01-numpy.ipynb 94.1 KB
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{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
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    "# Numpy\n",
    "\n",
    "## Qu'est-ce?"
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   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
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    "Les listes python sont extensibles à volonté et peuvent contenir n'importe quoi. On les créer avec les crochets:`uneListe=[]`. Dans un cadre scientifique, pour des questions de performances, on travaille essentiellement avec `numpy` qui crée des tenseurs, c.à.d:\n",
    "\n",
    "* des tableaux multi-dimensionnel \n",
    "* de taille fixée\n",
    "* qui ne peuvent contenir qu'un seul type d'objet : `int` ou `float` (numpy gère d'autre types, mais on ne les utisera pas)\n",
    "* rapidement manipulable dans leur ensemble\n",
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    "\n",
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    "L'ordre d'un tenseur, c'est son nombre d'indices:\n",
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    "\n",
    "* tenseur d'ordre 1 = vecteur, ex: pour stocker du son\n",
    "* tenseur d'ordre 2 = matrice, ex: pour stocker une image\n",
    "* tenseur d'ordre 3 = volume, ex: pour stocker une vidéo\n",
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    "* tenseur d'ordre 4 = très utile quand on fait du deep-learning\n"
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   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 32,
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   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On précise l'affichage des tableau numpy :\n",
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    "\n",
    "* 3 décimales après la virgule\n",
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    "* on supprime la notation scientifique des nombres\n",
    "* on veut de grandes lignes\n"
   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 33,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
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   "outputs": [],
   "source": [
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    "np.set_printoptions(precision=3,suppress=True,linewidth=5000)"
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   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
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    "## Type des données\n",
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    "\n",
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    "Pour les proba-stats, nous utiliserons essentiellement des tableaux contenant des flottant de 64 bit ou des d'entier de 64 bits. Mais par exemple, quand on manipule des images, les pixels sont des entiers de 8 bits. "
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   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Tableau de flotant"
   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 34,
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   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "vec\n",
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      " [1. 2. 3. 4.]\n",
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      "mat\n",
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      " [[1. 2. 0.]\n",
      " [3. 4. 0.]]\n"
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     ]
    }
   ],
   "source": [
    "\"\"\"un vecteur initialisé par une liste de valeurs. \"\"\"\n",
    "vec=np.array([1.,2,3,4])\n",
    "\"\"\"une matrice\"\"\"\n",
    "mat=np.array([ [1.,2,0],[3,4,0]])\n",
    "\"notez qu'on a mis au moins un coefficient flottant (avec un point)\"\n",
    "\n",
    "print(\"vec\\n\",vec)\n",
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    "print(\"mat\\n\",mat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Les principales caractéristiques d'un tableau sont:\n",
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    "\n",
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    "* `dtype` = data-type\n",
    "* `shape` = sa forme"
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   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 35,
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   "metadata": {},
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "vec: float64 (4,)\n",
      "mat: float64 (2, 3)\n"
     ]
    }
   ],
   "source": [
    "print(\"vec:\",vec.dtype,vec.shape)\n",
    "print(\"mat:\",mat.dtype,mat.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " vec_zero\n",
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      " [0. 0. 0. 0. 0.]\n",
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      "\n",
      " mat_un \n",
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      " [[1. 1. 1. 1.]\n",
      " [1. 1. 1. 1.]\n",
      " [1. 1. 1. 1.]\n",
      " [1. 1. 1. 1.]\n",
      " [1. 1. 1. 1.]]\n"
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     ]
    }
   ],
   "source": [
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    "\"\"\"vecteur de flotant initialisé à 0, matrice de flottant initialisé à 1\"\"\"\n",
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    "vec_zero=np.zeros(shape=[5])\n",
    "mat_un=np.ones(shape=[5,4])\n",
    "print(\"\\n vec_zero\\n\",vec_zero)\n",
    "print(\"\\n mat_un \\n\",mat_un)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Tableau d'entier\n",
    "\n",
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    "Attention, depuis python 3, les indices de tableau doivent nécessairement être des entier. Par exemple `a[3.0]` renvoie une erreur.  \n",
    "\n",
    "Du coup, si l'on veut faire un tableau qui contiens plusieurs indices, il faut créer un tableau d'entier. C'est d'ailleurs le seul cas où l'on va utiliser de tels tableaux."
