presentation : 2 juillet Version

presentation/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}

presentation/presentation.tex

@@ -12,11 +12,14 @@
 \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{tikz} % For graph plots
 \usepackage[outline]{contour}
+\usepackage{appendixnumberbeamer} % Cacher la numérotation des slides d'appendices
+\usepackage{beamerappendixnote}
 \usepackage{adjustbox} % To resize tikzpictures
 \usepackage{fontawesome5}
 \usepackage{makecell}
@@ -28,11 +31,18 @@
     \end{beamercolorbox}%
 }%
 \beamertemplatenavigationsymbolsempty % Pas de bar de navigation
-\setbeamerfont{caption}{size=\scriptsize} % Petit titre de figures
 
-% bibliographie
-\usepackage[style=apa,sorting=none]{biblatex}
+% Beamer settings
+\setbeamertemplate{caption}[numbered]
+\setbeamertemplate{note page}[plain] % Notes
+\setbeamerfont{caption}{size=\scriptsize} % Taille des légendes
+
+
+
+% Bibliographie
+\usepackage[style=apa, sorting=none, backend=biber]{biblatex}
 \addbibresource{../references.bib}
+\AtEveryCite{\color{blue}}
 
 % Images
 \graphicspath{{../img/}{../figure/}}
@@ -84,8 +94,10 @@
 
 
 \subtitle{Séminaire des stagiaires}
-\title[Collections de réseaux bipartites]{Détection de structure dans des réseaux bipartites}
-\author[L. Lacoste]{Louis \textsc{Lacoste}} % Sous la supervision de Pierre
+\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{4 juillet 2024}
 
 \begin{document}
@@ -95,271 +107,25 @@
     \maketitle
 \end{frame}
 
-\section{Contexte du modèle}
-\label{sec:contexte-du-modele}
+\begin{refsection}
+    \include{principal}
 
-\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 & 1 & 1 \\
-                                0 & 0 & 0 \\
-                                1 & 0 & 0 \\
-                                0 & 0 & 0
-                            \end{pmatrix}
-                        \end{align*}
-                        \footnotesize
-                        Matrice d'adjacence associée
-                    \end{column}
-                \end{columns}
-            \end{block}
+    \renewcommand{\pgfuseimage}[1]{\scalebox{.75}{\includegraphics{#1}}}
+    \begin{frame}[noframenumbering,plain,allowframebreaks]
+        \frametitle{Bibliographie}
+        \printbibliography
+    \end{frame}
+\end{refsection}
 
-        \end{column}
-        \onslide<2>{
-            \begin{column}{0.45\textwidth}
-                \begin{block}{Contexte mathématique}
-                    Pour un unique réseau~: variables latentes,
-                    \emph{embedding}, \dots
+\appendix
 
-                    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
-    Maximisation d'une borne inférieure de la log-vraisemblance des données observées.
-    \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*}
-
-    \begin{block}{Approximation variationnelle}
-        $\tau_{i,q}^{1,m} = P(Z_i = q | X^m_{ij})$ et $\tau_{j,r}^{2,m} = P(W_j = r | X^m_{ij})$ tels que $P(Z_i = q, W_j = r | X^m_{ij}) = \tau_{i,q}^{1,m}\times\tau_{j,r}^{2,m}$
-    \end{block}
-
-\end{frame}
-\section{Sélection de modèle}
-
-\begin{frame}
-    \frametitle{Choix de $(Q_1,Q_2)$ - Approche gloutonne}
-    \underline{L'estimation de paramètres se fait à $Q_1, Q_2$ fixés}, il faut donc déterminer les \enquote*{meilleures} coordonnées.
-    Nous maximisons un critère\footnote{
-        \emph{Bayesian Information Criterion - Like},
-        vraisemblance pénalisée en adaptant les formules
-        de~\cite{chabert-liddellLearningCommonStructures2024a}}.
-
-    \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}
-
-
-\section{Application}
-\label{sec:application}
-
-\begin{frame}
-    \frametitle{Clustering de réseaux}
-    \begin{tikzpicture}
-        \input{../tikz/clustering.tex}
-    \end{tikzpicture}
-\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 Partitionner un ensemble de réseaux selon leurs structures.
-        \end{itemize}
-    \end{block}
-
-    \begin{block}{Perspectives}
-        \begin{itemize}
-            \item Investiguer stabilité à la \emph{graine}.
-            \item Preuve d'identifiabilité du modèle $\pi\rho$.
-        \end{itemize}
-    \end{block}
-
-    \bigskip
-    \centering
-    Merci pour votre attention~!
-
-\end{frame}
-\renewcommand{\pgfuseimage}[1]{\scalebox{.75}{\includegraphics{#1}}}
-\begin{frame}[noframenumbering,plain,allowframebreaks]
-    \frametitle{Bibliographie}
-    \printbibliography
-\end{frame}
+\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}

