Polynomial Regression

Extending linear models to capture non-linear relationships in data

What is Polynomial Regression?

Polynomial regression is an extension of linear regression that models the relationship between the independent variable x and the dependent variable y as an nth degree polynomial. Unlike linear regression, which fits a straight line to the data, polynomial regression can capture more complex, non-linear patterns.

The general form of a polynomial regression model is:

y = β₀ + β₁x + β₂x² + β₃x³ + ... + βₙxⁿ + ε

where β₀, β₁, β₂, ..., βₙ are the regression coefficients and ε is the error term.

Key Concepts

Degree of Polynomial

The highest power of the independent variable in the polynomial equation. Higher degrees can fit more complex patterns but risk overfitting.

Basis Functions

Polynomial terms (x, x², x³, etc.) serve as basis functions that transform the original features into a higher-dimensional space.

Overfitting Risk

Higher-degree polynomials can lead to overfitting, where the model captures noise in the training data rather than the underlying pattern.

Feature Transformation

Polynomial regression can be implemented as a linear regression model after transforming the input features to include polynomial terms.

When to Use Polynomial Regression

When the relationship between variables follows a curvilinear pattern
When linear models show systematic errors in residual plots
When domain knowledge suggests non-linear relationships
When modeling phenomena with diminishing returns or saturation effects
As a simple approach to capture non-linearity before trying more complex models

Comparison with Other Regression Models

Model	Strengths	Weaknesses	Best Use Cases
Linear Regression	Simple and interpretable Computationally efficient Less prone to overfitting	Cannot capture non-linear relationships Limited flexibility	Simple linear relationships When interpretability is crucial
Polynomial Regression	Can model non-linear relationships Still relatively interpretable Flexible degree selection	Prone to overfitting with high degrees Sensitive to outliers Extrapolation can be unreliable	Curvilinear relationships When the pattern follows a polynomial form
Ridge/Lasso Regression	Prevents overfitting Handles multicollinearity Feature selection (Lasso)	Requires tuning regularization parameter Still limited to linear relationships	High-dimensional data When features are correlated
Spline Regression	Flexible for complex patterns Smooth transitions at knot points Better local fitting	More complex to implement Requires knot selection Less interpretable	Complex non-linear patterns When relationship changes across ranges

Linear Regression Ridge & Lasso Regression