# Sigmoid function

Many natural processes and complex system learning curves display a history dependent progression from small beginnings that accelerates and approaches a climax over time. For lack of complex descriptions 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.

## Properties

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. There are also two asymptotes, $t \rightarrow \pm \infty$.

The general case 1 / (1 + e-x) is particularly useful, used in Artificial Neural Networks partly because it has a simple derivative: if s(x) is the sigmoid function, then s'(x) = s(x) * (1 - s(x)).[1]

## Examples

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.