Nonlinear Regression
Nonlinear regression fits a model to data when the relationship between the independent variable(s) and the dependent variable is curved rather than a straight line. While simple linear regression models relationships with a straight line (for example, y = mx + b), nonlinear regression uses functions that produce curves (for example, logistic, exponential, or power functions).
Key ideas
- Purpose: model complex, curved relationships between variables.
- Goal: find parameter values that minimize the sum of squared differences between observed values and model predictions (the residual sum of squares, SSE).
- Difficulty: nonlinear models usually require iterative numerical methods and good starting values; they can converge to local minima or fail to converge.
Nonlinear vs. linear regression
- Linear regression: fits a linear function in parameters. Some curved-looking relationships can be handled by linear regression if the equation is linear in parameters (e.g., polynomial regression) or by transforming variables.
- Nonlinear regression: used when the model is nonlinear in parameters and cannot be expressed as a linear combination of parameters. Some nonlinear equations can be algebraically transformed to a linear form (“intrinsically linear”), but many cannot.
How nonlinear regression works
- Specify a functional form that describes the relationship (e.g., logistic: P(t) = K / (1 + A e^{-Bt}), exponential, Gompertz, power law).
- Choose initial guesses for parameters.
- Use an iterative optimization algorithm to adjust parameters to minimize SSE = Σ(y_i − f(x_i; θ))^2.
- Evaluate fit with residuals, goodness-of-fit measures, and diagnostics; refine model or starting values as needed.
Common algorithms for parameter estimation include Gauss–Newton and Levenberg–Marquardt, which combine gradient information and damping to navigate the parameter space.
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Choosing the function and starting values
- Base the model form on theory, domain knowledge, and visual examination of the data (scatterplots).
- Good starting values are critical. Poor starts may lead to non-convergence or convergence to a local, suboptimal solution.
- Scale variables to improve numerical stability and reduce problems with large or tiny parameter values.
Data requirements and variable types
- Independent and dependent variables should be quantitative. Categorical predictors must be encoded numerically (dummy/binary variables or suitable encodings) before inclusion.
- Check for outliers and leverage points; nonlinear fits can be especially sensitive to them.
Applications and examples
- Population growth: logistic or Gompertz models estimate carrying capacity and growth rates from time-series population data.
- Biology and pharmacokinetics: dose–response curves, enzyme kinetics.
- Engineering and physical sciences: stress–strain relationships, decay processes.
- Economics and finance: models with diminishing returns or saturation effects.
Example: If population over time shows an S-shaped pattern, a logistic model can estimate past unmeasured values and project future growth, capturing saturation where linear models would fail.
Practical tips
- Always plot the data first to assess linearity vs. curvature.
- Try simple transformations when appropriate (e.g., log transforms) to see if a linear model suffices.
- Compare multiple candidate nonlinear forms and use information criteria (AIC/BIC) or cross-validation to select among models.
- Inspect residuals and confidence intervals for parameters. Consider bootstrap methods when analytic standard errors are unreliable.
- Be aware of identifiability issues—ensure the data contain enough information to estimate the parameters uniquely.
Conclusion
Nonlinear regression is a powerful tool for modeling curved relationships that linear models cannot capture. Success depends on selecting an appropriate functional form, providing good starting values, applying robust optimization methods, and carefully evaluating fit and diagnostics. When applied thoughtfully, nonlinear models reveal dynamics—such as saturation, growth limits, and diminishing returns—that linear models miss.