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{{Short description|Mathematical function having a characteristic S-shaped curve or sigmoid curve}}
{{Use dmy dates|date=July 2022|cs1-dates=y}}
{{Use list-defined references|date=July 2022}}
[[File:Logistic-curve.svg|thumb|320px|right|The [[logistic curve]]]]
[[File:Error Function.svg|thumb|right|320px|Plot of the [[error function]]]]
A '''sigmoid function'''
:<math>\sigma(x) = \frac{1}{1 + e^{-x}} = \frac{e^x}{1 + e^x} = 1 - \sigma(-x).</math>▼
In many fields, especially in the context of [[Neural network (machine learning)|artificial neural networks]], the term "sigmoid function" is correctly recognized as a synonym for the logistic function. While other S-shaped curves, such as the [[Gompertz function|Gompertz curve]] or the [[Ogee|ogee curve]], may resemble sigmoid functions, but they are distinct mathematical functions with different properties and applications.
▲:<math>\sigma(x) = \frac{1}{1 + e^{-x}} = \frac{e^x}{1 + e^x}=1-\sigma(-x).</math>
Sigmoid functions, particularly the logistic function, have a domain of all [[Real number|real numbers]] and typically produce output values in the range from 0 to 1, although some variations, like the [[Hyperbolic functions|hyperbolic tangent]], output values between −1 and 1. These functions are commonly used as [[Activation function|activation functions]] in artificial neurons and as [[Cumulative distribution function|cumulative distribution functions]] in [[statistics]]. The logistic sigmoid is also invertible, with its inverse being the [[Logit|logit function]].
== Definition ==
A sigmoid function is a [[bounded function|bounded]], [[differentiable function|differentiable]], real function that is defined for all real input values and has a non-negative derivative at each point<ref name="Han-Morag_1995" /> <ref name="yibei" /> and exactly one [[inflection point]]
== Properties ==
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* and in a more general form<ref name="Dunning-Kensler-Coudeville-Bailleux_2015" /> <math display="block"> f(x) = \frac{x}{\left(1 + |x|^{k}\right)^{1/k}} </math>
* Up to shifts and scaling, many sigmoids are special cases of <math display="block"> f(x) = \varphi(\varphi(x, \beta), \alpha) , </math> where <math display="block"> \varphi(x, \lambda) = \begin{cases} (1 - \lambda x)^{1/\lambda} & \lambda \ne 0 \\e^{-x} & \lambda = 0 \\ \end{cases} </math> is the inverse of the negative [[Box–Cox transformation]], and <math>\alpha < 1</math> and <math>\beta < 1</math> are shape parameters.<ref name="grex" />
* [[Non-analytic_smooth_function#Smooth_transition_functions|Smooth
<!--
<math display="block"> f(x) = \begin{cases}
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<math display="block">\begin{align}f(x) &= \begin{cases}
{\displaystyle
\frac{2}{1+e^{-
\\
\sgn(x) & |x| \ge 1 \\
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&= \begin{cases}
{\displaystyle
\tanh\left(
\\
\sgn(x) & |x| \ge 1 \\
\end{cases}\end{align}</math> using the hyperbolic tangent mentioned above. Here, <math>m</math> is a free parameter encoding the slope at <math>x=0</math>, which must be greater than or equal to <math>\sqrt{3}</math> because any smaller value will result in a function with multiple inflection points, which is therefore not a true sigmoid. This function is unusual because it actually attains the limiting values of -1 and 1 within a finite range, meaning that its value is constant at -1 for all <math>x \leq -1</math> and at 1 for all <math>x \geq 1</math>. Nonetheless, it is [[Smoothness|smooth]] (infinitely differentiable, <math>C^\infty</math>) ''everywhere'', including at <math>x = \pm 1</math>.
== Applications ==
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{{Commons category|Sigmoid functions}}
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