S函数










Sigmoid function 2D plot




Sigmoid function complex plot


S函数得名因其形状像S字母。一种常见的S函数是逻辑函数:


S(t)=11+e−t.{displaystyle S(t)={frac {1}{1+e^{-t}}}.}{displaystyle S(t)={frac {1}{1+e^{-t}}}.}

其级数展开为：


s:=1/2+14t−148t3+1480t5−1780640t7+311451520t9−691319334400t11+O(t12){displaystyle 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})}{displaystyle 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})}



参考资料






  • 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.  Properties of the sigmoid, including how it can shift along axes and how its domain may be transformed.







参见


  • 函数



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