From 8a2a795441407eeee04d84af207ededd2f224f89 Mon Sep 17 00:00:00 2001 From: Louis Lacoste Date: Sun, 25 Aug 2024 20:05:49 +0200 Subject: [PATCH] =?UTF-8?q?soutenance=20:=20adaptation=20de=20la=20pr?= =?UTF-8?q?=C3=A9sentation=20des=20stagiaires?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- soutenance/annexe.tex | 61 ++++++++ soutenance/principal.tex | 286 ++++++++++++++++++++++++++++++++++++++ soutenance/soutenance.pdf | Bin 0 -> 446495 bytes soutenance/soutenance.tex | 119 ++++++++++++++++ 4 files changed, 466 insertions(+) create mode 100644 soutenance/annexe.tex create mode 100644 soutenance/principal.tex create mode 100644 soutenance/soutenance.pdf create mode 100644 soutenance/soutenance.tex diff --git a/soutenance/annexe.tex b/soutenance/annexe.tex new file mode 100644 index 0000000..207fb2f --- /dev/null +++ b/soutenance/annexe.tex @@ -0,0 +1,61 @@ +\section{Notes supplémentaires} +\printappxnotes + + +\section{Sélection de modèle} +\begin{frame} + \frametitle{Choix de $(Q_1,Q_2)$ - Approche gloutonne} + \begin{columns} + \begin{column}{0.5\linewidth} + \begin{tikzpicture} + \input{../tikz/greedy-exploration.tex} + \end{tikzpicture} + \end{column} + \begin{column}{0.35\linewidth} + \begin{itemize} + \item Modèle initialisé~:\\ + \begin{tikzpicture} + \draw[fill=gray, draw=gray] circle [radius=0.225cm]; + \end{tikzpicture} + \onslide<2->{ + \item Modèle après \emph{split}~: + \begin{tikzpicture} + \draw[fill=blueind, draw=blueind] circle [radius=0.225cm]; + \end{tikzpicture} + \item Modèle maximisant le critère~:\\ + \begin{tikzpicture} + \draw[fill=white, draw=green, very thick] circle [radius=0.225cm]; + \end{tikzpicture} + } + \onslide<3->{ + \item Modèle après \emph{merge}~: + \begin{tikzpicture} + \draw[fill=red, draw=red] circle [radius=0.225cm]; + \end{tikzpicture} + } + \end{itemize} + \end{column} + \end{columns} +\end{frame} +\begin{frame} + \frametitle{Choix de $(Q_1,Q_2)$ - Fenêtre glissante} + \begin{columns} + \begin{column}{0.6\textwidth} + \begin{figure} + \input{../tikz/moving-window.tex} + \caption{Fenêtre glissante} + \end{figure} + \end{column} + \begin{column}{0.4\textwidth} + \only<3>{\begin{block}{} + Initialisation du modèle si nécessaire + \end{block}} + \only<9>{\begin{block}{} + Localisation du nouveau mode + \end{block}} + \only<10>{\begin{block}{} + Déplacement sur le nouveau mode puis itération + \end{block}} + \end{column} + \end{columns} +\end{frame} \ No newline at end of file diff --git a/soutenance/principal.tex b/soutenance/principal.tex new file mode 100644 index 0000000..c66d2de --- /dev/null +++ b/soutenance/principal.tex @@ -0,0 +1,286 @@ +\section{Contexte du modèle} +\label{sec:contexte-du-modele} + +\begin{frame} + \begin{columns} + \begin{column}{0.5\textwidth} + \begin{block}{Contexte écologique} + \begin{itemize} + \small + \item Nombreux réseaux disponibles pour + interactions similaires. + \item Suivi biodiversité, robustesse et risque + d'effondrement \dots + \end{itemize} + \begin{columns} + \begin{column}{0.5\textwidth} + \begin{figure}[ht] + \centering + \begin{tikzpicture}[scale=.45,rotate=270] + \input{../tikz/plantpollinatornetwork.tex} + \end{tikzpicture} + \caption{Exemple d'un réseau plantes-pollinisateurs} + \label{fig:plantes-pollin} + \end{figure} + \end{column} + \begin{column}{0.4\textwidth} + \centering + \begin{align*} + \begin{pmatrix} + 1 & 0 & 1 \\ + 1 & 0 & 0 \\ + 1 & 0 & 0 \\ + 1 & 1 & 0 + \end{pmatrix} + \end{align*} + \footnotesize + Matrice d'adjacence associée + \end{column} + \end{columns} + \end{block} + + \end{column} + \onslide<2>{ + \begin{column}{0.45\textwidth} + \begin{block}{Contexte mathématique} + Pour un unique réseau~: variables latentes, + \emph{embedding}, \dots + + Motivations pour proposer des méthodes adaptées aux collections + de réseaux~: + \begin{itemize} + \item Espèces différentes, rôles analogues. + \item Transfert d'informations grands vers petits réseaux. + \item Regrouper les réseaux selon leur similarité (\emph{clustering} + de réseaux). + \end{itemize} + \end{block} + \end{column} + } + \end{columns} +\end{frame} + +\begin{frame} + \addtocounter{footnote}{1} + \frametitle{Latent Block Model (LBM\footnotemark[\thefootnote])} + %DONE remplacer i \in bullet par Zi = \bullet + Proposé par~\cite{govaertEMAlgorithmBlock2005}. + \begin{columns} + \begin{column}{0.40\linewidth} + \begin{figure}[H] + \center + \begin{tikzpicture}[scale=0.35] + \input{../tikz/lbm.tex} + \end{tikzpicture} + \caption{Exemple de LBM\footnotemark[\thefootnote]} + \label{fig:LBMvisu} + \end{figure} + \end{column} + \begin{column}{0.51\linewidth} + Pour \begin{itemize} + \item $Q_1 = |\{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}|$ blocs fixés en ligne + \item $Q_2 = |\{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{peach}\bullet}\}|$ blocs fixés en colonne + \end{itemize} + \begin{block}{Paramètres} + \begin{itemize} + \item $\pi_{\bullet} = \mathbb{P}(Z_i = \bullet)$ en ligne et $\rho_{\bullet} = \mathbb{P}(W_j = \bullet)$ en colonne + \item $\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} = \mathbb{P}(X_{ij} = 1 | Z_i = {\color{blueind}\bullet}, W_j = {\color{burntorange}\bullet})$ + \end{itemize} + \end{block} + \end{column} + \end{columns} + + \footnotetext[\thefootnote]{Que j'appellerai par la suite BiSBM} + +\end{frame} + +\section{Modèle de collection de réseaux bipartites} +\label{sec:extension-de-colsbm-aux-reseaux-bipartites} +\begin{frame} + \frametitle{Collections bipartites} + \begin{tikzpicture}[scale=0.33] + \input{../tikz/collbm-iid.tex} + \end{tikzpicture} + + \begin{itemize} + \item $Q_1 = |\{{\color{blueind}\bullet},{\color{cyanind}\bullet},{\color{electricblue}\bullet}\}|$ blocs fixés en ligne + \item $Q_2 = |\{{\color{burntorange}\bullet},{\color{goldenyellow}\bullet},{\color{peach}\bullet}\}|$ blocs fixés en colonne + \end{itemize} + \begin{block}{Paramètres} + \begin{itemize} + \item $\pi_{\bullet} = \mathbb{P}(Z_i =\bullet)$ en ligne et $\rho_{\bullet} = \mathbb{P}(W_j = \bullet)$ en colonne + \item $\alpha_{{\color{blueind}\bullet}{\color{burntorange}\bullet}} = \mathbb{P}(X_{ij} = 1 | Z_i = {\color{blueind}\bullet}, W_j = {\color{burntorange}\bullet})$ + \end{itemize} + \end{block} +\end{frame} + +\begin{frame} + \frametitle{Différents modèles} + \only<1>{ + \begin{tikzpicture}[scale=0.33] + \input{../tikz/collbm-iid.tex} + \end{tikzpicture} + \begin{block}{\emph{iid-colBiSBM}} + $\bm{\pi} = (\pi_1, \dots \pi_{Q_1})$ et $\bm{\rho} = (\rho_1, \dots \rho_{Q_2})$ + \end{block} + } + \only<2>{ + \begin{tikzpicture}[scale=0.33] + \input{../tikz/collbm-pirho.tex} + \end{tikzpicture} + \begin{block}{\emph{$\pi\rho$-colBiSBM}} + $\bm{\pi} = ((\pi_{\color{black}1}^{\color{red}m}, \dots \pi_{\color{black}Q_1}^{\color{red}m}))_{m=1,\dots M}$ et $\bm{\rho} = ((\rho_{\color{black}1}^{\color{red}m}, \dots \rho_{\color{black}Q_2}^{\color{red}m}))_{m=1,\dots M}$ %{$\forall