Module regularized 

Diagnostics for regularized linear regression — the family the OLS diagnostics deliberately excluded, now provided as first-class types.

Regularization changes the fit in ways the OLS formulas can’t be reused for. Under ridge the hat matrix becomes H_λ = X(XᵀX + λI)⁻¹Xᵀ, so leverage, degrees of freedom, and everything built on them differ; lasso has no closed-form hat matrix at all. Each estimator therefore gets its own fitted type with the diagnostics that are actually well-defined for it:

• RidgeFit — closed-form ridge (via SVD), with effective degrees of freedom Σ dⱼ²/(dⱼ²+λ), ridge leverage, GCV, effective AIC/BIC, and a ridge_vif that generalizes the OLS VIF and reduces to it at λ = 0 — the “VIF before/after regularization” comparison.
• LassoFit — coordinate-descent lasso, whose natural degrees-of-freedom estimate is simply the size of the active set (Zou–Hastie–Tibshirani).
• ElasticNetFit — the lasso/ridge blend λ[α‖β‖₁ + ½(1−α)‖β‖²], with a shrinkage-aware effective df (the active-set trace under the ridge part) that interpolates between the two.
• PenalizedLogisticFit — a penalized GLM: ridge-penalized logistic regression by penalized IRLS, with effective df tr[(XᵀWX+λP)⁻¹XᵀWX], sandwich standard errors, and shrinkage-aware AIC/BIC.

Penalty conventions (read before comparing λ across estimators)

• Ridge penalizes the centered predictors on their given scale; the intercept (a detected constant column) is never penalized. Ridge is not scale-invariant, so standardizing predictors first is the usual practice.
• Lasso and elastic net standardize predictors internally and minimize the (1/2n)-scaled objective, so their λ is on a different scale than ridge’s. Elastic net’s α = 1 reproduces the lasso exactly.
• Penalized logistic penalizes the coefficients on their given scale (like ridge OLS), leaving a detected intercept unpenalized.

Neither is a drop-in for the other’s λ; they are documented per-type.

Structs

ElasticNetFit

A fitted elastic-net regression model and its diagnostics.

LassoFit

A fitted lasso-regression model and its diagnostics.

PenalizedLogisticFit

A fitted ridge-penalized logistic regression — a penalized GLM — and its shrinkage-aware diagnostics.

RidgeFit

A fitted ridge-regression model and its diagnostics.

Functions

select_lambda_gcv

Select the ridge penalty that minimizes GCV over a grid of candidate λs, returning the best RidgeFit.