S型函数

一些S型函數的比較，圖中的函數皆以原點斜率為1的方式歸一化。

• 逻辑斯谛函数

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

• 雙曲正切函數（等價於逻辑斯谛函数的平移與縮放）

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

• 反正切函數

${\displaystyle f(x)=\arctan x}$

• 古德曼函數

${\displaystyle f(x)=\operatorname {gd} (x)=\int _{0}^{x}{\frac {1}{\cosh t}}\,dt=2\arctan \left(\tanh \left({\frac {x}{2}}\right)\right)}$

• 误差函数

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

• 廣義邏輯斯諦函數

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

• 平滑階躍函數

${\displaystyle f(x)={\begin{cases}\displaystyle {\frac {\int _{0}^{x}{\bigl (}1-u^{2}{\bigr )}^{N}\ du}{\int _{0}^{1}{{\bigl (}1-u^{2}{\bigr )}^{N}\ du}}},&|x|\leq 1\\\operatorname {sgn}(x)&|x|\geq 1\\\end{cases}}\,\quad N\geq 1}$

• 一些代數函數, 例如

${\displaystyle f(x)={\frac {x}{\sqrt {1+x^{2}}}}}$

所有連續非負的凸形函數的積分都是S型函數，因此許多常見概率分布的累积分布函数會是S型函數。一個常見的例子是误差函数，它是正态分布的累积分布函数。

• Mitchell, Tom M. . WCB–McGraw–Hill. 1997. ISBN 0-07-042807-7.. 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.
• Humphrys, Mark. . [2015-02-01]. （原始内容存档于2015-02-02）. Properties of the sigmoid, including how it can shift along axes and how its domain may be transformed.