Polynomial Regression in Python: What It Is and How to Use It

Polynomial regression in Python is a type of regression analysis where the relationship between the input x and output y is modeled as an n-degree polynomial. Using sklearn, you transform features to polynomial features and then fit a linear model to capture curves in data.

How It Works

Polynomial regression works by extending simple linear regression to fit curved lines instead of just straight lines. Imagine you want to predict something that doesn’t follow a straight path, like the growth of a plant over time that speeds up or slows down. Instead of drawing a straight line, polynomial regression fits a curve by adding powers of the input features (like x², x³, etc.).

In Python, this is done by first transforming the original input data into new features that include these powers. Then, a regular linear regression model is trained on these new features. This way, the model can learn more complex patterns while still using simple math behind the scenes.

Example

This example shows how to use polynomial regression with sklearn. We create some data that follows a curve, transform it to polynomial features, and fit a linear model to it.

python

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

# Create sample data
x = np.array([1, 2, 3, 4, 5]).reshape(-1, 1)
y = np.array([1, 4, 9, 16, 25])  # y = x^2

# Transform features to polynomial features (degree 2)
poly = PolynomialFeatures(degree=2)
x_poly = poly.fit_transform(x)

# Fit linear regression on polynomial features
model = LinearRegression()
model.fit(x_poly, y)

# Predict
y_pred = model.predict(x_poly)

# Print results
print(f"Coefficients: {model.coef_}")
print(f"Intercept: {model.intercept_}")
print(f"Mean Squared Error: {mean_squared_error(y, y_pred):.2f}")
print(f"R2 Score: {r2_score(y, y_pred):.2f}")

Output

Coefficients: [0. 0. 1.]
Intercept: 0.0
Mean Squared Error: 0.00
R2 Score: 1.00

When to Use

Use polynomial regression when your data shows a curved relationship between the input and output, and a straight line won’t fit well. For example, it’s useful in predicting growth rates, trends in sales over time, or any situation where the effect of the input changes at different levels.

It’s important to choose the right degree for the polynomial: too low might miss the curve, too high might overfit and capture noise instead of the true pattern.

Key Points

• Polynomial regression fits curves by adding powers of input features.
• It uses linear regression on transformed polynomial features.
• Choosing the right polynomial degree is key to good predictions.
• It helps model non-linear relationships simply and effectively.

Key Takeaways

Polynomial regression models curved relationships by adding powers of input features.

In Python, use sklearn's PolynomialFeatures with LinearRegression to fit polynomial models.

Choosing the polynomial degree balances fitting the data well and avoiding overfitting.

It is useful when data patterns are not straight lines but curves.

Model evaluation metrics like mean squared error and R2 score help check fit quality.