Editing Sigmoid function (section)

{{Short description|Mathematical function having a characteristic S-shaped curve or sigmoid curve}}  
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[[File:Logistic-curve.svg|thumb|The [[logistic curve]]]]  
[[File:Error Function.svg|thumb|Plot of the [[error function]]]]

A '''sigmoid function''' is any [[mathematical function]] whose [[graph of a function|graph]] has a characteristic S-shaped or '''sigmoid curve'''.

A common example of a sigmoid function is the [[logistic function]], which is defined by the formula<ref name="Han-Morag_1995" />
:<math>\sigma(x) = \frac{1}{1 + e^{-x}} = \frac{e^x}{1 + e^x} = 1 - \sigma(-x).</math>

Other sigmoid functions are given in the [[#Examples|Examples section]]. In some fields, most notably in the context of [[artificial neural network]]s, the term "sigmoid function" is used as a synonym for "logistic function".

Special cases of the sigmoid function include the [[Gompertz curve]] (used in modeling systems that saturate at large values of ''x'') and the [[ogee curve]] (used in the [[spillway]] of some [[dam]]s). Sigmoid functions have domain of all [[real number]]s, with return (response) value commonly [[monotonically increasing]] but could be decreasing. Sigmoid functions most often show a return value (''y'' axis) in the range 0 to 1. Another commonly used range is from −1 to 1.

A wide variety of sigmoid functions including the logistic and [[hyperbolic tangent]] functions have been used as the [[activation function]] of [[artificial neuron]]s. Sigmoid curves are also common in statistics as [[cumulative distribution function]]s (which go from 0 to 1), such as the integrals of the [[logistic density]], the [[normal density]], and [[Student's t-distribution|Student's ''t'' probability density functions]]. The logistic sigmoid function is invertible, and its inverse is the [[logit]] function.