---
tag: C7E69V030F
kind: concept
name: Nonlinear regression
msc: [62J02]
sources:
  - url: https://en.wikipedia.org/wiki/Nonlinear_regression
same_as:
  wikidata: Q2755024
  wikipedia: Nonlinear regression
---

## Summary

Nonlinear regression fits a model in which the mean response is a nonlinear function of the unknown parameters.
Examples are an exponential decay or a growth curve.
The parameters are usually estimated by nonlinear [[least squares|0BYTQ4X1HC]], which has no closed formula and is solved by iterative methods such as Gauss–Newton or Levenberg–Marquardt.
Inference relies on a linear approximation of the model near the estimate, so it is approximate and asymptotic.
It is the natural tool when a scientific theory dictates the form of the mean function.

## Explanations

### L1 human

In [[linear regression|41FVC4P5JB]] the unknowns enter as coefficients of known terms, and one formula gives the fit.
In a model like $y = \theta_1 e^{\theta_2 x}$, the unknown $\theta_2$ sits inside the exponential, and no such formula exists.
One starts from a guess, approximates the model by a linear one near that guess, solves that, and repeats.
A polynomial $y = \theta_0 + \theta_1 x + \theta_2 x^2$ looks curved but is still linear regression, because it is linear in the parameters.