{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "" ] }, { "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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DateCountTemperaturePressureMalfunction
04/12/81666500
111/12/81670501
23/22/82669500
311/11/82668500
44/04/83667500
56/18/82672500
68/30/836731000
711/28/836701000
82/03/846572001
94/06/846632001
108/30/846702001
1110/05/846782000
1211/08/846672000
131/24/856532002
144/12/856672000
154/29/856752000
166/17/856702000
177/29/856812000
188/27/856762000
1910/03/856792000
2010/30/856752002
2111/26/856762000
221/12/866582001
\n", "
" ], "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", "\n", "* secon test avec uniquement les tempéraures <65" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
DateCountTemperaturePressureMalfunction
82/03/846572001
94/06/846632001
131/24/856532002
221/12/866582001
\n", "
" ], "text/plain": [ " Date Count Temperature Pressure Malfunction\n", "8 2/03/84 6 57 200 1\n", "9 4/06/84 6 63 200 1\n", "13 1/24/85 6 53 200 2\n", "22 1/12/86 6 58 200 1" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "#data = data[data.Malfunction>0] data.Temperature <= 65\n", "data2 = data[data.Temperature <= 65]\n", "data2" ] }, { "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": 9, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "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", "data2[\"Frequency\"]=data2.Malfunction/data2.Count\n", "data2.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": 10, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "
Generalized Linear Model Regression Results
Dep. Variable: Frequency No. Observations: 4
Model: GLM Df Residuals: 2
Model Family: Binomial Df Model: 1
Link Function: logit Scale: 1.0000
Method: IRLS Log-Likelihood: -1.3845
Date: Mon, 18 Aug 2025 Deviance: 0.040847
Time: 15:47:24 Pearson chi2: 0.0407
No. Iterations: 4 Covariance Type: nonrobust
\n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "\n", " \n", "\n", "
coef std err z P>|z| [0.025 0.975]
Intercept 4.3201 20.789 0.208 0.835 -36.425 45.066
Temperature -0.0985 0.364 -0.271 0.787 -0.812 0.615
" ], "text/plain": [ "\n", "\"\"\"\n", " Generalized Linear Model Regression Results \n", "==============================================================================\n", "Dep. Variable: Frequency No. Observations: 4\n", "Model: GLM Df Residuals: 2\n", "Model Family: Binomial Df Model: 1\n", "Link Function: logit Scale: 1.0000\n", "Method: IRLS Log-Likelihood: -1.3845\n", "Date: Mon, 18 Aug 2025 Deviance: 0.040847\n", "Time: 15:47:24 Pearson chi2: 0.0407\n", "No. Iterations: 4 Covariance Type: nonrobust\n", "===============================================================================\n", " coef std err z P>|z| [0.025 0.975]\n", "-------------------------------------------------------------------------------\n", "Intercept 4.3201 20.789 0.208 0.835 -36.425 45.066\n", "Temperature -0.0985 0.364 -0.271 0.787 -0.812 0.615\n", "===============================================================================\n", "\"\"\"" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import statsmodels.api as sm\n", "\n", "data2[\"Success\"]=data2.Count-data2.Malfunction\n", "data2[\"Intercept\"]=1\n", "\n", "logmodel=sm.GLM(data2['Frequency'], data2[['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" ] }, { "cell_type": "markdown", "metadata": {}, "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" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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DateCountTemperaturePressureMalfunctionFrequencySuccessIntercept
82/03/8465720010.16666751
94/06/8466320010.16666751
131/24/8565320020.33333341
221/12/8665820010.16666751
\n", "
" ], "text/plain": [ " Date Count Temperature Pressure Malfunction Frequency Success \\\n", "8 2/03/84 6 57 200 1 0.166667 5 \n", "9 4/06/84 6 63 200 1 0.166667 5 \n", "13 1/24/85 6 53 200 2 0.333333 4 \n", "22 1/12/86 6 58 200 1 0.166667 5 \n", "\n", " Intercept \n", "8 1 \n", "9 1 \n", "13 1 \n", "22 1 " ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "data2" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "data_pred = pd.DataFrame({'Temperature': np.linspace(start=25, stop=60, 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=data2[\"Temperature\"],y=data2[\"Frequency\"])\n", "plt.grid(True)" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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InterceptTemperatureFrequency
0125.0000000.864905
1125.2916670.861511
2125.5833330.858046
3125.8750000.854509
4126.1666670.850900
5126.4583330.847216
6126.7500000.843459
7127.0416670.839627
8127.3333330.835719
9127.6250000.831734
10127.9166670.827674
11128.2083330.823536
12128.5000000.819320
13128.7916670.815026
14129.0833330.810654
15129.3750000.806203
16129.6666670.801673
17129.9583330.797064
18130.2500000.792375
19130.5416670.787607
20130.8333330.782759
21131.1250000.777832
22131.4166670.772826
23131.7083330.767741
24132.0000000.762576
25132.2916670.757333
26132.5833330.752012
27132.8750000.746614
28133.1666670.741138
29133.4583330.735586
............