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   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 37,
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   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0 3 1]\n",
      "int64\n",
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      "[0. 6. 2.]\n"
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     ]
    }
   ],
   "source": [
    "indices=np.array([0,3,1])\n",
    "vec = np.array([0.,2,4,6,8,10])\n",
    "print(indices)\n",
    "print(indices.dtype)\n",
    "print(vec[indices])"
   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 38,
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   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
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      "[1.5 2.5 3.5 4.5]  dtype: float64\n",
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      "[1 2 3 4]  dtype: int64\n",
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      "[1. 2. 3. 4.]  dtype: float64\n"
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     ]
    }
   ],
   "source": [
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    "\"\"\"convertissons\"\"\"\n",
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    "vec=np.array([1.5,2.5,3.5,4.5])\n",
    "print(vec,\" dtype:\",vec.dtype)\n",
    "vec=vec.astype(np.int64)\n",
    "print(vec,\" dtype:\",vec.dtype)\n",
    "vec=vec.astype(np.float64)\n",
    "print(vec,\" dtype:\",vec.dtype)\n"
   ]
  },
  {
   "cell_type": "markdown",
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   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
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   "source": [
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    "***Exo :*** modifiez $(1\\heartsuit)$ le vecteur `indices` ci-dessus, pour qu'il contienne des flottant. Indiquez le message d'erreur qui en découle.  "
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   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
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    "## Opérations élément par élément"
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   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 39,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
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      "vec [1. 2. 3. 4.]\n",
      "vec_carre [ 1.  4.  9. 16.]\n",
      "vec2 [ 2.  6. 12. 20.]\n",
      "vec3 [ 7. 14. 21. 28.]\n",
      "vec4 [ 0.841  0.909  0.141 -0.757]\n",
      "vec5 [ 14.  84. 252. 560.]\n"
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     ]
    }
   ],
   "source": [
    "\"\"\"toutes ces opérations sur les tenseurs sont faites terme à terme.\"\"\"\n",
    "\n",
    "vec = np.array([1., 2, 3, 4])\n",
    "vec_carre = vec**2\n",
    "vec2= vec+vec_carre\n",
    "vec3=7*vec\n",
    "vec4=np.sin(vec)\n",
    "vec5=vec2*vec3\n",
    "\n",
    "\n",
    "print(\"vec\",vec)\n",
    "print(\"vec_carre\",vec_carre)\n",
    "print(\"vec2\",vec2)\n",
    "print(\"vec3\",vec3)\n",
    "print(\"vec4\",vec4)\n",
    "print(\"vec5\",vec5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
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    "Toutes ces fonctions numpy ne modifient pas l'argument de départ (une 'copie' est faite). "
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   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 40,
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   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
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      "[1. 2. 3. 4.]\n",
      "[ 1.  4.  9. 16.]\n",
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      "[ 0.841  0.909  0.141 -0.757]\n",
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      "vec [1. 2. 3. 4.]\n"
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     ]
    }
   ],
   "source": [
    "vec = np.array([1., 2, 3, 4])\n",
    "vec1=vec**2\n",
    "vec2=np.sin(vec)\n",
    "\n",
    "print(vec.round())\n",
    "print(vec1)\n",
    "print(vec2)\n",
    "\n",
    "\"\"\"vec n'a pas changer\"\"\"\n",
    "print(\"vec\",vec)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "De nombreuses fonctions numpy peuvent s'utiliser avec deux syntaxes différentes : \n",
    "        \n",
    "1. `np.fonction(tenseur)`\n",
    "        \n",
    "2. `tenseur.fonction()`\n",
    "\n",
    "Et même dans le second cas, une copie du tenseur est faites ; c'est un peu trompeur car la syntaxe objet `tenseur.fonction()` suggère en générale une opération modifiant l'objet (opération \"inPlace\")"
   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 10,
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   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
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      "vec1 [1.1 2.1 3.1 4.1]\n",
      "vec2 [1.1 2.1 3.1 4.1]\n",
      "vec [1.123 2.123 3.123 4.123]\n"
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     ]
    }
   ],
   "source": [
    "vec = np.array([1.1234, 2.1234, 3.1234, 4.1234])\n",
    "\n",
    "\"\"\"arrondir\"\"\"\n",
    "vec1= np.round(vec,1)\n",
    "\"\"\"idem avec syntaxe différente\"\"\"\n",
    "vec2= vec.round(1)\n",
    "\n",
    "print(\"vec1\", vec1)\n",