q \in \llbracket 1, Q_1 - 1\rrbracket, \pi_q > 0$ et $\forall r \in \llbracket 1, Q_2 - 1\rrbracket, \rho_r > 0$} + \small \\ + avec $\forall q,m \in \llbracket 1, Q_1 \rrbracket \times \llbracket 1, M \rrbracket, \pi_q^m \in \left[ 0,1 \right]$ + et $\forall r,m \in \llbracket 1, Q_2 \rrbracket \times \llbracket 1, M \rrbracket, \rho_r^m \in \left[ 0,1 \right]$ + \end{block} + } + Dans tous les modèles la structure de connectivité ($\bm{\alpha}$) est supposée identique au sein de la collection. +\end{frame} +\begin{frame} + \frametitle{Estimation des paramètres} + % DONE dire que tau i q m c' est la proba que Zim = q, approximation de la proba variationnelle. Parce qu on impose lindependance + % Par maximisation d'une borne inférieure variationnelle de la + % log-vraisemblance des données observées. + En adaptant \cite{chabert-liddellLearningCommonStructures2024a} qui se base + sur la méthode proposée par \cite{daudinMixtureModelRandom2008} utilisant + l'algorithme \emph{Variational EM}. + \begin{multline*} + \ell (\bm{X};\bm{\theta}) \geq \color{red}\sum_{m=1}^{M} \bigg( + \color{black} Q^m(\boldsymbol\theta\mid\boldsymbol\theta^{(t)}) + + \mathcal{H}(\mathcal{R}_{\mathbf{X}^m,\boldsymbol\theta^{(t)}} + (\mathbf{Z}^m, \mathbf{W}^m)) + \color{red}\bigg) \color{black} + =: J(\bm{\tau};\bm{\theta}) + \end{multline*} + où $Q^m(\boldsymbol\theta\mid\boldsymbol\theta^{(t)}) = + \operatorname{E}_{\mathbf{Z}^m,\mathbf{W}^m + \sim \mathcal{R}_{\mathbf{X}^m,\boldsymbol\theta^{(t)}}(.)} + \left[ \log p (\mathbf{X}^m,\mathbf{Z}^m,\mathbf{W}^m | \boldsymbol\theta) \right] \,$ + + \begin{block}{Approximation variationnelle} + $\mathcal{R}_{\mathbf{X}^m,\boldsymbol\theta^{(t)}}(\mathbf{Z}^m, \mathbf{W}^m) = + P(\mathbf{Z}^m | \mathbf{X}^m,\boldsymbol\theta^{(t)}) P(\mathbf{W}^m | \mathbf{X}^m,\boldsymbol\theta^{(t)})$, c'est à dire avoir + une indépendance lignes, colonnes. + \end{block} +\end{frame} + +\begin{frame}{Formule développée de l'EM variationnel} + \begin{multline*} + \ell (\bm{X};\bm{\theta}) \geq \color{red}\sum_{m=1}^{M} \bigg( \color{black} \sum_{i = 1}^{n_1^m}\sum_{j=1}^{n_2^m}\sum_{q \in \mathcal{Q}_{1,m}} \sum_{r \in \mathcal{Q}_{2,m}} \tau^{1,m}_{i,q} \tau^{2,m}_{j,r} \log f(X^{m}_{ij}; \alpha_{qr}) \\ + + \sum_{i=1}^{n_1^m} \sum_{q \in \mathcal{Q}_{1,m}} \tau^{1,m}_{i,q} \log \pi_{\color{black}q}^{\color{gray}m} + \sum_{j=1}^{n_2^m} \sum_{r \in \mathcal{Q}_{2,m}} \tau^{2,m}_{j,r} \log \rho_{\color{black}r}^{\color{gray}m} \\ + - \sum_{i=1}^{n_1} \tau^{1,m}_{i,q} \log \tau^{1,m}_{i,q} - \sum_{j=1}^{n_2} \tau^{2,m}_{j,r} \log \tau^{2,m}_{j,r} \color{red}\bigg) \color{black} =: J(\bm{\tau};\bm{\theta}), + \end{multline*} + où $\ell$ désigne la $\log$ vraisemblance. + + \begin{block}{Approximation variationnelle} + $\tau_{i,q}^{1,m} = P_{\mathcal{R}_m}(Z_{iq}^m = 1|X_{i\bullet}^m)$ + et $\tau_{j,r}^{2,m} = P_{\mathcal{R}_m}(W_{jr}^m = 1|X_{\bullet j}^m)$ + \end{block} +\end{frame} + +\section{Sélection de modèle} +\begin{frame} + \frametitle{Problème de choix de $(Q_1, Q_2)$} + \underline{L'estimation de paramètres se fait à $Q_1, Q_2$ blocs fixés}, il faut donc déterminer les \enquote*{meilleures} coordonnées. + Nous maximisons un critère, le \emph{Bayesian Information Criterion - Like} + (BIC-L), de vraisemblance pénalisée en adaptant les formules + de~\cite{chabert-liddellLearningCommonStructures2024a}. + + \begin{alertblock}{Problèmes de l'exploration} + \begin{itemize} + \item Exploration de