91151.5416670.318910
92151.8333330.312700
93152.1250000.306557
94152.4166670.300481
95152.7083330.294475
96153.0000000.288539
97153.2916670.282675
98153.5833330.276884
99153.8750000.271166
100154.1666670.265524
101154.4583330.259956
102154.7500000.254466
103155.0416670.249052
104155.3333330.243715
105155.6250000.238457
106155.9166670.233277
107156.2083330.228176
108156.5000000.223154
109156.7916670.218211
110157.0833330.213348
111157.3750000.208564
112157.6666670.203859
113157.9583330.199234
114158.2500000.194689
115158.5416670.190222
116158.8333330.185834
117159.1250000.181525
118159.4166670.177294
119159.7083330.173141
120160.0000000.169064
\n", "

121 rows × 3 columns

\n", "
" ], "text/plain": [ " Intercept Temperature Frequency\n", "0 1 25.000000 0.864905\n", "1 1 25.291667 0.861511\n", "2 1 25.583333 0.858046\n", "3 1 25.875000 0.854509\n", "4 1 26.166667 0.850900\n", "5 1 26.458333 0.847216\n", "6 1 26.750000 0.843459\n", "7 1 27.041667 0.839627\n", "8 1 27.333333 0.835719\n", "9 1 27.625000 0.831734\n", "10 1 27.916667 0.827674\n", "11 1 28.208333 0.823536\n", "12 1 28.500000 0.819320\n", "13 1 28.791667 0.815026\n", "14 1 29.083333 0.810654\n", "15 1 29.375000 0.806203\n", "16 1 29.666667 0.801673\n", "17 1 29.958333 0.797064\n", "18 1 30.250000 0.792375\n", "19 1 30.541667 0.787607\n", "20 1 30.833333 0.782759\n", "21 1 31.125000 0.777832\n", "22 1 31.416667 0.772826\n", "23 1 31.708333 0.767741\n", "24 1 32.000000 0.762576\n", "25 1 32.291667 0.757333\n", "26 1 32.583333 0.752012\n", "27 1 32.875000 0.746614\n", "28 1 33.166667 0.741138\n", "29 1 33.458333 0.735586\n", ".. ... ... ...\n", "91 1 51.541667 0.318910\n", "92 1 51.833333 0.312700\n", "93 1 52.125000 0.306557\n", "94 1 52.416667 0.300481\n", "95 1 52.708333 0.294475\n", "96 1 53.000000 0.288539\n", "97 1 53.291667 0.282675\n", "98 1 53.583333 0.276884\n", "99 1 53.875000 0.271166\n", "100 1 54.166667 0.265524\n", "101 1 54.458333 0.259956\n", "102 1 54.750000 0.254466\n", "103 1 55.041667 0.249052\n", "104 1 55.333333 0.243715\n", "105 1 55.625000 0.238457\n", "106 1 55.916667 0.233277\n", "107 1 56.208333 0.228176\n", "108 1 56.500000 0.223154\n", "109 1 56.791667 0.218211\n", "110 1 57.083333 0.213348\n", "111 1 57.375000 0.208564\n", "112 1 57.666667 0.203859\n", "113 1 57.958333 0.199234\n", "114 1 58.250000 0.194689\n", "115 1 58.541667 0.190222\n", "116 1 58.833333 0.185834\n", "117 1 59.125000 0.181525\n", "118 1 59.416667 0.177294\n", "119 1 59.708333 0.173141\n", "120 1 60.000000 0.169064\n", "\n", "[121 rows x 3 columns]" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "data_pred\n" ] }, { "cell_type": "markdown", "metadata": { "hideCode": false, "hidePrompt": false, "scrolled": true }, "source": [ "\n", "\n", "En ne prenant que la partie des température basse, on voit une influence notable de la baisse de température avec une probabilité d'échec de 0.77\n" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "data p : 0.06521739130434782\n", "data p^2 : 0.004253308128544423\n", "data 1−(1−𝑝2)3 : 0.01270572944054793 soit 1.27%\n", "\n", "data2 p : 0.20833333333333334\n", "data2 p^2 : 0.04340277777777778\n", "data2 1−(1−𝑝2)3 : 0.1246386921781899 soit 12.46%\n", "\n" ] } ], "source": [ "data = pd.read_csv(\"shuttle.csv\")\n", "p_data = np.sum(data.Malfunction)/np.sum(data.Count)\n", "p_data2 = np.sum(data2.Malfunction)/np.sum(data2.Count)\n", "print(\"data p : \",p_data)\n", "print('data p^2 : ',p_data**2)\n", "print('data 1−(1−𝑝2)3 : ',1-(1-p_data**2)**3,f' soit {(1-(1-p_data**2)**3)*100:.2f}%')\n", "print()\n", "print(\"data2 p : \",p_data2)\n", "print('data2 p^2 : ',p_data2**2)\n", "print('data2 1−(1−𝑝2)3 : ',1-(1-p_data2**2)**3,f' soit {(1-(1-p_data2**2)**3)*100:.2f}%')\n", "print (\"\")" ] }, { "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." ] } ], "metadata": { "celltoolbar": "Hide code", "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": "3.6.4" } }, "nbformat": 4, "nbformat_minor": 2 }