What Is Curve Regression (Nonlinear Curve Fitting) and How Do You Use It?

If you have ever tried to model data that does not follow a straight line—such as a dose‑response curve, a calibration plot, or a decay process—you have likely needed curve regression. Also called nonlinear curve fitting, this method finds the mathematical equation that best describes the relationship between an independent variable \(x\) and a dependent variable \(y\) when that relationship is curved.

What It Is

Curve regression is a form of least‑squares regression applied to nonlinear models. Instead of forcing data into a straight line \(y = a + bx\), you fit a predefined function such as an exponential, logarithmic, power, or polynomial equation. The goal is to estimate the model parameters that minimize the sum of squared differences between the observed values and the values predicted by the equation.

This technique is standard practice in analytical chemistry and biology for building calibration curves—for example, relating instrument response to analyte concentration. The concepts of fitting, residual analysis, and parameter uncertainty are aligned with the calibration and detection‑limit framework described in standards such as ISO 11843 (which deals with capability of detection and calibration concepts).

How It Works: Formula and Steps

The core idea is the same as for linear regression, but the model equation is nonlinear in the parameters.

General model:
\(y = f(x, \theta) + \varepsilon\)

Where:

• \(y\) = measured response
• \(x\) = predictor (e.g., concentration)
• \(\theta\) = vector of parameters to estimate (e.g., slope, intercept, rate constant)
• \(\varepsilon\) = random error (assumed independent, with constant variance)

Least‑squares objective:
Minimize \(S(\theta) = \sum_{i=1}^{n} \big(y_i - f(x_i, \theta)\big)^2\)

Because the model is nonlinear, there is no closed‑form solution. Instead, iterative algorithms (such as Gauss–Newton or Levenberg–Marquardt) adjust the parameter values step by step until \(S(\theta)\) stops decreasing.

Typical steps in practice:

1. Choose a model based on theory or a scatter plot (e.g., \(y = a \cdot e^{bx}\) for exponential decay).
2. Provide starting guesses for the parameters.
3. Run the iterative fit to minimize the residual sum of squares.
4. Evaluate goodness of fit using \(R^2\), residual plots, and the standard error of the parameters.
5. Report parameter estimates with confidence intervals (typically 95% CI, using the \(t\)-distribution with \(n - p\) degrees of freedom, where \(p\) is the number of parameters).

A Worked Illustrative Example

Example data (illustrative only):
Suppose you measure the absorbance \(y\) of a standard solution at five concentrations \(x\):

\(x\) (mg/L) | \(y\) (absorbance)

• 0.0 | 0.00
• 1.0 | 0.25
• 2.0 | 0.48
• 4.0 | 0.90
• 8.0 | 1.70

The scatter plot suggests a linear relationship, but for many analytical methods the response is curved at higher concentrations. You decide to fit a quadratic calibration model:

\(y = a + b x + c x^2\)

Using least‑squares curve fitting, you obtain:

\(y = 0.01 + 0.24 x + 0.002 x^2\)

The fitted values are close to the observed ones, and \(R^2 = 0.999\). The residuals (observed minus fitted) are small and show no obvious pattern, indicating a good fit. The 95% confidence interval for the slope \(b\) might be, for example, \(0.24 \pm 0.02\) (illustrative values only).

You can now use this equation to predict the concentration of an unknown sample from its measured absorbance—provided you stay within the calibrated range.

Common Pitfalls

• Overfitting: Adding too many parameters (e.g., a 6th‑degree polynomial) will fit the noise, not the trend. Keep the model as simple as the data and theory justify.
• Ignoring residuals: A high \(R^2\) does not guarantee a good model. Always plot residuals versus \(x\) to check for systematic patterns.
• Extrapolation: Using the fitted curve outside the range of the calibration data is risky and can lead to large errors.
• Poor starting values: Nonlinear fitting algorithms can converge to a local minimum. Try several starting points to ensure a global solution.

---

For a quick and reliable curve fit without writing code, try the free Curve Regression tool at https://www.6sq.com/tools/curve_fit/. Paste your data, choose a model, and get parameter estimates, \(R^2\), and residual diagnostics in seconds.