dot-get_B_gee_gaussian: Path 1a: Gaussian Identity + GEE (Closed-Form Full-System...

In lgspline: Lagrangian Multiplier Smoothing Splines for Smooth Function Estimation

Path 1a: Gaussian Identity + GEE (Closed-Form Full-System Solve)

Description

Computes the constrained penalized GLS estimate for Gaussian response with identity link when a correlation structure is present. Because \mathbf{V}^{-1/2} couples all partitions, fitting must operate on the full P-dimensional whitened system rather than partition-wise.

Usage

.get_B_gee_gaussian(
  X_block,
  X_tilde,
  y_tilde,
  VhalfInv_perm,
  Lambda_block,
  A,
  K,
  p_expansions,
  constraint_value_vectors,
  family,
  return_G_getB,
  quadprog,
  qp_Amat,
  qp_bvec,
  qp_meq,
  qp_score_function,
  order_list,
  observation_weights,
  obs_wt_precomp = NULL,
  Gram_full_precomp = NULL,
  Xy_tilde_precomp = NULL,
  parallel_qr = FALSE,
  cl = NULL,
  ...
)

Arguments

X_block	Full N \times P unwhitened block-diagonal design.
X_tilde	Whitened design \mathbf{V}^{-1/2}\mathbf{X}.
y_tilde	Whitened response \mathbf{V}^{-1/2}\mathbf{y}.
VhalfInv_perm	\mathbf{V}^{-1/2} permuted to partition ordering.
Lambda_block	Full P \times P block-diagonal penalty.
A	Constraint matrix (P \times R).
K, p_expansions	Integer dimensions.
constraint_value_vectors	Constraint RHS list encoding \mathbf{A}^{\top}\boldsymbol{\beta} = \mathbf{c}.
family	GLM family object.
return_G_getB	Logical; return covariance components.
quadprog	Logical; apply QP refinement.
qp_Amat, qp_bvec, qp_meq	QP constraint specification.
qp_score_function	Score function for QP step.
order_list, observation_weights	Standard partition arguments.
obs_wt_precomp, Gram_full_precomp, Xy_tilde_precomp	Optional pre-computed tuning quantities for repeated correlated Gaussian fits.
...	Passed to score function.

Details

The unconstrained GLS estimate is:

\hat{\boldsymbol{\beta}} = \mathbf{G}(\mathbf{X}^{*\top} \mathbf{y}^{*}), \quad \mathbf{G} = (\mathbf{X}^{*\top}\mathbf{X}^{*} + \boldsymbol{\Lambda})^{-1}

where \mathbf{X}^{*} = \mathbf{V}^{-1/2}\mathbf{X} and \mathbf{y}^{*} = \mathbf{V}^{-1/2}\mathbf{y}. The constrained estimate is then obtained via the \mathbf{G}^{1/2}\mathbf{r}^* trick in the full P-dimensional space.

Value

If return_G_getB = TRUE: list with B, G_list, and qp_info. Otherwise: list of B and qp_info.

lgspline documentation built on May 8, 2026, 5:07 p.m.