Révisions exp-log.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Révisions \n",
    "\n",
    "- fonction exponentielle\n",
    "- fonction logarithme\n",
    "- régression linéaire"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Fonction exponentielle et logarithme\n",
    "\n",
    "Développement parallèle de ces deux fonctions : l'une pour permettre de transformer des multiplications en addition ; l'autre pour rendre compte de croissance spécifiques.\n",
    "\n",
    "Exponentielle et logarithme sont récirpoques, ce qui signifie que $\\exp(\\ln(x)) = x$ et $\\ln(\\exp(x))=x$ (la composition des deux fonctions est l'identité, la fonction x qui à x associe x)\n",
    "\n",
    "### Exponentielle\n",
    "\n",
    "La fonction exponentielle est une fonction f notée $\\exp$ définie sur R telle que :\n",
    "\n",
    "$$ f(x) = \\exp(x) = e^x$$\n",
    "\n",
    "Mais il y a quoi dans $\\exp$ ? En fait, la valeur exponentielle est le résultat d'une suite $e^x = \\lim_{n=+\\infty{}}(1+\\frac{x}{n})^n$\n",
    "\n",
    "Cette fonction à certaines propriétés.\n",
    "\n",
    "Soit a et b deux réels :\n",
    "\n",
    "\n",
    "- $e^0=1$\n",
    "- $e^x > 0$\n",
    "- $\\lim_{x=+\\infty{}}e^x = +\\infty{}$\n",
    "- $\\lim_{x=-\\infty{}}e^x = 0$\n",
    "\n",
    "- $e^{a + b} = e^a.e^b$\n",
    "- $e^{a-b} = \\frac{e^a}{e^b}$\n",
    "- $(e^a)^b = e^{a.b}$\n",
    "\n",
    "La fonction exponentielle est dérivable et \n",
    "\n",
    "$$ \\exp'(x) =  \\exp(x) $$\n",
    "\n",
    "La conséquence quand on a une fonction composée :\n",
    "\n",
    "$$ \\exp'(g(x)) =  g'(x).\\exp((g(x)) $$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "soit $f(x) = 10.\\exp(x^2)$, quelle est sa dérivée ?\n",
    "\n",
    "C'est la même chose que f(x) = $10.\\exp(g(x))$ avec $g(x)=x^2$\n",
    "\n",
    "Donc $f'(x) = g'(x).f(x) = 2*x*10*\\exp(x^2) = 20*x*\\exp(x^2)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Y')"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 1440x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-10,2,1000)\n",
    "y = np.exp(x)\n",
    "\n",
    "fix,ax = plt.subplots(figsize=(20,10))\n",
    "ax.plot(x,y,\".:\")\n",
    "plt.xlabel(\"X\")\n",
    "plt.ylabel(\"Y\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Quelques remarques générales\n",
    "\n",
    "- les fonctions exponentielles sont souvent des solutions d'équations différentielles impliquant une croissance proportionnelle à la valeur d'une fonction à un moment ;\n",
    "- les fonctions logarithmes permettent de faire des transformations intéressantes passant d'une multiplication à une somme, ce qui historiquement a eu un rôle important quand les ordinateurs n'étaient pas disponibles"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "### Logarithme\n",
    "\n",
    "C'est la fonction réciproque (bijective) le logarithme népérien. Elle est définie sur $R^+_*$ et appellée $\\ln$\n",
    "\n",
    "Comme c'est la réciproque de exponentielle, on a $ \\ln(e^x) = x$ et $e^{\\ln(x)} = x$ ou écrit autrement $\\exp ○ \\ln = Id$\n",
    "\n",
    "Soit a et b deux réels :\n",
    "\n",
    "- $\\ln(1) = 0$\n",
    "- $\\lim_{x=+\\infty{}}\\ln(x) = +\\infty{}$\n",
    "- $\\lim_{x=0}\\ln(x) = -\\infty{}$\n",
    "- $\\ln(a.b) = \\ln(a) + \\ln(b)$\n",
    "- $\\ln(\\frac{a}{b}) = \\ln(a) - \\ln(b)$\n",
    "- $\\ln(\\frac{1}{a}) = - \\ln(a)$\n",
    "- $\\ln(a^b) = b.\\ln(a)$\n",
    "\n",
    "- $a^b = e^{\\ln{(a)}.b}$\n",
    "\n",
    "Parce que $e^{\\ln{(a)}*b} = (e^{\\ln{(a)}})^b = a^b$\n",
    "\n",
    "\n",
    "\n",
    "Et la dérivée est :\n",
    "\n",
    "$$ ln'(x) = \\frac{1}{x}$$\n",
    "\n",
    "$$\\ln(g(x))' = \\frac{g'(x)}{x}$$\n",
    "\n",
    "\n",
    "Par exemple, $x^{10} = e^{10*\\ln(x)}$\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "### Logarithme décimal\n",
    "\n",
    "\n",
    "On définit souvent une fonction de logarithme décimal, de la manière suivante :\n",
    "\n",
    "$$ f(x) = \\log_{10}(x) = \\frac{\\ln(x)}{\\ln(10)}$$\n",
    "\n",
    "Pourquoi ? Parce que cette fonction permet d'avoir la propriété suivante :\n",
    "\n",
    "Si on prend x, quelle est la valeur du $log_{10}$ de 10*x :\n",
    "\n",
    "$$  \\log_{10}(10 * x) = \\frac{\\ln(10.x)}{\\ln(10)} = \\frac{\\ln(x)}{\\ln(10)} + \\frac{\\ln(10)}{\\ln(10)} = \\frac{\\ln(x)}{\\ln(10)} + 1 = \\log_{10}(x) + 1$$\n",
    "\n",
