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La première partie présente un modèle fondé sur le raisonnement à partir de cas et les méthodes d'ensemble avec un double empilement itératif. Nous l'avons baptisé \textit{ESCBR} (\textit{Ensemble Stacking Case Based Reasoning}). Ce processus se fait en deux étapes pour trouver des solutions approximatives à des problèmes de régression unidimensionnels et multidimensionnels. Une partie de cette proposition est publiée dans \cite{10.1007/978-3-031-63646-2_11}. La seconde partie montre la conception et l'implémentation d'un SMA intégré à l'ESCBR présenté dans la première partie. Nous considérons dans ce chapitre que le choix des exercices les plus adaptés revient à résoudre un problème de régression. C'est la raison pour laquelle nous testons notre approche sur des jeux de données classiques de régression. -Le modèle présenté dans cette seconde partie associe un raisonnement à partir de cas à des systèmes multi-agents cognitifs implémentant un raisonnement Bayésien. Cette association, combinant échange d'informations entre agents et apprentissage par renforcement, permet l'émergence de prédictions plus précises et améliore ainsi les performances d'ESCBR \cite{10.1007/978-3-031-63646-2_11}.\\ - -Avant de présenter nos propositions, rappelons quelques caractéristiques importantes des outils combinés dans notre approche. - -Le système de raisonnement à partir de cas ne nécessite pas d'entrainement, et peut fonctionner avec des données dynamiques au moment de l'exécution. Les solutions proposées par notre système de RàPC sont générées à l'aide d'algorithmes stochastiques guidant l'exploration de l'espace des solutions. L'évaluation des solutions possibles est effectuée en transformant le problème de régression en un problème d'optimisation et utilise une fonction \textit{objectif}. -\\ \colorbox{yellow}{Il faut expliquer en quelques mots ce que fait cette fonction objectif}\\ -Les prédictions de ce nouvel algorithme ont été comparées à celles de neuf algorithmes de régression classiques sur dix jeux de données pour la régression extraits du site de l'UCI\textcolor{red}{ref?}. En évaluant les résultats obtenus avec la métrique RMSE, ce nouvel algorithme se classe parmi les six meilleurs. Avec la métrique MAE il est le troisième meilleur algorithme des dix évalués, ce qui suggère que les résultats produits par ESCBR sont raisonnablement saitsfaisants. - -Les méthodes d'ensemble permettent de résoudre des problèmes de classification et de régression en combinant les résultats de plusieurs modèles exécutés indépendamment. Certaines de ces méthodes utilisent des modèles différents et des ensembles de données différents tandis que d'autres utilisent les mêmes modèles avec des paramètres différents. La combinaison des résultats provenant de multiples modèles peut être réalisée selon différentes stratégies comme l'application de règles simples ou des approches plus complexes \cite{BAKUROV2021100913}. Plusieurs travaux de recherche explorent la possibilité d'utiliser cette méthode d'ensemble en la combinant à des outils d'apprentissage automatique. - -Les méthodes d'apprentissage automatique appliquées à la régression permettent quant à elles de prédire des valeurs pour différents types de problèmes en construisant, évaluant et formant des modèles linéaires et non linéaires complexes. Mais s'il est possible d'en améliorer la précision en les associant. Les stratégies d'intégration les plus courantes utilisées pour l'apprentissage d'ensemble sont le \textit{Stacking} (empilement), le \textit{Boosting} et le \textit{Bagging} \cite{Liang}. Le \textit{Stacking} est un type de méta-modèle d'apprentissage profond d'ensemble dont l'objectif est d'utiliser diverses techniques d'apprentissage automatique pour surmonter les limites des modèles individuels. Plusieurs études démontrent que l'association de ces techniques permet d'améliorer la précision des résultats \cite{cmc.2023.033417}. Dans ces méthodes d'empilement, les algorithmes de base sont appelés \textit{niveau-0}. Il s'agit généralement de modèles ou d'algorithmes d'apprentissage automatique hétérogènes qui travaillent tous avec les mêmes données. Le méta-algorithme (appelé \textit{niveau-1}) qui unifie les résultats peut être une autre technique d'apprentissage automatique ou un ensemble de règles qui reçoit en entrée les résultats des algorithmes de \textit{niveau-0} \cite{10.3389/fgene.2021.600040}. - -La modélisation de systèmes implémentant différentes techniques d'apprentissage nous amène tout naturellement à considérer les avantages et inconvénients proposés par un système multi-agents. Un SMA est système décentralisé composé de plusieurs entités appelées agents qui ont la capacité d'effectuer des actions spécifiques et de réagir à l'environnement en fonction des informations partielles à leur disposition. Ils peuvent également collaborer les uns avec les autres et coopérer pour atteindre un objectif spécifique. À partir des informations qu'ils perçoivent et échangent, les agents peuvent apprendre de manière autonome et atteindre leurs objectifs efficacement \cite{KAMALI2023110242}. Les actions d'un système multi-agents peuvent être exécutées en parallèle grâce à l'indépendance des agents. Ils sont également robustes aux problèmes présentant une incertitude \cite{DIDDEN2023338}. - -Le raisonnement Bayésien clos ce premier tour d'horizon des techniques implémentées dans l'approche présentée dans ce chapitre. Celui-ci est fondé sur l'observation du raisonnement humain et la relation de l'humain avec l'environnement. Son principe repose sur le postulat que les expériences passées permettent de déduire les états futurs, guidant ainsi ses actions et ses décisions \cite{HIPOLITO2023103510}. Des informations peuvent également être déduites à partir d'informations incomplètes \cite{ZHANG2023110564}. - -\section{Apprentissage par empilement et raisonnement à partir de cas} - -Cette section présente la première version de l'algorithme que nous avons conçu pour résoudre des problèmes de régression. Celui-ci est fondé sur le raisonnement à partir de cas et l'empilement. - -\subsection{Modèle Proposé} - -L'algorithme proposé, ESCBR (\textit{Ensemble Stacking Case-Based Reasoning}), est fondé sur le paradigme générique du RàPC combiné à plusieurs algorithmes de recherche de voisins et de génération de solutions. Ces algorithmes ont été intégrés selon une variation du modèle d'empilement en deux étapes itératives. Cette intégration donne à l'algorithme la capacité de s'adapter à différents types de problèmes, d'éviter les biais et le surentraînement. Les résultats de l'exécution des niveaux d'empilement stockent dans la mémoire des conteneurs de connaissances du système de RàPC des informations qui aident non seulement à l'apprentissage de l'algorithme au fil des itérations, mais facilitent également la génération de solutions à divers problèmes sur différents ensembles de données sans nécessité d'entraînement préalable. La conception itérative en deux cycles améliore la capacité du système de RàPC à travailler et à s'adapter à des problèmes dynamiques en cours d'exécution, comme le montre la figure \ref{figNCBR1}. - -\begin{figure}[!ht] -\centering -\includegraphics[width=\textwidth]{./Figures/NCBR0.png} -\caption{Les deux cycles proposés pour le RàPC} -\label{figNCBR1} -\end{figure} -\colorbox{yellow}{Attention, cette figure apparaît deux fois dans ce chapitre.}\\ -\colorbox{yellow}{J'ai l'impression que ce n'est pas la seule.} - -L'étape de récupération utilise les algorithmes de recherche et la base de données de cas (conteneurs $C1$ et $C3$ de la figure \ref{figNCBR1}) pour trouver les voisins les plus proches d'un problème cible. Puis l'étape de réutilisation utilise les algorithmes de génération de solutions (conteneur $C2$). L'étape de révision évalue ensuite les solutions générées et permet d'en générer de nouvelles itérativement en fonction des paramètres stockés dans le conteneur $C4$. Vient ensuite l'étape de révision prenant en considération la solution sélectionnée. Faisant suite à cette révision, l'étape de renouvellement met à jour les paramètres et les données du conteneur. Enfin, dans l'étape de capitalisation, la base de cas est mise à jour avec l'intégration du nouveau cas. Les flux d'information de l'algorithme proposé sont présentés sur la figure~\ref{figFlowCBR}, tandis que le tableau~\ref{tabVarPar} présente l'ensemble des variables et paramètres de l'algorithme proposé. - -\begin{figure}[!ht] -\centering -\includegraphics[scale=0.6]{./Figures/FlowCBR0.png} -\caption{Flux du \textit{Stacking} RàPC} -\label{figFlowCBR} -\end{figure} - -\begin{table}[!ht] -\centering -\begin{tabular}{cccc} -ID&Type&Description&Domain\\ -\hline -$it$&p&Nombre d'itérations&$\mathbb{N}, it>0$\\ -$np$&p&Nombre de processus&$\mathbb{N}, np>2$\\ -$nl$&p&Nombre maximal de voisins locaux&$\mathbb{N}, nl>0$\\ -$ng$&p&Nombre de voisins globaux&$\mathbb{N}, ng>2$\\ -$n$&v&Dimension de l'espace du problème&$\mathbb{N}, n>0$\\ -$m$&v&Dimension de l'espace de solution&$\mathbb{N}, m>0$\\ -$z$&v&Taille de la base de données&$\mathbb{N}, z>0$\\ -$p$&v&Description du problème&$\mathbb{R} ^ n$\\ -$s$&v&Description de la solution&$\mathbb{R} ^ m$\\ -$r_a$&v&Nombre de modèles pour l'étape rétrouver&$\mathbb{N}, r_a>2$\\ -$r_b$&v&Nombre de modèles pour l'étape de réutilisation&$\mathbb{N}, r_b>2$\\ -$at$&v&Identificateur des actions&$[0,2] \in \mathbb{N}$\\ -$nl_i$&v&Nombre de voisins locaux pour le modèle $i$&$\mathbb{N}, nl_i \le nl$\\ -$g$&v&Description de la meilleure solution globale&$\mathbb{R} ^ m$\\ -$v$&v&Évaluation de la meilleure solution