Autres_Dimensions_1.ipynb

{"nbformat":4,"nbformat_minor":0,"metadata":{"colab":{"name":"Autres_Dimensions_1.ipynb","provenance":[],"collapsed_sections":[]},"kernelspec":{"name":"python3","display_name":"Python 3"}},"cells":[{"cell_type":"markdown","metadata":{"id":"37LNuwykzt3X","colab_type":"text"},"source":["# **Les aventuriers des Axes Perdus**\n","\n","#Epoch 1\n","\n","***Olivier Thöni*** *(novice en IA, et sinon... professeur de mathématiques)*\n","\n","*School of AI - Le Mans*\n","\n","![Les Aventuriers de l'aXe perdu...£](https://drive.google.com/uc?id=1Q4g3DpbT1X49U9m2nr8hPy9vfQXIFCU8)\n","\n","\n","\n","Lien vers le présent NoteBook : https://frama.link/OT_dimensions  \n","Lien de téléchargement du DataSet : https://frama.link/OT_dimension1_DataSet\n","\n","\n","## **Intro**\n","\n","En Machine Learning, le prétraitement des data est un travail incontournable : sans nettoyage, préparation, manipulation préalables des données, point de Machine Learning !\n","\n"]},{"cell_type":"markdown","metadata":{"id":"f2eSWLim7oNR","colab_type":"text"},"source":["![Le difficile parcours des données](https://drive.google.com/uc?id=117kYdf8gKMWnK401q_WLTsf9PkqFH0EJ)"]},{"cell_type":"markdown","metadata":{"id":"ZOittzqg8RR5","colab_type":"text"},"source":["Une des difficultés réside dans le volume de ces données, souvent énorme.\n","\n","Par \"volume\", on entend deux choses : le nombre de dimensions, et la taille.\n","\n","**--> Ne pas confondre \"taille\" et \"dimension\"**\n","\n","![Dimension vs Taille](https://drive.google.com/uc?id=1eB5GYH-qcqvpzmHkSml7aAJY7P4hd9s1)\n","\n","\n","En *Data Science*, chaque variable (chaque \"champ\") de la population étudiée constitue une dimension. Par exemple, s'il s'agit d'étudier les performances de décathloniens, le résultat à chacune des 10 épreuves constituera une dimension.\n","\n","Mais visualiser des données en 10 dimensions nous est impossible. C'est pourquoi on commence par réduire le nombre de dimensions.\n","\n","C'est la raison d'être, par exemple, de :\n","* la regression linéaire, polynomiale ou logistique, \n","* l'Analyse en Composantes Principales (\"ACP\" in french in ze text, ou \"PCA in english), et sa version simplifiée utilisée dans la compression d'image : la SVD (Singular Value Decomposition)\n","* la clusterisation (segmentation) via l'algorithme *K-means*, \n","* Le réseau de convolution (CNN) pour la classification d'images (en passant, dans un réseau de neurones convolutionnel, chaque \"tuilage\" revient à faire diminuer la **taille** des matrices en présence)\n","* etc.\n","\n"]},{"cell_type":"markdown","metadata":{"id":"RHFOf5DvAS5A","colab_type":"text"},"source":["Mais qui dit diminuer le nombre de dimensions dit forcément perdre de l'information !\n","Y aurait-il un moyen de récupérer l'information perdue ?\n","\n","*Taddammmm...*\n","\n","Ben oui, ça s'appelle **\"Auto-Encoder\"**\n","\n","En remontant aux causes, sur les couches précédentes, qui ont pu activer tel ou tel neurone d'une couche donnée (en maths, ça s'appelle \"Naive Bayes\") : en analysant les corrélations entre une couche et la couche précédente, on arrive à reconstituer de l'information... via un réseau de neurone, of course...\n","\n","Voyons tout ça !\n","... et sans trop de maths, si possible...\n","\n","\n","![…](https://drive.google.com/uc?id=1Vucwh4V_HPPCsYNagVqqZNeylZhvev1L)\n","\n","\n","---\n","\n","\n"]},{"cell_type":"markdown","metadata":{"id":"EigAUqpVFdPl","colab_type":"text"},"source":["# **1. Diminuer la dimensionnalité**\n","\n","## **1.1. la régression linéaire**\n","Pour démarrer simplement : on passe de deux à une dimension...\n","C'est simpliste, mais c'est en comprenant ce qui se passe avec des données de petite dimensionnalité et de petite taille que l'on pourra faire confiance aux boîtes noires que sont les modules utiles en ML.\n","\n","**Par exemple**, pour une agence immobilière, il peut être intéressant de disposer d'un outil permettant, connaissant la surface d'un appartement, de prédire le prix du marché, en fonction de la base de données déjà existante.\n","\n","En Data Science, il faut toujours commencer par visualiser les données, quand c'est possible.\n","Ici, à partir des données existantes, on dresse le **\"nuage de points** : en abscisses, la surface en m², en ordonnées, le prix en €.\n","Comme ce nuage est plutôt rectiligne, il est sensé de faire une **\"regression *linéaire* du prix en fonction de la surface\"**, c'est-à-dire que l'on va chercher à approcher le nuage par une droite, qui le représentera \"au mieux\". \n","\n","![\"au mieux\"…](https://poissonisfish.files.wordpress.com/2017/01/q7hip.gif?w=1108)\n","\n","Cette idée de \"au mieux\" étant très subjective, précisons : on va chercher à rendre la plus petite possible la somme totale des distances verticales entre chaque point du nuage et le point de la droite ayant la même abscisse.