From db834875606ff46bda83fe185e0c506da0f827b9 Mon Sep 17 00:00:00 2001 From: Samuel Loury Date: Mon, 20 Jul 2020 11:27:37 +0200 Subject: [PATCH] Add confidence intervals and analyse pressure --- module2/exo5/exo5_python_fr.html | 518 +++++++++++++++++++++++++ module2/exo5/exo5_python_fr.org | 332 +++++++--------- module2/exo5/freq_pressure_python.png | Bin 0 -> 89090 bytes module2/exo5/freq_temp_python.png | Bin 12556 -> 88330 bytes module2/exo5/proba_estimate_python.png | Bin 14660 -> 172897 bytes 5 files changed, 669 insertions(+), 181 deletions(-) create mode 100644 module2/exo5/exo5_python_fr.html create mode 100644 module2/exo5/freq_pressure_python.png diff --git a/module2/exo5/exo5_python_fr.html b/module2/exo5/exo5_python_fr.html new file mode 100644 index 0000000..76b846f --- /dev/null +++ b/module2/exo5/exo5_python_fr.html @@ -0,0 +1,518 @@ + + + + + + + +Analyse du risque de défaillance des joints toriques de la navette Challenger + + + + + + + + + + + + + + +
+

Analyse du risque de défaillance des joints toriques de la navette Challenger

+ +

+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. +

+ +
+

1 Chargement des données

+
+

+Nous commençons donc par charger ces données: +

+
+
import numpy as np
+import pandas as pd
+data = pd.read_csv("shuttle.csv")
+data
+
+
+ +
+		Date  Count  Temperature  Pressure  Malfunction
+0    4/12/81      6           66        50            0
+1   11/12/81      6           70        50            1
+2    3/22/82      6           69        50            0
+3   11/11/82      6           68        50            0
+4    4/04/83      6           67        50            0
+5    6/18/82      6           72        50            0
+6    8/30/83      6           73       100            0
+7   11/28/83      6           70       100            0
+8    2/03/84      6           57       200            1
+9    4/06/84      6           63       200            1
+10   8/30/84      6           70       200            1
+11  10/05/84      6           78       200            0
+12  11/08/84      6           67       200            0
+13   1/24/85      6           53       200            2
+14   4/12/85      6           67       200            0
+15   4/29/85      6           75       200            0
+16   6/17/85      6           70       200            0
+17   7/29/85      6           81       200            0
+18   8/27/85      6           76       200            0
+19  10/03/85      6           79       200            0
+20  10/30/85      6           75       200            2
+21  11/26/85      6           76       200            0
+22   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 lanceur principal), la +température (en Fahrenheit) et la pression (en psi), et enfin le +nombre de dysfonctionnements relevés. +

+
+
+ +
+

2 Inspection graphique des données

+
+

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

+
+
import matplotlib.pyplot as plt
+
+plt.clf()
+data["Frequency"]=data.Malfunction/data.Count
+data.plot(x="Temperature",y="Frequency",kind="scatter",ylim=[0,1])
+plt.grid(True)
+
+plt.savefig(matplot_lib_filename)
+print(matplot_lib_filename)
+
+
+ + +
+

freq_temp_python.png +

+
+ +

+Comment la fréquence d'échecs varie-t-elle avec la pression ? +

+
+
import matplotlib.pyplot as plt
+
+plt.clf()
+data["Frequency"]=data.Malfunction/data.Count
+data.plot(x="Pressure",y="Frequency",kind="scatter",ylim=[0,1])
+plt.grid(True)
+
+plt.savefig(matplot_lib_filename)
+print(matplot_lib_filename)
+
+
+ + +
+

freq_pressure_python.png +

+
+ +

+Avec si peu de points de pression, on ne peut pas étudier grand chose. De +plus, visuellement, aucune relation n'a l'air de sauter aux yeux. +

+ +

+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. +

+
+
+ +
+

3 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. +

+ +
+
import statsmodels.api as sm
+
+data["Intercept"]=1
+
+logmodel=sm.GLM(data['Frequency'], data[ ['Intercept', 'Temperature'] ], family=sm.families.Binomial(sm.families.links.logit)).fit()
+
+logmodel.summary()
+
+
+ +
+				 Generalized Linear Model Regression Results
+==============================================================================
+Dep. Variable:              Frequency   No. Observations:                   23
+Model:                            GLM   Df Residuals:                       21
+Model Family:                Binomial   Df Model:                            1
+Link Function:                  logit   Scale:                          1.0000
+Method:                          IRLS   Log-Likelihood:                -3.9210
+Date:                Mon, 20 Jul 2020   Deviance:                       3.0144
+Time:                        11:26:47   Pearson chi2:                     5.00
+No. Iterations:                     6
+Covariance Type:            nonrobust
+===============================================================================
+				  coef    std err          z      P>|z|      [0.025      0.975]
+-------------------------------------------------------------------------------
+Intercept       5.0850      7.477      0.680      0.496      -9.570      19.740
+Temperature    -0.1156      0.115     -1.004      0.316      -0.341       0.110
+===============================================================================
+
+ +