    "Multiplier par 10 revient à juste ajouter une unité. Ce qui signifie que vous allez avoir une courbe similaire mais décalée vers le haut (+1) si vous étudiez un phénomène qui a une variation proportionnelle mais juste plus amplifiée."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$ln(\\frac{a}{a}) = ln(\\frac{1}{a}*a) = ln(1) = 0 = ln(a) + ln(\\frac{1}{a})$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$f(x) = \\exp(10*x+5)$$\n",
    "\n",
    "$$\\ln(f(x)) = 10*x+5$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tracons ces fonctions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Fonction logarithme"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x11b110ed0>]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1440x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(0.01,2,1000)\n",
    "y = np.log(x)\n",
    "\n",
    "fix,ax = plt.subplots(figsize=(20,10))\n",
    "\n",
    "ax.plot(x,y,\".:\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ \\ln(a_0*r^n) = \\ln(a_0) + n*\\ln(r)$$ "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.log(100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Régression linéaire\n",
    "\n",
    "La philosophie est très simple : vous vous intéressez à une grandeur (souvent notée Y) et vous aimeriez mieux comprendre comment sa variation dépend d'un ensemble de facteurs. Dans le cas où vous avez de bonne raison de penser qu'une autre grandeur (X) est responsable de sa variation (quand X varie, alors Y varie), vous voulez écrire cette relation.\n",
    "\n",
    "La forme la plus simple qui relie deux grandeurs est la relation linéaire : si on a une variation $\\Delta X$ alors Y varie aussi, d'une quantité $\\Delta Y$, et la relation entre ces deux valeurs est $\\Delta Y = a.\\Delta X$\n",
    "\n",
    "Dit autrement, la manière la plus simple de relier Y à X est de faire l'hypothèse que les deux sont reliés par une équation de droite.\n",
    "\n",
    "Or, ce n'est jamais tout à fait vrai. Vos données ne sont jamais reliées par une relation exacte linéaire. C'est une hypothèse. Et même s'il y a bien une \"vraie\" relation linéaire entre vos données, le hasard et les erreurs de mesure font que les points ne sont pas exactement là où vous les attendez.\n",
    "\n",
    "L'objectif est alors de trouver la meilleure droite qui permet de s'adapter à vos mesures : **c'est ça une régression linéaire**.\n",
    "\n",
    "Comment ? L'idée est de prendre toutes les droites possibles et de garder celle qui est la plus proche des données réelles."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Avec des données"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = [3,20,2,78,56,9,38,5,100]\n",
    "Y = [8,34,1,56,13,56,10,4,78]\n",
    "\n",
    "fix,ax = plt.subplots(figsize=(10,5))\n",
    "ax.plot(X,Y,\"o\")\n",
    "plt.xlabel(\"X\")\n",
    "plt.ylabel(\"Y\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Définisson une droite au hasard"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = 1\n",
    "b = 3\n",
    "Y_d = [a*x + b for x in X] + [b]\n",
    "\n",
    "fix,ax = plt.subplots(figsize=(10,5))\n",
    "ax.plot(X,Y,\"o\")\n",
    "ax.plot(X + [0],Y_d)\n",
    "\n",
    "plt.xlabel(\"X\")\n",
    "plt.ylabel(\"Y\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On peut montrer par le calcul que si on cherche la droite qui rend l'erreur quadratique la plus petite (la somme des carrés des distances entre la droite et le point), on a une expression \n",
    "\n",
    "$$  b = \\frac{ \\sum_{i=1}^n (x_i - \\bar{x})(y_i - \\bar{y}) }{ \\sum_{i=1}^n (x_i - \\bar{x})^2 } $$\n",
    " \n",
    " et \n",
    " \n",
    " $$  a = \\bar{y} - b.\\bar{x}\n",
    "$$\n",
    "\n",
    "Dans les faits, les logiciels calculent directement ces valeurs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Calculer la droite de régression\n",
    "from scipy import stats\n",
    "a, b, r_value, p_value, std_err = stats.linregress(X,Y)\n",
    "\n",
    "\n",
    "# Visualiser la droite\n",
    "\n",
    "Y_d = [a*x + b for x in X] + [b]\n",
    "\n",
    "fix,ax = plt.subplots(figsize=(10,5))\n",
    "ax.plot(X,Y,\"o\")\n",
    "ax.plot(X + [0],Y_d)\n",
    "\n",
    "plt.xlabel(\"X\")\n",
    "plt.ylabel(\"Y\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Formules trigonométriques"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
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