TRES (version 1.1.5)

MenvU_sim: Generate matrices \(M\) and \(U\)

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 \(\Omega\) in \(M=\Gamma\Omega\Gamma^T+\Gamma_0\Omega_0\Gamma_0^T\). The default is \(\Omega=AA^T\) where the elements in \(A\) are generated from Uniform(0,1) distribution.

Omega0

The positive definite matrix \(\Omega_0\) in \(M=\Gamma\Omega\Gamma^T+\Gamma_0\Omega_0\Gamma_0^T\). The default is \(\Omega_0=AA^T\) where the elements in \(A\) are generated from Uniform(0,1) distribution.

Phi

The positive definite matrix \(\Phi\) in \(U=\Gamma\Phi\Gamma^T\). The default is \(\Phi=AA^T\) 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.

Value

M

The \(p\)-by-\(p\) matrix M.

U

The \(p\)-by-\(p\) matrix U.

Gamma

The \(p\)-by-\(u\) envelope basis.

Details

The matrices \(M\) and \(U\) are in forms of $$M = \Gamma \Omega \Gamma^T + \Gamma_0\Omega_0\Gamma_0^T, U = \Gamma \Phi \Gamma^T.$$

The envelope basis \(\Gamma\) is randomly generated from the Uniform (0, 1) distribution elementwise and then transformed to a semi-orthogonal matrix. \(\Gamma_0\) is the orthogonal completion of \(\Gamma\).

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.

References

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

Examples

# NOT RUN {
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

# }