(c'est-à-dire, dans notre exemple, pour une surface donnée, entre le prix réel et le prix du marché)...\n","\n","\n","\n","![Régression linéaire de y en x](https://drive.google.com/uc?id=1nf5PUc_0vTparoXruDoxNz8xL6SVVBk7)\n","\n","\n","Bon, certes, il existe une formule toute simple qui permet de calculer l'équation de cette droite : y = a.x + b\n","\n","![droite de régression](https://drive.google.com/uc?id=1-r5i_0lK36WLOJjeyF5vh2o7Sd8qCf8K)\n","\n","(Les grandeurs en présence : **moyenne**, **variance** (capacité qu'a une variable à s'écarter de sa moyenne), **covariance** (effet qu'a la variation d'une variable sur l'autre) sont essentielles à la compréhension derrière le ML, ce sont sur elles que s'appuient les métriques utilisées pour mesurer les écarts entre le réel, le théorique, et le prédit)\n","\n","Pour info, le calcul avec cette formule exacte donne approximativement a=25.6 et b=-68.6, si bien que pour un appartement de 250m², le loyer serait d'environ 6383€.\n","\n","\n","* *voir ici pour le développement mathématique complet : https://frama.link/OT_RegressionLineaire*\n","\n","* *voir notamment la fin, révélatrice d'une erreur fréquente en régression linéaire : si on cherche à inférer sur la surface connaissant le prix, il ne faut pas calculer l'erreur \"verticale\" mais l'erreur \"horizontale\", qu'on fasse turbiner les maths ou le Machine Learning...*\n","\n","\n","... Mais se contenter d'une bête formule mathématique ne serait pas rigolo, on peut entraîner un modèle *from scratch* pour approcher, par une petite descente de grandient, l'équation de cette droite..."]},{"cell_type":"markdown","metadata":{"id":"kzk3B8n3nlVv","colab_type":"text"},"source":["### **Code Python**\n","sources : \n","* *Linear Regression with one or more variables* - Luca Zammataro\n","\n","Traduit de l'anglais par [DeepL](https://www.deepl.com/fr/translator)\n","\n","\n","**1. import des bibliothèques à utiliser, import des données brutes, premiers traitement et visualisation**\n","\n","Tout d'abord, nous importons tous les paquets nécessaires pour le code Python  : **NumPy**, **Pandas**, et **matplotlib**. Ces paquets appartiennent à **SciPy.org**, qui est un écosystème de logiciels libres pour les mathématiques, les sciences et l'ingénierie basé sur Python.\n","Numpy est nécessaire pour les calculs d'Algèbre Linéaire.\n","Pandas est une bibliothèque open-source qui fournit des structures de données et des outils d'analyse de données de haute performance pour Python. Une DataFrame de Pandas est une structure de données tabulaires bidimensionnelle à taille variable et potentiellement hétérogène. Elle se compose de trois composants principaux, les données, les lignes et les colonnes, avec des axes étiquetés (lignes et colonnes). \n","Matplotlib est fondamental pour la création de tous les tracés.\n","\n"]},{"cell_type":"code","metadata":{"id":"wUPopxY7kcME","colab_type":"code","colab":{}},"source":["import numpy as np \n","import matplotlib.pyplot as plt \n","import pandas as pd \n","\n","from __future__ import division\n"],"execution_count":0,"outputs":[]},{"cell_type":"markdown","metadata":{"id":"E5OiuX4b724d","colab_type":"text"},"source":["**import des données** :\n","\n","Le code suivant utilise également le package **PyDrive** permettant d'accéder dans **Colaboratory**, à un fichier présent dans le Drive de l'utilisateur : ici, le fichier \"APPART.csv\", qui contient les données.\n","La première utilisation de la cellule demandera une procédure d'authentification : un lien à cliquer, qui envoie vers un code que l'on copie et colle dans la box prévue."]},{"cell_type":"code","metadata":{"id":"gy_bTTczzGJF","colab_type":"code","outputId":"9f77fdf6-0d40-45ab-a4ef-752427b0db23","executionInfo":{"status":"ok","timestamp":1581670612681,"user_tz":-60,"elapsed":37299,"user":{"displayName":"olivier thoni","photoUrl":"","userId":"04225242464974929302"}},"colab":{"base_uri":"https://localhost:8080/","height":917}},"source":["# Code to read csv file into Colaboratory:\n","!pip install -U -q PyDrive\n","from pydrive.auth import GoogleAuth\n","from pydrive.drive import GoogleDrive\n","from google.colab import auth\n","from oauth2client.client import GoogleCredentials\n","\n","# Authenticate and create the PyDrive client.