semaine2.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style=\"text-align: right\"><small>pierre-henri.wuillemin@lip6.fr<br/>git clone https://gitlab.lip6.fr/phw/projetMdC2018.git\n",
    "<br/>last version : <b>30/03/2018</b><br/>2018-fev</small></div>\n",
    "# 3i005 -Projet 3 - Chaîne de Markov - semaine 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Le  but de cette semaine est l'étude du comportement asymptotique d'une chaîne de Markov."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Il s'agira donc de réaliser un ensemble de codes qui permettront de calculer l'ergodicité, la convergence suivant différentes méthodes :\n",
    "1. Simulation\n",
    "2. Convergence de $\\pi_n$\n",
    "3. Convergence de $T^{(n)}$\n",
    "4. Résolution directe\n",
    "\n",
    "Ainsi que d'étudier le comportement des différentes méthodes en fonction de la taille de la châine de Markov."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Préambule technique\n",
    "Le projet prendra la forme d'un ensemble de fichiers `python` (**python3**) et un rapport. Il utilisera fortement la programmation objet. Même si ce n'est pas forcément très _pythonesque_, on s'attachera à ce sujet à définir une classe dans un fichier de même nom (sauf si spécification contraire). \n",
    "\n",
    "Ce notebook (`semaine2.ipynb`) est normalement directement exécutable néanmoins, si vous le préférez, vous pouvez également recopier vos codes de test dans un fichier python classique `semaine2.py`. Si vous êtes y êtes habitués ou si vous voulez vous y habituer, n'hésitez pas à utiliser des _frameworks_ de test comme [`nosetest`](http://pythontesting.net/framework/nose/nose-introduction/) , etc. Le rapport devra bien évidemment contenir un préambule technique pour expliquer vos choix et la manière d'exécuter vos codes.\n",
    "\n",
    "> L'énoncé du projet et les sources se trouvent dans le même repository gitlab. Il suffit donc de le mettre à jour :\n",
    "> ```bash\n",
    "> git pull --allow-unrelated-histories\n",
    "> ```\n",
    "\n",
    "Les codes source qui vous sont fournis sont \n",
    "- `CdM.py` qui contient une version à compléter de la classe `CdM`,\n",
    "- `FeuRouge.py` qui contient une version de la classe ` FeuRouge`,\n",
    "- `utils.py` qui contient quelques fonctions utilitaires.\n",
    "- `CdMSampler.py` qui contient une version à compléter de la classe de simulation `CdMSampler`\n",
    "- `coll*.py` qui contiennent des exemples de `Collector` (voir plus bas)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simulation d'une chaîne de Markov\n",
    "\n",
    "Le premier outil pour étudier le comportement d'une chaîne de Markov est la simulation. La simulation consiste à générer une séquence d'états suivant les caractéristiques ($\\pi_0$ et $P(X_t|X_{t-1})$). Cette séquence 'assez longue' permet de calculer toute statistique associée à la CdM. Nous allons donc mettre en place un framework d'échantillonnage suivant la Chaîne de Markov de manière assez générique afin de pouvoir y insérer tout code de calcul le long de la simulation.\n",
    "\n",
    "Nous allons, pour cela, suivre (à peu près) le design pattern d'**`Observer`**. Ce design pattern consiste à indiquer qu'une classe (le **`Subject`**) est _observée_ par un ensemble d'**`Observer`** qui sont notifiés à chaque fois qu'un évènement intéressant se produit dans le **`Subject`**.\n",
    "\n",
    "![Design Pattern Observer (from wikipedia)](images/observerDesignPattern.png)\n",
