unconstrained_fit_default: Unconstrained Generalized Linear Model Estimation

In lgspline: Lagrangian Multiplier Smoothing Splines for Smooth Function Estimation

Unconstrained Generalized Linear Model Estimation

Description

Fits generalized linear models without smoothing constraints using penalized maximum likelihood estimation. This is applied to each partition to obtain the unconstrained estimates, prior to imposing the smoothing constraints.

Hot-start estimates are initialized by treating the matrix square-root inverse as pseudo-observations, appending to the rows of the design matrix, and calling glm.fit replacing the response for pseudo-observations with the inverse link function applied to the value of eta = XB = 0. For log link, exp(0) = 1 is valid for most families. For cases like inverse link, dividing by 0 is obviously not possible, so this is replaced with 1/tol where tol is the convergence tolerance argument.

Usage

unconstrained_fit_default(
  X,
  y,
  LambdaHalf,
  Lambda,
  keep_weighted_Lambda,
  family,
  tol,
  K,
  parallel,
  cl,
  chunk_size,
  num_chunks,
  rem_chunks,
  order_indices,
  weights,
  ...
)

Arguments

X	Design matrix of predictors
y	Response variable vector
LambdaHalf	Square root of penalty matrix (\boldsymbol{\Lambda}^{1/2})
Lambda	Penalty matrix (\boldsymbol{\Lambda})
keep_weighted_Lambda	Logical flag to control penalty matrix handling: - TRUE: Return coefficients directly from weighted penalty fitting - FALSE: Apply damped Newton-Raphson optimization to refine estimates
family	Distribution family specification
tol	Convergence tolerance
K	Number of partitions minus one (K)
parallel	Flag for parallel processing
cl	Cluster object for parallel computation
chunk_size	Processing chunk size
num_chunks	Number of computational chunks
rem_chunks	Remaining chunks
order_indices	Observation ordering indices
weights	Optional observation weights
...	Additional arguments passed to glm.fit

Value

Numeric column vector of unconstrained coefficient estimates.

For fitting non-canonical GLMs, use keep_weighted_Lambda = TRUE since the score and Hessian equations below are no longer valid.

For Gamma(link='log') using keep_weighted_Lambda = TRUE is misleading. The information is weighted by a constant (shape parameter) rather than some mean-variance relationship. So keep_weighted_Lambda = TRUE is highly recommended for log-link Gamma models. This constant flushes into the penalty terms, and so the formulation of the information matrix is valid.

For other scenarios, like probit regression, there will be diagonal weights incorporated into the penalty matrix for providing initial MLE estimates, which technically imposes a prior distribution on beta coefficients that isn't by intent.

Heuristically, it shouldn't affect much, as these will be updated to their proper form when providing estimates under constraint; lgspline otherwise does use the correct form of score and information afterwards, regardless of canonical/non-canonical status, as long as 'glm_weight_function' and 'qp_score_function' are properly specified.

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