\n","auth.authenticate_user()\n","gauth = GoogleAuth()\n","gauth.credentials = GoogleCredentials.get_application_default()\n","drive = GoogleDrive(gauth)\n","\n","link = 'https://drive.google.com/open?id=1uhgOwfDnduwHrJPt8qo0fAuutufQqvjd' # The shareable link of the data  csv file\n","fluff, id = link.split('=')# récupérer l'id seulement (ce qu'il y a derrière le sigen \"=\")\n","\n","downloaded = drive.CreateFile({'id':id}) \n","downloaded.GetContentFile('APPART.csv')  \n","df = pd.read_csv('APPART.csv', sep =\";\") # Fabrication du DataFrame Pandas, le séaparateur du fichier csv étant \";\"\n","# Dataset is now stored in a Pandas Dataframe\n","\n","\n","df"],"execution_count":0,"outputs":[{"output_type":"execute_result","data":{"text/html":["<div>\n","<style scoped>\n","    .dataframe tbody tr th:only-of-type {\n","        vertical-align: middle;\n","    }\n","\n","    .dataframe tbody tr th {\n","        vertical-align: top;\n","    }\n","\n","    .dataframe thead th {\n","        text-align: right;\n","    }\n","</style>\n","<table border=\"1\" class=\"dataframe\">\n","  <thead>\n","    <tr style=\"text-align: right;\">\n","      <th></th>\n","      <th>num_appart</th>\n","      <th>Surface</th>\n","      <th>Loyer</th>\n","      <th>Localisation</th>\n","    </tr>\n","  </thead>\n","  <tbody>\n","    <tr>\n","      <th>0</th>\n","      <td>a01</td>\n","      <td>28</td>\n","      <td>650</td>\n","      <td>censier 1</td>\n","    </tr>\n","    <tr>\n","      <th>1</th>\n","      <td>a02</td>\n","      <td>50</td>\n","      <td>1400</td>\n","      <td>contrescarpe</td>\n","    </tr>\n","    <tr>\n","      <th>2</th>\n","      <td>a03</td>\n","      <td>106</td>\n","      <td>3250</td>\n","      <td>saint simon</td>\n","    </tr>\n","    <tr>\n","      <th>3</th>\n","      <td>a04</td>\n","      <td>196</td>\n","      <td>4000</td>\n","      <td>rapp</td>\n","    </tr>\n","    <tr>\n","      <th>4</th>\n","      <td>a05</td>\n","      <td>55</td>\n","      <td>1340</td>\n","      <td>saint andre des art</td>\n","    </tr>\n","    <tr>\n","      <th>5</th>\n","      <td>a06</td>\n","      <td>190</td>\n","      <td>3950</td>\n","      <td>pres quais du 5eme</td>\n","    </tr>\n","    <tr>\n","      <th>6</th>\n","      <td>a07</td>\n","      <td>110</td>\n","      <td>2500</td>\n","      <td>gobelins 1</td>\n","    </tr>\n","    <tr>\n","      <th>7</th>\n","      <td>a08</td>\n","      <td>60</td>\n","      <td>1600</td>\n","      <td>gobelins 2</td>\n","    </tr>\n","    <tr>\n","      <th>8</th>\n","      <td>a09</td>\n","      <td>48</td>\n","      <td>1250</td>\n","      <td>censier</td>\n","    </tr>\n","    <tr>\n","      <th>9</th>\n","      <td>a10</td>\n","      <td>35</td>\n","      <td>1250</td>\n","      <td>pantheon 1</td>\n","    </tr>\n","    <tr>\n","      <th>10</th>\n","      <td>a11</td>\n","      <td>86</td>\n","      <td>1750</td>\n","      <td>rue madame</td>\n","    </tr>\n","    <tr>\n","      <th>11</th>\n","      <td>a12</td>\n","      <td>65</td>\n","      <td>1500</td>\n","      <td>rue de seine</td>\n","    </tr>\n","    <tr>\n","      <th>12</th>\n","      <td>a13</td>\n","      <td>32</td>\n","      <td>775</td>\n","      <td>pantheon 2</td>\n","    </tr>\n","    <tr>\n","      <th>13</th>\n","      <td>a14</td>\n","      <td>52</td>\n","      <td>1225</td>\n","      <td>sevres babylone</td>\n","    </tr>\n","    <tr>\n","      <th>14</th>\n","      <td>a15</td>\n","      <td>40</td>\n","      <td>1000</td>\n","      <td>montparnasse 1</td>\n","    </tr>\n","    <tr>\n","      <th>15</th>\n","      <td>a16</td>\n","      <td>260</td>\n","      <td>7500</td>\n","      <td>rue d'assas</td>\n","    </tr>\n","    <tr>\n","      <th>16</th>\n","      <td>a17</td>\n","      <td>70</td>\n","      <td>1625</td>\n","      <td>boulevard saint ger</td>\n","    </tr>\n","    <tr>\n","      <th>17</th>\n","      <td>a18</td>\n","      <td>117</td>\n","      <td>4750</td>\n","      <td>ile saint louis</td>\n","    </tr>\n","    <tr>\n","      <th>18</th>\n","      <td>a19</td>\n","      <td>90</td>\n","      <td>1890</td>\n","      <td>jussieu</td>\n","    </tr>\n","    <tr>\n","      <th>19</th>\n","      <td>a20</td>\n","      <td>30</td>\n","      <td>390</td>\n","      <td>quartier latin</td>\n","    </tr>\n","    <tr>\n","      <th>20</th>\n","      <td>a21</td>\n","      <td>105</td>\n","      <td>1875</td>\n","      <td>montparnasse 2</td>\n","    </tr>\n","    <tr>\n","      <th>21</th>\n","      <td>a22</td>\n","      <td>52</td>\n","      <td>1000</td>\n","      <td>rue mazarine</td>\n","    </tr>\n","    <tr>\n","      <th>22</th>\n","      <td>a23</td>\n","      <td>80</td>\n","      <td>1350</td>\n","      <td>censier 2</td>\n","    </tr>\n","    <tr>\n","      <th>23</th>\n","      <td>a24</td>\n","      <td>60</td>\n","      <td>1475</td>\n","      <td>assas luxembourg</td>\n","    </tr>\n","    <tr>\n","      <th>24</th>\n","      <td>a25</td>\n","      <td>140</td>\n","      <td>4950</td>\n","      <td>jardins observatoir</td>\n","    </tr>\n","    <tr>\n","      <th>25</th>\n","      <td>a26</td>\n","      <td>20</td>\n","      <td>425</td>\n","      <td>rue de savoie</td>\n","    </tr>\n","    <tr>\n","      <th>26</th>\n","      <td>a27</td>\n","      <td>100</td>\n","      <td>2475</td>\n","      <td>pres luxembourg</td>\n","    </tr>\n","    <tr>\n","      <th>27</th>\n","      <td>a28</td>\n","      <td>28</td>\n","      <td>425</td>\n","      <td>metro gobelins</td>\n","    </tr>\n","  </tbody>\n","</table>\n","</div>"],"text/plain":["   num_appart  Surface  Loyer          Localisation\n","0         a01       28    650             censier 1\n","1         a02       50   1400          contrescarpe\n","2         a03      106   3250           saint simon\n","3         a04      196   4000                  rapp\n","4         a05       55   1340   saint andre des art\n","5         a06      190   3950    pres quais du 5eme\n","6         a07      110   2500            gobelins 1\n","7         a08       60   1600            gobelins 2\n","8         a09       48   1250               censier\n","9         a10       35   1250            pantheon 1\n","10        a11       86   1750            rue madame\n","11        a12       65   1500          rue de seine\n","12        a13       32    775            pantheon 2\n","13        a14       52   1225       sevres babylone\n","14        a15       40   1000        montparnasse 1\n","15        a16      260   7500           rue d'assas\n","16        a17       70   1625   boulevard saint ger\n","17        a18      117   4750       ile saint louis\n","18        a19       90   1890               jussieu\n","19        a20       30    390        quartier latin\n","20        a21      105   1875        montparnasse 2\n","21        a22       52   1000          rue mazarine\n","22        a23       80   1350             censier 2\n","23        a24       60   1475      assas luxembourg\n","24        a25      140   4950   jardins observatoir\n","25        a26       20    425         rue de savoie\n","26        a27      100   2475       pres luxembourg\n","27        a28       28    425        metro gobelins"]},"metadata":{"tags":[]},"execution_count":2}]},{"cell_type":"code","metadata":{"id":"XupXOlEBEEti","colab_type":"code","outputId":"062a7216-e807-49e9-b25e-c98e1519ed8e","executionInfo":{"status":"ok","timestamp":1581670612685,"user_tz":-60,"elapsed":37287,"user":{"displayName":"olivier thoni","photoUrl":"","userId":"04225242464974929302"}},"colab":{"base_uri":"https://localhost:8080/","height":35}},"source":["m = len(df) # taille des données\n","m"],"execution_count":0,"outputs":[{"output_type":"execute_result","data":{"text/plain":["28"]},"metadata":{"tags":[]},"execution_count":3}]},{"cell_type":"markdown","metadata":{"id":"PqCIr_MstF6a","colab_type":"text"},"source":["**extraction des deux colonnes utiles**"]},{"cell_type":"code","metadata":{"id":"_Jzi_SJ684ce","colab_type":"code","colab":{}},"source":["# Fabrique le numpy array X, contenant les valeurs de Surface.\n","# et le vecteur de sortie y, contenant les valeurs de Loyer.\n","X = np.array(df.Surface.values)\n","y = np.array(df.Loyer.values)\n","\n"],"execution_count":0,"outputs":[]},{"cell_type":"code","metadata":{"id":"L8p7YodXHvG_","colab_type":"code","outputId":"2ffc39c5-1bf9-4f74-e02a-81f49a489e0a","executionInfo":{"status":"ok","timestamp":1581670612688,"user_tz":-60,"elapsed":37266,"user":{"displayName":"olivier thoni","photoUrl":"","userId":"04225242464974929302"}},"colab":{"base_uri":"https://localhost:8080/","height":71}},"source":["X"],"execution_count":0,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([ 28,  50, 106, 196,  55, 190, 110,  60,  48,  35,  86,  65,  32,\n","        52,  40, 260,  70, 117,  90,  30, 105,  52,  80,  60, 140,  20,\n","       100,  28])"]},"metadata":{"tags":[]},"execution_count":5}]},{"cell_type":"code","metadata":{"id":"871EUPsYHyWr","colab_type":"code","outputId":"9ee11054-0c00-4cf0-b656-9225798be70b","executionInfo":{"status":"ok","timestamp":1581670612689,"user_tz":-60,"elapsed":37250,"user":{"displayName":"olivier thoni","photoUrl":"","userId":"04225242464974929302"}},"colab":{"base_uri":"https://localhost:8080/","height":71}},"source":["y"],"execution_count":0,"outputs":[{"output_type":"execute_result","data":{"text/plain":["array([ 650, 1400, 3250, 4000, 1340, 3950, 2500, 1600, 1250, 1250, 1750,\n","       1500,  775, 1225, 1000, 7500, 1625, 4750, 1890,  390, 1875, 1000,\n","       1350, 1475, 4950,  425, 2475,  425])"]},"metadata":{"tags":[]},"execution_count":6}]},{"cell_type":"markdown","metadata":{"id":"HO3U57Wy-Ylh","colab_type":"text"},"source":["**visualisation du nuage de