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The logistic curve

Plot of the error function

Sigmoid functionFrom Wikipedia, the free encyclopedia

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A sigmoid function is a mathematical function having an"S" s hape ( sigmoid curve ). Often, sigmoid functionrefer s to the special case of t he logistic function s hownon the right, but a sigmoid function is also the Gompertzcurve. It is used in modeling systems that saturate atlarge values of the argument, or another example theogee curve as used in the s pillway of some dams. Awide variety of sigmoid functions have been used as theactivation function of artificial neurons, including thelogistic and hyperbolic tangent functions. Sigmoid curvesare also common in statistics as cumulative distributio nfunctions, such as the integrals of the logistic distributi on,the normal distribution , and Student's t probabilitydensity func tions.

Contents

1 Definition2 Properties3 Examples4 See also5 Ref erences

efinition

A sigmoid function is a bound ed differentia ble real fun ction that is defined for all real input values and has a positive(PS) derivative everywhe re.[1]

PropertiesIn general, a sigmoid function is real-valued and differentiable, having either a non-negative or non-positive firstderivative which is bell shaped. There are also a pair of horizontal asymptotes as . The differentialequation , with the inclusion of a boundary condition providing a third degree

of freedom, , provides a class of functions of this type.

Examples

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Some sigmoid functions compared. In the drawing allfunctions are normalized in such a way that their slopeat the origin is 1.

Many natural processes, including those of complexsystem learning curves, exhibit a progression from small

beginnings that accelerates and approaches a climaxover time. When a detailed description is lacking, asigmoid function is often used [2] .

Besides the logistic function, sigmoid functions includethe ordinary arctangent, the hyperbolic tangent, and theerror function, but also the generalised logistic functionand algebraic functions like .

The integral of any smooth, positive, "bump-shaped"function will be sigmoidal, thus the cumulativedistribution functions for many common probabilitydistributions are sigmoidal. The most famous such example is the error function, which is related to the CumulativeDistribution Function (CDF) of a normal distribution.

See alsoCumulative distribution functionGeneralized logistic curveLogistic distributionLogistic regressionLogitHyperbolic functionWeibull distributionHeaviside step function

References

1. ^ Han, Jun; Morag, Claudio (1995). "The influence of the sigmoid function parameters on the speed of backpropagation learning" (http://dx.doi.org/10.1007/3-540-59497-3_175). In Mira, José; Sandoval, Francisco. From Natural to Artif icial Neural Computation . pp. 195–201.

2. ^ Gibbs, M.N. (Nov 2000). "Variational Gaussian process classifiers". IEEE Transactions on Neural Networks 11(6): 1458–1464. doi:10.1109/72.883477 (http://dx.doi.org/10.1109%2F72.883477).

Mitchell, Tom M. (1997). Machine Learning . WCB–McGraw–Hill. ISBN 0-07-042807-7.. In particular see "Chapter 4: Artificial Neural Networks" (in particular pp. 96–97) where Mitchell uses the word "logisticfunction" and the "sigmoid function" synonymously – this function he also calls the "squashing function" – andthe sigmoid (aka logistic) function is used to compress the outputs of the "neurons" in multi-layer neural nets.Humphrys, Mark. "Continuous output, the sigmoid function"(http://www.computing.dcu.ie/~humphrys/Notes/Neural/sigmoid.html). Properties of the sigmoid, includinghow it can shift along axes and how its domain may be transformed.

Retrieved from "http://en.wikipedia.org/w/index.php?title=Sigmoid_function&oldid=563110772"Categories: Elementary special functions Neural networks Probability distributions

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