svdLSplus {WALS}

Internal function: Uses SVD components to compute final estimate via Sherman-Morrison-Woodbury formula.

Description

Solves the equation system in walsNB via Sherman-Morrison-Woodbury formula for the unrestricted estimator \hat{\gamma}_{u}.

Usage

svdLSplus(U, V, singularVals, y, ell, geB)

Arguments

U

Left singular vectors of \bar{Z} or \bar{Z}_{1} from svd.

V

Right singular vectors of \bar{Z} or \bar{Z}_{1} from svd.

singularVals

Singular values of \bar{Z} or \bar{Z}_{1} from svd.

y

"Pseudo"-response, see details.

ell

Vector \bar{\ell} from section "Simplification for computing \tilde{\gamma}_{u}" Huynh (2024b)

geB

Scalar \bar{g} \bar{\epsilon} / (1 + B). See section "Simplification for computing \tilde{\gamma}_{u}" Huynh (2024b) for definition of \bar{g}, \bar{\epsilon} and B.

Details

The function can be reused for the computation of the fully restricted estimator \tilde{\gamma}_{1r} and the model averaged estimator \hat{\gamma}_{1}.

For \tilde{\gamma}_{1r} and \hat{\gamma}_{1} use U, V and singularVals from SVD of \bar{Z}_{1}.

For \hat{\gamma}_{u} and \tilde{\gamma}_{1r} use same pseudo-response \bar{y_{0}} - \bar{t} \bar{\epsilon} \bar{\Psi}^{-1/2} \bar{q} in argument y.

For \hat{\gamma}_{1} use pseudo-response \bar{y_{0}} - \bar{t} \bar{\epsilon} \bar{\Psi}^{-1/2} \bar{q} - (\bar{Z}_{2} + \bar{g} \bar{\epsilon} \bar{\Psi}^{-1/2} \bar{q} \bar{q}^{\top} Z_{2}) \hat{\gamma}_{2}.

See section "Note on function svdLSplus from WALS" in Huynh (2024b).

References

Huynh K (2024b). “WALS: Weighted-Average Least Squares Model Averaging in R.” University of Basel. Mimeo.