+L'estimateur le plus probable du paramètre de température est 0.0014 +et l'erreur standard de cet estimateur est de 0.122, autrement dit on +ne peut pas distinguer d'impact particulier et il faut prendre nos +estimations avec des pincettes. +

+
+
+ +
+

4 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: +

+ +
+
import matplotlib.pyplot as plt
+
+data_pred = pd.DataFrame({'Temperature': np.linspace(start=30, stop=90, num=121), 'Intercept': 1})
+prediction = logmodel.get_prediction(data_pred[ ['Intercept','Temperature'] ]).summary_frame()
+data_pred['Frequency'] = prediction["mean"]
+data_pred.plot(x="Temperature",y="Frequency",kind="line",ylim=[0,1])
+plt.scatter(x=data["Temperature"], y=data["Frequency"])
+plt.fill_between(
+    data_pred["Temperature"],
+    prediction["mean_ci_lower"],
+    prediction["mean_ci_upper"],
+    alpha=0.2,
+)
+plt.grid(True)
+
+plt.savefig(matplot_lib_filename)
+print(matplot_lib_filename)
+
+
+ + +
+

proba_estimate_python.png +

+
+ +

+Hmm, l'intervalle de confiance semble très grand. Peut-on déduire quoi que ce +soit de ces données ? +

