Traduction anglaise de challenger.org

module2/exo5/challenger-en.org

+#+TITLE: Analysis of the risk of failure of the toric joints of the space shuttle Challenger
+#+AUTHOR: Konrad Hinsen, Arnaud Legrand, Christophe Pouzat
+#+DATE: Juin 2018
+#+LANGUAGE: en
+
+#+OPTIONS: H:3 creator:nil timestamp:nil skip:nil toc:nil num:t ^:nil ~:~ 
+# #+OPTIONS: author:nil title:nil date:nil
+
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+
+#+LATEX_HEADER: \usepackage[utf8]{inputenc}
+#+LATEX_HEADER: \usepackage[T1]{fontenc}
+#+LATEX_HEADER: \usepackage{textcomp}
+#+LATEX_HEADER: \usepackage[a4paper,margin=.8in]{geometry}
+#+LATEX_HEADER: \usepackage[usenames,dvipsnames,svgnames,table]{xcolor}
+#+LATEX_HEADER: \usepackage{palatino}
+#+LATEX_HEADER: \usepackage{svg}
+#+LATEX_HEADER: \let\epsilon=\varepsilon
+
+*Forword:* The explanations given in this document about the context
+of the study have been taken from the excellent book /Visual
+Explanations: Images and Quantities, Evidence and Narrative/ by Edward
+R. Tufte, published in 1997 by /Graphics Press/ and re-edited in 2005,
+and from the article /Risk Analysis of the Space Shuttle:
+Pre-Challenger Prediction of Failure/ by Dalal et al., published in
+1989 in the /Journal of the American Statistical Association/.
+
+* Context
+In this study, we propose a re-examination of the [[https://en.wikipedia.org/wiki/Space_Shuttle_Challenger_disaster][space shuttle
+Challenger disaster]]. On January 28th, 1986, the space shuttle
+Challenged exploded 73 seconds after launch (see Figure [[fig:photo]]),
+causing the death of the seven astronauts on board. The explosion was
+caused by the failure of the two O-ring seals between the upper and
+lower parts of the boosters (see Figure [[fig:oring]]). The seals had
+lost their efficiency because of the exceptionally cold weather at the
+time of launch. The temperature on that morning was just below 0°C,
+whereas the preceding flights hat been launched at temperatures at
+least 7 to 10°C higher.
+
+#+NAME:   fig:photo
+#+ATTR_LATEX: :width 10cm
+#+CAPTION: Photographs of the Challenger catastrophe.
+file:challenger5.jpg
+
+
+#+NAME:   fig:oring
+#+ATTR_LATEX: :width 10cm
+#+CAPTION: Diagram of the boosters of space shuttle Challenger. The rubber O-ring seals (a principal and a secondary one) of more than 11 meter circumference prevent  leaks between the upper and lower parts.
+file:o-ring.png
+# From
+# https://i0.wp.com/www.kylehailey.com/wp-content/uploads/2014/01/Screen-Shot-2013-12-30-at-12.05.04-PM-1024x679.png?zoom=2&resize=594%2C393
+
+What is most astonishing is that the precise cause of the accident had
+been intensely debated several days before and ws still under
+discussion the day before the launch, during a three-hour
+teleconference involving engineers from Morton Thiokol (the supplier
+of the engines) and from NASA. Whereas the immediate cause of the
+accident, the failure of the O-ringe, was quickly identified, the
+underlying causes of the disaster have regularly served as a case
+study, be it in management training (work organisation, decision
+taking in spite of political pressure, communication problems),
+statistics (risk evaluation, modelisation, data visualization), or
+sociology (history, bureaucracy, conforming to organisational norms).
+
+In the study that we propose, we are mainly concerned with the
+statistical aspect, which however is only one piece of the puzzle. We
+invite you to read the documentes cited in the foreword for more
+information. The following study takes up a part of the analyses that
+were done that night with the goal of evaluating the potential impact
+of temperature and air pressure on the probability of O-ring
+malfunction. The starting point is experimental results obtained by
+NASA engineers over the six years preceding the Challenger launch.
+
+In the directory ~module2/exo5/~ of your GitLab workspace, you will
+find the original data as welas an analysis for each of the paths we
+offer. This analysis consists of four steps:
+
+1. Loading the data
+2. Visual inspection
+3. Estimation of the influence of temperature
+4. Estimation of the probability of O-ring malfunction
+
+The first two steps require only a basic knowledge of R or Python. The
+third step assumes some familiarity with logistic regression, and the
+fourth a basic knowledge of probability. In the next section, we give
+an introduction to logistic regression that skips the details of the
+computations and focuses instead on the interpretation of the results.
