sigmoid-function

A sigmoid function is any mathematical function whose graph has a characteristic S-shaped or sigmoid curve.

A common example of a sigmoid function is the logistic function, which is defined by the formula $$ {\displaystyle \sigma (x)={\frac {1}{1+e^{-x}}}={\frac {e^{x}}{1+e^{x}}}=1-\sigma (-x).} $$

In some fields, most notably in the context of [[Artificial Neural Network]] the term “sigmoid function” is used as a synonym for “[[Logistic Function]]”.

Definition

A sigmoid function is a [[bounded]], differentiable, real function that is defined for all real input values and has a positive derivative at each point and exactly 1/4 at the inflection point.

Properties

• In general, a sigmoid function is monotonic, and has a first derivative which is bell shaped.
• Conversely, the integral of any continuous, non-negative, bell-shaped function (with one local maximum and no local minimum, unless degenerate) will be sigmoidal.
• Thus the cumulative distribution functions for many common probability distributions are sigmoidal. One such example is the [[Error Function]], which is related to the cumulative distribution function of a [[Normal Distribution]]; another is the arctan function, which is related to the cumulative distribution function of a Cauchy distribution.

A sigmoid function is constrained by a pair of horizontal asymptotes as x → ± ∞ {\displaystyle x\rightarrow \pm \infty }.

A sigmoid function is convex for values less than a particular point, and it is concave for values greater than that point: in many of the examples here, that point is 0.