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Les résultats montrent l'adaptation dynamique du modèle proposé, les adaptations obtenues aidant le système à évoluer plus rapidement et identifier les faiblesses des apprenants dans les différents niveaux de complexité ainsi que la génération de recommandations pertinentes dans des cas spécifiques pour chaque capacité d'apprenant. -Le module mis en Å“uvre pour AI-VT est classé dans la catégorie des systèmes de recommandation. Les systèmes de recommandation dans les environnements d'apprentissage prennent en compte les exigences, les besoins, le profil, les talents, les intérêts et l'évolution de l'apprenant pour adapter et recommander des ressources ou des exercices dans le but d'améliorer l'acquisition et la maîtrise des concepts et des connaissances en général. L'adaptation de ces systèmes peut être de deux types, l'adaptation de la présentation qui montre aux apprenants des ressources d'étude en fonction de leurs faiblesses et l'adaptation de la navigation qui change la structure du cours en fonction du niveau et du style d'apprentissage de chaque apprenant \cite{MUANGPRATHUB2020e05227}. +\colorbox{yellow}{Il y a des différences sémantiques à bien différencier pour éviter les confusions : }\\ +\colorbox{yellow}{Il faut harmoniser : à certains moment, on parle de 3 tests, à d'autres de 3 scénarios.}\\ +\colorbox{yellow}{J'ai tout transformé en séances d'entrainement 1, 2, 3, mais je me suis peut-être trompé}\\ +\colorbox{yellow}{S'il s'agit de scénarios, il faut les décrire.}\\ +\\ -Les techniques de recommandation sont utiles dans les EIAH car elles peuvent détecter les changements et évoluer vers un état optimal, comme l'algorithme d'échantillonnage de Thompson (TS), qui est un algorithme de type probabiliste appartenant à la catégorie des algorithmes d'apprentissage par renforcement, où l'algorithme choisit au temps $t$ une action $a$ à partir d'un ensemble $A$, obtient une récompense pour l'action $a$ et, en fonction de la valeur de la récompense, ajuste sa stratégie de décision pour choisir au temps $t + 1$ une autre action $a$, dans le but de maximiser la récompense. Il est fondé sur le principe Bayésien, où il y a une distribution de probabilité a priori et avec les données obtenues une distribution de probabilité a posteriori est générée qui vise à maximiser l'estimation de la valeur attendue. Pour la variante de Bernoulli, où la récompense n'a que deux valeurs possibles 0 et 1 ou succès et échec, la distribution de base utilisée est la distribution Beta qui est définie sur [0, 1] et paramétrée par deux valeurs $\alpha$ et $\beta$ \cite{9870279}. +\colorbox{yellow}{Idem pour "modèle", "algorithme", "système", "module", "programme"}\\ +\colorbox{yellow}{Un système est constitué de modules,}\\ +\colorbox{yellow}{Un module est constitué de programmes, }\\ +\colorbox{yellow}{Un programme suit un algorithme,}\\ +\colorbox{yellow}{Un algorithme fonctionne selon un modèle}\\ +\colorbox{yellow}{Il faudrait revoir cette sémantique en particulier dans le chapitre précédent.} +\\ -La deuxième partie de ce chapitre présente l'intégration de tous les algorithmes développés et explicités dans les chapitres précédents. Le modèle intégré est appliqué au système AI-VT sur une base de données générée et une base de données réelle. Plusieurs types de test sont exécutés pour montrer que le modèle final permet en effet d'améliorer les capacités d'identification et adaptation. +\colorbox{yellow}{Idem pour "métrique", "équation", "mesure", "résultat"}\\ +\colorbox{yellow}{Une métrique permet de définir une graduation (comme le mètre ou le litre),}\\ +\colorbox{yellow}{Une équation donne la formule mathématique ou la fonction}\\ +\colorbox{yellow}{permettant de calculer un résultat selon une métrique}\\ +\colorbox{yellow}{Une mesure est la valeur numérique obtenue, résultant de l'application de l'équation}\\ +\colorbox{yellow}{Ainsi, on compare les valeurs dans une métrique, valeurs obtenues en appliquant une équation.}\\ -Les contributions de la deuxième partie sont : -\begin{itemize} - \item Vérification de l'efficacité du modèle de raisonnement à partir de cas pour la prédiction avec une base de données de notes d'apprenants par rapport à d'autres algorithmes. - \item Calcul explicite de l'évolution de l'acquisition des connaissances en analysant le changement des distributions de probabilité générées par le modèle de recommandation stochastique. - \item Intégration du modèle de recommandation stochastique à la prédiction par raisonnement à partir de cas pour améliorer la personnalisation de l'EIAH. -\end{itemize} +L'un des principaux modules d'un EIAH est le système de recommandation visant à trouver les faiblesses et à réviser la séance d'entraînement proposée intialement par celui-ci. Ce type de module permet donc au système de personnaliser les contenus et les exercices en fonction des besoins et des résultats de chacun des apprenants. Certains auteurs n'hésitent pas à considérer que l'efficacité d'un EIAH dans l'acquisition des connaissances et l'adaptation aux différents types d'apprentissage dépend de ce type de module fondé sur la recommandation \cite{Liu2023}. -L'un des principaux modules des EIAH est le système de recommandation, qui vise à trouver les faiblesses et à adapter la plateforme localement ou globalement pour faciliter le processus d'apprentissage et l'acquisition des connaissances, ce module est très important car il permet d'adapter le système et de personnaliser les contenus et les exercices en fonction des besoins et des résultats de chacun des apprenants, l'efficacité du système dans l'acquisition des connaissances et l'adaptation aux différents types d'apprentissage dépend de ce module \cite{Liu2023}. Il est donc nécessaire de trouver des techniques et des algorithmes capables d'exploiter les données disponibles et d'explorer les options d'apprentissage de manière dynamique, afin d'améliorer les performances globales des EIAH. +Les systèmes de recommandation dans les environnements d'apprentissage prennent en compte les exigences, les besoins, le profil, les talents, les intérêts et l'évolution de l'apprenant pour adapter et recommander des ressources ou des exercices dans le but d'améliorer l'acquisition et la maîtrise des concepts et des connaissances en général. Dans ces systèmes, l'adaptation peut être de deux types : l'adaptation de la présentation qui montre aux apprenants des ressources d'étude en fonction de leurs faiblesses et/ou l'adaptation de la navigation qui change la structure du cours en fonction du niveau et du style d'apprentissage de chaque apprenant \cite{MUANGPRATHUB2020e05227}. -Dans la troisième partie seront détaillées les contributions réalisées avec l'incorporation du processus de Hawkes : +Parmi les algorithmes les plus prometteurs pouvant aider à proposer des recommandations, nous avons identifié l'algorithme d'échantillonnage de Thompson (TS). Il s'agit d'un algorithme probabiliste appartenant à la catégorie des algorithmes d'apprentissage par renforcement. À l'instant $t$, TS choisit l'action $a_t$ d'un ensemble $A$ d'actions possibles, et obtient une récompense pour celle-ci. À $t+1$, une action $a_{t+1}$ est sélectionnée en tenant compte de la récompense précédente. L'objectif consiste à maximiser la récompense. Selon le principe bayésien, cette maximisation itérative est opérée en suivant une distribution de probabilité évoluant à chaque itération. Cette évolution peut être calculée selon la variante de Bernoulli où la récompense n'a que deux valeurs possibles 0 ou 1 (échec ou succès), ou selon une distribution \textit{Beta} définie sur l'intervalle $[0, 1]$ et calculée en fonction de deux valeurs $\alpha$ et $\beta$ \cite{9870279}. -\begin{itemize} - \item Simulation de la courbe d'oubli dans le processus d'apprentissage à l'aide du processus stochastique de Hawkes. - \item Intégration du raisonnement par cas, des systèmes multi-agents et du processus de Hawkes dans un algorithme de recommandation. - \item Vérification de la progression, de la stabilité, la précision et évolution de l'algorithme de recommandation stochastique proposé à l'aide de bases de données simulées et hétérogènes d'étudiants réels.\\\\ -\end{itemize} +Ce chapitre est divisé en trois parties, la première partie présente un algorithme delivrant des recommandations en fonction des résultats produits par l'apprenant en temps réel. Une partie de cette proposition est publiée dans \cite{Soto2}. Cet algorithme permet l'adaptation automatique en temps réel d'une séance prédéterminée dans l'EIAH AI-VT. Nous considérons qu'elle intervient durant la phase de révision du cylce classique du raisonnement à partir de cas (RàPC). Le modèle proposé est stochastique et il a été testé selon trois scénarios différents. Les résultats montrent de quelle manière AI-VT peut proposer des recommandations pertinentes selon les faiblesses identifiées de l'apprenant. + +La deuxième partie de ce chapitre montre l'intégration de tous les algorithmes présentés dans les chapitres précédents à AI-VT. Le modèle intégré est appliqué au système AI-VT sur des données générées et des données réelles. Plusieurs types de test sont exécutés pour montrer que le modèle final permet en effet d'améliorer les capacités d'identification et d'adaptation. Les performances de ce nouveau modèle sont comparées à celles d'autres algorithmes. Enfin, l'évolution de l'acquisition des connaissances induites par ces nouveaux modèles de recommandation stochastiques est analysée dans cette deuxième partie du présent chapitre. + +Enfin, dans la troisième partie de ce chapitre, nous présentons une évolution de ce système de recommandation intégrant le processus de Hawkes. L'intérêt de ce dernier réside dans le fait qu'il utilise une courbe d'oubli, nous permettant ainsi que tenir compte du fait que certaines connaissances et certains mécanismes doivent être rappelés aux apprenants. Cette troisième partie intègre une étude des performances du système de recommandation incluant ce processus stochastique de Hawkes. \section{Système de recommandation stochastique fondé sur l'échantillonnage de Thompson} \sectionmark{Système de recommandation fondé sur TS} \subsection{Modèle Proposé} -Le modèle proposé, en tant que système de recommandation, prend en compte les notes antérieures des apprenants pour estimer leurs connaissances et leur maîtrise des différentes compétences, sous-compétences et niveaux de complexité au sein du système AI-VT, puis adapte les séances pour maximiser l'acquisition des connaissances et la maîtrise des différents domaines contenus dans la même compétence définie. Le modèle est conçu comme une modification de l'algorithme d'échantillonnage de Thompson avec l'intégration de l'échantillonnage stratifié pour obtenir l'adaptation. +Le modèle proposé, en tant que système de recommandation, prend en compte les notes antérieures des apprenants pour estimer leurs connaissances et leur maîtrise des différentes compétences, sous-compétences et niveaux de complexité au sein du système AI-VT, puis adapte les séances pour maximiser l'acquisition des connaissances et la maîtrise des différents domaines contenus dans la même compétence définie. -La famille de distributions de probabilité Betq est utilisée pour définir dynamiquement le nouveau niveau de complexité (équation \ref{eqBeta}) inspiré de l'algorithme d'échantillonnage de Thompson. Cette version du modèle permet de recommander des niveaux de complexité non contigus, mais la priorité est de recommander les niveaux dans lesquels des défauts ont été détectés. La paramétrisation initiale de toutes les distributions de probabilité peut forcer le modèle à recommander des niveaux de complexité contigus plus élémentaires. +La famille de distributions de probabilité Beta est utilisée pour définir dynamiquement le niveau de complexité (équation \ref{eqBeta}) à proposer à l'apprenant. Cet algorithme permet de recommander des niveaux de complexité non contigus et dans lesquels des défauts ont été détectés. Les paramètres initiaux des distributions de probabilité peuvent forcer le système à recommander des