diff --git a/Rapport.ipynb b/Rapport.ipynb index ff5895cc18a6eb351aaff235abd504c182439eb6..9bf5fef3fe1b6a7a0aa2e3b91389e4e8feda7b64 100644 --- a/Rapport.ipynb +++ b/Rapport.ipynb @@ -466,18 +466,21 @@ "\n", "## Réseaux de Neurones Artificiels\n", "\n", - "$$Z^{L+1}=W^{L+1}A^{L}+B^{L+1}$$\n", - "\n", - "$$A^{L+1}=\\sigma(Z^{L+1})$$\n", + "$Z^{L+1}=W^{L+1}A^{L}+B^{L+1}\\\\\n", + "A^{L+1}=\\sigma(Z^{L+1})$\n", "\n", "$$C = \\frac{1}{N_{out}}\\sum_{i=1}^{N_{out}} (\\hat{y_i} - y_i)^2$$\n", "\n", "### 1\n", "\n", "$$\\sigma(x)=\\frac{1}{1+e^{-x}} $$\n", + "\n", "$$\\Rightarrow \\sigma'(x)=\\frac{-e^{-x}}{-(1+e^{-x})^2} $$\n", + "\n", "$$\\Rightarrow \\sigma'(x)=\\frac{1}{1+e^{-x}}(\\frac{1+e^{-x}-1}{1+e^{-x}}) $$\n", + "\n", "$$\\Rightarrow \\sigma'(x)=\\sigma(\\frac{1+e^{-x}}{1+e^{-x}}-\\frac{1}{1+e^{-x}}) $$\n", + "\n", "$$\\Rightarrow \\sigma'(x)=\\sigma(1-\\sigma)$$\n", "\n", "### 2\n",