machine_learningヘッダファイルパッケージで用いている計算式
1. 数学関数の定義と微分形
関数sigmoid
(Formula used in machine_learning header file package;
1. Definitions and differential forms of mathematical functions
— function sigmoid)
◆定義(Definition)
\[\begin{equation}
f(y)=\frac{1}{1+\exp(-y)}
\end{equation}\]
◆微分形(Derivative)
\[\begin{eqnarray}
f’(y)
&=& -\frac{-\exp(-y)}{[1+\exp(-y)]^2}
\nonumber \\
&=& \frac{\exp(-y)}{[1+\exp(-y)]^2}
\nonumber \\
&=& \frac{1}{1+\exp(-y)}\frac{\exp(-y)}{1+\exp(-y)}
\nonumber \\
&=& \frac{1}{1+\exp(-y)}\frac{[1+\exp(-y)]-1}{1+\exp(-y)}
\nonumber \\
&=& \frac{1}{1+\exp(-y)}\left[1-\frac{1}{1+\exp(-y)}\right]
\nonumber \\
&=& f(y)\left[1-f(y)\right]
\end{eqnarray}\]