points**"]},{"cell_type":"code","metadata":{"id":"YpbQs-8d-cyQ","colab_type":"code","outputId":"753ea565-9c45-4e4e-a8be-e63a2c4301d9","executionInfo":{"status":"ok","timestamp":1581670613450,"user_tz":-60,"elapsed":37997,"user":{"displayName":"olivier thoni","photoUrl":"","userId":"04225242464974929302"}},"colab":{"base_uri":"https://localhost:8080/","height":279}},"source":["# Plot the data (plt is a matplotlib object)\n","plt.scatter(X, y,  color='blue')\n","# Put labels\n","plt.xlabel(\"Surface(m²)\")\n","plt.ylabel(\"Loyer(€)\")\n","\n","plt.show()"],"execution_count":0,"outputs":[{"output_type":"display_data","data":{"image/png":"iVBORw0KGgoAAAANSUhEUgAAAYsAAAEGCAYAAACUzrmNAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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ScvS8o+RPVByqn6VI08tNZs0h4V1u9cAQ5IekbSdI2Mgfwz4vKQy2bWEev0XstHOvg/8oKL+QeCd\n6chlG3BaRDwCfIRsRMTtwBayC+4AX2P8o5J6vTNty2xK3HXWrA2lMQ1uiYjzp7CNBcD/jIhzG9cy\n61Y+sjBrQ5EN6HP9RG7Kq2Ip8McNapJ1OR9ZmJlZLh9ZmJlZLoeFmZnlcliYmVkuh4WZmeVyWJiZ\nWa7/Dyd1PuxNvQ8FAAAAAElFTkSuQmCC\n","text/plain":["<Figure size 432x288 with 1 Axes>"]},"metadata":{"tags":[]}}]},{"cell_type":"markdown","metadata":{"id":"QRr-XBt3AUQO","colab_type":"text"},"source":["\n","Le modèle\n","\n"]},{"cell_type":"code","metadata":{"id":"z3klN3VIAC6o","colab_type":"code","colab":{}},"source":["def predict(w, b, X):\n","    \"\"\" A method that runs our linear model\n","    w : np.ndarray (1, num_dimensions)\n","        A weight matrix which will be fit to the data\n","    b : float (1, 1)\n","        The bias unit for our linear model\n","    X : np.ndarray (num_dimensions, num_examples)\n","        The m input examples for which we would like to predict a y value\n","    \"\"\"\n","    return w * X + b\n"," \n"],"execution_count":0,"outputs":[]},{"cell_type":"markdown","metadata":{"id":"xCDAmEqzAZfC","colab_type":"text"},"source":["Le coût"]},{"cell_type":"markdown","metadata":{"id":"j5cGour6QnOV","colab_type":"text"},"source":["**Calcul du coût pour une droite donnée**\n","\n","Le coût, également appelé **\"risque quadratique\"** : c'est la moyenne des carrés des **\"résidus\"**, distances verticales entre chaque point du nuage et la droite supposée approcher le nuage, qui est la représentation graphique de la fonction appelée **\"hypothèse\"**, h(X) =  W.X+b (y prédit par l'hypothèse)\n","\n","Voici par exemple ce que cela donne si l'on prend \n","b  = 1000\n","et W = 15\n","ou bien si l'on prend \n","b  = -68.6\n","et W = 26.5, qui sont les valeurs idéales calculées par les mathématiques"]},{"cell_type":"code","metadata":{"id":"QCM8PKnGSvL0","colab_type":"code","outputId":"0fc958ad-f5a7-4c10-fc57-efc690e294bf","executionInfo":{"status":"ok","timestamp":1581670613453,"user_tz":-60,"elapsed":37977,"user":{"displayName":"olivier thoni","photoUrl":"","userId":"04225242464974929302"}},"colab":{"base_uri":"https://localhost:8080/","height":279}},"source":["# Plot the Hypothesis: set parameters manually\n","b = 1000.0 \n","#b = -68.6\n","w = 15.0\n","#w = 26.5\n","\n","# Plot the data\n","plt.scatter(X, y,  color='blue')\n","# Plot a red line corrensponding to our Hypothesis model.\n","y_pred = predict(w,b,X)\n","plt.plot(X, y_pred, color='red', linewidth=1)\n","# Plot costs\n","for k in range(m):\n","    plt.plot([X[k], X[k]],[y[k],y_pred[k]],'g--', linewidth=1 )\n","# Mean Point\n","plt.scatter(np.mean(X),np.mean(y), color='black')\n","# Put labels\n","plt.xlabel(\"Surface(m²)\")\n","plt.ylabel(\"Loyer(€)\")\n","\n","plt.show()\n"],"execution_count":0,"outputs":[{"output_type":"display_data","data":{"image/png":"iVBORw0KGgoAAAANSUhEUgAAAYsAAAEGCAYAAACUzrmNAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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size 432x288 with 1 Axes>"]},"metadata":{"tags":[]}}]},{"cell_type":"code","metadata":{"id":"wIBhLEFLAdrI","colab_type":"code","colab":{}},"source":["def cost(y, y_pred):\n","    \"\"\"\n","    y : np.ndarray (num_examples, 1)\n","        The ground truth results\n","    y_pred : np.ndarray (num_examples, 1)\n","    \"\"\"\n","    return np.sqrt(((y_pred - y) ** 2).mean())\n"],"execution_count":0,"outputs":[]},{"cell_type":"code","metadata":{"id":"SB9JGCYVYFPN","colab_type":"code","outputId":"e374415e-94ed-41f4-aa83-41a1324296d1","executionInfo":{"status":"ok","timestamp":1581670613455,"user_tz":-60,"elapsed":37956,"user":{"displayName":"olivier thoni","photoUrl":"","userId":"04225242464974929302"}},"colab":{"base_uri":"https://localhost:8080/","height":35}},"source":["loss = cost(y,y_pred)\n","print(\"L'erreur moyenne de l'hypothèse testée est :\",loss, \"€\")\n"],"execution_count":0,"outputs":[{"output_type":"stream","text":["L'erreur moyenne de l'hypothèse testée est : 904.892949469715 €\n"],"name":"stdout"}]},{"cell_type":"markdown","metadata":{"id":"l3EAGnHcefK9","colab_type":"text"},"source":["### **2. Implémentation de la descente de gradient**\n","On veut observer l'effet d'une variation des paramètres sur le coût, en mathématiques, cela s'appelle calculer la dérivée de la fonction de coût ! (pas compliquée ici puisqu'on dérive un carré...)