from sklearn.linear_model import Ridge, RidgeCV
from sklearn.model_selection import cross_val_score
import numpy as np
── Ridge adds L2 penalty to OLS ─────────────────────#
Minimises: Σ(y - ŷ)² + α * Σβᵢ²#
α (alpha): regularisation strength#
α = 0: same as linear regression (no regularisation)#
α → ∞: all coefficients shrink toward zero#
Effect: coefficients shrink proportionally (none become exactly zero)#
Best for: multicollinear features, more features than samples#
alphas = [0.01, 0.1, 1.0, 10, 100, 1000]
Manual grid search with cross-validation#
results = []
for alpha in alphas:
pipe = Pipeline([(“scaler”, StandardScaler()),
(“model”, Ridge(alpha=alpha))])
cv_r2 = cross_val_score(pipe, X_train, y_train, cv=5, scoring=“r2”)
results.append({“alpha”: alpha, “r2_mean”: cv_r2.mean(), “r2_std”: cv_r2.std()})
import pandas as pd
results_df = pd.DataFrame(results)
print(results_df)
best_alpha = results_df.loc[results_df[“r2_mean”].idxmax(), “alpha”]
print(f”Best alpha: {best_alpha}”)
Built-in CV (faster)#
alphas_to_try = np.logspace(-3, 4, 50)  # log-spaced from 0.001 to 10,000
ridge_cv = RidgeCV(alphas=alphas_to_try, cv=5, scoring=“r2”)
ridge_cv.fit(StandardScaler().fit_transform(X_train), y_train)
print(f”RidgeCV best alpha: {ridge_cv.alpha_:.4f}”)
print(f”RidgeCV R²: {ridge_cv.score(StandardScaler().fit_transform(X_test), y_test):.4f}”)
── Visualise coefficient shrinkage ──────────────────#
fig, ax = plt.subplots(figsize=(10, 5))
coefs = []
for alpha in np.logspace(-3, 4, 100):
r = Ridge(alpha=alpha)
r.fit(StandardScaler().fit_transform(X_train), y_train)
coefs.append(r.coef_)
coefs = np.array(coefs)
for i, name in enumerate(X.columns):
ax.plot(np.logspace(-3, 4, 100), coefs[:, i], label=name)
ax.set_xscale(“log”)
ax.set(xlabel=“Alpha (log scale)”, ylabel=“Coefficient Value”,
title=“Ridge: Coefficient Shrinkage”)
ax.axvline(best_alpha, color=“red”, linestyle=”—”, label=“Best alpha”)
ax.legend()
plt.tight_layout()

from sklearn.metrics import (mean_absolute_error, mean_squared_error,
                             root_mean_squared_error, r2_score,
                             mean_absolute_percentage_error)
import numpy as np
y_true = y_test
y_pred = pipeline.predict(X_test)
── MAE: Mean Absolute Error ──────────────────────────#
Average absolute difference. Same units as target.#
Robust to outliers (vs MSE/RMSE which square errors)#
Easy to explain: “on average, predictions are off by $X”#
mae = mean_absolute_error(y_true, y_pred)
print(f”MAE:  ${mae:,.0f}”)
── MSE: Mean Squared Error ───────────────────────────#
Squares errors → large errors penalised more heavily#
NOT in target units (hard to interpret directly)#
Used as training loss (differentiable)#
mse = mean_squared_error(y_true, y_pred)
print(f”MSE:  ${mse:,.0f}”)
── RMSE: Root Mean Squared Error ────────────────────#
sqrt(MSE) — back in target units#
Most common metric for regression#
More sensitive to large errors than MAE#
rmse = root_mean_squared_error(y_true, y_pred)
print(f”RMSE: ${rmse:,.0f}”)
── R²: Coefficient of Determination ─────────────────#
Proportion of variance explained: R² = 1 - SS_res/SS_tot#
R² = 1.0: perfect prediction#
R² = 0.0: as good as predicting the mean (useless model)#
R² < 0.0: WORSE than predicting the mean (seriously bad)#
r2 = r2_score(y_true, y_pred)
print(f”R²:   {r2:.4f}”)
── MAPE: Mean Absolute Percentage Error ─────────────#
Percentage error. Intuitive but problematic near zero.#
“Predictions are X% off on average”#
mape = mean_absolute_percentage_error(y_true, y_pred)
print(f”MAPE: {mape:.2%}”)
── When to use each ──────────────────────────────────#
print("""
MAE:  Use when outliers exist and you want robust metric
Use to communicate to stakeholders (“off by $X”)
RMSE: Use when large errors are unacceptable (safety, medical)
Standard metric for Kaggle regression competitions
R²:   Use to compare models on same dataset
Useful relative metric: 0.85 >> 0.70
MAPE: Use for business reporting when % errors matter
AVOID when target can be near zero
""")