    "<div style=\"text-align: center\"><small>--Observer design pattern from wikipedia--</small></div>\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nous allons adapter ce pattern à notre cas : Le `Subject` sera notre simulateur qu'on appelera `CdMSampler` dans le fichier `CdMSampler.py`. Ce simulateur contient une liste d'objets de type `Collector` (nos `Observers`) qui ont pour but de recueillir l'information générée par le `CdMSampler`, de la traiter et, si nécessaire, de pouvoir influencer l'arrêt de la simulation.\n",
    "![Collectors hiérachie](images/hierarchyCollectors.png)\n",
    "\n",
    "Il suffira ensuite de définir des classes spécialisant `Collector` et effectuant les calculs ou les actions attendues. Le code des collectors `CollProgresser`, `CollTimeOut` et `CollSingleStateCounter` vous est fourni pour exemple."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 9\n",
    "\n",
    "Compléter la classe CdMSampler. Il faut en particulier :\n",
    "\n",
    "1) Fournir le corps de la méthode de classe `CdMSampler.draw_from_distribution(distribution)`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Probabilité estimé de A : 0.3988\n"
     ]
    }
   ],
   "source": [
    "from CdMSampler import CdMSampler\n",
    "nbrA=0\n",
    "nbr_iteration=10000\n",
    "for i in range(nbr_iteration):\n",
    "    # draw_from_distribution est une méthode de classe : pas besoin de créer d'objet MdCSampler.\n",
    "    if CdMSampler.draw_from_distribution({\"A\":0.4,\"B\":0.6})==\"A\":\n",
    "        nbrA+=1\n",
    "print(\"Probabilité estimé de A : {}\".format(nbrA/nbr_iteration))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2) Fournir le corps de la méthode `CdMSampler.run(max_iter)`. Elle appellera les `notify_*` pour communiquer avec les `collectors` et doit :\n",
    "> a. initialiser les `collectors` et la génération\n",
    ">\n",
    "> b. générer aléatoirement la séquence d'états en notifiant à chaque fois les `collecters` et en s'arrêtant si un `collector` le demande ou si le nombre d'itérations maximum est atteint. \n",
    ">\n",
    "> c. finaliser les `collectors`\n",
    ">\n",
    "> d. retourner le résultat calcule par `collect_results()`\n",
    "\n",
    "Par exemple :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "- Sampler sans collector\n",
      "{'nbr_iterations': 10}\n",
      "\n",
      "- Sampler avec CollProgresser (voir CollProgresser.py)\n",
      "run(67): #..#..#..#..#..#..#..#. <-- stop with 67 iterations\n",
      "{'nbr_iterations': 67}\n",
      "\n",
      " - Sampler avec CollProgresser et CollSinleStateCounter (voir CollSingleStateCounter.py)\n",
      "run(150): #..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#. <-- stop with 150 iterations\n",
      "État <2> visité 2 fois\n",
      "État <5> visité 144 fois\n",
      "{'nbr_iterations': 150, 'count': {2: 2, 5: 144}}\n",
      "\n",
      " - Sampler avec CollProgresser, CollSinleStateCounter et ColTimeOut (voir CollTimeOut.py)\n",
      "run(150000000000): #..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#..#. [Time Out] <-- stop with 3101 iterations\n",
      "État <2> visité 5 fois\n",
      "État <5> visité 3087 fois\n",
      "Durée : 1.0249733924865723s\n",
      "{'nbr_iterations': 3101, 'count': {2: 5, 5: 3087}, 'duration': 1.0249733924865723}\n"
     ]
    }
   ],
   "source": [
    "from CdMSampler import CdMSampler\n",
    "from MouseInMaze import MouseInMaze\n",
    "m=MouseInMaze()\n",
    "\n",
    "s=CdMSampler(m)\n",
    "print(\"- Sampler sans collector\")\n",
    "print(s.run(10))\n",
    "\n",
    "print(\"\\n- Sampler avec CollProgresser (voir CollProgresser.py)\")\n",
    "from CollProgresser import CollProgresser\n",
    "s.add_collector(CollProgresser(3,9))\n",
