From 3675a7838bc4e0767661642e9bbe12ad5697ada9 Mon Sep 17 00:00:00 2001 From: Marc Oudart Date: Wed, 8 Apr 2020 14:19:00 +0200 Subject: [PATCH] =?UTF-8?q?R=C3=A9solution=20exercice=205=20module=202?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- module2/exo5/exo5_fr.Rmd | 20 +- module2/exo5/exo5_fr.html | 532 ++++++++++++++++++++++++++++++++++++++ module2/exo5/exo5_fr.pdf | Bin 0 -> 240656 bytes 3 files changed, 546 insertions(+), 6 deletions(-) create mode 100644 module2/exo5/exo5_fr.html create mode 100644 module2/exo5/exo5_fr.pdf diff --git a/module2/exo5/exo5_fr.Rmd b/module2/exo5/exo5_fr.Rmd index 479d782..aeb1d82 100644 --- a/module2/exo5/exo5_fr.Rmd +++ b/module2/exo5/exo5_fr.Rmd @@ -2,9 +2,12 @@ title: "Analyse du risque de défaillance des joints toriques de la navette Challenger" author: "Arnaud Legrand" date: "28 juin 2018" -output: html_document +output: + html_document: default + pdf_document: default --- - +# __J'ai mis mes remarques en gras__ + Le 27 Janvier 1986, veille du décollage de la navette _Challenger_, eu lieu une télé-conférence de trois heures entre les ingénieurs de la Morton Thiokol (constructeur d'un des moteurs) et de la NASA. La @@ -39,13 +42,14 @@ nombre de dysfonctionnements relevés. Les vols où aucun incident n'est relevé n'apportant aucun information sur l'influence de la température ou de la pression sur les dysfonctionnements, nous nous concentrons sur les expériences où au -moins un joint a été défectueux. +moins un joint a été défectueux. +__Je me demande pourquoi est-ce qu'on a enlevé des données ...__ ```{r} -data = data[data$Malfunction>0,] -data +#data = data[data$Malfunction>0,] +#data ``` - +__J'ai enlevé la partie où seules les malfonctions sont comptées__ Très bien, nous avons une variabilité de température importante mais la pression est quasiment toujours égale à 200, ce qui devrait simplifier l'analyse. @@ -55,6 +59,7 @@ Comment la fréquence d'échecs varie-t-elle avec la température ? plot(data=data, Malfunction/Count ~ Temperature, ylim=c(0,1)) ``` +__Ah bah oui !!! On dirait qu'il y a moins de malfonctions pour les hautes températures !!__ À première vue, ce n'est pas flagrant mais bon, essayons quand même d'estimer l'impact de la température $t$ sur la probabilité de dysfonctionnements d'un joint. @@ -75,6 +80,7 @@ logistic_reg = glm(data=data, Malfunction/Count ~ Temperature, weights=Count, summary(logistic_reg) ``` +__Ah bah en fait l'effet de la température a été sous-estimé d'un facteur 10 ... Elle est de 0.014 maintenant donc significatif ...__ L'estimateur le plus probable du paramètre de température est 0.001416 et l'erreur standard de cet estimateur est de 0.049, autrement dit on ne peut pas distinguer d'impact particulier et il faut prendre nos @@ -93,6 +99,7 @@ plot(tempv,rmv,type="l",ylim=c(0,1)) points(data=data, Malfunction/Count ~ Temperature) ``` +__C'est horrible ... Il y a maintenant 80% de chance qu'il y ait une défaillance à 31°F ...__ Comme on pouvait s'attendre au vu des données initiales, la température n'a pas d'impact notable sur la probabilité d'échec des joints toriques. Elle sera d'environ 0.2, comme dans les essais @@ -105,6 +112,7 @@ data_full = read.csv("shuttle.csv",header=T) sum(data_full$Malfunction)/sum(data_full$Count) ``` +__Oui ça c'est sans compter la température ...__ Cette probabilité est donc d'environ $p=0.065$, sachant qu'il existe un joint primaire un joint secondaire sur chacune des trois parties du lançeur, la probabilité de défaillance des deux joints d'un lançeur diff --git a/module2/exo5/exo5_fr.html b/module2/exo5/exo5_fr.html new file mode 100644 index 0000000..e221ad0 --- /dev/null +++ b/module2/exo5/exo5_fr.html @@ -0,0 +1,532 @@ + + + + + + + + + + + + + + +Analyse du risque de défaillance des joints toriques de la navette Challenger + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ + + + + + + +
+

