Editing Logistic function (section)

=== In statistics and machine learning ===
Logistic functions are used in several roles in statistics. For example, they are the [[cumulative distribution function]] of the [[Logistic distribution|logistic family of distributions]], and they are, a bit simplified, used to model the chance a chess player has to beat their opponent in the [[Elo rating system]]. More specific examples now follow.

==== Logistic regression ====
{{Main|Logistic regression}}

Logistic functions are used in [[logistic regression]] to model how the probability <math>p</math> of an event may be affected by one or more [[explanatory variables]]: an example would be to have the model

<math display="block">p = f(a + bx),</math>

where <math>x</math> is the explanatory variable, <math>a</math> and <math>b</math> are model parameters to be fitted, and <math>f</math> is the standard logistic function.

Logistic regression and other [[log-linear model]]s are also commonly used in [[machine learning]]. A generalisation of the logistic function to multiple inputs is the [[softmax activation function]], used in [[multinomial logistic regression]].

Another application of the logistic function is in the [[Rasch model]], used in [[item response theory]]. In particular, the Rasch model forms a basis for [[maximum likelihood]] estimation of the locations of objects or persons on a [[Continuum (theory)|continuum]], based on collections of [[categorical variable|categorical data]], for example the abilities of persons on a continuum based on responses that have been categorized as correct and incorrect.

==== Neural networks ====
Logistic functions are often used in [[artificial neural network]]s to introduce [[nonlinearity]] in the model or to clamp signals to within a specified [[interval (mathematics)|interval]].  A popular [[artificial neuron|neural net element]] computes a [[linear combination]] of its input signals, and applies a bounded logistic function as the [[activation function]] to the result; this model can be seen as a "smoothed" variant of the classical [[perceptron|threshold neuron]].<!--

A reason for its popularity in neural networks is because the logistic function satisfies the differential equation

<math display="block">y' = y(1-y).</math>

The right hand side is a low-degree polynomial. Furthermore, the polynomial has factors <math>y</math> and <math>1 − y</math>, both of which are simple to compute. Given <math>y = sig(t)</math> at a particular <math>t</math>, the derivative of the logistic function at that <math>t</math> can be obtained by multiplying the two factors together. -->

A common choice for the activation or "squashing" functions, used to clip large magnitudes to keep the response of the neural network bounded,<ref name="Gershenfeld-1999">Gershenfeld 1999, p. 150.</ref> is

<math display="block">g(h) = \frac{1}{1 + e^{-2 \beta h}},</math>

which is a logistic function.

These relationships result in simplified implementations of [[artificial neural network]]s with [[artificial neuron]]s. Practitioners caution that sigmoidal functions which are [[Odd functions|antisymmetric]] about the origin (e.g. the [[hyperbolic tangent]]) lead to faster convergence when training networks with [[backpropagation]].<ref name="LeCun-1998">{{cite book | author1 = LeCun, Y. | author2 = Bottou, L. | author3 = Orr, G. | author4 = Muller, K. | editor = Orr, G. | editor2 = Muller, K. | year = 1998 | contribution = Efficient BackProp | title = Neural Networks: Tricks of the trade | isbn = 3-540-65311-2 | publisher = Springer | contribution-url = http://yann.lecun.com/exdb/publis/pdf/lecun-98b.pdf | archive-date = 31 August 2018 | access-date = 16 September 2009 | archive-url = https://web.archive.org/web/20180831075352/http://yann.lecun.com/exdb/publis/pdf/lecun-98b.pdf | url-status = dead }}</ref>

The logistic function is itself the derivative of another proposed activation function, the [[softplus]].