Description Usage Arguments Details Value References Examples
This function generates the matrices M and U with envelope structure.
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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 |
wishart |
Logical. If it is |
n |
The sample size. If |
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.
M |
The p-by-p matrix |
U |
The p-by-p matrix |
Gamma |
The p-by-u envelope basis. |
Cook, R.D. and Zhang, X., 2018. Fast envelope algorithms. Statistica Sinica, 28(3), pp.1179-1197.
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