Editing Nonlinear regression (section)

==General  ==
In nonlinear regression, a [[statistical model]] of the form,

<math display="block"> \mathbf{y} \sim f(\mathbf{x}, \boldsymbol\beta)</math>

relates a vector of [[independent variables]], <math>\mathbf{x}</math>, and its associated observed [[dependent variables]], <math>\mathbf{y}</math>. The function <math>f</math> is nonlinear in the components of the vector of parameters <math>\beta</math>, but otherwise arbitrary. For example, the [[Michaelis–Menten]] model for enzyme kinetics has two parameters and one independent variable, related by <math>f</math> by:{{efn|This model can also be expressed in the conventional biological notation: <math display="block"> v = \frac{V_\max\ [\mathrm{S}]}{K_m + [\mathrm{S}]} </math>}}

<math display="block"> f(x,\boldsymbol\beta)= \frac{\beta_1 x}{\beta_2 + x} </math>

This function, which is a rectangular hyperbola, is {{em|nonlinear}} because it cannot be expressed as a [[linear combination]] of the two <math>\beta</math>s.

[[Systematic error]] may be present in the independent variables but its treatment is outside the scope of regression analysis. If the independent variables are not error-free, this is an [[errors-in-variables model]], also outside this scope.

Other examples of nonlinear functions include [[exponential function]]s, [[Logarithmic growth|logarithmic functions]], [[trigonometric functions]], [[Exponentiation|power functions]], [[Gaussian function]], and [[Cauchy distribution|Lorentz distribution]]s. Some functions, such as the exponential or logarithmic functions, can be transformed so that they are linear. When so transformed, standard linear regression can be performed but must be applied with caution. See {{mslink||Linearization|Transformation}}, below, for more details.

In general, there is no closed-form expression for the best-fitting parameters, as there is in [[linear regression]]. Usually numerical [[Optimization (mathematics)|optimization]] algorithms are applied to determine the best-fitting parameters. Again in contrast to linear regression, there may be many [[local maximum|local minima]] of the function to be optimized and even the global minimum may produce a [[Bias of an estimator|biased]] estimate. In practice, [[guess value|estimated values]] of the parameters are used, in conjunction with the optimization algorithm, to attempt to find the global minimum of a sum of squares.

For details concerning nonlinear data modeling see [[least squares]] and [[non-linear least squares]].