    "print(\"vec2\", vec2)\n",
    "\n",
    "\n",
    "\"\"\"vec n'a pas changer (cependant il ne s'affiche qu'avec 3 décimales à cause de notre entête)\"\"\"\n",
    "print(\"vec\", vec)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
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    "## Changer des éléments dans un tenseur\n",
    "\n",
    "### Directement"
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   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 11,
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   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "vec et mat après modification\n",
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      "[10.  2.  3.  4.]\n",
      "[[ 1.  2.]\n",
      " [ 3. 10.]]\n",
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      "\n",
      "vec_zero\n",
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      " [5. 5. 5. 0. 0.]\n",
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      "\n",
      "mat_un\n",
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      " [[7. 7. 7. 1.]\n",
      " [7. 7. 7. 1.]\n",
      " [1. 1. 1. 1.]\n",
      " [1. 1. 1. 1.]\n",
      " [1. 1. 1. 1.]]\n"
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     ]
    }
   ],
   "source": [
    "vec = np.array([1., 2, 3, 4])\n",
    "mat = np.array([[1., 2], [3, 4]])\n",
    "\n",
    "\"\"\"modifier un élément\"\"\"\n",
    "vec[0] = 10.  # la numérotation commence à zéro\n",
    "mat[1, 1] = 10.\n",
    "\n",
    "\"\"\" attention, on a ainsi modifier les objets initiaux\"\"\"\n",
    "print(\"vec et mat après modification\")\n",
    "print(vec)\n",
    "print(mat)\n",
    "\n",
    "\"\"\"modifier un bloc d'élément\"\"\"\n",
    "vec_zero = np.zeros(shape=[5])\n",
    "vec_zero[0:3]=5 # [0:3] indique les indices dans l'intervalle [0,3[\n",
    "\n",
    "mat_un = np.ones(shape=[5, 4])\n",
    "mat_un[0:2,0:3]=7\n",
    "\n",
    "print(\"\\nvec_zero\\n\", vec_zero)\n",
    "print(\"\\nmat_un\\n\", mat_un)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
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    "### Via une 'vue' sur les données\n",
    "\n",
    "Attention, on touche ici à une des grandes difficultés de `numpy`; difficulté qu'on retrouve partout en informatique: le passage de données par copie ou par référence.  "
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   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 42,
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   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      " mat_un\n",
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      " [[7. 7. 7. 1.]\n",
      " [7. 7. 7. 1.]\n",
      " [1. 1. 1. 1.]\n",
      " [1. 1. 1. 1.]\n",
      " [1. 1. 1. 1.]]\n",
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      "\n",
      " part\n",
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      " [[7. 7. 7.]\n",
      " [7. 7. 7.]]\n"
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     ]
    }
   ],
   "source": [
    "mat_un = np.ones(shape=[5, 4])\n",
    "\n",
    "part=mat_un[0:2, 0:3]\n",
    "\"\"\"toute la partie est mise à 7\"\"\"\n",
    "part[:,:]=7\n",
    "\n",
    "\"\"\"et là, observez : c'est à la fois logique et surprenant:\"\"\"\n",
    "print(\"\\n mat_un\\n\", mat_un)\n",
    "print(\"\\n part\\n\", part)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Comment ça marche :\n",
    "\n",
    "        mat_un = np.ones(shape=[5, 4])\n",
    "        \n",
    "L'ordi crée un tableau dans la mémoire de l'ordinateur. la variable 'mat_un' est une vue (=référence) sur ce tableau.\n",
    "        \n",
    "        part=mat_un[0:2, 0:3]\n",
    "        \n",
    "la variable 'part' est une vue sur une partie de ce tableau en mémoire. en effectuant :\n",
    "        \n",
    "        part[:,:]=7\n",
    "        \n",
    "on modifie une partie du tableau en mémoire. \n",
    "\n",
    "Par contre quand on fait :\n",
    "        \n",
    "        a=mat_un[0,0]\n",
    "        \n",
    "la variable `a` contient une copie du coefficient `mat_un[0,0]`. Modifier `a` ne modifie pas le tableau en mémoire. La preuve :"
   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 13,
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   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a= 7\n",
      "\n",
      " mat_un\n",
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      " [[1. 1. 1. 1.]\n",
      " [1. 1. 1. 1.]\n",
      " [1. 1. 1. 1.]\n",
      " [1. 1. 1. 1.]\n",
      " [1. 1. 1. 1.]]\n"
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     ]
    }
   ],
   "source": [
    "mat_un = np.ones(shape=[5, 4])\n",
    "a=mat_un[0,0]\n",
    "a=7\n",
    "print('a=',a)\n",
    "print(\"\\n mat_un\\n\", mat_un)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
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    "***Exo :*** Que sont les tenseurs a,b,c,.... ci-dessous $(4\\diamondsuit)$?\n",
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    "Mettez la réponse sur du papier, faites bien la différence entre les vecteurs et les matrices\n",
    "puis vérifiez avec un `print`\n",
    "Indiquez pour chaque cas, s'il s'agit d'un nombre, d'un vecteur ou d'une matrice."