niveaux de complexité contigus plus ou moins importants. \begin{equation} B(x, \alpha, \beta) = @@ -45,8 +54,6 @@ B(x, \alpha, \beta) = \label{eqBeta} \end{equation} -Les variables qui font partie du modèle sont spécifiées dans le tableau \ref{tabPar}. - \begin{table}[!ht] \centering \begin{tabular}{ccc} @@ -75,17 +82,19 @@ $y_c$&Proportion de questions dans le niveau de complexité $c$&$[0, 1] \in \mat $r$&Valeur totale de la métrique définie pour l'adaptabilité&$[0, c_n] \in \mathbb{R}$\\ $sc$&Valeur totale de la métrique de similarité cosinus&$[-1, 1] \in \mathbb{R}$\\ \end{tabular} -\caption{Variables et paramètres du modèle proposé} +\caption{Variables et paramètres du système de recommandation proposé} \label{tabPar} \end{table} -Dans ce cas, il est nécessaire d'utiliser la variable de seuil de grade $g_t$ pour déterminer la variabilité de la distribution de probabilité pour chaque niveau de complexité. Les équations \ref{eqsMg}, \ref{eqgtc} et \ref{eqltc} montrent les règles de mise à jour corrélées, ces règles modifient les valeurs par récompense inverse. Chaque niveau de complexité est associé à une distribution de probabilité Beta avec des valeurs initiales prédéfinies pour les paramètres $\alpha$ et $\beta$. +Le tableau \ref{tabPar} présente les variables de l'algorithme de recommandation. Nous avons considéré que les notes $g_c$ obtenues au niveau de complexité $c$ tiennent compte du temps de réponse. C'est la raison pour laquelle, nous avons défini le grade de l'apprenant $ng_c$ au niveau de complexité $c$. Ce grade calculé selon l'équation \ref{eqsGT}, tient compte d'un poids de pénalisation temporelle $\lambda$. \begin{equation} -ng_c=g_c -\label{eqsMg} + ng_c=g_c- \left(g_c * \lambda * \frac{t_c}{t_m} \right) + \label{eqsGT} \end{equation} +Dans cet algorithme, la variable de seuil de grade $g_t$ détermine la variabilité de la distribution de probabilité pour chaque niveau de complexité. Les niveaux de complexité des exercices proposés à l'apprenant sont calculés par récompense inverse selon les équations \ref{eqgtc} et \ref{eqltc}. Chaque niveau de complexité est associé à une distribution de probabilité Beta avec des valeurs initiales $\alpha$ et $\beta$ prédéfinies. + \begin{equation} ng_c \ge g_t \rightarrow \begin{cases} @@ -106,37 +115,20 @@ ng_c < g_t \rightarrow \label{eqltc} \end{equation} -Le nouveau niveau de complexité est l'indice de la valeur aléatoire maximale (générée à partir de la distribution Beta de chaque niveau de complexité, équation \ref{eqBRnd}) pour tous les niveaux de complexité (équation \ref{eqsncl}). - -\begin{equation} -\theta_c = Beta(\alpha_c, \beta_c) -\label{eqBRnd} -\end{equation} - -\begin{equation} -ncl=max_x(\mathbb{E}[\theta_x]), 0<=x<=c_n -\label{eqsncl} -\end{equation} - -La note des apprenants peut considérer aussi le temps de réponse comme une pénalité, et dans ce cas-là la note est calcule comme la équation \ref{eqsGT}. - -\begin{equation} -ng_c=g_c- \left(g_c * \lambda * \frac{t_c}{t_m} \right) -\label{eqsGT} -\end{equation} +Pour chaque niveau de complexité $c$, $Beta(\alpha_c, \beta_c)$ fournit une distribution de probabilité $\theta_c$ dont nous calculons l'espérance $\mathbb{E}[\theta_c]$. Le nouveau niveau de complexité $ncl$ correspond à l'espérance maximale obtenue. Le détail des pas d'exécution du modèle proposé sont dans l'algorithme \ref{alg2}. \begin{algorithm} -\caption{Stochastic Recommendation Model} +\caption{Algorithme de recommandation stochastique} \begin{algorithmic} -\State Initialize the a-priori distributions of probability -\For {\textbf{each} questions $q$} - \State With $i$ as actual complexity level $c$ - \State Calculate $ng_i$ \Comment{eq \ref{eqsMg} or eq \ref{eqsGT}} - \State Update parameters $\alpha_i$ and $\beta_i$ \Comment{eq \ref{eqgtc} and eq \ref{eqltc}} - \State Get random values $\theta_c$ with Beta distribution\Comment{$\forall c$, eq \ref{eqBRnd}} - \State Get $ncl$ \Comment{eq \ref{eqsncl}} +\State Initialisation de la distribution de probabilité +\For {\textbf{each} question $q$} + \State Soit le niveau de complexité $i$ + \State $ng_i=g_i- \left(g_i * \lambda * \frac{t_i}{t_m} \right)$ \Comment{eq \ref{eqsGT}} + \State Calculs des paramètres $\alpha_i$ et $\beta_i$ \Comment{eq \ref{eqgtc} et eq \ref{eqltc}} + \State Choisir $\theta_c$ selon la distribution de probabilité Beta \Comment{$\forall c, \theta_c = Beta(\alpha_c, \beta_c)$} + \State $ncl = max(\mathbb{E}[\theta_c]), \forall c$ \EndFor \end{algorithmic} \label{alg2} @@ -144,13 +136,13 @@ Le détail des pas d'exécution du modèle proposé sont dans l'algorithme \ref{ \subsection{Résultats} -Le comportement du modèle a été testé avec un jeu de données généré, ce jeu de données contient les notes et les temps de réponse de 1000 apprenants pour 5 niveaux de complexité différents, la description des données est indiquée dans le Tableau \ref{tabDataSet}. Les notes des apprenants sont générées avec la distribution logit-normale de probabilité, car c'est expérimentalement le meilleur modèle de représentation \cite{Arthurs}. +Le comportement du module de recommandation a été testé avec un des données générées contenant les notes et les temps de réponse de mille apprenants pour cinq niveaux de complexité différents. Ces données sont décrites dans le tableau \ref{tabDataSet}. Les notes des apprenants sont générées selon la loi de probabilité logit-normale considérée comme la plus fidèle dans ce contexte par \cite{Arthurs}. \colorbox{yellow}{<== OK pour toi, Daniel ?} -L'ensemble de données généré est une simulation des notes des apprenants pour les réponses à quinze questions à chacun des cinq niveaux de complexité. L'ensemble de données simule, via la distribution de probabilité logit-normale, une faiblesse dans chaque niveau de complexité pour 70\% des apprenants dans les dix premières questions. La difficulté de la complexité est également simulée en réduisant le score moyen et en augmentant la variance. La figure \ref{figData} montre la distribution de l'ensemble de données des notes de 1000 apprenants par niveau de complexité. +L'ensemble de données générées résulte d'une simulation des notes obtenues par des apprenants virtuels ayant répondu à quinze questions réparties sur cinq niveaux de complexité. L'ensemble de données simule, via la distribution de probabilité logit-normale, une faiblesse dans chaque niveau de complexité pour 70\% des apprenants sur les dix premières questions. La difficulté de la complexité est quant à elle simulée en réduisant le score moyen et en augmentant la variance. La figure \ref{figData} montre la manière dont sont réparties les notes selon le niveau de complexité. \begin{figure} \includegraphics[width=\textwidth]{./Figures/dataset.png} -\caption{Boîte à moustaches pour la base de données générée} +\caption{Répartition des notes générées selon le niveau de complexité.} \label{figData} \end{figure} @@ -180,31 +172,39 @@ Valeur&5&10&120&3&2&0.25&6 & 2 & 1 & 1 & 1 & 0.3 & 0.5 & 0.7\\ \label{tabgm1} \end{table} -Les résultats de la première comparaison sans données historiques (démarrage à froid) entre le modèle proposé, un système de recommandation déterministe et le système original (RàPC) sont présentés dans la figure \ref{figCmp2}, où apparaissent différents nombres et échelles de transitions, le système original ne présente pas de transitions, tous les apprenants sont évalués au niveau de complexité 0, les notes obtenues pendant la séance ne sont pas prises en compte. Le système avec des modèles de recommandation tente d'adapter le niveau de complexité en fonction des notes obtenues. Le modèle déterministe génère quatre grandes transitions avec un grand nombre d'apprenants dans les questions 5, 6, 8 et 12, toutes entre des niveaux de complexité contigus, la tendance est à la baisse pour les niveaux 0, 1 et 2 après la huitième question et à la hausse pour les niveaux 1 et 3. Le modèle proposé (stochastique), commence par proposer tous les niveaux de complexité possibles mais se concentre sur le niveau 0, les transitions sont constantes mais pour un petit nombre d'apprenants, la tendance après la dixième question est à la baisse pour les niveaux 0 et 4 et à la hausse pour les niveaux 1, 2 et 3. La tendance est à la baisse pour les niveaux 0 et 4 et à la hausse pour les niveaux 1, 2 et 3. La tendance est à la hausse pour les niveaux 1, 2 et 3. +La figure \ref{figCmp2} permet de comparer les résultats obtenus par le module proposé, un système de recommandation déterministe et le système AI-VT initial lors d'un \textit{démarrage à froid} (c'est-à-dire sans données historiques ni informations préalables sur le profile de l'apprenant). Sur les graphiques de cette figure, les numéros des questions posées sont reportées en abscisse selon l'ordre chronologique d'apparition durant la séance d'entrainement, le niveau de complexité de chaque question posée est représenté par une couleur différente, et le nombre d'apprenants ayant eu des questions de ce niveau de complexité sont reportés en ordonnées. Ainsi, le système AI-VT initial (premier graphique de la figure) et le système de recommandation déterministe (deuxième graphique) ont tous deux proposé trois questions de niveau de complexité 0 (le plus faible) à tous les apprenants au démarrage de la séance d'entrainement. Nous pouvons remarquer que le système initial est resté sur ce niveau de complexité durant toute la séance (pour les 15 questions du test), tandis que le système de recommandation déterministe a progressivement mixé les complexités des questions posées. Le système de recommandation stochastique décrit dans ce chapitre a quant à lui mixé ces niveaux de complexité dès la première question. + +Ainsi, les système de recommandation permettent de proposer une adaptation progressive du niveau de complexité en fonction des notes obtenues. Le modèle déterministe génère quatre grandes transitions avec un grand nombre d'apprenants dans les questions 5, 6, 8 et 12, toutes entre des niveaux de complexité contigus. La tendance est à la baisse pour les niveaux 0, 1 et 2 après la huitième question et à la hausse pour les niveaux 1 et 3. Le modèle stochastique commence par proposer tous les niveaux de complexité possibles tout en privilégiant le niveau 0. Avec ce système, les transitions sont constantes mais pour un petit nombre d'apprenants. La tendance après la dixième question est à la baisse pour les niveaux 0 et 4 et à la hausse pour les niveaux 1, 2 et 3. + +\colorbox{yellow}{Figure figCmp2 à corriger : ce n'est pas un nombre de question, }\\ +\colorbox{yellow}{mais un numéro de question.}\\ +\colorbox{yellow}{Idem pour les figures Cmp3 et CMP4} \begin{figure} \includegraphics[width=\textwidth]{./Figures/comp2.png} -\caption{Résultats pour le premier test} +\caption{Niveaux de complexité des questions posées aux apprenants par les trois systèmes testés lors de la première séance avec un démarrage à froid (sans données initiales sur les apprentants).