dot-bf_case_gauss_gee: Gaussian Identity + GEE: Closed-Form Solve (Case a)

In lgspline: Lagrangian Multiplier Smoothing Splines for Smooth Function Estimation

Gaussian Identity + GEE: Closed-Form Solve (Case a)

Description

When the response is Gaussian with identity link and a working correlation structure is present, the whitened system \mathbf{V}^{-1/2}\mathbf{X} is not block-diagonal, so partition-wise backfitting does not apply. This function performs a single closed-form solve that replicates get_B Path 1a but in the backfitting context.

Usage

.bf_case_gauss_gee(
  X,
  y,
  K,
  p_expansions,
  order_list,
  VhalfInv,
  Lambda,
  L_partition_list,
  unique_penalty_per_partition,
  A,
  constraint_values,
  spline_cols,
  flat_cols,
  observation_weights,
  quadprog = FALSE,
  qp_Amat = NULL,
  qp_bvec = NULL,
  qp_meq = NULL,
  qp_score_function = NULL,
  tol = 1e-08,
  parallel_eigen = FALSE,
  parallel_qr = FALSE,
  cl = NULL,
  chunk_size = NULL,
  num_chunks = NULL,
  rem_chunks = NULL,
  initial_active_ineq = integer(0),
  include_warnings = TRUE,
  verbose = FALSE,
  ...
)

Arguments

X	List of K+1 unwhitened design matrices.
y	List of K+1 response vectors.
K	Integer; number of interior knots.
p_expansions	Integer; columns per partition.
order_list	List of observation-index vectors by partition.
VhalfInv	\mathbf{V}^{-1/2} matrix in the original observation ordering.
Lambda	Shared penalty matrix.
L_partition_list	Partition-specific penalty matrices.
unique_penalty_per_partition	Logical.
A	Full constraint matrix.
constraint_values	List of constraint RHS vectors.
spline_cols, flat_cols	Integer vectors of column indices.
quadprog	Logical; whether inequality refinement is active.
qp_Amat, qp_bvec, qp_meq	QP constraint specification.
tol	Numeric tolerance.
parallel_eigen, cl, chunk_size, num_chunks, rem_chunks	Parallel arguments used by the Woodbury gate.
include_warnings, verbose	Logical.

Details

In other words, this begins from the dense correlated solve on the pooled coefficient space. When inequality constraints are present and Woodbury acceleration is available, the helper then attempts the same partition-wise Woodbury active-set refinement used by get_B(). If that path is unavailable or fails, it falls back to dense solve.QP on the whitened system.

Value

A named list:

beta_spline

List of K+1 spline coefficient vectors.

beta_flat

Numeric vector of flat coefficients.

result

List of K+1 full per-partition coefficient vectors.

qp_info

QP or active-set metadata, or NULL.

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