Performing Nonlinear Least Squares and Nonlinear Regression in R

Regression analysis lies at the core of statistical modeling and data science. It is used to explore and quantify the relationship between variables — typically between one dependent (response) variable and one or more independent (explanatory) variables. While linear regression has been a cornerstone technique for decades, not all real-world relationships follow a straight line. Many natural, social, and economic processes behave in a nonlinear manner — and that’s where Nonlinear Regression and Nonlinear Least Squares (NLS) come into play.

Origins of Nonlinear Regression
The concept of regression analysis dates back to Sir Francis Galton in the late 19th century, who studied the relationship between parents’ and children’s heights — giving rise to the term “regression toward the mean.” While early applications focused on linear relationships, mathematicians and statisticians soon realized that many natural phenomena didn’t fit neatly into linear equations.

The Nonlinear Least Squares method emerged as an extension of the least squares approach developed by Carl Friedrich Gauss and Adrien-Marie Legendre. It generalized the minimization of squared residuals to handle equations where parameters appear in nonlinear forms. Over time, with the advent of computers and numerical optimization techniques, nonlinear regression became widely used across biology, chemistry, economics, and engineering — fields where exponential growth, saturation effects, and decay patterns are common.

Understanding Nonlinear Regression
Linear regression assumes that the relationship between the independent variable xxx and dependent variable yyy is linear — that is, y=a+bx+ϵy = a + bx + epsilony=a+bx+ϵ, where ϵepsilonϵ is the error term. However, in nonlinear regression, the relationship can take more complex forms, such as exponential, logarithmic, logistic, or polynomial.

For example:

y=a×e(b×x)+cy = a times e^{(b times x)} + cy=a×e(b×x)+c

Here, the parameters aaa, bbb, and ccc are not linearly related to yyy, and the curve that best fits the data is not a straight line.

In R, linear regression is performed using the lm() function, while nonlinear regression uses the nls() (Nonlinear Least Squares) function. The nls() function estimates the parameters of a nonlinear model by minimizing the sum of squared residuals between observed and predicted values.

Implementing Nonlinear Regression in R
Let’s consider a simple example to illustrate nonlinear regression in R. Suppose we generate an exponential dataset using random values and fit both linear and nonlinear models to it.

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)

Plotting the data shows a clear exponential curve. A linear model (lm(y~x)) would produce a poor fit, while a nonlinear model of the form y=a×e(b×x)y = a times e^{(b times x)}y=a×e(b×x) gives a much better representation:

nonlin_mod <- nls(y ~ a * exp(b * x), start = list(a = 13, b = 0.1)) lines(x, predict(nonlin_mod), col = "red")

By comparing the root mean squared error (RMSE) between the two models, we find the nonlinear model’s error (≈1.52) is much lower than the linear model’s error (≈5.96), confirming a significantly better fit.

Understanding the nls() Function
The nls() function in R requires two key inputs:

1. Formula: Defines the nonlinear relationship (e.g., y ~ a * exp(b * x)).
2. Start Parameters: Provides initial guesses for the coefficients (e.g., a = 13, b = 0.1).

Choosing appropriate starting values is crucial. If they are too far from the true solution, the model might not converge. R iteratively adjusts these parameters to minimize the residual sum of squares until an optimal solution is found.

In our example, the fitted model returned:

a = 13.60391, b = 0.01911

Hence, the estimated model becomes:

y=13.60391×e0.01911xy = 13.60391 times e^{0.01911x}y=13.60391×e0.01911x

These values closely approximate the true coefficients used to generate the data.

Self-Starting Functions in R
Estimating starting parameters can be challenging, especially with complex biological or chemical data. To simplify this, R provides self-starting functions, which automatically determine initial parameter values.

For instance, consider the Puromycin dataset, which measures the reaction rate of the antibiotic Puromycin with varying substrate concentrations. The data follows the Michaelis–Menten equation, a well-known nonlinear relationship in enzyme kinetics:

rate=Vmax×concK+concrate = frac{V_{max} times conc}{K + conc}rate=K+concVmax​×conc​

Using nls() directly requires manual starting values:

mm1 <- nls(rate ~ vmax * conc / (k + conc), data = Puromycin, start = c(vmax = 50, k = 0.05), subset = state == "treated")

However, using R’s self-starting function SSmicmen() eliminates the need for guesses:

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

Both models converge to nearly identical parameter estimates:

vmax = 212.68, k = 0.064

but the self-starting function converges faster and more reliably.

To explore other available self-starting models in R, one can use:

apropos("^SS")

This lists models like SSasymp, SSlogis, SSweibull, and SSgompertz — each designed for different nonlinear relationships such as logistic growth, asymptotic regression, and Weibull distributions.

Goodness of Fit
To evaluate model performance, we can examine the correlation between actual and predicted values. For our earlier examples:

cor(y, predict(nonlin_mod)) # 0.9976

Similarly, for the Puromycin models, the correlations range between 0.96 and 0.98 — indicating strong model fits. High correlation values confirm that nonlinear regression successfully captures the underlying relationship.

Real-Life Applications of Nonlinear Regression
Nonlinear regression is widely used in scientific research and applied industries. Here are some prominent examples:

1. Pharmacology and Biochemistry
The Michaelis–Menten model, as demonstrated, is used to estimate enzyme kinetics — understanding how reaction rates vary with substrate concentration. This helps in determining drug efficiency, binding affinity, and dosage optimization.

2. Economics and Market Analysis
Nonlinear models capture diminishing returns, saturation effects, and exponential growth patterns — such as modeling GDP growth, inflation dynamics, or consumer behavior. For instance, logistic models are used to forecast market saturation levels for new products.

3. Environmental Science
Exponential decay and logistic growth models describe phenomena like pollutant degradation, population growth, and carbon absorption rates in ecosystems.

4. Engineering and Physics
Nonlinear regression helps model material stress-strain relationships, system dynamics, and temperature-dependent reaction rates, where relationships are often exponential or logarithmic.

5. Marketing and Customer Analytics
Marketers use nonlinear models to understand customer lifetime value (CLV), ad response curves, and conversion saturation, where the impact of advertising diminishes after a threshold.

Case Study: Enzyme Kinetics in Biomedical Research
A practical case involves modeling enzyme kinetics using the Puromycin dataset. By applying the Michaelis–Menten model through R’s nls() and SSmicmen() functions, researchers accurately estimate the maximum reaction rate (Vmax) and the Michaelis constant (K). These parameters are critical in pharmacology for designing effective enzyme inhibitors or understanding how a drug interacts at different concentrations.

The accuracy and convergence speed of self-starting functions demonstrate how R simplifies complex nonlinear modeling tasks that were once computationally intensive.

Summary
Nonlinear regression is a versatile and powerful tool for modeling complex relationships that cannot be captured through linear methods. By using Nonlinear Least Squares (nls) in R, analysts can estimate parameters of nonlinear equations efficiently and accurately.

The inclusion of self-starting functions like SSmicmen and SSlogis further enhances usability, making nonlinear modeling accessible even to beginners. From biological reactions to economic forecasting, nonlinear regression finds applications in every field where systems exhibit curvature, thresholds, or saturation.

Ultimately, mastering nonlinear regression in R empowers data scientists and researchers to uncover deeper insights, model real-world complexity more faithfully, and make data-driven predictions with higher precision.

Keywords Recap: Nonlinear Regression in R, Nonlinear Least Squares, nls() function in R, R regression modeling, SSmicmen example, nonlinear model fitting, exponential regression, Michaelis-Menten model, data science R programming.

This article was originally published on Perceptive Analytics.

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