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| [[Image:Logistic-curve.svg|thumb|320px|right|The [[logistic curve]]]]
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| [[Image:Error Function.svg|thumb|right|320px|Plot of the [[error function]]]]
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| A '''sigmoid function''' is a [[function (mathematics)|mathematical function]] having an "S" shape ('''sigmoid curve'''). Often, ''sigmoid function'' refers to the special case of the [[logistic function]] shown in the first figure and defined by the formula
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| :<math>S(t) = \frac{1}{1 + e^{-t}}.</math>
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| Another example is the [[Gompertz curve]]. It is used in modeling systems that saturate at large values of t.
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| Another example is the [[ogee curve]] as used in the [[spillway]] of some [[dam]]s.
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| A wide variety of sigmoid functions have been used as the [[activation function]] of [[artificial neuron]]s, including the logistic and [[hyperbolic tangent]] functions. Sigmoid curves are also common in statistics as [[cumulative distribution function]]s, such as the integrals of the [[logistic distribution]], the [[normal distribution]], and [[Student's t-distribution|Student's ''t'' probability density functions]].
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| ==Definition==
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| A sigmoid function is a bounded differentiable real function that is defined for all real input values and has a positive derivative at each point.<ref>{{Cite book |last1=Han |first1=Jun |last2=Morag |first2=Claudio |title=From Natural to Artificial Neural Computation |chapter=The influence of the sigmoid function parameters on the speed of backpropagation learning |editor1-last=Mira |editor1-first=José |editor2-last=Sandoval |editor2-first=Francisco |pages=195–201 |year=1995 |url=http://dx.doi.org/10.1007/3-540-59497-3_175}}</ref>
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| ==Properties==
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| In general, a sigmoid function is [[real number|real]]-valued and [[differentiable]], having either a [[non-negative]] or [[non-positive]] first [[derivative]]{{citation needed|date=November 2013}} which is bell shaped. There are also a pair of [[horizontal asymptotes]] as <math>t \rightarrow \pm \infty</math>. The differential equation <math> \tfrac{d}{dt} S(t) = c_1 S(t) \left( c_2 - S(t) \right)</math>, with the inclusion of a [[boundary condition]] providing a third [[degree of freedom]], <math>c_3</math>, provides a class of functions of this type.
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| ==Examples==
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| [[File:Gjl-t(x).svg|thumb|320px|right|Some sigmoid functions compared. In the drawing all functions are normalized in such a way that their slope at the origin is 1.]]
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| Many natural processes, including those of complex system [[learning curve]]s, exhibit a progression from small beginnings that accelerates and approaches a climax over time. When a detailed description is lacking, a sigmoid function is often used<ref>{{cite journal
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| | last = Gibbs
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| | first = M.N.
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| |date=Nov 2000
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| | title = Variational Gaussian process classifiers
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| | journal = IEEE Transactions on Neural Networks
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| | volume = 11
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| | issue = 6
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| | pages = 1458–1464
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| | doi = 10.1109/72.883477
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| | accessdate = 21 Aug 2012
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| }}</ref>
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| .
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| Besides the [[logistic function]], sigmoid functions include the ordinary [[Inverse trigonometric function|arctangent]], the [[Hyperbolic function|hyperbolic tangent]], the [[Gudermannian function]], and the [[error function]], but also the [[generalised logistic function]] and [[algebraic function]]s like <math>f(x)=\tfrac{x}{\sqrt{1+x^2}}</math>.
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| The [[integral]] of any smooth, positive, "bump-shaped" function will be sigmoidal, thus the [[cumulative distribution function]]s for many common [[probability distribution]]s are sigmoidal. The most famous such example is the [[error function]], which is related to the [[Cumulative distribution function|Cumulative Distribution Function (CDF)]] of a [[normal distribution]].
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| ==See also==
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| {{commons category|Sigmoid functions}}
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| * [[Cumulative distribution function]]
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| * [[Generalized logistic curve]]
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| * [[Logistic distribution]]
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| * [[Logistic regression]]
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| * [[Logit]]
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| * [[Hyperbolic function]]
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| * [[Weibull distribution]]
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| * [[Heaviside step function]]
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| * [[Gompertz function]]
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| == References ==
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| {{reflist}}
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| * {{ cite book | first1=Tom M. |last1= Mitchell | title=Machine Learning | publisher=WCB–McGraw–Hill |year=1997
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| |isbn=0-07-042807-7}}. In particular see "Chapter 4: Artificial Neural Networks" (in particular pp. 96–97) where Mitchell uses the word "logistic function" and the "sigmoid function" synonymously – this function he also calls the "squashing function" – and the sigmoid (aka logistic) function is used to compress the outputs of the "neurons" in multi-layer neural nets.
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| * {{cite web|first1= Mark | last1= Humphrys | url =http://www.computing.dcu.ie/~humphrys/Notes/Neural/sigmoid.html
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| |title= Continuous output, the sigmoid function}} Properties of the sigmoid, including how it can shift along axes and how its domain may be transformed.
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| [[Category:Elementary special functions]]
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| [[Category:Neural networks]]
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| [[Category:Probability distributions]]
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I'm Mazie and I live in a seaside city in northern Germany, Hammelburg. I'm 26 and I'm will soon finish my study at Computing and Information Science.
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