{ "metadata": { "name": "" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Processus de Poisson\n", "\n", "Bas\u00e9 sur [Poisson Process 1](https://fr.khanacademy.org/video?lang=fr&format=lite&v=31MoWKnPDaU) et [Poisson Process 2](https://fr.khanacademy.org/video?lang=fr&format=lite&v=sWAqWRPqLok). On va commencer avec la premi\u00e8re vid\u00e9o." ] }, { "cell_type": "code", "collapsed": false, "input": [ "from IPython.display import YouTubeVideo\n", "YouTubeVideo(\"31MoWKnPDaU\")" ], "language": "python", "metadata": {}, "outputs": [ { "html": [ "\n", " \n", " " ], "metadata": {}, "output_type": "pyout", "prompt_number": 1, "text": [ "" ] } ], "prompt_number": 1 }, { "cell_type": "markdown", "metadata": {}, "source": [ "## D\u00e9finitions\n", "\n", "$X =$ le nombre de voitures qui passent en une heure. $X$ est une variable al\u00e9atoire.
\n", "$E(X) = \\lambda =$ l'esp\u00e9rance de X, ou le nombre moyen de voitures qui passent en une heure.\n", "\n", "## Calculs\n", "\n", "$\\lambda = np$, ou $n =$ le nombre de p\u00e9riodes et $p =$ le nombre moyen de voitures qui passent en une p\u00e9riode.\n", "\n", "Si $n = 60$, $\\lambda = 60\\frac{\\lambda}{60}$. Donc on a une distribution binomiale.\n", "\n", "$$P(X=k) = {60 \\choose k} \\left(\\frac{\\lambda}{60}\\right)^k \\left(1 - \\frac{\\lambda}{60}\\right)^{(60-k)}$$\n", "\n", "Mais $\\frac{\\lambda}{60}$ n'est pas une probabilit\u00e9, parce qu'il peut \u00eatre plus qu'un. Alors \u00e7a marche comment ?\n", "\n", "## Un peu de calcul infinit\u00e9simal\n", "\n", "Et maintenant, la vid\u00e9o a une \u00e9quation :\n", "\n", "$$\\lim_{x \\rightarrow \\infty}{\\left(1 + \\frac{a}{x}\\right)^x} = e^a$$\n", "\n", "Franchement, j'ai oubli\u00e9 tout \u00e7a. Il me faut r\u00e9viser cette limite avant de continuer.\n", "\n", "D'abord, je vais charger la biblioth\u00e8que `sympy`, qui peut faire les calculs formels (ou \u00ab symboliques \u00bb)." ] }, { "cell_type": "code", "collapsed": false, "input": [ "import sympy as sym\n", "sym.init_printing()" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 2 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Puis on va construire une expression formelle :" ] }, { "cell_type": "code", "collapsed": false, "input": [ "a, x, y, z = sym.symbols(\"a, x, y, z\")\n", "expr = (1 + a/x)**x\n", "expr" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left(\\frac{a}{x} + 1\\right)^{x}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAFMAAAAoBAMAAABwedsxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n", "prompt_number": 3, "text": [ " x\n", "\u239ba \u239e \n", "\u239c\u2500 + 1\u239f \n", "\u239dx \u23a0 " ] } ], "prompt_number": 3 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Et prendre la limite :" ] }, { "cell_type": "code", "collapsed": false, "input": [ "sym.limit(expr, x, sym.oo)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$e^{a}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAQBAMAAAAG6llRAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAAa0lEQVQIHWNgAAMR410QBgNjFddHKJPFgasAyuSYwLwBypQP4DsoAGFzLFifAxVl\n2zzZB8hkVDYJgIqkB7AqQJisnQwiCyBM7h7jgxAWAz/MQAYGfgWoGAMDB5BpAOExX2AQgRlmYnMA\nIggAUuMRKLBYefMAAAAASUVORK5CYII=\n", "prompt_number": 4, "text": [ " a\n", "\u212f " ] } ], "prompt_number": 4 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Merci, c'est tr\u00e8s gentil, mais je veux la comprendre. Il y a [une \u00e9xplication en anglais](http://au.answers.yahoo.com/question/index?qid=20100915005550AAFRaMD) sur Yahoo. Je vais utiliser \u00e7a.