S型函数
（重定向自S函数）
S型函数（英语：sigmoid function，或称乙状函数）是一种函数，因其函数图像形状像字母S得名。其形状曲线至少有2个焦点，也叫“二焦点曲线函数”。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。
- <math>S(t) = \frac{1}{1 + e^{-t}}.</math>
其级数展开为：
- <math>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})</math>
其他S型函数案例见下。在一些学科领域，特别是人工神经网络中，S型函数通常特指逻辑斯谛函数。
常见的S型函数[编辑]
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- <math> f(x) = \frac{1}{1 + e^{-x}} </math>
- <math> f(x) = \tanh x = \frac{e^x-e^{-x}}{e^x+e^{-x}} </math>
- <math> f(x) = \arctan x </math>
- <math> f(x) = \operatorname{gd}(x) = \int_{0}^{x} \frac{1}{\cosh t} \, dt = 2\arctan\left(\tanh\left(\frac{x}{2}\right)\right)
</math>
- <math> f(x) = \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2} \, dt </math>
- <math> f(x) = (1+e^{-x})^{-\alpha}, \quad \alpha > 0 </math>
- <math> 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| \le 1 \\ \sgn(x) & |x| \ge 1 \\ \end{cases} \, \quad N \ge 1 </math>
- 一些代数函数, 例如
- <math> f(x) = \frac{x}{\sqrt{1+x^2}} </math>
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。
参考文献[编辑]
- ^ 1.0 1.1 Han, Jun; Morag, Claudio. The influence of the sigmoid function parameters on the speed of backpropagation learning. Mira, José; Sandoval, Francisco (编). From Natural to Artificial Neural Computation. Lecture Notes in Computer Science 930. 1995: 195–201. ISBN 978-3-540-59497-0. doi:10.1007/3-540-59497-3_175.
- 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.
