diff --git a/module2/exo5/exo5_R_fr.ipynb b/module2/exo5/exo5_R_fr.ipynb new file mode 100644 index 0000000000000000000000000000000000000000..4a60719f46be949abde4b8de40513cbcc8ff0a47 --- /dev/null +++ b/module2/exo5/exo5_R_fr.ipynb @@ -0,0 +1,502 @@ +{ + "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." + ] + }, + { + "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": [ + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\n", + "
DateCountTemperaturePressureMalfunction
4/12/81 6 66 50 0
11/12/816 70 50 1
3/22/82 6 69 50 0
11/11/826 68 50 0
4/04/83 6 67 50 0
6/18/82 6 72 50 0
8/30/83 6 73 100 0
11/28/836 70 100 0
2/03/84 6 57 200 1
4/06/84 6 63 200 1
8/30/84 6 70 200 1
10/05/846 78 200 0
11/08/846 67 200 0
1/24/85 6 53 200 2
4/12/85 6 67 200 0
4/29/85 6 75 200 0
6/17/85 6 70 200 0
7/29/85 6 81 200 0
8/27/85 6 76 200 0
10/03/856 79 200 0
10/30/856 75 200 2
11/26/856 76 200 0
1/12/86 6 58 200 1
\n" + ], + "text/latex": [ + "\\begin{tabular}{r|lllll}\n", + " Date & Count & Temperature & Pressure & Malfunction\\\\\n", + "\\hline\n", + "\t 4/12/81 & 6 & 66 & 50 & 0 \\\\\n", + "\t 11/12/81 & 6 & 70 & 50 & 1 \\\\\n", + "\t 3/22/82 & 6 & 69 & 50 & 0 \\\\\n", + "\t 11/11/82 & 6 & 68 & 50 & 0 \\\\\n", + "\t 4/04/83 & 6 & 67 & 50 & 0 \\\\\n", + "\t 6/18/82 & 6 & 72 & 50 & 0 \\\\\n", + "\t 8/30/83 & 6 & 73 & 100 & 0 \\\\\n", + "\t 11/28/83 & 6 & 70 & 100 & 0 \\\\\n", + "\t 2/03/84 & 6 & 57 & 200 & 1 \\\\\n", + "\t 4/06/84 & 6 & 63 & 200 & 1 \\\\\n", + "\t 8/30/84 & 6 & 70 & 200 & 1 \\\\\n", + "\t 10/05/84 & 6 & 78 & 200 & 0 \\\\\n", + "\t 11/08/84 & 6 & 67 & 200 & 0 \\\\\n", + "\t 1/24/85 & 6 & 53 & 200 & 2 \\\\\n", + "\t 4/12/85 & 6 & 67 & 200 & 0 \\\\\n", + "\t 4/29/85 & 6 & 75 & 200 & 0 \\\\\n", + "\t 6/17/85 & 6 & 70 & 200 & 0 \\\\\n", + "\t 7/29/85 & 6 & 81 & 200 & 0 \\\\\n", + "\t 8/27/85 & 6 & 76 & 200 & 0 \\\\\n", + "\t 10/03/85 & 6 & 79 & 200 & 0 \\\\\n", + "\t 10/30/85 & 6 & 75 & 200 & 2 \\\\\n", + "\t 11/26/85 & 6 & 76 & 200 & 0 \\\\\n", + "\t 1/12/86 & 6 & 58 & 200 & 1 \\\\\n", + "\\end{tabular}\n" + ], + "text/markdown": [ + "\n", + "Date | Count | Temperature | Pressure | Malfunction | \n", + "|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|\n", + "| 4/12/81 | 6 | 66 | 50 | 0 | \n", + "| 11/12/81 | 6 | 70 | 50 | 1 | \n", + "| 3/22/82 | 6 | 69 | 50 | 0 | \n", + "| 11/11/82 | 6 | 68 | 50 | 0 | \n", + "| 4/04/83 | 6 | 67 | 50 | 0 | \n", + "| 6/18/82 | 6 | 72 | 50 | 0 | \n", + "| 8/30/83 | 6 | 73 | 100 | 0 | \n", + "| 11/28/83 | 6 | 70 | 100 | 0 | \n", + "| 2/03/84 | 6 | 57 | 200 | 1 | \n", + "| 4/06/84 | 6 | 63 | 200 | 1 | \n", + "| 8/30/84 | 6 | 70 | 200 | 1 | \n", + "| 10/05/84 | 6 | 78 | 200 | 0 | \n", + "| 11/08/84 | 6 | 67 | 200 | 0 | \n", + "| 1/24/85 | 6 | 53 | 200 | 2 | \n", + "| 4/12/85 | 6 | 67 | 200 | 0 | \n", + "| 4/29/85 | 6 | 75 | 200 | 0 | \n", + "| 6/17/85 | 6 | 70 | 200 | 0 | \n", + "| 7/29/85 | 6 | 81 | 200 | 0 | \n", + "| 8/27/85 | 6 | 76 | 200 | 0 | \n", + "| 10/03/85 | 6 | 79 | 200 | 0 | \n", + "| 10/30/85 | 6 | 75 | 200 | 2 | \n", + "| 11/26/85 | 6 | 76 | 200 | 0 | \n", + "| 1/12/86 | 6 | 58 | 200 | 1 | \n", + "\n", + "\n" + ], + "text/plain": [ + " Date Count Temperature Pressure Malfunction\n", + "1 4/12/81 6 66 50 0 \n", + "2 11/12/81 6 70 50 1 \n", + "3 3/22/82 6 69 50 0 \n", + "4 11/11/82 6 68 50 0 \n", + "5 4/04/83 6 67 50 0 \n", + "6 6/18/82 6 72 50 0 \n", + "7 8/30/83 6 73 100 0 \n", + "8 11/28/83 6 70 100 0 \n", + "9 2/03/84 6 57 200 1 \n", + "10 4/06/84 6 63 200 1 \n", + "11 8/30/84 6 70 200 1 \n", + "12 10/05/84 6 78 200 0 \n", + "13 11/08/84 6 67 200 0 \n", + "14 1/24/85 6 53 200 2 \n", + "15 4/12/85 6 67 200 0 \n", + "16 4/29/85 6 75 200 0 \n", + "17 6/17/85 6 70 200 0 \n", + "18 7/29/85 6 81 200 0 \n", + "19 8/27/85 6 76 200 0 \n", + "20 10/03/85 6 79 200 0 \n", + "21 10/30/85 6 75 200 2 \n", + "22 11/26/85 6 76 200 0 \n", + "23 1/12/86 6 58 200 1 " + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "data = read.csv(\"shuttle.csv\",header=T)\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." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\n", + "\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\t\n", + "\n", + "
A data.frame: 7 × 5
DateCountTemperaturePressureMalfunction
<fct><int><int><int><int>
211/12/81670 501
92/03/84 6572001
104/06/84 6632001
118/30/84 6702001
141/24/85 6532002
2110/30/856752002
231/12/86 6582001
\n" + ], + "text/latex": [ + "A data.frame: 7 × 5\n", + "\\begin{tabular}{r|lllll}\n", + " & Date & Count & Temperature & Pressure & Malfunction\\\\\n", + " & & & & & \\\\\n", + "\\hline\n", + "\t2 & 11/12/81 & 6 & 70 & 50 & 1\\\\\n", + "\t9 & 2/03/84 & 6 & 57 & 200 & 1\\\\\n", + "\t10 & 4/06/84 & 6 & 63 & 200 & 1\\\\\n", + "\t11 & 8/30/84 & 6 & 70 & 200 & 1\\\\\n", + "\t14 & 1/24/85 & 6 & 53 & 200 & 2\\\\\n", + "\t21 & 10/30/85 & 6 & 75 & 200 & 2\\\\\n", + "\t23 & 1/12/86 & 6 & 58 & 200 & 1\\\\\n", + "\\end{tabular}\n" + ], + "text/markdown": [ + "\n", + "A data.frame: 7 × 5\n", + "\n", + "| | Date <fct> | Count <int> | Temperature <int> | Pressure <int> | Malfunction <int> |\n", + "|---|---|---|---|---|---|\n", + "| 2 | 11/12/81 | 6 | 70 | 50 | 1 |\n", + "| 9 | 2/03/84 | 6 | 57 | 200 | 1 |\n", + "| 10 | 4/06/84 | 6 | 63 | 200 | 1 |\n", + "| 11 | 8/30/84 | 6 | 70 | 200 | 1 |\n", + "| 14 | 1/24/85 | 6 | 53 | 200 | 2 |\n", + "| 21 | 10/30/85 | 6 | 75 | 200 | 2 |\n", + "| 23 | 1/12/86 | 6 | 58 | 200 | 1 |\n", + "\n" + ], + "text/plain": [ + " Date Count Temperature Pressure Malfunction\n", + "2 11/12/81 6 70 50 1 \n", + "9 2/03/84 6 57 200 1 \n", + "10 4/06/84 6 63 200 1 \n", + "11 8/30/84 6 70 200 1 \n", + "14 1/24/85 6 53 200 2 \n", + "21 10/30/85 6 75 200 2 \n", + "23 1/12/86 6 58 200 1 " + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "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 ?" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/plain": [ + "plot without title" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "plot(data=data, Malfunction/Count ~ Temperature, ylim=c(0,1))" + ] + }, + { + "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." + ] + }, + { + "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": 5, + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "\n", + "Call:\n", + "glm(formula = Malfunction/Count ~ Temperature, family = binomial(link = \"logit\"), \n", + " data = data, weights = Count)\n", + "\n", + "Deviance Residuals: \n", + " 2 9 10 11 14 21 23 \n", + "-0.3015 -0.2836 -0.2919 -0.3015 0.6891 0.6560 -0.2850 \n", + "\n", + "Coefficients:\n", + " Estimate Std. Error z value Pr(>|z|)\n", + "(Intercept) -1.389528 3.195752 -0.435 0.664\n", + "Temperature 0.001416 0.049773 0.028 0.977\n", + "\n", + "(Dispersion parameter for binomial family taken to be 1)\n", + "\n", + " Null deviance: 1.3347 on 6 degrees of freedom\n", + "Residual deviance: 1.3339 on 5 degrees of freedom\n", + "AIC: 18.894\n", + "\n", + "Number of Fisher Scoring iterations: 4\n" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "logistic_reg = glm(data=data, Malfunction/Count ~ Temperature, weights=Count, \n", + " family=binomial(link='logit'))\n", + "summary(logistic_reg)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "L'estimateur le plus probable du paramètre de température est 0.001416\n", + "et l'erreur standard de cet estimateur est de 0.049, autrement dit on\n", + "ne peut pas distinguer d'impact particulier et il faut prendre nos\n", + "estimations avec des pincettes." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Estimation de la probabilité de dysfonctionnant des joints toriques\n", + "\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 :" + ] + }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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+ "text/plain": [ + "plot without title" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "tempv = seq(from=30, to=90, by = .5)\n", + "rmv <- predict(logistic_reg,list(Temperature=tempv),type=\"response\")\n", + "plot(tempv,rmv,type=\"l\",ylim=c(0,1))\n", + "points(data=data, Malfunction/Count ~ Temperature)" + ] + }, + { + "cell_type": "markdown", + "metadata": { + "hideCode": false, + "hidePrompt": false, + "scrolled": true + }, + "source": [ + "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 :" + ] + }, + { + "cell_type": "code", + "execution_count": 7, + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "0.0652173913043478" + ], + "text/latex": [ + "0.0652173913043478" + ], + "text/markdown": [ + "0.0652173913043478" + ], + "text/plain": [ + "[1] 0.06521739" + ] + }, + "metadata": {}, + "output_type": "display_data" + } + ], + "source": [ + "data_full = read.csv(\"shuttle.csv\",header=T)\n", + "sum(data_full$Malfunction)/sum(data_full$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." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "celltoolbar": "Hide code", + "kernelspec": { + "display_name": "R", + "language": "R", + "name": "ir" + }, + "language_info": { + "codemirror_mode": "r", + "file_extension": ".r", + "mimetype": "text/x-r-source", + "name": "R", + "pygments_lexer": "r", + "version": "3.5.1" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +}