Tensor Regression with Envelope Structure
ECD algorithm for estimating the envelope subspace
The Objective function and its gradient
The covariance estimation of tensor normal distribution
Estimate the envelope subspace (ManifoldOptim 1D)
Estimate the envelope subspace (ManifoldOptim FG)
Generate matrices and
Envelope dimension selection based on 1D-BIC
Estimate the envelope subspace (OptM 1D)
Estimate the envelope subspace (OptM FG)
Optimization on Stiefel manifold
Plot coefficients and p-value for Tenv object.
Prediction and mean squared error.
Predict method for Tenv object.
SIMPLS-type algorithm for estimating the envelope subspace
Elementwise standard error.
The distance between two subspaces.
Summarize method for Tenv object.
The -value and standard error of coefficient in tensor response reg...
Tensor predictor regression
Envelope dimension by cross-validation for tensor predictor regression...
Generate simulation data for tensor predictor regression (TPR)
Tensor Regression with Envelope Structure
Tensor response regression
Generate simulation data for tensor response regression (TRR)
Matrix product of two tensors
Provides three estimators for tensor response regression (TRR) and tensor predictor regression (TPR) models with tensor envelope structure. The three types of estimation approaches are generic and can be applied to any envelope estimation problems. The full Grassmannian (FG) optimization is often associated with likelihood-based estimation but requires heavy computation and good initialization; the one-directional optimization approaches (1D and ECD algorithms) are faster, stable and does not require carefully chosen initial values; the SIMPLS-type is motivated by the partial least squares regression and is computationally the least expensive. For details of TRR, see Li L, Zhang X (2017) <doi:10.1080/01621459.2016.1193022>. For details of TPR, see Zhang X, Li L (2017) <doi:10.1080/00401706.2016.1272495>. For details of 1D algorithm, see Cook RD, Zhang X (2016) <doi:10.1080/10618600.2015.1029577>. For details of ECD algorithm, see Cook RD, Zhang X (2018) <doi:10.5705/ss.202016.0037>. For more details of the package, see Zeng J, Wang W, Zhang X (2021) <doi:10.18637/jss.v099.i12>.