Sigmoid function

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File:Logistic-curve.svg

The logistic curve

File:Error Function.svg

Plot of the error function

Many natural processes, including those of complex system learning curves, 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. A sigmoid curve is produced by a mathematical function having an "S" shape. Often, sigmoid function refers to the special case of the logistic function shown at right and defined by the formula

$P(t) = \frac{1}{1 + e^{-t}}.$

Another example is the Gompertz curve. It is used in modeling systems that saturate at large values of t. Another example is the ogee curve as used in the spillway of some dams. A wide variety of sigmoid functions have been used as the activation function of artificial neurons, including the logistic function and tanh(x).

Properties

In general, a sigmoid function is real-valued and differentiable, having either a non-negative or non-positive first derivative which is bell shaped. There are also a pair of horizontal asymptotes as $t \rightarrow \pm \infty$. The logistic functions are sigmoidal and are characterized as the solutions of the differential equation

$P'(t) = \frac{r}{k}P(t)(b - P(t)).$

Examples

File:Gjl-t(x).svg

Some sigmoid functions compared. In the drawing all functions are normalized in such a way that their slope at 0 is 1.

Besides the logistic function, sigmoid functions include the ordinary arctangent, the hyperbolic tangent, and the error function, but also the generalised logistic function and algebraic functions like $f(x)=\tfrac{x}{\sqrt{1+x^2}}$.

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 most famous such example is the error function.

See also

Wikimedia Commons has media related to:

Sigmoid functions

• Logistic function
• Logistic distribution
• Logistic regression
• Logit
• Hyperbolic function
• Weibull distribution

References

• Tom M. Mitchell, Machine Learning, WCB–McGraw–Hill, 1997, ISBN 0-07-042807-7. In particular see "Chapter 4: Artificial Neural Networks" (in particular pp. 96–97) where Mitchel 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.
• http://www.computing.dcu.ie/~humphrys/Notes/Neural/sigmoid.html Properties of the sigmoid, including how it can shift along axes and how its domain may be transformed.