globale&$\mathbb{R}$\\ -$d(x_1,x_2)$&f&Fonction de distance entre $x_1$ et $x_2$ &$\mathbb{R}$\\ -$MP(x_1^z,x_2,a)$&f&Fonction du modèle pour retrouver entre $x_1$ et $x_2$&$\mathbb{R}^{a \times z}$\\ -$MS(x_1^m)$&f&Fonction du modèle pour réutiliser avec $x_1$ &$\mathbb{R}^m$\\ -$f_s(p^n,s^m)$&f&Évaluation des solutions&$\mathbb{R}$\\ -\end{tabular} -\caption{Variables et paramètres du modèle proposé (Type: p - paramètre, v - variable, f - fonction)} -\label{tabVarPar} -\end{table} - -\subsubsection{Rechercher} -La première étape de l'algorithme consiste à trouver les cas les plus similaires à un nouveau cas (appelé \textit{cas cible}). Pour ce faire, le modèle d'empilement de différents processus présenté sur la figure \ref{figSta1} est utilisé. Au niveau-0, chaque processus sélectionne et exécute un algorithme de recherche de voisins différents choisi parmi $r_a$ modèles dans le conteneur $C3$, avec un nombre de voisins $nl_i$ choisi aléatoirement dans l'intervalle $[0,nl]$. Puis au niveau-1, les résultats sont unifiés en construisant un ensemble global de cas similaires. Cinq algorithmes pour le niveau-0 ont été mis en Å“uvre pour l'étape de récupération : KNN (\textit{K Nearest Neighbors} - K plus proches voisins), KMeans, GMM (\textit{Gaussian Mixture Model}), FuzzyC et KNN Ponderation. - -\begin{figure} -\centering -\includegraphics[width=\textwidth]{./Figures/Stacking1.png} -\caption{\textit{Stacking} pour chercher les plus proches voisins} -\label{figSta1} -\end{figure} - -Formellement, le premier modèle d'empilement proposé fonctionne avec deux paramètres : \textcolor{red}{quels sont les 2 paramètres?}une base de données de $z$ cas, un cas est composé de la description du problème et de la description de la solution ${(p^n,s^m)}^{z}$ et un nouveau cas sans solution $p_w^n$. Le but de tous les algorithmes de niveau-0 est de générer une liste locale de cas similaires au cas cible. Ainsi, pour chaque algorithme $j$ exécuté, l'ensemble $X_j=\{x_1,x_2,. ..,x_z \; | \; x_i=MP_i((p^n)^z, p_w^n,nl_i) \}$ est généré. Au niveau-1, un ensemble global est créé à l'aide de tous les ensembles locaux $j$ $X_g=\cup_{n=1}^{ng} min \ ; ((\cup_{j=1}^{np} X_j)-X_g)$. Le résultat du premier modèle d'empilement est l'ensemble $X_g$ avec les $ng$ voisins les plus proches. - -\subsubsection{Réutiliser} - -Une fois la liste globale des cas similaires établie, les informations correspondant aux solutions de chacun de ces cas sont extraites et utilisées pour générer une nouvelle solution qui s'adapte au cas cible, en suivant le processus et les flux d'informations représentés sur la figure \ref{figAuto}. La génération est effectuée avec un deuxième modèle d'empilement de différents processus. Ce modèle d'empilement est représenté sur la figure \ref{figSta2}. Au niveau-0, chaque processus sélectionne et exécute un algorithme de génération différent à partir des modèles $r_b$ dans le conteneur $C2$. Au niveau-1, toutes les solutions générées sont stockées dans une mémoire globale. - -Neuf algorithmes ont été mis en Å“uvre pour l'étape de réutilisation au niveau-0 : moyenne avec probabilité, moyenne sans probabilité, valeurs médianes, sélection aléatoire avec probabilité, copie et changement, vote, interpolation, ACP (analyse en composantes principales) et marche aléatoire. Tous les algorithmes proposés pour générer une nouvelle solution combinent et modifient l'ensemble de solutions construit dans l'étape de récupération. - -\begin{figure}[!ht] -\centering -\includegraphics[width=\textwidth]{./Figures/AutomaticS.png} -\caption{Génération et vérification automatique des solutions} -\label{figAuto} -\end{figure} - -\colorbox{yellow}{Il faut revoir ce qui suit car ne comprenant pas, j'ai complètement ré-écrit.} -La moyenne pondérée avec probabilité permet de construire une solution. La pondération est calculée en fonction de la proximité \colorbox{yellow}{du cas ? lequel ? de la solution ? On la connait ?} avec le cas cible. La probabilité est calculée en fonction de la distance entre les problèmes selon l'équation \ref{gen00} et la solution globale est construite à partir des solutions proposées par les $nl$ algorithmes selon l'équation \ref{gen01}. - -\begin{equation} - \alpha_j=\frac{d(p_j^n,p_w^n)}{\sum_{i=0}^{nl} d(p_i^n,p_w^n)} - \label{gen00} -\end{equation} - -\begin{equation} - s^m_{j,w}= \frac{1}{nl} \sum_{i=0}^{nl} \alpha_j s_{i,j} - \label{gen01} -\end{equation} - -La moyenne sans probabilité copie \colorbox{yellow}{copie ?} aléatoirement les informations des solutions selon la distribution uniforme en donnant la même probabilité à tous les éléments de toutes les solutions proches. Formellement la génération de la solution est calculée selon l'équation \ref{gen2}, où chaque position $j$ est calculée de la même façon \colorbox{yellow}{?}. - -\begin{equation} - s^m_{w,j}= \frac{1}{nl} \sum_{i=0}^{nl} s_{i,j} - \label{gen2} -\end{equation} - -La génération par valeurs médianes construit la solution en utilisant la valeur médiane de toutes les solutions pour chaque dimension. $X_j$ représente l'ensemble des valeurs ordonnées de toutes les solutions $S$ dans la position $j$ \colorbox{yellow}{position ?}. - -\begin{equation} - s_{j,w}^m= - \begin{cases} - x_{j,\frac{m+1}{2}}, \; si \; m \in \{{2k+1 : k \in \mathbb{Z}}\}\\ - \frac{x_{j,\frac{m}{2}}+x_{j,\frac{m}{2}+1}}{2}, \; sinon\\ - \end{cases} -\end{equation} - -La sélection aléatoire avec probabilité génère une solution en copiant aléatoirement l'information des solutions dans chaque position ayant une plus grande probabilité vers le cas de problème associé le plus proche. -\colorbox{yellow}{Il est difficile à comprendre à quel moment intervient l'aléatoire dans ce processus}\\ -\colorbox{yellow}{À ce stade, les équations ne sont plus citées.} - -\begin{equation} -s_{j,w}^m=s_{j,k}^m; k \sim \left(\frac{d(p_k,p_w)}{\sum_{i=1}^nl(p_k,p_i)} \right) -\end{equation} - -"Copie/changement" copie les informations d'une solution aléatoire et en remplace une partie par celles d'une autre solution sélectionnée aléatoirement. - -\begin{equation} -s_{j,w}^m=s_{j,k}^m; k \sim \left(\frac{1}{nl} \right) -\end{equation} - -Le vote permet de copier l'information la plus fréquente de toutes. - -\begin{equation} -s_{j,w}^m=s_{j,k}^m; k=argmax_i \; \mathbb{P}(X=s_{j,i}^m), 1\le i \le nl -\end{equation} - -L'interpolation utilise des informations aléatoires à partir d'une fonction d'interpolation lineaire 1D calculée. -\\ \colorbox{yellow}{Pourquoi aléatoire ? Une interpolation ne rend pas un résultat aléatoire} - -\begin{equation} - s_{w,j}^m \sim \left( \left( \frac{y_i-y_{i+1}}{x_i-x_{i+1}} \right) (x-x_i)+y_i \right), \forall i \; 0 \le i < nl -\end{equation} - -Dans l'analyse en composantes principales (ACP), si la dimension de l'espace des solutions $m$ est plus petit que la dimension de l'espace des problèmes $n$ ($m0$\\ -$np$&p&Nombre d'agents&$\mathbb{N}, np>2$\\ -$nl$&p&Nombre maximal de voisins locaux&$\mathbb{N}, nl>0$\\ -$ng$&p&Nombre de voisins globaux&$\mathbb{N}, ng>2$\\ -$n$&v&Dimension de l'espace du problème&$\mathbb{N}, n>0$\\ -$m$&v&Dimension de l'espace de solution&$\mathbb{N}, m>0$\\ -$p$&v&Description du problème&$\mathbb{R} ^ n$\\ -$s$&v&Description de la solution&$\mathbb{R} ^ m$\\ -$p_w$&v&Description du nouveau problème&$\mathbb{R} ^ n$\\ -$s_w$&v&Description de la nouvelle solution&$\mathbb{R} ^ m$\\ -$n_{rt}$&v&Nombre d'algorithmes por l'étape de rétrouver&$\mathbb{Z}$\\ -$n_{rs}$&v&Nombre d'algorithmes de réutilisation&$\mathbb{Z}$\\ -$d(x_1,x_2)$&f&Fonction de distance entre $x_1$ et $x_2$ &$\mathbb{R}$\\ -$rn(x,y)$&f& - -\begin{tabular}{@{}c@{}}Valeur aléatoire avec distribution normale\\ $x$ moyenne, $y$ écart-type -\end{tabular} - -&$\mathbb{R}_+$\\ -$rnp(x,y)$&f& - -\begin{tabular}{@{}c@{}}Valeur aléatoire discrète, $x$ nombre d'options \\ $y$ vecteur discret de probabilités -\end{tabular} -&$\mathbb{Z}$\\ -$f_s(p^n,s^m)$&f&Évaluation des solutions&$\mathbb{R}$\\ -\end{tabular} -\caption{Variables et paramètres du modèle proposé (Type: p - paramètre, v - variable, f - fonction)} -\label{tabVarPar} -\end{table} - -\subsubsection{Algorithmes} - -Cette section présente de manière plus détaillée les comportements des agents d'ESCBR-SMA. - -Lors de la première étape de l'empilement, chaque agent peut sélectionner l'un des algorithmes à mettre en Å“uvre pour rechercher des problèmes similaires au nouveau problème posé. Les algorithmes possibles sont les suivants : KNN (K-Nearest Neighbor), KMeans, GMM (Gaussian Mixture Model), FuzzyC et KNN pondéré. - -Lors de la deuxième étape de l'empilement, les agents peuvent choisir un algorithme parmi les suivants : pondération avec probabilité, pondération sans probabilité, valeurs médianes, Copier/Changer, vote, interpolation, PCA (analyse en composantes principales) et pas aléatoire\textcolor{red}{et non aléatoire?}. - -L'algorithme de construction de solution par pondération avec probabilité utilise la probabilité que le problème considéré soit le plus proche du problème cible. Cet algorithme construit une solution en copiant certaines informations de la ou des solutions ayant la probabilité la plus élevée. Le choix des informations à copier est aléatoire. Pour sa part, la sélection aléatoire pondérée sans probabilité copie certaines informations (choisies aléatoirement) des solutions sans tenir compte de la probabilité que le problème associé soit le plus proche du problème cible. - -Les valeurs médianes construisent une solution à partir des valeurs médianes des informations de toutes les solutions. - -L'algorithme Copie/Changer copie les informations d'une solution aléatoire -\\ \colorbox{yellow}{c'est le contenu de la solution qui est généré avec des valeurs aléatoires ou} -\\ \colorbox{yellow}{cette solution est choisie aléatoirement ?