Polynomial & Nonlinear Regression Calculator

Free polynomial regression calculator. Fit curves of degree 1-5 with step-by-step solutions, R², and graphs. Browser-based privacy.

지금 계산

Polynomial & Nonlinear Regression Calculator

Free polynomial regression calculator. Fit curves of degree 1-5 with step-by-step solutions, R², and graphs. Browser-based privacy.

Polynomial Degree Minimum points needed = degree + 1

Enter your data points

#	X	Y

Results

Equation

R²

Degree

Predicted Y

For X = —

Coefficients

Term	Value

Statistics

Statistic	Value
Standard Error
Sample Size (n)
Degrees of Freedom

Chart

Step-by-Step Solution

사용 방법 Polynomial & Nonlinear Regression Calculator

성장과 감소

Fit polynomial curves of degree 1 through 5 to match your data's complexity.

입력 데이터

Enter paired X and Y values. Minimum points = degree + 1.

통계 출력

Get all coefficients, R², standard error, and prediction diagnostics.

Higher-degree polynomials can overfit with small datasets. Prefer the lowest degree that captures the pattern.

What Is Polynomial Regression?

📐 Polynomial regression generalizes linear and quadratic regression to fit a polynomial of any chosen degree n: y = aₙxⁿ + ... + a₂x² + a₁x + a₀. By selecting the degree (1 through 5), you control how flexibly the curve can bend to match your data.

📊 Key applications include: (1) Physics — projectile trajectories and drag forces, (2) Finance — yield curves and option pricing models, (3) Biology — dose-response curves with multiple thresholds, and (4) Engineering — thermal expansion and vibration modes.

⚠️ Higher degrees increase flexibility but also risk overfitting — the curve may wiggle to fit noise rather than signal. Choose the lowest degree that captures the underlying pattern.

How Polynomial Regression Works

1
Choose the degree: Select 1 (linear) through 5 (quintic). The degree determines how many bends the curve can have — degree n allows up to n−1 turning points.
2
Build the design matrix: Create matrix X where each row is [1, xᵢ, xᵢ², ..., xᵢⁿ] for each data point, and vector y of observed values.
3
Solve normal equations: Compute (XᵀX)·β = Xᵀy using Gaussian elimination with partial pivoting. The solution vector β contains coefficients [a₀, a₁, ..., aₙ].
4
Evaluate predictions and fit: Calculate predicted ŷ for every xᵢ, then compute residuals, SSE, SST, and R² = 1 − SSE/SST.
5
Assess model complexity: Compare R² across degrees, but favor simpler models. Standard error = sqrt(SSE/(n − degree − 1)) penalizes overfitting.

When to Use Polynomial Regression

Your data follows a curved but not strictly exponential or logistic pattern
You need to model multiple turning points (peaks, troughs, inflections)
You want to compare different complexity levels quickly (degree 1 vs 3 vs 5)
Domain knowledge suggests a polynomial relationship (physics, engineering)

When to Avoid Polynomial Regression

When you have very few data points relative to the chosen degree
When the relationship is naturally exponential, logistic, or periodic
When higher-degree wiggles are not physically interpretable
When a simpler model (linear, quadratic, cubic) achieves nearly identical R²

Polynomial & Nonlinear Regression Calculator