X: Tensor predictor: a list of observations, or an order-(m + 1) tensor with observations in the last mode.
Y: Response: numeric vector (length n) or n x r matrix.
u: Envelope dimension per mode: an integer vector of length m, or a scalar recycled across modes. Each u[k] must satisfy 1 <= u[k] <= p_k. The default NULL picks each u[k] from the largest consecutive eigenvalue ratio of that mode's signal matrix, the same rule spgtr() uses; supply u explicitly when you know the rank.
ridge: Small ridge added when inverting covariance / normal-equation matrices for numerical stability (default 1e-8).

Value

An object of class tepls: a list with elements coef (the coefficient Tensor, order m for scalar response or order m + 1 with a trailing response mode otherwise), W (list of factor matrices), intercept, Xbar, dims, u, r, and fitted.

Details

The predictor may be supplied either as a list of n Tensor/array observations that share the same dimensions, or as a single order-(m + 1) Tensor/array whose last mode indexes the n observations. The response Y is a length-n numeric vector (scalar response) or an n x r matrix (multivariate response).

Each mode's factor matrix W_k has u[k] columns; u is the envelope dimension per mode (recycled if a scalar). The mode-k marginal covariance uses the moment estimator (their eq. 1), the mode-k cross-covariance is standardized as in Algorithm 4, and the reduced regression of Y on the latent tensor is fit by (regularized) least squares (their Step 6).

The mode-k second-moment matrix maximized in the SIMPLS step, C_(k) (Sigma_Y^-1 kron ... kron Sigma_1^-1) C_(k)^T (excluding Sigma_k), is identical to the U U^T matrix in the reference implementation TEReg::TensPLS_fit. Two deliberate divergences: (1) that package estimates each mode's basis with an envelope (EnvMU) optimizer, whereas this function uses the SIMPLS deflation of Algorithm 4 exactly; (2) the estimated factor directions and the reduced-regression fit are both invariant to the per-mode scale ambiguity of the separable covariance, avoiding the need to pin the Kronecker scale.

References

Zhang, X. and Li, L. (2017). Tensor envelope partial least-squares regression. Technometrics 59(4), 426-436.

Examples

set.seed(1)
p <- c(8, 6)
B <- outer(c(1, rep(0, 7)), c(1, rep(0, 5))) # rank-1, envelope dim 1 per mode
X <- lapply(1:80, function(i) matrix(rnorm(prod(p)), p[1], p[2]))
y <- vapply(X, function(xi) sum(B * xi), numeric(1)) + rnorm(80, sd = 0.1)
fit <- tepls(X, y, u = c(1, 1))
tepls(X, y) # envelope dimensions chosen automatically
#> <tepls: tensor envelope PLS regression>
#> Predictor dims:  8 x 6 
#> Envelope dims (u):  1 1 
#> Response dim (r):  1