} -\\et en modifie une partie avec les informations d'une autre solution sélectionnée de manière aléatoire. - -L'algorithme fondé sur le vote construit une solution en considérant les informations indépendemment. Chaque information aura la valeur apparaissant leplus fréquemment dans l'ensemble des solutions possibles. - -L'algorithme fondé sur l'interpolation utilise des informations aléatoires à partir d'une fonction d'interpolation calculée. -\\ \colorbox{yellow}{Qu'est-ce qui est aléatoire ?} -\\ \colorbox{yellow}{Qu'est-ce qui est interpolé ?} -\\ \colorbox{yellow}{Qu'est-ce qui est calculé ?} - -Le générateur avec PCA transforme la description du problème en une description de la solution et établit ainsi une relation entre les valeurs des descripteurs de la partie problème et celles des descripteurs de la patie solution. - -Enfin, l'algorithme de pas aléatoire sélectionne une solution et modifie la valeur d'une seule de ses composantes choisie au hasard selon un certain pas. - -\subsubsection{Structure des agents} - -Tous les agents ont une structure siùilaire, mais chaque agent suit un processus cognitif individuel qui lui permet d'obtenir un comportement indépendant et différent de tous les autres. La figure \ref{figAgent} montre les actions et les variables nécessaires à l'exécution de l'ensemble du processus. - -Chaque agent peut exécuter trois actions différentes : -\begin{itemize} - \item « Récupérer et réutiliser », - \item « Réutiliser » et - \item « Échanger ». -\end{itemize} - -Chaque agent accède aux valeurs de huit variables qui lui sont propres : -\begin{itemize} - \item le nombre de voisins définissant le nombre de problèmes similaires au nouveau problème posé que l'agent doit rechercher dans la base de connaissancesÃ’, - \item la liste des voisins les plus proches établit la liste des agents avec lesquels des informations peuvent être échangées, - \item l'identifiant de l'algorithme de récupération que l'agent exécutera pour trouver les problèmes similaires au nouveau problème, - \item l'identifiant de l'algorithme de réutilisation que l'agent exécutera pour générer une solution candidate au cas cible, - \item la description de la solution générée, - \item l'évaluation de la solution renseignant sur la qualité de la solution générée et calculée selon une fonction d'optimisation (eq \ref{eqOpt1} et eq \ref{eqOpt2}), - \item le vecteur bayésien nécessaire aux algorithmes de récupération contenant les valeurs de probabilité, et - \item le vecteur bayésien nécessaire aux algorithmes de réutilisation contenant les valeurs de probabilité. -\end{itemize} - -\begin{equation} -min \; \left( f_s(p_w^n, s_w^m) \right) = min \left( \sum_{i=1}^{ng} \frac{d(s_w^m,s_i^t)}{d(p_w^n,p_i^n)^2} \right) -\label{eqOpt1} -\end{equation} - -\begin{equation} -s_i^t=s_i^m+rn(0,d(p_w^n,p_i^n)) -\label{eqOpt2} -\end{equation} - -\begin{figure}[!ht] -\centering -\includegraphics[scale=0.7]{Figures/agent.png} -\caption{Structure interne des agents} -\label{figAgent} -\end{figure} - -\subsubsection{Apprentissage des agents} - -Tous les agents considèrent un vecteur bayésien de probabilité pour la phase de récupération (conteneur $C3$) et un vecteur bayésien de probabilité pour la phase de réutilisation (conteneur $C2$). Initialement, ces vecteurs sont configurés avec une probabilité uniforme pour tous les algorithmes de récupération et de réutilisation (probabilité \textit{a priori}). À chaque itération, les vecteurs sont mis à jour selon l'équation bayésienne \colorbox{yellow}{ref ? Numéro d'équation ?} en utilisant les meilleurs résultats du même agent comme paramètre de vraisemblance. L'agent apprend ainsi de son expérience et sélectionne les algorithmes les plus aptes à fournir les meilleures réponses au regard de l'objectif défini. - -Par conséquent, d'un point plus global au niveau du SMA, l'apprentissage est fondé sur un raisonnement bayésien où les vecteurs de récupération et de réutilisation évoluent. Un exemple d'évolution est présenté sur la figure \ref{fig:bayev}. Au cours d'une itération, si un algorithme a contribué à la construction de la meilleure solution, il reçoit un point pour chaque agent qui l'a utilisé. Ces informations sont stockées dans un vecteur normalisé qui est ensuite utilisé comme vecteur de vraisemblance $P(A)$ pour calculer les nouvelles probabilités dans l'équation bayésienne \ref{eqBay} (dans cette équation, $P(B)$ est le terme de normalisation global). - -\begin{figure} - \centering - \includegraphics[scale=0.5]{Figures/BayesianEvolution.png} - \caption{Exemple d'évolution Bayésienne des vecteurs pour un agent. a) Initialisation des probabilités $P(B|A)$ vecteurs pour Retrieve et Reuse, b) Probabilités après quelques itérations $P(A|B)$ vecteurs pour Retrieve et Reuse} - \label{fig:bayev} -\end{figure} - -\begin{equation} - P(A|B)=\frac{P(B|A)P(A)}{P(B)} - \label{eqBay} -\end{equation} - -La sélection d'un algorithme de recherche $a_{rt}$ se fait au moyen d'une valeur aléatoire -\\ \colorbox{yellow}{La valeur est aléatoire ? Ou le choix de la valeur est aléatoire ?} \\ -tirée du vecteur de distribution de recherche discrète obtenu par raisonnement bayésien et calculé selon l'équation \ref{eqRta}. - -\begin{equation} - a_{rt}=rnp(n_{rt},P(A|B)_{rt}) - \label{eqRta} -\end{equation} - -La sélection d'un algorithme de réutilisation $a_{rs}$ se fait au moyen d'une valeur aléatoire -\\ \colorbox{yellow}{La valeur est aléatoire ? Ou le choix de la valeur est aléatoire ?} \\ -tirée du vecteur de distribution de réutilisation discrète obtenu par raisonnement bayésien et calculé selon l'équation \ref{eqRsa}. - -\begin{equation} - a_{rs}=rnp(n_{rs},P(A|B)_{rs}) - \label{eqRsa} -\end{equation} - -Le processus d'apprentissage est réalisé individuellement par chaque agent, mais comporte un aspect collaboratif puisque les agents échangent les informations qu'ils ont calculées, les optimisations locales qu'ils ont réalisées, ainsi que les algorithmes et les paramètres qu'ils ont utilisés. - -\subsubsection{Échanges entre les agents} - -Un agent peut modifier ses informations et leur affecter les valeurs de celles d'un voisin au cours de chaque itération en choisissant au hasard un voisin dans sa liste de voisins les plus proches. L'ensemble de ces changements permettent de propager les paramètres menant aux meilleurs résultats et d'effectuer des actions rétrospectivement. Les informations modifables sont choisies aléatoirement : il peut s'agir du nombre de voisins, de la liste des voisins les plus proches, de l'algorithme de récupération, de l'algorithme de réutilisation ou même des vecteurs bayésiens. - -\subsection{Résultats} - -Des expérimentations sur onze jeux de données relatifs à des problèmes de régression et présentant des caractéristiques différentes ont été réalisées. Les jeux de données et leurs caractéristiques sont consignés dans le tableau \ref{tabBases}. Les valeurs des paramètres de l'algorithme sont les suivantes $it=100$, $np=50$, $nl=10$ et $ng=10$. -\\ \colorbox{yellow}{Quel algorithme ?} -\\ \colorbox{yellow}{À quoi correspondent ces paramètres ? Quelles sont les unités ?}\\ -Le tableau \ref{AlgsPar} présente les valeurs des paramètres de tous les autres algorithmes. - -\begin{table}[!ht] -\footnotesize -\centering -\begin{tabular}{c|ccccccccccc} -&A1&A2&A3&A4&A5&A6&A7&A8&A9&A10\\ -\hline -%DS1&9.081&12.301&1.228&1.066&7.763&9.081&9.078&9.764&0.750&8.263&8.225\\ -DS1&9.081&12.301&1.228&1.066&7.763&9.081&9.078&9.764&0.750&8.225\\ -%DS2&0.022&0.025&0.019&0.013&0.017&0.022&0.022&0.037&0.011&0.016&0.016\\ -DS2&0.022&0.025&0.019&0.013&0.017&0.022&0.022&0.037&0.011&0.016\\%DS3&8.756&8.465&9.656&7.665&8.716&8.756&9.005&9.177&7.369&8.148&7.991\\ -DS3&8.756&8.465&9.656&7.665&8.716&8.756&9.005&9.177&7.369&7.991\\ -%DS4&0.647&0.752&0.794&0.602&0.688&0.647&0.646&0.798&0.616&0.628&0.607\\ -DS4&0.647&0.752&0.794&0.602&0.688&0.647&0.646&0.798&0.616&0.607\\ -%DS5&0.767&0.824&0.877&0.66&0.826&0.767&0.775&0.87&0.703&0.690&0.662\\ -DS5&0.767&0.824&0.877&0.66&0.826&0.767&0.775&0.87&0.703&0.662\\ -%DS6&10.525&9.174&6.93&5.372&6.662&10.525&10.525&10.527&5.131&9.413&9.070\\ -DS6&10.525&9.174&6.93&5.372&6.662&10.525&10.525&10.527&5.131&9.070\\ -%DS7&2.961&2.451&0.589&0.528&3.955&2.961&3.009&4.083&0.490&3.031&2.941\\ -DS7&2.961&2.451&0.589&0.528&3.955&2.961&3.009&4.083&0.490&2.941\\ -%DS8&1.298&1.125&1.360&1.197&1.486&1.298&1.298&1.306&1.128&2.752&2.553\\ -DS8&1.298&1.125&1.360&1.197&1.486&1.298&1.298&1.306&1.128&2.553\\ -%DS9&2.256&2.565&3.174&2.377&2.817&2.256&2.255&2.468&2.293&2.747&2.468\\ -DS9&2.256&2.565&3.174&2.377&2.817&2.256&2.255&2.468&2.293&2.468\\ -%DS10&3.136&3.415&4.173&3.165&3.710&3.136&3.135&3.161&3.108&3.897&3.621\\ -DS10&3.136&3.415&4.173&3.165&3.710&3.136&3.135&3.161&3.108&3.621\\ -DS11&0.625&0.565&0.741&0.56&0.606&0.626&0.626&0.681& -0.541&0.54\\ -\hline -%Avg. Rank&5.8&6.5&7.2&2.3&6.7&5.7&5.6&8.65&1.9&4.65\\ -Avg. Rank&6.4&6.9&8.2&2.6&7.2&6.45&6.35&9.55&2.1&4.75\\ -\end{tabular} -\caption{Résultat selon la métrique RMSE (Root Mean Squared Error) pour les jeux de données évalués avec les différents algorithmes de régression considérés.