    "print(s.run(67))\n",
    "\n",
    "print(\"\\n - Sampler avec CollProgresser et CollSinleStateCounter (voir CollSingleStateCounter.py)\")\n",
    "from CollSingleStateCounter import CollSingleStateCounter\n",
    "s.add_collector(CollSingleStateCounter(m.get_states()[1]))\n",
    "s.add_collector(CollSingleStateCounter(m.get_states()[4]))\n",
    "print(s.run(150))\n",
    "\n",
    "print(\"\\n - Sampler avec CollProgresser, CollSinleStateCounter et ColTimeOut (voir CollTimeOut.py)\")\n",
    "from CollTimeOut import CollTimeOut\n",
    "s.add_collector(CollTimeOut(1)) # time out de 1 seconde\n",
    "print(s.run(150000000000))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Collectors\n",
    "\n",
    "Les `collectors` permettent d'effectuer des actions et des calculs le long de l'exécution de la simulation. Le deisgn pattern `Observer` permet de les cumuler. Par exemple, voici le code d'un `Collector` très simple ne faisant qu'afficher les états générés :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[debut]-> 21-> 2-> 1-> 1-> 2-> 1-> 2-> 4-> 3-> 1-> 1-> 2-> 4-> 3-> 2-> 1-> 1-> 2-> 1[fin : 20]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "{'nbr_iterations': 20}"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from Collector import Collector\n",
    "from CdMSampler import CdMSampler\n",
    "from MouseInMaze import MouseInMaze\n",
    "\n",
    "class BasicCollector(Collector):\n",
    "  def __init__(self):\n",
    "    pass\n",
    "\n",
    "  def initialize(self, cdm, max_iter):\n",
    "    print(\"[debut]\", end=\"\", flush=True)\n",
    "\n",
    "  def receive(self, cdm, iter, state):\n",
    "    if iter==1:\n",
    "        print(state, end=\"\", flush=True)\n",
    "    else:\n",
    "        print(\"-> \"+str(state), end=\"\", flush=True)\n",
    "    return False # True pour arrêter la génération\n",
    "\n",
    "  def finalize(self, cdm, iteration):\n",
    "    print(\"[fin : {}]\".format(iteration))\n",
    "\n",
    "  def get_results(self, cdm):\n",
    "    return None\n",
    "\n",
    "m=MouseInMaze()\n",
    "sampler=CdMSampler(m)\n",
    "sampler.add_collector(BasicCollector())\n",
    "sampler.run(20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "La cellule ci-dessus exécutée plusieurs fois, donnera des trajectoires différentes de la chaîne de Markov `MouseInMaze`, toujours de taille 20."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 10 \n",
    "Écrire dans `CollGetDistribution.py` une classe `CollGetDistribution` qui permet de calculer la probabilité de chaque état et qui arrête la simulation à la convergence. Pour tester la convergence, il s'agit de faire la différence entre 2 distributions de probabilité. On utilisera par exemple, la norme max. Le critère d'arrêt se fera par rapport à un argument `epsilon`.\n",
    "\n",
    "Par ailleurs, un autre argument `pas` indiquera de présenter la distribution courante toutes les `pas` itérations (en commençant par afficher l'itération initiale). Si `pas` est nul, il n'y aura pas d'affichage en cours de simulation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4 premières itérations de Monobestiole\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "{'erreur': 0.08333333333333337,\n",
       " 'nbr_iterations': 4,\n",
       " 'proba': {1: 0.75, 2: 0.25}}"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0QAAABZCAYAAAAeueT2AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\npG5ZiCRJkiR1y0IkSZIkqVv/BSqdn96CiCyeAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f45726b9ba8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0QAAABZCAYAAAAeueT2AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nSZK6ZSGSJEmS1C0LkSRJkqRu/RdpC5/eXMYVIwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f456a7edba8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0QAAABZCAYAAAAeueT2AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f456a7b84a8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0QAAABZCAYAAAAeueT2AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f456a788d68>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "from CdMSampler import CdMSampler\n",
    "from MonoBestiole import MonoBestiole\n",
    "from CollGetDistribution import CollGetDistribution\n",
    "\n",
    "print(\"4 premières itérations de Monobestiole\")\n",
    "m=MonoBestiole(15,0.5,0.5)\n",
    "sampler=CdMSampler(m)\n",
    "sampler.add_collector(CollGetDistribution(epsilon=1e-5,pas=1))\n",
    "sampler.run(4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Itérations de monobestiole jusque convergence\n",
      "Durée : 1.2607321739196777s\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "{'duration': 1.2607321739196777,\n",
       " 'erreur': 9.9997176374877483e-06,\n",
       " 'nbr_iterations': 66034,\n",
       " 'proba': {1: 0.33967864552572197,\n",
       "  2: 0.22375176048339468,\n",
       "  3: 0.14733542319749215,\n",
       "  4: 0.09692123635152121,\n",
       "  5: 0.06404373570790363,\n",
       "  6: 0.043160238062786786,\n",
       "  7: 0.028697772325958232,\n",
       "  8: 0.01964169430436297,\n",
       "  9: 0.013765844350551996,\n",
       "  10: 0.009177229567034664,\n",
       "  11: 0.005769842351551497,\n",
       "  12: 0.0035739705904623445,\n",
       "  13: 0.0021201520451895263,\n",
       "  14: 0.0013780988293731922,\n",
       "  15: 0.0009843563066951372}}"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0QAAABZCAYAAAAeueT2AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\npG5ZiCRJkiR1y0IkSZIkqVv/BSqdn96CiCyeAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f456a74ae10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0QAAABZCAYAAAAeueT2AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nJEmSJKlbFiJJkiRJ3fo/1qWf3s9hXpcAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f456a693b00>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0QAAABZCAYAAAAeueT2AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\nAAALEgAACxIB0t1+/AAAADl0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uIDIuMS4wLCBo\ndHRwOi8vbWF0cGxvdGxpYi5vcmcvpW3flQAACbxJREFUeJzt3W2sZWV5xvH/xYzjzCDNyJsVZ1qs\nMSghCpQQlJRQqS0gGXwJCUYNUSOa+IJNTVtKojWmSRvb0qYfTFqxQyLSUJS0oUogvn5C4yAIOqBR\nQd5kMLZFJWmx3H7Yawxl5gBzztrn2Xs9/19ysveZ7GSuO+ucvda19rPWSVUhSZIkST06pHUASZIk\nSWrFQiRJkiSpWxYiSZIkSd2yEEmSJEnqloVIkiRJUrcsRJIkSZK6ZSGSJEmS1C0LkSRJkqRuWYgk\nSZIkdWvjwbx409attXXbtnllaeY5W7a0jjA3mzdtah1hLjZt2NA6wtxsmOhshxwyzfMvhyStI8xN\nJjrbNKcauM0k6Vd2797946o66uled1CFaOu2bfzOxRevPtWCOv3441tHmJuX7NjROsJcbD/iiNYR\n5ubwww5rHWEutm7e3DrCXGyZ6EkHgE0bD2oXsTSmWvQANk71hMpEt9lUTxRJiyLJPc/kdf4mSpIk\nSeqWhUiSJElStyxEkiRJkrplIZIkSZLULQuRJEmSpG5ZiCRJkiR1y0IkSZIkqVsWIkmSJEndshBJ\nkiRJ6paFSJIkSVK3LESSJEmSumUhkiRJktQtC5EkSZKkblmIJEmSJHXLQiRJkiSpWxYiSZIkSd2y\nEEmSJEnqloVIkiRJUrcsRJIkSZK6ZSGSJEmS1C0LkSRJkqRuWYgkSZIkdctCJEmSJKlbFiJJkiRJ\n3bIQSZIkSeqWhUiSJElStyxEkiRJkrplIZIkSZLULQuRJEmSpG5ZiCRJkiR1y0IkSZIkqVsWIkmS\nJEndshBJkiRJ6paFSJIkSVK3LESSJEmSumUhkiRJktQtC5EkSZKkblmIJEmSJHXLQiRJkiSpWxYi\nSZIkSd2yEEmSJEnqloVIkiRJUrcsRJIkSZK6ZSGSJEmS1C0LkSRJkqRuWYgkSZIkdctCJEmSJKlb\nFiJJkiRJ3bIQSZIkSeqWhUiSJElStyxEkiRJkrplIZIkSZLULQuRJEmSpG6lqp75i5OHgXvmF+f/\nORL48Tr9X+tpqnPBdGdzruUz1dmca/lMdTbnWj5TnW2qc8F0Z1vPuX6zqo56uhcdVCFaT0m+XlWn\ntM4xtqnOBdOdzbmWz1Rnc67lM9XZnGv5THW2qc4F051tEedyyZwkSZKkblmIJEmSJHVrkQvRP7YO\nMCdTnQumO5tzLZ+pzuZcy2eqsznX8pnqbFOdC6Y728LNtbDXEEmSJEnSvC3yJ0SSJEmSNFcLV4iS\nfCLJ3iR3tM4ypiQ7knwxyZ4k30pySetMY0iyOcnXktw2zPXh1pnGlGRDkm8kub51ljEluTvJ7Ulu\nTfL11nnGkmRbkmuT3Dn8rr2idaYxJDlu2Fb7vh5J8v7WucaQ5A+H9447klydZHPrTGNIcskw07eW\nfVsdaL+c5PAkNyX57vD43JYZV2OFuS4YttnjSRbqLlgHY4XZPjq8N34zyXVJtrXMuBorzPWRYaZb\nk9yY5JiWGVfjqY59k3wgSSU5skW2tVphm/15kvufsE87t2VGWMBCBOwCzm4dYg5+AfxRVb0UOA14\nd5LjG2caw/8Ar6qqlwMnAmcnOa1xpjFdAuxpHWJOfreqTly0W1+u0d8DN1TVS4CXM5FtV1V3Ddvq\nROC3gUeB6xrHWrMkLwDeB5xSVScAG4AL26ZauyQnAO8ATmX2c3hekhe3TbUmu9h/v/ynwOer6sXA\n54fvl80u9p/rDuD1wFfWPc24drH/bDcBJ1TVy4DvAJeud6gR7GL/uT5aVS8b3h+vBz647qnWbhcH\nOPZNsgN4NfDD9Q40ol0c+Lj+8n37tar67Dpn2s/CFaKq+grwk9Y5xlZVD1bVLcPznzI7UHtB21Rr\nVzM/G7591vA1iQvTkmwHXgN8vHUWPb0kvwacAVwBUFX/W1X/1TbVXJwFfK+q1uuPZM/bRmBLko3A\nVuCBxnnG8FLg5qp6tKp+AXwZeF3jTKu2wn75fODK4fmVwGvXNdQIDjRXVe2pqrsaRRrNCrPdOPw8\nAtwMbF/3YGu0wlyPPOHbQ1nCY5CnOPa9HPhjlnCmfZbluH7hClEPkhwLnAR8tW2ScQzLym4F9gI3\nVdUk5gL+jtkb0eOtg8xBATcm2Z3k4tZhRvJbwMPAPw/LHD+e5NDWoebgQuDq1iHGUFX3A3/N7Ozn\ng8B/V9WNbVON4g7gjCRHJNkKnAvsaJxpbM+rqgdhdsIPOLpxHh2ctwGfax1iLEn+Ism9wJtYzk+I\n9pNkJ3B/Vd3WOsucvGdY6viJRVhyayFaZ0meA3waeP+Tzmosrar6v+Gj6u3AqcNykaWW5Dxgb1Xt\nbp1lTk6vqpOBc5gt3zyjdaARbAROBj5WVScBP2c5l/GsKMkmYCfwr62zjGHYCZ4PvBA4Bjg0yZvb\nplq7qtoD/BWzJUo3ALcxWzYtNZfkMmY/j1e1zjKWqrqsqnYwm+k9rfOs1XAi5TImUu4O4GPAi5hd\navEg8Ddt41iI1lWSZzErQ1dV1Wda5xnbsDzpS0zjGrDTgZ1J7gb+BXhVkk+2jTSeqnpgeNzL7FqU\nU9smGsV9wH1P+ITyWmYFaUrOAW6pqodaBxnJ7wE/qKqHq+ox4DPAKxtnGkVVXVFVJ1fVGcyWi3y3\ndaaRPZTk+QDD497GefQMJLkIOA94U03z7658CnhD6xAjeBGzE0W3Dcch24Fbkvx601QjqaqHhpPp\njwP/xAIcg1iI1kmSMLu2YU9V/W3rPGNJctS+O9Uk2cLsAOfOtqnWrqourartVXUssyVKX6iqpT9z\nDZDk0CSH7XsO/D6zJT5Lrap+BNyb5Ljhn84Cvt0w0jy8kYkslxv8EDgtydbhPfIsJnIjjCRHD4+/\nwewi/SltN4B/By4anl8E/FvDLHoGkpwN/Amws6oebZ1nLE+6YclOpnEMcntVHV1Vxw7HIfcBJw/7\nuaW372TK4HUswDHIxtYBnizJ1cCZwJFJ7gM+VFVXtE01itOBtwC3D9fbAPzZItxZY42eD1yZZAOz\ngn1NVU3qFtUT9DzgutnxJxuBT1XVDW0jjea9wFXD0rLvA29tnGc0wxKKVwPvbJ1lLFX11STXArcw\nW8LzDRbwL5iv0qeTHAE8Bry7qv6zdaDVOtB+GfhL4Jokb2dWbC9ol3B1VpjrJ8A/AEcB/5Hk1qr6\ng3YpV2eF2S4Fng3cNLz/31xV72oWchVWmOvc4UTY48A9wFLNBJM+9l1pm52Z5ERm1zPfzQLs1zLN\nT0wlSZIk6em5ZE6SJElStyxEkiRJkrplIZIkSZLULQuRJEmSpG5ZiCRJkiR1y0IkSZIkqVsWIkmS\nJEndshBJkiRJ6tYvAcddn96W2YBOAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f456a5eec18>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0QAAABZCAYAAAAeueT2AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nSeqWhUiSJElSt/4X282f3u/ahyIAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f456a5e7748>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from CollTimeOut import CollTimeOut\n",
    "\n",
    "print(\"Itérations de monobestiole jusque convergence\")\n",
    "m=MonoBestiole(15,0.4,0.6)\n",
    "sampler=CdMSampler(m)\n",
    "sampler.add_collector(CollGetDistribution(epsilon=1e-5,pas=20000))\n",
    "sampler.add_collector(CollTimeOut(10)) # time out de 10 seconde\n",
    "sampler.run(100000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Convergence et ergodicité\n",
    "Une Chaîne de Markov est dite convergente si le processus de simulation ci-dessus converge vers une distribution $\\pi^*$. \n",
    "\n",
    "Dans ce cadre, une propriété importante des CdM est l'**ergodicité**. Une CdM ergodique converge, vers la même distribution $\\pi^*$ quelque soit la distribution initiale $\\pi_0$.\n",
    "\n",
    "Pour une CdM finie, ergodicité est équivalent à irréductible+apériodique.\n",
    "\n",
    "### Question 11\n",
    "Ajouter une méthode dans `CdM.py` une méthode `CdM.is_ergodic()` qui vérifie si une chaîne de Markov est ergodique. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "done\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 12 \n",
    "Vérifier les propriétés d'ergodicité en analysant les résultats de simulation pour les Cdm \n",
    ":\n",
    "> 1. MouseInMaze\n",
    "> 2. MonoBestiole\n",
    "> 3. PeriodiCdM dont le code se trouve dans `PeriodicCdm.py`"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> 1. MouseInMaze\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Apériodique : True\n",
      "Périodicité : 1\n",
      "Irréductible : False\n",
      "Ergodicité False\n"
     ]
    }
   ],
   "source": [
    "m1=MouseInMaze()\n",
    "print(\"Apériodique : \"+str(m1.is_aperiodic()))\n",
    "print(\"Périodicité : \"+str(m1.get_periodicity()))\n",
    "print(\"Irréductible : \"+str(m1.is_irreducible()))\n",
    "print(\"Ergodicité\",m1.is_ergodic())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Itérations de mouse jusque convergence\n",
      "Durée : 7.414817810058594e-05s\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "{'duration': 7.414817810058594e-05,\n",
       " 'erreur': 0.0,\n",
       " 'nbr_iterations': 1,\n",
       " 'proba': {2: 1}}"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(\"Itérations de mouse jusque convergence\")\n",
    "sampler=CdMSampler(m1)\n",
    "sampler.add_collector(CollGetDistribution(epsilon=1e-5,pas=20000))\n",
    "sampler.add_collector(CollTimeOut(10))\n",
    "sampler.run(100000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> 2. MonoBestiole"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Apériodique : True\n",
      "Périodicité : 1\n",
      "Irréductible : True\n",
      "Ergodicité True\n"
     ]
    }
   ],
   "source": [
    "m2=MonoBestiole(15,0.5,0.5)\n",
    "print(\"Apériodique : \"+str(m2.is_aperiodic()))\n",
    "print(\"Périodicité : \"+str(m2.get_periodicity()))\n",
    "print(\"Irréductible : \"+str(m2.is_irreducible()))\n",
    "print(\"Ergodicité\",m2.is_ergodic())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Itérations de mono jusque convergence\n",
      "Durée : 1.7377772331237793s\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "{'duration': 1.7377772331237793,\n",
       " 'erreur': 9.9947710775066811e-06,\n",
       " 'nbr_iterations': 93006,\n",
       " 'proba': {1: 0.07263050373635827,\n",
       "  2: 0.07136175474436858,\n",
       "  3: 0.07042632116552873,\n",
       "  4: 0.06910381162303102,\n",
       "  5: 0.06736196978657061,\n",
       "  6: 0.06651255308854363,\n",
       "  7: 0.06750174721789151,\n",
       "  8: 0.06772754153002526,\n",
       "  9: 0.06614698134508898,\n",
       "  10: 0.06611472501478415,\n",
       "  11: 0.06479221547228643,\n",
       "  12: 0.06310413418633407,\n",
       "  13: 0.062405247029729584,\n",
       "  14: 0.06222246115800226,\n",
       "  15: 0.06258803290145691}}"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0QAAABZCAYAAAAeueT2AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f45726b9828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0QAAABZCAYAAAAeueT2AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n1C0LkSRJkqRuWYgkSZIkdeunj0jB+mL+jNsAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f456a586828>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0QAAABZCAYAAAAeueT2AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n1fLUAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f456a52ef60>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0QAAABZCAYAAAAeueT2AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\ntyxEkiRJkrr1f+CXsfBp6og0AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f456a74ad68>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0QAAABZCAYAAAAeueT2AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f456a836c88>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print(\"Itérations de mono jusque convergence\")\n",
    "sampler=CdMSampler(m2)\n",
    "sampler.add_collector(CollGetDistribution(epsilon=1e-5,pas=20000))\n",
    "sampler.add_collector(CollTimeOut(10)) \n",
    "sampler.run(100000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> 3. PeriodiCdM"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Apériodique : False\n",
      "Périodicité : 4\n",
      "Irréductible : True\n",
      "Ergodicité False\n"
     ]
    }
   ],
   "source": [
    "from PeriodicCdM import PeriodicCdM\n",
    "period=PeriodicCdM()\n",
    "\n",
    "print(\"Apériodique : \"+str(period.is_aperiodic()))\n",
    "print(\"Périodicité : \"+str(period.get_periodicity()))\n",
    "print(\"Irréductible : \"+str(period.is_irreducible()))\n",
    "print(\"Ergodicité\",period.is_ergodic())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Itérations de period jusque convergence\n",
      "Durée : 8.58306884765625e-05s\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "{'duration': 8.58306884765625e-05,\n",
       " 'erreur': 0.0,\n",
       " 'nbr_iterations': 1,\n",
       " 'proba': {'A': 1}}"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(\"Itérations de period jusque convergence\")\n",
    "sampler=CdMSampler(period)\n",
    "sampler.add_collector(CollGetDistribution(epsilon=1e-5,pas=20000))\n",
    "sampler.add_collector(CollTimeOut(10)) \n",
    "sampler.run(100000)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Calcul de $\\pi*$\n",
    "En notant $\\pi_n$ la distribution de la CdM à l'état $n$, on se souvient qu'on a :\n",
    "$$\\pi_{n+1}=\\pi_t \\cdot M = \\pi_0 \\cdot M^n$$\n",
    "où $M$ est la matrice de transition de la CdM.\n",
    "\n",
    "\n",
    "Lorsqu'une chaîne de Markov est irréductible, outre la simulation, on a donc plusieurs méthodes qui permettent de calculer $\\pi$ :\n",
    "\n",
    "1. Convergence de $\\pi_n$ : en itérant l'équation $\\pi_{n+1}=\\pi_n\\cdot M$ et en s'arrêtant quand la distance entre $\\pi_n$ et $\\pi_{n+1}$ est assez faible.\n",
    "\n",
    "2. Convergence de $M^n$ : La suite des puissances de $M$ converge (vers quoi ?).\n",
    "\n",
    "3. Point fixe : $\\pi^*$ est un point fixe et vérifié donc $$\\pi^*=\\pi^*\\cdot M$$ $\\pi^*$ est donc un vecteur propre de $M$ pour la valeur propre $1$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Question 13\n",
    "Écrire une classe `CdMConvergence` dans `CdMConvergence.py` qui calcule pour une CdM ergodique la distribution $\\pi^*$ selon les 4 méthodes en relevant pour chacune la valeur calculée pour $\\pi^*$, le nombre d'itérations nécessaires (si c'est un processus itératif), le temps nécessaire."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "La chaine de Markov n'est ergodique\n",
      "[[ 0.5   0.5   0.    0.    0.    0.  ]\n",
      " [ 0.5   0.    0.    0.5   0.    0.  ]\n",
      " [ 0.25  0.25  0.    0.    0.25  0.25]\n",
      " [ 0.    0.    1.    0.    0.    0.  ]\n",
      " [ 0.    0.    0.    0.    1.    0.  ]\n",
      " [ 0.    0.    0.    0.    0.    1.  ]]\n",
      "{'proba': array([ 0.+0.j,  0.+0.j,  0.+0.j,  0.+0.j,  1.+0.j,  0.+0.j]), 'temps_conver': 0.10573601722717285}\n"
     ]
    }
   ],
   "source": [
    "from CdMConvergence import CdMConvergence\n",
    "c=CdMConvergence.Convergence4Methodes(m1,10,100000,0.000000000001)\n",
    "print(CdMConvergence.ConvergencePtfixe(m1,10))"
   ]
  }
 ],
 "metadata": {
  "hide_input": false,
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}