+
+* Introduction to logistic regression
+
+Suppose we have the following dataset that indicates for a group of
+people of varying age if they suffer from a specific illness or not. I
+will present the analysis in R but Python code would look quite
+similar. The data are stored in a data frame that is summarized as:
+
+#+begin_src R :results output :session *R* :exports none
+library(Hmisc) # pour calculer un intervalle de confiance sur des données binomiales
+library(ggplot2)
+library(dplyr)
+set.seed(42)
+proba = function(age) {
+    val=(age-50)/4
+    return(exp(val)/(1+exp(val)))
+}
+df = data.frame(Age = runif(400,min=22,max=80))
+df$Ill = sapply(df$Age, function(x) rbinom(n=1,size=1,prob=proba(x)))
+#+end_src
+
+#+RESULTS:
+
+#+begin_src R :results output :session *R* :exports both
+summary(df)
+str(df)
+#+end_src
+
+#+RESULTS:
+#+begin_example
+      Age             Ill       
+ Min.   :22.01   Min.   :0.000  
+ 1st Qu.:35.85   1st Qu.:0.000  
+ Median :50.37   Median :1.000  
+ Mean   :50.83   Mean   :0.515  
+ 3rd Qu.:65.37   3rd Qu.:1.000  
+ Max.   :79.80   Max.   :1.000
+'data.frame':	400 obs. of  2 variables:
+ $ Age: num  75.1 76.4 38.6 70.2 59.2 ...
+ $ Ill: int  1 1 0 1 1 1 0 0 1 1 ...
+#+end_example
+
+Here is a plot that provides a better indication of the link that
+could exist between age and illness:
+
+#+begin_src R :results output graphics :file fig1.svg :exports both :width 4 :height 3 :session *R* 
+ggplot(df,aes(x=Age,y=Ill)) + geom_point(alpha=.3,size=3) + theme_bw()
+#+end_src
+
+#+ATTR_LATEX: :width 8cm
+#+RESULTS:
+[[file:fig1.svg]]
+
+Clearly the probability of suffering from this illness increases with
+age. But how can we estimate this probability based only on this
+binary data ill/not ill? For each age slice (of, for example, 5
+years), we could look at the frequency of the illness. The following
+code is a bit complicated because the computation of the confidence
+interval for this kind of data requires a particular treatment using
+the function =binconf=.
+
+#+begin_src R :results output graphics :file fig1bis.svg :exports both :width 4 :height 3 :session *R* 
+age_range=5
+df_grouped = df %>% mutate(Age=age_range*(floor(Age/age_range)+.5)) %>%
+                    group_by(Age) %>% summarise(Ill=sum(Ill),N=n()) %>% 
+                    rowwise() %>% 
+                    do(data.frame(Age=.$Age, binconf(.$Ill, .$N, alpha=0.05))) %>%
+                    as.data.frame()
+
+ggplot(df_grouped,aes(x=Age)) + geom_point(data=df,aes(y=Ill),alpha=.3,size=3)  +
+    geom_errorbar(data=df_grouped,
+                  aes(x=Age,ymin=Lower, ymax=Upper, y=PointEst), color="darkred") +
+    geom_point(data=df_grouped, aes(x=Age, y=PointEst), size=3, shape=21, color="darkred") +
+    theme_bw() 
+#+end_src
+
+#+ATTR_LATEX: :width 8cm
+#+RESULTS:
+[[file:fig1bis.svg]]
+
+A disadvantage of this method is that the computation is done
+independently for each age slice, which moreover has been chosen
+arbitrarily. For describing the evolution in a more continuous
+fashion, we could apply a linear regression (which is the simplest
+model for taking into account the influence of a parameter) and thus estimate the impact of age on the probability of illness:
+
+#+begin_src R :results output graphics :file fig2.svg :exports both :width 4 :height 3 :session *R* 
+ggplot(df,aes(x=Age,y=Ill)) + geom_point(alpha=.3,size=3) + 
+    theme_bw() + geom_smooth(method="lm")
+#+end_src
+
+#+ATTR_LATEX: :width 8cm
+#+RESULTS:
+[[file:fig2.svg]]
+
+The blue line is the linear regression in the sense of least squares,
+and the grey zone is the 95% confidence interval of this
+estimation. In other words, given the dataset and the hypothesis of
+linearity, the blue line is the most probable one and there is a 95%
+chance that the true line is in the grey zone.