l'espace $\mathbb{N}^2$ coûteux, besoin d'une + stratégie. + \item Sensibilité aux initialisations et à l'aléatoire. + \end{itemize} + + \end{alertblock} +\end{frame} + +\section{Application} +\label{sec:application} + +\begin{frame} + \frametitle{Clustering de réseaux} + \centering + \begin{tikzpicture} + \input{../tikz/clustering.tex} + \end{tikzpicture} +\end{frame} + +\begin{frame}[allowframebreaks] + \frametitle{Application à~\cite{baldockDailyTemporalStructure2011, + baldockSystemsApproachReveals2019a}} + \begin{figure}[t] + \centering + \begin{subfigure}{0.5\textwidth} + \centering + \includegraphics[scale=0.2,angle=-90]{backup-app-iid.png} + \caption{Modèle $iid$,\\ + séparent réseau africain et réseaux anglais} + \end{subfigure}% + ~ + \begin{subfigure}{0.5\textwidth} + \centering + \includegraphics[scale=0.2,angle=-90]{backup-app-pirho.png} + \caption{Modèle $\pi\rho$,\\ + fusionnent réseaux africain et anglais} + \end{subfigure}% + \caption{Partitionnement des réseaux + de~\cite{baldockDailyTemporalStructure2011, + baldockSystemsApproachReveals2019a}} + \end{figure} + + \begin{figure}[t] + \centering + \begin{subfigure}{0.5\textwidth} + \centering + \includegraphics[scale=0.1]{backup-app-iid-struct1.png} + \includegraphics[scale=0.2]{backup-app-iid-struct2.png} + \caption{Modèle $iid$,\\ + séparent réseau africain et réseaux anglais} + \end{subfigure}% + ~ + \begin{subfigure}{0.5\textwidth} + \centering + \includegraphics[scale=0.2]{backup-app-pirho-struct.png} + \caption{Modèle $\pi\rho$,\\ + fusionnent réseaux africain et anglais} + \end{subfigure}% + \caption{Structures détectées pour les réseaux + de~\cite{baldockDailyTemporalStructure2011, + baldockSystemsApproachReveals2019a}} + \end{figure} +\end{frame} + +\section{Conclusion} +\label{sec:conclusion} +\begin{frame} + \frametitle{Conclusion et perspectives} + % DONE Ajouter une slide conclusion perspective + % Rappeler les modeles avec clustering + % Evoquer l'analyse de reseaux corrigés pour l'échantillonnage + % Lien vers le package + \begin{block}{Capacités} + \begin{itemize} + \item 4 modèles dont 3 qui ont une flexibilité sur au moins une des + dimensions (adaptabilité aux données). + \item Détecter structures classiques et moins classique de façon + agnostique. + \item Partitionner un ensemble de réseaux selon leurs structures. + \end{itemize} + \end{block} + + \begin{block}{Perspectives} + \begin{itemize} + \item Investiguer stabilité à l'aléatoire. + \item Intégration au package \texttt{colSBM} et publication CRAN + \item Preuve d'identifiabilité du modèle $\pi\rho$. + \end{itemize} + \end{block} + + \bigskip + \centering + Merci pour votre attention~! + +\end{frame} \ No newline at end of file diff --git a/soutenance/soutenance.pdf b/soutenance/soutenance.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b4f1b82947e1862178773d57871302a6fbd2582f GIT binary patch literal 446495 zcmd431yo#HwuXzlySsZ8?(Py?gS)%CJ3)dIg1ZNIC%6+J1a~KRu$P?fzTM|`zj4n= zcly0O230kxHZ^Llz1P3in*TQ!rJ|SwBY=q=j&f>lU=a?