MenvU_sim: Generate matrices M and U

In TRES: Tensor Regression with Envelope Structure

Description

This function generates the matrices M and U with envelope structure.

Usage

MenvU_sim(
  p,
  u,
  Omega = NULL,
  Omega0 = NULL,
  Phi = NULL,
  jitter = FALSE,
  wishart = FALSE,
  n = NULL
)

Arguments

p	Dimension of p-by-p matrix M.
u	The envelope dimension. An integer between 0 and p.
Omega	The positive definite matrix Ω in M=ΓΩΓ^T+Γ_0Ω_0Γ_0^T. The default is Ω=AA^T where the elements in A are generated from Uniform(0,1) distribution.
Omega0	The positive definite matrix Ω_0 in M=ΓΩΓ^T+Γ_0Ω_0Γ_0^T. The default is Ω_0=AA^T where the elements in A are generated from Uniform(0,1) distribution.
Phi	The positive definite matrix Φ in U=ΓΦΓ^T. The default is Φ=AA^T where the elements in A are generated from Uniform(0,1) distribution.
jitter	Logical or numeric. If it is numeric, the diagonal matrix diag(jitter, nrow(M), ncol(M)) is added to matrix M to ensure the positive definiteness of M. If it is TRUE, then it is set as 1e-5 and the jitter is added. If it is FALSE (default), no jitter is added.
wishart	Logical. If it is TRUE, the sample estimator from Wishart distribution W_p(M/n, n) and W_p(U/n, n) are generated as the output matrices M and U.
n	The sample size. If wishart is FALSE, then n is ignored.

Details

The matrices M and U are in forms of

M = Γ Ω Γ^T + Γ_0Ω_0Γ_0^T, U = Γ Φ Γ^T.

The envelope basis Γ is randomly generated from the Uniform (0, 1) distribution elementwise and then transformed to a semi-orthogonal matrix. Γ_0 is the orthogonal completion of Γ.

In some cases, to guarantee that M is positive definite which is required by the definition of envelope, a jitter should be added to M.

If wishart is TRUE, after the matrices M and U are generated, the samples from Wishart distribution W_p(M/n, n) and W_p(U/n, n) are output as matrices M and U. If so, n is required.

Value

M	The p-by-p matrix M.
U	The p-by-p matrix U.
Gamma	The p-by-u envelope basis.

References

Cook, R.D. and Zhang, X., 2018. Fast envelope algorithms. Statistica Sinica, 28(3), pp.1179-1197.

Examples

data1 <- MenvU_sim(p = 20, u = 5)
M1 <- data1$M
U1 <- data1$U

# Sample version from Wishart distribution
data2 <- MenvU_sim(p = 20, u = 5, wishart = TRUE, n = 200)
M2 <- data2$M
U2 <- data2$U

TRES documentation built on Oct. 20, 2021, 9:06 a.m.