J’ai mis mes remarques en gras

+

Le 27 Janvier 1986, veille du décollage de la navette Challenger, eu lieu une télé-conférence de trois heures entre les ingénieurs de la Morton Thiokol (constructeur d’un des moteurs) et de la NASA. La discussion portait principalement sur les conséquences de la température prévue au moment du décollage de 31°F (juste en dessous de 0°C) sur le succès du vol et en particulier sur la performance des joints toriques utilisés dans les moteurs. En effet, aucun test n’avait été effectué à cette température.

+

L’étude qui suit reprend donc une partie des analyses effectuées cette nuit là et dont l’objectif était d’évaluer l’influence potentielle de la température et de la pression à laquelle sont soumis les joints toriques sur leur probabilité de dysfonctionnement. Pour cela, nous disposons des résultats des expériences réalisées par les ingénieurs de la NASA durant les 6 années précédant le lancement de la navette Challenger.

+
+
+

Chargement des données

+

Nous commençons donc par charger ces données:

+
data = read.csv("shuttle.csv",header=T)
+data
+
##        Date Count Temperature Pressure Malfunction
+## 1   4/12/81     6          66       50           0
+## 2  11/12/81     6          70       50           1
+## 3   3/22/82     6          69       50           0
+## 4  11/11/82     6          68       50           0
+## 5   4/04/83     6          67       50           0
+## 6   6/18/82     6          72       50           0
+## 7   8/30/83     6          73      100           0
+## 8  11/28/83     6          70      100           0
+## 9   2/03/84     6          57      200           1
+## 10  4/06/84     6          63      200           1
+## 11  8/30/84     6          70      200           1
+## 12 10/05/84     6          78      200           0
+## 13 11/08/84     6          67      200           0
+## 14  1/24/85     6          53      200           2
+## 15  4/12/85     6          67      200           0
+## 16  4/29/85     6          75      200           0
+## 17  6/17/85     6          70      200           0
+## 18  7/29/85     6          81      200           0
+## 19  8/27/85     6          76      200           0
+## 20 10/03/85     6          79      200           0
+## 21 10/30/85     6          75      200           2
+## 22 11/26/85     6          76      200           0
+## 23  1/12/86     6          58      200           1
+

Le jeu de données nous indique la date de l’essai, le nombre de joints toriques mesurés (il y en a 6 sur le lançeur principal), la température (en Farenheit) et la pression (en psi), et enfin le nombre de dysfonctionnements relevés.

+
+
+

Inspection graphique des données

+

Les vols où aucun incident n’est relevé n’apportant aucun information sur l’influence de la température ou de la pression sur les dysfonctionnements, nous nous concentrons sur les expériences où au moins un joint a été défectueux.
+Je me demande pourquoi est-ce qu’on a enlevé des données …

+
#data = data[data$Malfunction>0,]
+#data
+

J’ai enlevé la partie où seules les malfonctions sont comptées
+Très bien, nous avons une variabilité de température importante mais la pression est quasiment toujours égale à 200, ce qui devrait simplifier l’analyse.

+

Comment la fréquence d’échecs varie-t-elle avec la température ?

+
plot(data=data, Malfunction/Count ~ Temperature, ylim=c(0,1))
+

+

Ah bah oui !!! On dirait qu’il y a moins de malfonctions pour les hautes températures !!
+À première vue, ce n’est pas flagrant mais bon, essayons quand même d’estimer l’impact de la température \(t\) sur la probabilité de dysfonctionnements d’un joint.