} -\label{tabRes1} -\end{table} - -\begin{table}[!ht] -\scriptsize -\centering -\begin{tabular}{llccp{1.4cm}p{1.2cm}p{1.2cm}} -ID&DataSet&Features&Instances&Output. Dimension&Input. Domain&Output Domain\\ -\hline -DS1&Yatch Hydrodynamics&6&308&1&$\mathbb{R}$&$\mathbb{R}$\\ -DS2&Electrical Grid Stability&12&10000&1&$\mathbb{R}$&$\mathbb{R}$\\ -DS3&Real State Valuation&6&414&1&$\mathbb{R_+}$&$\mathbb{R_+}$\\ -DS4&Wine Quality (Red)&11&1598&1&$\mathbb{R_+}$&$\mathbb{N}$\\ -DS5&Wine Quality (White)&11&4897&1&$\mathbb{R_+}$&$\mathbb{N}$\\ -DS6&Concrete Compressive Strength&8&1030&1&$\mathbb{R_+}$&$\mathbb{R_+}$\\ -DS7&Energy Efficiency&8&768&2&$\mathbb{R_+}$&$\mathbb{R_+}$\\ -DS8&Gas Turbine CO, NOx Emission (2015)&9&7384&2&$\mathbb{R_+}$&$\mathbb{R_+}$\\ -DS9&Student Performace Portuguese&30&649&3&$\mathbb{N*}$&$\mathbb{N}$\\ -DS10&Student Performance Math&30&395&3&$\mathbb{N*}$&$\mathbb{N}$\\ -DS11&Generated Student Performance&5&1000&1&$\mathbb{R}_+$&$\mathbb{R}_+$\\ -\end{tabular} -\caption{Description des jeux de données évalués. (* après codification comme -\textit{String})} -\label{tabBases} -\end{table} - -\begin{table}[!ht] -\centering -\begin{tabular}{ccc|ccc} -ID&Parameter&Value&ID&Parameter&Value\\ -\hline -A1&Intercept&True&A6&Degree&4\\ -&Positive&True&&Bias&True\\ -\hline -A2&Neighbors&5&A7&Fit Intercept&True\\ -&Weights&Uniform&&alpha&0.2\\ -&Metric&Minkowsky&&tol&1e-4\\ -&Power Minkowsky&2\\ -\hline -A3&Error&Squared Error&A8&Fit Intercept&True\\ -&Min samples split&2&&alpha&[0.00001, 0.4]\\ -&&&&Max iter&1000\\ -&&&&tol&1e-4\\ -\hline -A4&Estimators&10&A9&Error&Squarred Error\\ -&Error&Squared Error&&Learning Rate&0.1\\ -&Min samples split&2&&Estimators&100\\ -&Bootstrap&True&&Min Split&2\\ -\hline -A5&Hidden Layers&100\\ -&Activation&Relu\\ -&Solver&Adam\\ -&alpha&0.0001\\ -&Learning Rate&0.001\\ -&Max Iter&200\\ -&beta1&0.9\\ -&beta2&0.999\\ -&epsilon&1e-8\\ -\end{tabular} -\caption{Paramètres de tous les algorithmes comparés} -\label{AlgsPar} -\end{table} - -\begin{table}[!ht] -\footnotesize -\centering -\begin{tabular}{c|ccccccccccc} -Dataset&A1&A2&A3&A4&A5&A6&A7&A8&A9&A10\\ -\hline -%DS1&6.776&2.385&0.231&0.207&3.632&6.778&6.307&5.186&0.162&1.193&1.218\\ -DS1&6.776&2.385&0.231&0.207&3.632&6.778&6.307&5.186&0.162&1.218\\ -%DS2&0.015&0.017&0.012&0.008&0.012&0.015&0.015&0.030&0.007&0.011&0.010\\ -DS2&0.015&0.017&0.012&0.008&0.012&0.015&0.015&0.030&0.007&0.010\\ -%DS3&5.092&4.320&4.1&3.632&4.435&5.092&5.20&5.132&3.504&3.90&3.771\\ -DS3&5.092&4.320&4.1&3.632&4.435&5.092&5.20&5.132&3.504&3.771\\ -%DS4&0.413&0.495&0.18&0.325&0.451&0.413&0.412&0.544&0.387&0.154&0.135\\ -DS4&0.413&0.495&0.18&0.325&0.451&0.413&0.412&0.544&0.387&0.135\\ -%DS5&0.509&0.548&0.285&0.374&0.550&0.509&0.509&0.633&0.456&0.113&0.085\\ -DS5&0.509&0.548&0.285&0.374&0.550&0.509&0.509&0.633&0.456&0.085\\ -%DS6&6.989&5.709&3.134&2.839&4.306&6.989&6.989&6.986&3.084&5.439&5.072\\ -DS6&6.989&5.709&3.134&2.839&4.306&6.989&6.989&6.986&3.084&5.072\\ -%DS7&1.393&1.372&0.217&0.218&2.523&1.393&1.529&2.346&0.243&1.008&1.006\\ -DS7&1.393&1.372&0.217&0.218&2.523&1.393&1.529&2.346&0.243&1.006\\ -%DS8&0.549&0.297&0.365&0.289&0.742&0.549&0.549&0.540&0.309&0.861&0.794\\ -DS8&0.549&0.297&0.365&0.289&0.742&0.549&0.549&0.540&0.309&0.794\\ -%DS9&1.496&1.788&2.080&1.612&2.005&1.496&1.496&1.714&1.538&1.721&1.556\\ -DS9&1.496&1.788&2.080&1.612&2.005&1.496&1.496&1.714&1.538&1.556\\ -%DS10&2.344&2.534&2.910&2.331&2.543&2.344&2.344&2.481&2.258&2.602&2.371\\ -DS10&2.344&2.534&2.910&2.331&2.543&2.344&2.344&2.481&2.258&2.371\\ -DS11&0.387&0.35&0.46&0.338&0.384&0.387&0.387&0.453& -0.327&0.347\\ -\hline -Avg. Rank&7.15&6.9&5.35&2.6&7.95&7.25&7.3&9.0&2.5 -&4.5\\ -%Avg. Rank&6.45&6.5&4.35&2.4&7.45&6.55&6.6&8.1&2.4&4.2\\ -\end{tabular} -\caption{Comparaison des résultats MAE (Median Absolute Error) pour les bases de données évaluées avec des algorithmes de régression} -\label{tabRes2} -\end{table} - -Les tableaux \ref{tabRes1} et \ref{tabRes2} présentent respectivement les RMSE et les MAE obtenues pour chaque jeu de données et chaque algorithme testé. - -La figure \ref{figBox} représente graphiquement la dispersion globale, la médiane et les valeurs aberrantes obtenues durant ces tests. Cette figure montre qu'ESCBR-SMA génère parfois plus de valeurs aberrantes que d'autres algorithmes. Toutefois, la variance est très faible. De plus, la convergence est proche de la valeur réelle et même meilleure que celle de la plupart des autres algorithmes testés. Il est également possible de remarquer que les valeurs aberrantes sont plus élevées que les valeurs réelles.\textcolor{red}{plus élevées??, pourquoi?} - -\begin{figure}[!ht] -\includegraphics[width=\textwidth]{Figures/boxplot2.png} -\caption{Résultats selon la métrique MAE (Median Absolute Error) pour les dix algorithmes testés} -\label{figBox} -\end{figure} - -Les résultats montrent que l'algorithme ESCBR-SMA est compétitif pour la plupart des jeux de données considérés dans cette étude. Il obtient en effet les meilleurs résultats sur $DS2$, $DS4$, $DS5$ et $DS9$. Globalement, ESCBR-SMA est le troisième meilleur algorithme. Il se classe donc à proximité d'autres algorithmes d'ensemble. Par rapport à l'ESCBR, une amélioration des résultats et une réduction de la variance ont été obtenues, démontrant ainsi que les systèmes multi-agents et le raisonnement stochastique bayésien contribuent à l'apprentissage et à la convergence vers des solutions plus proches de la solution réelle. - -\subsection{Conclusion} -L'ECBR intégrant un SMA propose d'utiliser les avantages des systèmes multi-agents pour améliorer la qualité des solutions proposées et également le processus d'apprentissage global en couvrant un plus large spectre de possibilités et en le couvrant de manière intelligente à l'aide d'un raisonnement bayésien. ESCBR-SMA permet ainsi d'obtenir de bonnes approximations avec peu de données. - -Ce travail a démontré la capacité d'ESCBR-SMA à trouver des solutions proches de l'optimum global pour la majorité des ensembles de données analysés. Ces jeux de données présentent une certaine diversité, ils peuvent être déséquilibrés et ils sont de tailles différentes. Grâce aux caractéristiques inhérentes aux systèmes multi-agents (possibilités de rétroactions, d'échanges d'informations, l'émergence d'une intelligence collective, utilisation d'un raisonnement cognitif), les performances d'ESCBR-SMA sont meilleures que celles d'ESCBR. +\chapter{Le raisonnement à partir de cas (RàPC) pour la régression} +\chaptermark{RàPC POUR RÉGRESSION} + +\section{Introduction} + +Ce chapitre est divisé en deux parties. La première partie présente un modèle fondé sur le raisonnement à partir de cas et les méthodes d'ensemble avec un double empilement itératif. Nous l'avons baptisé \textit{ESCBR} (\textit{Ensemble Stacking Case Based Reasoning}). Ce processus se fait en deux étapes pour trouver des solutions approximatives à des problèmes de régression unidimensionnels et multidimensionnels. Une partie de cette proposition est publiée dans \cite{10.1007/978-3-031-63646-2_11}. La seconde partie montre la conception et l'implémentation d'un SMA intégré à l'ESCBR présenté dans la première partie. Nous considérons dans ce chapitre que le choix des exercices les plus adaptés revient à résoudre un problème de régression. C'est la raison pour laquelle nous testons notre approche sur des jeux de données classiques de régression. +Le modèle présenté dans cette seconde partie associe un raisonnement à partir de cas à des systèmes multi-agents cognitifs implémentant un raisonnement Bayésien. Cette association, combinant échange d'informations entre agents et apprentissage par renforcement, permet l'émergence de prédictions plus précises et améliore ainsi les performances d'ESCBR \cite{10.1007/978-3-031-63646-2_11}.\\ + +Avant de présenter nos propositions, rappelons quelques caractéristiques importantes des outils combinés dans notre approche. + +Le système de raisonnement à partir de cas ne nécessite pas d'entrainement, et peut fonctionner avec des données dynamiques au moment de l'exécution. Les solutions proposées par notre système de RàPC sont générées à l'aide d'algorithmes stochastiques guidant l'exploration de l'espace des solutions. L'évaluation des solutions possibles est effectuée en transformant le problème de régression en un problème d'optimisation et utilise une fonction \textit{objectif}. +\\ \colorbox{yellow}{Il faut expliquer en quelques mots ce que fait cette fonction objectif}\\ +Les prédictions de ce nouvel algorithme ont été comparées à celles de neuf algorithmes de régression classiques sur dix jeux de données pour la régression extraits du site de l'UCI\textcolor{red}{ref?