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zy3u`!sj5Y8@!1KVr0di#oZT{eUh@*YWB=o`=FcwI`*GH^g!{?G($~zJH&zr}_czWI z{IKJ0RsF=B0q>V*zYI(Gxbo}MrW-4ij{9G~__kxx<7=z$&3dd{H_KJHL+xC2{r@V< zu=t*zFROMjTYb5Anr)ZxuTP7o=S2B@&Q!eXR_++D`}6GG3+CV7{N1=@y3vdF-(hxl zzXb8kow3E}=JTaYzkberukz*gg1L3?zinc-`g-A%Pw(`-7jD{V7t6+dbF7cq`f2m= z({(S_M~BsYuUgr1?*6w}*WfMF<d%HhGSTefV%IMgwv2x_XXXEUYyLlUmUm|TX3<?W zN56Zvyx96am?P@NZr+7IX1@1pnbh*0ujQ5K{d9q#={4m?;~iJma_7do{A2Df3O+Xd Vc`K_dkBO;~r6rfDs;j>n7XU4ePx}A> diff --git a/rapport.tex b/rapport.tex index 697b614..4b0850e 100644 --- a/rapport.tex +++ b/rapport.tex @@ -6,7 +6,7 @@ \usepackage{graphicx} \graphicspath{ {./images/} } -\title{Rapport de Mathématiques \\ TP1: Optimisation} +\title{HEPIA \\ Rapport de Mathématiques \\ TP1: Optimisation} \author{Gaël Cartier-Michaud - Yannis Chamot} \date{2021} @@ -17,10 +17,23 @@ \end{titlepage} \section{Introduction} - Lors de cette étude, nous avons réalisé un programme permettant de générer une régression linéaire. Dit autrement nous souhaitons + Lors de cette étude, nous souhaitons réaliser un programme permettant de définir la tendance des données collectées lors de l'observation + d'un phénomène. Ces données comportant souvent une part de bruit, nous souhaitons aussi vérifier que ce dernier n'est pas trop important, + et que nos données sont suffisament précises pour être fiables et exploitables.\\ + Pour atteindre cet objectif nous allons générer une régression linéaire. Dit autrement nous allons implémenter un programme permettant de trouver la "meilleure droite" passant par le maximum de point d'un échantillon. - Pour ce travail, nous nous sommes appuyé sur la méthode de la descente de gradient. \\ - Ce rapport décrit les différentes étapes de ce travail.\\ + Dans la vie courante ces techniques sont utilisées dans différentes situations. A titre d'exemple la regression linéaire peut être utile pour + des gérants d’une chaine de magasins pour comparer la performance de différentes boutiques. Dans ce cas là, le chiffre + d’affaires serait comparé au nombre de clients potentiel d’un magasin. La régression linéaire peut aussi être très utile + pour un particulier pour savoir si le prix d’une voiture d’occasion est correct en comparant le prix et le kilométrage + pour un modèle donné. Cela peut aussi être vrai sur un véhicule neuf si on veut comparer l’autonomie d’une voiture + électrique par rapport au prix.\\ + Pour ce travail, nous nous sommes appuyé sur la méthode de la descente de gradient. + Cette méthode peut être utilisé dans des domaines très divers tel que machine learning par exemple. Dans ce cadre la descente de gradient permet + de gagner du temps lorsque le nombre de paramètres devient très élevé s'il on cherche à minimiser un paramètre. + Pour finir, la fiabilité de nos résultats sera vérifiée à l'aide de la validation croiée. Cette méthode est aussi fréquemment utilisée dans le domaine + du machine learning pour l'entrainement des algorithmes. + Ce rapport décrit et explique les différentes étapes de ce travail.\\ Dans une première partie, nous présenterons les concepts théoriques ayant permis la réalisation de ce programme.\\ Nous présenterons ensuite dans une seconde partie les résultats obtenus. Celle-ci contiendra par exemple les coefficients des équations calculés à l'aide de la descente de gradient, les visuels de nos échantillons, @@ -131,6 +144,9 @@ Les erreurs moyennes obtenues sont ensuite comparées entre elles et avec les erreurs obtenues lors de l'analyse numérique pour vérifier que les valeurs sont cohérentes. + + \clearpage + \newpage \section{Résultats} @@ -201,16 +217,16 @@ gradient. L'erreur moyenne est ensuite calculée sur la base de ces deux dernières valeurs. - \begin{figure}[h]x + \begin{figure}[h!] \centering \caption{Valeurs obtenues à partir de la méthode analytique, à partir de la descente de gradient et erreur moyenne de l'échantillon} \includegraphics[scale=0.6]{First_result} \end{figure} - \begin{figure}[h] + \begin{figure}[h!] \centering \caption{Représentation de l'échantillon, régression linéaire et droite obtenue grâce à la méthode linéaire} - \includegraphics[scale=0.5]{Full_sample} + \includegraphics[scale=0.4]{Full_sample} \end{figure} Comme on peut le voir sur ces deux figures, l'erreur de l'échantillon est très faible de façon absolue par rapport aux valeurs théoriques de la régression @@ -275,7 +291,7 @@ \section{Conclusion} Lors de la collecte de données pour une étude statistique et/ou expérimentale, un biais humain ou matériel est souvent compris dans les données. Ce biais est du 'bruit', une erreur souvent comprise dans un interval de valeurs plus ou moins grand et plus ou moins fixe.\\ -Cette collecte de données est réalisé en général pour connaître la tendance que suis le phénomène que l'on observe. Pour définir cette tendance on traduit donc +Cette collecte de données est réalisée en général pour connaître la tendance que suis le phénomène que l'on observe. Pour définir cette tendance on traduit donc notre nuage de point en une fonction représentant la meilleure droite (cas de la régression linéaire) ou courbe passant par le maximum de point de notre échantillon. Notre étude a pu montrer qu'il est possible d'obtenir les paramètres $a$ et $b$ d'une droite d'équation $y = a \cdot x + b$ de façon analytique, ou à l'aide de la méthode de la descente de gradient pour optimiser la fonction d'erreur quadratique d'un nuage de point. La droite obtenue est la régression linéaire d'un nuage de point. -- GitLab