Performing Nonlinear Least Squares & Nonlinear Regression in R

Performing Nonlinear Least Squares & Nonlinear Regression in R (2025 Edition)

Nonlinear regression remains a powerful extension of linear models—capable of fitting complex, real-world relationships with a parametric curve rather than a straight line. In R, the nls() function still serves as the foundational tool for nonlinear least squares modeling. Below, we walk through its use, illustrate improvements, and introduce contemporary alternatives to make your analysis more accurate, robust, and efficient.

1. The Classic nls() Workflow

Start with a foundational example: generate synthetic exponential data, compare linear vs. nonlinear fits, and observe performance differences:

set.seed(23)
x <- seq(0, 100, 1)
y <- runif(1, 0, 20) * exp(runif(1, 0.005, 0.075) * x) + runif(101, 0, 5)

lin_mod <- lm(y ~ x)
nonlin_mod <- nls(y ~ a * exp(b * x), start = list(a = 13, b = 0.1))

The nonlinear model often reduces error significantly compared to linear regression. However, it's highly sensitive to starting values—choosing them wisely is vital.

2. Simplifying Start Values with Self-Starting Functions

R now offers robust self-starting models, eliminating manual guesswork. For enzymatic kinetics (e.g., Michaelis–Menten), use:

mm3 <- nls(rate ~ SSmicmen(conc, vmax, k),
data = Puromycin, subset = state == "treated")

The result is identical to manually started models—but simpler and faster.

There are now many built-in self-starting model families, including logistic, Gompertz, Weibull, and more—ideal for biologically and ecologically inspired curves.

3. Enhanced Optimization with Modern Solvers

Modern practice favors more robust solvers, such as:

• nlsLM() from minpack.lm — Levenberg–Marquardt algorithm, offering improved convergence.
• nls2() — supports grid searches over multiple starting points, helping avoid local minima.
• gslnls package — leverages advanced optimization routines with multi-start algorithms, trust-region methods, and robust loss functions for complex or noisy data.

4. Diagnosing & Improving Fits

Prior to model fitting, inspect and preprocess data:

• Clean data: handle missing values, outliers, or noise through imputation or smoothing.
• Visualize trend and residuals.
• Use transformations or standardization to stabilize estimations.

Post-fit, validate with:

• Residual plots and predicted vs. actual comparisons
• Parameter confidence intervals
• Cross-validation or bootstrapping for predictive reliability
• Correlation metrics between predictions and observations for goodness of fit

5. Navigating Common Pitfalls

• Convergence failure: Poor starting values or overly complex models can cause non-meaningful estimates.
• Local minima: Try multiple starting points or use global optimization methods (multi-start or pattern search).
• Identifiability issues: Ensure unique parameter estimation—check sensitivity or Jacobian diagnostics.
• Parameter correlations: Exponential or logistic forms can suffer from highly correlated parameters—evaluate and constrain where needed.
• Weighted or robust fitting: For heteroskedastic or noisy data, use weighted least squares or robust loss functions.

6. Theory That Powers Better Fits

Nonlinear least squares iteratively refines parameter estimates using local linear approximations (like Gauss–Newton or Levenberg–Marquardt methods). These rely on the Jacobian matrix to guide convergence.

Convergence criteria typically include minimal change in sum of squares or in parameter values. In challenging or noisy datasets, trust-region methods and acceleration techniques can improve stability and speed.

7. Why These Updates Matter

• Greater reliability with self-starting models and advanced solvers like nlsLM() and gslnls
• Better performance and fit via multi-start and robust optimization strategies
• Deeper diagnostics using residual, sensitivity, and goodness-of-fit evaluations
• Stronger theoretical grounding, understanding algorithmic behavior helps craft stable models

Sample Modernized Workflow — Full Example
library(minpack.lm)
library(gslnls)

Exploratory data

plot(x, y)

Fit with nlsLM

mod_lm <- nlsLM(y ~ a * exp(b * x),
start = list(a = 10, b = 0.02),
control = nls.lm.control(maxiter = 50))

Multi-start fit with gslnls

mod_gsl <- gsl_nls(y ~ a * exp(b * x), data = data.frame(x, y),
start = list(a = 10, b = 0.02),
algorithm = "dogleg", loss = "huber")

Compare residuals and predictions

pred1 <- predict(mod_lm)
pred2 <- predict(mod_gsl)

plot(x, y)
lines(x, pred1, col = "blue")
lines(x, pred2, col = "green")

Summaries and diagnostics

summary(mod_lm)
summary(mod_gsl)
cor(y, pred1)
cor(y, pred2)

By incorporating these modern solutions—self-starting models, Levenberg–Marquardt optimization, multi-start global search, and deeper diagnostics—you can elevate your nonlinear regression workflow in R to be faster, more reliable, and better suited for today’s complex data challenges.

This article was originally published on Perceptive Analytics.

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