}. En évaluant les résultats obtenus avec la métrique RMSE, ce nouvel algorithme se classe parmi les six meilleurs. Avec la métrique MAE il est le troisième meilleur algorithme des dix évalués, ce qui suggère que les résultats produits par ESCBR sont raisonnablement saitsfaisants. + +Les méthodes d'ensemble permettent de résoudre des problèmes de classification et de régression en combinant les résultats de plusieurs modèles exécutés indépendamment. Certaines de ces méthodes utilisent des modèles différents et des ensembles de données différents tandis que d'autres utilisent les mêmes modèles avec des paramètres différents. La combinaison des résultats provenant de multiples modèles peut être réalisée selon différentes stratégies comme l'application de règles simples ou des approches plus complexes \cite{BAKUROV2021100913}. Plusieurs travaux de recherche explorent la possibilité d'utiliser cette méthode d'ensemble en la combinant à des outils d'apprentissage automatique. + +Les méthodes d'apprentissage automatique appliquées à la régression permettent quant à elles de prédire des valeurs pour différents types de problèmes en construisant, évaluant et formant des modèles linéaires et non linéaires complexes. Mais s'il est possible d'en améliorer la précision en les associant. Les stratégies d'intégration les plus courantes utilisées pour l'apprentissage d'ensemble sont le \textit{Stacking} (empilement), le \textit{Boosting} et le \textit{Bagging} \cite{Liang}. Le \textit{Stacking} est un type de méta-modèle d'apprentissage profond d'ensemble dont l'objectif est d'utiliser diverses techniques d'apprentissage automatique pour surmonter les limites des modèles individuels. Plusieurs études démontrent que l'association de ces techniques permet d'améliorer la précision des résultats \cite{cmc.2023.033417}. Dans ces méthodes d'empilement, les algorithmes de base sont appelés \textit{niveau-0}. Il s'agit généralement de modèles ou d'algorithmes d'apprentissage automatique hétérogènes qui travaillent tous avec les mêmes données. Le méta-algorithme (appelé \textit{niveau-1}) qui unifie les résultats peut être une autre technique d'apprentissage automatique ou un ensemble de règles qui reçoit en entrée les résultats des algorithmes de \textit{niveau-0} \cite{10.3389/fgene.2021.600040}. + +La modélisation de systèmes implémentant différentes techniques d'apprentissage nous amène tout naturellement à considérer les avantages et inconvénients proposés par un système multi-agents. Un SMA est système décentralisé composé de plusieurs entités appelées agents qui ont la capacité d'effectuer des actions spécifiques et de réagir à l'environnement en fonction des informations partielles à leur disposition. Ils peuvent également collaborer les uns avec les autres et coopérer pour atteindre un objectif spécifique. À partir des informations qu'ils perçoivent et échangent, les agents peuvent apprendre de manière autonome et atteindre leurs objectifs efficacement \cite{KAMALI2023110242}. Les actions d'un système multi-agents peuvent être exécutées en parallèle grâce à l'indépendance des agents. Ils sont également robustes aux problèmes présentant une incertitude \cite{DIDDEN2023338}. + +Le raisonnement Bayésien clos ce premier tour d'horizon des techniques implémentées dans l'approche présentée dans ce chapitre. Celui-ci est fondé sur l'observation du raisonnement humain et la relation de l'humain avec l'environnement. Son principe repose sur le postulat que les expériences passées permettent de déduire les états futurs, guidant ainsi ses actions et ses décisions \cite{HIPOLITO2023103510}. Des informations peuvent également être déduites à partir d'informations incomplètes \cite{ZHANG2023110564}. + +\section{Apprentissage par empilement et raisonnement à partir de cas} + +Cette section présente la première version de l'algorithme que nous avons conçu pour résoudre des problèmes de régression. Celui-ci est fondé sur le raisonnement à partir de cas et l'empilement. + +\subsection{Modèle Proposé} + +L'algorithme proposé, ESCBR (\textit{Ensemble Stacking Case-Based Reasoning}), est fondé sur le paradigme générique du RàPC combiné à plusieurs algorithmes de recherche de voisins et de génération de solutions. Ces algorithmes ont été intégrés selon une variation du modèle d'empilement en deux étapes itératives. Cette intégration donne à l'algorithme la capacité de s'adapter à différents types de problèmes, d'éviter les biais et le surentraînement. Les résultats de l'exécution des niveaux d'empilement stockent dans la mémoire des conteneurs de connaissances du système de RàPC des informations qui aident non seulement à l'apprentissage de l'algorithme au fil des itérations, mais facilitent également la génération de solutions à divers problèmes sur différents ensembles de données sans nécessité d'entraînement préalable. La conception itérative en deux cycles améliore la capacité du système de RàPC à travailler et à s'adapter à des problèmes dynamiques en cours d'exécution, comme le montre la figure \ref{figNCBR1}. + +\begin{figure}[!ht] +\centering +\includegraphics[width=\textwidth]{./Figures/NCBR0.png} +\caption{Les deux cycles proposés pour le RàPC} +\label{figNCBR1} +\end{figure} +%\colorbox{yellow}{Attention, cette figure apparaît deux fois dans ce chapitre.}\\ +%\colorbox{yellow}{J'ai l'impression que ce n'est pas la seule.} + +L'étape de récupération utilise les algorithmes de recherche et la base de données de cas (conteneurs $C1$ et $C3$ de la figure \ref{figNCBR1}) pour trouver les voisins les plus proches d'un problème cible. Puis l'étape de réutilisation utilise les algorithmes de génération de solutions (conteneur $C2$). L'étape de révision évalue ensuite les solutions générées et permet d'en générer de nouvelles itérativement en fonction des paramètres stockés dans le conteneur $C4$. Vient ensuite l'étape de révision prenant en considération la solution sélectionnée. Faisant suite à cette révision, l'étape de renouvellement met à jour les paramètres et les données du conteneur. Enfin, dans l'étape de capitalisation, la base de cas est mise à jour avec l'intégration du nouveau cas. Les flux d'information de l'algorithme proposé sont présentés sur la figure~\ref{figFlowCBR}, tandis que le tableau~\ref{tabVarPar} présente l'ensemble des variables et paramètres de l'algorithme proposé. + +\begin{figure}[!ht] +\centering +\includegraphics[scale=0.6]{./Figures/FlowCBR0.png} +\caption{Flux du \textit{Stacking} RàPC} +\label{figFlowCBR} +\end{figure} + +\begin{table}[!ht] +\centering +\begin{tabular}{cccc} +ID&Type&Description&Domain\\ +\hline +$it$&p&Nombre d'itérations&$\mathbb{N}, it>0$\\ +$np$&p&Nombre de processus&$\mathbb{N}, np>2$\\ +$nl$&p&Nombre maximal de voisins locaux&$\mathbb{N}, nl>0$\\ +$ng$&p&Nombre de voisins globaux&$\mathbb{N}, ng>2$\\ +$n$&v&Dimension de l'espace du problème&$\mathbb{N}, n>0$\\ +$m$&v&Dimension de l'espace de solution&$\mathbb{N}, m>0$\\ +$z$&v&Taille de la base de données&$\mathbb{N}, z>0$\\ +$p$&v&Description du problème&$\mathbb{R} ^ n$\\ +$s$&v&Description de la solution&$\mathbb{R} ^ m$\\ +$r_a$&v&Nombre de modèles pour l'étape rétrouver&$\mathbb{N}, r_a>2$\\ +$r_b$&v&Nombre de modèles pour l'étape de réutilisation&$\mathbb{N}, r_b>2$\\ +$at$&v&Identificateur des actions&$[0,2] \in \mathbb{N}$\\ +$nl_i$&v&Nombre de voisins locaux pour le modèle $i$&$\mathbb{N}, nl_i \le nl$\\ +$g$&v&Description de la meilleure solution globale&$\mathbb{R} ^ m$\\ +$v$&v&Évaluation de la meilleure solution globale&$\mathbb{R}$\\ +$d(x_1,x_2)$&f&Fonction de distance entre $x_1$ et $x_2$ &$\mathbb{R}$\\ +$MP(x_1^z,x_2,a)$&f&Fonction du modèle pour retrouver entre $x_1$ et $x_2$&$\mathbb{R}^{a \times z}$\\ +$MS(x_1^m)$&f&Fonction du modèle pour réutiliser avec $x_1$ &$\mathbb{R}^m$\\ +$f_s(p^n,s^m)$&f&Évaluation des solutions&$\mathbb{R}$\\ +\end{tabular} +\caption{Variables et paramètres du modèle proposé (Type: p - paramètre, v - variable, f - fonction)} +\label{tabVarPar} +\end{table} + +\subsubsection{Rechercher} +La première étape de l'algorithme consiste à trouver les cas les plus similaires à un nouveau cas (appelé \textit{cas cible}). Pour ce faire, le modèle d'empilement de différents processus présenté sur la figure \ref{figSta1} est utilisé. Au niveau-0, chaque processus sélectionne et exécute un algorithme de recherche de voisins différents choisi parmi $r_a$ modèles dans le conteneur $C3$, avec un nombre de voisins $nl_i$ choisi aléatoirement dans l'intervalle $[0,nl]$. Puis au niveau-1, les résultats sont unifiés en construisant un ensemble global de cas similaires. Cinq algorithmes pour le niveau-0 ont été mis en Å“uvre pour l'étape de récupération : KNN (\textit{K Nearest Neighbors} - K plus proches voisins), KMeans, GMM (\textit{Gaussian Mixture Model}), FuzzyC et KNN Ponderation. + +\begin{figure} +\centering +\includegraphics[width=\textwidth]{./Figures/Stacking1.png} +\caption{\textit{Stacking} pour chercher les plus proches voisins} +\label{figSta1} +\end{figure} + +Formellement, le premier modèle d'empilement proposé fonctionne avec deux paramètres : %\textcolor{red}{quels sont les 2 paramètres?} +une base de données de $z$ cas où un cas est composé de la description du problème et de la description de la solution ${(p^n,s^m)}^{z}$ et un nouveau cas sans solution $p_w^n$. Le but de tous les algorithmes de niveau-0 est de générer une liste locale de cas similaires au cas cible. Ainsi, pour chaque algorithme $j$ exécuté, l'ensemble $X_j=\{x_1,x_2,. ..,x_z \; | \; x_i=MP_i((p^n)^z, p_w^n,nl_i) \}$ est généré. Au niveau-1, un ensemble global est créé à l'aide de tous les ensembles locaux $j$ $X_g=\cup_{n=1}^{ng} min \ ; ((\cup_{j=1}^{np} X_j)-X_g)$. Le résultat du premier modèle d'empilement est l'ensemble $X_g$ avec les $ng$ voisins les plus proches. + +\subsubsection{Réutiliser} + +Une fois la liste globale des cas similaires établie, les informations correspondant aux solutions de chacun de ces cas sont extraites et utilisées pour générer une nouvelle solution qui s'adapte au cas cible, en suivant le processus et les flux d'informations représentés sur la figure \ref{figAuto}. La génération est effectuée avec un deuxième modèle d'empilement de différents processus. Ce modèle d'empilement est représenté sur la figure \ref{figSta2}. Au niveau-0, chaque processus sélectionne et exécute un algorithme de génération différent à partir des modèles $r_b$ dans le conteneur $C2$. Au niveau-1, toutes les solutions générées sont stockées dans une mémoire globale. + +Neuf algorithmes ont été mis en Å“uvre pour l'étape de réutilisation au niveau-0 : moyenne avec probabilité, moyenne sans probabilité, valeurs médianes, sélection aléatoire avec probabilité, copie et changement, vote, interpolation, ACP (analyse en composantes principales) et marche aléatoire. Tous les algorithmes proposés pour générer une nouvelle solution combinent et modifient l'ensemble de solutions construit dans l'étape de récupération. + +\begin{figure}[!ht] +\centering +\includegraphics[width=\textwidth]{./Figures/AutomaticS.png} +\caption{Génération et vérification automatique des solutions} +\label{figAuto} +\end{figure} + +\colorbox{yellow}{Il faut revoir ce qui suit car ne comprenant pas, j'ai complètement ré-écrit.