From 61a70002625796c9a6a31ac42378cd76369f0bf5 Mon Sep 17 00:00:00 2001 From: ExtraDev <scott.birner@gmail.com> Date: Fri, 10 Dec 2021 15:58:27 +0100 Subject: [PATCH] fix data quality and advance documentation --- create_document.sh | 2 +- readme.md | 55 ++++++++++++++++++++++++++------- src/C/descente_radiant | Bin 183472 -> 186840 bytes src/C/equation.c | 2 +- src/C/group.c | 14 +++++++-- src/C/group.h | 1 + src/C/main.c | 45 +++++++++++++++++++++------ 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 318 -> 246 bytes src/points_Y.vec | Bin 318 -> 246 bytes 17 files changed, 95 insertions(+), 24 deletions(-) diff --git a/create_document.sh b/create_document.sh index 040231a..79efdd1 100755 --- a/create_document.sh +++ b/create_document.sh @@ -2,4 +2,4 @@ filename=$1 output="${filename%.*}.pdf" -pandoc --number-sections --table-of-contents -V geometry:margin=1in rapport.md -o rapport.pdf \ No newline at end of file +pandoc --number-sections --table-of-contents -V geometry:margin=1in readme.md -o readme.pdf diff --git a/readme.md b/readme.md index aef4cc4..59ff6c8 100644 --- a/readme.md +++ b/readme.md @@ -19,6 +19,7 @@ Ci-dessous un exemple du résultat attendu: {width=400px} # Solution analytique + ## Minimisation de la fonction de coût En premier lieux, nous avons cherchons à minimiser la fonction de coût (Erreur quadratique): @@ -92,7 +93,7 @@ $b = -a\frac{B}{N} + \frac{D}{N} = -(\frac{CN - BD}{AN - {B}^2})\frac{B}{N} + \ $b = -(\frac{CNB - {B^2}D}{{AN^2} - {B^2}N}) + \frac{D}{N}$ -## Vérifiaction +## Vérifications On peux tester nos deux formules dans un excel par exemple pour s'assurer de la justesse des transformation. On vas donc génerer un nuage de points suivant une droite, puis calculer nos deux formules en rentrant comme paramètre les differentes sommes de points. @@ -107,6 +108,7 @@ De cette façon, nous avons pû confirmer la qualité de nos calculs, mais aussi 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 générés autour d'une fonction de droite tel que f() = ax+b avec une marge d'erreur entre 0 et 1 sur l'axe y. @@ -117,8 +119,29 @@ En effet, de cette façon on s'assure que la droite que nous souhaitons retrouve ## Descente de gradient Pour trouver la meilleure droite passant par le nuage de points, nous devont appliquer la descente de gradient. En effet, c'est cette formule qui nous permettra de retrouver l'ordonnée à l'origine ainsi que la pente de la meilleur droite. -Sur notre nuage de points nous avons appliqué la formule suivante: -**Formule descente de gradient** +Sur notre nuage de points nous avons appliqué la formule suivante : + +$\begin{pmatrix} ai+1 \\ bi+1 \end{pmatrix} = \begin{pmatrix} ai \\ bi \end{pmatrix}-\lambda*\vec{\nabla}E(ai,bi)$ + +Chez nous, nous l'avons appliquée tel que : + +$\begin{pmatrix} ai+1 \\ bi+1 \end{pmatrix} = \begin{pmatrix} ai \\ bi \end{pmatrix} - \lambda * \vec{\nabla}E\begin{pmatrix} aA + bB -C \\ aB + Nb - D \end{pmatrix}$ + +Pour rappel : + +$A = \sum{{x_{j}}^2}$ + +$B = \sum{x_{j}}$ + +$C = \sum{x_{j}}\sum{y_{j}}$ + +$D = \sum{y_{j}}$ + +et + +$\lambda$ = 0,001 + +N = Nombre de point Et nous nous sommes arrêté quand le $\epsilon$ était inférieur à 0.001 ce qui représente environ 385 itération sur 30 points. @@ -133,22 +156,32 @@ Il a donc fallu en premier temps séparer de manière aléatoire notre nuage de L'erreur quadratique va nous donner un indice de précision 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** +E(a,b) = $\sum_{j=1}^N (a * x_{j} + b - y_{j})^2$ # Résultats Comme nous avons pu le voir dans les figures précédentes, nous avons obtenu des résultats concluents. 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 obtenues sur une génération aléatoire: +Voici un tableau des données obtenues: + +| Nuage de 30 points | Descente de gradient | Erreur quadratique | Groupe comparaison | Descente de gradient | Erreur quadratique | +| :-- | :-- | :-- | :-- | :-- | :-- | +| Complet | 1.009172; 0.458943 | 3.338631 | - | - | - | +| Groupe 0-1 | 1.050928; 0.451662 | 4.045180 | G2 | 1.050411; 0.313806 | 0.505211 | +| Groupe 0-2 | 1.050928; 0.451662 | 1.265707 | G1 | 0.928786; 0.625360 | 0.717456 | +| Groupe 1-2 | 0.953815; 0.847161 | 3.387789 | G0 | 1.050928; 0.451662 | 0.562148 | + +**Comparaison suite** -| 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 | +| Groupes | Différence Gradient | Différence d'erreur quadratique | +| :-- | :-- | :-- | +| Groupe 0-1 et G2 | -0.038394; -0.009297 | 1.092668 | +| Groupe 0-2 et G1 | -0.024588; -0.042307 | 2.104142 | +| Groupe 1-2 et G0 | 0.024588; 0.042307 | 0.654723 | 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. +\newpage + # Conclusion Pour conclure, nous sommes parvenu à retrouvé la meilleure droite passant par un nuage de point aléatoire. Ce qui était l'objectif final ce travail. Ce qui veut dire que les étapes intermédiaire pour arriver à ce point sont fonctionnels, à savoir: **la minisation de la fonction de coup** et la **descente de gradient**. 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int cpt = 0; - while(norme(soustraction_vecteur(vnew,vold)) > 0.001) //0.00001 + while(norme(soustraction_vecteur(vnew,vold)) > 0.00001) //0.00001 { vold.x = vnew.x; vold.y = vnew.y; diff --git a/src/C/group.c b/src/C/group.c index 83d2479..91f7bb4 100644 --- a/src/C/group.c +++ b/src/C/group.c @@ -82,8 +82,7 @@ point_t* regroup(group_t *group, int cadran_a, int cadran_b) { point_t* result = malloc(sizeof(point_t) * nb_points); - for (int i = 0; i < nb_points/2; i++) - { + for (int i = 0; i < nb_points/2; i++) { result[i] = group->points[cadran_a][i]; result[nb_points/2+i] = group->points[cadran_b][i]; } @@ -91,6 +90,17 @@ point_t* regroup(group_t *group, int cadran_a, int cadran_b) { return result; } +point_t* peek_group(group_t *group, int cadran) { + point_t* result = malloc(sizeof(point_t) * group->n_points_in_group); + + for (int i = 0; i < group->n_points_in_group; i++) + { + result[i] = group->points[cadran][i]; + } + + return result; +} + void print_group(group_t *group){ printf("nb points %d ", group->n_points); printf("nb groups %d ", group->n_groups); diff --git a/src/C/group.h b/src/C/group.h index ee89a5d..02b60f1 100644 --- a/src/C/group.h +++ b/src/C/group.h @@ -9,6 +9,7 @@ typedef struct _group_t { void create_group(group_t *group, point_t* points, int n_points, int n_groups); void create_group_random(group_t *group, point_t* points, int n_points, int n_groups); +point_t* peek_group(group_t *group, int cadran); point_t* regroup(group_t *group, int cadran_a, int cadran_b); void print_group(group_t *group); void destroy_group(group_t *group); diff --git a/src/C/main.c b/src/C/main.c index b24e567..d493f8e 100644 --- a/src/C/main.c +++ b/src/C/main.c @@ -61,31 +61,58 @@ int main(int argc, char* argv[]) { point_t* groupe_0_1 = regroup(&groups, 0, 1); point_t* groupe_0_2 = regroup(&groups, 0, 2); point_t* groupe_1_2 = regroup(&groups, 1, 2); + + point_t* G0 = peek_group(&groups, 0); + point_t* G1 = peek_group(&groups, 1); + point_t* G2 = peek_group(&groups, 2); vector_t droite_nuage = descente_gradient(nuage_points, nb_points); vector_t droite_0_1 = descente_gradient(groupe_0_1, groups.n_points_in_group); vector_t droite_0_2 = descente_gradient(groupe_0_2, groups.n_points_in_group); vector_t droite_1_2 = descente_gradient(groupe_1_2, groups.n_points_in_group); + vector_t gradient_G0 = descente_gradient(G0, groups.n_points_in_group); + vector_t gradient_G1 = descente_gradient(G1, groups.n_points_in_group); + vector_t gradient_G2 = descente_gradient(G2, groups.n_points_in_group); + // Faire la comparaison des gradients printf("----- Calcul des gradients -----\n"); printf("Gradient nuage complet: %f; %f ", droite_nuage.x, droite_nuage.y); printf("Erreur quadratique: %f\n", erreur_quadratique(droite_nuage.x, droite_nuage.y, nuage_points, nb_points)); - printf("Gradient groupe 0-1: %f; %f ", droite_0_1.x, droite_0_1.y); - printf("Erreur quadratique: %f\n", erreur_quadratique(droite_0_1.x, droite_0_1.y, groupe_0_1, groups.n_points_in_group)); + printf("Gradient groupe 0-1: %f; %f ", droite_0_1.x, droite_0_1.y); + float erreur01 = erreur_quadratique(droite_0_1.x, droite_0_1.y, groupe_0_1, groups.n_points_in_group*2); + printf("Erreur quadratique: %f\n", erreur01); + + printf("Gradient groupe 0-2: %f; %f ", droite_0_2.x, droite_0_2.y); + float erreur02 = erreur_quadratique(droite_0_2.x, droite_0_2.y, groupe_0_2, groups.n_points_in_group*2); + printf("Erreur quadratique: %f\n", erreur02); - printf("Gradient groupe 0-2: %f; %f ", droite_0_2.x, droite_0_2.y); - printf("Erreur quadratique: %f\n", erreur_quadratique(droite_0_2.x, droite_0_2.y, groupe_0_2, groups.n_points_in_group)); + printf("Gradient groupe 1-2: %f; %f ", droite_1_2.x, droite_1_2.y); + float erreur12 = erreur_quadratique(droite_1_2.x, droite_1_2.y, groupe_1_2, groups.n_points_in_group*2); + printf("Erreur quadratique: %f\n\n", erreur12); - printf("Gradient groupe 1-2: %f; %f ", droite_1_2.x, droite_1_2.y); - printf("Erreur quadratique: %f\n", erreur_quadratique(droite_1_2.x, droite_1_2.y, groupe_1_2, groups.n_points_in_group)); + printf("Gradient groupe G0: %f; %f ", gradient_G0.x, gradient_G0.y); + float erreurG0 = erreur_quadratique(gradient_G0.x, gradient_G0.y, G0, groups.n_points_in_group); + printf("Erreur quadratique: %f\n", erreurG0); + printf("Gradient groupe G1: %f; %f ", gradient_G1.x, gradient_G1.y); + float erreurG1 = erreur_quadratique(gradient_G1.x, gradient_G1.y, G1, groups.n_points_in_group); + printf("Erreur quadratique: %f\n", erreurG1); + printf("Gradient groupe G2: %f; %f ", gradient_G2.x, gradient_G2.y); + float erreurG2 = erreur_quadratique(gradient_G2.x, gradient_G2.y, G2, groups.n_points_in_group); + printf("Erreur quadratique: %f\n", erreurG2); printf("\n"); printf("----- Calcul des différence gradients en fonction des groupes -----\n"); - printf("Différence gradient complet ET groupe 0-1: %f; %f \n", droite_nuage.x - droite_0_1.x, droite_nuage.y - droite_0_1.y); - printf("Différence gradient complet ET groupe 0-2: %f; %f \n", droite_nuage.x - droite_0_2.x, droite_nuage.y - droite_0_2.y); - printf("Différence gradient complet ET groupe 1-2: %f; %f \n", droite_nuage.x - droite_1_2.x, droite_nuage.y - droite_1_2.y); + printf("Différence gradient G2 ET groupe 0-1: %f; %f \n", gradient_G2.x - droite_0_1.x, gradient_G2.y - droite_0_1.y); + printf("Différence gradient G1 ET groupe 0-2: %f; %f \n", gradient_G1.x - droite_0_2.x, gradient_G1.y - droite_0_2.y); + printf("Différence gradient G0 ET groupe 1-2: %f; %f \n", gradient_G0.x - droite_1_2.x, gradient_G0.y - droite_1_2.y); + + printf("\n"); + printf("----- Calcul des différence des erreurs quadratique -----\n"); + printf("Différence erreurs quadratique G2 ET groupe 0-1: %f \n", fabs(erreurG2 - erreur01)); + printf("Différence erreurs quadratique G1 ET groupe 0-2: %f \n", fabs(erreurG1 - erreur02)); + printf("Différence erreurs quadratique G0 ET groupe 1-2: %f \n", fabs(erreurG0 - erreur12)); // Préparation à l'export point_t* nuage_AB = malloc(sizeof(point_t)); diff --git a/src/droite_nuage_0_1_A.vec b/src/droite_nuage_0_1_A.vec index badc478a34e09966f1ee5b65a5667a6639f10c15..842287c2b65b171ec3da9cfa73420a787ff35a9d 100644 GIT binary patch literal 14 VcmZSNU}Rum2;9@`#$Nfs9sm$h1CRg! literal 14 VcmZSNU}RumFrRsE|AUMV_5c<=1n&R< diff --git a/src/droite_nuage_0_1_B.vec b/src/droite_nuage_0_1_B.vec index 2ab48e35a0296fd62ac37740c4d1c89f20ec6724..d605680f9da71044bb0e249177aab3f4d8a4e11b 100644 GIT binary patch literal 14 VcmZSNU}Rum*u1TL!Hq-r>;V|91wsG- literal 14 VcmZSNU}Rumn0VgkjYarvdjJ$;1Tp{s diff --git a/src/droite_nuage_0_2_A.vec b/src/droite_nuage_0_2_A.vec index badc478a34e09966f1ee5b65a5667a6639f10c15..842287c2b65b171ec3da9cfa73420a787ff35a9d 100644 GIT binary patch literal 14 VcmZSNU}Rum2;9@`#$Nfs9sm$h1CRg! literal 14 VcmZSNU}RumFrRsE|AUMV_5c<=1n&R< diff --git a/src/droite_nuage_0_2_B.vec b/src/droite_nuage_0_2_B.vec index 2ab48e35a0296fd62ac37740c4d1c89f20ec6724..d605680f9da71044bb0e249177aab3f4d8a4e11b 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