   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 44,
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   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "mat \n",
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      " [[0. 1. 2. 3.]\n",
      " [1. 2. 3. 4.]\n",
      " [2. 3. 4. 5.]]\n"
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     ]
    }
   ],
   "source": [
    "\"\"\"remplissons une matrice par une double boucle\"\"\"\n",
    "nbLigne = 3\n",
    "nbColonne = 4\n",
    "mat = np.zeros([nbLigne, nbColonne])\n",
    "for i in range(nbLigne):\n",
    "    for j in range(nbColonne):\n",
    "        mat[i, j] = i + j\n",
    "\n",
    "print(\"\\nmat \\n\",mat)\n",
    "\n",
    "a = mat[0, :]\n",
    "aBis = mat[0:1, :]\n",
    "b = mat[:,0]\n",
    "c = mat[:, nbColonne - 1]\n",
    "d = mat[:,  - 1]  # ça: c'est idem que la ligne précédente\n",
    "e = mat[-1,-1]\n",
    "f = mat[0:2,:]\n",
    "g = mat[0:2, 0:1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
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    "## Sélection par valeurs\n",
    "\n",
    "### Via les tableaux de booléens"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.  1.  2.  3.  4. -1.]\n",
      "[False False False False False  True]\n",
      "[ True  True  True  True  True False]\n",
      "[-1.]\n",
      "[0. 1. 2. 3. 4.]\n",
      "[0.]\n",
      "[1. 2. 3.]\n",
      "[ 0.  1.  2.  3.  4. -1.]\n"
     ]
    }
   ],
   "source": [
    "a=np.array([0.,1.,2,3,4,-1])\n",
    "print(a)\n",
    "print(a<0)\n",
    "print(a>=0)\n",
    "print(a[a<0])\n",
    "print(a[a>=0])\n",
    "print(a[a==0])\n",
    "\"\"\" le 'and' et 'or' dans les indices n'est pas le même que les 'and' et 'or' classiques\"\"\"\n",
    "print(a[(a>0)&(a<4)])\n",
    "print(a[(a>0)|(a<4)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Contrairement à la selection par indice (ex : `a[:,2]`), la selection par valeur créer une \"copie\", donc préserve l'originale."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "b [10. 10. 10. 10. 10.]\n",
      "a [ 0.  1.  2.  3.  4. -1.]\n"
     ]
    }
   ],
   "source": [
    "a=np.array([0.,1.,2,3,4,-1])\n",
    "b = a[a >= 0]\n",
    "b[:] = 10\n",
    "print('b',b)\n",
    "\"\"\" 'a'  n'a pas changé, ouf. \"\"\"\n",
    "print('a',a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***Important:*** Quand on a un doute sur 'copie' ou 'référence', on teste, comme ci-dessus."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Via des conditions\n",
    "\n",
    "Pour sélectionner des éléments dans un tableau selon un critère complexe,\n",
    "il faut faire une boucle, et utiliser une liste car on ne sait jamais combien d'élément on va sélectionner à l'avance. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[21  0 42 84  0  0 42 63 21 84 42 84 84 42 42 21  0 84  0 21 63  0 63 63 63 84 63 42 21 84 42  0  0 84 42 21 84 63 63 21 42]\n"
     ]
    }
   ],
   "source": [
    "nbSimu=1000\n",
    "a=np.random.randint(0,100,nbSimu)\n",
    "res=[]\n",
    "for i in range(nbSimu):\n",
    "    if a[i]%3==0 and a[i]%7==0: res.append(a[i])\n",
    "\"\"\"à la fin, on repasse le résultat en tableau numpy\"\"\"\n",
    "res=np.array(res)\n",
    "print(res)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Algèbre linéaire\n",
    "\n",
    "Attention, en numpy les vecteurs, les matrices lignes et les matrices colonnes sont des objets différents:\n",
    "\n",
    "* vecteur.shape = (?)\n",
    "* matrice_ligne.shape = (1,?)\n",
    "* matrice_colonne.shape = (?,1)\n",
    "* matrice_quelconque.shape = (?,?)\n",
    "\n",
    "Observez bien les sorties consoles: les vecteurs s'écrivent avec 1 crochet, les matrices avec 2 crochets."