} \label{figCmp2} \end{figure} -Après la génération de la première séance, le système peut continuer avec la liste suivante d'exercices, dans ce cas les trois modèles ont été initialisés avec les mêmes données, et des valeurs égales pour tous les apprenants. La figure \ref{figCmp3} permet de voir la première transition du système original, cette transition montre que le système agit uniquement avec les notes obtenues dans le passé et les transitions sont très lentes, même si les notes sont différentes au cours de la séance, tous les apprenants doivent suivre le même chemin. Cependant, les modèles de recommandation changent, le modèle déterministe présente trois transitions dans les questions 3, 5 et 12. Les tendances sont statiques pour le niveau 3, variables pour le niveau 2 et fortement descendantes pour le niveau 0. Le modèle stochastique continue avec des transitions douces mais essaie toujours de préférer le niveau le plus faible, dans ce cas le modèle a identifié le niveau de complexité 1. Ici, les niveaux 0 et 1 sont descendants, le niveau 2 est statique et les niveaux 3 et 4 sont ascendants. +Après la génération de la première séance, le système peut continuer avec une deuxième liste d'exercices. Pour cette partie des tests, les trois modèles ont été initialisés avec les mêmes données, et des valeurs égales pour tous les apprenants. La figure \ref{figCmp3} permet de voir la première transition du système original ne réagissant qu'aux notes obtenues dans les séances précédentes et non à celles de la séance en cours. Les transitions sont très lentes, et tous les apprenants doivent suivre un chemin identique même s'ils obtiennent des notes différentes au cours de la séance. + +Pour leur part, les deux autres systèmes de recommandation testés proposent un fonctionnement différent. Le modèle déterministe présente trois transitions aux questions 3, 5 et 12. Les tendances sont y relativement homogènes et progressives pour le niveau 3, très variables pour le niveau 2 et fortement décroissantes pour le niveau 0. Le modèle stochastique quant à lui, propose des transitions douces mais il a tendance à toujours privilégier le niveau le plus faible. Nous pouvons observer une prépondérance du niveau 1 avec ce système. Ici, les niveaux 0 et 1 sont décroissants, le niveau 2 est statique et les niveaux 3 et 4 sont ascendants. \begin{figure} \includegraphics[width=\textwidth]{./Figures/comp3.png} -\caption{Résultats pour le deuxième Test} +\caption{Niveaux de complexité des questions posées aux apprenants par les trois systèmes testés lors de la deuxième séance.} \label{figCmp3} \end{figure} -Enfin, les données d'initialisation considèrent comme évalués deux niveaux de complexité 0 et 1, alors naturellement le système doit commencer avec le niveau 1 ou 2. Comme le système original est très lent à passer d'un niveau à l'autre, ce système commence par le niveau de complexité 1, comme le montre la figure \ref{figCmp4}, comme les deux autres comparaisons, les changements dans ce système ne sont pas progressifs, mais directs pour tous. Dans ce cas, le modèle de recommandation déterministe adopte la même stratégie et propose un changement direct pour tous les apprenants autour de la cinquième question. Le modèle stochastique continue avec des changements faibles mais constants, mais avec une préférence pour le niveau 2, la tendance est très stable sauf pour 1 (à la hausse) et 2 (à la baisse) niveaux. +Les questions de la première et la deuxième séance étant de niveaux 0 et 1, le système a proposé des niveaux de complexité 1 ou 2 pour la troisième séance. La figure \ref{figCmp4} montre que le système original est très lent à passer d'un niveau à l'autre et les changements sont abrupts pour tous. Dans ce cas, le modèle de recommandation déterministe adopte la même stratégie et propose un changement brutal à tous les apprenants autour de la cinquième question. Le modèle stochastique continue avec des changements progressifs tout en privilégiant le niveau 2. \begin{figure} \includegraphics[width=\textwidth]{./Figures/comp4.png} -\caption{Résultats pour le troisième Test} +\caption{Niveaux de complexité des questions posées aux apprenants par les trois systèmes testés lors de la troisième séance.} \label{figCmp4} \end{figure} -Pour comparer numériquement le système original, le modèle déterministe et le modèle de recommandation proposé, un ensemble d'équations a été défini (équation \ref{eqMetric1} et équation \ref{eqMetric2}) qui décrit le système de recommandation idéal si l'objectif de l'apprenant est l'apprentissage standard, la métrique calcule une valeur pour chaque niveau de complexité en fonction de la moyenne des notes et du nombre de questions recommandées dans ce niveau de complexité. L'objectif de cette métrique est d'attribuer un score élevé aux systèmes de recommandation qui proposent plus d'exercices au niveau de complexité où l'apprenant a obtenu une note moyenne plus basse, dans l'idée de renforcer les connaissances à ce niveau de complexité, de même s'ils proposent moins d'exercices aux niveaux de complexité où la note moyenne est élevée, puisqu'il est supposé que l'étudiant a déjà acquis des connaissances suffisantes à ces niveaux de complexité. Les scores faibles sont attribués aux systèmes qui recommandent peu d'exercices à des niveaux de complexité dont les notes moyennes sont faibles et, inversement, s'ils proposent beaucoup d'exercices à des niveaux de complexité dont les notes moyennes sont élevées. +Pour comparer numériquement le système original, le modèle déterministe et le modèle de recommandation proposé, un ensemble d'équations a été défini (équation \ref{eqMetric1} et équation \ref{eqMetric2}). Celles-ci premettent de décrire le système de recommandation idéal si l'objectif de l'apprenant est de suivre un apprentissage standard. Une valeur est calculée pour chaque niveau de complexité en fonction de la moyenne des notes et du nombre de questions recommandées dans ce niveau de complexité. L'objectif de cette mesure est d'attribuer un score élevé aux systèmes de recommandation qui proposent plus d'exercices au niveau de complexité où l'apprenant a obtenu une note moyenne plus basse, lui permettant ainsi de renforcer ses connaissances à ce niveau de complexité. De la même manière, il est attendu que le système de recommandation propose moins d'exercices aux niveaux de complexité pour lesquels les notes moyennes sont élevées, l'étudiant ayant acquis des connaissances suffisantes à ces niveaux de complexité. Les scores faibles sont attribués aux systèmes qui recommandent peu d'exercices à des niveaux de complexité dont les notes moyennes sont faibles et, inversement, s'ils proposent beaucoup d'exercices à des niveaux de complexité dont les notes moyennes sont élevées. \begin{equation} %r_c=x+y-2xy @@ -225,7 +225,7 @@ Les propriétés de la métrique sont : \item $min(rp_c(x))=0.1353; \; if \; \left ( \sum_{i=1}^2 x_{i,c}=0 \; \lor \; \sum_{i=1}^2 x_{i,c} = 2 \right )$\\ \end{itemize} -Dans l'équation \ref{eqMetric1}, $x_{0,c}$ est la moyenne normalisée des notes dans le niveau de complexité $c$ (équation \ref{eqXc}), et $x_{1,c}$ est le nombre normalisé de questions répondues dans le niveau de complexité $c$ (équation \ref{eqYc}). +Dans l'équation \ref{eqMetric1}, $x_{0,c}$ est la moyenne normalisée des notes dans le niveau de complexité $c$ (équation \ref{eqXc}), et $x_{1,c}$ est le nombre normalisé de questions auxquelles des réponses ont été données dans le niveau de complexité $c$ (équation \ref{eqYc}). Ainsi, plus la valeur de $r$ est élevée, meilleure est la recommandation. \begin{equation} x_{0,c}=\frac{_{G_c}}{g_m} @@ -237,15 +237,15 @@ x_{1,c}=\frac{ny_c}{n_c} \label{eqYc} \end{equation} -La figure \ref{figMetric} montre l'équation globale pour la métrique $rp$ dans le domaine de deux variables $x_{0,c}$ et $x_{1,c}$. La valeur maximale de $r$ dans un niveau de complexité spécifique est de 1, la valeur maximale globale pour les scénarios testés est de 5. Un bon système de recommandation devrait donc avoir une valeur $r$ élevée. +La figure \ref{figMetric} représente la fonction $rp_c(x)$. La valeur maximale de $r$ dans un niveau de complexité spécifique étant égale à $1$, la valeur maximale globale pour les scénarios testés est égale à 5. \begin{figure} \includegraphics[width=\textwidth]{./Figures/metric.png} -\caption{Métrique pour le parcours standard} +\caption{Fonction d'évaluation de la qualité de la recommandation pour un parcours standard} \label{figMetric} \end{figure} -Les résultats des calculs de la métrique établie pour le système original et les deux modèles dans les trois scénarios définis sont présentés dans le tableau \ref{tabRM}. +Les résultats des calculs de la métrique $rp_c(x)$ établie pour le système original et les deux modèles dans les trois scénarios testés sont présentés dans le tableau \ref{tabRM}. \begin{table}[!ht] \centering @@ -271,11 +271,11 @@ RàPC&-&0.8559&0.7377&-&-&1.5936&31.872 DM&-&-&0.5538&\textbf{0.7980}&-&1.3518&27.036\\ SM&0.9089&\textbf{0.9072}&\textbf{0.9339}&0.7382&0.6544&\textbf{4.1426}&\textbf{82.852}\\ \end{tabular} -\caption{Résultats de la métrique $rp_c(x)$ (RàPC - Système sans modèle de recommandation, DM - Modèle deterministique, SM - Modèle stochastique)} +\caption{Résultats de la métrique $rp_c(x)$ (RàPC - Système sans module de recommandation, DM - Module de recommandation deterministe, SM - Module de recommandation stochastique)} \label{tabRM} \end{table} -Une métrique pour l'apprentissage en douceur est définie dans l'équation \ref{eqMetricS1} et l'équation \ref{eqMetricS2}, avec cette métrique un score élevé est attribué aux systèmes qui proposent plus d'exercices dans un niveau de complexité où les notes moyennes sont d'environ 0,4 et qui sont plus flexibles avec des notes moyennes plus basses, également si le nombre d'exercices proposés est faible pour des notes moyennes élevées. Les scores faibles sont attribués aux systèmes qui recommandent un nombre élevé de questions dans un niveau de complexité avec des notes moyennes élevées et si le nombre d'exercices recommandés est trop élevé ou trop faible pour les notes inférieures. +Les équations \ref{eqMetricS1} et \ref{eqMetricS2} permettent de caractériser un apprentissage progressif. Dans ce cas, un score élevé est attribué aux systèmes proposant plus d'exercices dans un niveau de complexité où les notes moyennes sont légèrement insuffisantes (4/10), plus flexibles avec des notes moyennes plus basses, et un petit nombre d'exercices pour des notes moyennes élevées. Les scores faibles sont attribués aux systèmes qui recommandent de nombreuses questions dans un niveau de complexité avec des notes moyennes élevées ou faibles. \begin{equation} rs_c(x)=e^{-\frac{2}{100}(32x_{0,c}^2-28x_{0,c}+10x_{1,c}-4)^2} ; \{x \in \mathbb{R}^2 | 0<=x<=1\} @@ -293,14 +293,14 @@ Les propriétés de la métrique sont : \item $max(rs_c(x))=1; \; if \; 16x_{0,c}^2-14x_{0,c}+5x_{1,c}-2=0$\\ \end{itemize} -La figure \ref{figMetric2} montre l'équation globale pour la métrique $rs$ dans le domaine de deux variables $x_{0,c}$ et $x_{1,c}$. La valeur maximale de $r$ dans un niveau de complexité spécifique est de 1, la valeur maximale globale pour les scénarios testés est de 5, un bon système de recommandation doit donc avoir une valeur $r$ élevée. +La figure \ref{figMetric2} représente la fonction $rs_c(x)$. Comme pour $rp_c$, la valeur maximale de $r$ dans un niveau de complexité spécifique étant égale à $1$, la valeur maximale globale pour les scénarios testés est égale à 5. Les résultats du calcul des métriques pour le système original et les deux modèles dans les trois scénarios définis sont présentés dans le tableau \ref{tabRM2}. \begin{figure}[!ht] \centering \includegraphics[width=\textwidth]{./Figures/metric2.png} -\caption{Fonction d'évaluation métrique à chaque niveau de complexité (Soft learning)} +\caption{Fonction d'évaluation de la qualité de la recommandation pour un apprentissage progressif.} \label{figMetric2} \end{figure} @@ -309,30 +309,30 @@ Les résultats du calcul des métriques pour le système original et les deux mo \begin{tabular}{cccccccc} &$c_0$&$c_1$&$c_2$&$c_3$&$c_4$&Total ($r$)&Total ($\%$)\\ \hline -Test 1\\ +Séance 1\\ \hline RàPC&\textbf{0.9979}&-&-&-&-&0.9979&19.96\\ DM&0.8994&0.1908&\textbf{0.3773}&\textbf{0.2990}&-&1.7665&35.33\\ SM&0.8447&\textbf{0.3012}&0.2536&0.2030&\textbf{0.1709}&\textbf{1.7734}&\textbf{35.47}\\ \hline -Test 2\\ +Séance 2\\ \hline RàPC&\textbf{0.4724}&\textbf{0.7125}&-&-&-&1.1849&23.70\\ DM&-&0.6310&\textbf{0.3901}&\textbf{0.4253}&-&1.4464&28.93\\ SM&0.2697&0.7089&0.2634&0.2026&\textbf{0.1683}&\textbf{1.6129}&\textbf{32.26}\\ \hline -Test3\\ +Séance 3\\ \hline RàPC&-&\textbf{0.9179}&0.2692&-&-&1.1871&23.74 \\ DM&-&-&0.2236&\textbf{0.9674}&-&1.191&23.82\\ SM&0.1873&0.3038&\textbf{0.6345}&0.2394&\textbf{0.1726}&\textbf{1.5376}&\textbf{30.75}\\ \end{tabular} -\caption{Résultats de la métrique $rs_c(x)$ (RàPC - Système sans modèle de recommandation, DM - Modèle deterministique, SM - Modèle stochastique)} +\caption{Évaluation des recommandations proposées selon $rs_c(x)$ par les différents systèmes de recommandation testés : RàPC - Système sans module de recommandation, DM - Modèle deterministique, SM - Modèle stochastique} \label{tabRM2} \end{table} -Pour comparer le système original et le modèle de recommandation, est utilisée la métrique de la diversité des propositions, avec la similarité en cosinus (La similarité en cosinus entre un vecteur $A$ et un vecteur $B$, Équation \ref{eqCS}) entre toutes les propositions des apprenants. Les résultats de la similarité cosinus moyenne sont présentés dans le tableau \ref{tabCS}. +En complément, le tableau \ref{tabCS} présente les similarités entre toutes les recommandations faites aux apprenants par les trois systèmes et les trois séances d'entrainement. Pour ce faire, nous avons choisi d'appliquer l'équation \ref{eqCS} permettant de calculer une similarité cosinus entre deux vecteurs $A$ et $B$. \begin{equation} sc=\frac{\sum_{i=1}^n A_i B_i}{\sqrt{\sum_{i=1}^n A_i^2} \sqrt{\sum_{i=1}^n B_i^2}} @@ -342,35 +342,39 @@ sc=\frac{\sum_{i=1}^n A_i B_i}{\sqrt{\sum_{i=1}^n A_i^2} \sqrt{\sum_{i=1}^n B_i^ \begin{table}[!ht] \centering \begin{tabular}{cccc} -Model&Scenario 1&Scenario 2&Scenario 3\\ +Système de recommandation & Séance 1 & Séance 2 & Séance 3\\ \hline RàPC&1&1&1\\ DM&0.9540&0.9887&0.9989\\ SM&\textbf{0.8124}&\textbf{0.8856}&\textbf{0.9244}\\ \end{tabular} -\caption{Moyenne de la diversité des propositions pour tous les apprenants. Une valeur plus faible représente une plus grande diversité. (RàPC - Système sans modèle de recommandation, DM - Modèle deterministique, SM - Modèle stochastique)} +\caption{Moyenne de la diversité des propositions pour tous les apprenants. Une valeur plus faible représente une plus grande diversité. (RàPC - Système sans module de recommandation, DM - Module deterministe, SM - Module stochastique)} \label{tabCS} \end{table} -\subsection{Discussion et Conclusions} -Avec la génération d'exercices RàPC, le système propose les mêmes exercices à tous les apprenants, et l'évolution des niveaux de complexité est très lente, presque un changement toutes les 3 ou 4 séances, ceci parce que le système ne prend pas en compte les notes obtenues pendant la séance. Les systèmes de recommandation sont plus dynamiques et les évolutions sont plus rapides mais en considérant les notes des apprenants, le modèle déterministe suggère des changements de niveaux à un grand nombre d'apprenants soudainement parce qu'ils sont regroupés à l'intérieur d'un intervalle de taux de maîtrise, alors que le modèle stochastique est plus axé sur la personnalisation individuelle et les changements de niveau de complexité sont produits pour un petit nombre d'apprenants. Les deux modèles proposés ont la capacité de détecter les faiblesses des apprenants et d'adapter la séance à leurs besoins particuliers. +\subsection{Discussion et Conclusion} +Avec la génération d'exercices par le système de RàPC initial, AI-VT propose les mêmes exercices à tous les apprenants, et l'évolution des niveaux de complexité est très lente, un changement toutes les trois ou quatre séances environ. En effet, le système ne prend pas en compte les notes obtenues pendant la séance. Les systèmes intégrant l'un des modules de recommandation testés sont plus dynamiques et les évolutions sont plus rapides. En considérant les notes des apprenants, le modèle déterministe suggère des changements de niveaux à un grand nombre d'apprenants de manière soudaine, tandis que le modèle stochastique est plus axé sur la personnalisation individuelle et les changements de niveau de complexité sont produits pour un petit nombre d'apprenants. Les deux modules de recommandation proposés ont la capacité de détecter les faiblesses des apprenants et d'adapter la séance à leurs besoins particuliers. -La base de données générée a permis de simuler diverses situations avec les notes de 1000 apprenants, permettant ainsi d'évaluer le comportement des systèmes de recommandation avec différentes configurations. +Les données générées ont permis de simuler diverses situations avec les notes de mille apprenants, permettant ainsi d'évaluer le comportement des systèmes de recommandation avec différentes configurations. -Les résultats numériques utilisant la métrique définie montrent que les distributions des questions dans une séance par les deux versions du modèle de recommandation sont différentes mais avec une tendance générale similaire pour tous les apprenants. Le modèle proposé tente de répartir les questions dans tous les niveaux de complexité définis. Globalement, avec la métrique définie, le modèle stochastique a obtenu un meilleur score. Par rapport au système original, le modèle de recommandation (versions déterministe et stochastique) obtient une augmentation globale de l'adaptabilité comprise entre 15\% et 68\% pour tous les niveaux de complexité. +Les résultats numériques montrent que les distributions des questions dans une séance par les deux modules de recommandation sont différentes bien que la tendance générale soit similaire. Les modules de recommandation proposés tentent de répartir les questions dans tous les niveaux de complexité définis. Globalement, le module de recommandation stochastique a obtenu un meilleur score. En comparaison du système original, les modules de recommandation (déterministe et stochastique) proposent 15\% à 68\% d'adaptations pour tous les niveaux de complexité. -Selon la métrique de la similarité cosinus, le modèle de recommandation proposé augmente la diversité des propositions par rapport au système original dans les trois scénarios évalués, ce qui indique qu'en plus d'atteindre l'adaptabilité, des propositions personnalisées sont générées tout en maintenant l'objectif de faire progresser les apprenants entre les niveaux de complexité. La diversité des propositions est une caractéristique essentielle du modèle de recommandation dans ses deux versions. +Selon la métrique de la similarité cosinus, le module de recommandation stochastique augmente la diversité des propositions par rapport au système original dans les trois séances d'entrainement testées, ce qui indique qu'en plus d'atteindre l'adaptabilité, des propositions personnalisées sont générées tout en maintenant l'objectif de progression des niveaux de compétence des apprenants. La diversité des propositions est une caractéristique essentielle du modèle de recommandation dans ses deux versions. -Les modules de recommandation sont une pièce essentielle pour certaines EIAH car ils aident à guider le processus d'apprentissage individuel, permettent d'identifier les faiblesses et de réorienter le processus complet afin d'améliorer les connaissances et les compétences. Les versions du modèle proposé peuvent détecter en temps réel les faiblesses de l'apprenant et tentent de réorienter la séance vers le meilleur niveau de complexité possible afin d'aider l'apprenant à acquérir et à maîtriser les connaissances avant de passer aux niveaux de complexité supérieurs, car généralement les connaissances des niveaux de complexité inférieurs sont nécessaires pour compléter les niveaux supérieurs. Même si l'ensemble de données généré est une simulation des temps de réponse et des notes des apprenants, les tests qui l'utilisent permettent de voir la flexibilité et la robustesse du modèle de recommandation proposé, car les données relatives aux apprenants présentent une grande diversité et obligent le système à s'adapter à différents types de configurations. Par conséquent, il est possible de conclure que le modèle de recommandation proposé a la capacité de fonctionner dans différentes situations et dans chaque cas de proposer des chemins alternatifs pour améliorer le processus d'apprentissage global, même si l'objectif d'apprentissage est différent pour chaque apprenant, comme le démontrent les résultats obtenus dans l'évaluation des deux métriques proposées. Le modèle proposé permet également la diversité et la personnalisation du système, puisque selon les résultats de la comparaison avec la similarité cosinus entre toutes les recommandations générées pour chaque apprenant, il y a une augmentation par rapport au système original.\\\\ +Les modules de recommandation sont une pièce essentielle pour certaines EIAH car ils aident à guider le processus d'apprentissage individuel, permettent d'identifier les faiblesses et de réorienter le processus complet afin d'améliorer les connaissances et les compétences. Les deux modules de recommandation proposés peuvent détecter en temps réel les faiblesses de l'apprenant et tentent de réorienter la séance vers le niveau de complexité le plus adapté. Même si l'ensemble de données générées est une simulation de temps de réponse et de notes fictives d'apprenants fictifs, les tests démontrent la flexibilité et la robustesse des modules de recommandation proposés : les données relatives aux apprenants présentent en effet une grande diversité et obligent le système à s'adapter à différents types de configurations. Par conséquent, il est possible de conclure que les modules de recommandation proposés ont la capacité de fonctionner dans différentes situations et de proposer des chemins alternatifs et personnalisés pour améliorer le processus d'apprentissage global. Leur intégration à l'EIAH AI-VT originel constituent donc une améliortation notable du système. -\section{ESCBR-SMA et Échantillonnage de Thompson} +\section{ESCBR-SMA et échantillonnage de Thompson} \sectionmark{ESCB-SMA et TS} +La section précédente a démontré l'intérêt de l'intégration d'un module de recommandation afin de proposer des exercices d'un niveau de difficulté adapté aux besoin de l'apprenant en fonction des difficultés décelées au cours de la séance d'entraînement. Le système AI-VT originel fondé sur le cycle du raisonnement à partir de cas et ne proposant que des adaptations entre deux séances d'entrainement consécutives, a été supplanté par l'intégration de modules de recommandation utilisés durant la phase de révision du cycle classique du RàPC. Les deux modules de recommandation testés dans ala section précedente étaient l'un déterministe, l'autre stochastique. + +La section précédente a également démontré qu'il était possible et intéressant que les niveaux de complexité des exercices proposés puissent suivre des fonctions permettant de les faire fluctuer de manière progressive au cours de la séance, et ce afin que les apprenants ne soient pas confrontés à des difficultés changeant de manière trop abrupte durant l'entraînement. Cette étude nous amène donc à considérer la résolution de la génération d'une séance d'exercices sous l'angle de la régression. Nous proposons donc dans cette partie de montrer de quelle manière nous avons intégré et vérifié l'intérêt des outils définis dans le chapitre précédent dans l'EIAH AI-VT. + \subsection{Concepts Associés} -Cette section présente les concepts, les définitions et les algorithmes nécessaires à la compréhension du modèle proposé, ainsi que les modèles et les mesures fondamentaux. Le premier paradigme fondamental utilisé dans ce travail est le raisonnement à partir de cas (RàPC), qui permet d'exploiter les connaissances historiquement acquises et l'expérience accumulée en ce qui concerne un problème spécifique. Ce paradigme est utilisé pour générer des solutions émergentes pour un nouveau problème en utilisant une base de données de connaissances. L'idée principale est de rechercher des situations antérieures similaires et d'utiliser l'expérience acquise pour résoudre de nouveaux problèmes. Le RàPC est particulièrement utile lorsque les causes sous-jacentes d'un problème ne sont pas bien comprises. Le raisonnement à base de cas définit un cycle de quatre étapes pour améliorer la solution d'inférence \cite{jmse11050890}. +Cette section présente les concepts, les définitions et les algorithmes nécessaires à la compréhension du module proposé. Le paradigme fondamental utilisé dans ce travail est le raisonnement à partir de cas (RàPC), qui permet d'exploiter les connaissances acquises et l'expérience accumulée pour résoudre un problème spécifique. L'idée principale est de rechercher des situations antérieures similaires et d'utiliser l'expérience acquise pour résoudre de nouveaux problèmes. Le RàPC suit classiquement un cycle de quatre étapes pour améliorer la solution d'inférence \cite{jmse11050890}. -Puisque l'objectif ici est d'adapter les exercices proposés par AI-VT, il est nécessaire de connaître le fonctionnement de l'un des algorithmes les plus utilisés pour effectuer l'adaptation du contenu et des exercices dans certains STI, afin de comparer les résultats avec l'algorithme proposé et de voir dans quelle mesure il permet d'obtenir une amélioration de l'adaptation et de la performance des apprenants. L'un des modèles les plus couramment utilisés dans les EIAH pour adapter le contenu et estimer la progression du niveau de connaissance des apprenants est le BKT (Bayesian Knowledge Tracing) \cite{ZHANG2018189}. Ce modèle utilise quatre paramètres pour estimer la progression des connaissances. $P(k)$ estime la probabilité de connaissance dans une compétence spécifique. $P(w)$, est la probabilité que l'apprenant démontre ses connaissances. $P(s)$, est la probabilité que l'apprenant fasse une erreur.$P(g)$, est la probabilité que l'apprenant ait deviné une réponse. La valeur estimée de la connaissance est mise à jour avec les équations \ref{eqbkt1}, \ref{eqbkt2} et \ref{eqbkt3}. Si la réponse de l'apprenant est correcte, l'équation \ref{eqbkt1} est utilisée, mais si la réponse est incorrecte, l'équation \ref{eqbkt2} est utilisée. +L'un des modèles les plus couramment utilisés dans les EIAH pour adapter le contenu et estimer la progression du niveau de connaissance des apprenants est le BKT (\textit{Bayesian Knowledge Tracing}) \cite{ZHANG2018189}. Ce modèle utilise quatre paramètres pour estimer la progression des connaissances. $P(k)$ estime la probabilité de connaissance dans une compétence spécifique. $P(w)$, est la probabilité que l'apprenant démontre ses connaissances. $P(s)$, est la probabilité que l'apprenant fasse une erreur.$P(g)$, est la probabilité que l'apprenant ait deviné une réponse. La valeur estimée de la connaissance est mise à jour selon les équations \ref{eqbkt1}, \ref{eqbkt2} et \ref{eqbkt3}. Si la réponse de l'apprenant est correcte, l'équation \ref{eqbkt1} est utilisée, mais si la réponse est incorrecte, l'équation \ref{eqbkt2} est utilisée. \begin{equation} P(k_{t-1}|Correct_t)=\frac{P(k_{t-1})(1-P(s))}{P(k_{t-1})(1-P(s))+(1-P(k_{t-1}))P(g)} @@ -387,7 +391,7 @@ Puisque l'objectif ici est d'adapter les exercices proposés par AI-VT, il est n \label{eqbkt3} \end{equation} -Le modèle de recommandation proposé, associé à AI-VT, est fondé sur le paradigme de l'apprentissage par renforcement. L'apprentissage par renforcement est une technique d'apprentissage automatique qui permet, par le biais d'actions et de récompenses, d'améliorer les connaissances du système sur une tâche spécifique \cite{NEURIPS2023_9d8cf124}. L'algorithme utilisé pour l'adaptation est un algorithme d'apprentissage par renforcement appelé échantillonnage de Thompson, qui, par le biais d'une distribution de probabilité initiale (distribution a priori) et d'un ensemble de règles de mise à jour prédéfinies, peut adapter et améliorer les estimations initiales d'un processus analysé spécifique \cite{pmlr-v238-ou24a}. La distribution de probabilité initiale est généralement définie comme une distribution spécifique de la famille des distributions Bêta (équation \ref{fbeta}) avec des valeurs initiales prédéterminées pour $\alpha$ et $\beta$ \cite{math12111758}, \cite{NGUYEN2024111566}. +Le module de recommandation proposé, associé à AI-VT, est fondé sur le paradigme de l'apprentissage par renforcement. L'apprentissage par renforcement est une technique d'apprentissage automatique qui permet, par le biais d'actions et de récompenses, d'améliorer les connaissances du système sur une tâche spécifique \cite{NEURIPS2023_9d8cf124}. Nous nous intéressons ici plus particulièrement à l'échantillonnage de Thompson, qui, par le biais d'une distribution de probabilité initiale (distribution a priori) et d'un ensemble de règles de mise à jour prédéfinies, peut adapter et améliorer les estimations initiales d'un processus \cite{pmlr-v238-ou24a}. La distribution de probabilité initiale est généralement définie comme une distribution spécifique de la famille des distributions Beta (équation \ref{fbeta}) avec des valeurs initiales prédéterminées pour $\alpha$ et $\beta$ \cite{math12111758}, \cite{NGUYEN2024111566}. %\begin{equation} % Beta(x,\alpha,\beta)=\begin{cases} @@ -401,7 +405,7 @@ Le modèle de recommandation proposé, associé à AI-VT, est fondé sur le para \label{fbeta} \end{equation} -En utilisant la definition formelle de la fonction Gamma $\Gamma$ (équation \ref{eqGamma1}) et en remplaçant des variables, une nouvelle expression de la fonction Beta est obtenue (équation \ref{f2beta}). +En utilisant la definition formelle de la fonction $\Gamma$ (équation \ref{eqGamma1}) et en remplaçant certaines variables, une nouvelle expression de la fonction Beta est obtenue (équation \ref{f2beta}). \begin{equation} \Gamma(z)=\int_0^\infty e^{-x} x^{z-1} dx @@ -420,7 +424,7 @@ En exprimant les deux intégrales du denominateur comme une seule intégrale, l' \label{f3Beta} \end{equation} -Après, sont remplacées $u=st$, $v=s(1-t)$, $s=u+v$ et $t=u/(u+v)$, avec le résultat du Jacobien \ref{eqJac}, alors l'expression finale est comme montre l'équation \ref{f4Beta}. +$u=st$, $v=s(1-t)$, $s=u+v$ et $t=u/(u+v)$ sont ensuite remplacées par le résultat du Jacobien \ref{eqJac}, menant ainsi à l'expression finale définie par l'équation \ref{f4Beta}. \begin{equation} \left ( @@ -443,22 +447,20 @@ Après, sont remplacées $u=st$, $v=s(1-t)$, $s=u+v$ et $t=u/(u+v)$, avec le ré \label{f4Beta} \end{equation} -Si les intégrales sont exprimées en fonction des variables indépendantes $s$ et $t$ l'équation \ref{f5Beta} est générée. +Viennent ensuite les équations \ref{f5Beta} et \ref{f6Beta} en exprimant les intégrales en fonction des variables indépendantes $s$ et $t$. \begin{equation} \int_{s=0}^\infty e^{-s}s^{\alpha+\beta-1}ds \int_{t=0}^1 t^{\alpha-1}(1-t)^{\beta-1}dt \label{f5Beta} \end{equation} -En plaçant les termes dans l'équation le résultat est l'équation \ref{f6Beta}. - \begin{equation} Beta(\theta | \alpha, \beta) = \frac{\int_0^\infty e^{-s} s^{\alpha+\beta-1}ds}{\int_{s=0}^\infty e^{-s}s^{\alpha+\beta-1}ds \int_{t=0}^1 t^{\alpha-1}(1-t)^{\beta-1}dt }\theta^{\alpha-1}(1-\theta)^{\beta-1} \label{f6Beta} \end{equation} -Finalement, la famille de fonctions de distribution Beta peut être exprimée comme l'équation \ref{f7Beta}. Les métriques utilisées dans ce chapitre s'expriment en fonction de cette définition. +Finalement, la famille de fonctions de distribution Beta peut être calculée selon l'équation \ref{f7Beta}. \begin{equation} Beta(\theta | \alpha, \beta) = \frac{\theta^{\alpha-1}(1-\theta)^{\beta-1}}{\int_{0}^1 t^{\alpha-1}(1-t)^{\beta-1}dt @@ -466,14 +468,16 @@ Finalement, la famille de fonctions de distribution Beta peut être exprimée co \label{f7Beta} \end{equation} -L'évolution de l'algorithme de recommandation TS est établie par le changement des distributions de probabilité, mais au moment de quantifier l'évolution, le changement et la variabilité doivent être calculés en fonction du temps. Les distributions de probabilités peuvent être comparées pour déterminer leur degré de similitude, sous la forme d'une métrique qui détermine numériquement les différences entre elles. L'apprentissage automatique utilise la divergence de Kullback-Liebler, qui décrit l'entropie relative de deux distributions de probabilités. Cette fonction est fondée sur le concept d'entropie et le résultat peut être interprété comme la quantité d'information nécessaire pour obtenir la distribution de probabilité $q$ à partir de la distribution de probabilité $p$. La divergence de Kullback-Liebler (équation \ref{dkl}) est largement utilisée, mais elle présente l'inconvénient de ne pas pouvoir être utilisée comme métrique dans certains cas, car il ne s'agit pas d'une mesure symétrique, $D_{KL}(p,q) \neq D_{KL}(q,p)$, elle ne satisfait pas à l'inégalité triangulaire et elle n'est pas bornée \cite{Li_2024}. Pour remédier à cette difficulté, il est possible d'utiliser la divergence de Jensen-Shannon. +L'évolution de l'algorithme de recommandation TS résulte du changement des distributions de probabilité. Il est à noter qu'au moment de quantifier l'évolution, le changement et la variabilité doivent être calculés en fonction du temps. Les distributions de probabilités peuvent être comparées pour déterminer leur degré de similitude. + +Par ailleurs, l'apprentissage automatique utilise la divergence de Kullback-Liebler, qui décrit l'entropie relative de deux distributions de probabilités. Cette fonction est fondée sur le concept d'entropie et le résultat peut être interprété comme la quantité d'informations nécessaires pour obtenir la distribution de probabilité $q$ à partir de la distribution de probabilité $p$. Bien que largement utilisée, la divergence de Kullback-Liebler (équation \ref{dkl}) présente toutefois l'inconvénient de ne pas être une mesure symétrique car elle ne satisfait pas à l'inégalité triangulaire et n'est pas bornée \cite{Li_2024}. Pour remédier à cette difficulté, il est possible d'utiliser la divergence de Jensen-Shannon. \begin{equation} D_{KL}(p(x),q(x))=\int_{-\infty}^{\infty}p(x) log \left(\frac{p(x)}{q(x)} \right)dx \label{dkl} \end{equation} -La divergence de Jenser-Shannon est fondée sur la divergence de Kullback-Liebler, à la différence qu'une distribution de probabilité auxiliaire $m$ est créée dont la définition est fondée sur les distributions initiales $p$ et $q$ \cite{Kim2024}. L'équation \ref{djs} montre la définition formelle de la divergence de Jensen-Shannon, où $m(x)$ est une distribution de mélange de probabilités fondée sur $p(x)$ et $q(x)$, l'équation \ref{djs2} montre comment elle est calculée. La divergence de Jensen-Shannon est un mélange de distributions de probabilités fondé sur $p(x)$ et $q(x)$. +La divergence de Jenser-Shannon est fondée sur la divergence de Kullback-Liebler. Une distribution de probabilité auxiliaire $m$ est créée dont la définition est fondée sur les distributions initiales $p$ et $q$ \cite{Kim2024}. L'équation \ref{djs} montre la définition formelle de la divergence de Jensen-Shannon, où $m(x)$ est une distribution de mélange de probabilités fondée sur $p(x)$ et $q(x)$. Celle-ci est calculée selon l'équation \ref{djs2}. Les distributions de probabilité à comparer doivent être continues et définies dans le même domaine. %Jensen-Shannon Divergence (equations \ref{djs}, \ref{djs2}).\\ @@ -487,24 +491,25 @@ La divergence de Jenser-Shannon est fondée sur la divergence de Kullback-Lieble \label{djs2} \end{equation} -Les distributions de probabilité à comparer doivent être continues et définies dans le même domaine. +La prédiction utilisée dans le module proposé est fondée sur les travaux de \cite{10.1007/978-3-031-63646-2_11}. Il s'agit d'un modèle d'empilement de raisonnement à partir de cas mettant en Å“uvre deux niveaux d'intégration. Le module utilise globalement la stratégie d'empilement pour exécuter plusieurs algorithmes afin de rechercher des informations dans un ensemble de données et générer des solutions à différents problèmes génériques. En outre une étape d'évaluation permet de sélectionner la solution la plus optimale pour un problème donné en fonction d'une métrique adaptative définie pour les problèmes de régression. -La prédiction utilisée dans le modèle proposé est fondée sur les travaux de Soto \textit{et al.} \cite{10.1007/978-3-031-63646-2_11}, il s'agit d'un modèle d'empilage de raisonnement à partir de cas qui met en Å“uvre deux niveaux d'intégration, le modèle utilise globalement la stratégie d'empilage pour exécuter plusieurs algorithmes afin de rechercher des informations dans un ensemble de données et de générer des solutions à différents problèmes génériques, en outre il y a une étape d'évaluation qui permet de sélectionner la solution la plus optimale pour un problème donné en fonction d'une métrique adaptative définie pour les problèmes de régression. Il a été décidé de mettre en Å“uvre le modèle fondé sur l'empilement car il s'agit d'une méthode d'ensemble qui permet fondé sur le paradoxe de Stein puisqu'elle combine les points de vue de différents estimateurs à des étapes de récupération et de réutilisation dans le raisonnement à partir de cas. +\subsection{Module Proposé} +\label{Sec:TS-ESCBR-SMA} -\subsection{Modèle Proposé} +Nous proposons ici une intégration du module d'adaptation stochastique (fondé sur l'échantillonnage de Thompson) avec le raisonnement à partir de cas d'ensemble (ESCBR-SMA). Dans ce cas, le module de recommandation révise la séance en fonction des notes de l'apprenant et l'ESCBR-SMA effectue une prédiction pour valider l'adaptation générée. -Le modèle proposé est une intégration du modèle d'adaptation stochastique (fondé sur l'échantillonnage de Thompson) avec le raisonnement à partir de cas d'ensemble (ESCBR-SMA). Dans ce cas, le modèle de recommandation produit une adaptation en fonction des notes de l'apprenant et l'ESCBR-SMA effectue une prédiction pour valider l'adaptation générée. +L'idée est d'unifier les deux modules en se fondant à la fois sur des informations locales (recommandation fondée sur l'échantillonnage de Thompson et les informations propres à l'apprenant), et sur des informations globales (cas similaires de la base de connaissances du système de RàPC). -L'idée d'unifier les deux modèles est d'obtenir des informations du point de vue local où une recommandation est obtenue en utilisant uniquement sur les informations des apprenants individuels (modèle fondé sur l'échantillonnage de Thompson) et la prédiction globale où les informations sont obtenues à partir de tous les apprenants qui ont des résultats similaires (filtre collaboratif avec RàPC). L'architecture du modèle est présentée dans la figure \ref{fig:Amodel1}, où l'on peut voir que les deux modèles 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 modèle sont obtenus, les résultats sont unifiés par le biais d'une fonction de pondération, la recommandation finale est celle qui est calculée avec l'expression \ref{eqMixModels_}. La consolidation des résultats des deux modèles permet d'atténuer l'effet du paradoxe de Simpson \cite{10.1145/3578337.3605122}. Ce paradox décrit l'effet qui se présente lorsque les données sont grouppes de différents manières et montrent tendances divergentes \cite{lei2024analysis}. +L'architecture du modèle est présentée sur la figure \ref{fig:Amodel}, où l'on peut voir que les deux modèles TS et RàPC sont exécutés en parallèle et indépendamment. Des synchronisations sont faites après obtention des résultats de chaque module sont obtenus. Ces résultats sont unifiés via d'une fonction de pondération. La recommandation finale est calculée selon l'équation \ref{eqMixModels_}. La consolidation des résultats permet d'atténuer l'effet \textit{paradoxe de Simpson} \cite{10.1145/3578337.3605122} caractérisant des tendances divergentes lors de l'unification d'ensembles de données différents \cite{lei2024analysis}. \colorbox{yellow}{<== Contresens ?} \begin{figure} \centering \includegraphics[width=0.7\linewidth]{Figures/Model.png} \caption{Schéma de l'architecture du modèle proposé} - \label{fig:Amodel1} + \label{fig:Amodel} \end{figure} -La première étape est l'adaptation avec l'échantillonnage de Thompson, puis la prédiction ECBR-SMA et enfin la prise de décision à envoyer à l'apprenant. Le système de recommandation obtient une valeur de probabilité pour tous les niveaux de complexité de l'apprenant et l'ECBR-SMA évalue la proposition avec une prédiction pour chaque niveau de complexité. Le tableau \ref{tabvp} présente les variables et les paramètres du modèle proposé ainsi que les mesures employées. Le tableau \ref{tabvp} présente les variables et les paramètres du modèle proposé ainsi que les mesures employées. +La première étape est l'adaptation avec l'échantillonnage de Thompson. Vient ensuite la prédiction via ECBR-SMA. Enfin, le processus se termine par la prise de décision concernant la suite de la séance à délivrer à l'apprenant. Le système de recommandation obtient une valeur de probabilité pour tous les niveaux de complexité et l'ECBR-SMA évalue la proposition avec une prédiction pour chaque niveau de complexité. Le tableau \ref{tabvp} présente les variables et les paramètres du module proposé ainsi que les mesures employées. \begin{table}[!ht] \centering @@ -512,27 +517,27 @@ La première étape est l'adaptation avec l'échantillonnage de Thompson, puis l \begin{tabular}{c|c|>{\centering\arraybackslash}p{8cm}|c} ID&Type&Description&Domain\\ \hline - $\alpha$&p&Paramètre de la distribution bêta&$[1, \infty] \in \mathbb{R}$\\ - $\beta$&p&Paramètre de la distribution bêta&$[1, \infty] \in \mathbb{R}$\\ + $\alpha$&p&Paramètre de la distribution beta&$[1, \infty] \in \mathbb{R}$\\ + $\beta$&p&Paramètre de la distribution beta&$[1, \infty] \in \mathbb{R}$\\ $t$&p&Temps défini comme itérations&$\mathbb{N}$\\ $c$&p&Niveau de complexité&$\mathbb{N}$\\ - $x_c$&p&Mean grades for complexity level $c$&$\mathbb{R}$\\ - $y_c$&p&Number of questions for complexity level $c$&$\mathbb{N}$\\ - $r$&f&Recommender metric function&$[0,1] \in \mathbb{R}$\\ + $x_c$&p&Notes moyennes par niveau de complexité $c$&$\mathbb{R}$\\ + $y_c$&p&Nombre de questions par niveau de complexité $c$&$\mathbb{N}$\\ + $r$&f&Fonction suivie pour la recommandation&$[0,1] \in \mathbb{R}$\\ $k_{t,c}$&v&Évolution de la connaissance dans le temps $t$ pour le niveau de complexité $c$&$[0,1] \in \mathbb{R}$\\ $vk_{t,c}$&v&Évolution de la connaissance pour chaque niveau de complexité $c$&$\mathbb{R}$\\ $TS_c$&v&Récompense d'échantillonnage de Thompson pour un niveau de complexité $c$&$[0,1] \in \mathbb{R}$\\ - $TSN_c$&v&Normalization de $TS_c$ avec d'autres niveaux de complexité&$[0,1] \in \mathbb{R}$\\ + $TSN_c$&v&Normalisation de $TS_c$ avec d'autres niveaux de complexité&$[0,1] \in \mathbb{R}$\\ $ESCBR_c$&v&Prédiction de la note pour un niveau de complexité $c$&$\mathbb{R}_+$\\ $p_c$&f&Fonction de densité de probabilité pour le niveau de complexité $c$&$\mathbb{R}_+$\\ $D_{JS}$&f&Divergence de Jensen-Shannon&$[0,1] \in \mathbb{R}$\\ \end{tabular} - \caption{Paramètres (p), variables (v) et fonctions (f) du modèle proposé et les métriques utilisées} + \caption{Paramètres (p), variables (v) et fonctions (f) du modèle proposé et métriques utilisées} \label{tabvp} \end{table} -L'intégration se fait en trois étapes. Tout d'abord, il est nécessaire d'avoir des valeurs aléatoires pour chaque niveau de complexité $c$ en utilisant les distributions de probabilité générées avec le modèle TS (équation \ref{IntEq1_}), une fois que toutes les valeurs de probabilité correspondant à tous les niveaux de complexité ont été obtenues, la normalisation de toutes ces valeurs est calculée comme indiqué dans l'équation \ref{IntEq2_}. Les valeurs de normalisation servent de paramètres de priorité pour les prédictions effectuées par le modèle ESCBR-SMA, comme le montre l'équation \ref{eqMixModels_}. +Comme exprimé précedemment, le processus de recommandation se fait en trois étapes. Tout d'abord, il est nécessaire d'avoir des valeurs aléatoires pour chaque niveau de complexité $c$ en utilisant les distributions de probabilité générées avec le modèle TS (équation \ref{IntEq1_}). Une fois que toutes les valeurs de probabilité correspondant à tous les niveaux de complexité ont été obtenues, la normalisation de toutes ces valeurs est calculée selon l'équation \ref{IntEq2_}. Les valeurs de normalisation servent de paramètres de priorité pour les prédictions effectuées par ESCBR-SMA (équation \ref{eqMixModels_}). La recommandation finalement proposée est celle dont la valeur est la plus élevée. \begin{equation} TS_c=rand(Beta(\alpha_c, \beta_c)) @@ -549,41 +554,42 @@ L'intégration se fait en trois étapes. Tout d'abord, il est nécessaire d'avoi \label{eqMixModels_} \end{equation} -Avec les valeurs finales calculées pour chaque niveau de complexité, le niveau de complexité qui a la valeur la plus élevée est proposé comme recommandation finale (équation \ref{eqMixModels_}). Le niveau de complexité qui a la valeur la plus élevée est proposé comme recommandation finale (équation \ref{eqMixModels_}). - \subsection{Résultats et Discussion} -Cette section présente la description de la base de données et les paramètres utilisés pour mesurer la précision, la performance et la progression des connaissances, les résultats individuels du modèle de recommandation, le modèle de prédiction ainsi que leur intégration finale pour améliorer la personnalisation du système d'AI-VT. Cette section présente les résultats individuels du modèle de recommandation, le modèle de prédiction ainsi que leur intégration finale pour améliorer la personnalisation du système d'AI-VT. - -La base de données a été générée avec la distribution logit-normale pour simuler les notes des apprenants, car il s'agit d'un bon modèle pour se rapprocher du monde réel. La base de données représente les notes et les temps de réponse d'un apprenant pour cinq questions à chaque niveau de complexité. +Le principal inconvénient posé par la validation d'un tel système « en situation réelle » est la difficulté à collecter des données et à évaluer des systèmes différents dans des conditions strictement similaires. Cette difficulté est accentuée dans les contextes d'apprentissage autorégulés, puisque les apprenants peuvent quitter la plateforme d'apprentissage à tout moment rendant ainsi les données incomplètes \cite{badier:hal-04092828}. -Le principal inconvénient de ce système de validation « en situation réelle » est la difficulté de la collecte des données. Cette difficulté est accentuée dans les contextes d'apprentissage autorégulé, puisque les apprenants peuvent quitter la plateforme d'apprentissage à tout moment et que les données peuvent être incomplètes \cite{badier:hal-04092828}. +Pour cette raison, les différentes approches proposées ont été testées sur des données générées : les notes et les temps de réponse de 1000 apprenants fictifs et cinq questions par niveau de complexité. Les notes des apprenants ont été créés en suivant la loi de distribution logit-normale que nous avons jugée proche de la réalité de la progression d'un apprentissage. -Quatre tests différents ont été effectués pour démontrer les avantages de l'intégration de la TS et de la RàPC dans les EIAH. Le premier est l'utilisation du RàPC pour la régression avec une base de données d'apprenants afin de démontrer la capacité du modèle à prédire les notes à différents niveaux de complexité, le deuxième est l'évaluation de la progression des connaissances avec TS afin de déterminer l'efficacité du modèle dans la recommandation personnalisée pour chaque apprenant, La troisième est la comparaison entre les modèles de recommandation BKT et TS afin d'établir la performance du modèle TS en utilisant BKT comme modèle de base et enfin, la comparaison entre TS seul et TS avec ESCBR-SMA pour démontrer que l'intégration entre les deux modèles améliore l'ensemble du système de recommandation dans AI-VT. +Quatre séries de tests ont été effectués. La première série a été menée sur le système AI-VT intégrant le système de RàPC pour la régression afin de démontrer la capacité du modèle à prédire les notes à différents niveaux de complexité. +La deuxième série de tests a évalué la progression des connaissances avec TS afin d'analyser la capacité du module à proposer des recommandations personnalisées. Lors de la troisième série de tests, nous avons comparé les modèles de recommandation BKT et TS. Enfin, lors de la quatrième série de tests, nous avons comparé TS seul et TS avec ESCBR-SMA. -\subsubsection{Régression dans la base de données des apprenants avec ESCBR-SMA} +\subsubsection{Régression avec ESCBR-SMA pour l'aide à l'apprentissage humain} -Le SMA utilise le raisonnement bayésien, ce qui permet aux agents d'apprendre des données et des interactions au cours de l'exécution et de l'exploration. +Le SMA que nous avons implémenté utilise un raisonnement bayésien, ce qui permet aux agents d'apprendre des données et d'intéragir au cours de l'exécution et de l'exploration. -L'algorithme utilise une fonction noyau pour obtenir la meilleure approximation de la solution du nouveau problème, le problème de l'obtention de la meilleure solution est un problème NP, car la formulation est similaire au problème de Fermat-Weber à N dimensions. Le problème de l'obtention de la meilleure solution est un problème NP, car la formulation est similaire au problème de Fermat-Weber à N dimensions \cite{doi:10.1137/23M1592420}. +ESCBR-SMA utilise une fonction noyau pour obtenir la meilleure approximation de la solution du problème cible. Dans notre cas, l'obtention de la meilleure solution est un problème NP\colorbox{yellow}{NP-complet ? / NP-Difficile ?