\n", "\n", "$$f = \\lim_{x \\rightarrow \\infty}{\\left(1 + \\frac{a}{x}\\right)^x}$$\n", "\n", "$$\\log{f} = \\log{\\lim_{x \\rightarrow \\infty}{\\left(1 + \\frac{a}{x}\\right)^x}}$$\n", "\n", "$$\\log{f} = \\lim_{x \\rightarrow \\infty}{x \\log{\\left(1 + \\frac{a}{x}\\right)}}$$\n", "\n", "Soit $y$ \u00e9gal \u00e0 $1/x$ :\n", "\n", "$$\\log{f} = \\lim_{y \\rightarrow 0}{\\frac{1}{y} \\log{\\left(1 + ay\\right)}}$$\n", "\n", "Et maintenant il nous faut la s\u00e9rie Taylor. Soit $z = -ay$ :" ] }, { "cell_type": "code", "collapsed": false, "input": [ "approx = sym.series(sym.log(1 - z))\n", "approx" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- z - \\frac{z^{2}}{2} - \\frac{z^{3}}{3} - \\frac{z^{4}}{4} - \\frac{z^{5}}{5} + \\mathcal{O}\\left(z^{6}\\right)$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAATUAAAAwBAMAAACCi/AwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nSUVORK5CYII=\n", "prompt_number": 5, "text": [ " 2 3 4 5 \n", " z z z z \u239b 6\u239e\n", "-z - \u2500\u2500 - \u2500\u2500 - \u2500\u2500 - \u2500\u2500 + O\u239dz \u23a0\n", " 2 3 4 5 " ] } ], "prompt_number": 5 }, { "cell_type": "code", "collapsed": false, "input": [ "approx = approx.subs({z: -(a*y)})\n", "approx" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$a y - \\frac{a^{2} y^{2}}{2} + \\frac{a^{3} y^{3}}{3} - \\frac{a^{4} y^{4}}{4} + \\frac{a^{5} y^{5}}{5} + \\mathcal{O}\\left(a^{6} y^{6}\\right)$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAZkAAAAwBAMAAAAx7xpvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nWf7skabLHbrYSsuT3I3rThgU/6v0P2+/RdLlnTDMAAAAAElFTkSuQmCC\n", "prompt_number": 6, "text": [ " 2 2 3 3 4 4 5 5 \n", " a \u22c5y a \u22c5y a \u22c5y a \u22c5y \u239b 6 6\u239e\n", "a\u22c5y - \u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500 + O\u239da \u22c5y \u23a0\n", " 2 3 4 5 " ] } ], "prompt_number": 6 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Et maintenant, utilisons l'$1/y$ dans :\n", "\n", "$$\\log{f} = \\lim_{y \\rightarrow 0}{\\frac{1}{y} \\log{\\left(1 + ay\\right)}}$$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "sym.simplify((1/y) * approx)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$a - \\frac{a^{2} y}{2} + \\frac{a^{3} y^{2}}{3} - \\frac{a^{4} y^{3}}{4} + \\frac{a^{5} y^{4}}{5} + \\mathcal{O}\\left(a^{6} y^{5}\\right)$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAYQAAAAwBAMAAADjvxA3AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n8jqSMXG+qcMRtnJd3DdjjMD/H/4Xi0Qh/iS+2EkAAAAASUVORK5CYII=\n", "prompt_number": 7, "text": [ " 2 3 2 4 3 5 4 \n", " a \u22c5y a \u22c5y a \u22c5y a \u22c5y \u239b 6 5\u239e\n", "a - \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500 + O\u239da \u22c5y \u23a0\n", " 2 3 4 5 " ] } ], "prompt_number": 7 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Mais la limite de tous les termes avec $y$ est 0, et donc on a :\n", "\n", "$$log{f} = \\lim_{y \\rightarrow \\infty}{a} = a$$\n", "\n", "$$e^{log{f}} = f = e^a$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "D'accord, alors j'y crois.\n", "\n", "## La d\u00e9rivation (enfin)\n", "\n", "On reprend la d\u00e9rivation avec [la duxi\u00e8me vid\u00e9o](https://fr.khanacademy.org/video?lang=fr&format=lite&v=sWAqWRPqLok)." ] }, { "cell_type": "code", "collapsed": false, "input": [ "YouTubeVideo(\"sWAqWRPqLok\")" ], "language": "python", "metadata": {}, "outputs": [ { "html": [ "\n", " \n", " " ], "metadata": {}, "output_type": "pyout", "prompt_number": 8, "text": [ "" ] } ], "prompt_number": 8 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Des \u00e9quations de la vid\u00e9o :\n", "\n", "$$E(X) = \\lambda = np$$\n", "\n", "(Il faut r\u00e9viser l'esp\u00e8rance d'une distribution binomiale.)\n", "\n", "$$p = \\text{probabilit\u00e9 d'une r\u00e9ussite} = \\frac{\\lambda}{n}$$\n", "\n", "La probabilit\u00e9 que $k$ voitures passent :\n", "\n", "$$\\begin{aligned}\n", "P(X=k) & = \\lim_{n \\rightarrow\\infty}{{n \\choose k} \\left(\\frac{\\lambda}{n}\\right)^k \\left(1-\\frac{\\lambda}{n}\\right)^{n-k}} \\\\\n", " & = \\lim_{n \\rightarrow\\infty}{\n", " \\frac{n!}{(n-k)!k!}\n", " \\frac{\\lambda^k}{n^k}\n", " \\left(1-\\frac{\\lambda}{n}\\right)^n\n", " \\left(1-\\frac{\\lambda}{n}\\right)^{-k}} \\\\\n", " & = \\lim_{n \\rightarrow\\infty}{\n", " \\frac{n!}{(n-k)!n^k}\n", " \\frac{\\lambda^k}{k!}\n", " e^{-\\lambda} 1^{-k}} \\\\\n", " & = \\lim_{n \\rightarrow\\infty}{\n", " \\frac{n(n-1)(n-2) \\cdots (n-k+1)}{n^k}\n", " \\frac{\\lambda^k}{k!