+
+
+

Estimation de l’influence de la température

+

Supposons que chacun des 6 joints toriques est endommagé avec la même probabilité et indépendamment des autres et que cette probabilité ne dépend que de la température. Si on note \(p(t)\) cette probabilité, le nombre de joints \(D\) dysfonctionnant lorsque l’on effectue le vol à température \(t\) suit une loi binomiale de paramètre \(n=6\) et \(p=p(t)\). Pour relier \(p(t)\) à \(t\), on va donc effectuer une régression logistique.

+
logistic_reg = glm(data=data, Malfunction/Count ~ Temperature, weights=Count, 
+                   family=binomial(link='logit'))
+summary(logistic_reg)
+
## 
+## Call:
+## glm(formula = Malfunction/Count ~ Temperature, family = binomial(link = "logit"), 
+##     data = data, weights = Count)
+## 
+## Deviance Residuals: 
+##      Min        1Q    Median        3Q       Max  
+## -0.95227  -0.78299  -0.54117  -0.04379   2.65152  
+## 
+## Coefficients:
+##             Estimate Std. Error z value Pr(>|z|)  
+## (Intercept)  5.08498    3.05247   1.666   0.0957 .
+## Temperature -0.11560    0.04702  -2.458   0.0140 *
+## ---
+## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+## 
+## (Dispersion parameter for binomial family taken to be 1)
+## 
+##     Null deviance: 24.230  on 22  degrees of freedom
+## Residual deviance: 18.086  on 21  degrees of freedom
+## AIC: 35.647
+## 
+## Number of Fisher Scoring iterations: 5
+

Ah bah en fait l’effet de la température a été sous-estimé d’un facteur 10 … Elle est de 0.014 maintenant donc significatif …
+L’estimateur le plus probable du paramètre de température est 0.001416 et l’erreur standard de cet estimateur est de 0.049, autrement dit on ne peut pas distinguer d’impact particulier et il faut prendre nos estimations avec des pincettes.

+
+
+

Estimation de la probabilité de dysfonctionnant des joints toriques

+

La température prévue le jour du décollage est de 31°F. Essayons d’estimer la probabilité de dysfonctionnement des joints toriques à cette température à partir du modèle que nous venons de construire:

+
# shuttle=shuttle[shuttle$r!=0,] 
+tempv = seq(from=30, to=90, by = .5)
+rmv <- predict(logistic_reg,list(Temperature=tempv),type="response")
+plot(tempv,rmv,type="l",ylim=c(0,1))
+points(data=data, Malfunction/Count ~ Temperature)
+

+

C’est horrible … Il y a maintenant 80% de chance qu’il y ait une défaillance à 31°F …
+Comme on pouvait s’attendre au vu des données initiales, la température n’a pas d’impact notable sur la probabilité d’échec des joints toriques. Elle sera d’environ 0.2, comme dans les essais précédents où nous il y a eu défaillance d’au moins un joint. Revenons à l’ensemble des données initiales pour estimer la probabilité de défaillance d’un joint:

+
data_full = read.csv("shuttle.csv",header=T)
+sum(data_full$Malfunction)/sum(data_full$Count)
+
## [1] 0.06521739
+

Oui ça c’est sans compter la température …
+Cette probabilité est donc d’environ \(p=0.065\), sachant qu’il existe un joint primaire un joint secondaire sur chacune des trois parties du lançeur, la probabilité de défaillance des deux joints d’un lançeur est de \(p^2 \approx 0.00425\). La probabilité de défaillance d’un des lançeur est donc de \(1-(1-p^2)^3 \approx 1.2%\). Ça serait vraiment pas de chance… Tout est sous contrôle, le décollage peut donc avoir lieu demain comme prévu.

+

Seulement, le lendemain, la navette Challenger explosera et emportera avec elle ses sept membres d’équipages. L’opinion publique est fortement touchée et lors de l’enquête qui suivra, la fiabilité des joints toriques sera directement mise en cause. Au delà des problèmes de communication interne à la NASA qui sont pour beaucoup dans ce fiasco, l’analyse précédente comporte (au moins) un petit problème… Saurez-vous le trouver ? Vous êtes libre de modifier cette analyse et de regarder ce jeu de données sous tous les angles afin d’expliquer ce qui ne va pas.

+
+ + + + +
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