+
+
+
+
+

Auteur: Arnaud Legrand

+

Created: 2020-07-20 Mon 11:26

+

Validate

+
+ + diff --git a/module2/exo5/exo5_python_fr.org b/module2/exo5/exo5_python_fr.org index afff5e0..4a907b8 100644 --- a/module2/exo5/exo5_python_fr.org +++ b/module2/exo5/exo5_python_fr.org @@ -32,191 +32,161 @@ de la NASA durant les 6 années précédant le lancement de la navette Challenger. * Chargement des données -Nous commençons donc par charger ces données: -#+begin_src python :results value :session *python* :exports both -import numpy as np -import pandas as pd -data = pd.read_csv("shuttle.csv") -data -#+end_src - -#+RESULTS: -#+begin_example - Date Count Temperature Pressure Malfunction -0 4/12/81 6 66 50 0 -1 11/12/81 6 70 50 1 -2 3/22/82 6 69 50 0 -3 11/11/82 6 68 50 0 -4 4/04/83 6 67 50 0 -5 6/18/82 6 72 50 0 -6 8/30/83 6 73 100 0 -7 11/28/83 6 70 100 0 -8 2/03/84 6 57 200 1 -9 4/06/84 6 63 200 1 -10 8/30/84 6 70 200 1 -11 10/05/84 6 78 200 0 -12 11/08/84 6 67 200 0 -13 1/24/85 6 53 200 2 -14 4/12/85 6 67 200 0 -15 4/29/85 6 75 200 0 -16 6/17/85 6 70 200 0 -17 7/29/85 6 81 200 0 -18 8/27/85 6 76 200 0 -19 10/03/85 6 79 200 0 -20 10/30/85 6 75 200 2 -21 11/26/85 6 76 200 0 -22 1/12/86 6 58 200 1 -#+end_example - -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 Fahrenheit) et la pression (en psi), et enfin le -nombre de dysfonctionnements relevés. + Nous commençons donc par charger ces données: + #+begin_src python :results value :session *python* :exports both + import numpy as np + import pandas as pd + data = pd.read_csv("shuttle.csv") + data + #+end_src + + #+RESULTS: + #+begin_example + Date Count Temperature Pressure Malfunction + 0 4/12/81 6 66 50 0 + 1 11/12/81 6 70 50 1 + 2 3/22/82 6 69 50 0 + 3 11/11/82 6 68 50 0 + 4 4/04/83 6 67 50 0 + 5 6/18/82 6 72 50 0 + 6 8/30/83 6 73 100 0 + 7 11/28/83 6 70 100 0 + 8 2/03/84 6 57 200 1 + 9 4/06/84 6 63 200 1 + 10 8/30/84 6 70 200 1 + 11 10/05/84 6 78 200 0 + 12 11/08/84 6 67 200 0 + 13 1/24/85 6 53 200 2 + 14 4/12/85 6 67 200 0 + 15 4/29/85 6 75 200 0 + 16 6/17/85 6 70 200 0 + 17 7/29/85 6 81 200 0 + 18 8/27/85 6 76 200 0 + 19 10/03/85 6 79 200 0 + 20 10/30/85 6 75 200 2 + 21 11/26/85 6 76 200 0 + 22 1/12/86 6 58 200 1 + #+end_example + + Le jeu de données nous indique la date de l'essai, le nombre de joints + toriques mesurés (il y en a 6 sur le lanceur principal), la + température (en Fahrenheit) 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 aucune 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. - -#+begin_src python :results value :session *python* :exports both -data = data[data.Malfunction>0] -data -#+end_src - -#+RESULTS: -: Date Count Temperature Pressure Malfunction -: 1 11/12/81 6 70 50 1 -: 8 2/03/84 6 57 200 1 -: 9 4/06/84 6 63 200 1 -: 10 8/30/84 6 70 200 1 -: 13 1/24/85 6 53 200 2 -: 20 10/30/85 6 75 200 2 -: 22 1/12/86 6 58 200 1 - -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 ? -#+begin_src python :results output file :var matplot_lib_filename="freq_temp_python.png" :exports both :session *python* -import matplotlib.pyplot as plt - -plt.clf() -data["Frequency"]=data.Malfunction/data.Count -data.plot(x="Temperature",y="Frequency",kind="scatter",ylim=[0,1]) -plt.grid(True) - -plt.savefig(matplot_lib_filename) -print(matplot_lib_filename) -#+end_src - -#+RESULTS: -[[file:freq_temp_python.png]] - -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. + + Comment la fréquence d'échecs varie-t-elle avec la température ? + #+begin_src python :results output file :var matplot_lib_filename="freq_temp_python.png" :exports both :session *python* + import matplotlib.pyplot as plt + + plt.clf() + data["Frequency"]=data.Malfunction/data.Count + data.plot(x="Temperature",y="Frequency",kind="scatter",ylim=[0,1]) + plt.grid(True) + + plt.savefig(matplot_lib_filename) + print(matplot_lib_filename) + #+end_src + + #+RESULTS: + [[file:freq_temp_python.png]] + + Comment la fréquence d'échecs varie-t-elle avec la pression ? + #+begin_src python :results output file :var matplot_lib_filename="freq_pressure_python.png" :exports both :session *python* + import matplotlib.pyplot as plt + + plt.clf() + data["Frequency"]=data.Malfunction/data.Count + data.plot(x="Pressure",y="Frequency",kind="scatter",ylim=[0,1]) + plt.grid(True) + + plt.savefig(matplot_lib_filename) + print(matplot_lib_filename) + #+end_src + + #+RESULTS: + [[file:freq_pressure_python.png]] + + Avec si peu de points de pression, on ne peut pas étudier grand chose. De + plus, visuellement, aucune relation n'a l'air de sauter aux yeux. + + 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. - -#+begin_src python :results value :session *python* :exports both -import statsmodels.api as sm - -data["Success"]=data.Count-data.Malfunction -data["Intercept"]=1 - - -# logit_model=sm.Logit(data["Frequency"],data[["Intercept","Temperature"]]).fit() -logmodel=sm.GLM(data['Frequency'], data[['Intercept','Temperature']], family=sm.families.Binomial(sm.families.links.logit)).fit() - -logmodel.summary() -#+end_src - -#+RESULTS: -#+begin_example - Generalized Linear Model Regression Results -============================================================================== -Dep. Variable: Frequency No. Observations: 7 -Model: GLM Df Residuals: 5 -Model Family: Binomial Df Model: 1 -Link Function: logit Scale: 1.0 -Method: IRLS Log-Likelihood: -3.6370 -Date: Fri, 20 Jul 2018 Deviance: 3.3763 -Time: 16:56:08 Pearson chi2: 0.236 -No. Iterations: 5 -=============================================================================== - coef std err z P>|z| [0.025 0.975] -------------------------------------------------------------------------------- -Intercept -1.3895 7.828 -0.178 0.859 -16.732 13.953 -Temperature 0.0014 0.122 0.012 0.991 -0.238 0.240 -=============================================================================== -#+end_example - -L'estimateur le plus probable du paramètre de température est 0.0014 -et l'erreur standard de cet estimateur est de 0.122, autrement dit on -ne peut pas distinguer d'impact particulier et il faut prendre nos -estimations avec des pincettes. + 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. + + #+begin_src python :results value :session *python* :exports both + import statsmodels.api as sm + + data["Intercept"]=1 + + logmodel=sm.GLM(data['Frequency'], data[ ['Intercept', 'Temperature'] ], family=sm.families.Binomial(sm.families.links.logit)).fit() + + logmodel.summary() + #+end_src + + #+RESULTS: + #+begin_example + Generalized Linear Model Regression Results + ============================================================================== + Dep. Variable: Frequency No. Observations: 23 + Model: GLM Df Residuals: 21 + Model Family: Binomial Df Model: 1 + Link Function: logit Scale: 1.0000 + Method: IRLS Log-Likelihood: -3.9210 + Date: Mon, 20 Jul 2020 Deviance: 3.0144 + Time: 11:22:26 Pearson chi2: 5.00 + No. Iterations: 6 + Covariance Type: nonrobust + =============================================================================== + coef std err z P>|z| [0.025 0.975] + ------------------------------------------------------------------------------- + Intercept 5.0850 7.477 0.680 0.496 -9.570 19.740 + Temperature -0.1156 0.115 -1.004 0.316 -0.341 0.110 + =============================================================================== + #+end_example + + L'estimateur le plus probable du paramètre de température est 0.0014 + et l'erreur standard de cet estimateur est de 0.122, 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: - -#+begin_src python :results output file :var matplot_lib_filename="proba_estimate_python.png" :exports both :session *python* -import matplotlib.pyplot as plt - -data_pred = pd.DataFrame({'Temperature': np.linspace(start=30, stop=90, num=121), 'Intercept': 1}) -data_pred['Frequency'] = logmodel.predict(data_pred[['Intercept','Temperature']]) -data_pred.plot(x="Temperature",y="Frequency",kind="line",ylim=[0,1]) -plt.scatter(x=data["Temperature"],y=data["Frequency"]) -plt.grid(True) - -plt.savefig(matplot_lib_filename) -print(matplot_lib_filename) -#+end_src - -#+RESULTS: -[[file:proba_estimate_python.png]] - -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: - -#+begin_src python :results output :session *python* :exports both -data = pd.read_csv("shuttle.csv") -print(np.sum(data.Malfunction)/np.sum(data.Count)) -#+end_src - -#+RESULTS: -: 0.06521739130434782 - -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. - + 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: + + #+begin_src python :results output file :var matplot_lib_filename="proba_estimate_python.png" :exports both :session *python* + import matplotlib.pyplot as plt + + data_pred = pd.DataFrame({'Temperature': np.linspace(start=30, stop=90, num=121), 'Intercept': 1}) + prediction = logmodel.get_prediction(data_pred[ ['Intercept','Temperature'] ]).summary_frame() + data_pred['Frequency'] = prediction["mean"] + data_pred.plot(x="Temperature",y="Frequency",kind="line",ylim=[0,1]) + plt.scatter(x=data["Temperature"], y=data["Frequency"]) + plt.fill_between( + data_pred["Temperature"], + prediction["mean_ci_lower"], + prediction["mean_ci_upper"], + alpha=0.2, + ) + plt.grid(True) + + plt.savefig(matplot_lib_filename) + print(matplot_lib_filename) + #+end_src + + #+RESULTS: + [[file:proba_estimate_python.png]] + + Hmm, l'intervalle de confiance semble très grand. Peut-on déduire quoi que ce + soit de ces données ? diff --git a/module2/exo5/freq_pressure_python.png b/module2/exo5/freq_pressure_python.png new file mode 100644 index 0000000000000000000000000000000000000000..56a2173f56455377a2bbe82c8131900f59098ba0 GIT binary patch literal 89090 zcmeFa2UJy8*Dk!(7?aq74HW@{1q*_R(wmhiDk>l#&437sfb?FB1r&vd3ZgVcML|J8 zdNoo+kfIdn2-17+oc~#9BHQHu?-<{BzwzJi-gCxKk#o-8YtJ>;TyxF&%x7LcbWmpQ zPs@LzC~EHBUw0j$sQLJDw)79*R@c$pr{;Fb1QEGe0 zpJ|J3?%a-F3Y+a#Hj_6xX=eGGi4JA(o7vejMrLPDX|1%-F)=-5WGKM9g_oaarJkAD zSy4W|ukZ63ndtHb_K22I)JkgauI&m|K5gav)2NkmQa7ijG#YcARhxQBV7v@kXrc{`^GPx?4D+pp>+7$?aX>BQrxd%ZJm^(9sz;qJe^D$i~ zrbCIfKrx*J2!ZJ&KnP4Hf$1a=i@5pZppZs!-vhlljb|E72NE?l3s*;WTBhU`4Zewya@zG>Dp7#1uRmO< zZ=zQVSiTK#aZNw|^$X;uUw3^Qo4WkNx8W<6Y2QS+R&#wD1Y-{7vsjsu`7DQ(Or7~G zi2u-P`c*b*7Ipm8aIGs}ZMjj(H38k+v@44@-VX`cC}>!*l%1WupJVRlJAVE<;1?|8 z<@pz9v(~0vUdX>?%Ys#_RvF`GVapD+;X&2tGnWHp+<9B}+YWYIi>b2Nn30*;I2IZg 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