presentation/principal.tex

@@ -0,0 +1,278 @@
+\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 & 1 & 1 \\
+                                0 & 0 & 0 \\
+                                1 & 0 & 0 \\
+                                0 & 0 & 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 $Q^m(\boldsymbol\theta\mid\boldsymbol\theta^{(t)}) =
+        \operatorname{E}_{\mathbf{Z}^m,\mathbf{W}^m
+        \sim \mathcal{R}_{\mathbf{X}^m,\boldsymbol\theta^{(t)}}(.)}
+        \left[ \log p (\mathbf{X}^m,\mathbf{Z}^m,\mathbf{W}^m | \boldsymbol\theta) \right] \,$
+
+    \appxnote{Formule développée 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}
+        $\mathcal{R}_{\mathbf{X}^m,\boldsymbol\theta^{(t)}}(\mathbf{Z}^m, \mathbf{W}^m) =
+            P(\mathbf{Z}^m | \mathbf{X}^m,\boldsymbol\theta^{(t)}) P(\mathbf{W}^m | \mathbf{X}^m,\boldsymbol\theta^{(t)})$, c'est à dire avoir
+        une indépendance lignes, colonnes.
+    \end{block}
+
+\end{frame}
+\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 $iid$}
+        \end{subfigure}%
+        ~
+        \begin{subfigure}{0.5\textwidth}
+            \centering
+            \includegraphics[scale=0.2,angle=-90]{backup-app-pirho.png}
+            \caption{Modèle $\pi\rho$}
+        \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 $iid$}
+        \end{subfigure}%
+        ~
+        \begin{subfigure}{0.5\textwidth}
+            \centering
+            \includegraphics[scale=0.2]{backup-app-pirho-struct.png}
+            \caption{Modèle $\pi\rho$}
+        \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 la \emph{graine}.
+            \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}

references.bib

@@ -16,7 +16,7 @@
   urldate = {2023-06-14},
   abstract = {The structure of a bipartite interaction network can be described by providing a clustering for each of the two types of nodes. Such clusterings are outputted by fitting a Latent Block Model (LBM) on an observed network that comes from a sampling of species interactions in the field. However, the sampling is limited and possibly uneven. This may jeopardize the fit of the LBM and then the description of the structure of the network by detecting structures which result from the sampling and not from actual underlying ecological phenomena. If the observed interaction network consists of a weighted bipartite network where the number of observed interactions between two species is available, the sampling efforts for all species can be estimated and used to correct the LBM fit. We propose to combine an observation model that accounts for sampling and an LBM for describing the structure of underlying possible ecological interactions. We develop an original inference procedure for this model, the efficiency of which is demonstrated in simulation studies. The practical interest in ecology of our model is highlighted on a large dataset of plant-pollinator network.},
   langid = {english},
-  pubstate = {preprint},
+  pubstate = {prepublished},
   keywords = {Statistics - Methodology},
   file = {/home/polarolouis/Zotero/storage/LQ3FINZG/Anakok et al. - 2022 - Disentangling the structure of ecological bipartit.pdf}
 }
@@ -39,6 +39,43 @@
   file = {/home/polarolouis/Zotero/storage/A4V9MJAF/Aubert et al. - 2021 - Model-based biclustering for overdispersed count d.pdf}
 }
 