} +La moyenne pondérée avec probabilité permet de construire une solution. La pondération est calculée en fonction de la proximité \colorbox{yellow}{du cas ? lequel ? de la solution ? On la connait ?} avec le cas cible. La probabilité est calculée en fonction de la distance entre les problèmes selon l'équation \ref{gen00} et la solution globale est construite à partir des solutions proposées par les $nl$ algorithmes selon l'équation \ref{gen01}. + +\begin{equation} + \alpha_j=\frac{d(p_j^n,p_w^n)}{\sum_{i=0}^{nl} d(p_i^n,p_w^n)} + \label{gen00} +\end{equation} + +\begin{equation} + s^m_{j,w}= \frac{1}{nl} \sum_{i=0}^{nl} \alpha_j s_{i,j} + \label{gen01} +\end{equation} + +La moyenne sans probabilité copie \colorbox{yellow}{copie ?} aléatoirement les informations des solutions selon la distribution uniforme en donnant la même probabilité à tous les éléments de toutes les solutions proches. Formellement la génération de la solution est calculée selon l'équation \ref{gen2}, où chaque position $j$ est calculée de la même façon \colorbox{yellow}{?}. + +\begin{equation} + s^m_{w,j}= \frac{1}{nl} \sum_{i=0}^{nl} s_{i,j} + \label{gen2} +\end{equation} + +La génération par valeurs médianes construit la solution en utilisant la valeur médiane de toutes les solutions pour chaque dimension. $X_j$ représente l'ensemble des valeurs ordonnées de toutes les solutions $S$ dans la position $j$ \colorbox{yellow}{position ?}. + +\begin{equation} + s_{j,w}^m= + \begin{cases} + x_{j,\frac{m+1}{2}}, \; si \; m \in \{{2k+1 : k \in \mathbb{Z}}\}\\ + \frac{x_{j,\frac{m}{2}}+x_{j,\frac{m}{2}+1}}{2}, \; sinon\\ + \end{cases} +\end{equation} + +La sélection aléatoire avec probabilité génère une solution en copiant aléatoirement l'information des solutions dans chaque position ayant une plus grande probabilité vers le cas de problème associé le plus proche. +\colorbox{yellow}{Il est difficile à comprendre à quel moment intervient l'aléatoire dans ce processus}\\ +\colorbox{yellow}{À ce stade, les équations ne sont plus citées.} + +\begin{equation} +s_{j,w}^m=s_{j,k}^m; k \sim \left(\frac{d(p_k,p_w)}{\sum_{i=1}^nl(p_k,p_i)} \right) +\end{equation} + +"Copie/changement" copie les informations d'une solution aléatoire et en remplace une partie par celles d'une autre solution sélectionnée aléatoirement. + +\begin{equation} +s_{j,w}^m=s_{j,k}^m; k \sim \left(\frac{1}{nl} \right) +\end{equation} + +Le vote permet de copier l'information la plus fréquente de toutes. + +\begin{equation} +s_{j,w}^m=s_{j,k}^m; k=argmax_i \; \mathbb{P}(X=s_{j,i}^m), 1\le i \le nl +\end{equation} + +L'interpolation utilise des informations aléatoires à partir d'une fonction d'interpolation lineaire 1D calculée. +\\ \colorbox{yellow}{Pourquoi aléatoire ? Une interpolation ne rend pas un résultat aléatoire} + +\begin{equation} + s_{w,j}^m \sim \left( \left( \frac{y_i-y_{i+1}}{x_i-x_{i+1}} \right) (x-x_i)+y_i \right), \forall i \; 0 \le i < nl +\end{equation} + +Dans l'analyse en composantes principales (ACP), si la dimension de l'espace des solutions $m$ est plus petit que la dimension de l'espace des problèmes $n$ ($m0$\\ +$np$&p&Nombre d'agents&$\mathbb{N}, np>2$\\ +$nl$&p&Nombre maximal de voisins locaux&$\mathbb{N}, nl>0$\\ +$ng$&p&Nombre de voisins globaux&$\mathbb{N}, ng>2$\\ +$n$&v&Dimension de l'espace du problème&$\mathbb{N}, n>0$\\ +$m$&v&Dimension de l'espace de solution&$\mathbb{N}, m>0$\\ +$p$&v&Description du problème&$\mathbb{R} ^ n$\\ +$s$&v&Description de la solution&$\mathbb{R} ^ m$\\ +$p_w$&v&Description du nouveau problème&$\mathbb{R} ^ n$\\ +$s_w$&v&Description de la nouvelle solution&$\mathbb{R} ^ m$\\ +$n_{rt}$&v&Nombre d'algorithmes por l'étape de rétrouver&$\mathbb{Z}$\\ +$n_{rs}$&v&Nombre d'algorithmes de réutilisation&$\mathbb{Z}$\\ +$d(x_1,x_2)$&f&Fonction de distance entre $x_1$ et $x_2$ &$\mathbb{R}$\\ +$rn(x,y)$&f& + +\begin{tabular}{@{}c@{}}Valeur aléatoire avec distribution normale\\ $x$ moyenne, $y$ écart-type +\end{tabular} + +&$\mathbb{R}_+$\\ +$rnp(x,y)$&f& + +\begin{tabular}{@{}c@{}}Valeur aléatoire discrète, $x$ nombre d'options \\ $y$ vecteur discret de probabilités +\end{tabular} +&$\mathbb{Z}$\\ +$f_s(p^n,s^m)$&f&Évaluation des solutions&$\mathbb{R}$\\ +\end{tabular} +\caption{Variables et paramètres du modèle proposé (Type: p - paramètre, v - variable, f - fonction)} +\label{tabVarPar} +\end{table} + +\subsubsection{Algorithmes} + +Cette section présente de manière plus détaillée les comportements des agents d'ESCBR-SMA. + +Lors de la première étape de l'empilement, chaque agent peut sélectionner l'un des algorithmes à mettre en Å“uvre pour rechercher des problèmes similaires au nouveau problème posé. Les algorithmes possibles sont les suivants : KNN (K-Nearest Neighbor), KMeans, GMM (Gaussian Mixture Model), FuzzyC et KNN pondéré. + +Lors de la deuxième étape de l'empilement, les agents peuvent choisir un algorithme parmi les suivants : pondération avec probabilité, pondération sans probabilité, valeurs médianes, Copier/Changer, vote, interpolation, PCA (analyse en composantes principales) et non aléatoire. + +L'algorithme de construction de solution par pondération avec probabilité utilise la probabilité que le problème considéré soit le plus proche du problème cible. Cet algorithme construit une solution en copiant certaines informations de la ou des solutions ayant la probabilité la plus élevée. Le choix des informations à copier est aléatoire. Pour sa part, la sélection aléatoire pondérée sans probabilité copie certaines informations (choisies aléatoirement) des solutions sans tenir compte de la probabilité que le problème associé soit le plus proche du problème cible. + +Les valeurs médianes construisent une solution à partir des valeurs médianes des informations de toutes les solutions. + +L'algorithme Copie/Changer copie les informations d'une solution choisie de façon aléatoire +%\colorbox{yellow}{c'est le contenu de la solution qui est généré avec des valeurs aléatoires ou}\colorbox{yellow}{cette solution est choisie aléatoirement ?} +et en modifie une partie avec les informations d'une autre solution sélectionnée aussi de manière aléatoire. + +L'algorithme fondé sur le vote construit une solution en considérant les informations indépendamment. Chaque information aura la valeur apparaissant le plus fréquemment dans l'ensemble des solutions possibles. + +L'algorithme fondé sur l'interpolation utilise des informations aléatoires à partir d'une fonction d'interpolation calculée. +\\ \colorbox{yellow}{Qu'est-ce qui est aléatoire ?} +\\ \colorbox{yellow}{Qu'est-ce qui est interpolé ?} +\\ \colorbox{yellow}{Qu'est-ce qui est calculé ?} + +Le générateur avec PCA transforme la description du problème en une description de la solution et établit ainsi une relation entre les valeurs des descripteurs de la partie problème et celles des descripteurs de la patie solution. + +Enfin, l'algorithme de pas aléatoire sélectionne une solution et modifie la valeur d'une seule de ses composantes choisie au hasard selon un certain pas. + +\subsubsection{Structure des agents} + +Tous les agents ont une structure siùilaire, mais chaque agent suit un processus cognitif individuel qui lui permet d'obtenir un comportement indépendant et différent de tous les autres. La figure \ref{figAgent} montre les actions et les variables nécessaires à l'exécution de l'ensemble du processus. + +Chaque agent peut exécuter trois actions différentes : +\begin{itemize} + \item « Récupérer et réutiliser », + \item « Réutiliser » et + \item « Échanger ». +\end{itemize} + +Chaque agent accède aux valeurs de huit variables qui lui sont propres : +\begin{itemize} + \item le nombre de voisins définissant le nombre de problèmes similaires au nouveau problème posé que l'agent doit rechercher dans la base de connaissancesÃ’, + \item la liste des voisins les plus proches établit la liste des agents avec lesquels des informations peuvent être échangées, + \item l'identifiant de l'algorithme de récupération que l'agent exécutera pour trouver les problèmes similaires au nouveau problème, + \item l'identifiant de l'algorithme de réutilisation que l'agent exécutera pour générer une solution candidate au cas cible, + \item la description de la solution générée, + \item l'évaluation de la solution renseignant sur la qualité de la solution générée et calculée selon une fonction d'optimisation (eq \ref{eqOpt1} et eq \ref{eqOpt2}), + \item le vecteur bayésien nécessaire aux algorithmes de récupération contenant les valeurs de probabilité, et + \item le vecteur bayésien nécessaire aux algorithmes de réutilisation contenant les valeurs de probabilité. +\end{itemize} + +\begin{equation} +min \; \left( f_s(p_w^n, s_w^m) \right) = min \left( \sum_{i=1}^{ng} \frac{d(s_w^m,s_i^t)}{d(p_w^n,p_i^n)^2} \right) +\label{eqOpt1} +\end{equation} + +\begin{equation} +s_i^t=s_i^m+rn(0,d(p_w^n,p_i^n)) +\label{eqOpt2} +\end{equation} + +\begin{figure}[!ht] +\centering +\includegraphics[scale=0.7]{Figures/agent.png} +\caption{Structure interne des agents} +\label{figAgent} +\end{figure} + +\subsubsection{Apprentissage des agents} + +Tous les agents considèrent un vecteur bayésien de probabilité pour la phase de récupération (conteneur $C3$) et un vecteur bayésien de probabilité pour la phase de réutilisation (conteneur $C2$). Initialement, ces vecteurs sont configurés avec une probabilité uniforme pour tous les algorithmes de récupération et de réutilisation (probabilité \textit{a priori}). À chaque itération, les vecteurs sont mis à jour selon l'équation bayésienne \colorbox{yellow}{ref ? Numéro d'équation ?