FGfun {TRES}R Documentation

The Objective function and its gradient

Description

Calculates the objective function and its gradient for estimating the MM-envelope of span(UU), where MM is positive definite and UU is positive semi-definite.

Usage

FGfun(Gamma, M, U)

Arguments

Gamma

Γ\Gamma matrix in the envelope objective function. A pp-by-uu matrix.

M

The pp-by-pp positive definite matrix MM in the envelope objective function.

U

The pp-by-pp positive semi-definite matrix UU in the envelope objective function.

Details

The generic objective function F(Γ)F(\Gamma) and its gradient G(Γ)G(\Gamma) are listed below for estimating MM-envelope of span(UU). For the detailed description, see Cook, R. D., & Zhang, X. (2016).

F(Γ)=logΓTMΓ+logΓT(M+U)1ΓF(\Gamma)=\log|\Gamma^T M \Gamma|+\log| \Gamma^T(M+U)^{-1}\Gamma|

G(Γ)=dF/dΓ=2MΓ(ΓTMΓ)1+2(M+U)1Γ(ΓT(M+U)1Γ)1G(\Gamma) = dF/d \Gamma = 2 M \Gamma (\Gamma^T M \Gamma)^{-1} + 2 (M + U)^{-1} \Gamma (\Gamma^T (M + U)^{-1} \Gamma)^{-1}

Value

F

The value of the objective function at Gamma.

G

The value of the gradient function at Gamma.

References

Cook, R.D. and Zhang, X., 2016. Algorithms for envelope estimation. Journal of Computational and Graphical Statistics, 25(1), pp.284-300.


[Package TRES version 1.1.5 Index]