S型函数

• 逻辑斯谛函数

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

• 双曲正切函数（等价于逻辑斯谛函数的平移与缩放）

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

• 反正切函数

${\displaystyle f(x)=\arctan x}$ 

• 古德曼函数

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

• 误差函数

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

• 广义逻辑斯谛函数（英语：Generalised logistic function）

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

• 平滑阶跃函数（英语：Smoothstep）

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

• 一些代数函数, 例如

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

所有连续非负的凸形函数的积分都是S型函数，因此许多常见概率分布的累积分布函数会是S型函数。一个常见的例子是误差函数，它是正态分布的累积分布函数。

• Mitchell, Tom M. Machine Learning. 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. Continuous output, the sigmoid function. [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.