} car la formulation est similaire au problème de Fermat-Weber à $N$ dimensions \cite{doi:10.1137/23M1592420}. -La première série de tests est définie sous la forme de différents scénarios, comme le montre le tableau \ref{tab:scenarios}. Dans le scénario 1 (E1), il s'agit de prédire la note d'un apprenant au premier niveau de complexité, après 3 questions. Le scénario 2 (E2) contient les notes de 8 questions et l'objectif est de prédire la note de 9 questions dans le même niveau de complexité. Le scénario 3 (E3) contient les données permettant de prédire le passage à un niveau de complexité supérieur après 4 questions. Le scénario 4 (E4) contient 4 questions et la prédiction de 2 notes dans un niveau de complexité supérieur. +Les différents scénarios du tableau \ref{tab:scenarios} ont été considérés dans un premier temps. Dans le scénario $E_1$, il s'agit de prédire la note d'un apprenant au premier niveau de complexité, après 3 questions. Le scénario $E_2$ considère les notes de 8 questions et l'objectif est de prédire la note de 9 questions dans le même niveau de complexité. Le scénario $E_3$ considère les données permettant de prédire le passage à un niveau de complexité supérieur après 4 questions. +\colorbox{yellow}{Je ne comprends pas ce scénario E3. La reformulation suivante est-elle correcte ? }\\ +\colorbox{yellow}{Dans $E_3$, l'objectif est d'interpoler le fait qu'il faille passer ou non }\\ +\colorbox{yellow}{à un niveau de complexité supérieur après 4 questions.}\\ +Le scénario $E_4$ considère 4 questions et le système doit interpoler 2 notes dans un niveau de complexité supérieur. \begin{table}[!ht] \centering \begin{tabular}{ccc} - Scenario&Features&Output Dimension\\ + Scenario&Caractéristiques du problème&Dimension de la solution\\ \hline - E1 & 5 & 1\\ - E2 & 15& 1\\ - E3 & 9 & 1\\ - E4 & 9 & 2\\ + $E_1$ & 5 & 1\\ + $E_2$ & 15& 1\\ + $E_3$ & 9 & 1\\ + $E_4$ & 9 & 2\\ \end{tabular} \caption{Description des scénarios} \label{tab:scenarios} \end{table} -Le modèle a été comparé à neuf algorithmes bien connus utilisés pour résoudre les problèmes de régression. La liste des algorithmes est présentée dans le tableau \ref{tabAlgs}. +ESCBR-SMA a été comparé aux neuf outils classiquement utilisés pour résoudre la régression consignés dans le tableau \ref{tabAlgs} et selon l'erreur quadratique moyenne (RMSE - \textit{Root Mean Squared Error}), l'erreur médiane absolue (MedAE - \textit{Median Absolute Error}) et l'erreur moyenne absolue (MAE - \textit{Mean Absolute Error}). \begin{table}[!ht] \centering @@ -601,7 +607,7 @@ A5&Multi Layer Perceptron&A10&Proposed Ensemble Stacking RàPC\\ \label{tabAlgs} \end{table} -Les algorithmes ont été évalués à l'aide de trois mesures (Root Mean Squared Error - RMSE, Median Absolute Error - MedAE, Mean Absolute Error - MAE), dont les résultats figurent dans le tableau \ref{tab:results}, où l'on constate que l'algorithme proposé obtient de meilleurs résultats que les autres algorithmes avec lesquels il a été comparé, sauf dans les cas E1(MedAE), E1(MAE), E2(MedAE), E2(MAE), E3 et E4(MedAE) où les meilleurs résultats sont obtenus par l'algorithme A9, mais l'algorithme proposé occupe la deuxième place dans ces cas avec des résultats très proches. Il est possible de conclure que l'intégration de plusieurs algorithmes de recherche et de génération de solutions dans le cadre des paradigmes RàPC et Stacking est efficace dans le cas de l'application à la prédiction des notes des apprenants. +Le tableau \ref{tab:results} présente les résultats obtenus par les 10 algorithmes sur les quatre scénarios. Ces résultats montrent qu'ESCBR-SMA (A10) et le \textit{Gradient Boosting} (A9) obtiennent toujours les deux meilleurs résultats. Si l'on considère uniquement la RMSE, ESCBR-SMA occupe toujours la première place sauf pour $E_3$ où il est deuxième. Inversement, en considérant l'erreur médiane absolue ou l'erreur moyenne absolue, A10 se classe juste après A9. ESCBR-SMA et le \textit{Gradient Boosting} sont donc efficaces pour interpoler les notes des apprenants. \begin{table}[!ht] \centering @@ -612,29 +618,29 @@ Les algorithmes ont été évalués à l'aide de trois mesures (Root Mean Square & A1&A2&A3&A4&A5&A6&A7&A8&A9&A10\\ \textbf{Scenario (Metrique)}\\ \hline -E1 (RMSE)&0.625&0.565&0.741&0.56&0.606&0.626&0.626&0.681&0.541&\textbf{0.54}\\ - E1 (MedAE) & 0.387&0.35&0.46&0.338&0.384&0.387&0.387&0.453&\textbf{0.327}&0.347\\ - E1 (MAE) &0.485&0.436&0.572&0.429&0.47&0.485&0.485&0.544&\textbf{0.414}&0.417\\ + $E_1$ (RMSE)&0.625&0.565&0.741&0.56&0.606&0.626&0.626&0.681&0.541&\textbf{0.54}\\ + $E_1$ (MedAE) & 0.387&0.35&0.46&0.338&0.384&0.387&0.387&0.453&\textbf{0.327}&0.347\\ + $E_1$ (MAE) &0.485&0.436&0.572&0.429&0.47&0.485&0.485&0.544&\textbf{0.414}&0.417\\ \hline - E2 (RMSE)& 0.562&0.588&0.78&0.571&0.61&0.562&0.562&0.622&0.557&\textbf{0.556}\\ - E2 (MedAE)&0.351&0.357&0.464&0.344&0.398&0.351&0.351&0.415&\textbf{0.334}&0.346\\ - E2 (MAE)&0.433&0.448&0.591&0.437&0.478&0.433&0.433&0.495&\textbf{0.422}&0.429\\ + $E_2$ (RMSE)& 0.562&0.588&0.78&0.571&0.61&0.562&0.562&0.622&0.557&\textbf{0.556}\\ + $E_2$ (MedAE)&0.351&0.357&0.464&0.344&0.398&0.351&0.351&0.415&\textbf{0.334}&0.346\\ + $E_2$ (MAE)&0.433&0.448&0.591&0.437&0.478&0.433&0.433&0.495&\textbf{0.422}&0.429\\ \hline - E3 (RMSE)&0.591&0.59&0.79&0.57&0.632&0.591&0.591&0.644&\textbf{0.555}&0.558\\ - E3 (MedAE)&0.367&0.362&0.474&0.358&0.404&0.367&0.367&0.433&\textbf{0.336}&0.349\\ - E3 (MAE)&0.453&0.45&0.598&0.441&0.49&0.453&0.453&0.512&\textbf{0.427}&0.43\\ + $E_3$ (RMSE)&0.591&0.59&0.79&0.57&0.632&0.591&0.591&0.644&\textbf{0.555}&0.558\\ + $E_3$ (MedAE)&0.367&0.362&0.474&0.358&0.404&0.367&0.367&0.433&\textbf{0.336}&0.349\\ + $E_3$ (MAE)&0.453&0.45&0.598&0.441&0.49&0.453&0.453&0.512&\textbf{0.427}&0.43\\ \hline - E4 (RMSE)&0.591&0.589&0.785&0.568&0.613&0.591&0.591&0.644&0.554&\textbf{0.549}\\ - E4 (MedAE)&0.367&0.362&0.465&0.57&0.375&0.367&0.367&0.433&\textbf{0.336}&0.343\\ - E4 (MAE)&0.453&0.45&0.598&0.438&0.466&0.453&0.453&0.512&0.426&\textbf{0.417}\\ + $E_4$ (RMSE)&0.591&0.589&0.785&0.568&0.613&0.591&0.591&0.644&0.554&\textbf{0.549}\\ + $E_4$ (MedAE)&0.367&0.362&0.465&0.57&0.375&0.367&0.367&0.433&\textbf{0.336}&0.343\\ + $E_4$ (MAE)&0.453&0.45&0.598&0.438&0.466&0.453&0.453&0.512&0.426&\textbf{0.417}\\ \end{tabular} - \caption{Résultats de la régression pour la base de données des apprenants avec 100 exécutions} + \caption{Erreurs moyennes et médianes des interpolations des 10 algorithmes sélectionnés sur les 4 scénarios considérés et obtenues après 100 exécutions.} \label{tab:results} \end{table} \subsubsection{Progression des connaissances} -Le modèle de recommandation TS est fondé sur le paradigme bayésien, ce qui est très utile lorsque les données sont limitées et l'incertitude forte. Afin de quantifier la connaissance et de voir sa progression dans le temps avec TS, la divergence de Jensen-Shannon avec la famille de distribution Beta en $t$ et $t-1$ fois a été utilisée comme second test. L'équation \ref{eqprog1} décrit formellement le calcul à effectuer avec les distributions de probabilité en un temps $t$ pour un niveau de complexité $c$, en utilisant la définition $m$ (équation \ref{eqprog2}). +Le modèle de recommandation TS est fondé sur le paradigme bayésien le rendant ainsi particulièrement adapté au problèmes liés à la limitation de la quantité de données et à une incertitude forte. Afin de quantifier la connaissance et de voir sa progression dans le temps avec TS, la divergence de Jensen-Shannon avec la famille de distribution Beta en $t$ et $t-1$ a été mesurée. L'équation \ref{eqprog1} décrit formellement le calcul à effectuer avec les distributions de probabilité en un temps $t$ pour un niveau de complexité $c$, en utilisant la définition $m$ (équation \ref{eqprog2}). %\begin{equation} \begin{multline} @@ -654,7 +660,7 @@ Le modèle de recommandation TS est fondé sur le paradigme bayésien, ce qui es \label{eqprog2} \end{multline} -La progression totale des connaissances en $t$ est la somme des différences entre $t$ et $t-1$ pour tous les $c$ niveaux de complexité calculés avec la divergence de Jensen-Shannon (équation \ref{eqTEK}). en utilisant l'évaluation de la progression de la variabilité (équation \ref{eqVarP}). +La progression du nombre total de connaissances en $t$ est la somme des différences entre $t$ et $t-1$ pour tous les $c$ niveaux de complexité calculés avec la divergence de Jensen-Shannon (équation \ref{eqTEK}). Pour ce faire, nous évaluons la progression de la variabilité données par équation \ref{eqVarP}. \begin{equation} vk_{t,c}=\begin{cases} @@ -680,37 +686,37 @@ La progression totale des connaissances en $t$ est la somme des différences ent \label{fig:evolution} \end{figure} -La figure \ref{fig:evolution} montre la progression cumulative des connaissances sur les quinze questions d'une seule séance de formation. Entre la première et la dernière question de la même séance, tous les apprenants ont statistiquement augmenté leur niveau de connaissance puisque la moyenne a augmenté, la variabilité augmente à partir de la première question jusqu'à la question neuf, où le système a acquis plus d'informations sur les apprenants, à partir de là la variabilité diminue et la moyenne augmente. +La figure \ref{fig:evolution} montre la progression cumulée des connaissances sur les quinze questions d'une même séance d'entrainement. L'augmentation de la moyenne du niveau de connaissance entre la première et la dernière question de la même séance montre que tous les apprenants ont statistiquement augmenté leur niveau de connaissance. La variabilité augmente à partir de la première question jusqu'à la question neuf, où le système a acquis plus d'informations sur les apprenants. À ce stade, la variabilité diminue et la moyenne augmente. \subsubsection{Comparaison entre TS et BKT} -L'évolution du système de recommandation TS est testée en comparaison avec BKT, la figure \ref{fig:EvGrades} montre l'évolution des notes des apprenants en fonction du nombre de questions auxquelles ils répondent dans la même séance. Dans ce cas, le modèle TS génère moins de variabilité que BKT, mais si est faite la comparaison des moyennes générées par chaque question, l'évolution est très similaire. La figure \ref{fig:EvGrades} montre l'évolution des notes des apprenants en fonction du nombre de questions auxquelles ils répondent au cours de la même séance. +La figure \ref{fig:EvGrades} permet de comparer la recommandation fondé sur l'échantillonnage de Thompson et celle fondé sur BKT. Cette figure montre l'évolution des notes des apprenants en fonction du nombre de questions auxquelles ils répondent dans la même séance. Dans ce cas, le TS génère moins de variabilité que BKT, mais les évolutions induites par les deux systèmes restent globalement très similaires. \begin{figure}[!ht] \centering \includegraphics[width=\textwidth]{Figures/GradesEv.jpg} - \caption{Comparaison de l'évolution des notes entre les algorithmes BKT et TS} + \caption{Comparaison de l'évolution des notes entre les systèmes fondés sur TS et BKT.} \label{fig:EvGrades} \end{figure} -Mais, si les résultats obtenus sont comparés par rapport à l'évolution du niveau de complexité recommandé (figure \ref{fig:EvCL}), le modèle TS fait évoluer le niveau de complexité des apprenants, alors que le modèle BKT a tendance à laisser les apprenants au même niveau de complexité, c'est-à-dire qu'avec le modèle BKT, il est difficile d'apprendre de nouveaux sujets ou des concepts plus complexes au sein du même domaine. En examinant les résultats des deux figures (figures \ref{fig:EvGrades} et \ref{fig:EvCL}) et en établissant des comparaisons, le modèle TS permet de progresser en moyenne dans la valeur des notes et facilite l'évolution des niveaux de complexité.\N- Le modèle TS permet de progresser en moyenne dans la valeur des notes et facilite l'évolution des niveaux de complexité. Le modèle TS permet de progresser en moyenne dans la valeur des notes et facilite l'évolution des niveaux de complexité. +Toutefois, si l'on considère l'évolution du niveau de complexité recommandé (figure \ref{fig:EvCL}), TS fait évoluer le niveau de complexité des apprenants, alors que BKT a tendance à laisser les apprenants au même niveau de complexité. Autrement dit, avec BKT, il est difficile d'apprendre de nouveaux sujets ou des concepts plus complexes au sein du même domaine. En comparant les résultats des deux figures (figures \ref{fig:EvGrades} et \ref{fig:EvCL}), TS permet de faire progresser la moyenne des notes et facilite l'évolution des niveaux de complexité. \begin{figure}[!ht] \centering \includegraphics[width=\textwidth]{Figures/LevelsEv.jpg} - \caption{Comparaison de l'évolution des niveaux entre les algorithmes BKT et TS} + \caption{Comparaison de l'évolution des niveaux entre les systèmes de recommandation fondés sur BKT et TS} \label{fig:EvCL} \end{figure} \subsubsection{Système de recommandation avec ESCBR-SMA} -Le troisième test est l'intégration entre deux modèles. Cette combinaison est faite fondée sur le paradoxe de Stein, en essayant de combiner des observations qui ne sont pas directement liées l'une à l'autre, c'est-à-dire en utilisant l'information individuelle (Thomson sampling recommender) et le filtre collaboratif (Case-base reasoning prediction) pour améliorer la personnalisation. Le test est une comparaison entre le système de recommandation TS et le système de recommandation TS avec la prédiction ESCBR-SMA afin de déterminer si l'intégration des deux modèles permet d'améliorer l'évolution du processus d'apprentissage proposé par le système AI-VT. +Le troisième étape est l'association deux modèles afin de combiner des observations qui ne sont pas directement liées l'une à l'autre, c'est-à-dire en utilisant l'information individuelle (recommandation avec Thomson) et le filtre collaboratif (interpolation avec ESCBR-SMA). Cette partie présente une comparaison entre le système de recommandation TS et le système de recommandation TS intégrée à la prédiction ESCBR-SMA. -La comparaison est effectuée après la question 6 pour tous les apprenants, car il est nécessaire de disposer d'informations préalables pour utiliser l'algorithme ESCBR-SMA et prédire les notes dans tous les niveaux de complexité pour la question suivante. +Les résultats proposés sont comparés après réponse des apprenants à six questions car il est nécessaire de disposer d'informations préalables pour utiliser l'algorithme ESCBR-SMA et prédire les notes dans tous les niveaux de complexité à la question suivante. \subsubsection{Progression des connaissances TS vs TS et ESCBR-SMA} -Pour établir la différence entre le modèle de recommandation TS et le même modèle associé à la prédiction fondée sur le raisonnement à partir de cas ESCBR-SMA, le quatrième test est défini en utilisant la métrique de Jensen-Shannon, mais dans ce cas la comparaison est faite entre les différents modèles (TS, TS-ESCBR) sur le même niveau de complexité dans le même temps $t$. La définition formelle de la métrique est exprimée par les équations \ref{eqjs4} et \ref{eqjs5}. +Pour établir la différence entre le système de recommandation TS et ce même système associé à la prédiction fondée sur le raisonnement à partir de cas ESCBR-SMA, nous utiliserons la métrique de Jensen-Shannon (définie suivant les équations \ref{eqjs4} et \ref{eqjs5}). Dans ce cas la comparaison est faite sur le même niveau de complexité et dans le même temps $t$. \begin{multline} k_{t,c}=\frac{1}{2} @@ -730,40 +736,52 @@ log \left(\frac{p_c(\alpha_{p1,t},\beta_{p1,t},x)}{m(p_c(\alpha_{p1,t},\beta_{p1 %\end{equation} \end{multline} -La comparaison entre l'évolution des connaissances présente une bifurcation après la septième question, et l'intégration de l'échantillonnage de Thompson et du raisonnement à partir de cas permet d'améliorer l'évolution des connaissances par rapport au seul modèle d'échantillonnage de Thompson (Figure \ref{fig_cmp2}). Pour toutes les questions de la même séance, en moyenne, la progression est supérieure par rapport à l'utilisation du seul modèle d'échantillonnage de Thompson comme modèle de recommandation. +La figure \ref{fig_cmp2} présente une visualisation normalisée des différences de progression entre TS et TS intégré à ESCBR-SMA. \\ +\colorbox{yellow}{Il faut expliquer les axes, ce que représentent les points rouge, les carrés bleu de cette figure}\\ +\colorbox{yellow}{Il manque une légende sur cette figure. Que représentent les axes ? Quelles sont les unités ?}\\ +Cette figure montre une bifurcation après la septième question.\\ +\colorbox{yellow}{La suite est à reformuler ensemble, je ne vois pas en quoi TS-ESCBR-SMA est meilleur ou supérieur} + , et l'intégration de l'échantillonnage de Thompson et du raisonnement à partir de cas permet d'améliorer l'évolution des connaissances par rapport au seul modèle d'échantillonnage de Thompson (Figure \ref{fig_cmp2}). Pour toutes les questions de la même séance, en moyenne, la progression est supérieure par rapport à l'utilisation du seul modèle d'échantillonnage de Thompson comme modèle de recommandation. \begin{figure}[!h] \centering \includegraphics[width=\textwidth]{Figures/TS-ESCBR.png} - \caption{Normalisation de la différence de progression entre l'échantillonnage de Thompson et l'échantillonnage de Thompson avec ESCBR pour 1000 apprenants} + \caption{Normalisation de la différence de progression entre l'échantillonnage de Thompson et l'échantillonnage de Thompson avec ESCBR-SMA pour 1000 apprenants} \label{fig_cmp2} \end{figure} \subsection{Conclusion} -Cette partie a présenté un modèle intégré entre deux modèles développés précédemment, un système de recommandation fondé sur l'algorithme d'échantillonnage de Thompson et un modèle de régression d'ensemble fondé sur le raisonnement par cas. Le modèle intégré est appliqué au système AI-VT, les résultats montrent en effet que l'intégration permet d'améliorer la performance des deux modèles utilisés séparément, en outre il montre de meilleurs résultats dans la révision/adaptation des étapes de solutions pour chaque apprenant, en fonction des métriques utilisées et des tests définis, donnant une meilleure personnalisation du système et facilitant l'acquisition de connaissances. -%Le modèle intégré est appliqué à un ITS appelé AI-VT.\\ +Cette partie montre que l'intégration du module d'échantillonnage de Thompson au module de régression ESCBR-SMA est profitable à l'EIAH AI-VT dans la mesure où leur utilisation conjointe permet de réviser une séance d'entrainement en proposant des exercices de difficulté adaptée à l'apprenant en fonction des notes obtenues aux réponses précédentes. -Les avantages du modèle proposé sont les suivants : i) il permet de générer des recommandations personnalisées pour chaque apprenant avec relativement peu de données historiques, ii) l'intégration de plusieurs perspectives de solution sur une même problématique, selon le principe du paradoxe de Stein, permet de limiter le risque d'être confronté au paradoxe de Simpson, iii) les deux modèles se complètent mutuellement en améliorant les résultats finaux d'une manière généralisée. Le modèle proposé a été conçu pour être utilisé dans le cadre d'un projet de recherche et de développement en cours.\\\\ +Le système proposé permet de générer des recommandations personnalisées pour chaque apprenant avec relativement peu de données historiques. Il permet de réviser une séance en se fondant sur des informations générales (progression standardisée) et locales (niveau acquis par l'apprenant). +%les deux modèles se complètent mutuellement en améliorant les résultats finaux d'une manière généralisée. Le modèle proposé a été conçu pour être utilisé dans le cadre d'un projet de recherche et de développement en cours.\\\\ -\section{ESCBR-SMA, Échantillonnage de Thompson et Processus de Hawkes} +Toutefois, un apprentissage n'est pas linéaire et il est nécessaire de proposer des rappels de cours ou de notions à l'apprenant. La partie suivante de ce chapitre tente d'apporter une solution à cette observation pouvant limiter les performances d'AI-VT en introduisant le processus de Hawkes. + +\section{ESCBR-SMA, échantillonnage de Thompson et processus de Hawkes} \sectionmark{ESCBR-SMA, TS et Hawkes} +Dans la partie précédente, nous avons proposé un système capable de permettre à l'apprenant de suivre une progression relativement linéaire. Toutefois, certaines notions sont parfois oubliées par les apprenants et des rappels doivent être proposés afin de mieux consolider les connaissances. C'est la raison pour laquelle nous nous intéressons dans cette partie à l'intégration d'un processus de Hawkes qui permet de simuler une courbe d'oubli. + \subsection{Modèle Proposé} + L'algorithme proposé est une intégration de la recommandation stochastique (fondée sur l'échantillonnage de Thompson), du raisonnement à partir de cas (ESCBR-SMA) et du processus de Hawkes. 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 consolidation des deux résultats globaux permet d'atténuer l'effet du paradoxe de Simpson \cite{10.1145/3578337.3605122}. +L'idée de cette unification est d'obtenir des informations d'un point de vue local où une recommandation est obtenue en se fondant sur les informations des apprenants individuels (rôle de l'échantillonnage de Thompson), la prédiction globale fondée sur les expériences acquises précédemment par le système (le système de RàPC fournit des exercices ayant permis par le passé à d'autres apprenants aux profiles similaires d'acquérir les connaissances visées), et le processus d'apprentissage dynamique avec le processus de Hawkes. + +La figure \ref{fig:Amodel} montre de quelle manière sont organisés ces différents modules les uns par rapport aux autres. TS et ESCBR-SMA sont exécutés en parallèle et indépendamment avec les informations extraites de la même base de connaissances. Une fois que les résultats de chaque modules sont obtenus, ils 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. Une dernière étape permet de consolider et d'atténuer l'effet du paradoxe de Simpson \cite{10.1145/3578337.3605122}. \begin{figure}[!ht] \centering \includegraphics[width=0.6\linewidth]{Figures/ModelHawkes.png} - \caption{Architecture de modèle proposé avec Hawkes} + \caption{Organisation des modules TS, ESCBR-SMA et processus de Hawkes.} \label{fig:Amodel} \end{figure} -La première étape est l'adaptation avec l'échantillonnage de Thompson et la prédiction de l'ECBR-SMA, et enfin la prise de décision pour l'envoi à l'apprenant. Le système de recommandation obtient une valeur de probabilité pour tous les niveaux de complexité pour l'apprenant et l'ECBR-SMA évalue la proposition avec une prédiction pour chaque niveau de complexité. Le tableau \ref{tabvp} présente les variables et les paramètres de l'algorithme proposé et les mesures employées. +La première étape est l'adaptation avec l'échantillonnage de Thompson et la prédiction de l'ECBR-SMA. Celle-ci est suivie par la prise de décision pour l'envoi à l'apprenant. Le système de recommandation obtient une valeur de probabilité pour tous les niveaux de complexité pour l'apprenant et l'ECBR-SMA évalue la proposition avec une prédiction pour chaque niveau de complexité. Le tableau \ref{tabvp} présente les variables et les paramètres du module proposé et les mesures employées. -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. +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}), afin de simuler la courbe d'oubli dans l'évolution de la distribution de probabilité Beta. \begin{equation} \lambda(t)=\mu(t)+\sum_{t_i +<./Figures/dataset.png, id=1294, 15.13687pt x 8.08058pt> File: ./Figures/dataset.png Graphic file (type png) -Package pdftex.def Info: ./Figures/dataset.png used on input line 152. +Package pdftex.def Info: ./Figures/dataset.png used on input line 144. (pdftex.def) Requested size: 427.43153pt x 228.35583pt. 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