semiorthogonalize: Internal function: Semiorthogonal-type transformation of X2...
In WALS: Weighted-Average Least Squares Model Averaging
semiorthogonalize R Documentation
Internal function: Semiorthogonal-type transformation of X2 to Z2
Description
Uses the matrix Z2s (called \bar{\Xi} in eq. (9) of
\insertCitedeluca2018glm;textualWALS) to transform \bar{X}_2 to
\bar{Z}_2, i.e. to perform \bar{Z}_2 = \bar{X}_2 \bar{\Delta}_2 \bar{\Xi}^{-1/2}.
For WALS in the linear regression model, the variables do not have a "bar".
Usage
semiorthogonalize(Z2s, X2, Delta2, SVD = TRUE, postmult = FALSE)
Arguments
Z2s
Matrix for which we take negative square root in
X2 * Delta2 * Z2s^{1/2}.
X2
Design matrix of auxiliary regressors to be transformed to Z2
Delta2
Scaling matrix such that diagonal of
\bar{\Delta}_2 \bar{X}_2^{\top} \bar{M}_1 \bar{X}_2 \Delta_{2} is one
(ignored scaling by n because not needed in code).
See \insertCitedeluca2018glm;textualWALS
SVD
If TRUE, uses svd to compute eigendecomposition
of Z2s, otherwise uses eigen.
postmult
If TRUE, then it uses
Z2s^{-1/2} = T \Lambda^{-1/2} T^{\top}, where T contains
the eigenvectors of Z2s in its columns and \Lambda the corresponding
eigenvalues. If FALSE it uses Z2s^{-1/2} = T \Lambda^{-1/2}.
On the "semiorthogonal-type" transformation
For WALS GLM (and WALS in the linear regression model),
the transformation is semiorthogonal (ignored scaling by n for clarity
and because it is not needed in the code)
in the sense that \bar{M}_{1} \bar{Z}_{2} is semiorthogonal since
\bar{Z}_{2}^{\top} \bar{M}_1 \bar{Z}_{2} =
(\bar{Z}_{2}^{\top} \bar{M}_1) (\bar{M}_{1} \bar{Z}_{2}) = I_{k_2},
where \bar{M}_1 is an idempotent matrix.
For WALS in the NB2 regression model, \bar{M}_{1} \bar{Z}_{2} is not
semiorthogonal anymore due to the rank-1 perturbation in \bar{M}_1 which
causes \bar{M}_1 to not be idempotent anymore, see
the section "Transformed model" in \insertCitehuynhwalsnb;textualWALS.
On the use of postmult = TRUE
The transformation of the auxiliary regressors Z_2 for linear WALS in
eq. (12) of \insertCitemagnus2016wals;textualWALS differs from the
transformation for WALS GLM (and WALS NB) in eq. (9) of
\insertCitedeluca2018glm;textualWALS:
In \insertCitemagnus2016wals;textualWALS the transformed auxiliary
regressors are
Z_{2} = X_2 \Delta_2 T \Lambda^{-1/2},
where T contains the eigenvectors of
\Xi = \Delta_2 X_{2}^{\top} M_{1} X_{2} \Delta_2 in the columns and
\Lambda the respective eigenvalues. This definition is used when
postmult = FALSE.
In contrast, \insertCitedeluca2018glm;textualWALS defines
Z_2 = X_2 \Delta_2 T \Lambda^{-1/2} T^{\top},
where we ignored scaling by n and the notation with "bar" for easier
comparison. This definition is used when postmult = TRUE and is
strongly preferred for walsGLM and walsNB.
See \insertCitehuynhwals;textualWALS for more details.
References
\insertAllCited
WALS documentation built on June 22, 2024, 9:42 a.m.
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| semiorthogonalize | R Documentation |
Internal function: Semiorthogonal-type transformation of X2 to Z2
Description
Uses the matrix Z2s (called \bar{\Xi} in eq. (9) of
\insertCitedeluca2018glm;textualWALS) to transform \bar{X}_2 to
\bar{Z}_2, i.e. to perform \bar{Z}_2 = \bar{X}_2 \bar{\Delta}_2 \bar{\Xi}^{-1/2}.
For WALS in the linear regression model, the variables do not have a "bar".
Usage
semiorthogonalize(Z2s, X2, Delta2, SVD = TRUE, postmult = FALSE)
Arguments
Z2s |
Matrix for which we take negative square root in
|
X2 |
Design matrix of auxiliary regressors to be transformed to Z2 |
Delta2 |
Scaling matrix such that diagonal of
|
SVD |
If |
postmult |
If |
On the "semiorthogonal-type" transformation
For WALS GLM (and WALS in the linear regression model),
the transformation is semiorthogonal (ignored scaling by n for clarity
and because it is not needed in the code)
in the sense that \bar{M}_{1} \bar{Z}_{2} is semiorthogonal since
\bar{Z}_{2}^{\top} \bar{M}_1 \bar{Z}_{2} =
(\bar{Z}_{2}^{\top} \bar{M}_1) (\bar{M}_{1} \bar{Z}_{2}) = I_{k_2},
where \bar{M}_1 is an idempotent matrix.
For WALS in the NB2 regression model, \bar{M}_{1} \bar{Z}_{2} is not
semiorthogonal anymore due to the rank-1 perturbation in \bar{M}_1 which
causes \bar{M}_1 to not be idempotent anymore, see
the section "Transformed model" in \insertCitehuynhwalsnb;textualWALS.
On the use of postmult = TRUE
The transformation of the auxiliary regressors Z_2 for linear WALS in
eq. (12) of \insertCitemagnus2016wals;textualWALS differs from the
transformation for WALS GLM (and WALS NB) in eq. (9) of
\insertCitedeluca2018glm;textualWALS:
In \insertCitemagnus2016wals;textualWALS the transformed auxiliary regressors are
Z_{2} = X_2 \Delta_2 T \Lambda^{-1/2},
where T contains the eigenvectors of
\Xi = \Delta_2 X_{2}^{\top} M_{1} X_{2} \Delta_2 in the columns and
\Lambda the respective eigenvalues. This definition is used when
postmult = FALSE.
In contrast, \insertCitedeluca2018glm;textualWALS defines
Z_2 = X_2 \Delta_2 T \Lambda^{-1/2} T^{\top},
where we ignored scaling by n and the notation with "bar" for easier
comparison. This definition is used when postmult = TRUE and is
strongly preferred for walsGLM and walsNB.
See \insertCitehuynhwals;textualWALS for more details.
References
\insertAllCitedR Package Documentation
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