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Prédictions}\label{pruxe9dictions}} sigmoide de manière à retourner un nombre entre 0 et 2 (avec les poids et bais entraînés). -\hypertarget{ruxe9gression-logistique-multinomiale}{% \paragraph{2.2.4.2 -- Régression logistique -multinomiale}\label{ruxe9gression-logistique-multinomiale}} +multinomiale}\label{ruxe9gression-logistique-multinomiale} Pour la prédiction de la régression logistique multinomiale, nous avons utilisé la fonction suivante: @@ -805,13 +790,11 @@ \subsubsection{2.2.4 -- Prédictions}\label{pruxe9dictions}} \newpage{} -\hypertarget{ruxe9sultats}{% -\subsubsection{2.2.5 -- Résultats}\label{ruxe9sultats}} +\subsubsection{2.2.5 -- Résultats}\label{ruxe9sultats} -\hypertarget{ruxe9sultats-obtenus-par-la-ruxe9gression-logistique-avec-la-fonction-sigmouxefde}{% \paragraph{2.2.5.1 -- Résultats obtenus par la régression logistique avec la fonction -sigmoïde}\label{ruxe9sultats-obtenus-par-la-ruxe9gression-logistique-avec-la-fonction-sigmouxefde}} +sigmoïde}\label{ruxe9sultats-obtenus-par-la-ruxe9gression-logistique-avec-la-fonction-sigmouxefde} Suite à l'apprentissage , nous avons obtenu les résultats suivants: \begin{align*} @@ -874,9 +857,8 @@ \subsubsection{2.2.5 -- Résultats}\label{ruxe9sultats}} modèles sont efficaces, avec un penchant pour la régression logistique qui semble être plus efficace que Naive Bayes. -\hypertarget{ruxe9sultats-obtenus-pour-la-ruxe9gression-logistique-multinomiale}{% \paragraph{2.2.5.2 Résultats obtenus pour la régression logistique -multinomiale}\label{ruxe9sultats-obtenus-pour-la-ruxe9gression-logistique-multinomiale}} +multinomiale}\label{ruxe9sultats-obtenus-pour-la-ruxe9gression-logistique-multinomiale} Suite à l'apprentissage du modèle multinomiale, on obtien les performances suivantes: @@ -916,15 +898,13 @@ \subsubsection{2.2.5 -- Résultats}\label{ruxe9sultats}} \newpage{} -\hypertarget{naive-bayes-1}{% -\subsection{2.3 -- Naive Bayes}\label{naive-bayes-1}} +\subsection{2.3 -- Naive Bayes}\label{naive-bayes-1} Dans cette section, une implémentation d'un classifieur linéaire bayesien (naive bayes) a été réalisée. -\hypertarget{extraction-des-distributions}{% \subsubsection{2.3.1 -- Extraction des -distributions}\label{extraction-des-distributions}} +distributions}\label{extraction-des-distributions} Dans cette implémentation, étant données que toutes nos features sont continues, nous avons considéré que \emph{sepal length}, \emph{sepal @@ -946,8 +926,7 @@ \subsubsection{2.3.1 -- Extraction des paramètres des distributions conditionnelles (normales) des features pour cette classe. -\hypertarget{pruxe9dictions-1}{% -\subsubsection{2.3.2 -- Prédictions}\label{pruxe9dictions-1}} +\subsubsection{2.3.2 -- Prédictions}\label{pruxe9dictions-1} Deux fonctions de prédictions ont été implémenté, @@ -1008,8 +987,7 @@ \subsubsection{2.3.2 -- Prédictions}\label{pruxe9dictions-1}} qui est ce que calculent les fonctions dont on a donné la signature ci-dessus. -\hypertarget{ruxe9sultats-1}{% -\subsubsection{2.3.3 -- Résultats}\label{ruxe9sultats-1}} +\subsubsection{2.3.3 -- Résultats}\label{ruxe9sultats-1} Dans cette section, nous allons simplement reprendre ce qui a été fait dit dans la \href{#ruxe9sultats}{section 2.2.4} et remontrer les mêmes @@ -1038,8 +1016,7 @@ \subsubsection{2.3.3 -- Résultats}\label{ruxe9sultats-1}} \newpage{} -\hypertarget{analyse}{% -\section{3. -- Analyse}\label{analyse}} +\section{3. -- Analyse}\label{analyse} Pour chaque classe y, on peut tracer les fonctions de distribution de probabilité pour chaque donnée \(X_k\) sachant la classe y afin @@ -1111,9 +1088,8 @@ \section{3. -- Analyse}\label{analyse}} faible. De plus, les variables indépendantes ne sont pas séparables les unes des autres. -\hypertarget{phuxe9nomuxe8ne-de-sur-apprentissage}{% \subsection{3.1 -- Phénomène de -sur-apprentissage}\label{phuxe9nomuxe8ne-de-sur-apprentissage}} +sur-apprentissage}\label{phuxe9nomuxe8ne-de-sur-apprentissage} Dans cette partie, nous allons voir les problèmes de sur-apprentissage du modèle de