Table of Contents
What Is Cubic Regression?
Cubic regression fits a third-degree polynomial to data using least squares. The equation y = ax³ + bx² + cx + d can model curves with up to two turning points (inflection points), making it more flexible than linear or quadratic models while remaining interpretable.
It is widely used in engineering for material stress-strain curves, economics for supply/demand modeling, biology for growth patterns, and physics for trajectory analysis.
The Model
The four coefficients are found by solving a 4x4 system of normal equations derived from minimizing the sum of squared residuals.
Least Squares Method
We minimize Σ(yi - axi³ - bxi² - cxi - d)². Taking partial derivatives and setting to zero yields a system solved by Gaussian elimination.
| Degree | Max Turns | Min Points |
|---|---|---|
| 1 (Linear) | 0 | 2 |
| 2 (Quadratic) | 1 | 3 |
| 3 (Cubic) | 2 | 4 |
When to Use Cubic Regression
Use cubic regression when data shows an S-shape or two bending points. If coefficient a is not significantly different from zero, a simpler model may be more appropriate. Always check residual plots for model adequacy.
Frequently Asked Questions
How many data points do I need?
Minimum 4, but 12-20 points provide reliable estimates. More points reduce overfitting risk.
Can I extrapolate?
Extrapolation is risky with cubic models because the x³ term dominates for large |x|, potentially giving unrealistic predictions. Only interpolate within the data range.
How to compare with quadratic?
Compare R-squared values and use an F-test to determine if the cubic term adds significant explanatory power. If not, prefer the simpler model.