Residual Analysis — Diagnosing Linear Regression

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression
from sklearn.datasets import fetch_california_housing
from sklearn.model_selection import train_test_split
from sklearn.preprocessing import StandardScaler
from scipy import stats

housing = fetch_california_housing()
X = pd.DataFrame(housing.data, columns=housing.feature_names)
y = housing.target

X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)
scaler = StandardScaler()
model = LinearRegression()
model.fit(scaler.fit_transform(X_train), y_train)
y_pred = model.predict(scaler.transform(X_test))
residuals = y_test - y_pred

fig, axes = plt.subplots(2, 2, figsize=(12, 10))

# PLOT 1: Residuals vs Fitted Values
axes[0, 0].scatter(y_pred, residuals, alpha=0.3, color="steelblue", s=20)
axes[0, 0].axhline(0, color="red", linestyle="--", linewidth=2)
axes[0, 0].set_xlabel("Fitted Values")
axes[0, 0].set_ylabel("Residuals")
axes[0, 0].set_title("Residuals vs Fitted\n(should be random scatter around 0)")

# PLOT 2: Distribution of residuals (should be roughly normal)
axes[0, 1].hist(residuals, bins=50, color="steelblue", edgecolor="white", density=True)
x_range = np.linspace(residuals.min(), residuals.max(), 100)
axes[0, 1].plot(x_range, stats.norm.pdf(x_range, residuals.mean(), residuals.std()),
                color="red", linewidth=2, label="Normal fit")
axes[0, 1].set_title("Residual Distribution\n(should be bell-shaped)")
axes[0, 1].legend()

# PLOT 3: Q-Q plot (normality check)
stats.probplot(residuals, plot=axes[1, 0])
axes[1, 0].set_title("Q-Q Plot\n(points on diagonal = normal residuals)")

# PLOT 4: Actual vs Predicted
axes[1, 1].scatter(y_test, y_pred, alpha=0.3, color="steelblue", s=20)
min_val, max_val = min(y_test.min(), y_pred.min()), max(y_test.max(), y_pred.max())
axes[1, 1].plot([min_val, max_val], [min_val, max_val], "r--", linewidth=2, label="Perfect prediction")
axes[1, 1].set_xlabel("Actual Values")
axes[1, 1].set_ylabel("Predicted Values")
axes[1, 1].set_title("Actual vs Predicted\n(diagonal = perfect model)")
axes[1, 1].legend()

plt.tight_layout()
plt.savefig("residual_plots.png", dpi=100, bbox_inches="tight")
plt.show()

# RESIDUAL STATISTICS
print("Residual diagnostics:")
print(f"  Mean:        {residuals.mean():.4f}  (should be ~0)")
print(f"  Std:         {residuals.std():.4f}")
print(f"  Skewness:    {pd.Series(residuals).skew():.3f}  (|<0.5| = good, >1.0 = problematic)")
print(f"  Max error:   {abs(residuals).max():.3f}  (worst single prediction)")
print(f"  Within 0.5:  {(abs(residuals) < 0.5).mean():.1%}  (predictions within $50k)")

# HETEROSCEDASTICITY DETECTION -- Breusch-Pagan-like test
corr_w_residuals = np.corrcoef(y_pred, residuals**2)[0, 1]
print(f"\nResiduals^2 vs fitted correlation: {corr_w_residuals:.3f}")
print("  (near 0 = homoscedastic, large value = heteroscedastic -> consider log(y))")