1. Your first fit¶

The smallest interesting thing skein can do: fit a sparse linear regression and see which features are active.

What we’re building¶

A linear model y = X β + ε where most components of β are zero. The job is to recover the support — the set of features with nonzero coefficients — and estimate their values. We’ll use MCPRegressor because the MCP penalty is unbiased on truly active coefficients, unlike plain lasso.

A toy problem¶

import numpy as np
import skein_glm

rng = np.random.default_rng(0)

# 200 samples, 50 features. Only features 0, 4, 9, 23 actually matter.
n, p = 200, 50
X = rng.standard_normal((n, p))

true_beta = np.zeros(p)
true_beta[[0, 4, 9, 23]] = [1.5, -1.0, 0.8, -1.2]

y = X @ true_beta + 0.3 * rng.standard_normal(n)

Four features carry signal; 46 are pure noise. A regular OLS fit would give noisy estimates for every coefficient — useless if you want to know which features actually matter.

Fit the model¶

model = skein_glm.MCPRegressor(lambda_=0.05, gamma=3.0).fit(X, y)

That’s it. lambda_ is the penalty strength; gamma controls how sharply MCP transitions from “shrink the estimate” to “leave it alone.” The default gamma=3.0 is what ncvreg recommends.

What you get back¶

model.coef_                # ndarray (p,)
model.intercept_           # float
model.n_features_in_       # 50

active = np.flatnonzero(np.abs(model.coef_) > 1e-6)
print(active)              # [ 0  4  9 23]
print(model.coef_[active]) # ~[1.5, -1.0, 0.8, -1.2]

The exact-zero pattern outside the active set is the headline of sparse regression — non-zero entries identify the variables that matter; zeros let you drop the rest.

Predict and score¶

y_pred = model.predict(X)
r2 = model.score(X, y)
print(f"R² = {r2:.3f}")

predict returns Xβ + α. score returns the sklearn-default R² (since MCPRegressor inherits from RegressorMixin).

What’s the catch?¶

Picking lambda_=0.05 was a guess. Try lambda_=0.5 and watch every coefficient collapse to zero (over-regularized); try lambda_=0.001 and the noise features creep back in. The next tutorial covers the three principled ways to pick λ: the full λ-path (and warm starts), cross-validation, and information criteria.

Recap¶

What	API
Fit a sparse linear model	`skein_glm.MCPRegressor(lambda_, gamma).fit(X, y)`
Read coefficients	`model.coef_`, `model.intercept_`
Predictions	`model.predict(X)`
R² score	`model.score(X, y)`

Same Regressor / fit / predict / score shape as scikit-learn — every skein estimator follows it.

Next¶

→ 2. Picking λ — paths, CV, and AIC/BIC/EBIC.