Polynomial Regression Calculator

About the Polynomial Regression Calculator

Polynomial regression is the most flexible single-equation curve fitting method: choose degree 1 (linear), 2 (quadratic), 3 (cubic), or anything up to 8, and the calculator fits the best polynomial to your data. The degree comparison table with AIC/BIC helps you select the optimal polynomial — the simplest one that captures the data's shape.

Enter your data, select a degree, and get the full equation, coefficients, R², adjusted R², standard error, AIC, and BIC. The coefficient magnitude bars show which terms dominate the equation. Residual analysis with visual bars reveals where the model fits well and where it struggles.

The degree comparison table is the key feature: it shows R², adjusted R², and AIC for degrees 1 through the selected degree + 2, highlighting the AIC-optimal choice. This prevents both underfitting (too few terms) and overfitting (too many terms). Use it to compare how the residual pattern and information criteria change as you move up or down in degree.

When This Page Helps

This is the Swiss Army knife of curve fitting. When you don't know the theoretical form of a relationship, polynomial regression explores degrees 1 through 8 and lets the data tell you the right complexity level.

The AIC/BIC degree comparison is what makes This calculator powerful beyond naive curve fitting. It automates the bias-variance tradeoff: too few parameters underfit (high bias), too many overfit (high variance). The table shows exactly where the sweet spot is.

How to Use the Inputs

1. Enter X and Y values (comma-separated).
2. Or select a preset for common polynomial patterns.
3. Choose the polynomial degree (1-8).
4. Review the equation, coefficients, and R².
5. Check the degree comparison table — find the AIC-minimizing degree.
6. If AIC suggests a different degree, adjust and re-fit.
7. Enter an X value for prediction.

Example Calculation

Tips & Best Practices

• Start low (degree 1-2) and increase only if residuals show patterns.
• AIC-optimal degree is usually the right choice — trust it over raw R².
• BIC tends to prefer simpler models — use it when parsimony matters (e.g., for extrapolation).
• Centering X (subtracting mean) improves numerical stability for high degrees.
• If R² at degree d and d+1 are nearly identical, use degree d (Occam's razor).
• Degrees above 5-6 rarely add genuine insight — they typically fit noise.

Normal Equations for Polynomial Regression

A degree-d polynomial yields a (d+1)×(d+1) system of normal equations involving power sums S₀ through S₂d. The matrix has Vandermonde structure: entry (i,j) is S_{i+j}. This structure can become ill-conditioned (near-singular) when X values span many orders of magnitude. Solutions: center and scale X, use orthogonal polynomials (Legendre, Chebyshev), or use QR decomposition instead of Cramer's rule.

The Bias-Variance Tradeoff

Degree 1 has high bias (underfits curves) but low variance (stable predictions). Degree n-1 has zero bias (passes through every point) but maximum variance (wildly unstable). AIC/BIC formalize this tradeoff: they add a penalty proportional to the number of parameters, favoring simpler models unless additional parameters significantly reduce error.

Polynomial Alternatives

When polynomials struggle: splines (piecewise polynomials with smooth joins) for irregular shapes, Fourier series for periodic data, logarithmic/exponential for specific growth patterns, and kernel regression for nonparametric flexibility. Polynomials are best for smooth, moderately curved data within a limited X range.

Frequently Asked Questions

• How do I choose the right polynomial degree?

  Use the degree comparison table: pick the degree that minimizes AIC (or BIC for a stricter penalty). If adjusted R² stops improving, you've likely found the right degree. Never use degree ≥ n-1 (interpolation captures noise).

• What's the difference between AIC and BIC?

  Both penalize model complexity, but BIC penalizes more heavily (log(n) vs. 2 per parameter). BIC prefers simpler models, especially with large datasets. When AIC and BIC disagree, BIC's choice is safer against overfitting.

• Can I use polynomial regression for extrapolation?

  Generally no. High-degree polynomials oscillate wildly outside the data range (Runge's phenomenon). Even within-range accuracy doesn't guarantee out-of-range validity. For extrapolation, use theoretical models or low-degree polynomials with caution.

• Why is my high-degree polynomial giving nonsensical predictions?

  Overfitting. With degree ≥ n-1, the polynomial passes through every point but oscillates wildly between them. Check adjusted R² (not raw R²) and AIC — they penalize unnecessary complexity. Use the lowest degree that fits adequately.

• Is polynomial regression the same as multiple regression?

  Polynomial regression is a special case of multiple regression. Degree 3: Y = a₃X³ + a₂X² + a₁X + a₀ is identical to multiple regression with predictors X, X², X³. The normal equations and R² calculations are the same.

• What numerical issues arise with high degrees?

  The normal equations become ill-conditioned (near-singular) with high degrees and wide X ranges. Our solver uses partial pivoting to mitigate this, but degrees above 6-7 may produce unreliable coefficients. Center X data (subtract mean) for better stability.