Supply unpenalized columns through always.include, for example a low-df
B-spline basis for a smooth chromatographic background. With nonnegative=TRUE,
L0glm.fit() and L0nnpois() refine these coefficients in a small constrained
block, separately from sparse predictor selection.
Details
Write the linear predictor as $$\eta_j=X\beta_j+B\alpha_j+o_j,\qquad \alpha_j\geq0,$$ where \(B\) contains the compulsory columns. For an identity link, \(B\alpha_j\) is an additive background in response units. For other links it is additive on the link scale. Binomial and quasibinomial intercepts retain their unrestricted bounds; Cox models have no identifiable intercept.
Conditional on \(\beta\), the compulsory block minimizes the likelihood loss subject to its bounds. Gaussian identity models use an exact weighted nonnegative quadratic solve. Poisson identity models with nonnegative designs and fixed signals use the observed likelihood curvature. Other scalar families and links use safeguarded Fisher scoring: $$\alpha^{+}=\arg\min_{a\in\mathcal C} (z-X\beta-o-Ba)^T W(z-X\beta-o-Ba).$$ An active-set solver handles both unrestricted and nonnegative coefficients. Backtracking keeps the family's mean domain and checks the actual likelihood loss. Cox uses its partial-likelihood score and information. Negative-binomial fits also update estimated shape parameters. Multinomial models retain their unconstrained CLR formulation.
The fitter initializes jointly, then alternates the compulsory solve with
sparse fitting conditional on its contribution. At most two outer cycles and
a final compulsory-block polish are used. Each sparse refinement uses two
grouped passes unless group.maxit was supplied explicitly. Updates are
accepted using the original-data likelihood plus the requested complexity
penalty; validation stopping instead uses the held-out deviance. Quasi families
use scaled deviance plus complexity. The original predictor count determines
the preset penalty throughout. Compulsory parameters retain their usual
contribution to information-criterion degrees of freedom and estimated scale.
For positive-mean identity-link distributions (Poisson, Gamma, inverse Gaussian
and negative binomial) with nonnegative designs, automatic starts use a positive
signal backprojection above a low channel-wise background. The pilot accounts
for predictor scales, observation weights and known offsets, avoiding excessive
initial shrinkage of peaks relative to unpenalized background terms.
Gaussian models retain their ridge pilot, rescaled to the response units.
Supplied starts keep their meaning;
zero starts exclude selectable coefficients while compulsory terms remain
eligible. maxit=1 retains a single ridge-pilot iteration.
Both dense and sparse designs retain their storage type. Native Poisson and
Gaussian compulsory solves use OpenMP across channels; native Poisson sparse
fitting also retains its OpenMP backend. With precluster=TRUE, screening uses
summed Poisson channels and every refinement is scored on the original channels.
The fitted object contains nuisance, with compulsory indices, candidate rows,
initialization, accepted/rejected scores, conditional KKT residuals and timings.