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propos du calcul de \(\pi\) + + + + + + + + +
+

À propos du calcul de \(\pi\)

+
+ +
+

1 En demandant à la lib maths

+
+

+Mon ordinateur m'indique que \(\pi\) vaut approximativement +

+ +
+
pi
+
+
+ +
+[1] 3.141593
+
+
+
+
+ +
+

2 En utilisant la méthode des aiguilles de Buffon

+
+

+Mais calculé avec la méthode des aiguilles de Buffon, on obtiendrait comme approximation : +

+ +
+
set.seed(42)
+N = 100000
+x = runif(N)
+theta = pi/2*runif(N)
+2/(mean(x+sin(theta)>1))
+
+
+ +
+[1] 3.14327
+
+
+
+
+ +
+

3 Avec un argument "fréquentiel" de surface

+
+

+Sinon, une méthode plus simple à comprendre et ne faisant pas +intervenir d'appel à la fonction sinus se base sur le fait que si +\(X∼U(0,1)\) et \(Y∼U(0,1)\) alors \(P[X^2+Y^2 \leq 1]=\pi/4\) (voir méthode +de Monte Carlo sur Wikipedia). Le code suivant illustre ce fait : +

+ +
+
set.seed(42)
+N = 1000
+df = data.frame(X = runif(N), Y = runif(N))
+df$Accept = (df$X**2 + df$Y**2 <=1)
+library(ggplot2)
+ggplot(df, aes(x=X,y=Y,color=Accept)) + geom_point(alpha=.2) + coord_fixed() + theme_bw()
+
+
+ + +
+

figure_pi_mc1.png +

+
+ +

+Il est alors aisé d'obtenir une approximation (pas terrible) de \(\pi\) en +comptant combien de fois, en moyenne, \(X^2+Y^2\) est inférieur à 1 : +

+ +
+
4*mean(df$Accept)
+
+
+ +
+[1] 3.156
+
+
+
+
+
+
+

Date: 10/06/2020

+

Auteur: Julien EMMANUEL

+

Created: 2020-06-10 Wed 15:07

+

Validate

+
+ + diff --git a/module2/exo1/toy_document_orgmode_R_fr.org b/module2/exo1/toy_document_orgmode_R_fr.org index 1bb8f61..b596f94 100644 --- a/module2/exo1/toy_document_orgmode_R_fr.org +++ b/module2/exo1/toy_document_orgmode_R_fr.org @@ -1,84 +1,60 @@ -#+TITLE: Votre titre -#+AUTHOR: Votre nom -#+DATE: La date du jour +#+TITLE: À propos du calcul de $\pi$ +#+AUTHOR: Julien EMMANUEL +#+DATE: 10/06/2020 #+LANGUAGE: fr # #+PROPERTY: header-args :eval never-export -#+HTML_HEAD: -#+HTML_HEAD: -#+HTML_HEAD: -#+HTML_HEAD: -#+HTML_HEAD: -#+HTML_HEAD: +* En demandant à la lib maths -* Quelques explications - -Ceci est un document org-mode avec quelques exemples de code -R. Une fois ouvert dans emacs, ce document peut aisément être -exporté au format HTML, PDF, et Office. Pour plus de détails sur -org-mode vous pouvez consulter https://orgmode.org/guide/. - -Lorsque vous utiliserez le raccourci =C-c C-e h o=, ce document sera -compilé en html. Tout le code contenu sera ré-exécuté, les résultats -récupérés et inclus dans un document final. Si vous ne souhaitez pas -ré-exécuter tout le code à chaque fois, il vous suffit de supprimer -le # et l'espace qui sont devant le ~#+PROPERTY:~ au début de ce -document. - -Comme nous vous l'avons montré dans la vidéo, on inclut du code -R de la façon suivante (et on l'exécute en faisant ~C-c C-c~): +Mon ordinateur m'indique que $\pi$ vaut /approximativement/ #+begin_src R :results output :exports both -print("Hello world!") +pi #+end_src #+RESULTS: -: [1] "Hello world!" +: [1] 3.141593 -Voici la même chose, mais avec une session R (c'est le cas le -plus courant, R étant vraiment un langage interactif), donc une -persistance d'un bloc à l'autre (et on l'exécute toujours en faisant -~C-c C-c~). +* En utilisant la méthode des aiguilles de Buffon -#+begin_src R :results output :session *R* :exports both -summary(cars) +Mais calculé avec la *méthode* des [[https://fr.wikipedia.org/wiki/Aiguille_de_Buffon][aiguilles de Buffon]], on obtiendrait comme *approximation* : + +#+begin_src R :results output :exports both +set.seed(42) +N = 100000 +x = runif(N) +theta = pi/2*runif(N) +2/(mean(x+sin(theta)>1)) #+end_src #+RESULTS: -: speed dist -: Min. : 4.0 Min. : 2.00 -: 1st Qu.:12.0 1st Qu.: 26.00 -: Median :15.0 Median : 36.00 -: Mean :15.4 Mean : 42.98 -: 3rd Qu.:19.0 3rd Qu.: 56.00 -: Max. :25.0 Max. :120.00 - -Et enfin, voici un exemple de sortie graphique: -#+begin_src R :results output graphics :file "./cars.png" :exports results :width 600 :height 400 :session *R* -plot(cars) +: [1] 3.14327 + +* Avec un argument "fréquentiel" de surface + +Sinon, une méthode plus simple à comprendre et ne faisant pas +intervenir d'appel à la fonction sinus se base sur le fait que si +$X∼U(0,1)$ et $Y∼U(0,1)$ alors $P[X^2+Y^2 \leq 1]=\pi/4$ (voir [[https://fr.wikipedia.org/wiki/M%C3%A9thode_de_Monte-Carlo#D%C3%A9termination_de_la_valeur_de_%CF%80][méthode +de Monte Carlo sur Wikipedia]]). Le code suivant illustre ce fait : + +#+begin_src R :results output graphics :file figure_pi_mc1.png :session *R* :exports both +set.seed(42) +N = 1000 +df = data.frame(X = runif(N), Y = runif(N)) +df$Accept = (df$X**2 + df$Y**2 <=1) +library(ggplot2) +ggplot(df, aes(x=X,y=Y,color=Accept)) + geom_point(alpha=.2) + coord_fixed() + theme_bw() #+end_src #+RESULTS: -[[file:./cars.png]] - -Vous remarquerez le paramètre ~:exports results~ qui indique que le code -ne doit pas apparaître dans la version finale du document. Nous vous -recommandons dans le cadre de ce MOOC de ne pas changer ce paramètre -(indiquer ~both~) car l'objectif est que vos analyses de données soient -parfaitement transparentes pour être reproductibles. +[[file:figure_pi_mc1.png]] -Attention, la figure ainsi générée n'est pas stockée dans le document -org. C'est un fichier ordinaire, ici nommé ~cars.png~. N'oubliez pas -de le committer si vous voulez que votre analyse soit lisible et -compréhensible sur GitLab. +Il est alors aisé d'obtenir une approximation (pas terrible) de $\pi$ en +comptant combien de fois, en moyenne, $X^2+Y^2$ est inférieur à 1 : -Enfin, pour les prochains exercices, nous ne vous fournirons pas -forcément de fichier de départ, ça sera à vous de le créer, par -exemple en repartant de ce document et de le commiter vers -gitlab. N'oubliez pas que nous vous fournissons dans les ressources de -ce MOOC une configuration avec un certain nombre de raccourcis -claviers permettant de créer rapidement les blocs de code R (en -faisant ~