Polynomial Regression | Fera Analytics

Introduction

Polynomial regression is a form of linear regression in which the relationship between the independent variable x and the dependent variable y is modeled as an n-th degree polynomial. This lesson explores the concept of polynomial regression, its purpose, methods, practical considerations, and implementation in Python.

Purpose of Polynomial Regression

Modeling Non-linear Relationships: Capturing non-linear relationships between variables that cannot be adequately modeled with simple linear regression.
Flexibility in Curve Fitting: Providing a flexible curve that can fit more complex patterns in the data.
Trade-off between Bias and Variance: Balancing bias and variance in modeling by adjusting the polynomial degree.

Methods of Polynomial Regression

The polynomial regression model is defined as:

\[ y = \beta_0 + \beta_1 x + \beta_2 x^2 + \ldots + \beta_n x^n + \epsilon \]

where:

\begin{align*}
y & : \text{is the dependent variable}, \\
x & : \text{is the independent variable}, \\
\beta_0, \beta_1, \ldots, \beta_n & : \text{are the coefficients}, \\
\epsilon & : \text{is the error term}.
\end{align*}

The model can be extended to include higher-degree terms (e.g., \( x^2, x^3, \ldots \)) to better fit the data.

Practical Considerations

Choosing the Degree: Selecting the appropriate degree of the polynomial is crucial. Higher-degree polynomials can overfit the data, capturing noise instead of underlying patterns.
Model Evaluation: Evaluate model performance using metrics such as the coefficient of determination and mean squared error (MSE) to assess goodness of fit.
Feature Scaling: Polynomial features should be scaled to avoid numerical instability and ensure all features contribute equally.

Implementing Polynomial Regression in Python

Here’s an example of implementing polynomial regression using Python’s numpy and sklearn libraries:

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

# Generate example data
np.random.seed(0)
X = 2 * np.random.rand(100, 1)
y = 4 + 3 * X + np.random.randn(100, 1)

# Fit polynomial features (degree 2)
poly_features = PolynomialFeatures(degree=2, include_bias=False)
X_poly = poly_features.fit_transform(X)

# Fit polynomial regression model
model = LinearRegression()
model.fit(X_poly, y)

# Visualize the polynomial regression
plt.scatter(X, y, color='blue', label='Data')
plt.plot(X, model.predict(X_poly), color='red', label='Polynomial Regression (degree 2)')
plt.xlabel('X')
plt.ylabel('y')
plt.title('Polynomial Regression')
plt.legend()
plt.show()

Practical Applications

Polynomial regression is applied in various domains:

Engineering: Modeling physical processes where relationships are inherently non-linear.
Economics: Analyzing relationships between variables in economic models.
Social Sciences: Modeling complex relationships in social and behavioral research.
Finance: Forecasting trends in financial markets based on historical data.

Conclusion

Polynomial regression is a powerful technique for capturing non-linear relationships in data, providing flexibility in modeling compared to simple linear regression. By understanding its principles, selecting appropriate polynomial degrees, and evaluating model performance, data scientists can effectively apply polynomial regression to extract valuable insights and make informed decisions.