dot-compute_gcvu: Evaluate Modified GCV_u Criterion at a Given Penalty...
In lgspline: Lagrangian Multiplier Smoothing Splines for Smooth Function Estimation

.compute_gcvu

R Documentation

Evaluate Modified GCV_u Criterion at a Given Penalty Configuration

Description

Computes the modified GCV_u criterion for a given set of penalty parameters. This is the objective function minimized during penalty tuning; gcv_gamma = 1 recovers ordinary GCV.

Usage

.compute_gcvu(par, log_penalty_vec, env, ...)

Arguments

`par`	Numeric vector; log-scale penalty parameters. First two elements are log(wiggle_penalty) and log(flat_ridge_penalty). Remaining elements (if any) are log-scale predictor/partition penalties.
`log_penalty_vec`	Numeric vector; log-scale predictor/partition penalties (passed separately for compatibility with the grid search).
`env`	List; pre-computed objects and tuning configuration. Contains: X, y, X_gram, Xy Design matrices, response, Gram matrices, cross-products. A, R_constraints, K, p_expansions, N_obs Constraint matrix and dimensions. smoothing_spline_penalty Integrated second-derivative penalty matrix. custom_penalty_mat Optional custom penalty matrix. colnm_expansions Character vector of expansion column names. unique_penalty_per_predictor, unique_penalty_per_partition Logicals. family GLM family object. delta Pseudocount for link function stabilization. gcv_gamma Modified-GCV inflation factor applied to the effective-degrees-of-freedom term in the denominator. meta_penalty Meta-penalty coefficient. order_list List of observation indices per partition. observation_weights, homogenous_weights Observation weighting. parallel flags, cl, chunk_size, num_chunks, rem_chunks Parallel config. unconstrained_fit_fxn, keep_weighted_Lambda, iterate Fitting config. qp_score_function, quadprog, qp_Amat, qp_bvec, qp_meq QP config. tol, sd_y Convergence tolerance and response scale. constraint_value_vectors Constraint values. glm_weight_function, schur_correction_function GLM functions. need_dispersion_for_estimation, dispersion_function Dispersion. blockfit, just_linear_without_interactions Blockfit config. use_blockfit Logical; pre-computed dispatch flag for blockfit. flat_cols Integer vector; pre-computed flat column indices. Vhalf, VhalfInv Correlation structure matrices. verbose, include_warnings Output control.
`...`	Additional arguments passed to fitting functions.

Value

List containing:

GCV_u: Numeric; GCV_u criterion value including meta-penalty.
B: List; fitted coefficient vectors by partition.
GXX: List; \mathbf{G}_{k} \mathbf{X}_{k}^{\top}\mathbf{X}_{k} matrices.
G_list: List; eigendecomposition results from compute_G_eigen.
mean_W: Numeric; \mathrm{tr}(\mathbf{H})/N, the average leverage entering the denominator of \mathrm{GCV}_u.
sum_W: Numeric; \mathrm{tr}(\mathbf{H}), the effective degrees of freedom.
Lambda: Matrix; combined penalty matrix \boldsymbol{\Lambda}.
L1: Matrix; baseline smoothness penalty component.
L2: Matrix; baseline ridge penalty component.
L_predictor_list: List; predictor-specific penalty matrices.
L_partition_list: List; partition-specific penalty matrices.
numerator: Numeric; sum of squared residuals.
denominator: Numeric; modified-GCV denominator N(1 - \gamma\bar{W})^{2}.
residuals: List; residual vectors by partition.
denom_sq: Numeric; squared denominator.
AGAInv: Matrix; (\mathbf{A}^{\top}\mathbf{G}\mathbf{A})^{-1}.