régression logistique multinomial. @@ -1303,9 +1279,8 @@ \subsection{3.1 -- Phénomène de les données de test comptent plus que les performances sur les données d'apprentissage\ldots{} -\hypertarget{fonctions-et-signatures}{% \subsubsection{3.1.1 -- Fonctions et -signatures}\label{fonctions-et-signatures}} +signatures}\label{fonctions-et-signatures} Pour bruiter les données, une fonction a été créée dans \code{overfitting.py}. Cette fonction possède la signature suivante: @@ -1347,12 +1322,10 @@ \subsubsection{3.1.1 -- Fonctions et \newpage{} -\hypertarget{comparaisons}{% -\section{4 -- Comparaisons}\label{comparaisons}} +\section{4 -- Comparaisons}\label{comparaisons} -\hypertarget{vraisemblance-et-classification-des-uxe9chantillons}{% \subsection{4.1 - Vraisemblance et classification des -échantillons}\label{vraisemblance-et-classification-des-uxe9chantillons}} +échantillons}\label{vraisemblance-et-classification-des-uxe9chantillons} Une fois que les paramètres des classes sont obtenus en supposant l'indépendance des variables, on échantillone de nouvelles données afin @@ -1455,12 +1428,10 @@ \subsection{4.1 - Vraisemblance et classification des Les nombres et les graphiques montrent que les échantillons sont très proches des données réelles, donc on a bien la vraisemblance. -\hypertarget{comparaison-avec-sklearn}{% \subsection{4.2 - Comparaison avec -SKLearn}\label{comparaison-avec-sklearn}} +SKLearn}\label{comparaison-avec-sklearn} -\hypertarget{nauxefve-bayes}{% -\subsubsection{4.2.1 - Naïve Bayes}\label{nauxefve-bayes}} +\subsubsection{4.2.1 - Naïve Bayes}\label{nauxefve-bayes} Notre implémentation de Naïve Bayes a été comparée avec celle de SKLearn dans le fichier \texttt{sampling.py}, avec un split des données @@ -1494,9 +1465,8 @@ \subsubsection{4.2.1 - Naïve Bayes}\label{nauxefve-bayes}} f1\_score: 0.9752738654147106 \end{itemize} -\hypertarget{ruxe9gression-logistique-2}{% \subsubsection{4.2.2 - Régression -Logistique}\label{ruxe9gression-logistique-2}} +Logistique}\label{ruxe9gression-logistique-2} Notre implémentation de Régression Logistique a été comparée avec celle de SKLearn dans le fichier \texttt{sampling.py}, avec un split des @@ -1533,9 +1503,8 @@ \subsubsection{4.2.2 - Régression F1\_score: 0.9752738654147106 \end{itemize} -\hypertarget{conclusion-sur-les-comparaisons}{% \subsection{4.3 - Conclusion sur les -comparaisons}\label{conclusion-sur-les-comparaisons}} +comparaisons}\label{conclusion-sur-les-comparaisons} On a vu qu'on avait la vraisemblance, puisque les échantillons sont très proches des données réelles, comme le montrent les graphiques. @@ -1558,11 +1527,9 @@ \subsection{4.3 - Conclusion sur les \newpage{} -\hypertarget{contributions}{% -\section{5 -- Contributions}\label{contributions}} +\section{5 -- Contributions}\label{contributions} -\hypertarget{noah-munz}{% -\subsection{Noah Munz}\label{noah-munz}} +\subsection{Noah Munz}\label{noah-munz} \begin{itemize} \tightlist @@ -1596,8 +1563,7 @@ \subsection{Noah Munz}\label{noah-munz}} \end{itemize} \end{itemize} -\hypertarget{gregory-sedyhk}{% -\subsection{Gregory Sedyhk}\label{gregory-sedyhk}} +\subsection{Gregory Sedykh}\label{gregory-sedykh} \begin{itemize} \tightlist @@ -1607,8 +1573,7 @@ \subsection{Gregory Sedyhk}\label{gregory-sedyhk}} Auteur de la section 4 \end{itemize} -\hypertarget{noah-petershmitt}{% -\subsection{Noah Petershmitt}\label{noah-petershmitt}} +\subsection{Noah Petershmitt}\label{noah-petershmitt} \begin{itemize} \tightlist @@ -1618,8 +1583,7 @@ \subsection{Noah Petershmitt}\label{noah-petershmitt}} Auteur de la section 3 (jusqu'à 3.1) \end{itemize} -\hypertarget{luxe9andre-catogni}{% -\subsection{Léandre Catogni}\label{luxe9andre-catogni}} +\subsection{Léandre Catogni}\label{luxe9andre-catogni} \begin{itemize} \tightlist @@ -1627,8 +1591,7 @@ \subsection{Léandre Catogni}\label{luxe9andre-catogni}} Implémentation de \code{metrics.py} \end{itemize} -\hypertarget{michel-donnet}{% -\subsection{Michel Donnet}\label{michel-donnet}} +\subsection{Michel Donnet}\label{michel-donnet} \begin{itemize} \tightlist