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

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

Another example is the Gompertz curve.

1 Members of the sigmoid family
2 See also
3 References
4 External links

[edit] Members of the sigmoid family

The logistic curve

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 the Gompertz function, 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.

[edit] See also

[edit] 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 p. 96-97) where Mitchel uses the word "logistic function" and the "sigmoid function" synonomously -- 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.