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   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
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    "### Multiplication matricielle\n",
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    "\n",
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    "`np.matmul()` s'applique uniquement entre matrices."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mat_col\n",
      " [[1.]\n",
      " [1.]\n",
      " [1.]]\n",
      "mat_lin\n",
      " [[1. 1. 1.]]\n",
      "mat_square\n",
      " [[1. 1. 1.]\n",
      " [1. 1. 1.]\n",
      " [1. 1. 1.]]\n",
      "mat_square × mat_col\n",
      " [[3.]\n",
      " [3.]\n",
      " [3.]]\n",
      "mat_lin × mat_square\n",
      " [[3. 3. 3.]]\n",
      "mat_lin × mat_square × mat_col\n",
      " [[9.]]\n"
     ]
    }
   ],
   "source": [
    "size=3\n",
    "mat_col=np.ones(shape=[size,1])\n",
    "mat_lin=np.ones(shape=[1,size])\n",
    "mat_square=np.ones(shape=[size,size])\n",
    "\n",
    "print(\"mat_col\\n\",mat_col)\n",
    "print(\"mat_lin\\n\",mat_lin)\n",
    "print(\"mat_square\\n\",mat_square)\n",
    "\n",
    "print(\"mat_square × mat_col\\n\",np.matmul(mat_square,mat_col) )\n",
    "print(\"mat_lin × mat_square\\n\",np.matmul(mat_lin,mat_square))\n",
    "print(\"mat_lin × mat_square × mat_col\\n\", np.matmul(np.matmul(mat_lin,mat_square),mat_col))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "sinon on peut utiliser `np.dot()` qui permet les multiplications matrice $\\times$ vecteur, matrice $\\times$ matrice, vecteur $\\times$  vecteur (=produit scalaire)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "vec0 * vec1\n",
      " 6.0\n",
      "vec0 * mat_square\n",
      " [3. 3. 3.]\n",
      "mat_square * vec0\n",
      " [3. 3. 3.]\n",
      "mat_square * mat_square\n",
      " [[3. 3. 3.]\n",
      " [3. 3. 3.]\n",
      " [3. 3. 3.]]\n"
     ]
    }
   ],
   "source": [
    "size=3\n",
    "vec0=np.ones(shape=[size])\n",
    "vec1=2*np.ones(shape=[size])\n",
    "mat_square=np.ones(shape=[size,size])\n",
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    "\n",
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    "print(\"vec0 * vec1\\n\",np.dot(vec0,vec1))\n",
    "print(\"vec0 * mat_square\\n\",np.dot(vec0,mat_square))\n",
    "print(\"mat_square * vec0\\n\",np.dot(mat_square,vec0))\n",
    "print(\"mat_square * mat_square\\n\",np.dot(mat_square,mat_square))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "et si vous avez python 3.5+, vous pouver utilisez l'opérateur @ qui rend les codes plus lisibles"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "vec0 × vec1\n",
      " 6.0\n",
      "vec0 × mat_square\n",
      " [3. 3. 3.]\n",
      "mat_square × vec0\n",
      " [3. 3. 3.]\n",
      "mat_square × mat_square\n",
      " [[3. 3. 3.]\n",
      " [3. 3. 3.]\n",
      " [3. 3. 3.]]\n"
     ]
    }
   ],
   "source": [
    "print(\"vec0 * vec1\\n\",vec0 @ vec1)\n",
    "print(\"vec0 * mat_square\\n\",vec0 @ mat_square)\n",
    "print(\"mat_square * vec0\\n\",mat_square @ vec0)\n",
    "print(\"mat_square * mat_square\\n\",mat_square @  mat_square)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***Exo:*** Multipliez $(2\\heartsuit)$ matriciellement des matrices de taille non-compatible.\n",
    "Extrayez la partie intéressante du message d'erreur."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### inverse et transposée \n",
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    "\n",
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    "Notez que le pseudo-inverse permet d'inverser les matrices non-inversibles (testez). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "mat\n",
      " [[1. 0. 2.]\n",
      " [0. 1. 0.]\n",
      " [0. 0. 1.]]\n",
      "mat^T\n",
      " [[1. 0. 0.]\n",
      " [0. 1. 0.]\n",
      " [2. 0. 1.]]\n",
      "mat^(-1)\n",
      " [[ 1.  0. -2.]\n",
      " [ 0.  1.  0.]\n",
      " [ 0.  0.  1.]]\n",
      "mat^(-1)-pseudo-inverse\n",
      " [[ 1.  0. -2.]\n",
      " [ 0.  1.  0.]\n",
      " [ 0.  0.  1.]]\n"
     ]
    }
   ],
   "source": [
    "size=3\n",
    "mat=np.zeros(shape=[size,size])\n",
    "for i in range(size):\n",
    "    mat[i,i]=1\n",
    "mat[0,size-1]=2\n",
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    "\n",
    "\n",
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    "print(\"mat\\n\",mat )\n",