} en utilisant les meilleurs résultats du même agent comme paramètre de vraisemblance. L'agent apprend ainsi de son expérience et sélectionne les algorithmes les plus aptes à fournir les meilleures réponses au regard de l'objectif défini. + +Par conséquent, d'un point plus global au niveau du SMA, l'apprentissage est fondé sur un raisonnement bayésien où les vecteurs de récupération et de réutilisation évoluent. Un exemple d'évolution est présenté sur la figure \ref{fig:bayev}. Au cours d'une itération, si un algorithme a contribué à la construction de la meilleure solution, il reçoit un point pour chaque agent qui l'a utilisé. Ces informations sont stockées dans un vecteur normalisé qui est ensuite utilisé comme vecteur de vraisemblance $P(A)$ pour calculer les nouvelles probabilités dans l'équation bayésienne \ref{eqBay} (dans cette équation, $P(B)$ est le terme de normalisation global). + +\begin{figure} + \centering + \includegraphics[scale=0.5]{Figures/BayesianEvolution.png} + \caption{Exemple d'évolution Bayésienne des vecteurs pour un agent. a) Initialisation des probabilités $P(B|A)$ vecteurs pour Retrieve et Reuse, b) Probabilités après quelques itérations $P(A|B)$ vecteurs pour Retrieve et Reuse} + \label{fig:bayev} +\end{figure} + +\begin{equation} + P(A|B)=\frac{P(B|A)P(A)}{P(B)} + \label{eqBay} +\end{equation} + +La sélection d'un algorithme de recherche $a_{rt}$ se fait au moyen d'une valeur aléatoire +\\ \colorbox{yellow}{La valeur est aléatoire ? Ou le choix de la valeur est aléatoire ?} \\ +tirée du vecteur de distribution de recherche discrète obtenu par raisonnement bayésien et calculé selon l'équation \ref{eqRta}. + +\begin{equation} + a_{rt}=rnp(n_{rt},P(A|B)_{rt}) + \label{eqRta} +\end{equation} + +La sélection d'un algorithme de réutilisation $a_{rs}$ se fait au moyen d'une valeur aléatoire +\\ \colorbox{yellow}{La valeur est aléatoire ? Ou le choix de la valeur est aléatoire ?} \\ +tirée du vecteur de distribution de réutilisation discrète obtenu par raisonnement bayésien et calculé selon l'équation \ref{eqRsa}. + +\begin{equation} + a_{rs}=rnp(n_{rs},P(A|B)_{rs}) + \label{eqRsa} +\end{equation} + +Le processus d'apprentissage est réalisé individuellement par chaque agent, mais comporte un aspect collaboratif puisque les agents échangent les informations qu'ils ont calculées, les optimisations locales qu'ils ont réalisées, ainsi que les algorithmes et les paramètres qu'ils ont utilisés. + +\subsubsection{Échanges entre les agents} + +Un agent peut modifier ses informations et leur affecter les valeurs de celles d'un voisin au cours de chaque itération en choisissant au hasard un voisin dans sa liste de voisins les plus proches. L'ensemble de ces changements permettent de propager les paramètres menant aux meilleurs résultats et d'effectuer des actions rétrospectivement. Les informations modifables sont choisies aléatoirement : il peut s'agir du nombre de voisins, de la liste des voisins les plus proches, de l'algorithme de récupération, de l'algorithme de réutilisation ou même des vecteurs bayésiens. + +\subsection{Résultats} + +Des expérimentations sur onze jeux de données relatifs à des problèmes de régression et présentant des caractéristiques différentes ont été réalisées. Les jeux de données et leurs caractéristiques sont consignés dans le tableau \ref{tabBases}. Les valeurs des paramètres de l'algorithme sont les suivantes $it=100$, $np=50$, $nl=10$ et $ng=10$. +\\ \colorbox{yellow}{Quel algorithme ?} +\\ \colorbox{yellow}{À quoi correspondent ces paramètres ? Quelles sont les unités ?}\\ +Le tableau \ref{AlgsPar} présente les valeurs des paramètres de tous les autres algorithmes. + +\begin{table}[!ht] +\footnotesize +\centering +\begin{tabular}{c|ccccccccccc} +&A1&A2&A3&A4&A5&A6&A7&A8&A9&A10\\ +\hline +%DS1&9.081&12.301&1.228&1.066&7.763&9.081&9.078&9.764&0.750&8.263&8.225\\ +DS1&9.081&12.301&1.228&1.066&7.763&9.081&9.078&9.764&0.750&8.225\\ +%DS2&0.022&0.025&0.019&0.013&0.017&0.022&0.022&0.037&0.011&0.016&0.016\\ +DS2&0.022&0.025&0.019&0.013&0.017&0.022&0.022&0.037&0.011&0.016\\%DS3&8.756&8.465&9.656&7.665&8.716&8.756&9.005&9.177&7.369&8.148&7.991\\ +DS3&8.756&8.465&9.656&7.665&8.716&8.756&9.005&9.177&7.369&7.991\\ +%DS4&0.647&0.752&0.794&0.602&0.688&0.647&0.646&0.798&0.616&0.628&0.607\\ +DS4&0.647&0.752&0.794&0.602&0.688&0.647&0.646&0.798&0.616&0.607\\ +%DS5&0.767&0.824&0.877&0.66&0.826&0.767&0.775&0.87&0.703&0.690&0.662\\ +DS5&0.767&0.824&0.877&0.66&0.826&0.767&0.775&0.87&0.703&0.662\\ +%DS6&10.525&9.174&6.93&5.372&6.662&10.525&10.525&10.527&5.131&9.413&9.070\\ +DS6&10.525&9.174&6.93&5.372&6.662&10.525&10.525&10.527&5.131&9.070\\ +%DS7&2.961&2.451&0.589&0.528&3.955&2.961&3.009&4.083&0.490&3.031&2.941\\ +DS7&2.961&2.451&0.589&0.528&3.955&2.961&3.009&4.083&0.490&2.941\\ +%DS8&1.298&1.125&1.360&1.197&1.486&1.298&1.298&1.306&1.128&2.752&2.553\\ +DS8&1.298&1.125&1.360&1.197&1.486&1.298&1.298&1.306&1.128&2.553\\ +%DS9&2.256&2.565&3.174&2.377&2.817&2.256&2.255&2.468&2.293&2.747&2.468\\ +DS9&2.256&2.565&3.174&2.377&2.817&2.256&2.255&2.468&2.293&2.468\\ +%DS10&3.136&3.415&4.173&3.165&3.710&3.136&3.135&3.161&3.108&3.897&3.621\\ +DS10&3.136&3.415&4.173&3.165&3.710&3.136&3.135&3.161&3.108&3.621\\ +DS11&0.625&0.565&0.741&0.56&0.606&0.626&0.626&0.681& +0.541&0.54\\ +\hline +%Avg. Rank&5.8&6.5&7.2&2.3&6.7&5.7&5.6&8.65&1.9&4.65\\ +Avg. Rank&6.4&6.9&8.2&2.6&7.2&6.45&6.35&9.55&2.1&4.75\\ +\end{tabular} +\caption{Résultat selon la métrique RMSE (Root Mean Squared Error) pour les jeux de données évalués avec les différents algorithmes de régression considérés.} +\label{tabRes1} +\end{table} + +\begin{table}[!ht] +\scriptsize +\centering +\begin{tabular}{llccp{1.4cm}p{1.2cm}p{1.2cm}} +ID&DataSet&Features&Instances&Output. Dimension&Input. Domain&Output Domain\\ +\hline +DS1&Yatch Hydrodynamics&6&308&1&$\mathbb{R}$&$\mathbb{R}$\\ +DS2&Electrical Grid Stability&12&10000&1&$\mathbb{R}$&$\mathbb{R}$\\ +DS3&Real State Valuation&6&414&1&$\mathbb{R_+}$&$\mathbb{R_+}$\\ +DS4&Wine Quality (Red)&11&1598&1&$\mathbb{R_+}$&$\mathbb{N}$\\ +DS5&Wine Quality (White)&11&4897&1&$\mathbb{R_+}$&$\mathbb{N}$\\ +DS6&Concrete Compressive Strength&8&1030&1&$\mathbb{R_+}$&$\mathbb{R_+}$\\ +DS7&Energy Efficiency&8&768&2&$\mathbb{R_+}$&$\mathbb{R_+}$\\ +DS8&Gas Turbine CO, NOx Emission (2015)&9&7384&2&$\mathbb{R_+}$&$\mathbb{R_+}$\\ +DS9&Student Performace Portuguese&30&649&3&$\mathbb{N*}$&$\mathbb{N}$\\ +DS10&Student Performance Math&30&395&3&$\mathbb{N*}$&$\mathbb{N}$\\ +DS11&Generated Student Performance&5&1000&1&$\mathbb{R}_+$&$\mathbb{R}_+$\\ +\end{tabular} +\caption{Description des jeux de données évalués. (* après codification comme +\textit{String})} +\label{tabBases} +\end{table} + +\begin{table}[!ht] +\centering +\begin{tabular}{ccc|ccc} +ID&Parameter&Value&ID&Parameter&Value\\ +\hline +A1&Intercept&True&A6&Degree&4\\ +&Positive&True&&Bias&True\\ +\hline +A2&Neighbors&5&A7&Fit Intercept&True\\ +&Weights&Uniform&&alpha&0.2\\ +&Metric&Minkowsky&&tol&1e-4\\ +&Power Minkowsky&2\\ +\hline +A3&Error&Squared Error&A8&Fit Intercept&True\\ +&Min samples split&2&&alpha&[0.00001, 0.4]\\ +&&&&Max iter&1000\\ +&&&&tol&1e-4\\ +\hline +A4&Estimators&10&A9&Error&Squarred Error\\ +&Error&Squared Error&&Learning Rate&0.1\\ +&Min samples split&2&&Estimators&100\\ +&Bootstrap&True&&Min Split&2\\ +\hline +A5&Hidden Layers&100\\ +&Activation&Relu\\ +&Solver&Adam\\ +&alpha&0.0001\\ +&Learning Rate&0.001\\ +&Max Iter&200\\ +&beta1&0.9\\ +&beta2&0.999\\ +&epsilon&1e-8\\ +\end{tabular} +\caption{Paramètres de tous les algorithmes comparés} +\label{AlgsPar} +\end{table} + +\begin{table}[!ht] +\footnotesize +\centering +\begin{tabular}{c|ccccccccccc} +Dataset&A1&A2&A3&A4&A5&A6&A7&A8&A9&A10\\ +\hline +%DS1&6.776&2.385&0.231&0.207&3.632&6.778&6.307&5.186&0.162&1.193&1.218\\ +DS1&6.776&2.385&0.231&0.207&3.632&6.778&6.307&5.186&0.162&1.218\\ +%DS2&0.015&0.017&0.012&0.008&0.012&0.015&0.015&0.030&0.007&0.011&0.010\\ +DS2&0.015&0.017&0.012&0.008&0.012&0.015&0.015&0.030&0.007&0.010\\ +%DS3&5.092&4.320&4.1&3.632&4.435&5.092&5.20&5.132&3.504&3.90&3.771\\ +DS3&5.092&4.320&4.1&3.632&4.435&5.092&5.20&5.132&3.504&3.771\\ +%DS4&0.413&0.495&0.18&0.325&0.451&0.413&0.412&0.544&0.387&0.154&0.135\\ +DS4&0.413&0.495&0.18&0.325&0.451&0.413&0.412&0.544&0.387&0.135\\ +%DS5&0.509&0.548&0.285&0.374&0.550&0.509&0.509&0.633&0.456&0.113&0.085\\ +DS5&0.509&0.548&0.285&0.374&0.550&0.509&0.509&0.633&0.456&0.085\\ +%DS6&6.989&5.709&3.134&2.839&4.306&6.989&6.989&6.986&3.084&5.439&5.072\\ +DS6&6.989&5.709&3.134&2.839&4.306&6.989&6.989&6.986&3.084&5.072\\ +%DS7&1.393&1.372&0.217&0.218&2.523&1.393&1.529&2.346&0.243&1.008&1.006\\ +DS7&1.393&1.372&0.217&0.218&2.523&1.393&1.529&2.346&0.243&1.006\\ +%DS8&0.549&0.297&0.365&0.289&0.742&0.549&0.549&0.540&0.309&0.861&0.794\\ +DS8&0.549&0.297&0.365&0.289&0.742&0.549&0.549&0.540&0.309&0.794\\ +%DS9&1.496&1.788&2.080&1.612&2.005&1.496&1.496&1.714&1.538&1.721&1.556\\ +DS9&1.496&1.788&2.080&1.612&2.005&1.496&1.496&1.714&1.538&1.556\\ +%DS10&2.344&2.534&2.910&2.331&2.543&2.344&2.344&2.481&2.258&2.602&2.371\\ +DS10&2.344&2.534&2.910&2.331&2.543&2.344&2.344&2.481&2.258&2.371\\ +DS11&0.387&0.35&0.46&0.338&0.384&0.387&0.387&0.453& +0.327&0.347\\ +\hline +Avg. Rank&7.15&6.9&5.35&2.6&7.95&7.25&7.3&9.0&2.5 +&4.5\\ +%Avg. Rank&6.45&6.5&4.35&2.4&7.45&6.55&6.6&8.1&2.4&4.2\\ +\end{tabular} +\caption{Comparaison des résultats MAE (Median Absolute Error) pour les bases de données évaluées avec des algorithmes de régression} +\label{tabRes2} +\end{table} + +Les tableaux \ref{tabRes1} et \ref{tabRes2} présentent respectivement les RMSE et les MAE obtenues pour chaque jeu de données et chaque algorithme testé. + +La figure \ref{figBox} représente graphiquement la dispersion globale, la médiane et les valeurs aberrantes obtenues durant ces tests. Cette figure montre qu'ESCBR-SMA génère parfois plus de valeurs aberrantes que d'autres algorithmes. Toutefois, la variance est très faible. De plus, la convergence est proche de la valeur réelle et même meilleure que celle de la plupart des autres algorithmes testés. Il est également possible de remarquer que les valeurs aberrantes sont plus élevées que les valeurs réelles.\textcolor{red}{plus élevées??, pourquoi?} + +\begin{figure}[!ht] +\includegraphics[width=\textwidth]{Figures/boxplot2.png} +\caption{Résultats selon la métrique MAE (Median Absolute Error) pour les dix algorithmes testés} +\label{figBox} +\end{figure} + +Les résultats montrent que l'algorithme ESCBR-SMA est compétitif pour la plupart des jeux de données considérés dans cette étude. Il obtient en effet les meilleurs résultats sur $DS2$, $DS4$, $DS5$ et $DS9$. Globalement, ESCBR-SMA est le troisième meilleur algorithme. Il se classe donc à proximité d'autres algorithmes d'ensemble. Par rapport à l'ESCBR, une amélioration des résultats et une réduction de la variance ont été obtenues, démontrant ainsi que les systèmes multi-agents et le raisonnement stochastique bayésien contribuent à l'apprentissage et à la convergence vers des solutions plus proches de la solution réelle. + +\subsection{Conclusion} +L'ECBR intégrant un SMA propose d'utiliser les avantages des systèmes multi-agents pour améliorer la qualité des solutions proposées et également le processus d'apprentissage global en couvrant un plus large spectre de possibilités et en le couvrant de manière intelligente à l'aide d'un raisonnement bayésien. ESCBR-SMA permet ainsi d'obtenir de bonnes approximations avec peu de données. + +Ce travail a démontré la capacité d'ESCBR-SMA à trouver des solutions proches de l'optimum global pour la majorité des ensembles de données analysés. Ces jeux de données présentent une certaine diversité, ils peuvent être déséquilibrés et ils sont de tailles différentes. Grâce aux caractéristiques inhérentes aux systèmes multi-agents (possibilités de rétroactions, d'échanges d'informations, l'émergence d'une intelligence collective, utilisation d'un raisonnement cognitif), les performances d'ESCBR-SMA sont meilleures que celles d'ESCBR. diff --git a/chapters/TS.aux b/chapters/TS.aux index 8ca764b..c0c7640 100644 --- a/chapters/TS.aux +++ b/chapters/TS.aux @@ -135,12 +135,12 @@ \newlabel{eqBetaH}{{7.45}{99}{Modèle Proposé}{equation.7.4.45}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {7.4.2}Résultats et Discussion}{99}{subsection.7.4.2}\protected@file@percent } \@writefile{toc}{\contentsline {subsubsection}{\numberline {7.4.2.1}Système de recommandation avec une base de données d'étudiants réels (TS avec Hawkes)}{99}{subsubsection.7.4.2.1}\protected@file@percent } -\@writefile{toc}{\contentsline {subsubsection}{\numberline {7.4.2.2}Base de données simulée (ESCBR, TS avec Hawkes)}{100}{subsubsection.7.4.2.2}\protected@file@percent } -\newlabel{metric1}{{7.46}{100}{Base de données simulée (ESCBR, TS avec Hawkes)}{equation.7.4.46}{}} -\@writefile{lot}{\contentsline {table}{\numberline {7.12}{\ignorespaces Comparison entre ESCBR-TS et ESCBR-TS-Hawkes\relax }}{100}{table.caption.68}\protected@file@percent } -\newlabel{tab:my_label}{{7.12}{100}{Comparison entre ESCBR-TS et ESCBR-TS-Hawkes\relax }{table.caption.68}{}} -\@writefile{lof}{\contentsline {figure}{\numberline {7.13}{\ignorespaces Nombre de recommandations par niveau de complexité (à gauche processus d'apprentissage statique, à droite processus d'apprentissage dynamique avec processus de Hawkes)\relax }}{101}{figure.caption.67}\protected@file@percent } -\newlabel{fig:stabilityBP}{{7.13}{101}{Nombre de recommandations par niveau de complexité (à gauche processus d'apprentissage statique, à droite processus d'apprentissage dynamique avec processus de Hawkes)\relax }{figure.caption.67}{}} +\@writefile{lof}{\contentsline {figure}{\numberline {7.13}{\ignorespaces Nombre de recommandations par niveau de complexité (à gauche processus d'apprentissage statique, à droite processus d'apprentissage dynamique avec processus de Hawkes)\relax }}{100}{figure.caption.67}\protected@file@percent } +\newlabel{fig:stabilityBP}{{7.13}{100}{Nombre de recommandations par niveau de complexité (à gauche processus d'apprentissage statique, à droite processus d'apprentissage dynamique avec processus de Hawkes)\relax }{figure.caption.67}{}} +\@writefile{toc}{\contentsline {subsubsection}{\numberline {7.4.2.2}Base de données simulée (ESCBR, TS avec Hawkes)}{101}{subsubsection.7.4.2.2}\protected@file@percent } +\newlabel{metric1}{{7.46}{101}{Base de données simulée (ESCBR, TS avec Hawkes)}{equation.7.4.46}{}} +\@writefile{lot}{\contentsline {table}{\numberline {7.12}{\ignorespaces Comparison entre ESCBR-TS et ESCBR-TS-Hawkes\relax }}{101}{table.caption.68}\protected@file@percent } +\newlabel{tab:my_label}{{7.12}{101}{Comparison entre ESCBR-TS et ESCBR-TS-Hawkes\relax }{table.caption.68}{}} \@writefile{lof}{\contentsline {figure}{\numberline {7.14}{\ignorespaces Évolution de la variance pour la distribution de probabilité bêta et tous les niveaux de complexité (en haut : processus d'apprentissage statique. 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Dans ce cas, l'algorithme de recommandation produit une adaptation en fonction des notes de l'apprenant et l'ESCBR-SMA effectue une prédiction pour valider l'adaptation générée, le processus de Hawkes simule la courbe d'oubli dans le processus d'apprentissage. -L'idée de l'unification est d'obtenir des informations d'un point de vue local où une recommandation est obtenue en se basant uniquement sur les informations des apprenants individuels (modèle fondé sur l'échantillonnage de Thompson), la prédiction globale où les informations sont obtenues à partir de tous les apprenants qui ont des résultats similaires (filtre collaboratif avec RàPC) et le processus d'apprentissage dynamique avec le processus de Hawkes. L'architecture de l'algorithme est présentée dans la figure \ref{fig:Amodel}, où TS et RàPC sont exécutés en parallèle et indépendamment avec les informations extraites de la même base de données, une fois que les résultats de chaque algorithme sont obtenus, les résultats sont unifiés par une fonction de pondération et une distribution de probabilité mises à jour dynamiquement en fonction des événements passés et du niveau de complexité sélectionné, la recommandation finale est celle qui maximise l'expression \ref{eqMixModels}. La consolidation des deux résultats globaux permet d'atténuer l'effet du paradoxe de Simpson \cite{10.1145/3578337.3605122}. +L'idée de l'unification est d'obtenir des informations d'un point de vue local où une recommandation est obtenue en se basant uniquement sur les informations des apprenants individuels (modèle fondé sur l'échantillonnage de Thompson), la prédiction globale où les informations sont obtenues à partir de tous les apprenants qui ont des résultats similaires (filtre collaboratif avec RàPC) et le processus d'apprentissage dynamique avec le processus de Hawkes. L'architecture de l'algorithme est présentée dans la figure \ref{fig:Amodel}, où TS et RàPC sont exécutés en parallèle et indépendamment avec les informations extraites de la même base de données, une fois que les résultats de chaque algorithme sont obtenus, les résultats sont unifiés par une fonction de pondération et une distribution de probabilité mises à jour dynamiquement en fonction des événements passés et du niveau de complexité sélectionné. La consolidation des deux résultats globaux permet d'atténuer l'effet du paradoxe de Simpson \cite{10.1145/3578337.3605122}. \begin{figure}[!ht] \centering @@ -781,7 +781,7 @@ La première étape est l'adaptation avec l'échantillonnage de Thompson et la p \label{tabvp} \end{table} -L'intégration se fait en trois étapes comme est décrit dans la section 7.3.2. Après la sélection du niveau de complexité, toutes les distributions de probabilité sont mises à jour selon le processus de Hawkes (équation \ref{hp1}) pour chaque paramètre $\alpha$ et $\beta$ en utilisant la fonction d'intensité définie constante (équations \ref{hp21} et \ref{hp22}) et la fonction d'excitation (équation \ref{hp31}). En simulant la courbe d'oubli dans lévolution de la distribution de probabilité Beta. +L'intégration se fait en trois étapes comme est décrit dans la section 7.3.2. Après la sélection du niveau de complexité, toutes les distributions de probabilité sont mises à jour selon le processus de Hawkes (équation \ref{hp1}) pour chaque paramètre $\alpha$ et $\beta$ en utilisant la fonction d'intensité définie constante (équations \ref{hp21} et \ref{hp22}) et la fonction d'excitation (équation \ref{hp31}). En simulant la courbe d'oubli dans l'évolution de la distribution de probabilité Beta. \begin{equation} \lambda(t)=\mu(t)+\sum_{t_i not available ] [6] Overfull \hbox (1.29184pt too wide) detected at line 88 - [][]\T1/phv/m/n/10.95 100[] + [][]\T1/phv/m/n/10.95 101[] [] @@ -1650,143 +1650,114 @@ File: ./Figures/FlowCBR0.png Graphic file (type png) Package pdftex.def Info: ./Figures/FlowCBR0.png used on input line 44. (pdftex.def) Requested size: 222.23195pt x 396.8858pt. - -Underfull \vbox (badness 10000) has occurred while \output is active [] - [54 <./Figures/FlowCBR0.png>] <./Figures/Stacking1.png, id=1078, 743.77875pt x 414.54875pt> File: ./Figures/Stacking1.png Graphic file (type png) Package pdftex.def Info: ./Figures/Stacking1.png used on input line 83. 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