\n","\n","![Formule de la descente de gradient](https://drive.google.com/uc?id=1TaSb_ce-rny-1asU3YNQFDA-6Q6uwKpB)\n","\n","a (alpha) désigne le **\"taux d'apprentissage\"** (*Learning Ratio*), trop lent, on ne convergera pas assez vers la solution optimale, trop rapide, on oscillera autour..."]},{"cell_type":"markdown","metadata":{"id":"5HLptlxMA5Jj","colab_type":"text"},"source":["Le gradient"]},{"cell_type":"code","metadata":{"id":"07n-vVaEA0Ne","colab_type":"code","colab":{}},"source":["def gradient(X, y, y_pred):\n","    \"\"\"\n","    X : np.ndarray (num_dimensions, num_examples)\n","        The m input examples for which we would like to predict a y value\n","    y : np.ndarray (num_examples, 1)\n","        The ground truth results\n","    y_pred : np.ndarray (num_examples, 1)\n","        Our model's predictions\n","    \"\"\"\n","    dw = ((y_pred - y) * X).mean()/cost(y,y_pred)\n","    db = (y_pred - y).mean()/cost(y,y_pred)\n","    return dw, db\n"," \n"],"execution_count":0,"outputs":[]},{"cell_type":"markdown","metadata":{"id":"hWhfzNumBKKI","colab_type":"text"},"source":["L'optimisation par descente de gradient\n"]},{"cell_type":"code","metadata":{"id":"iVV2A7G4BAVW","colab_type":"code","colab":{}},"source":["def gradient_descent(X, y, w, b, epochs, learning_ratio):\n","    \"\"\"\n","    X : np.ndarray (num_dimensions, num_examples)\n","        The m input examples for which we would like to predict a y value\n","    y : np.ndarray (num_examples, 1)\n","        The ground truth results\n","    w : np.ndarray (1, num_dimensions)\n","        A weight matrix which will be fit to the data\n","    b : float\n","        The bias unit for our linear model\n","    epochs : int\n","        The number of times our model iterates over the entire training set\n","    learning_rate : float\n","        The rate at which our model tends to adjust it's parameters\n","    \"\"\"\n","\n","    for i in range(epochs):\n","        # predict the value of y with our current model\n","        y_pred = predict(w, b, X)\n","        \n","        # compute the gradient with respect to w and b\n","        dw, db = gradient(X, y, y_pred)\n","        \n","        # adjust w and b as to minimize the cost function\n","        w = w - learning_ratio * dw\n","        b = b - learning_ratio * db\n","\n","        # cost\n","        l = cost(y,y_pred)\n","        loss_history.append(l)\n","        w_history.append(w)\n","        b_history.append(b)\n","        absc.append(i)\n","\n","    return w, b\n"],"execution_count":0,"outputs":[]},{"cell_type":"markdown","metadata":{"id":"omtTP1_pAtEY","colab_type":"text"},"source":["Initialisation des paramètres"]},{"cell_type":"code","metadata":{"id":"Zu782EHMAvmm","colab_type":"code","colab":{}},"source":["def initialize_parameters(n):\n","    \"\"\"\n","    n : int\n","        The dimensions of the data\n","    \"\"\"\n","    return np.zeros((1, n)), -60.0\n"," \n"],"execution_count":0,"outputs":[]},{"cell_type":"markdown","metadata":{"id":"6-55efG0Bn3Z","colab_type":"text"},"source":["Exécution\n","\n","![regression par descente de gradient…](https://miro.medium.com/max/2000/1*XGaA7KWUlhWZLIezYEBIHA.gif)"]},{"cell_type":"code","metadata":{"id":"WxAfM8TkBr1k","colab_type":"code","outputId":"ef4f04bb-fabf-42d0-ecc1-31539cbbb681","executionInfo":{"status":"ok","timestamp":1581670613802,"user_tz":-60,"elapsed":38270,"user":{"displayName":"olivier thoni","photoUrl":"","userId":"04225242464974929302"}},"colab":{"base_uri":"https://localhost:8080/","height":406}},"source":["# initialize w and b\n","w, b = initialize_parameters(m)\n","\n","#initialize histories\n","loss_history = []\n","w_history = []\n","b_history = []\n","absc = []\n","        \n","# run gradient descent in order to find optimal values for w and b\n","epochs=10\n","learning_ratio=0.1\n","w, b = gradient_descent(X, y, w, b, epochs, learning_ratio)\n","\n","# graphics\n","plt.subplot(1,3,1)\n","plt.plot(range(epochs),loss_history,'r')\n","plt.xlabel(\"Iterations\")\n","plt.ylabel(\"loss\")\n","\n","plt.subplot(1,3,2)\n","plt.plot(range(epochs), np.mean(w_history, axis = 1))\n","plt.xlabel(\"Iterations\")\n","plt.ylabel(\"poids\")\n","\n","plt.subplot(1,3,3)\n","plt.plot(range(epochs), b_history)\n","plt.xlabel(\"Iterations\")\n","plt.ylabel(\"biais\")\n","        \n","# calculate the final cost function\n","c = cost(y, predict(w, b, X))\n","    \n","# print out the model details\n","print('w:', w)\n","print('b:', b)\n","print('cost:', c)\n"],"execution_count":0,"outputs":[{"output_type":"stream","text":["w: [[26.00451204 26.00451204 26.00451204 26.00451204 26.00451204 26.00451204\n","  26.00451204 26.00451204 26.00451204 26.00451204 26.00451204 26.00451204\n","  26.00451204 26.00451204 26.00451204 26.00451204 26.00451204 26.00451204\n","  26.00451204 26.00451204 26.00451204 26.00451204 26.00451204 26.00451204\n","  26.00451204 26.00451204 26.00451204 26.00451204]]\n","b: -59.822684881798416\n","cost: 594.392338750875\n"],"name":"stdout"},{"output_type":"display_data","data":{"image/png":"iVBORw0KGgoAAAANSUhEUgAAAYsAAAEHCAYAAABfkmooAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4xLjMsIGh0\ndHA6Ly9tYXRwbG90bGliLm9yZy+AADFEAAAgAElEQVR4nOydeXiU5dX/P2dmkhAgIWyBkARDLQgY\nkhTilqoFZBGKWgtttVaRsAguv4Ig1b61pW9rXzektljL5sZbsW4UaiMaVF5bW5REI2CUAkINJBIi\nELLPTHL//nieCQNMQpaZeSYz9+e65ppn7meZM5nMnLnPfc73iFIKjUaj0Whaw2a1ARqNRqMJfbSz\n0Gg0Gs050c5Co9FoNOdEOwuNRqPRnBPtLDQajUZzTrSz0Gg0Gs05cVhtQCDo16+fSktLs9oMDVBY\nWFihlOrvr+vp99Y/lJaWUlFRgcNhfAUkJyfTq1cvmpqa+OKLL6ipqUFESE1NJS4u7qzzCwsLG4DP\nzIcJwAmlVJaIRAFrgdEY3y/PKaX+51z26Pc1NGjt8xqWziItLY2CggKrzdAAIvIff15Pv7f+Ydmy\nZfTs2ZMlS5acNv7EE09QUFDA008/TXl5OVOmTOGDDz7AZjs9CCEiu5VS2eb2cqDS3PU9IEYpNUpE\nugPFIrJBKXWwNXv0+xoatPZ51WEojUbTTHFxMePHjwcgMTGRhISEVr/ERUSA7wMbzCEF9BARBxAL\nOIGTATVaExS0s9BoIpSVK1eSkZFBbm4ux48fByAzM5PNmzfjdrs5cOAAhYWFlJSUtHaZK4AjSqm9\n5uOXgRqgDPgCeFQpdczXiSIyT0QKRKTg6NGjfntdmsCgnYVGE6ZMmDCB9PT0s26bNm1iwYIF7N+/\nn6KiIpKSkli8eDEAubm5pKSkkJ2dzcKFC8nJycFut7f2NDdyalYBcDHQCAwChgCLReRrvk5USq1W\nSmUrpbL79/fbspYmQITlmoVGo4GtW7e26bi5c+cybdo0ABwOBytWrGjel5OTw7Bhw3yeZ4aavguM\n8Rr+IbBFKeUCykXkPSAb+LwDL0ETQuiZhUYTgZSVlTVvb9y4kfT0dABqa2upqakBID8/H4fDwciR\nI1u6zATgM6XUIa+xL4DxACLSA7iUU1lTmi6MnlloNBHI0qVLKSoqQkRIS0tj1apVAJSXlzN58mRs\nNhvJycmsX7+++Zw5c+Ywf/58srOzPUM3cHoICuAJ4GkR+QQQ4Gml1M6AvyBNwNHOQqOJQLydgDdp\naWns2bPH5761a9ee9lgpdeuZxyilqjHSZzVhRmSEoZ54Al54wWorNCHOqkX3sWrRT602Q+Nn9pVX\n89anR6w2o8sTGc5i3Tpo4ZeURuOhrkcC5fEx/HHRfVabovEjf3hnH/PWF/L50WqrTenSRIazSE2F\n1nPFNRrq7C4aRXE8vgcr77nHanM0fuLQiToamxTL8/9ttSldmshxFocOnfs4TcTy1xeepUHcJNX3\nQAEN3fqw9qH/ttosjR8oq6zDYRP+trOM3Ycrz32CxieR4SxSUuD4cTBTAjWaM/lPwS4AHM56+p6s\no8bmpP5kN97b+kaHrre7YAdP3f0b1t39K3YX7PCnqZp20NSk+LKynhsuTiWhexQPv+F78V5zbiLD\nWaSmGvc6FKVpgejGGAAaVQO3rXiA5CobFVG1fLJ1J2XtnJX+8Zc/4/82vsMX8U5K4ht5Y9NW1i36\nJQf3FgfCdE0rVFQ34GpUXDAgjtvHns+7/z7Kv/Z/ZbVZXZLISJ31OItDh2D4cGtt0YQkdukGNOK0\n1wMw+7H7WXfPQ5T0qCFvxf8ye/m9bbrOH+++l5M946l31DH4pIMm5eZkT6Gkl2Ljc5uJq36Bby+e\nR1JKSptte2LJErrRn5OxiqgmG7FORZOrGkdKDLOW/KwjL5eDe4t57ZlnSUjsy49+vLRD1+gKlFYa\n72dSr1guH9qPp/5xkIff+IxXF+RgaCBq2kpkOAvPB1PPLDQtYYsCGkka/vXmoak/volNT2ygJK6G\ntYuWMWfFshZPLzt0iLzHnqU8LpZYBf2r6sh97EHACEm9/6fXORanONTLxot//BOxNVXMW/HrFq+3\n8t57iHX1oaq7cKJnT1B19G3sjtumONS9DiV2qHKz4v4H6em0YXPWUxd9kul3LTzNEb2y9o8cKf6c\nGNUD7N1wRdmodzRSbXPijoql4ngtD/ziV3RrchDTZCeqUbA3NiGNbpRy4aKexpgmLrtmKmO+Obaz\nf+WgU3aiDoCkhG50i7Lz4wlDue/VXWz9tJyJIwdYbF3XIjKcRXKyca8XuTUt0GS3E6McTJ8zv3ks\nKSWF5MxEnLu/pCy+gScX3cuCFQ+ede4ra/9Ixb4qyuJd9Hd1xx5TxfzHTh2Xnn0R6dkX8d7WN/gs\nbwdf9WzkeC8HK//rYRy1x5m/wugN9Ief3Ue32p5Ud4/iWLce0K2B3u5YBldFUxtdyZ3/swyA559Y\nQfW+r7Db46iLEY50a8Ad2wT05LnVzxLviqbJpqi1u6kRJ8R3BxSoOnqoaLo3OhjY0A2b24USG012\nO2674LI3URntpl5cNIkyre8GQNHLf++SzuKw6SySE2IB+N6YFNa8+zmPvPEZ44cnYrfp2UVbiQxn\nERMDAwbomYWmRVx2Rbemsz8O19wwkyeX/RfdmnpQGR/Hyp8s4c6HHm3ev/KeJbi69aMypp7Umm5c\n8oNxpGdf5PM5vjlhMt+cMJm/vvAsRz84THmci/peMTzx04dRAl85YlDxLuIb7aRWd6NOKrjzkWVn\nXeeHdyw67fHugh28+/zLdCMeZ0w0VVFuopSNOFcUfVw2aGygQWoYNPoCvvOj2ef8WxzcW0ze/z4H\nVYooFYNdoqm1nzjneaFIWWU9sVF2esVGAeCw27h70jDufP4jNhUd5ruj2x4OjHQiw1mAEYrSzkLT\nAg32JmIafed7LFj2AE8suYcTPeJxRvfj6Ud/zawlP2PNol9QGd8LwU3yyUZmr2jbusY1N8yEG+DP\nq35P9b9rONLTSZSyk1IbS33TV9yxfFm7bPfMXPxF2tCR3P7Ls2dQXZGyyjqSErqdtj4xNT2J9OT9\nPJb/b6ZlDCLaERl5Pp0lcpxFairs22e1FZoQpc7mpqczusX9dzz6CKsX/Yyy+Caij8ey7p6HONxL\nSHBH43AdZe6KR1s8tyV+cNtdAM3ZVu1Z9Na0jdIT9QzqFXvamM0m3DN5ODOf+oANH3zBzJw0a4zr\nYkSOS9VV3JoW+POq3+MUNzZ3Y6vHzVvxa1JOGim1JT3qGFTfg/PTB54WluoISSkp2lEEiNITdQxK\n6HbW+JVD+3Hp1/rw+7f3UdPgtsCyrkfkOIuUFKishKoqqy3RhBgVe80e9U2ucx47e8X9pJ50kFIp\nzHvwHiOkpAlJnO4mjlY3kHTGzAJARFh69XAqqht4+r0DFljX9QiYsxCRVBF5R0SKReQTEfmxOb5M\nRA6LSJF5m+p1zn0isk9E9ojIZK/xq82xfSLStsDwmXjXWmg0XkQ1Gr883TS06fjZj/2MOSt+EUiT\nNH7gyMl6lMLnzAJg9ODeTBw5gFX/9znHa5xBtq7rEciZhRtYrJQaidEt6w4R8bTcWqGUyjJveQDm\nvhuAC4GrgT+IiF1E7BgNVaYAI4Ebva7TdnQVt6YF7BjV2y6H/sIIJ0o9NRY+ZhYelky6gGqnmz/+\n3/5gmdVlCZizUEqVKaU+NLergE+B5FZOuQ54QSnVoJQ6AOzDaP5+MbBPKfW5UsoJvGAe2z50YZ6m\nJexGWuUFl445x4GarkSZWb09KKFlZ3HBwDiu/0Yyz/zzIF+ax2t8E5Q1CxFJA74BvG8O3SkiO0Xk\nKRHpbY4lA97f5IfMsZbGz3yOeSJSICIFR48ePduIQYNARIeh/EhJSQnjxo1j5MiRXHjhhTz++OMA\nLFu2jOTkZLKysgBGeocavfFLeNEPNDrsdFNRTJx+g1UmaAJAaaUxs2gpDOVh0YRhNCnF797eGwyz\nuiwBT50VkZ7AK8BCpdRJEXkS+BWgzPvlQG5nn0cptRpYDZCdna3OOiA6Whfm+RmHw8Hy5csZPXo0\nVVVVjBkzhokTJwKwaNEilixZgogUe0KN3niFFydi/ADYISKblVJBV9tz2RXdGiMnizxSKDtRT6/Y\nKLpHt/7epvbpzk2XnMf67f9h7hVfY0i/HrgbmyirrKf0RB2HT9Rx+HgdpZV1HDpuPB4zuDcPTs+I\nqArwgH5CRCQKw1H8SSn1KoBS6ojX/jXAa+bDw0Cq1+kp5hitjLcPnT7rV5KSkkhKSgIgLi6OESNG\ncPhwm9+a5vAigIh4wotBdxYNtiZiG+3BflpNgDHSZlsOQXlzx7iv82JBCT9csx0BvjxZT9MZPzn7\n9YwmOSGW5IRYXio8RO8e0fx06gj/Gx6iBMxZiFEyuQ74VCn1mNd4klKqzHx4PbDb3N4MPC8ijwGD\ngKHAB4AAQ0VkCIaTuAH4YYeMSk2Fzz7r0Kma1jl48CAfffQRl1xyCe+99x4rV67kueeeA0gTkd5K\nqeNnnOIrvHiJr2uLyDxgHsDgwYP9bnudzUVcQ+RkkYMRKlyzZg39+/cH4De/+Q1Tp07F6XRy2223\nUVBQgM1m4/HHH2fs2LG+LhErItsxxKPcwO1KqQ/Mz/3jwFSgFrjVs3YZbEor6xnUq/UQlIf+cTH8\n17dH8NePSxlkOoTkhFiSe8c2P+4WdeoHxc837Wb1u59zwYA4po+JjBqZQM4svgncDOwSkSJz7Kc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DLPIXly6623cuedd562yPrnP/+5eXvx4sU89thjrRVrjFNKVXTY6HMQDlIfZyYg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\n2hKND1z2JqKb9BdKuKGlP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WrHDFLUlxn7xZwvAI+k+/uAO9NVUTcCJ0unq8zMbA404shi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\nszAzs5rcWZiZWU3uLMzMrKb/A2fz0/cM17oAAAAAAElFTkSuQmCC\n","text/plain":["<Figure size 432x288 with 3 Axes>"]},"metadata":{"tags":[]}}]},{"cell_type":"code","metadata":{"id":"ESZ8kRnia5In","colab_type":"code","outputId":"dc2f0cbe-9910-4553-fce0-49a484c27b58","executionInfo":{"status":"ok","timestamp":1581670613804,"user_tz":-60,"elapsed":38256,"user":{"displayName":"olivier thoni","photoUrl":"","userId":"04225242464974929302"}},"colab":{"base_uri":"https://localhost:8080/","height":35}},"source":["epochs"],"execution_count":0,"outputs":[{"output_type":"execute_result","data":{"text/plain":["10"]},"metadata":{"tags":[]},"execution_count":16}]},{"cell_type":"markdown","metadata":{"id":"5sZb5MhQjE7R","colab_type":"text"},"source":["**Affichage de l'hypothèse finale**"]},{"cell_type":"code","metadata":{"id":"r3zaBQyEjLJ0","colab_type":"code","outputId":"39d46878-f9c4-4bf0-a98a-12f640745c8b","executionInfo":{"status":"ok","timestamp":1581670614604,"user_tz":-60,"elapsed":39041,"user":{"displayName":"olivier thoni","photoUrl":"","userId":"04225242464974929302"}},"colab":{"base_uri":"https://localhost:8080/","height":387}},"source":["\n","# Plot training set\n","plt.scatter(X, y ,  color='blue')\n","\n","# Plot Hypothesis (theta were calculated with the Gradient Descent)\n","\n","y_pred =w*X+b\n","print(y_pred)\n","\n","plt.plot(X, y_pred.T, color='red', linewidth=1)\n","# Plot costs\n","#for k in range(m):\n","#   plt.plot([X[k], X[k]],[y[k],y_pred[k]],'g--', linewidth=1 )\n","\n","# Mean Point\n","plt.scatter(np.mean(X),np.mean(y), color='black')\n","\n","# Put labels\n","plt.xlabel(\"Surface(m²)\")\n","plt.ylabel(\"Loyer(€)\")\n","\n","plt.show()\n"],"execution_count":0,"outputs":[{"output_type":"stream","text":["[[ 668.30365227 1240.40291718 2696.65559149 5037.0616752  1370.42547739\n","  4881.03460296 2800.67363966 1500.44803759 1188.3938931   850.33523656\n","  2176.56535067 1630.4705978   772.32170044 1292.41194126  980.35779677\n","  6701.35044584 1760.49315801 2982.70522394 2280.58339883  720.31267636\n","  2670.65107945 1292.41194126 2020.53827842 1500.44803759 3580.80900089\n","   460.26755594 2540.62851924  668.30365227]]\n"],"name":"stdout"},{"output_type":"display_data","data":{"image/png":"iVBORw0KGgoAAAANSUhEUgAAAYsAAAEGCAYAAACUzrmNAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\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size 432x288 with 1 Axes>"]},"metadata":{"tags":[]}}]},{"cell_type":"markdown","metadata":{"id":"4U1AMWGaGAhW","colab_type":"text"},"source":["## **1.2. L' \"Analyse en Composantes Principales\" (ACP)**\n","\n","\n","\n","### **1.2.a/ la classification des iris**\n","\n","https://frama.link/OT_dimension2  \n"]},{"cell_type":"markdown","metadata":{"id":"gfBrGD5QGm4-","colab_type":"text"},"source":["### **1.2.b/ Les décathloniens**\n","\n","C'est en R, mais ça devrait calmer un peu ceux qui pensent que R, c'est pas bien... C'est super bien fait, et facile à comprendre : (et si quelqu'un sait faire la même chose en Python, je suis preneur...)\n","\n","https://www.kaggle.com/capedcrusader/pca-on-a-random-dataset/notebook \n","\n","\n","### **1.2.c/ Un usage d'une ACP simplifiée : la SVD pour la compression d'image**\n","\n","https://frama.link/OT_dimension3\n","\n","\n","\n","\n","\n"]},{"cell_type":"markdown","metadata":{"id":"_PSHUhZKIb6x","colab_type":"text"},"source":["## **1.3. \"Réseau de Neurones Convolutionnel\" (CNN)**\n","\n","https://frama.link/OT_dimension4 \n"]},{"cell_type":"markdown","metadata":{"id":"Qr4MIOqNJFre","colab_type":"text"},"source":["# **2. Récupérer l'information perdue : \"AutoEncoder\"**\n","\n","https://frama.link/OT_dimension5 "]}]}