}\n", " e^{-\\lambda}} \\\\\n", " & = \\lim_{n \\rightarrow\\infty}{\n", " \\frac{n^k + \\mathcal{O}(n^{k-1})}{n^k}\n", " \\frac{\\lambda^k}{k!}\n", " e^{-\\lambda}} \\\\\n", " & = \\frac{\\lambda^k}{k!}e^{-\\lambda}\n", "\\end{aligned}$$\n", "\n", "## Un graphique\n", "\n", "Enfin, c'est fini. On peut jouer avec les graphiques maintenant." ] }, { "cell_type": "code", "collapsed": false, "input": [ "%pylab inline --no-import-all" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Populating the interactive namespace from numpy and matplotlib\n" ] } ], "prompt_number": 9 }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "\n", "def poisson(lambda_, k):\n", " return lambda_**k * math.exp(-lambda_) / math.factorial(k)\n", "\n", "# Nos param\u00e8tres.\n", "lambda_ = 7 # Le nombre moyen de voitures qui passent.\n", "k = range(0,21) # Les probabilit\u00e9s de 0 \u00e0 20 voitures.\n", "\n", "fig = plt.figure()\n", "axes = fig.add_axes([0.1, 0.1, 0.8, 0.8])\n", "axes.plot(k, map(lambda k: poisson(lambda_, k), k))\n", "axes.set_xlabel(\"Voitures\")\n", "axes.set_ylabel(\"Prob\")\n", "axes.axvline(3, color='r')\n", "axes.set_title(\"Distribution Poisson\")\n", "None" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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Xl9oTbPVqvSMRovnUWf117bXXkp2dzdChQ2nbtq26k8FAWlpaswRYG1csqSgK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\nVUKCmlhuvFFNLN7eekckmsP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"text": [ "" ] } ], "prompt_number": 10 }, { "cell_type": "markdown", "metadata": {}, "source": [ "## SciPy.stats : une biblioteque avec beaucoup de distributions\n", "\n", "SciPy est une biblioth\u00e8que tr\u00e8s populaire chez les scientifiques. SciPy fournit des outils pour toutes les distributions communes. " ] }, { "cell_type": "code", "collapsed": false, "input": [ "from scipy import stats\n", "\n", "# Cr\u00e9er une distribution Poission avec lambda \u00e9gal \u00e0 7. \n", "nv = stats.poisson(7)\n", "# Trouver la moyenne et l'\u00e9cart type.\n", "nv.mean(), nv.std() " ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\begin{pmatrix}7.0, & 2.64575131106\\end{pmatrix}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 11, "text": [ "(7.0, 2.64575131106)" ] } ], "prompt_number": 11 }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Fonction de masse (Probability mass function)\n", "\n", "Voil\u00e0 la m\u00eame fonction qu'on a \u00e9crite ci-dessus." ] }, { "cell_type": "code", "collapsed": false, "input": [ "def dessiner(titre, fonc, x):\n", " \"\"\"Dessiner fonc(x).\"\"\"\n", " fig = plt.figure()\n", " axes = fig.add_axes([0.1, 0.1, 0.8, 0.8])\n", " axes.plot(x, fonc(x))\n", " axes.set_title(titre)\n", "\n", "dessiner(\"Distribution Poisson\", nv.pmf, range(0, 21))" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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LJAYH+/FH+PZbVZUk3M+YMWpa7m3b9I5ECPuRxOBgW7fCqFEyaZ67Mhjgz3+G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\nz4mWVhF1aOj3rNskBk2DO++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"text": [ "" ] } ], "prompt_number": 12 }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Fonction de r\u00e9partition (Cummulative distribution function)\n", "\n", "Cette fonction est l'integral de la fonction de masse. Elle donne la probabilit\u00e9 qu'on a vu entre 0 et $k$ voitures, inclusif." ] }, { "cell_type": "code", "collapsed": false, "input": [ "dessiner(\"Distribution Poisson (cdf)\", nv.cdf, range(0, 21))" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 13 }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Fonction de survie (Survival function)\n", "\n", "Cette fonction est 1 moins la fonction de r\u00e9partition. Elle donne la probabilit\u00e9 qu'on a vu plus que $k$ voitures." ] }, { "cell_type": "code", "collapsed": false, "input": [ "dessiner(\"Distribution Poisson (sf)\", nv.sf, range(0, 21))" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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