General
The data consist of error-free independent variables (explanatory variables), x, and their associated observed dependent variables (response variables), y. Each y is modeled as a random variable with a mean given by a nonlinear function f(x,β). Systematic error may be present 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.
For example, the Michaelis–Menten model for enzyme kinetics
can be written as
where is the parameter, is the parameter and is the independent variable, x. This function is nonlinear because it cannot be expressed as a linear combination of the s.
Other examples of nonlinear functions include exponential functions, logarithmic functions, trigonometric functions, power functions, Gaussian function, and Lorenz curves. 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 Linearization, 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 algorithms are applied to determine the best-fitting parameters. Again in contrast to linear regression, there may be many local minima of the function to be optimized and even the global minimum may produce a biased estimate. In practice, 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.
Read more about this topic: Nonlinear Regression
Famous quotes containing the word general:
“You have lived longer than I have and perhaps may have formed a different judgment on better grounds; but my observations do not enable me to say I think integrity the characteristic of wealth. In general I believe the decisions of the people, in a body, will be more honest and more disinterested than those of wealthy men.”
—Thomas Jefferson (1743–1826)
“We have seen over and over that white male historians in general have tended to dismiss any history they didn’t themselves write, on the grounds that it is unserious, unscholarly, a fad, too “political,” “merely” oral and thus unreliable.”
—Adrienne Rich (b. 1929)
“According to the historian, they escaped as by a miracle all roving bands of Indians, and reached their homes in safety, with their trophies, for which the General Court paid them fifty pounds. The family of Hannah Dustan all assembled alive once more, except the infant whose brains were dashed out against the apple tree, and there have been many who in later time have lived to say that they have eaten of the fruit of that apple tree.”
—Henry David Thoreau (1817–1862)