1. What is Polynomial Regression?

Polynomial regression is a form of regression analysis where the relationship between the independent variable x and the dependent variable y is modeled as an nth degree polynomial. It fits a nonlinear relationship between x and y.

2. How Does the Calculator Work?

The calculator uses the least squares method:

Where:

• \( X \) — Vandermonde matrix of x values
• \( X^T \) — Transpose of X
• \( y \) — Vector of y values
• \( coeffs \) — Coefficients of the polynomial (from x⁰ to xⁿ)

Explanation: The method minimizes the sum of squared residuals between observed and predicted values.

3. Importance of Polynomial Regression

Details: Polynomial regression can model nonlinear relationships and is more flexible than linear regression, though it requires careful selection of polynomial degree to avoid overfitting.

4. Using the Calculator

Tips: Enter polynomial degree, comma-separated x values and y values. The number of data points should be greater than the polynomial degree.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between linear and polynomial regression?
A: Linear regression fits a straight line, while polynomial regression can fit curves by including higher-order terms.

Q2: How do I choose the right polynomial degree?
A: Start with degree 2 (quadratic) and increase only if needed. Higher degrees may overfit the data.

Q3: What are the limitations of polynomial regression?
A: It can be sensitive to outliers and may produce unrealistic predictions outside the data range (extrapolation).

Q4: When should I use polynomial regression?
A: When the relationship between variables appears curved in scatter plots or when theory suggests a nonlinear relationship.

Q5: What's the Vandermonde matrix?
A: A matrix where each row represents x values raised to successive powers (x⁰, x¹, x², ..., xⁿ).