Polynomial vs. Linear: Choosing the Right Method

When working with data prediction, choosing the right extrapolation method can make or break your results. Here’s a practical guide to deciding between polynomial and linear approaches.

Linear: Simple and Stable

Linear extrapolation shines when:

• Your data has a roughly constant rate of change
• You need quick, interpretable results
• You’re extrapolating a short distance

The main advantage is stability — linear models rarely produce wildly unrealistic predictions, even far from your data range.

Polynomial: Flexible but Risky

Polynomial extrapolation (especially higher degrees) can fit complex curves beautifully within your data range. However, outside that range, polynomials can diverge dramatically — producing extremely large or small values that make no physical sense.

Rule of thumb: Use the lowest polynomial degree that gives an acceptable R² score.

The R² Decision Framework

1. Fit a linear model first and check R²
2. If R² < 0.7, try quadratic (degree 2)
3. If still low, try cubic (degree 3)
4. Compare R² scores — if a higher degree barely improves R², stick with the simpler model
5. Always sanity-check extrapolated values against domain knowledge

Warning Signs

• R² increases dramatically with degree → likely overfitting
• Extrapolated values are orders of magnitude beyond your data → model divergence
• The equation has very large coefficients → numerical instability

The best extrapolation method is the simplest one that adequately fits your data.