What Is Quartic Regression?
Quartic regression fits a fourth-degree polynomial to a set of data points. This model can capture complex, multi-curved relationships with up to three turning points (local maxima or minima). The general form is y = ax⁴ + bx³ + cx² + dx + e, where five coefficients determine the shape of the curve.
Quartic models are used when data exhibits double-S curves, multiple inflection points, or behaviors that simpler polynomial models cannot adequately capture. They appear in advanced physics, signal processing, and complex engineering applications. However, quartic models require more data points and can overfit if used carelessly.
The Normal Equations
The coefficients are found by solving a 5x5 system of normal equations from the least squares method:
This requires computing sums of powers of x up to x⁸ and solving the resulting system simultaneously. The R² value measures the goodness of fit.
Polynomial Regression Comparison
| Degree | Name | Max Turning Points | Min Data Points |
|---|---|---|---|
| 1 | Linear | 0 | 2 |
| 2 | Quadratic | 1 | 3 |
| 3 | Cubic | 2 | 4 |
| 4 | Quartic | 3 | 5 |
Frequently Asked Questions
When should I use quartic instead of quadratic regression?
Use quartic regression when your data shows more than one change in curvature direction (more than one turning point). If a quadratic (single parabola) does not fit well and the residuals show a systematic pattern, try cubic first, then quartic. Always compare R² values and residual plots between models.
What is overfitting and how do I avoid it?
Overfitting occurs when a model is too complex for the data, fitting noise rather than the underlying trend. A quartic model with only 6 data points may overfit. Use adjusted R² rather than regular R² to compare models with different numbers of parameters. As a rule of thumb, have at least 2-3 times as many data points as coefficients.