{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Analyse du risque de défaillance des joints toriques de la navette Challenger"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Le 27 Janvier 1986, veille du décollage de la navette *Challenger*, eu\n",
    "lieu une télé-conférence de trois heures entre les ingénieurs de la\n",
    "Morton Thiokol (constructeur d'un des moteurs) et de la NASA. La\n",
    "discussion portait principalement sur les conséquences de la\n",
    "température prévue au moment du décollage de 31°F (juste en dessous de\n",
    "0°C) sur le succès du vol et en particulier sur la performance des\n",
    "joints toriques utilisés dans les moteurs. En effet, aucun test\n",
    "n'avait été effectué à cette température.\n",
    "\n",
    "L'étude qui suit reprend donc une partie des analyses effectuées cette\n",
    "nuit là et dont l'objectif était d'évaluer l'influence potentielle de\n",
    "la température et de la pression à laquelle sont soumis les joints\n",
    "toriques sur leur probabilité de dysfonctionnement. Pour cela, nous\n",
    "disposons des résultats des expériences réalisées par les ingénieurs\n",
    "de la NASA durant les 6 années précédant le lancement de la navette\n",
    "Challenger.\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Chargement des données\n",
    "Nous commençons donc par charger ces données:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Date</th>\n",
       "      <th>Count</th>\n",
       "      <th>Temperature</th>\n",
       "      <th>Pressure</th>\n",
       "      <th>Malfunction</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>4/12/81</td>\n",
       "      <td>6</td>\n",
       "      <td>66</td>\n",
       "      <td>50</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>11/12/81</td>\n",
       "      <td>6</td>\n",
       "      <td>70</td>\n",
       "      <td>50</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>3/22/82</td>\n",
       "      <td>6</td>\n",
       "      <td>69</td>\n",
       "      <td>50</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>11/11/82</td>\n",
       "      <td>6</td>\n",
       "      <td>68</td>\n",
       "      <td>50</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>4/04/83</td>\n",
       "      <td>6</td>\n",
       "      <td>67</td>\n",
       "      <td>50</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>6/18/82</td>\n",
       "      <td>6</td>\n",
       "      <td>72</td>\n",
       "      <td>50</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>8/30/83</td>\n",
       "      <td>6</td>\n",
       "      <td>73</td>\n",
       "      <td>100</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>11/28/83</td>\n",
       "      <td>6</td>\n",
       "      <td>70</td>\n",
       "      <td>100</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>2/03/84</td>\n",
       "      <td>6</td>\n",
       "      <td>57</td>\n",
       "      <td>200</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>4/06/84</td>\n",
       "      <td>6</td>\n",
       "      <td>63</td>\n",
       "      <td>200</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>8/30/84</td>\n",
       "      <td>6</td>\n",
       "      <td>70</td>\n",
       "      <td>200</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>10/05/84</td>\n",
       "      <td>6</td>\n",
       "      <td>78</td>\n",
       "      <td>200</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>11/08/84</td>\n",
       "      <td>6</td>\n",
       "      <td>67</td>\n",
       "      <td>200</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>1/24/85</td>\n",
       "      <td>6</td>\n",
       "      <td>53</td>\n",
       "      <td>200</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>4/12/85</td>\n",
       "      <td>6</td>\n",
       "      <td>67</td>\n",
       "      <td>200</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>4/29/85</td>\n",
       "      <td>6</td>\n",
       "      <td>75</td>\n",
       "      <td>200</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>6/17/85</td>\n",
       "      <td>6</td>\n",
       "      <td>70</td>\n",
       "      <td>200</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>7/29/85</td>\n",
       "      <td>6</td>\n",
       "      <td>81</td>\n",
       "      <td>200</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>8/27/85</td>\n",
       "      <td>6</td>\n",
       "      <td>76</td>\n",
       "      <td>200</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>10/03/85</td>\n",
       "      <td>6</td>\n",
       "      <td>79</td>\n",
       "      <td>200</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>10/30/85</td>\n",
       "      <td>6</td>\n",
       "      <td>75</td>\n",
       "      <td>200</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>11/26/85</td>\n",
       "      <td>6</td>\n",
       "      <td>76</td>\n",
       "      <td>200</td>\n",
       "      <td>0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>1/12/86</td>\n",
       "      <td>6</td>\n",
       "      <td>58</td>\n",
       "      <td>200</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "         Date  Count  Temperature  Pressure  Malfunction\n",
       "0     4/12/81      6           66        50            0\n",
       "1    11/12/81      6           70        50            1\n",
       "2     3/22/82      6           69        50            0\n",
       "3    11/11/82      6           68        50            0\n",
       "4     4/04/83      6           67        50            0\n",
       "5     6/18/82      6           72        50            0\n",
       "6     8/30/83      6           73       100            0\n",
       "7    11/28/83      6           70       100            0\n",
       "8     2/03/84      6           57       200            1\n",
       "9     4/06/84      6           63       200            1\n",
       "10    8/30/84      6           70       200            1\n",
       "11   10/05/84      6           78       200            0\n",
       "12   11/08/84      6           67       200            0\n",
       "13    1/24/85      6           53       200            2\n",
       "14    4/12/85      6           67       200            0\n",
       "15    4/29/85      6           75       200            0\n",
       "16    6/17/85      6           70       200            0\n",
       "17  7/29/85      6           81       200            0\n",
       "18    8/27/85      6           76       200            0\n",
       "19   10/03/85      6           79       200            0\n",
       "20   10/30/85      6           75       200            2\n",
       "21   11/26/85      6           76       200            0\n",
       "22    1/12/86      6           58       200            1"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "data = pd.read_csv(\"shuttle.csv\")\n",
    "data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Le jeu de données nous indique la date de l'essai, le nombre de joints\n",
    "toriques mesurés (il y en a 6 sur le lançeur principal), la\n",
    "température (en Farenheit) et la pression (en psi), et enfin le\n",
    "nombre de dysfonctionnements relevés. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Inspection graphique des données\n",
    "Les vols où aucun incident n'est relevé n'apportant aucun information\n",
    "sur l'influence de la température ou de la pression sur les\n",
    "dysfonctionnements, nous nous concentrons sur les expériences où au\n",
    "moins un joint a été défectueux.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "      <th>Date</th>\n",
       "      <th>Count</th>\n",
       "      <th>Temperature</th>\n",
       "      <th>Pressure</th>\n",
       "      <th>Malfunction</th>\n",
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       "      <th>1</th>\n",
       "      <td>11/12/81</td>\n",
       "      <td>6</td>\n",
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       "      <td>1</td>\n",
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       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>2/03/84</td>\n",
       "      <td>6</td>\n",
       "      <td>57</td>\n",
       "      <td>200</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>4/06/84</td>\n",
       "      <td>6</td>\n",
       "      <td>63</td>\n",
       "      <td>200</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>8/30/84</td>\n",
       "      <td>6</td>\n",
       "      <td>70</td>\n",
       "      <td>200</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>1/24/85</td>\n",
       "      <td>6</td>\n",
       "      <td>53</td>\n",
       "      <td>200</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>10/30/85</td>\n",
       "      <td>6</td>\n",
       "      <td>75</td>\n",
       "      <td>200</td>\n",
       "      <td>2</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>1/12/86</td>\n",
       "      <td>6</td>\n",
       "      <td>58</td>\n",
       "      <td>200</td>\n",
       "      <td>1</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "        Date  Count  Temperature  Pressure  Malfunction\n",
       "1   11/12/81      6           70        50            1\n",
       "8    2/03/84      6           57       200            1\n",
