ECD: ECD algorithm for estimating the envelope subspace

In kusakehan/TEReg: An R package for tensor envelope regression with tensor response and tensor predictor

Description

Estimate the envelope subspace with specified dimension based on ECD algorithm that described in Cook, R. D., & Zhang, X. (2018).

Usage

ECD(M, U, d, maxiter=500, epsilon=1e-08)

Arguments

M	M matrix in the envelope objective function. An r-by-r positive semi-definite matrix.
U	U matrix in the envelope objective function. An r-by-r positive semi-definite matrix.
d	Envelope dimension. An integer between 0 and r.
maxiter	Maximum number of iterations.
epsilon	Convergence criterion. |F_k - F_{k-1}|< ε, where F_k is the objective function.

Details

Estimate M-envelope contains span(U) where M > 0 and is symmetric. The dimension of the envelope is d.

Value

Ghat	The orthogonal basis of the envelope subspace.

References

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

Examples

##simulate two matrices M and U with an envelope structure#
p <- 20
u <- 5

##randomly generate a semi-orthogonal p-by-u basis matrix (Gamma) for the
##envelope and its orthogonal completion (Gamma0) of dimension p-by-(p-u)
Gamma <- matrix(runif(p*u), p, u)

###make Gamma semi-orthogonal
Gamma <- qr.Q(qr(Gamma))
Gamma0 <- qr.Q(qr(Gamma),complete=TRUE)[,(u+1):p]

## randomly generated symmetric positive definite matrices, M and U, to have
## an exact u-dimensional envelope structure

Phi <- matrix(runif(u^2), u, u)
Phi <- Phi %*% t(Phi)
Omega <- matrix(runif(u^2), u, u)
Omega <- Omega %*% t(Omega)
Omega0 <- matrix(runif((p-u)^2),p-u,p-u)
Omega0 <- Omega0 %*% t(Omega0)
M <- Gamma %*% Omega %*% t(Gamma) + Gamma0 %*% Omega0 %*% t(Gamma0)
U <- Gamma %*% Phi %*% t(Gamma)

# randomly generate symmetric positive definite matrices, Mhat and Uhat, as
# root-n consistent sample estimators for M and U

n=200
X <- mvrnorm(n, mu = rep(0, p), Sigma = M)
Y <- mvrnorm(n, mu = rep(0, p), Sigma = U)
Mhat <- (t(X) %*% X)/n
Uhat <- (t(Y) %*% Y)/n

Ghat_ECD <- ECD(Mhat, Uhat, u)

kusakehan/TEReg documentation built on May 30, 2019, 7:17 a.m.