S型函数

S型函数（Template:Lang-en，或稱乙狀函數）是一種函数，因其函數圖像形状像字母S得名。其形狀曲線至少有2個焦點，也叫“二焦點曲線函數”。S型函数是有界、可微的实函数，在实数范围内均有取值，且导数恒为非负^[1]，有且只有一个拐点。S型函数和S型曲线指的是同一事物。

逻辑斯谛函数是一种常见的S型函数，其公式如下：^[1]

S (t) = \frac{1}{1 + e^{- t}} .

其级数展开为：

s : = 1 / 2 + \frac{1}{4} t - \frac{1}{48} t^{3} + \frac{1}{480} t^{5} - \frac{17}{80640} t^{7} + \frac{31}{1451520} t^{9} - \frac{691}{319334400} t^{11} + O (t^{12})

其他S型函數案例見下。在一些學科領域，特別是人工神经网络中，S型函數通常特指邏輯斯諦函數。

常見的S型函數

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

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

f (x) = \arctan x

f (x) = gd (x) = \int_{0}^{x} \frac{1}{\cosh t} d t = 2 \arctan (\tanh (\frac{x}{2}))

f (x) = \erf (x) = \frac{2}{\sqrt{π}} \int_{0}^{x} e^{- t^{2}} d t

f (x) = (1 + e^{- x})^{- α}, α > 0

f (x) = {\begin{matrix} \frac{\int_{0}^{x} (1 - u^{2})^{N} d u}{\int_{0}^{1} (1 - u^{2})^{N} d u}, & | x | \leq 1 \\ sgn (x) & | x | \geq 1 \end{matrix} N \geq 1

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。

Template:Cite book. 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.
Template:Cite web Properties of the sigmoid, including how it can shift along axes and how its domain may be transformed.