    "print(\"mat^T\\n\",mat.T)\n",
    "print(\"mat^(-1)\\n\",np.linalg.inv(mat))\n",
    "print(\"mat^(-1)-pseudo-inverse\\n\",np.linalg.pinv(mat))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Vecteur propre et valeur propre\n",
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    "\n",
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    "Souvenez-vous qu'en anglais valeur/vecteur 'propre' c'est 'eigen' value/vector"
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   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "mat\n",
      " [[0.331 0.366 0.175 0.128]\n",
      " [0.308 0.003 0.303 0.386]\n",
      " [0.001 0.557 0.129 0.313]\n",
      " [0.316 0.221 0.025 0.438]]\n",
      "\n",
      "eigVal\n",
      " [ 1.     0.157  0.157 -0.413]\n",
      "\n",
      "eigVec\n",
      " [[ 0.5   -0.244 -0.244  0.174]\n",
      " [ 0.5    0.153  0.153 -0.712]\n",
      " [ 0.5    0.628  0.628  0.673]\n",
      " [ 0.5   -0.215 -0.215  0.101]]\n"
     ]
    }
   ],
   "source": [
    "size=4\n",
    "\"\"\"une matrice aléatoire auquel on fait subir une opération de normalisation (laquelle?)\"\"\"\n",
    "mat=np.random.uniform(low=0,high=10,size=[size,size])\n",
    "rowSum=np.sum(mat,axis=1)\n",
    "\"\"\" mat[i,j]=mat[i,j]/rowSum(i)  \"\"\"\n",
    "for i in range(size): mat[i,:]/=rowSum[i]\n",
    "\n",
    "eigVal,eigVec=np.linalg.eig(mat)\n",
    "print(\"\\nmat\\n\", mat )\n",
    "print(\"\\neigVal\\n\",np.real(eigVal))\n",
    "print(\"\\neigVec\\n\", np.real(eigVec))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***A vous:*** \n",
    "\n",
    "* Pour la `mat` précédent: pourquoi 1 est-il valeur propre $(2\\heartsuit)$? Quel est le vecteur propre associé $(1\\heartsuit)$? \n",
    "\n",
    "* `np.linalg.eig` donne-t-il les vecteurs propres à droite ou à gauche $(1\\heartsuit)$? Sont-ils disposés en ligne ou bien en colonne $(1\\heartsuit)$?\n",
    "\n",
    "* De manière générale, toutes les matrices sont-elles diagonalisables $(1\\heartsuit)$?  A quoi servent les `np.real()` dans le programme précédent $(1\\heartsuit)$?  \n",
    "\n",
    "* laquelle de ces deux formules est la bonne $(1\\heartsuit)$?\n",
    "\n",
    "        Diag = P^-1 @ mat @ P   \n",
    "\n",
    "ou\n",
    "\n",
    "        Diag = P @ mat @ P^-1\n",
    "        \n",
    "Vérifiez avec python. Aide: Pour transformez un vecteur en matrice diagonale, utilisez `np.diag(vecteur)`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Application :   loi forte des grands nombres\n",
    "\n",
    "### Illustration graphique\n",
    "***Théorème:*** Soit $(X_n)$ une suite de v.a, qui admet une espérance, on a: \n",
    "$$\n",
    "\\lim_n \\frac 1 n  (X_0+....+X_{n-1}) =  \\mathbf E[X_0]\n",
    "$$\n",
    "\n",
    "Remarque : pour illustrer ce théorème, on va tracer la courbe \n",
    "$$\n",
    "   n \\to  \\frac 1 n  (X_0+....+X_{n-1})\n",
    "$$\n",
    "mais on n'est pas obligé de l'évaluer pour tous les $n$ (ex: on peut aller par pas de 40)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
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   "metadata": {},
   "outputs": [
    {
     "data": {
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      "image/png": 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\n",
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      "text/plain": [
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       "<Figure size 432x288 with 1 Axes>"
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "% matplotlib inline\n",
    "\n",
    "\"\"\" np.arange c'est comme range() sauf que cela donne un tableau numpy\"\"\"\n",
    "n_max=10000\n",
    "ns=np.arange(10,n_max,40)\n",
    "means=[]\n",
    "for n in ns:\n",
    "    simus=np.random.uniform(0,10,size=n)\n",
    "    means.append(np.mean(simus))\n",
    "\n",
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    "fig,ax=plt.subplots()\n",
    "ax.plot(ns,means)\n",
    "ax.plot([0,n_max],[5,5]);"
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   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
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    "***A vous:*** $(2\\heartsuit)$ J'ai volontairement fait une erreur dans le programme précédent, une erreur que font beaucoup d'étudiant chaque année. Relisez attentivement la loi des grands nombre, comparez cet énoncé avec le programme ci-dessus. Trouvez le bug."
