Choose exponential, logistic, power, or Michaelis-Menten curves here. Tune starting guesses and iteration controls safely. Export results, fitted table, and charts for sharing anywhere.
This sample follows a logistic-shaped curve with small noise.
| x | y | x | y | x | y |
|---|---|---|---|---|---|
| 0 | 0.55 | 2 | 1.31 | 4 | 2.55 |
| 6 | 4.36 | 8 | 6.67 | 10 | 8.43 |
| 12 | 9.33 | 14 | 9.62 | 16 | 9.83 |
The calculator estimates parameters p by minimizing the sum of squared errors:
It uses a damped least-squares update (Levenberg–Marquardt style):
Here, r is the residual vector and J is the Jacobian, estimated using finite differences.
Nonlinear regression helps you describe curved relationships when straight lines fail. This calculator fits common nonlinear forms to paired x and y observations, returning parameters and prediction values. Use it when growth saturates, decay accelerates, or responses bend with scale. Because optimization is iterative, clean input matters. Remove obvious entry errors, keep units consistent, and consider rescaling very large x values to reduce numerical strain. Wider x ranges improve stability.
Choosing a model is a mathematical hypothesis about shape. Exponential terms capture rapid change, power laws capture scaling, logistic curves capture saturation, and Michaelis–Menten describes diminishing returns against a limited capacity. If two models fit similarly, prefer the simpler one or the one with meaningful parameters for your context. Always check whether the domain rules apply, such as requiring positive x for power fits. Plot your points before selecting.
The fitting engine minimizes squared residuals by adjusting parameters until improvement stalls. It uses a damped least squares update where the Jacobian approximates how each parameter changes the curve. The damping value λ starts small and grows when a step worsens the error, then shrinks after successful steps. Good starting guesses speed convergence and reduce the risk of landing in poor local minima. Set λ higher when measurement noise is heavy. Carefully.
Metrics translate the fit into decision signals. RMSE summarizes typical error size in y units, while MAE is less sensitive to outliers. R-squared compares the fitted error against total variance, but can be misleading when y variance is tiny. Residual plots are essential: patterns, funnels, or waves indicate model mismatch or heteroscedastic noise. If residuals cluster randomly around zero, the curve is capturing structure well. Compare metrics across competing models.
A practical workflow is to paste points, choose a curve family, and run with automatic guesses first. If the fit stops early, increase iterations, relax tolerance slightly, or enter more informed initial parameters. Use the prediction box to estimate y at a specific x, then export the fitted table for reporting. Download CSV for analysis pipelines and PDF for quick sharing in reviews or classrooms. Save settings for reproducible comparisons.
Enter one x,y pair per line. Separate values with a comma, space, tab, or semicolon. Blank or invalid lines are ignored, so review your paste for typos and missing numbers.
Start by plotting your points. Use exponential for rapid change, logistic for saturation, power for scaling, and Michaelis–Menten for diminishing returns. Prefer models whose parameters have meaning for your problem, not only the highest R².
Nonlinear fitting can be sensitive to starting guesses and scaling. Try automatic guesses, then provide better initial values if needed. Increase iterations, adjust tolerance, or raise the damping start. Also remove extreme outliers that dominate the error.
λ balances cautious gradient steps against faster Gauss–Newton steps. If a trial update increases error, λ rises to shrink steps. When updates improve error, λ falls so the solver can move faster toward the minimum.
RMSE estimates typical error size in y units and penalizes large misses. MAE is more robust to outliers. R² compares fitted error to total variance, but it can be misleading when y varies little, so always check residuals.
Exponential, logistic, and Michaelis–Menten can accept negative x. The power model requires x greater than zero because it computes x^b in real numbers. If your data includes nonpositive x, choose another model or shift x values.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.