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 $\mathrm{\Omega }$ in $M=\mathrm{\Gamma }\mathrm{\Omega }{\mathrm{\Gamma }}^T+{\mathrm{\Gamma }}_0{\mathrm{\Omega }}_0{\mathrm{\Gamma }}_0^T$. The default is $\mathrm{\Omega }=A{A}^T$ where the elements in $A$ are generated from Uniform(0,1) distribution.

Omega0

The positive definite matrix ${\mathrm{\Omega }}_0$ in $M=\mathrm{\Gamma }\mathrm{\Omega }{\mathrm{\Gamma }}^T+{\mathrm{\Gamma }}_0{\mathrm{\Omega }}_0{\mathrm{\Gamma }}_0^T$. The default is ${\mathrm{\Omega }}_0=A{A}^T$ where the elements in $A$ are generated from Uniform(0,1) distribution.

Phi

The positive definite matrix $\mathrm{\Phi }$ in $U=\mathrm{\Gamma }\mathrm{\Phi }{\mathrm{\Gamma }}^T$. The default is $\mathrm{\Phi }=A{A}^T$ 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 $$M=\mathrm{\Gamma }\mathrm{\Omega }{\mathrm{\Gamma }}^T+{\mathrm{\Gamma }}_0{\mathrm{\Omega }}_0{\mathrm{\Gamma }}_0^T,U=\mathrm{\Gamma }\mathrm{\Phi }{\mathrm{\Gamma }}^T.$$

The envelope basis $\mathrm{\Gamma }$ is randomly generated from the Uniform (0, 1) distribution elementwise and then transformed to a semi-orthogonal matrix. ${\mathrm{\Gamma }}_0$ is the orthogonal completion of $\mathrm{\Gamma }$.

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

# }