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   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
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    "### Illustration numérique"
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   ]
  },
  {
   "cell_type": "code",
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   "execution_count": 51,
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   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
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      "0.7005 valeur attendue: 0.7\n",
      "0.08349414764097192 valeur attendue: 0.08333333333333333\n",
      "0.5014 valeur attendue: 0.5\n"
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     ]
    }
   ],
   "source": [
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    "nbSimu = 10000\n",
    "\"\"\"un échantillon de va de loi uniforme sur [0,1]\"\"\"\n",
    "X1 = np.random.random(size=nbSimu)\n",
    "\"\"\" un échantillon de va de loi normale (=gaussienne), centrée réduite par défaut.   \"\"\"\n",
    "X2 = np.random.normal(size=nbSimu)\n",
    "\"\"\" un échantillon de va de loi de Poisson\"\"\"\n",
    "X3 = np.random.poisson(1, size=nbSimu)\n",
    "\n",
    "print(len(X1[X1<0.7])/nbSimu,\"valeur attendue:\",0.7)\n",
    "print(sum( (X1-0.5)**2)/nbSimu,\"valeur attendue:\",1/12)\n",
    "print(len(X2[X2<0.])/nbSimu,\"valeur attendue:\",0.5)\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Donnez ci-dessus les valeurs attendue, et révisez ainsi votre loi de Poisson $(3\\times 1\\heartsuit)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.9885952712068063 valeur attendue:\n",
      "0.0611 valeur attendue:\n",
      "0.6317 valeur attendue:\n"
     ]
    }
   ],
   "source": [
    "print(np.sum(X2**2)/nbSimu,\"valeur attendue:\")\n",
    "print(len(X3[X3==3])/nbSimu,\"valeur attendue:\")\n",
    "print(len(X3[X3!=0])/nbSimu,\"valeur attendue:\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Monte carlo\n",
    "\n",
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    "\n",
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    "***A vous:*** $(3\\times 2\\heartsuit)$ à l'aide de variables uniformes, en utilisant la loi forte des grands nombres, estimez  \n",
    "$$\n",
    " \\int_0^1  x \\ dx  \\qquad \\text{et} \\qquad  \\int_0^{\\frac 12 }  \\frac x {\\ln(x)}\\  dx \\qquad \\text{et} \\qquad  \\int_0^1 \\frac  1 x\\  dx\n",
    "$$   \n",
    "Cette méthode pour calculer des intégrales est appelée : Méthode de Monte Carlo\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Application : moyenne et écart-type\n",
    "\n",
    "Simulons des échantillons:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "nbData=100\n",
    "X1=np.random.random(size=nbData)\n",
    "X2=np.random.normal(size=nbData);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "* calculons leur moyenne, qu'on peut  appeler : mu_hat (mu pour espérance, et hat pour estimation)\n",
    "* calculons leur écart type empirique : sigma_hat (sigma pour écart type, et hat pour estimation)\n",
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    "\n",
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    "La formule écart type empirique  est ..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "loi uniforme sur [0,1]\n",
      "* mu_hat 0.5012182491359483\n",
      "* sigma_hat 0.3025619268091313\n",
      "loi normale\n",
      "* mu_hat -0.13541845638430194\n",
      "* sigma_hat 1.0389979183900737\n"
     ]
    }
   ],
   "source": [
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    "print(\"loi uniforme sur [0,1]\")\n",
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    "print(\"* mu_hat\",X1.mean())\n",
    "print(\"* sigma_hat\",X1.std())\n",
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    "\n",
    "print(\"loi normale\")\n",
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    "print(\"* mu_hat\", X2.mean())\n",
    "print(\"* sigma_hat\", X2.std())"
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   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
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    "A l'aide des estimateurs, vérifiez expérimentallement les résultats suivants $(4\\times 2\\heartsuit)$:\n",
    "\n",
    "* l'espérance de la somme de deux variables aléatoires c'est la somme des espérances.\n",
    "* la variance de la somme de deux variables aléatoires c'est ...\n",
    "* soient $X$ une v.a, $a$ et $b$ des constantes. On a: $\\mathbf E[aX+b]= ...$ et $\\mathbf V[aX+b]= ...$. \n",
    "* L'espérance du carré d'une v.a est toujours plus... que le carré de l'espérance. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Numpy or not numpy\n",
    "\n",
    "Quand on écrit un programme il faut toujours se demander si on va utiliser des tenseurs `numpy` ou pas. Par exemple, considérons la fonction suivante:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(x):\n",