+@article{baldockDailyTemporalStructure2011,
+  title = {Daily Temporal Structure in {{African}} Savanna Flower Visitation Networks and Consequences for Network Sampling},
+  author = {Baldock, Katherine C. R. and Memmott, Jane and Ruiz-Guajardo, Juan Carlos and Roze, Denis and Stone, Graham N.},
+  date = {2011},
+  journaltitle = {Ecology},
+  volume = {92},
+  number = {3},
+  pages = {687--698},
+  issn = {1939-9170},
+  doi = {10.1890/10-1110.1},
+  url = {https://onlinelibrary.wiley.com/doi/abs/10.1890/10-1110.1},
+  urldate = {2024-07-02},
+  abstract = {Ecological interaction networks are a valuable approach to understanding plant–pollinator interactions at the community level. Highly structured daily activity patterns are a feature of the biology of many flower visitors, particularly provisioning female bees, which often visit different floral sources at different times. Such temporal structure implies that presence/absence and relative abundance of specific flower–visitor interactions (links) in interaction networks may be highly sensitive to the daily timing of data collection. Further, relative timing of interactions is central to their possible role in competition or facilitation of seed set among coflowering plants sharing pollinators. To date, however, no study has examined the network impacts of daily temporal variation in visitor activity at a community scale. Here we use temporally structured sampling to examine the consequences of daily activity patterns upon network properties using fully quantified flower–visitor interaction data for a Kenyan savanna habitat. Interactions were sampled at four sequential three-hour time intervals between 06:00 and 18:00, across multiple seasonal time points for two sampling sites. In all data sets the richness and relative abundance of links depended critically on when during the day visitation was observed. Permutation-based null modeling revealed significant temporal structure across daily time intervals at three of the four seasonal time points, driven primarily by patterns in bee activity. This sensitivity of network structure shows the need to consider daily time in network sampling design, both to maximize the probability of sampling links relevant to plant reproductive success and to facilitate appropriate interpretation of interspecific relationships. Our data also suggest that daily structuring at a community level could reduce indirect competitive interactions when coflowering plants share pollinators, as is commonly observed during flowering in highly seasonal habitats.},
+  langid = {english},
+  keywords = {Africa,competition,ecological networks,facilitation,Kenya,mutualism,pollination,savanna,temporal structure,visitation webs},
+  file = {/home/polarolouis/Zotero/storage/4ALS9Y6W/10-1110.1.pdf.pdf;/home/polarolouis/Zotero/storage/4YSLVYC5/Baldock et al. - 2011 - Daily temporal structure in African savanna flower.pdf;/home/polarolouis/Zotero/storage/7PEDTWU9/10-1110.html}
+}
+
+@article{baldockSystemsApproachReveals2019a,
+  title = {A Systems Approach Reveals Urban Pollinator Hotspots and Conservation Opportunities},
+  author = {Baldock, Katherine C. R. and Goddard, Mark A. and Hicks, Damien M. and Kunin, William E. and Mitschunas, Nadine and Morse, Helen and Osgathorpe, Lynne M. and Potts, Simon G. and Robertson, Kirsty M. and Scott, Anna V. and Staniczenko, Phillip P. A. and Stone, Graham N. and Vaughan, Ian P. and Memmott, Jane},
+  date = {2019-03},
+  journaltitle = {Nat Ecol Evol},
+  volume = {3},
+  number = {3},
+  eprint = {30643247},
+  eprinttype = {pmid},
+  pages = {363--373},
+  issn = {2397-334X},
+  doi = {10.1038/s41559-018-0769-y},
+  abstract = {Urban areas are often perceived to have lower biodiversity than the wider countryside, but a few small-scale studies suggest that some urban land uses can support substantial pollinator populations. We present a large-scale, well-replicated study of floral resources and pollinators in 360 sites incorporating all major land uses in four British cities. Using a systems approach, we developed Bayesian network models integrating pollinator dispersal and resource switching to estimate city-scale effects of management interventions on plant-pollinator community robustness to species loss. We show that residential gardens and allotments (community gardens) are pollinator 'hotspots': gardens due to their extensive area, and allotments due to their high pollinator diversity and leverage on city-scale plant-pollinator community robustness. Household income was positively associated with pollinator abundance in gardens, highlighting the influence of socioeconomic factors. Our results underpin urban planning recommendations to enhance pollinator conservation, using increasing city-scale community robustness as our measure of success.},
+  langid = {english},
+  pmcid = {PMC6445365},
+  keywords = {Bayes Theorem,Biodiversity,Cities,Conservation of Natural Resources,England,Pollination,Scotland,Systems Analysis},
+  file = {/home/polarolouis/Zotero/storage/BSGKKFLX/s41559-018-0769-y.pdf.pdf;/home/polarolouis/Zotero/storage/NZR8WPUA/Baldock et al. - 2019 - A systems approach reveals urban pollinator hotspo.pdf}
+}
+
 @article{biernackiAssessingMixtureModel2000,
   title = {Assessing a Mixture Model for Clustering with the Integrated Completed Likelihood},
   author = {Biernacki, C. and Celeux, G. and Govaert, G.},
@@ -85,7 +122,7 @@
   url = {http://arxiv.org/abs/2206.00560},
   urldate = {2023-05-22},
   abstract = {Let a collection of networks represent interactions within several (social or ecological) systems. We pursue two objectives: identifying similarities in the topological structures that are held in common between the networks and clustering the collection into sub-collections of structurally homogeneous networks. We tackle these two questions with a probabilistic model based approach. We propose an extension of the Stochastic Block Model (SBM) adapted to the joint modeling of a collection of networks. The networks in the collection are assumed to be independent realizations of SBMs. The common connectivity structure is imposed through the equality of some parameters. The model parameters are estimated with a variational Expectation-Maximization (EM) algorithm. We derive an ad-hoc penalized likelihood criterion to select the number of blocks and to assess the adequacy of the consensus found between the structures of the different networks. This same criterion can also be used to cluster networks on the basis of their connectivity structure. It thus provides a partition of the collection into subsets of structurally homogeneous networks. The relevance of our proposition is assessed on two collections of ecological networks. First, an application to three stream food webs reveals the homogeneity of their structures and the correspondence between groups of species in different ecosystems playing equivalent ecological roles. Moreover, the joint analysis allows a finer analysis of the structure of smaller networks. Second, we cluster 67 food webs according to their connectivity structures and demonstrate that five mesoscale structures are sufficient to describe this collection.},
-  pubstate = {preprint},
+  pubstate = {prepublished},
   keywords = {Statistics - Applications,Statistics - Methodology},
   file = {/home/polarolouis/Zotero/storage/M74TXGCF/Chabert-Liddell et al. - 2023 - Learning common structures in a collection of netw.pdf;/home/polarolouis/Zotero/storage/A35M8KNP/2206.html}
 }