OptimballGBB1D: Estimate the envelope subspace (Feasi 1D)

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

Description

The 1D algorithm to estimate the envelope subspace with specified dimension based on Wen and Yin (2013).

Usage

OptimballGBB1D(M, U, d, opts=NULL)

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	Dimension of the envelope. An integer between 0 and r.
opts	Option structure with fields: "record = 0" – no print out. "mxitr" – max number of iterations. "xtol" – stop control for ||X_k - X_{k-1}||. "gtol" – stop control for the projected gradient. "ftol" – stop control for \frac{|F_k - F_{k-1}|}{(1+|F_{k-1}|)} usually with max{xtol, gtol} > ftol. The default values are: "xtol"=1e-08; "gtol"=1e-08; "ftol"=1e-12; "mxitr"=500.

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

Wen, Z., & Yin, W. (2013). A feasible method for optimization with orthogonality constraints. Mathematical Programming, 142(1-2), 397-434.

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_1D <- OptimballGBB1D(Mhat, Uhat, d=u)

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