+
+It is, however, clear from the plot that this estimation is
+meaningless. A probability must lie between 0 and 1, whereas a linear
+regression will inevitably lead to impossible values (negative or
+greater than 1) for somewhat extreme age values (young or old).  The
+reason is simply that a linear regression implies the hypothesis
+$\textsf{Ill} = \alpha.\textsf{Age} + \beta + \epsilon$, where
+$\alpha$ and $\beta$ are real numbers and $\epsilon$ is a noise (a
+random variable of mean zero), wihh $\alpha$ and $\beta$ estimated
+from the data. This doesn't make sense for estimating a probability,
+and therefore [[https://en.wikipedia.org/wiki/Logistic_regression][logistic regression]] is a better choice:
+
+#+begin_src R :results output graphics :file fig3.svg :exports both :width 4 :height 3 :session *R* 
+ggplot(df,aes(x=Age,y=Ill)) + geom_point(alpha=.3,size=3) + 
+    theme_bw() + 
+    geom_smooth(method = "glm", 
+        method.args = list(family = "binomial")) + xlim(20,80)
+#+end_src
+
+#+ATTR_LATEX: :width 8cm
+#+RESULTS:
+[[file:fig3.svg]]
+
+Here the =ggplot= library does all the computations for us and only
+shows the result graphically, but in the Challenger risk analysis we
+perform the regression and prediction "by hand" in =R= or =Python=
+(depending on the path you have chosen), so that we can inspect the
+results in more detail. Like before, the blue line indicates the
+estimation of the probability of being ill as a function of age, and
+the grey zone informs us about the uncertainty of this estimate, i.e.
+given the hypotheses and the dataset, there is a 95% chance for the
+true curve to lie somewhere in the grey zone.
+
+In this model, the assumption is $P[\textsf{Ill}] = \pi(\textsf{Age})$
+with $\displaystyle\pi(x)=\frac{e^{\alpha.x + \beta}}{1+e^{\alpha.x +
+\beta}}$. This at first look strange formule has the nice property of
+always yielding a value between zero and one, and to approach 0 and 1
+rapidly as the age tends to $-\infty$ or $+\infty$, but this is not
+the only motivation for this choice.
+
+In summary, when we have event-like data (binary) and we wish to
+estimate the influence of a parameter on the probability of the event
+occurring (illness, failure, ...), the most natural and simple model
+is logistic regression. Note that even if we restrain ourselves to a
+small part of the dta, e.g. only patients less than 50 years old, it
+is possible to get a reasonable estimate, even though, as is to be
+expected, the uncertainty grows rapidly.
+
+#+begin_src R :results output graphics :file fig4.svg :exports both :width 4 :height 3 :session *R* 
+ggplot(df[df$Age<50,],aes(x=Age,y=Ill)) + geom_point(alpha=.3,size=3) + 
+    theme_bw() + 
+    geom_smooth(method = "glm", 
+        method.args = list(family = "binomial"),fullrange = TRUE) + xlim(20,80)
+#+end_src
+
+#+ATTR_LATEX: :width 8cm
+#+RESULTS:
+[[file:fig4.svg]]
+
+* Emacs Setup 							   :noexport:
+  This document has local variables in its postembule, which should
+  allow Org-mode (9) to work seamlessly without any setup. If you're
+  uncomfortable using such variables, you can safely ignore them at
+  startup. Exporting may require that you copy them in your .emacs.
+
+# Local Variables:
+# eval: (add-to-list 'org-latex-packages-alist '("" "minted"))
+# eval: (setq org-latex-listings 'minted) 
+# eval:    (setq org-latex-minted-options '(("style" "Tango") ("bgcolor" "Moccasin") ("frame" "lines") ("linenos" "true") ("fontsize" "\\small")))
+# eval: (setq org-latex-pdf-process '("pdflatex -shell-escape -interaction nonstopmode -output-directory %o %f" "pdflatex -shell-escape -interaction nonstopmode -output-directory %o %f" "pdflatex -shell-escape -interaction nonstopmode -output-directory %o %f"))
+# End: