MenvU_sim {TRES}

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.

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.

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