A nonlinear model describes nonlinear relationships in experimental data. Nonlinear regression models are generally assumed to be parametric, where the model is described as a nonlinear equation. Typically machine learning methods are used for non-parametric nonlinear regression.
Parametric nonlinear regression models the dependent variable (also called the response) as a function of a combination of nonlinear parameters and one or more independent variables (called predictors). The model can be univariate (with a single response variable) or multivariate (with multiple response variables).
The parameters can take the form of an exponential, trigonometric, power, or any other nonlinear function. To determine the nonlinear parameter estimates, an iterative algorithm is typically used.
\[y = f(X,\beta) + \epsilon\]
where, \(\beta\) represents nonlinear parameter estimates to be computed and \(\epsilon\) represents the error terms.
Common algorithms for fitting a nonlinear model include:
- Gauss-Newton algorithm
- Gradient descent algorithm
- Levenberg-Marquardt algorithm
Parametric nonlinear regression can be used to:
- Fit a nonlinear model to data and compare different models
- Generate predictions
- Evaluate parameter confidence intervals
- Evaluate goodness-of-fit
For detail, see Statistics and Machine Learning Toolbox. To create a nonlinear model that fits curves, surfaces, and splines to data interactively, see Curve Fitting Toolbox. To create nonparametric models using Deep Learning Toolbox and decision trees, see the machine learning functions available with MATLAB. To create nonlinear models of dynamic systems from measured input-output data, see System Identification Toolbox.