$Lc&5~Z)63>#|Ou(>gixg z!mRnq$ja2%1&&$8)yUkcY~h$iMC?6Cbbg!v 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0000000..ff7ee7a --- /dev/null +++ b/soutenance/soutenance.tex @@ -0,0 +1,119 @@ +\documentclass{beamer} +\usetheme{Boadilla} + +% importations +\usepackage[french]{babel} % pour dire que le texte est en francais +\usepackage{csquotes} +\usepackage[T1]{fontenc} % pour les font postscript +\usepackage[cyr]{aeguill} % Police vectorielle TrueType, guillemets francais +\usepackage{epsfig} % pour gérer les images +\usepackage{amsmath,amsthm, stmaryrd} % très bon mode mathématique +\usepackage{amsfonts,amssymb,bm, bbold}% permet la definition des ensembles +\usepackage{algorithm2e} % pour les algorithmes +\usepackage{algpseudocode} % pour les algorithmes +\usepackage{graphicx} +\usepackage{subcaption} +\usepackage{float} % pour le placement des figure +\usepackage{url} % pour une gestion efficace des url +\usepackage{hyperref} % pour les hyperliens dans le document + +\usepackage{appendixnumberbeamer} % Cacher la numérotation des slides d'appendices +\usepackage{beamerappendixnote} +\usepackage{adjustbox} % To resize tikzpictures +\usepackage{fontawesome5} +\usepackage{makecell} + +% Images +\graphicspath{{../img/}{../figure/}} + +% Tikz +\usepackage{tikz} % For graph plots +\usepackage[outline]{contour} +%% Tikz Related +\usetikzlibrary{calc,shapes,backgrounds,arrows,automata,shadows,positioning} +\usetikzlibrary{arrows,shapes,positioning,shadows,trees,calc,backgrounds,automata,positioning} +\usetikzlibrary{decorations.pathreplacing,calligraphy,external,petri} + +%% Tikz sets +\tikzset{ + basic/.style = {draw, text width=3cm, font=\sffamily, rectangle}, + root/.style = {basic, rounded corners=2pt, thin, align=center, + fill=green!30}, + level 2/.style = {basic, rounded corners=6pt, thin,align=center, fill=green!60, + text width=8em}, + level 3/.style = {basic, thin, align=left, fill=pink!60, text width=3.5cm} +} + +% Couleurs +% pour tickz multilevel +\input{../shared/colors} +%% Biblio +\input{../shared/biblio.tex} + +% Footnote +\makeatletter +\newcommand\blfootnote[1]{% + \begingroup + \renewcommand{\@makefntext}[1]{\noindent\makebox[1.8em][r]#1} + \renewcommand\thefootnote{}\footnote{#1}% + \addtocounter{footnote}{-1}% + \endgroup +} +\makeatother + + +% Beamer +\setbeamertemplate{headline}{% + \begin{beamercolorbox}[ht=2.25ex,dp=3.75ex]{section in head/foot} + \insertnavigation{\paperwidth} + \end{beamercolorbox}% +}% +\beamertemplatenavigationsymbolsempty % Pas de bar de navigation + +% Beamer settings +\setbeamertemplate{caption}[numbered] +\setbeamertemplate{note page}[plain] % Notes +\setbeamerfont{caption}{size=\scriptsize} % Taille des légendes +\setbeamercolor{palette primary}{bg=cyanps, fg=black} +\setbeamercolor{palette secondary}{bg=blueps, fg=white} +\setbeamercolor{palette tertiary}{bg=bluefonceps, fg=white} +\setbeamercolor{structure}{fg=blueps} + + + +\subtitle{Soutenance de Master MathSV} +\title[Collections de réseaux bipartites]{Détection de structures et +\emph{clustering} dans des réseaux bipartites} +\author[L. Lacoste]{Louis \textsc{Lacoste}, encadré par Pierre Barbillon et +Sophie Donnet} % Sous la supervision de Pierre +\date{29 août 2024} + +\begin{document} + +% titre +\begin{frame}[noframenumbering,plain] + \maketitle +\end{frame} + +\begin{refsection} + \include{principal} + + \renewcommand{\pgfuseimage}[1]{\scalebox{.75}{\includegraphics{#1}}} + \begin{frame}[noframenumbering,plain,allowframebreaks] + \frametitle{Bibliographie} + \printbibliography + \end{frame} +\end{refsection} + +\appendix + +\begin{refsection} + \include{annexe} + \renewcommand{\pgfuseimage}[1]{\scalebox{.75}{\includegraphics{#1}}} + \begin{frame}[noframenumbering,plain,allowframebreaks] + \frametitle{Bibliographie des annexes} + \printbibliography + \end{frame} +\end{refsection} + +\end{document} \ No newline at end of file