    "    return x**2*(1-x)**2+1   "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Cette fonction fonctionne (!)  pour `x` scalaire, mais aussi pour `x` tenseur: car les opérations puissance `**`, somme `+` et produit `*` sont bien définies sur les tenseurs. Ainsi on peut tracer le graphe de `f` de deux manières différentes:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\"\"\"première méthode\"\"\"\n",
    "xs=np.linspace(0,1,100)\n",
    "ys=[]\n",
    "for x in xs:\n",
    "    ys.append(f(x))\n",
    "plt.plot(xs,ys);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "\"\"\" seconde méthode\"\"\"\n",
    "plt.plot(xs,f(xs));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La seconde méthode est plus élégante mais aussi beaucoup plus rapide.\n",
    "\n",
    "Par contre, considérons la fonction:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "def g(x):\n",
    "    if 0<x<0.5: return x\n",
    "    elif 0.5<x<2: return 2*x\n",
    "    else : return 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Cette fonction `g` ne fonctionne pas quand `x` est un tenseur numpy (essayez $(1\\heartsuit)$, observez le message d'erreur, vous le rencontrerez sans doute plus tard). On peut transformer cette fonction pour quelle marche avec `numpy` mais c'est un peu technique. \n",
    "\n",
    "Le conseil: dans un premier temps, ne vous prenez pas trop la tête. Faites au plus simple même si ce n'est pas le plus performant. Vous apprendrez les astuces au fur et à mesure. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Sous le capot de numpy\n",
    "\n",
    "### Des données à plats. \n",
    "\n",
    "Peu importe la `shape` ou le `dtype` (=data_type): dans la mémoire, les tableaux `numpy` sont stockés sous la forme d'un tableau 1D, par paquets-de-8bits. Quand vous changer la `shape`, les données ne change pas du tout, c'est uniquement la façon de les parcourir qui change. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "vec\n",
      " [0 1 2 3 4 5 6 7 8]\n",
      "mat_square\n",
      " [[0 1 2]\n",
      " [3 4 5]\n",
      " [6 7 8]]\n",
      "mat_col\n",
      " [[0]\n",
      " [1]\n",
      " [2]\n",
      " [3]\n",
      " [4]\n",
      " [5]\n",
      " [6]\n",
      " [7]\n",
      " [8]]\n",
      "mat_line\n",
      " [[0 1 2 3 4 5 6 7 8]]\n"
     ]
    }
   ],
   "source": [
    "\" ces 4 objets sont associées aux même données toutes plates.\"\n",
    "vec=np.arange(0,9,1)\n",
    "print(\"vec\\n\",vec)\n",
    "mat_square=vec.reshape([3,3])\n",
    "print(\"mat_square\\n\",mat_square)\n",
    "mat_col=mat_square.reshape([9,1])\n",
    "print(\"mat_col\\n\",mat_col)\n",
    "mat_line=mat_square.reshape([1,9])\n",
    "print(\"mat_line\\n\",mat_line)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### l'attribut 'strides'\n",
    "\n",
    "Quand on change la `shape`, on chage aussi l'attribut `strides`: il indique comment on passe d'une case à l'autre.\n",
    "\n",
    "### ex:  vecteur \n",
    "\n",
    "Dans ce cas,  `strides` est un scalaire: il indique de combien de paquet-de-8bits il faut glisser pour aller de `vec[i]` à `vec[i+1]`. Cela dépend uniquement du `dtype`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "np.uint8, strides: (1,)\n",
      "np.int32, strides: (4,)\n",
      "np.int64, strides: (8,)\n"
     ]
    }
   ],
   "source": [
    "a=np.zeros(shape=5,dtype=np.uint8)\n",
    "print(\"np.uint8, strides:\",a.strides)\n",
    "a=np.zeros(shape=5,dtype=np.int32)\n",
    "print(\"np.int32, strides:\",a.strides)\n",
    "a = np.zeros(shape=5, dtype=np.int64)\n",
    "print(\"np.int64, strides:\",a.strides)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### ex:  matrice\n",
    "\n",
    "Dans ce cas,  `strides` est une paire. \n",
    "\n",
    "* `strides[0]` indique de combien de paquet-de-8bits il faut glisser pour passer de la case `mat[i,j]` à la case `mat[i+1,j]` \n",
    "* `strides[1]` indique de combien de paquet-de-8bits il faut glisser pour passer de la case ``mat[i,j]`` à la case ``mat[i,j+1]``\n",
    "\n",
    "Illustrons cela avec uniquement `dtype=np.uint8`. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "shape=(2,4), strides: (4, 1)\n",
      "shape=(30,4), strides: (5, 1)\n"
     ]
    }
   ],
   "source": [
    "a = np.zeros(shape=(2,4), dtype=np.uint8)\n",
    "print(\"shape=(2,4), strides:\", a.strides)\n",
    "\n",
    "a = np.zeros(shape=(30, 5), dtype=np.uint8)\n",
    "print(\"shape=(30,4), strides:\", a.strides)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "***A vous:*** Essayez de répondres à ces 3 questions sans faire de `print` $(3\\heartsuit)$ puis avec un `print` $(3\\heartsuit)$. \n",
    "\n",
    "* quelle est la stride de `np.zeros(shape=(2,4), dtype=np.int32)`\n",
    "* quelle est la stride de `np.zeros(shape=(30,5), dtype=np.int32)`\n",
    "* A quoi correspond la stride d'un tenseur de dimension 3."
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  {
   "cell_type": "code",
   "execution_count": null,
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   "source": []
  }
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