Linear regression is often the first modeling technique analysts learn—and for good reason. It is simple, interpretable, and effective when relationships between variables are approximately linear. However, modern data problems rarely follow straight lines. Customer growth curves, biological reactions, system saturation, financial risk, and machine performance metrics often exhibit exponential, logistic, asymptotic, or other nonlinear patterns.
This is where nonlinear regression becomes essential.
Nonlinear regression extends the idea of linear regression by fitting curves that better reflect real-world processes. Instead of assuming a straight-line relationship, it estimates parameters of a nonlinear function that minimizes error using nonlinear least squares (NLS). Despite the rise of machine learning models, nonlinear regression remains highly relevant because it offers interpretability, parametric clarity, and strong theoretical grounding.
This article revisits nonlinear regression in R, modernizes the examples, and aligns them with current analytics and industry practices—while preserving the original learning intent.
What Is Nonlinear Regression?
In nonlinear regression, the expected value of the response variable is modeled as a nonlinear function of predictors:y=f(x,θ)+εy = f(x, \theta) + \varepsilony=f(x,θ)+ε
where:
f(⋅)f(\cdot)f(⋅) is a nonlinear function,
θ\thetaθ represents unknown parameters,
ε\varepsilonε is random error.
Unlike linear regression, these parameters cannot be solved analytically and must be estimated iteratively.
Typical real-world examples include:
Exponential growth/decay (marketing adoption, system degradation)
Logistic curves (population growth, churn saturation)
Michaelis–Menten kinetics (biochemistry, pharmacology)
Weibull curves (reliability and survival analysis)
Linear vs Nonlinear Regression: A Simple Illustration
Let’s begin with simulated exponential data to highlight why linear regression can fail on nonlinear patterns.
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)
plot(x, y, main = "Simulated Exponential Data")
Linear Model Fit
lin_mod <- lm(y ~ x)
plot(x, y)
abline(lin_mod, col = "blue")
The fitted line clearly misses the curvature of the data, resulting in high residual error.
Nonlinear Model Fit
nonlin_mod <- nls(
y ~ a * exp(b * x),
start = list(a = 13, b = 0.1)
)
plot(x, y)
lines(x, predict(nonlin_mod), col = "red", lwd = 2)
The nonlinear model captures the exponential trend far more effectively.
Model Accuracy Comparison
lm_error <- sqrt(mean(residuals(lin_mod)^2))
nls_error <- sqrt(mean((y - predict(nonlin_mod))^2))
lm_error
nls_error
Result:
The nonlinear model produces less than one-third the error of the linear model—demonstrating why nonlinear regression is indispensable when the data structure demands it.
Understanding the nls() Function
The nonlinear least squares function requires two key inputs:
Formula – The mathematical relationship you expect between variables
Starting values – Initial guesses for model parameters
nonlin_mod
Nonlinear regression model
model: y ~ a * exp(b * x)
a b
13.60391 0.01911
Residual sum-of-squares: 235.5
Why Starting Values Matter
Good starting values → fast convergence
Poor starting values → slow convergence or failure
Industry practice today often combines exploratory plots, domain knowledge, and automated initialization to choose starting values wisely
Self-Starting Functions: A Modern Best Practice
One of the biggest challenges in nonlinear modeling is parameter initialization. To address this, R provides self-starting models that automatically estimate reasonable starting values.
Example: Michaelis–Menten Kinetics
The built-in Puromycin dataset models enzyme reaction rates.
plot(Puromycin$conc, Puromycin$rate)
The Michaelis–Menten equation:
mm <- function(conc, vmax, k) vmax * conc / (k + conc)
Manual Starting Values
mm1 <- nls(
rate ~ mm(conc, vmax, k),
data = Puromycin,
start = c(vmax = 50, k = 0.05),
subset = state == "treated"
)
Self-Starting Version (Recommended)
mm2 <- nls(
rate ~ SSmicmen(conc, vmax, k),
data = Puromycin,
subset = state == "treated"
)
Both models converge to nearly identical estimates, but the self-starting model:
Requires no manual parameter tuning
Converges faster
Is more robust in automated pipelines
Built-in Self-Starting Models in R
apropos("^SS")
Commonly used models include:
SSlogis – Logistic growth
SSgompertz – Growth and diffusion modeling
SSweibull – Reliability and failure analysis
SSmicmen – Enzyme kinetics
SSfpl – Four-parameter logistic models (popular in bioanalytics)
These functions align well with modern workflows where models are trained repeatedly across segments or time windows.
Model Validation: Goodness of Fit
A simple yet effective validation step is measuring correlation between predicted and observed values.
cor(y, predict(nonlin_mod))
cor(subset(Puromycin$rate, state == "treated"), predict(mm2))
High correlations (>0.97) indicate excellent model fit, reinforcing that nonlinear regression can be both accurate and interpretable.
Where Nonlinear Regression Fits in Today’s Analytics Stack
While machine learning models like gradient boosting and neural networks dominate large-scale prediction tasks, nonlinear regression still plays a vital role when:
Interpretability matters
Physics- or biology-based relationships are known
Data is limited but domain knowledge is strong
Regulatory or scientific transparency is required
In practice, nonlinear regression often complements ML models rather than competing with them.
Summary
Nonlinear regression remains a powerful, relevant technique for modern data science. By explicitly modeling nonlinear relationships, it provides interpretable, mathematically grounded insights that black-box models cannot always deliver.
Key takeaways:
Use nonlinear regression when relationships are inherently curved
Choose meaningful starting values—or use self-starting functions
Validate models with residuals and correlation checks
Prefer nonlinear regression when explanation is as important as prediction
As datasets grow more complex, understanding when—and how—to apply nonlinear regression is a valuable skill for analysts, data scientists, and researchers alike.
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