sigmoid函数

$ \varsigma_\alpha:x\mapsto\frac{1}{1+e^{-\alpha x}}

$ \varsigma_1:x\mapsto\frac1{1+e^{-x}}

性質

$ \forall x\in\R:0<\varsigma_\alpha(x)<1

$ \varsigma_\alpha(x)+\varsigma_\alpha(-x)=1

$ \varsigma_\alpha(x)=\frac12\left(\tanh\frac12\alpha x+1\right)

$ \begin{dcases}y'&=\alpha y(1-y)\\y(0)&=2\end{dcases}\iff y=\varsigma_\alpha

$ \sigma(x)=\frac{1}{1+e^{-\alpha\frac xT}}

References