W Code - Pattern-Based DSA Learning Platform

Bhanu Bisht

Polynomial Regression

Handling non-linear data with polynomial features: w₀ + w₁x + w₂x²...

Topics

sklearn

Implementation

🌾 Indian Context

We model crop yield vs rainfall — showing the classic inverted-U pattern where moderate rainfall gives maximum yield, while drought and floods both reduce it!

When Linear Regression Fails

Foundation

Python Implementation

import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression
from sklearn.preprocessing import PolynomialFeatures
from sklearn.metrics import r2_score

# Generate curved data: y = 0.5x² + x + 2 + noise
np.random.seed(42)
X = np.linspace(-3, 3, 50).reshape(-1, 1)
y_true = 0.5 * X**2 + X + 2
y = y_true.ravel() + np.random.randn(50) * 0.5

print("=" * 60)
print("📈 LINEAR vs POLYNOMIAL REGRESSION")
print("=" * 60)

# 1. Linear Regression (Will fail on curved data!)
lin_reg = LinearRegression()
lin_reg.fit(X, y)
y_pred_linear = lin_reg.predict(X)
r2_linear = r2_score(y, y_pred_linear)

print(f"\n❌ Linear Regression:")
print(f"   R² Score: {r2_linear:.4f}")
print(f"   Equation: ŷ = {lin_reg.coef_[0]:.2f}x + {lin_reg.intercept_:.2f}")

# 2. Polynomial Regression (Degree 2 - perfect for quadratic!)
# Step 1: Transform X into [1, x, x²]
poly = PolynomialFeatures(degree=2)
X_poly = poly.fit_transform(X)
print(f"\n📐 Feature Transformation:")
print(f"   Original shape: {X.shape} → Transformed: {X_poly.shape}")
print(f"   Features: [1, x, x²]")

# Step 2: Fit Linear Regression on polynomial features
poly_reg = LinearRegression()
poly_reg.fit(X_poly, y)
y_pred_poly = poly_reg.predict(X_poly)
r2_poly = r2_score(y, y_pred_poly)

print(f"\n✅ Polynomial Regression (Degree 2):")
print(f"   R² Score: {r2_poly:.4f}")
coefs = poly_reg.coef_
intercept = poly_reg.intercept_
print(f"   Equation: ŷ = {coefs[2]:.2f}x² + {coefs[1]:.2f}x + {intercept:.2f}")

print(f"\n💡 Improvement: R² increased from {r2_linear:.2f} to {r2_poly:.2f}!")

Key Takeaways

Linear Regression fails when data has curved (non-linear) relationships

Polynomial features transform x → [1, x, x², x³, ...]

The model is still "linear" in terms of weights, just with transformed features

Degree 2 (quadratic) fits parabolic data, Degree 3 (cubic) fits S-curves

import numpy as np import matplotlib.pyplot as plt from sklearn.linear_model import LinearRegression from sklearn.preprocessing import PolynomialFeatures from sklearn.metrics import r2_score # Generate curved data: y = 0.5x² + x + 2 + noise np.random.seed(42) X = np.linspace(-3, 3, 50).reshape(-1, 1) y_true = 0.5 * X**2 + X + 2 y = y_true.ravel() + np.random.randn(50) * 0.5 print("=" * 60) print("📈 LINEAR vs POLYNOMIAL REGRESSION") print("=" * 60) # 1. Linear Regression (Will fail on curved data!) lin_reg = LinearRegression() lin_reg.fit(X, y) y_pred_linear = lin_reg.predict(X) r2_linear = r2_score(y, y_pred_linear) print(f"\n❌ Linear Regression:") print(f" R² Score: {r2_linear:.4f}") print(f" Equation: ŷ = {lin_reg.coef_[0]:.2f}x + {lin_reg.intercept_:.2f}") # 2. Polynomial Regression (Degree 2 - perfect for quadratic!) # Step 1: Transform X into [1, x, x²] poly = PolynomialFeatures(degree=2) X_poly = poly.fit_transform(X) print(f"\n📐 Feature Transformation:") print(f" Original shape: {X.shape} → Transformed: {X_poly.shape}") print(f" Features: [1, x, x²]") # Step 2: Fit Linear Regression on polynomial features poly_reg = LinearRegression() poly_reg.fit(X_poly, y) y_pred_poly = poly_reg.predict(X_poly) r2_poly = r2_score(y, y_pred_poly) print(f"\n✅ Polynomial Regression (Degree 2):") print(f" R² Score: {r2_poly:.4f}") coefs = poly_reg.coef_ intercept = poly_reg.intercept_ print(f" Equation: ŷ = {coefs[2]:.2f}x² + {coefs[1]:.2f}x + {intercept:.2f}") print(f"\n💡 Improvement: R² increased from {r2_linear:.2f} to {r2_poly:.2f}!")