Sigmoid function

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A sigmoid function is a mathematical function that produces a sigmoid curve — a curve having an "S" shape. Often, sigmoid function refers to the special case of the logistic function shown at right and defined by the formula

Derivative of the sigmoid function

The derivative of the sigmoid function can be written

Members of the sigmoid family

In general, a sigmoid function is real-valued and differentiable, having either a non-negative or non-positive first derivative and exactly one inflection point.

Besides the logistic function, sigmoid functions include the ordinary arc-tangent, the hyperbolic tangent, and the error function, but also algebraic functions like . The integral of any smooth, positive, "bump-shaped" function will be sigmoidal, thus the cumulative distribution functions for many common probability distributions are sigmoidal.

The logistic sigmoid function is related to the hyperbolic tangent, e.g. by

Sigmoid functions in neural networks

Sigmoid functions are often used in neural networks to introduce nonlinearity in the model and/or to clamp signals to within a specified range. A popular neural net element computes a linear combination of its input signals, and applies a bounded sigmoid function to the result; this model can be seen as a "smoothed" variant of the classical threshold neuron.

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

y' = y(1 − y).

The right hand side is a low order polynomial. Furthermore, the polynomial has factors y and 1 − y, both of which are simple to compute. Given y = sig(t) at a particular t, the derivative of the sigmoid function at that t can be obtained by multiplying the two factors together. These relationships result in simplified implementations of artificial neural networks with artificial neurons.

Double sigmoid function

The double sigmoid is a function similar to the sigmoid function with numerous applications. Its general formula is:

where d is its centre and s is the steepness factor.

It is based on the Gaussian curve and graphically it is similar to two identical sigmoids bonded together at the point x = d.

One of its applications is non-linear normalization of a sample, as it has the property of eliminating outliers.

External links

• Modeling Market Adoption in Excel with a simplified s-curve

See also

• Logistic distribution
• Logistic regression
• Logit
• Hyperbolic function
• Weibull distributionde:Sigmoidfunktion

fr:Sigmoïde (mathématiques) it:Funzione sigmoidea nl:Sigmoidfunctie ja:シグモイド関数

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Some of the initial content on this page may be incorporated in part from copyleft sources in the public domain including wikis such as Wikipedia and AskDrWiki. Drug information for patients came from the The National Library of Medicine. Infectious disease information may have come from the Centers for Disease Control (CDC). Differential Diagnoses are drawn from clinicians as well as an amalgamation of 3 sources: 1.The Disease Database; 2. Kahan, Scott, Smith, Ellen G. In A Page: Signs and Symptoms. Malden, Massachusetts: Blackwell Publishing, 2004:3; 3. Sailer, Christian, Wasner, Susanne. Differential Diagnosis Pocket. Hermosa Beach, CA: Borm Bruckmeir Publishing LLC, 2002:7 .