From b883b5478125803e4743b95139c4c3f22a3fdb5c Mon Sep 17 00:00:00 2001
From: ExtraDev <scott.birner@gmail.com>
Date: Wed, 8 Dec 2021 11:36:25 +0100
Subject: [PATCH] advance on documentation

---
 doc/screenshots/meilleure_droite.png     | Bin 0 -> 21145 bytes
 doc/screenshots/nuage_aleatoire.png      | Bin 0 -> 12879 bytes
 doc/screenshots/optimisation_tableur.png | Bin 0 -> 40108 bytes
 doc/screenshots/validation_croisee.png   | Bin 0 -> 42048 bytes
 readme.md                                |  77 ++++++++++++++++++-----
 src/Python/plot_groups.py                |   8 ++-
 src/droite_nuage_0_1_A.vec               | Bin 14 -> 14 bytes
 src/droite_nuage_0_1_B.vec               | Bin 14 -> 14 bytes
 src/droite_nuage_0_2_A.vec               | Bin 14 -> 14 bytes
 src/droite_nuage_0_2_B.vec               | Bin 14 -> 14 bytes
 src/droite_nuage_1_2_A.vec               | Bin 14 -> 14 bytes
 src/droite_nuage_1_2_B.vec               | Bin 14 -> 14 bytes
 src/droite_nuage_complet_A.vec           | Bin 14 -> 14 bytes
 src/droite_nuage_complet_B.vec           | Bin 14 -> 14 bytes
 src/points_X.vec                         | Bin 270 -> 246 bytes
 src/points_Y.vec                         | Bin 270 -> 246 bytes
 16 files changed, 66 insertions(+), 19 deletions(-)
 create mode 100644 doc/screenshots/meilleure_droite.png
 create mode 100644 doc/screenshots/nuage_aleatoire.png
 create mode 100644 doc/screenshots/optimisation_tableur.png
 create mode 100644 doc/screenshots/validation_croisee.png

diff --git a/doc/screenshots/meilleure_droite.png b/doc/screenshots/meilleure_droite.png
new file mode 100644
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diff --git a/readme.md b/readme.md
index 8649ee4..8f360ba 100644
--- a/readme.md
+++ b/readme.md
@@ -6,39 +6,84 @@ tof: true
 lof: true
 ---
 \newpage
-# Todo
-- Equation de base: yj = c*xj+d+rj
-- Créer un nuage de point autout de l'équation de base, le garder constant
-- Descente du radiant
-    - Validation croisée: Séparer le nuage de point en 3 groupe
-- Visualisation
 
 # Introduction
 Dans le cadre du cours de mathématiques en technologie de l’information nous avons pour mission de réaliser un travail pratique en C sur l'optimisation de fonction.
 
-En effet, le but est de mettre en pratique les connaissances acquises en cours durant le premier semestre 2021 qui portait sur les fonctions, dérivées, minimum et maximum d'une fonction et optimisation de celles-ci.
+En effet, le but est de mettre en pratique les connaissances acquises durant le premier semestre 2021 qui portait sur les fonctions, dérivées, minimum et maximum d'une fonction et optimisation de celles-ci.
 
-Plus précisément, notre objectif dans ce travail est de trouver la meilleur droite passant par un nuage de point(x,y) sur un plan 2D.
+Plus précisément, notre objectif dans ce travail est de trouver la meilleur droite passant par un nuage de point(x,y) généré de manière pseudo aléatoire sur un plan 2D.
 
-Ci-dessous un exemple du résultat attendu, à savoir, connaitre la meilleure droite passant par dans un nuage de points généré de manière aléatoire.
+Ci-dessous un exemple du résultat attendu:
 
 ![Exemple ](./doc/screenshots/exemple.png){width=400px}
 
-# Méthodes employée
-## Solution analytique
+# Solution analytique
 En premier lieux, nous avons cherchons à minimiser la fonction de coût:
 
-E(a,b) = $\sum_{j=1}^n (a * x_{j} + b - y_{i})^2$
+E(a,b) = $\sum_{j=1}^N(a * x_{j} + b - y_{i})^2$
 
-En résolvant le système de deux équations à deux inconnues.
+En résolvant le système de deux équations à deux inconnues. Ce sont avec ces équations que nous allons pouvoir réaliser notre solutions numérique.
 
-## Solution numérique
+Voici les equations obtenues en A et B:
 
+## A
+**Equation en A**
+## B
+**Equation en B**
 
-# Résultats
+## Passage par un tableur
+Afin de s'assurer que les equations obtenues sont correct, nous avons réaliser les calculs dans un tableur. De cette façon, nous avons pu confirmer la qualité de nos calculs, mais aussi de s'avoir à quoi s'attendre en therme de résultat par la suite.
+
+Sur l'image suivante, les points représentent les points aléatoires et les trétillés représentent la meilleure droite.
+![Résultat par un tableur](./doc/screenshots/optimisation_tableur.png)
+
+À partir de ce moment là, on en a déduit que les équations obtenues en **A et B** sont correcte.
+
+# Solution numérique
+## Génération des points pseudo aléatoire
+Nous parlons de points pseudo aléatoire car, ils sont en réalité généré autour d'une fonction de droite tel que f() = ax+b avec une marge d'erreur en 0 et 1 sur l'axe y.
+
+![Nuage de points pseudo aléatoire](./doc/screenshots/nuage_aleatoire.png){width=400px}
+
+En effet, de cette façon on s'assure que la droite que nous souhaitons retrouver resemble à celle qui a permis de générer les points...
+
+## Descente de gradient
+Pour trouver la meilleure droite passant par le nuage de points, nous devont appliqué la descente de gradient. En effet, c'est cette formule qui nous permettra de retrouver l'ordonée à l'origine ainsi que la pente de la meilleur droite passant par un nuage de points.
+
+Sur notre nuage de points nous avons appliqué la formule suivante:
+**Formule descente de gradient**
+
+Et nous nous sommes arrêté quand le $\epsilon$ était inférieur à 0.001 ce qui représente environ 385 itération sur 30 points
 
 ## Validation du modèle de régression
+La validation croisée consiste à valider la descente de dradient de notre droite sur plusieurs échantillons de données.
+
+Il a donc fallu en premier temps séparé de manière aléatoire notre nuage de points en 3 groupes distinct. Par la suite, le calcule de la descente du gradient et appliqué sur chacun des groupe. Le but de cette opération est de retrouvé dans chacun des groupes des droits similaires à celle trouvée au départ avec le nuage complet. Cela a pour but de valider nos données!
+
+![Validation croisée, descente de gradient](./doc/screenshots/validation_croisee.png){width=400px}
+
+## Erreur quadratique
+L'erreur quadratique va nous donner un indice pour savoir à quel point nous somme précis dans le calcul de la descente de gradient. Plus la valeur de l'erreur quadratique est petite, mieux c'est.
+
+En voici la formule :
+**Formule erreur quadratique**
+
+# Résultats
+Comme nous avons pu le voir dans les figures précédentes, nous avons optenu des résultats concluent. J'entends pas là que aussi bien la droite obtenue sur la descente de gradient avec le nuage complet **et** les droits obtenues sur la validation croisée se ressemble et semblent juste au niveau de nos nuage de points. 
+
+Voici un tableau des données obtenue sur une génération aléatoire ainsi que leur différence.
+
+| Nuage de 30 points | Descente de gradient | Différence | Erreur quadratique |
+| :-- | :-- | :-- | :-- |
+| Complet       | 1.012865; 0.437018 | - | 3.774071 |
+| Groupe 0 et 1 | 1.001448; 0.469333 | 0.011417; -0.032315 | 1.163377 |
+| Groupe 0 et 2 | 1.001448; 0.469333 | 0.011417; -0.032315 | 1.163377 |
+| Groupe 1 et 2 | 0.963636; 0.605260 | 0.049229; -0.168242 | 1.226093 |
+
+On peut constater une différence de précision au niveau de l'erreur quadratique. En effet, l'erreur quadratique obtenue pendant la validation croisée par chacun des groupes sont bien plus petites que cette optenue sur le nuage complet. On peut en déduire qu'il est préférable de réaliser la descente de gradient sur des lots données pas trop volumineux et de vérifier l'écart de chacun des résultats afin de s'assurer que le résultat obtenu soit le plus juste possible.
 
 # Conclusion
 
-# Remerciements
\ No newline at end of file
+
+# Remerciements
diff --git a/src/Python/plot_groups.py b/src/Python/plot_groups.py
index b0fbf3a..01eb40a 100644
--- a/src/Python/plot_groups.py
+++ b/src/Python/plot_groups.py
@@ -4,7 +4,7 @@ from load_vec import load_vector
 import numpy as np
 
 fig, (ax1, ax2, ax3) = plt.subplots(3)
-fig.suptitle('Vertically stacked subplots')
+fig.suptitle('Validation croisée')
 
 X_nuage_point = load_vector("../points_X.vec")
 Y_nuage_point = load_vector("../points_Y.vec")
@@ -22,16 +22,18 @@ x = np.array(range(int(max(X_nuage_point))+1))
 y = x * droite_nuage_0_1_A + droite_nuage_0_1_B
 ax1.scatter(X_nuage_point, Y_nuage_point, marker='.', label="Points")
 ax1.plot(x, y, label="Meilleure droite", color="red")
+ax1.set_title("Groupe 0 et 1")
 
 x = np.array(range(int(max(X_nuage_point))+1))
 y = x * droite_nuage_0_2_A + droite_nuage_0_2_B
 ax2.scatter(X_nuage_point, Y_nuage_point, marker='.', label="Points")
 ax2.plot(x, y, label="Meilleure droite", color="red")
+ax2.set_title("Groupe 0 et 2")
 
 x = np.array(range(int(max(X_nuage_point))+1))
 y = x * droite_nuage_1_2_A + droite_nuage_1_2_B
 ax3.scatter(X_nuage_point, Y_nuage_point, marker='.', label="Points")
 ax3.plot(x, y, label="Meilleure droite", color="red")
+ax3.set_title("Groupe 1 et 2")
 
-
-plt.show()
+plt.show()
\ No newline at end of file
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-- 
GitLab