       "9    4/06/84      6           63       200            1\n",
       "10   8/30/84      6           70       200            1\n",
       "13   1/24/85      6           53       200            2\n",
       "20  10/30/85      6           75       200            2\n",
       "22   1/12/86      6           58       200            1"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data = data[data.Malfunction>0]\n",
    "data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Très bien, nous avons une variabilité de température importante mais\n",
    "la pression est quasiment toujours égale à 200, ce qui devrait\n",
    "simplifier l'analyse.\n",
    "\n",
    "Comment la fréquence d'échecs varie-t-elle avec la température ?\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "[encoded data, 7604 characters]\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "pd.set_option('mode.chained_assignment',None) # this removes a useless warning from pandas\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "data[\"Frequency\"]=data.Malfunction/data.Count\n",
    "data.plot(x=\"Temperature\",y=\"Frequency\",kind=\"scatter\",ylim=[0,1])\n",
    "plt.grid(True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "À première vue, ce n'est pas flagrant mais bon, essayons quand même\n",
    "d'estimer l'impact de la température $t$ sur la probabilité de\n",
    "dysfonctionnements d'un joint. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Estimation de l'influence de la température\n",
    "\n",
    "Supposons que chacun des 6 joints toriques est endommagé avec la même\n",
    "probabilité et indépendamment des autres et que cette probabilité ne\n",
    "dépend que de la température. Si on note $p(t)$ cette probabilité, le\n",
    "nombre de joints $D$ dysfonctionnant lorsque l'on effectue le vol à\n",
    "température $t$ suit une loi binomiale de paramètre $n=6$ et\n",
    "$p=p(t)$. Pour relier $p(t)$ à $t$, on va donc effectuer une\n",
    "régression logistique."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table class=\"simpletable\">\n",
       "<caption>Generalized Linear Model Regression Results</caption>\n",
       "<tr>\n",
       "  <th>Dep. Variable:</th>      <td>Frequency</td>    <th>  No. Observations:  </th>  <td>     7</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model:</th>                 <td>GLM</td>       <th>  Df Residuals:      </th>  <td>     5</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Model Family:</th>       <td>Binomial</td>     <th>  Df Model:          </th>  <td>     1</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Link Function:</th>        <td>logit</td>      <th>  Scale:             </th> <td>  1.0000</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Method:</th>               <td>IRLS</td>       <th>  Log-Likelihood:    </th> <td> -2.5250</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Date:</th>           <td>Sat, 13 Apr 2019</td> <th>  Deviance:          </th> <td> 0.22231</td> \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Time:</th>               <td>19:11:24</td>     <th>  Pearson chi2:      </th>  <td> 0.236</td>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>No. Iterations:</th>         <td>4</td>        <th>  Covariance Type:   </th> <td>nonrobust</td>\n",
       "</tr>\n",
       "</table>\n",
       "<table class=\"simpletable\">\n",
       "<tr>\n",
       "       <td></td>          <th>coef</th>     <th>std err</th>      <th>z</th>      <th>P>|z|</th>  <th>[0.025</th>    <th>0.975]</th>  \n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Intercept</th>   <td>   -1.3895</td> <td>    7.828</td> <td>   -0.178</td> <td> 0.859</td> <td>  -16.732</td> <td>   13.953</td>\n",
       "</tr>\n",
       "<tr>\n",
       "  <th>Temperature</th> <td>    0.0014</td> <td>    0.122</td> <td>    0.012</td> <td> 0.991</td> <td>   -0.238</td> <td>    0.240</td>\n",
       "</tr>\n",
       "</table>"
      ],
      "text/plain": [
       "<class 'statsmodels.iolib.summary.Summary'>\n",
       "\"\"\"\n",
       "                 Generalized Linear Model Regression Results                  \n",
       "==============================================================================\n",
       "Dep. Variable:              Frequency   No. Observations:                    7\n",
       "Model:                            GLM   Df Residuals:                        5\n",
       "Model Family:                Binomial   Df Model:                            1\n",
       "Link Function:                  logit   Scale:                          1.0000\n",
       "Method:                          IRLS   Log-Likelihood:                -2.5250\n",
