diff --git a/Rapport.ipynb b/Rapport.ipynb index 876c42d4e155ee8f11afe0b62b3a6664bf07b7d4..d61c1ab23e37fa89342bdc3db482747555678f0f 100644 --- a/Rapport.ipynb +++ b/Rapport.ipynb @@ -466,20 +466,21 @@ "\n", "## Réseaux de Neurones Artificiels\n", "\n", - "$$Z^{L+1}=W^{L+1}A^{L}+B^{L+1}\\\\\n", - "A^{L+1}=\\sigma(Z^{L+1})$$\n", + "$$Z^{L+1}=W^{L+1}A^{L}+B^{L+1}$$ and $$A^{L+1}=\\sigma(Z^{L+1})$$\n", "\n", "$$C = \\frac{1}{N_{out}}\\sum_{i=1}^{N_{out}} (\\hat{y_i} - y_i)^2$$\n", "\n", "### 1\n", "\n", - "$`\n", - "\\sigma(x)=\\frac{1}{1+e^{-x}}\\\\\n", - "\\Rightarrow \\sigma'(x)=\\frac{-e^{-x}}{-(1+e^{-x})^2}\\\\\n", - "\\Rightarrow \\sigma'(x)=\\frac{1}{1+e^{-x}}(\\frac{1+e^{-x}-1}{1+e^{-x}})\\\\\n", - "\\Rightarrow \\sigma'(x)=\\sigma(\\frac{1+e^{-x}}{1+e^{-x}}-\\frac{1}{1+e^{-x}})\\\\\n", - "\\Rightarrow \\sigma'(x)=\\sigma(1-\\sigma)\n", - "`$\n", + "$$\n", + "\\begin{align*}\n", + "\\sigma(x)&=\\frac{1}{1+e^{-x}} \\\\\n", + "\\Rightarrow \\sigma'(x)&=\\frac{-e^{-x}}{-(1+e^{-x})^2} \\\\\n", + "\\Rightarrow \\sigma'(x)&=\\frac{1}{1+e^{-x}}(\\frac{1+e^{-x}-1}{1+e^{-x}}) \\\\\n", + "\\Rightarrow \\sigma'(x)&=\\sigma(\\frac{1+e^{-x}}{1+e^{-x}}-\\frac{1}{1+e^{-x}}) \\\\\n", + "\\Rightarrow \\sigma'(x)&=\\sigma(1-\\sigma)\n", + "\\end{align*}\n", + "$$\n", "\n", "### 2\n", "\n",