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 shown in the first figure and defined by the formula:^[1]

Other standard sigmoid functions are given in the Examples section. In some fields, most notably in the context of artificial neural networks, the term "sigmoid function" is used as an alias for the logistic function.

Special cases of the sigmoid function include the Gompertz curve (used in modeling systems that saturate at large values of x) and the ogee curve (used in the spillway of some dams). Sigmoid functions have domain of all real numbers, with return (response) value commonly monotonically increasing but could be decreasing. Sigmoid functions most often show a return value (y axis) in the range 0 to 1. Another commonly used range is from −1 to 1.

A wide variety of sigmoid functions including the logistic and hyperbolic tangent functions have been used as the activation function of artificial neurons. Sigmoid curves are also common in statistics as cumulative distribution functions (which go from 0 to 1), such as the integrals of the logistic density, the normal density, and Student's t probability density functions. The logistic sigmoid function is invertible, and its inverse is the logit function.

Definition[edit]

A sigmoid function is a bounded, differentiable, real function that is defined for all real input values and has a non-negative derivative at each point^[1] ^[2] and exactly one inflection point.

Properties[edit]

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 $x\rightarrow \pm \infty$ .

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.

$f(x)={\frac {1}{1+e^{-x}}}$

Logistic function

(shifted and scaled version of the logistic function, above) $f(x)=\tanh x={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}$

Hyperbolic tangent

$f(x)=\arctan x$

Arctangent function

$f(x)=\operatorname {gd} (x)=\int _{0}^{x}{\frac {dt}{\cosh t}}=2\arctan \left(\tanh \left({\frac {x}{2}}\right)\right)$

Gudermannian function

$f(x)=\operatorname {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt$

Error function

$f(x)=\left(1+e^{-x}\right)^{-\alpha },\quad \alpha >0$

Generalised logistic function

function $f(x)={\begin{cases}{\displaystyle \left(\int _{0}^{1}\left(1-u^{2}\right)^{N}du\right)^{-1}\int _{0}^{x}\left(1-u^{2}\right)^{N}\ du},&|x|\leq 1\\\\\operatorname {sgn}(x)&|x|\geq 1\\\end{cases}}\quad N\in \mathbb {Z} \geq 1$

Smoothstep

Some , for example $f(x)={\frac {x}{\sqrt {1+x^{2}}}}$

algebraic functions

and in a more general form $f(x)={\frac {x}{\left(1+|x|^{k}\right)^{1/k}}}$

[3]

Up to shifts and scaling, many sigmoids are special cases of $f(x)=\varphi (\varphi (x,\beta ),\alpha ),$ where $\varphi (x,\lambda )={\begin{cases}(1-\lambda x)^{1/\lambda }&\lambda \neq 0\\e^{-x}&\lambda =0\\\end{cases}}$ is the inverse of the negative , and $\alpha <1$ and $\beta <1$ are shape parameters.^[4]

Box–Cox transformation

^[5] normalized to (-1,1):

Smooth transition function

${\begin{aligned}f(x)&={\begin{cases}{\displaystyle {\frac {2}{1+e^{-2m{\frac {x}{1-x^{2}}}}}}-1},&|x|<1\\\\\operatorname {sgn}(x)&|x|\geq 1\\\end{cases}}\\&={\begin{cases}{\displaystyle \tanh \left(m{\frac {x}{1-x^{2}}}\right)},&|x|<1\\\\\operatorname {sgn}(x)&|x|\geq 1\\\end{cases}}\end{aligned}}$ using the hyperbolic tangent mentioned above. Here, $m$ is a free parameter encoding the slope at $x=0$ , which must be greater than or equal to ${\sqrt {3}}$ because any smaller value will result in a function with multiple inflection points, which is therefore not a true sigmoid. This function is unusual because it actually attains the limiting values of -1 and 1 within a finite range, meaning that its value is constant at -1 for all $x\leq -1$ and at 1 for all $x\geq 1$ . Nonetheless, it is smooth (infinitely differentiable, $C^{\infty }$ ) everywhere, including at $x=\pm 1$ .

Mitchell, Tom M. (1997). Machine Learning. WCB . ISBN 978-0-07-042807-2.. (NB. In particular see "Chapter 4: Artificial Neural Networks" (in particular pp. 96–97) where Mitchell 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.)

McGraw–Hill

Humphrys, Mark. . Archived from the original on 2022-07-14. Retrieved 2022-07-14. (NB. Properties of the sigmoid, including how it can shift along axes and how its domain may be transformed.)

"Continuous output, the sigmoid function"

. Archived from the original on 2022-07-14.