       "Date:                Sat, 13 Apr 2019   Deviance:                      0.22231\n",
       "Time:                        19:11:24   Pearson chi2:                    0.236\n",
       "No. Iterations:                     4   Covariance Type:             nonrobust\n",
       "===============================================================================\n",
       "                  coef    std err          z      P>|z|      [0.025      0.975]\n",
       "-------------------------------------------------------------------------------\n",
       "Intercept      -1.3895      7.828     -0.178      0.859     -16.732      13.953\n",
       "Temperature     0.0014      0.122      0.012      0.991      -0.238       0.240\n",
       "===============================================================================\n",
       "\"\"\""
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import statsmodels.api as sm\n",
    "\n",
    "data[\"Success\"]=data.Count-data.Malfunction\n",
    "data[\"Intercept\"]=1\n",
    "\n",
    "logmodel=sm.GLM(data['Frequency'], data[['Intercept','Temperature']], family=sm.families.Binomial(sm.families.links.logit)).fit()\n",
    "\n",
    "logmodel.summary()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "L'estimateur le plus probable du paramètre de température est 0.0014\n",
    "et l'erreur standard de cet estimateur est de 0.122, autrement dit on\n",
    "ne peut pas distinguer d'impact particulier et il faut prendre nos\n",
    "estimations avec des pincettes.\n"
   ]
  },
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   "source": [
    "## Estimation de la probabilité de dysfonctionnant des joints toriques\n",
    "La température prévue le jour du décollage est de 31°F. Essayons\n",
    "d'estimer la probabilité de dysfonctionnement des joints toriques à\n",
    "cette température à partir du modèle que nous venons de construire:\n"
   ]
  },
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   "execution_count": 5,
   "metadata": {},
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    {
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       "<Figure size 432x288 with 1 Axes>"
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   "source": [
    "%matplotlib inline\n",
    "data_pred = pd.DataFrame({'Temperature': np.linspace(start=30, stop=90, num=121), 'Intercept': 1})\n",
    "data_pred['Frequency'] = logmodel.predict(data_pred[['Intercept','Temperature']])\n",
    "data_pred.plot(x=\"Temperature\",y=\"Frequency\",kind=\"line\",ylim=[0,1])\n",
    "plt.scatter(x=data[\"Temperature\"],y=data[\"Frequency\"])\n",
    "plt.grid(True)"
   ]
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   "cell_type": "markdown",
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    "Comme on pouvait s'attendre au vu des données initiales, la\n",
    "température n'a pas d'impact notable sur la probabilité d'échec des\n",
    "joints toriques. Elle sera d'environ 0.2, comme dans les essais\n",
    "précédents où nous il y a eu défaillance d'au moins un joint. Revenons\n",
    "à l'ensemble des données initiales pour estimer la probabilité de\n",
    "défaillance d'un joint:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.06521739130434782\n"
     ]
    }
   ],
   "source": [
    "data = pd.read_csv(\"shuttle.csv\")\n",
    "print(np.sum(data.Malfunction)/np.sum(data.Count))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Cette probabilité est donc d'environ $p=0.065$, sachant qu'il existe\n",
    "un joint primaire un joint secondaire sur chacune des trois parties du\n",
    "lançeur, la probabilité de défaillance des deux joints d'un lançeur\n",
    "est de $p^2 \\approx 0.00425$. La probabilité de défaillance d'un des\n",
    "lançeur est donc de $1-(1-p^2)^3 \\approx 1.2%$.  Ça serait vraiment\n",
    "pas de chance... Tout est sous contrôle, le décollage peut donc avoir\n",
    "lieu demain comme prévu.\n",
    "\n",
    "Seulement, le lendemain, la navette Challenger explosera et emportera\n",
    "avec elle ses sept membres d'équipages. L'opinion publique est\n",
    "fortement touchée et lors de l'enquête qui suivra, la fiabilité des\n",
    "joints toriques sera directement mise en cause. Au delà des problèmes\n",
    "de communication interne à la NASA qui sont pour beaucoup dans ce\n",
    "fiasco, l'analyse précédente comporte (au moins) un petit\n",
    "problème... Saurez-vous le trouver ? Vous êtes libre de modifier cette\n",
    "analyse et de regarder ce jeu de données sous tous les angles afin\n",
    "d'expliquer ce qui ne va pas."
   ]
  }
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