From 9b9bdcb2e7b536fb6d3a513d47d9146bc950c2e7 Mon Sep 17 00:00:00 2001
From: Sucio <esteban.cosserat@gmail.com>
Date: Fri, 10 Nov 2023 08:28:33 +0100
Subject: [PATCH] Update Rapport.ipynb
---
Rapport.ipynb | 19 ++++++++++---------
1 file changed, 10 insertions(+), 9 deletions(-)
diff --git a/Rapport.ipynb b/Rapport.ipynb
index 876c42d..d61c1ab 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",
--
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