Polynomial Regression

What is Polynomial Regression?

Polynomial regression models the relationship between the dependent variable (y) and the independent variable (x) as an nth degree polynomial:

Where:

• y: Dependent variable (target)
• x: Independent variable (feature)
• β0,β1,...,βn: Coefficients of the polynomial terms
• ?: Error term

This allows for the modeling of curves, making it suitable for non-linear data.

Why Use Polynomial Regression?

While linear regression works well for linear relationships, real-world data often exhibits non-linear patterns. Polynomial regression can capture these patterns, making it ideal for:

• Economics: Modeling diminishing returns or exponential growth.
• Physics: Describing motion under varying forces.
• Biology: Modeling population growth or enzyme kinetics.

When to Use Polynomial Regression

• When visualizing data shows a clear non-linear trend.
• When a higher degree of accuracy is needed compared to linear regression.
• When the relationship between variables can be approximated by a polynomial curve.

However, avoid overfitting by using excessively high-degree polynomials.

Steps in Polynomial Regression

1. Data Preprocessing

• Scale or normalize features for better performance.
• Transform the input feature(s) into polynomial features.

2. Fit a Model

• Use techniques like least squares or optimization algorithms to determine coefficients.

3. Evaluate the Model

• Check metrics like R2, Mean Squared Error (MSE), and residual plots.

Hands-On Mini-Challenge: Fitting a Polynomial Curve

Let’s fit a polynomial regression model using synthetic data for better understanding.

Starter Code:

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

# Generate synthetic non-linear data
np.random.seed(42)
X = np.random.rand(100, 1) * 10  # Independent variable
y = 2 + 1.5 * X + 0.5 * X**2 + np.random.randn(100, 1) * 5  # Target with quadratic relationship

# Transform to polynomial features
poly = PolynomialFeatures(degree=2)  # Change degree to experiment
X_poly = poly.fit_transform(X)

# Train the model
model = LinearRegression()
model.fit(X_poly, y)

# Make predictions
y_pred = model.predict(X_poly)

# Evaluate the model
print("MSE:", mean_squared_error(y, y_pred))
print("R-squared:", r2_score(y, y_pred))

# Visualize the result
plt.scatter(X, y, color='blue', label='Data')
plt.plot(X, y_pred, color='red', label='Polynomial Fit')
plt.xlabel("X")
plt.ylabel("y")
plt.legend()
plt.title("Polynomial Regression Fit")
plt.show()
        

Key Considerations in Polynomial Regression

1. Choosing the Degree:
2. Regularization:
3. Feature Engineering:

Best Practices for Polynomial Regression

• Always visualize your data to identify trends and relationships.
• Use tools like scikit-learn’s PolynomialFeatures for easy implementation.
• Evaluate the model thoroughly to avoid overfitting or underfitting.
• Compare polynomial regression with other non-linear models like decision trees or neural networks.